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abstract: 'Parameters of LDPC codes, such as minimum distance, stopping distance, stopping redundancy, girth of the Tanner graph, and their influence on the frame error rate performance of the BP, ML and near-ML decoding over a BEC and an AWGN channel are studied. Both random and structured LDPC codes are considered. In particular, the BP decoding is applied to the code parity-check matrices with an increasing number of redundant rows, and the convergence of the performance to that of the ML decoding is analyzed. A comparison of the simulated BP, ML, and near-ML performance with the improved theoretical bounds on the error probability based on the exact weight spectrum coefficients and the exact stopping size spectrum coefficients is presented. It is observed that decoding performance very close to the ML decoding performance can be achieved with a relatively small number of redundant rows for some codes, for both the BEC and the AWGN channels.'
author:
- 'Irina E. Bocharova'
- 'Boris D. Kudryashov'
- Vitaly Skachek
- Yauhen Yakimenka
title: 'Distance Properties of Short LDPC Codes and their Impact on the BP, ML and Near-ML Decoding Performance'
---
Introduction
=============
[^1]
It is well-known that typically binary LDPC codes have minimum distances which are smaller than those of the best known linear codes of the same rate and length. It is not surprising, since minimum distance does not play an important role in iterative (belief propagation (BP)) decoding. On the other hand, a significant gap in the frame error rate (FER) performance of BP and maximum-likelihood (ML) decoding motivates developing near-ML decoding algorithms for LDPC codes.
There are two main approaches to improving the BP decoding performance. First one is based on post-processing in case of BP decoder failure. Different post-processing techniques for decoding of binary LDPC codes over additive white Gaussian noise (AWGN) channels are studied in [@pishro2007results; @varnica; @fang2010bp; @bocharova2016low]. A similar approach to decoding of nonbinary LDPC codes over extensions of the binary Galois field is considered in [@baldi2014hybrid]. Near-ML decoding algorithms for LDPC codes over binary erasure channel (BEC) can be found in [@pishro2004decoding; @hosoya2004; @olmos2010tree].
The second approach is based on identifying and destroying specific structural configurations such as trapping and stopping sets of the Tanner graph of the code. In particular, this can be done by adding redundant rows to the code parity-check matrix (see, for example, [@laendner2006cth02; @hehn2010improved; @mu2011]).
Suboptimality of the above modifications of BP decoding rises the following question: which properties of LDPC codes and to what extent influence their decoding FER performance? In this paper, we are trying to partially answer this question by studying short LDPC codes. We consider both binary LDPC codes and binary images of nonbinary random LDPC codes over extensions of the binary field, as well as quasi-cyclic (QC) LDPC codes constructed by using an optimization technique in [@Boch2016]. Parameters such as minimum distance, stopping distance, girth of the Tanner graph and estimates on the stopping redundancy are tabulated. Near-ML decoding based on adding redundant rows to the code parity-check matrix is analyzed. Simulated over the BEC and the AWGN channel, the FER performance of the BP, ML and near-ML decoding of these classes of LDPC codes is presented and compared to the improved upper bounds on the performance of ML decoding of regular LDPC codes [@bocharova2017performance] over the corresponding channels. The presented error probability bounds rely on precise average enumerators for given ensembles which makes these bounds tighter than known bounds (see e.g. [@sason2000]). By using an approach similar to that in [@bocharova2017performance], an improved upper bound on the performance of the BP decoding of binary images of nonbinary regular LDPC codes over BEC is presented.
The paper is organized as follows. In Section \[prelim\], all necessary notations and definitions are given. In Section \[redundancy\], a near-ML decoding method, which is based on adding redundant rows to the code parity-check matrix, is revisited. In Section \[bounds\], a recurrent procedure for computing the exact coefficients of the weight and stopping set size spectra is described. The improved upper bound on the ensemble average performance of the BP decoding over BEC is derived. Tables of the computed code parameters along with the simulation results for the BP, ML and near-ML decoding are presented in Section \[simulations\]. A comparison with the theoretical bounds is done and conclusions are drawn in Section \[discussion\].
Preliminaries {#prelim}
=============
Ensembles of binary and binary images of nonbinary regular LDPC codes
---------------------------------------------------------------------
For a binary linear $[n,k]$ code $\mathcal C$ of rate $R=k/n$ denote by $r=n-k$ its redundancy. We use a notation $\{A_{n,w} \}_{0\le w\le n}$ for a set of code weight enumerators, where $A_{n,w}$ is a number of codewords of weight $w$. Let ${{\mbox{\boldmath $H$}}}$ be an $r\times n$ parity-check matrix which defines $\mathcal C$.
By viewing ${{\mbox{\boldmath $H$}}}$ as a biadjacency matrix [@biadj], we obtain a corresponding bipartite Tanner graph. The girth $g$ is the length of the shortest cycle in the Tanner graph.
When decoded over a BEC, the FER performance of the BP decoding is determined by the size of the smallest stopping set called stopping distance $d_{\rm{stop}}$ (see, for example, [@di2002finite]). In turn, a stopping set is defined as a subset of indices of columns in a parity-check matrix, such that a matrix constructed from these columns does not have a row of weight one. The asymptotic behavior of a stopping set distribution for ensembles of binary LDPC codes is studied in [@orlitsky2005stopping]. In this paper, we study both the average performance of the ensembles of random LDPC codes and of QC LDPC codes widely used in practical schemes. Two ensembles of random regular LDPC codes are studied below. First we study the Gallager ensemble [@gallager] of $(J,K)$-regular LDPC codes, where $J$ and $K$ denote the number of ones in each column and in each row of the code parity-check matrix, respectively. Codes of this ensemble are determined by random parity-check matrices ${{\mbox{\boldmath $H$}}}$, which consist of the strips ${{\mbox{\boldmath $H$}}}_{i}$ of width $M=r/J$ rows each, $ i =1,2,\dots, J$. All strips are random column permutations of the strip where the $j$th row contains $K$ ones in positions $(j-1)K+1, (j-1)K+2, \ldots, jK$, for $j = 1, 2, \ldots, n/K$.
Next, we study the ensemble of binary $(J,K)$-regular LDPC codes, which is a special case of the ensemble described in . We refer to this ensemble as the Richardson-Urbanke (RU) ensemble of ($J,K$)-regular LDPC codes.
For $a\in\{1,2,...\}$ denote by $a^m$ a sequence $(a,a,...,a)$ of $m$ identical symbols $a$. In order to construct an $r \times n$ parity-check matrix ${{\mbox{\boldmath $H$}}}$ of an LDPC code from the RU ensemble, one does the following:
- construct the sequence ${\ensuremath{\boldsymbol{a}}}=(1^J,2^J,...,n^J)$;
- apply a random permutation ${\ensuremath{\boldsymbol{b}}} = \pi ({\ensuremath{\boldsymbol{a}}})$ to obtain a sequence ${\ensuremath{\boldsymbol{b}}}=(b_1,...,b_N)$, where $N=Kr=Jn$;
- set to one the entries in the first row of ${{\mbox{\boldmath $H$}}}$ in columns $b_1,...,b_K$, the entries in the second row of ${{\mbox{\boldmath $H$}}}$ in columns $b_{K+1},...,b_{2K}$, etc. The remaining entries of ${{\mbox{\boldmath $H$}}}$ are zeros.
In fact, an LDPC code from the RU ensemble is $(J,K)$-regular if for a given permutation $\pi$ all elements of subsequences $(b_{iK-K+1},...,b_{iK})$ are different for all $i=1,...,r$. It is shown in [@litsyn2002ensembles] that the fraction of regular codes among the RU LDPC codes is roughly $$\exp \left\{ -\frac{1}{2} (K-1)(J-1) \right\}$$ which means that most of the RU codes are irregular. In what follows, we ignore this fact and interpret the RU LDPC codes as the $(J,K$)-regular codes, and call them “almost regular”.
[ Generally, the design rate $R=1-J/K$ is a lower bound on the actual code rate since the rank of randomly constructed parity-check matrix can be smaller than the number of its rows. However, in our study the best generated almost regular RU codes always have the rate equal to the design rate. For this reason, we do not distinguish between the design rate and the actual rate. ]{}
In order to construct random binary images of nonbinary ($J,K$)-regular LDPC codes, we use the standard two-stage procedure. It consists of labeling a proper binary base parity-check matrix by random nonzero elements of the extension of the binary Galois field. In our work, we select a parity-check matrix of a binary LDPC code from the Gallager or the RU ensembles as the base matrix.
In what follows, the Gallager ensembles of binary regular LDPC codes and binary images of nonbinary regular LDPC codes are used only for the theoretical analysis, while for the simulations we use almost regular LDPC codes from the RU ensemble. The reason for this choice is that in the simulations, the RU LDPC ensembles outperform the Gallager LDPC codes with the same parameters.
QC LDPC codes
-------------
The QC LDPC codes represent a class of LDPC codes which is very intensively used in communication standards. Rate $R=b/c$ QC LDPC codes are determined by a $(c-b)\times c$ polynomial parity-check matrix of their parent convolutional code [@johannesson2015fundamentals] $${{\mbox{\boldmath $H$}}}(D)=\left(\begin{array}{cccc}
h_{11}(D)&h_{12}(D)&\dots&h_{1c}(D)\\
h_{21}(D)& h_{22}(D)&\dots&h_{2c}(D)\\
\vdots&\vdots&\ddots&\vdots\\
h_{(c-b)1}(D)& h_{(c-b)2}(D)&\dots&h_{(c-b)c}(D)
\end{array}
\right)\label{polynom_matr}$$ where $h_{ij}(D)$ is either zero or a monomial entry, that is, $h_{ij}(D)\in \{0,D^{w_{ij}}\}$ with $w_{ij}$ being a nonnegative integer, $w_{ij}\le \mu$, and $\mu=\max_{i,j} \{ w_{ij} \}$ is the syndrome memory.
[ The polynomial matrix (\[polynom\_matr\]) determines an $[Mc,Mb]$ QC LDPC block code using a set of polynomials modulo $D^{M}-1$. By tailbiting the parent convolutional code to length $M > \mu$, we obtain the binary parity-check matrix $$\small
\arraycolsep=3pt \def\arraystretch{1.2}
{\ensuremath{\boldsymbol{H}}}_{\rm TB}=\begin{pmatrix}
{\ensuremath{\boldsymbol{H}}}_{0}&{\ensuremath{\boldsymbol{H}}}_{1}&\dots&{\ensuremath{\boldsymbol{H}}}_{\mu-1}&{\ensuremath{\boldsymbol{H}}}_{\mu}&{{\ensuremath{\boldsymbol{0}}}}&\dots&{\ensuremath{\boldsymbol{0}}}\\
{{\ensuremath{\boldsymbol{0}}}}&{\ensuremath{\boldsymbol{H}}}_{0}&{\ensuremath{\boldsymbol{H}}}_{1}&\dots&{\ensuremath{\boldsymbol{H}}}_{\mu-1}&{\ensuremath{\boldsymbol{H}}}_{\mu}&\dots&{\ensuremath{\boldsymbol{0}}}\\
\vdots & &\ddots &\vdots&\vdots &\vdots &\ddots&\\
{\ensuremath{\boldsymbol{H}}}_{\mu}& {{\ensuremath{\boldsymbol{0}}}}&\dots&{\ensuremath{\boldsymbol{0}}}&{\ensuremath{\boldsymbol{H}}}_{0}&{\ensuremath{\boldsymbol{H}}}_{1}&\dots&{\ensuremath{\boldsymbol{H}}}_{\mu-1}\\
\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\
{\ensuremath{\boldsymbol{H}}}_{1}&\dots&{\ensuremath{\boldsymbol{H}}}_{\mu}&{{\ensuremath{\boldsymbol{0}}}}&\dots&{\ensuremath{\boldsymbol{0}}}&\dots&{\ensuremath{\boldsymbol{H}}}_{0}
\end{pmatrix},
\label{tb}$$ of an equivalent (in the sense of column permutation) TB code (see [@johannesson2015fundamentals Chapter 2]), where ${\ensuremath{\boldsymbol{H}}}_{i}$, $i=0,1,\dotsc,\mu$, are binary $(c-b)\times c$ matrices in the series expansion $${\ensuremath{\boldsymbol{H}}}(D)={\ensuremath{\boldsymbol{H}}}_{0}+{\ensuremath{\boldsymbol{H}}}_{1}D+\cdots+{\ensuremath{\boldsymbol{H}}}_{\mu}D^{\mu},$$ and ${\ensuremath{\boldsymbol{0}}}$ is the all-zero matrix of size $(c-b)\times c$. If each column of ${{\mbox{\boldmath $H$}}}(D)$ contains $J$ nonzero elements, and each row contains $K$ nonzero elements, the QC LDPC block code is $(J,K)$-regular. It is irregular otherwise. ]{}
Another form of the equivalent $[Mc,Mb]$ binary QC LDPC block code can be obtained by replacing the nonzero monomial elements of ${{\mbox{\boldmath $H$}}}(D)$ in (\[polynom\_matr\]) by the powers of the circulant $M\times M$ permutation matrix ${{\mbox{\boldmath $P$}}}$, whose rows are cyclic shifts by one position to the right of the rows of the identity matrix.
The polynomial parity-check matrix ${{\mbox{\boldmath $H$}}}(D)$ (\[polynom\_matr\]) can be interpreted as a $(c-b) \times c$ binary base matrix ${{\mbox{\boldmath $B$}}}$ labeled by monomials, where the entry in ${{\mbox{\boldmath $B$}}}$ is one if and only if the corresponding entry of ${{\mbox{\boldmath $H$}}}(D)$ is nonzero, i.e. $${{\mbox{\boldmath $B$}}}={{\mbox{\boldmath $H$}}}(D)|_{D=1}$$ All three matrices ${{\mbox{\boldmath $B$}}}$, ${{\mbox{\boldmath $H$}}}(D)$, and ${{\mbox{\boldmath $H$}}}$ can be interpreted as bi-adjacency matrices of the corresponding Tanner graphs.
Stopping redundancy and convergence to the ML decoding performance {#redundancy}
==================================================================
The idea to improve the performance of iterative decoding of linear codes over a BEC by using redundant parity checks was studied, for example, in [@Santhi; @Yedidia]. This approach was further explored in [@schwartz2006stopping] (for BEC) and in [@jens] (for BSC and AWGN). The idea of using redundant parity checks was also studied in the context of linear-programming decoding [@feldman], the reader can refer, for example, to [@pascal].
A straightforward method to extend a parity-check matrix of an LDPC code is based on appending a predetermined number of dual codewords to the parity-check matrix. In this approach, the BP decoder uses the redundant matrix instead of the original parity-check matrix. One of the strategies used to extend the parity-check matrix consists of appending dual codewords in the order of their increasing weights starting with the minimum weight $d_{\rm dual}$. A problem of searching for low-weight dual codewords has high computational complexity in general, yet for short LDPC codes it is feasible. We apply this approach in the sequel, and study the convergence of the FER of BP decoding of LDPC codes determined by their extended parity-check matrices to the FER of the ML decoding (for both BEC and AWGN channels).
The stopping redundancy is defined as the minimum number of rows in a parity-check matrix required to ensure that the stopping distance of the code $d_{\rm{stop}}$ is equal to the code minimum distance $d_{\min}$. For a set of the selected LDPC codes, we compute estimates on the minimum number of the rows required in order to ensure removal of stopping sets of a certain size. Next, we describe this approach in more detail. By *$\ell$-th stopping redundancy*, $\rho_\ell$, we denote the minimum number of rows in any parity-check matrix of the code, such that all ML-decodable stopping sets of size less than or equal to $\ell$ are removed. In particular, $\rho_r$ is the minimum number of rows in any parity-check matrix of the code, such that there are no ML-decodable stopping sets of size up to $r$ (incl.), i.e. no stopping sets which, if erased, still can be decoded by the ML decoder. Our definition of $\ell$-th stopping redundancy is analogous to its counterpart in [@hehn2008permutation].
However, we stress the difference between the updated definition of the $\ell$-th stopping redundancy for $\ell \geq d$ and its counterpart in [@hehn2008permutation]. In fact, the stopping sets of size $\ell \geq d$ that are *not* ML-decodable, are exactly the supports of the codewords.[^2]
In order to calculate the upper bounds on the $\ell$-th stopping redundancy with a method based on [@yakimenka2015refined], we first estimate by sampling $u_i$, the number of ML-decodable stopping sets of size $i$ in a particular parity-check matrix. Then, we use the estimates on $u_i$ ($i=1,2,\dotsc,r$) with the method similar to [@yakimenka2015refined Thm. 1,2] in order to obtain the approximate upper bounds on the *stopping redundancy hierarchy*, i.e. the stopping redundancies $\rho_1, \rho_2, \dotsc, \rho_r$. In Table \[t1\], we present estimates on $\rho_\ell$, $\ell=d_{\min}, d_{\min} +1,d_{\min}+2$, and $\ell=r$, along with $d_{\min}$, $d_{\rm stop}$, $d_{\rm dual}$ and $g$, for a set of selected LDPC codes. In Section \[simulations\], we also present the simulated FER performance of the BP and ML decoding over the BEC for this set of codes with varying number of redundant rows. The same set of LDPC codes with varying number of redundant rows in their parity-check matrices is also simulated over the AWGN channel.
The LDPC codes from the following four families were selected:
- [Random regular LDPC codes from the RU ensemble (rows 2 and 3 in Table \[t1\]) ]{}
- [QC LDPC codes (row 4)]{}
- [Binary images of nonbinary regular LDPC codes (row 5)]{}
- [Linear codes represented in a “sparse form” (row 1)]{}
Two random RU codes were selected by an exhaustive search among 100000 code candidates. As a search criteria, we used the minimum distance and the first spectrum coefficient $A_{d_{\min},n}$. The QC LDPC code was obtained by optimization of lifting degrees for a constructed base matrix in order to guarantee the best possible minimum distance under a given restriction on the girth value of the code Tanner graph. For comparison, we simulated the best linear code with the same length and dimension determined by a parity-check matrix with the lowest possible correlation between its rows. Next, we refer to this form of the parity-check matrix as a “sparse form”. Parameters of the selected codes are presented in Table \[t1\]. Here we use the notations ‘RU’ for random LDPC codes, ‘L’ for the best linear code with parity-check matrix in ‘sparse form’, ‘NB’ for the binary image of nonbinary regular LDPC code and ‘QC’ for QC LDPC code, respectively.
Code $d_{\min}$ $A_{d_{\min},n}$ $d_{{\rm stop}}$ $d_{\rm {dual}}$ $g$ $J$,$K$ $\rho_{d_{\min}},\rho_{d_{\min}+1},\rho_{d_{\min}+2}$ $\rho_r$ Type
------ ------------ ------------------ ------------------ ------------------ ----- --------- ------------------------------------------------------- ---------- ------
1 12 17296 4 12 4 6,12 6240,12151,23468 13761585 ’L’
2 8 13 4 6 4 6,12 261,581,1254 13683513 ’RU’
3 7 1 5 5 4 4,8 83,175,380 12549204 ’RU’
4 7 8 7 5 6 3,6 58,130,274 9876964 ’QC’
5 8 7 4 7 4 3,6 355,751,1551 13819276 ’NB’
: \[Table\_AWGN\] Parameters of studied $[48.24]$ codes
\[t1\]
Upper bounds on ML and BP decoding error probability for ensembles of LDPC codes {#bounds}
=================================================================================
In this section, we analyze the Gallager ensembles of binary and binary images of nonbinary ($J,K$)-regular LDPC codes. By following the approach in [@bocharova2017performance] we derive estimates on the decoding error probability of the ML and BP decoding by using precise coefficients of the average weight spectrum and average stopping set size spectrum, respectively. Additionally to the bounds on the performance of the ML decoding obtained in [@bocharova2017performance], in this paper we derive the improved bounds on the performance of BP decoding for both binary LDPC codes and binary images of nonbinary regular LDPC codes.
The main idea behind the approach in [@bocharova2017performance] is computing the average spectra coefficients recurrently with complexity linear in $n$. The resulting coefficients are substituted into the union-type upper bound on the error probability of the ML decoding over a BEC [@berlekamp1980technology] $$\label{gen_spectr}
P_e \le \sum_{i=d}^n \min \left\{ \binom{n}{i} ,
\sum_{w=d}^i S_w \binom{n-w}{i-w}
\right\} \epsilon^i(1-\epsilon)^{n-i}$$ where $S_{w}$ is the $w$-th weight (stopping set size) spectrum coefficient, $\varepsilon$ is the erasure probability and $d$ denotes the minimum distance (stopping distance). In order to upper-bound the error probability of the ML decoding over an AWGN channel, the average weight spectrum coefficients are substituted into the tangential-sphere bound [@poltyrev1994bounds].
Consider the Gallager ensemble of $q$-ary LDPC codes, where $q=2^{m}$, $m \ge 1$ is an integer. The weight generating function of $q$-ary sequences of length $n$ satisfying the nonzero part of one $q$-ary parity-check equation is given in [@gallager] as $$\label{eq:2stars}
g(s)=\frac{(1+(q-1)s)^{K}+(q-1)(1-s)^{K}}{q} \; .$$ It is easy to derive the weight generating function of $q$-ary sequences of length $K$ and $q$-ary weight not equal to 1: $$\label{stopping}
g_{\rm{stop}}(s)=\sum_{w=0,2,3,...,K}\binom{K}{w}(q-1)^{w}s^{w}=(1+(q-1)s)^{K}-K(q-1)s.\;$$ Each $q$-ary symbol can be represented as a binary sequence (image) of length $m$. It is easy to see that different representations of a finite field of characteristic two will lead to different generating functions of binary images for the same ensemble of nonbinary LDPC codes. Following the techniques in [@el2004bounds], we study an average binary weight spectrum for the ensemble of $m$-dimensional binary images. By assuming uniform distribution on the $m$-dimensional binary images of the non-zero $q$-ary symbols, we obtain the generating function of the average binary weights of a $q$-ary symbol in the form $$\label{eq:3stars}
\phi(s)= \frac{1}{q-1} \sum_{w=1}^m\binom{m}{w}s^{w}=\frac{(1+s)^{m}-1}{q-1} \; .$$ The average binary weight generating function for one strip is given by $$G(s)= \big( g( \phi(s)) \big)^{M}=\sum_{w=0}^{nm}N_{nm,w}s^{w} \; ,$$ where $N_{nm,w}$ denotes the average number of binary sequences ${\ensuremath{\boldsymbol{\beta}}}$ of weight $w$ and of length $nm$ satisfying ${{\ensuremath{\boldsymbol{\beta}}}}\mathcal{B}_{i}^{\rm T}={\ensuremath{\boldsymbol{0}}}$. Here, $\mathcal B_{i}$ denotes the average binary image of ${{\mbox{\boldmath $H$}}}_{i}$. We obtain the average binary weight enumerator of nonbinary regular LDPC code as $${\rm E} \{A_{nm,w} \}=\binom{nm}{w}\big(p(w)\big)^{J}
=\binom{nm}{w}^{1-J}N_{nm,w}^{J},\label{nonbin}$$ where $
p(w)={\binom{nm}{w}} ^{-1}{N_{nm,w}} .
$ By substituting (\[eq:3stars\]) into (\[stopping\]), similarly to (\[nonbin\]), we obtain the average binary stopping set size spectrum coefficient.
It is known that if the generating function is represented as a degree of another generating function it can be easily computed by applying a recurrent procedure. Details of the recurrent procedure for computing coefficients of the average weight spectra can be found in [@bocharova2017performance]. We proceed by computing $N_{nm,w}$ recursively.
Simulation results {#simulations}
==================
We simulate the BP and ML decoding over the BEC and AWGN channel for the five LDPC codes whose parameters are presented in Table \[t1\]. In Fig. \[comp\_BEC\_AWGN\], the FER performance of the BP and ML decoding over the BEC and the AWGN channel is compared. It is easy to see that the best BP decoding performance both over the BEC and over the AWGN channel (and at the same time the worse ML decoding performance) is shown by the QC LDPC code with the most sparse parity-check matrix and the largest girth value of its Tanner graph. We remark that the best linear \[48,24,12\] code determined by a parity-check matrix in a “sparse form”, as expected, has the best ML decoding performance over the both channels. Its BP decoding performance is worse than that of the selected LDPC codes except for the binary image of nonbinary LDPC code.
Fig. \[rpc\_BEC\_AWGN\] shows the BP decoding performance over the BEC and AWGN channel of the codes ‘QC’ and ‘L’ from Table \[t1\], when their parity-check matrices are extended. We call the corresponding decoding technique “redundant parity check” (RPC) decoding. The number next to “RPC” in Fig. \[rpc\_BEC\_AWGN\] indicates the number of redundant rows that was added. The best convergence of the FER performance of the BP decoding over the BEC to that of the ML decoding is demonstrated by the QC LDPC code, while the best linear code has the slowest convergence of its BP performance to the ML decoding performance. We observe that the obtained simulation results are consistent with the estimates on the stopping redundancy hierarchy given in Table \[t1\]. Surprisingly, similar behavior can also be observed for the FER performance of RPC decoding over the AWGN channel.
![\[comp\_BEC\_AWGN\] Comparison of the FER performance of BP and ML decoding over the BEC and the AWGN channel for LDPC codes of length $n=48$ and rate $R=1/2$ ](BPvsML.pdf){width="120mm"}
![\[rpc\_BEC\_AWGN\] FER performance of RPC decoding over the BEC and the AWGN channel for ‘L’ and ‘QC’ codes. ](RPC.pdf){width="120mm"}
Discussion
==========
In this section, we compare the simulated FER performance of the BP, ML and near-ML (RPC) decoding over the BEC and the AWGN channel with improved bounds on the ML and BP decoding performance. In Fig. \[NB\], the FER performance over the BEC for the binary image of nonbinary $(3,6)$-regular LDPC code over $GF(2^{4})$ (‘NB’ code in Table \[t1\]) and the corresponding bounds are shown.
![\[NB\] Comparison of the FER performance of BP and RPC decoding over the BEC with improved union-type bounds (\[gen\_spectr\]) on the ML and BP decoding performance. ](NB36.pdf){width="90mm"}
As it is shown in the presented plots, the ML performance of the ‘NB’ code is rather close to the ML performance of the ‘L’ code, but the convergence of the FER performance of the RPC decoding to the performance of the ML decoding for the ‘NB’ code is much faster than for the ‘L’ code.
In Fig. \[Bin\_b\], the FER performance of the BP, ML and RPC decoding over the BEC and the AWGN channel is compared to the corresponding upper and lower bounds on the performance of the ML decoding. In particular, for comparison of the performance over the BEC, we use the improved upper bound (\[gen\_spectr\]) computed for the precise ensemble average spectrum coefficients for both random linear code and $(3,6)$-regular random binary LDPC code. As a lower bound, we consider the tighten sphere-packing bound in [@our_MLbounds]. For comparison of the performance over the AWGN channel, we show the tangential-sphere upper bound [@poltyrev1994bounds] computed with the precise ensemble average spectrum coefficients for the same two ensembles and the Shannon lower bound [@Shannon1959].
![\[Bin\_b\]Comparison of the FER performance of BP, ML and RPC decoding with upper and lower bounds on the ML decoding performance. ](Bin_Bounds.pdf){width="120mm"}
Based on the presented results, we conclude the following:
- [Although it is commonly believed that the stopping sets influence the BP decoding performance over the BEC only, the behavior of the analyzed codes over the BEC and the AWGN channel is very similar. In particular, for short codes, the FER performance of the BP decoding over the AWGN channel can be significantly improved by adding redundant rows to the parity-check matrix.]{}
- [Convergence of the RPC decoding performance to the ML decoding performance is faster for those codes which are most suitable for iterative decoding, that is, codes with large girth of the Tanner graph.]{}
- [RPC decoding has a decoding threshold. When a small number of redundant rows is added, the FER performance rapidly improves, but after adding a certain number of redundant rows, the performance improvement becomes practically unjustified due to growing complexity.]{}
- [The FER performance of the RPC decoding achieves the FER performance of the ML decoding over the BEC with exponential (in length) complexity. However, a significant reduction in the FER compared to the FER of BP decoding can be achieved with a significantly lower complexity than that of the ML decoding.]{}
- [Binary images of nonbinary LDPC codes with RPC decoding demonstrate good FER performance over the BEC. In order to apply RPC decoding to these codes over the AWGN channel it is required to add $q$-ary parity-checks to their parity-check matrices. This method looks promising and is subject of our future research.]{}
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[^1]: This work is supported in part by the Norwegian-Estonian Research Cooperation Programme under the grant EMP133 and by the Estonian Research Council under the grant PUT405.
[^2]: We recall that a support of a codeword is a stopping set.
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{
"pile_set_name": "ArXiv"
}
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Shortly after the paper of Morris and Thorne [@motho] the fate of wormholes has rapidly grown into an active area of research. Intuitively speaking these wormholes are tunnels linking widely separated regions of spacetime from where in-going causal curves can pass through and become out-going on the other side. The spherically symmetric line element for such a spacetime reads, $$ds^2 = -\alpha^2 dt^2 + d\eta^2 + r^2 (d\theta^2 + \sin^2\theta d\phi^2)$$ where the non-monotonic coordinate $r$ decreases from $+\infty$ to a minimum value $r\th$, representing the location of the throat of the wormhole and then increases from $r\th$ to $+\infty$. The proper radial distance $\eta$ ranges from $-\infty$ to $\infty$ with $\eta=0$ at the throat. The redshift function $\alpha$ is positive everywhere, this ensures the absence of an event horizon.
It has long been known that when the gravitational field is strong, the Brans Dicke (BD) scalar may have some “exotic” effects on stellar configurations through a local modification of the gravitational constant by the matter energy distribution [@kiril]. In the context of traversable wormholes, Agnese and La Camera showed that a static spherically symmetric vacuum BD solution in the Jordan frame, supports a two-way traversable wormhole for values of the BD parameter $\omega < -2$ [@agnese]. The domain of the coupling constant was later extended to positive values [@nandi]. Nevertheless, if one tries to get back general relativity (GR) these wormholes are doomed. As direct consequence of the GR limiting process the BD scalar must become a constant, triggering a gravitational collapse of the throat. Moreover, the rigorous analysis performed by Nandi et al. (in the conformally rescaled Einstein frame) severely constrains the range of $\omega$ in the Jordan frame, while at the same time shows that wormhole solutions do not exist at all in the Einstein frame unless one is willing to violate the energy conditions by choice [@nandi2].
Whether or not BD theory goes over GR when the parameter $\omega$ has an infinitely large value is yet to see the light of day. The study of the limits of spacetimes depending on some parameter has been initiated by Geroch [@geroch], who called attention to the fact that the limit may depend on the coordinate system chosen to perform the calculations. More recently, based on the characterization of a given spacetime by the Cartan scalars, it was shown that the BD solutions might not have a unique limit as $\omega \rightarrow
\infty$ [@paiva]. As a matter of fact, an estimation of order of magnitude seems to indicate that BD theory always reduces to GR for large values of $\omega$ when the trace of the matter stress energy tensor is not zero, however BD solutions do not reduce to the corresponding GR limit in the trace-less case [@BS].
Non-vacuum wormhole solutions (without trace-less matter) in BD theory were already reported somewhere else [@bdwh]. The ones threaded with “ordinary” matter only exist in a very narrow interval of the coupling parameter. Although the limiting process $\omega
\rightarrow \pm \infty$ does not destroy these wormholes, the matter that threads their throats needs to be exotic.
The type of wormholes obtained by surgically grafting two identical copies of various well known spacetimes (Schwarzschild [@S], Reissner–Nordström [@RN], Friedman–Robertson–Walker [@FRW], Schwarzschild–de Sitter [@SdS]) provides a particularly elegant collection of exemplars which are not limited to be spherically symmetric [@visser]. In this report, as a natural extension of the Schwarzschild case, I work out a traversable wormhole solution in the modified BD theory with torsion [@kim]. This particular solution, with non-zero stress energy on the boundary layer between the two asymptotically vacuum flat regions, goes over the Schwarzschild case for infinite large values of the coupling parameter.
To construct the wormhole of interest, let me assume that the matter which creates the gravitational field of the wormhole is confined to a narrow region surrounding the throat $\eta
\in (-\epsilon, \epsilon)$. Namely, consider a thin shell of stress energy, let $$S_{\mu\nu} = \lim_{\epsilon \rightarrow 0} \int_{-\epsilon}^\epsilon
T_{\rho\sigma} \,h^\rho_ \mu \,h^\sigma _\mu \,d \eta$$ denote the surface matter energy momentum tensor of such a shell; where $T_{\mu\nu}$ stands for the matter stress energy tensor and $h_{\mu\nu}$ projects general tensors down onto the subspace spanned by the thin shell (remember that $\eta$ measures the geodesic distance in the direction normal to the throat) [@israel]. Associating negative values of $\eta$ to one side of the throat (say lower universe) and positive values to the other side, without loss of generality the metric in the neighborhood of the shell can be written as, $$g_{\mu\nu} = \Theta (\eta) \,\,g^{+}_{\mu\nu} +
\Theta (-\eta)\,\, g^{-}_{\mu\nu}.$$
Outside the shell the spacetime is described by the vacuum solution $$\alpha = \left(
1 - \frac{2 (2 \,\omega -1)}{(2\omega+3)} \frac{M}{r}\right)^{(2\omega+3)
/(4 \omega -2)},$$ $$\frac{dr}{d\eta} = \sqrt{1 - \frac{ 2 \,(2 \omega -1) }{(2\omega+3)}
\frac{M}{r}}$$ derived from the modified Brans–Dicke action $S= \int d^4x \sqrt{-g}
(- \phi R + \omega \phi^{,\mu} \phi_{,\mu} / \phi)$ [@kimcho]. It is worthy of notice that the differential spacetime manifold has a non-symmetric affine connection with the torsion field being generated by the gradient of the BD scalar $\phi$. The scalar curvature $R$ is that of $U_4$ theory, and $M$ is the mass of the wormhole as measured by distant observers. Note that I have previously used $\omega$ to denote the usual BD coupling parameter. The field equation of the torsion endowed BD case are equivalent to those of vacuum by making $ \omega_{_{\rm
torsion}} \leftrightarrow
-(\omega_{_{\rm vacuum}} + 3/2)$; see Eq. 25 of ref. [@kimcho] and Eq. 2.14 of ref. [@van]. Before going on, and for the sake of completeness, let me remark that the non-vanishing components of the torsion tensor are [@kimcho], $$\begin{aligned}
F_{01}^0 = - F_{12}^2 = -F_{13}^3 & = & \frac{2M}{(2\omega+3) r^2}
\nonumber \\
& \times & \left(
1 - \frac{ 2\,(2\omega-1)}{(2\omega+3)} \frac{M}{r} \right)^{-1/2}
\label{torsion}\end{aligned}$$ and the BD scalar is given by, $$\phi = \left( 1 - \frac{2\,(2\omega-1)}{(2\omega+3)} \frac{M}{r}
\right)^{-2/(2\omega-1)}.
\label{phi}$$ It follows from Eqs. (\[torsion\]) and (\[phi\]) that in the limit $\omega \rightarrow \infty$ the torsion tensor vanishes, and the BD scalar approaches to 1, the GR value.
Hereafter, I shall assume symmetry under interchange of asymptotically flat regions $\pm \leftrightarrow
\mp$. However, one should note that this requirement is not essential to the definition of a traversable wormhole. Actually, if the wormhole throat is taken to have different masses (or equivalently the behaviour of the BD scalar is different) in the upper and lower universe, the wormhole will be non-symmetric [@frono].
Since the stress energy tensor is of delta function type at the boundary of the two regions, $S_{\mu\nu}$ can be expressed in terms of the jump of the second fundamental forms $$S^\mu_\nu = ({\cal K}^\mu_\nu - \delta^\mu_\nu \,{\cal K})$$ where ${\cal K}^\mu_\nu = K^{\mu\,+}_\nu - K^{\mu\,-}_\nu$, being $K^{\mu\,\pm}_\nu$ the second fundamental forms evaluated above and below the thin shell, $$K^{\mu\,\pm}_\nu = \left. \frac{1}{2} g^{\mu\nu}
\frac{dg_{\mu\nu}}{d\eta} \right|_{\eta \rightarrow \pm 0}.$$ The surface stress energy has to satisfy a condition of pressure balance together with a constraint reflecting the fact that stress energy may be exchanged between the layers. For the static, spherically, and reflection symmetric cases under consideration these constraints are automatically satisfied. Concerning the role of the BD scalar on the layer, there exists some regions of the parameter space in which the scalar field behaves like a domain wall (the reader is referred to Section III part B of ref. [@letwang]). However, for hypothetical wormhole’s engineering considerations, one has nothing to show for this repulsive gravitational field. This could be checked by computing the surface matter energy density $$S_t^t = -\frac{\phi\th}{2\pi r\th} \sqrt{1 -
\frac{2 \,(2 \omega -1)}{(2\omega+3)}\frac{M}{r\th}}
\label{1}$$ and the surface matter tension $$S_\theta^\theta = S_\phi^\phi= -\frac{\phi\th}{4 \pi
r\th} \left[ \frac{ 1- (2\omega -
5)\,M/(2\omega+3)\,r\th}{\sqrt{1-2\,(2\omega-1)\,M/(2\omega+3)\, r\th}}
\right].
\label{2}$$ It follows from Eqs. (\[1\]) and (\[2\]) that one can not keep the surface matter stress energy from violating the energy conditions. It is important to stress that the expectation values of the stress energy tensor with respect to certain quantum states are known to violate these conditions [@qft]. On a purely classical side, it has been recently suggested that some of the gamma ray bursts detected by the [*Burst and Transient Source Experiment*]{} (BATSE) might be the telltale signature of gravitational negative anomalous compact objects [@grbwh].
It is straightforward to prove that the solution will survive a GR limiting process. Moreover, in such a limit the classical wormhole solution introduced by Visser and its equation of state are regained [@S].
Summing up, using the massive thin shell formalism I found out a wormhole solution of BD theory. The matter at the throat is necessarily exotic. The solution is well behaved for every value of the coupling constant, and in the limit $\omega \rightarrow \pm \infty$ one recovers GR which is believed to be the best theory of gravitation.
This work has been partially supported by FOMEC.
[99]{}
M. S. Morris and K. S. Thorne, Am. J. Phys. [**56**]{}, 395 (1988).
K. A. Bronnikov, Acta Phys. Pol. [**4**]{}, 231 (1973).
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It is interesting to recall that a wormhole with the two mouths embedded in a single asymptotically flat universe and the time running at different rates on either side of the throat posses causal anomalies in the form of closed timelike curves; V. P. Frolov, I. D. Novikov, Phys. Rev. D [**42**]{}, 1057 (1990).
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D. F. Torres, G. E. Romero and L. A. Anchordoqui, Sussex preprint: SUSSEX-AST 98/1-2.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'The Jarzynski Equality relates the free energy difference between two equilibrium states of a system to the average of the work over all irreversible paths to go from one state to the other. We claim that the derivation of this equality is flawed, introducing an ad hoc and unjustified weighting factor which handles improperly the heat exchange with a heat bath. Therefore Liphardt et al’s experiment cannot be viewed as a confirmation of this equality, although the numerical deviations between the two are small. However, the Jarzynski Equality may well be a useful approximation, e.g. in measurements on single molecules in solution.'
author:
- |
E. G. D. Cohen\
and\
David Mauzerall\
The Rockefeller University\
New York, NY 10021
title: Note on the Jarzynski Equality
---
=1.5
Introduction
============
Ever since Jarzynski derived a remarkable equality, the Jarzynski Equality$^{[1,2]}$, (JE), his results have been widely used: duplicated theoretically and tested experimentally. It is impossible to do justice to all publications that have appeared in connection with the JE: a survey article would be needed for that and, in fact, one covering part of the material has been written by Ritort$^{[3]}$. However, in this paper we want to scrutinize the original general derivation of the JE$^{[1,2]}$ and discuss briefly a crucial experiment by Liphardt et al$^{[4]}$ which was designed to check the JE. The communities accepting the JE consist overwhelmingly of chemists and biophysicists to which also one of us (DM) belongs, while the physicists (to which EGDC belongs) have divided opinions. We hope that this paper may clarify and unify the various aspects of the JE.
The main point of this paper is to argue that the JE is not an equality in any mathematical sense, but can be a useful approximate equality in certain important fields, like, e.g. the study of single molecules in solution.
The JE equates the difference between the free energy of two equilibrium states $A$ and $B$ of a system in contact with a heat bath, with an average over the irreversible work done over all paths from $A$ to $B$. This work is done in general by a set of external force centers, which can be characterized by a set of time dependent parameters $\{\lambda_j(t)\}$ in the Hamiltonian of the system: $H \equiv H(\{p_i(t)\}, \{q_i(t)\}; \{\lambda_j(t)\})
= H(\Gamma(t); \lambda(t))$. Here $\Gamma(t) \equiv
\{p_i(t)\},\{q_i(t)\}$, defines the microscopic state of the system in its phase space, by giving the values of the momenta $\{p_i(t)\}$ and coordinates $\{q_i(t)\}$ respectively, of the $i
= 1,2,..., N$ particles of the system at time $t$. For simplicity we will restrict ourselves to only one parameter $\lambda(t)$, so that $j = 1$. One is interested in the Helmholtz free energy difference $\Delta F=F_B-F_A$ between the initial and final equilibrium states $A$ and $B$. By considering the work $W$ done on the system when the external forces bring the system from state $A$ to state $B$ along all possible [*[irreversible]{}*]{} paths, with weights $e^{-\beta W_{irr}}$, the JE reads: $$<e^{-\beta W_{irr}}> = e^{-\beta \Delta F}$$ where $\beta = 1/k_BT$, with $T$ the temperature of the heat bath and $k_B$ Boltzmann’s constant. Since in (1) the average $< \; >$ is over the work $W_{irr}$ done over all possible irreversible paths in phase space from $A$ to $B$, the derivation of the left hand side of Eq.(1) proceeds microscopically via statistical mechanics.
We will consider here the case that the system is for all times $t
\geq 0$ in thermal contact with a heat bath. The heat exchange with the heat bath can be reversible or irreversible, depending on whether it takes place with the internal temperature $T_I$ of the system equal or not equal to the external temperature $T_E$ of the heat bath surrounding the system, respectively. In the former case the system will always be in a (canonical) equilibrium state during the work process. As is explained in textbooks of Thermodynamics, a slow or fast work process does not guarantee at all its reversibility or its irreversibility, only the equality of internal and external parameters does.
At $t=0$ the system in state $A$ is coupled to a heat bath of temperature $T$, so that its temperature is also $T$. It has then a canonical distribution given by$^{[1]}$: $$f(\Gamma(0),0) = \frac{1}{Z_0} \; e^{-\beta H (\Gamma(0),
\lambda(0))}$$ Here $Z_0 = Z_A$ is the canonical partition function at $t=0$, where $\lambda = \lambda (0) = \lambda_A$. We assume that work is done on the system during a total time $\tau$, so that $0 \leq t
\leq \tau$, while $\lambda(t)$ goes from $\lambda(0) = \lambda_A$ to $\lambda(\tau) = \lambda_B$, the final value of $\lambda$ which corresponds to the equilibrium state $B$. The (mechanical) work $W$ done on the system in an initial micro phase $\Gamma(0)$ over a time $t$ will be given by: $$W_t \equiv W(\Gamma,t) = \int^t_0 dt' \frac{\partial H (\Gamma
(t'), \lambda(t'))}{\partial \lambda} \; \; {\dot{\lambda}}(t')$$ where ${\dot{\lambda}} (t') = d \lambda(t')/dt'$.
We will assume here for simplicity that the rate of change of $\lambda$, i.e. ${\dot{\lambda}}$, is constant.
However, if the change of $\lambda (t)$ with $t$ is such that irreversible processes are induced in the system driving it possibly far from equilibrium, then, in order to reach an equilibrium final state $B, \lambda (t)$ must be such that the equilibrium state $B$ can actually be reached. During this transition, heat exchange will take place (and possibly work will be done). This will depend on the procedure of varying $\lambda$ with time and on the nature of the induced irreversible processes. The change $\Delta F$ is computed by Jarzynski considering only the mechanical work in going from $A$ to $B$.
JE for thermostatted system
===========================
Since the system is in constant contact with a heat bath, it becomes important whether the (constant) rate of change ${\dot{\lambda}}$ of $\lambda$ allows thermal equilibrium to be maintained at all times between the system and the heat bath. If it does, the process is reversible[[^1]]{}, if it does not then the process is irreversible, since the internal and external variables will then not always be equal, or may not even be definable in the system.
In addition, the reversibility can be characterized in a different way, viz., the strength of the coupling of the system to the heat reservoir or the rate of heat transfer, which we will call ${\dot{c}}$. It is this quantity together with the work rate ${\dot{w}} = {\dot{\lambda}} \frac{\partial H}{\partial \lambda}$, which will determine whether the system will remain in thermal equilibrium at all times i.e., whether a reversible or irreversible process takes place. We can consider now several cases.
- The rate of work done on the system ${\dot{w}}$ is very small $({\dot{\lambda}} \approx 0)$ and the coupling ${\dot{c}}
\approx 0$ also. Then depending on the ratio ${\dot{w}}/{\dot{c}}$, thermal equilibrium can be maintained between the system and the heat bath, at all times $0\leq t \leq
\tau$ when ${\dot{w}} < < {\dot{c}}$, i.e., there will be plenty of time for the system to exchange heat with the reservoir, so that the disturbance of the system due to the work done on the system and which brings it out of thermal equilibrium, can be readjusted for all $0 \leq t \leq \tau$ so that the system is always in thermal equilibrium with the heat bath at temperature $T$. If this obtains during the entire time $\tau$ that work is done on the system then the work process is isothermal and reversible.
- If the ${\dot{w}}$ is very small, but the coupling ${\dot{c}}$ is not, then the work process will also be isothermal and reversible.
- If, on the other hand, ${\dot{w}}$ is very large and the coupling ${\dot{c}}\approx 0$, then there is no way to maintain thermal equilibrium during the work time $\tau$ and the system is thermodynamically in a non-equilibrium state.
- However, if ${\dot{w}}$ is very large, then ${\dot{c}}$ has to be sufficiently larger to maintain thermal equilibrium, if at all possible, and in general one cannot assume that this is so.
Critique
========
The above considerations are relevant because, during the evolution of an initial phase $\Gamma(0)$ of the system, it will not only be subject to the mechanical work done by the external forces via the Hamiltonian $H$ on the system alone, but also to the simultaneous energy exchange with the heat bath. Therefore, to know the mechanical work and the heat separately along a phase space path, one has, in principle, to know the microscopic state of both the system, the heat bath and their coupling, so as to know whether, and if so, how much and in what direction, heat exchange between the system and the heat bath has taken place.[[^2]]{}
Consequently the introduction in phase space of the weight $e^{-\beta W_{irr}}$, for every irreversible (stochastic) path, where the inverse temperature $\beta$ of the heat bath is used in the weight for every $W_{irr}$, does not seem to make physical sense. In fact, even if the mechanical work $W_{irr}$ itself could be precisely determined, the canonical weight $e^{-\beta W_{irr}}$ associated with it, especially when no internal (system) temperature is known or can even be defined - a possibility of being far from equilibrium, which Jarzynski notes himself$^{[2]}$ - seems completely arbitrary and the use of the heat bath temperature $1/\beta$ - the only known temperature available - without foundation. As a consequence, an average over the microscopic $e^{-\beta W_{irr}}$, as carried out in Eq.(1), does not seem physically meaningful unless they are very close to reversible ones and provide then a good approximation to those. Therefore the JE is correct but trivial if $W=W_{rev}$ and it seems that the use of $W_{irr}$ instead of $W_{rev}$ is, in general, unfounded. A further discussion of this point can be found in points of section 5.
In the above we argued that the heat exchange has not been properly taken into account and an essentially only mechanical theory has been used to derive Eq.(1), while, however, also non-mechanical work e.g. thermal expansion due to heat energy has to be considered.
A striking example of this is Jarzynski’s remark$^{[2]}$ that it would suffice to let the system evolve from parameter values $\lambda_A$ to $\lambda_{B^*} = \lambda_B$ regardless of whether $B^*$ is the equilibrium state $B$, whose free energy difference $\Delta F$ with state $A$ one wants. As Jarzynski clearly states, the [*[non-equilibrium]{}*]{} state $B^*$ at $\lambda_{B^*} =
\lambda_B$, reached [*[before]{}*]{} the equilibrium state $B$, can easily be transformed into the desired equilibrium state $B$, while the amount of (mechanical) work done to get from $\lambda_A$ to $\lambda_B$ remains the same. The only way, however, to go from a non-equilibrium state $B^*$ with $\lambda_B$ to an equilibrium state at $B$, with no more work done, is through the contact with the heat bath at temperature $T$. Since the free energy $F$ will be a minimum in equilibrium, the non-equilibrium $\lambda_{B^*} =
\lambda_B$ state will, [*[only if properly chosen]{}*]{}, indeed go to the equilibrium state $B$, accompanied, however, with an exchange of an unknown amount of heat and possibly work ,which depend on $B^*$, all of which will change the free energy.
So far we have discussed the general physical theory of Jarzynski as found in refs.\[1,2\]. This theory is supposed to hold for systems of any size. As Jarzynski remarks$^{[2]}$ to obtain observable effects the systems have to be small, since otherwise the fluctuations of the work values become too small to be observed. In this connection measurements on single molecules in aqueous solutions seem very appropriate as a check on the JE.
Experiment
==========
Liphardt et al$^{[4,6]}$ have carried out such an experiment on the free energy difference (in their case the Gibbs free energy $\Delta G$ instead of $\Delta F$) between the unfolded and the folded conformations of a single P5abc RNA molecule, suspended between two handles, in an aqueous salt solution to check the JE. The experiments were carried out carefully, by stretching a single RNA molecule many times, between the folded and unfolded conformations $A$ and $B$. In this case the irreversibility of the procedure manifests itself in hysteresis curves associated with a cycle $A$ (folded) $\rightarrow B$ (unfolded) $\rightarrow A$ (folded). A number of constant stretching (switching) rates of 2 - 5 pN/s (slow) and 34 pN/s and 52 pN/s (fast), (the ${\dot{\lambda}}$ above), between folded and unfolded states were applied and histograms were made of the work done versus the extension of the molecule. To obtain statistics, seven independent sets of data were collected for seven different RNA molecules with a slow step between two fast steps and about 40 unfolding-refolding cycles per molecule were performed i.e. about 300 independent measurements were made. Results for the average work differences for these three switching rates relative to the reversible work were plotted in bins of about 0.7 $k_BT$. Plotting for each switching rate the most probable value (i.e. the maximum of the Gaussian fits to their histograms) of the work done on each molecule, the fast extension (switching) rates produced work values $\approx 2-3 k_BT$, above the expected thermodynamic free energy difference of about 60 $k_BT$, because of irreversible contributions (apart from measurement errors) to the work, in agreement with the Second Law of Thermodynamics. On the other hand, an average for each switching rate, using the JE Eq.(1) gave values for the expected free energy difference within their estimated error.
We note that the deviations from equilibrium for even the fastest switching time, are in fact only 5% (or about $3k_BT$), which would not be unusual for such measurements. In addition, the thermal equilibration time is of the order of picoseconds, while we estimate the structural equilibration time to be of the order of milliseconds$^{[7]}$ and the experimental switching times of the order of seconds (slow) to 0.1 seconds (fast). Therefore since ${\dot{w}} < {\dot{c}}$, i.e., essentially reversible isothermal measurements (cases a) and b) above) were performed. Considering the above mentioned time scales, the entire experiment could well be very close to an isothermal one for all switching rates used, so that the JE equality gives results very close to those of using only reversible paths in phase space.[[^3]]{}
A few additional remarks on this experiment follow.
1\. The Liphardt et al. data appear to fit better to the JE for large stretching than to the second cumulant or the “Fluctuation Dissipation Relation (FD)”: $<W_{irr}> = W_{rev} +
\beta \sigma^2/2$, where $\sigma^2=<W_{irr}^2> - <W_{irr}>^2$ is the variance of $W_{irr}$[[^4]]{} with $W_{irr} = W - W_{rev} = W_d$, the dissipative work. This, in spite of the fact the system is not far from equilibrium.
We note that in the table S1 of the supplementary material their $<W_d>$ (scaled with $\beta$) for various extensions and switching rates are rather close to $\sigma$, except $<W_d>$ in the reversible case (very slow switch) where $<W_d>$ is assumed to be zero. If one computes $\sigma^2/2$, using the $\sigma$ in the second column of $S$1, the resulting $<W_d>$ is too large. If, on the other hand, one corrects the $\sigma$’s in the second column for the measurement error by a simple subtraction of the $\sigma$’s at $<W_d>=0$, the estimates for $<W_d>$ are too small. This bracketing of $<W_d>$ indicates that the data are compatible with $<W_d>=\sigma^2/2$.
2\. We will now derive the JE under three assumptions:
1\) The $W_{irr}$ all have the weight $e^{-\beta W_{irr}}$, where $\beta$ is the temperature of the heat bath, not the system;
2\) a Gaussian assumption is made for the measured non-equilibrium distribution functions;
3\) a FD of $<W_d> = \frac{\beta \sigma^2}{2}$ has to be used.
Assuming that the distribution functions for $W_d$ are all Gaussian and using those to fit the data, the left hand side of the JE in eq.(1) can be written as: $$\begin{aligned}
<e^{-\beta W_d}> & = & \int d\beta W_d \exp-[\beta W_d - (\beta
W_d - \beta^2 \sigma^2/2)^2/2\sigma^2] \\ \nonumber & \cdot &
\{\int d\beta W_d \exp - [\beta W_d - \beta^2
\sigma^2/2)^2/2\sigma^2]\}^{-1}\end{aligned}$$
Carrying out the Gaussian integrals leads then to:\
$<e^{-\beta W_d}>$ = 1 or $<e^{-\beta W_{irr}}> = e^{-\beta
W_{rev}} = e^{-\beta \Delta F}$ i.e. the JE. This result depends crucially on the validity of $<W_d> = \frac{\beta \sigma^2}{2}$.\
We emphasize that two of these three assumptions, although approximately correct, are not justified, as explained above, since the first two are not satisfied in non equilibrium systems.\
3. We note that the often quoted claim $^{[4,11,12]}$ that the origin of the correctness of the JE is because of the over-weighting of the negative dissipative work balanced by the under weighting the positive dissipative work has no known basis. That the JE works here, in spite of the fact that it is based on unfounded assumptions, is because of a felicitous accident of Gaussian statistics rather than for reasons of thermodynamics and statistical mechanics.
Discussion
==========
1\. As said before, it is clearly impossible to do justice to all the publications that have appeared in connection with the JE. We have confined ourselves here to represent the contents of two articles on the [*[general foundations]{}*]{} of the JE on Statistical Mechanics by Jarzinski$^{[1,2]}$.\
2. In addition we discussed an important experiment which appears to confirm the JE, since the system is in aqueous solution, close to equilibrium and the processes are (quasi) quasi-static and therefore nearly reversible, in spite of some hysteresis.\
3. Although in Liphardts’ experiment clear hysteresis loops can be seen, they occur over relatively small parts of the trajectories, which mostly appear to be reversible. True irreversible processes, where e.g. no temperature can be defined in the system, have so far not been considered experimentally (because of the liquid surroundings used so far). Such irreversible processes would question severely the assignments of a canonical weights $e^{-\beta W_{irr}}$, with $\beta$ the temperature of the heat bath.
We note that although to observe the JE the systems have to be small, there is a gap between the experiments performed on biochemical systems (in aqueous solutions) and those on small physical (or chemical) systems without an aqueous heat bath, which have not yet been performed.\
4. There are many different theoretical [*[models]{}*]{}, to mimic the heat bath, which have been used to derive the JE. They are all stochastic and, as far as we are aware, all use Markovian and detailed balance properties, which place them, in our opinion, near predominantly reversible heat exchanges. We confine ourselves again to two examples.
a\) This holds for Crooks’$^{[8]}$ derivation of the JE for a simple Ising model. He discretizes there the phase space trajectory in sequences of two sub steps, whereby first a control parameter $\lambda$ does instantaneous work $W$ on the system followed by an exchange of thermal energy $E$ with a heat bath. Either the first forward work step is followed by a heat exchange step which lasts sufficiently long compared to the characteristic relaxation times of the system that a state of canonical equilibrium is reached (cf.Crooks’ eq.(8)) in which case the process is a discrete sequence of equilibrium states and therefore a reversible process (like in a reversible experiment) and the JE will be obtained as that for a reversible process. Or the time after the first forward step is not sufficiently long to obtain an equilibrium state, in which case an irreversible process takes place, which may be close to a reversible one, but no detailed balance obtains at any $\lambda_i$.[[^5]]{}
Similarly Jarzynski’s procedure applied to the Langevin Equation$^{[2]}$, because of the $e^{-\beta W_{irr}}$ weight assumption, is restricted to systems near equilibrium, with near reversible heat exchanges.
b\) Jarzynski made numerical calculations$^{[9]}$ of which we will discuss only that on a harmonic oscillator. He considered $10^5$ simulations of the work done on a single harmonic oscillator, whose frequency is switched from $\omega_0 =1.0$ to $\omega_1 =
2.0$ over a switching time $t_s$. The Hamiltonian of the harmonic oscillator is: $$H = \frac{p^2}{2} + \omega^2_\lambda \; \frac{x^2}{2}$$ where $\omega_\lambda$ switches from $\omega_0$ to $\omega_1 = 2
\omega_0$. If the change from $\omega_0$ to $\omega_1$ proceeds infinitely slowly ($t_s \rightarrow \infty$) and adiabatic invariance can be applied, i.e. $H/\omega$ = constant, then Jarzynski obtains, for a canonical distribution of initial energies a distribution function for the work $W$. Using this to compute the average of the exponential work $e^{-\beta W}$, gives $\beta^{-1} \ln (\omega_1/\omega_0) = 0.693 \beta^{-1} = \Delta
F$, since $\omega_1/\omega_0$ is just the ratio of the canonical partition functions (cf.Jarzynski in ref.9 eq.(59) and fig.3 ($W^x$ points)). One can get an idea of the average value of $\beta W$ for fast switchings ($t_s=1$), by using a similar calculation.
One finds then for the arithmetic average of $\beta W$ the result $(\frac{\omega_1}{\omega_0}- 1)= 1$. Adding then $W_{rev} = 1$, a value of 2 is obtained not far from the numerical result (ref.7, fig.3 ($W^a$). However, this result depends critically on the assumption of adiabatic invariance for $t_s=1$, while this actually only holds for $t_s \rightarrow \infty$. Moreover, if for example, the energy would change as $\omega^2$, the Jarzynski weighting would give $2\Delta F$ and the arithmetic weighting would give a value 4 instead of 2 for $\beta W$.
Jarzynski claims that the JE holds for all systems (not only harmonic oscillators), while we see that the JE is critically dependent on adiabatic invariance for finite $t_s$, even for the harmonic oscillator.\
5. Finally we note that eq.(1) implies that a remarkable new relation would follow from the $JE$, if it were an exact equality, viz.: $$<e^{-\beta W_{irr}}> = <e^{-\beta W_{rev}}>$$ In the literature, the eq.(6), or equivalently $<e^{-\beta W_d}> =
1$, is justified by invoking an apparently very fortuitous general cancellation due to the weighting of $W_d>0$ and $W_d<0$ with $e^{-\beta W_d}$. It is entirely unclear to us, how this can be achieved for irreversible processes in a system, for all $\lambda(t)$ and all corresponding phase space paths, using in all cases the (unconnected) heat bath temperature $\beta$ in the weights.
This would imply the equality of the average over the exponentiated work of all irreversible paths from $A$ to $B$, to the (average over all the) reversible path(s) from $A$ to $B$. Considering the unknown nature and wide variety of all irreversible paths this equality does not seem physically understandable. If true, it would incorporate a hitherto unknown symmetry for irreversible processes for any switching rate, i.e. a new extension of the Second Law. To be sure there are special cases, like those treated in section 4, sub 2 and in section 5, sub 4b, where this equality holds under certain specific assumptions for irreversible processes near equilibrium.
[**[References]{}**]{}
1\. C. Jarzynski, [*[Phys.Rev.Lett.]{}*]{} [**[78]{}**]{}, 2690 (1997).\
2. C. Jarzynski, in [*[Dynamics of Dissipation]{}*]{}, P. Garbaczewski, R. Olkiewicz, eds., (Springer, Berlin 2002) p.63 and references therein.\
3. F. Ritort, , 6/12/2003, p.63.\
4. J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, Jr., and C. Bustamante, [*[Science]{}*]{} [**[296]{}**]{}, 1832 (2002).\
5. L. D. Landau and E. M. Lifshitz, [*[Statistical Physics]{}*]{}, Addison Wesley, Reading, Mass. (1958) p.45, Eq.(15.1).\
6. S. K. Blau, [*[Physics Today]{}*]{}, September (2002) p.19.\
7. W. Zhang and S-J. Chen, [*[Proc. Natl. Acad. Sci.USA]{}*]{}, [**[99]{}**]{}, 1931 (2002).\
8. G. E. Crooks, [*[J. Stat. Phys.]{}*]{} , 1481 (1998).\
9. C. Jarzynski, [*[Phys. Rev. E]{}*]{} , 5018 (1997).\
10. Ref. 4, Supplementary Material.\
11. Ritort et al, PNAS , 13544 (2002).\
12. Gore et al, PNAS, . 12564 (2003).\
13. S. R. de Groot and P. Mazur, [*[Non-Equilibrium Thermodynamics]{}*]{}, Dover (1984) p.93.\
14. N. G. van Kampen, [*[Stochastic Processes in Physics and Chemistry]{}*]{}, Elseveier, Amsterdam, V.6 (1992).\
The authors are indebted to Professors R. F. Fox, J. M. Kincaid, and B. Widom and Drs. R. van Zon, E. van Nimwegen and T. Tuschl for helpful discussions. EGDC also gratefully acknowledges support from the Office of Basic Energy Sciences of the US Department of Energy under Grant number DE-FG02-88-ER13847.\
[^1]: This is usually formulated by requiring that the external changes induced in the system are slow when compared to the internal relaxation times needed to return to thermal equilibrium once a change of the system (e.g., its volume) has taken place. In that case the temperature of the system and that of the heat bath will always remain the same during a measurement.
[^2]: For a system in contact with a heat bath the work done on the system is not just the mechanical work, which will change the system’s internal energy $(dE)$, but also work associated with the heat (energy) exchange with the bath $(TdS)$, can already be seen in the isothermal reversible case, where $dW =
dE - dQ = dE - TdS = d(E-TS) = dF^{[5]}$.
[^3]: The remark by Ritort$^{[3]}$ that a system may be far from equilibrium even when close to an isothermal process because of its small size does not seem to be correct. The small size will allow the fluctuations to be measurable. However, they have no bearing on the fast change in the control parameter, needed for deviations far from equilibrium.
[^4]: We note that their FD is derived from the JE, thereby assuming its validity, which is supposed to be checked in the experiment. This manifests itself in the appearance of only work contributions to $\sigma^2$, while $\Delta F$ should also contain, in principle, entropic contributions.
[^5]: The assumed detailed balance at every step $\lambda_i$ is not a direct consequence of microscopic reversibility alone, and has been proved so far only for thermal equilibrium \[13,14\].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Vanadium dioxide (VO$_{2}$) undergoes a first-order metal-insulator transition (MIT) upon cooling near room temperature, concomitant with structural change from rutile to monoclinic. Accurate characterization of lattice vibrations is vital for elucidating the transition mechanism. To investigate the lattice dynamics and thermal transport properties of VO$_{2}$ across the MIT, we present a phonon renormalization scheme based on self-consistent phonon theory through iteratively refining vibrational free energy. Using this technique, we compute temperature-dependent phonon dispersion and lifetimes, and point out the importance of both magnetic and vibrational entropy in driving the MIT. The predicted phonon dispersion and lifetimes show quantitative agreement with experimental measurements. We demonstrate that lattice thermal conductivity of rutile VO$_{2}$ is nearly temperature independent as a result of strong intrinsic anharmonicity, while that of monoclinic VO$_{2}$ varies according to $1/T$. Due to phonon softening and enhanced scattering rates, the lattice thermal conductivity is deduced to be substantially lower in the rutile phase, suggesting that Wiedemann-Franz law might not be strongly violated in rutile VO$_{2}$.'
author:
- Yi Xia
- 'Maria K. Y. Chan'
bibliography:
- 'VO2.bib'
title: 'Renormalized Lattice Dynamics and Thermal Transport in VO$_{2}$'
---
Vanadium dioxide (VO$_{2}$) is a transition metal oxide under intense study, because it exhibits a first-order metal-insulator transition (MIT) from metallic rutile phase (R-VO$_{2}$) to semiconducting monoclinic phase (M-VO$_{2}$) upon cooling to $T_{\text{MIT}}$=340 K, which facilitates near-room-temperature switching of either electronic or thermal conductivity [@Pergament2017]. The mechanism underlying the MIT in VO$_{2}$ has been a longstanding subject of controversy due to the coupled changes in structural and electronic properties across the MIT. Various studies have focused on clarifying whether the MIT is driven by instabilities in electron-lattice dynamics or charge localization due to strong electron-electron correlation [@Wentzcovitch1994; @Hearn1972; @Zylbersztejn1975; @Eyert2002; @Goodenough1971]. A recent study [@Budai2014] suggested the importance of vibrational entropy in driving the MIT based on X-ray/neutron scattering measurements and *ab initio* molecular dynamics (AIMD) calculations [@Hellman2013]. By examining experimentally-measured temperature-dependent phonon density of states (DOS), a discontinuity in DOS shape across the MIT is identified, which is associated with the softening of transverse acoustic modes. The change in vibrational entropy across the MIT was found to be $\Delta S_{\text{ph}}$=1.02$\pm$0.09 $k_{\text{B}}$/VO$_{2}$, which is comparable to the total entropy change of 1.50$\pm$0.15 $k_{\text{B}}$/VO$_{2}$ from latent heat measurements [@Berglund1969], suggesting that phonons dominate over electrons to stabilize R-VO$_{2}$. More recently however, quantum Monte Carlo (QMC) calculations [@Huihuo2015] show that while both phases have antiferromagnetic ground states, a significant change in magnetic coupling strength across the transition leads to magnetic ordering and disordering in M- and R-VO$_{2}$ respectively, thus contributing additional entropy to stabilize R-VO$_{2}$ [@Huihuo2016]. Accurately determining phonon entropy change is therefore important towards resolving the transition mechanism.
While electrical conductivity increase by several orders of magnitude is well documented, only a few studies address the separate contributions from electrons and phonons to thermal transport in VO$_{2}$ across the MIT, primarily due to coupled electron and phonon transport in metallic R-VO$_{2}$. In a recent study, @Lee371 [@Lee371] attempted to decouple the thermal conductivity of R-VO$_{2}$ into electron ($\kappa_{e}$) and phonon ($\kappa_{l}$) contributions, by deducing $\kappa_{l}$ using experimentally-measured phonon linewidths and computed phonon dispersions. The resultant value of $\kappa_{e}$ in R-VO$_{2}$ was anomalously low, implying a significant violation of the Wiedemann-Franz (WF) law, which the authors attributed to unconventional quasiparticle dynamics, i.e. absence of quasiparticles in a strongly-correlated electron fluid where heat and charge diffuse independently [@Lee371].
Since $\kappa_{e}$ can only be indirectly determined experimentally, by subtracting $\kappa_{l}$ from the total thermal conductivity measurements, accurate modeling of $\kappa_{l}$ of R-VO$_{2}$ is crucial. From a theoretical point of view, R-VO$_{2}$ is challenging due to its strong anharmonicity and lattice metastability. We present a phonon renormalization scheme based on iteratively refining vibrational free energy, temperature-dependent interatomic force constants (IFCs) obtained using compressive sensing lattice dynamics (CSLD) [@csld]; and application of this scheme towards a comprehensive study of lattice dynamics and thermal transport properties of VO$_{2}$. Our study quantitatively clarifies the the changes in vibrational entropy and $\kappa_{l}$ across the MIT, demonstrates the importance of magnetic entropy, and suggests that WF law may not be strongly violated in R-VO$_{2}$.
Existing methods that are capable of treating strong anharmonic effects nonperturbatively are based on or related to self-consistent phonon theory [@Werthamer1970], such as self-consistent *ab initio* lattice dynamics (SCAILD) [@Souvatzis2009] and stochastic self-consistent harmonic approximation (SSCHA) [@Errea2014]. The present implementation of phonon renormalization is in the spirit of SCAILD, where vibrational free energy is iteratively refined based on temperature-dependent atomic displacements. As implemented by Roekeghem *et. al.* [@Roekeghem2016], an improvement over SCAILD can be achieved by using the full quantum mean square thermal displacement matrix, which allows for simultaneous update of both phonon frequencies and eigenvectors. The temperature-dependent atomic displacements $\{u_{a,\alpha}\}$ are generated according to the probability $\rho (\{u_{a,\alpha}\}) \propto \text{exp}( -\frac{1}{2} \mathbf{u} \Sigma^{-1} \mathbf{u} )$ to find a displaced configuration in the harmonic approximation, where $u_{a,\alpha}$ is the displacement of atom $a$ in $\alpha$ direction, and $\Sigma$ is known as the quantum covariance matrix of displacement vector [@Errea2014; @Roekeghem2016] $$\label{eq:covar}
\Sigma_{a\alpha, b\beta} = \frac{\hbar}{2\sqrt{M_{a}M_{b}}} \sum_{\lambda} \frac{\left(1+2n_{\omega_{\lambda}}\right) }{\omega_{\lambda}} \epsilon^{\lambda}_{a\alpha} \epsilon^{\ast\lambda}_{b\beta},$$ where $M$, $\omega$, $\epsilon$, $n$, $\lambda$ are atomic mass, phonon frequency, eigenvector, Bose-Einstein distribution and branch index respectively. Various sets of $\{u_{a,\alpha}\}$ can be generated using the covariance matrix following a given distribution, such as continuous Gaussian distribution or discrete Rademacher distribution. Note that the current scheme to generate temperature-dependent atomic displacements is superior to sampling trajectory from AIMD, which suffers from the absence of nuclear quantum effects and thus underestimates the thermal displacements for temperatures below the Debye temperature ($\approx$ 1000 K for VO$_{2}$). A self-consistent loop was formed by generating $\{u_{a,\alpha}\}$ using effective IFCs extracted from the previous step. Both $2^{\text{nd}}$- and $3^{\text{rd}}$-order IFCs were simultaneously extracted using a recently developed method named compressive sensing lattice dynamics (CSLD) [@csld], wherein DFT forces are expressed as a Taylor expansion in displacements and the coefficients are obtained from sparse regression. From the extracted effective IFCs, phonon lifetimes were evaluated using Fermi’s golden rule by treating 3$^{\text{rd}}$-order IFCs as perturbation to harmonic phonons [@ziman], and linearized Boltzmann transport equation (BTE) was solved in an iterative manner to account for the non-equilibrium phonon distributions [@omini1; @omini2; @broido; @wuli]. (See Supplemental Material [@VO2SI] for more details of CSLD and implementation/validation of our renormalization scheme, and structural and computational details.)
. \[fig:Disp\_DOS\]
To confirm the convergence of renormalized phonon dispersions, vibrational free energy of R-VO$_{2}$ computed in each iteration, as shown in Fig. \[fig:Disp\_DOS\](a), is examined and found to achieve good convergence to about $\pm$1 meV/atom after seven iterations. Without accounting for temperature effects, phonon dispersions of M-VO$_{2}$ computed in the harmonic approximation, as shown in Fig. \[fig:Disp\_DOS\](b), have well defined frequencies with real values, exhibiting normal harmonic behavior. In contrast, there is a large number of imaginary frequencies in phonon dispersions of R-VO$_{2}$ across the Brillouin zone, consistent with previous theoretical studies [@Kim2013; @Budai2014; @Lee371]. With temperature effects considered, the renormalized phonon dispersions of R-VO$_{2}$ at 425 K in Fig. \[fig:Disp\_DOS\](c) exhibit hardening of optical modes and achieve overall good agreement with experimental IXS measurements [@Budai2014], confirming the validity of the phonon renormalization approach. Specifically, the computed high-lying transverse acoustic (TA2), longitudinal acoustic (LA) and low-lying zone-center optical (ZCO) phonon frequencies are in excellent agreement with experiments, while low-lying transverse acoustic (TA1) phonon frequencies display some overestimation, particularly along $\Gamma$-R and $\Gamma$-M paths. Renormalized phonon dispersions of R-VO$_{2}$ were additionally computed at 360 and 600 K to investigate the temperature dependence. As shown in Fig. S6 [@VO2SI], the presence of a few imaginary frequencies at 360 K indicates that R-VO$_{2}$ tends to be unstable near MIT. Only the low-lying ZCO phonon mode is significantly hardened (from 17.8 meV to 23.3 meV) with temperature enhanced from 425 to 600 K, indicating that the ZCO phonon mode is severely anharmonic.
To quantitatively investigate the anharmonicity of the ZCO phonon mode, Fig. \[fig:Disp\_DOS\](d) shows its potential energy surface as a function of atomic displacement following the renormalized eigenvector, wherein V and O atoms move away from each other along the rutile $c$ axis. The resultant double-well potential of the ZCO phonon mode demonstrates the structural instability of R-VO$_{2}$, which appears in 0 K phonon dispersions. Indeed, by projecting the renormalized eigenvector onto those zone-center phonon modes computed at 0 K, it is found that ZCO phonon mode matches exactly the zone-center imaginary optical mode. Therefore, the structure of R-VO$_{2}$ is a dynamic average over symmetry-broken minima separated by relatively deep energetic barriers ($\approx$ 42 meV), and the positions of V and O atoms represent their averaged spatial occupations, consistent with experimental observations of large thermal displacements [@McWhan1974]. The severe anharmonicity of the ZCO mode is also reflected in the detailed shape of the energy profile, an accurate fitting of which requiring polynomials well beyond 6$^\text{th}$-order.
To confirm and explain the experimentally observed large phonon entropy change across the MIT [@Budai2014], the total phonon DOS of M- and R-VO$_{2}$ are compared in Fig. \[fig:Disp\_DOS\](e). It can be seen that temperature-induced phonon renormalization significantly alters the DOS of R-VO$_{2}$. Further increasing temperature from 425 K to 600 K tends to slightly harden low-lying modes while softening high-lying modes. Comparison of total DOS between M- and R-VO$_{2}$ displays an abrupt blueshift of low-lying DOS shoulders ($\approx$ 13 meV) with phonon modes notably hardened when temperature is reduced below the MIT. These phonon modes are mainly associated with the TA1 modes near zone boundary and constitute the majority of vibrational entropy change. To quantify it, temperature-dependent vibrational entropy is computed using phonon dispersions of M-VO$_{2}$ at 0 K and R-VO$_{2}$ at 425 K. As shown in Fig. \[fig:Disp\_DOS\](f), both PBE and PBEsol xc functionals are found to yield similar vibrational entropy change of 0.64 $k_{\text{B}}$/VO$_{2}$ at 340 K, which is smaller than 1.02 $k_{\text{B}}$/VO$_{2}$ reported by @Budai2014. As a result, total entropy change, which includes contributions of 0.64 and 0.25 $k_{\text{B}}$/VO$_{2}$ from lattice vibration and partial occupancy of electrons [@Budai2014] in R-VO$_{2}$ respectively, is smaller than experimental report of 1.50 $k_{\text{B}}$/VO$_{2}$ [@Berglund1969]. The additional entropy change may be explained by the magnetic contribution revealed by the aforementioned QMC study [@Huihuo2015; @Huihuo2016]. In M-VO$_{2}$, singlets are formed due to the strongly coupled magnetic moments within the dimers, which leads to zero entropy for such a state. Whereas above the transition temperature, R-VO$_{2}$ is in a magnetic disordered state, leading to an entropy change of $\Delta S = k_{\text{B}}\text{ln}(2) = 0.69$ $k_{\text{B}}$/VO$_{2}$ across the MIT. As shown in Fig. \[fig:Disp\_DOS\](f), the total entropy change accounting for electron, phonon and magnetic contributions is 1.58 $k_{\text{B}}$/VO$_{2}$, which is in excellent agreement with experimental value of 1.50 $k_{\text{B}}$/VO$_{2}$ [@Berglund1969], implying that magnetic ordering is also important in stabilizing the rutile phase.
Having validated the present phonon renormalization approach by comparing the computed phonon dispersions and vibrational entropy change against experiments, we calculate lattice thermal transport properties and compare them with experiments in Fig. \[fig:Kappa\_Ana\]. Considering that experiments found that both phonon lifetimes and $\kappa_{l}$ of R-VO$_{2}$ show no temperature dependence up to 425 K [@Budai2014; @Lee371], the $\kappa_{l}$ of R-VO$_{2}$ was computed using renormalized IFCs at 425 K. By comparing the computed $\kappa_{l}$ of polycrystalline VO$_{2}$ to experimental measurements reported by @ANDREEV1978 and @Chen2012, as shown in Fig. \[fig:Kappa\_Ana\](a), it is found that both theoretical and experimental results show $1/T$ dependence of $\kappa_{l}$ in M-VO$_{2}$. The $\kappa_{l}$ of R-VO$_{2}$ at 600 K was further evaluated to investigate its temperature dependence. The PBEsol results show a slightly decreased $\kappa_{l}$ from 2.95 to 2.60 Wm$^{-1}$K$^{-1}$ with temperature increased from 425 to 600 K, which is considerably slower than $1/T$ and consistent with the “amorphous" character of heat conduction without presence of significant structural disorder found in Ref. [@ANDREEV1978]. We find that this weak temperature dependence of $\kappa_{l}$ is due to strong anharmonicity, which significantly renormalizes 2$^{\text{nd}}$ and 3$^{\text{rd}}$-order IFCs.
Compared to experiments, PBE results agree very well with Ref. \[\], while both PBE and PBEsol results exhibit overestimation compared to Ref. \[\], the later of which is probably caused by the presence of defects such as grain boundaries and pores in R-VO$_{2}$ [@Chen2012]. Nevertheless, both PBE and PBEsol yield significant decrease of $\kappa_{l}$ across the MIT (1.4 and 1.0 Wm$^{-1}$K$^{-1}$ for PBE and PBEsol respectively), which is comparable with the value of 1.3 Wm$^{-1}$K$^{-1}$ reported in Ref. \[\]. Since $\kappa_{l}$ of R-VO$_{2}$ in Ref. \[\] is evaluated using standard Lorentz number $L_{0}$, the present theoretical results suggest that WF law is not strongly violated in polycrystalline VO$_{2}$. However, compared to the estimated $\kappa_{l}$ along the rutile $c$ axis using first-principles phonon dispersions and phonon scattering rates obtained from the IXS measurements [@Lee371] as shown in Fig. \[fig:Kappa\_Ana\](b), both PBE and PBEsol results display smaller $\kappa_{l}$ for M- and R-VO$_{2}$ and a much larger decrease of $\kappa_{l}$ across the MIT, which is primarily due to significant differences in the $\kappa_{l}$ of R-VO$_{2}$. Since $\kappa_{l}$ is subtracted from total thermal conductivity to estimate $\kappa_{e}$ in Ref. [@Lee371], which is subsequently utilized to identify a strong violation of WF law in R-VO$_{2}$, it is crucial to confirm and verify $\kappa_{l}$ of R-VO$_{2}$.
To explore the origin of our predicted lower $\kappa_{l}$ of R-VO$_2$, the full widths at half maximum (FWHM = $h / \tau$ with lifetime $\tau$) and mean free paths (MFP = $|\mathbf{v}| \tau$, where $\mathbf{v}$ is group velocity) of phonon modes at 425 K and varying energies are compared with those estimated using IXS measurements [@Lee371]. As shown in Fig. \[fig:Kappa\_Ana\](c), the predicted FWHMs are consistent with IXS data. Considering the fact that there are many different lifetimes for phonon modes with similar energies and IXS samples a limited space in the full Brillouin zone, a better comparison with experiments can be achieved by computing energy-dependent FWHM weighted by squared phonon group velocity, which is a more rigorous way to average the FWHM than linearly interpolating the phonon modes with small FWHMs as adopted in Ref. \[\]. Our predicted energy-dependent FWHM agrees quantitatively with IXS data from low to high phonon energies, further verifying the accuracy of computed phonon scattering rates. The same good agreement is also achieved in mode-resolved MFP, as shown in Fig. \[fig:Kappa\_Ana\](d). Our results imply that simple linear interpolation tends to significantly overestimate/underestimate FWHMs of phonon modes with low/high energies, in addition to neglecting the strong anisotropy of mode-dependent FWHM. Since the energy cumulative $\kappa_{l}$ (see Fig. S7 [@VO2SI]) shows that phonon modes with energies larger than 10 meV contribute about 70% of total $\kappa_{l}$, the $\kappa_{l}$ of R-VO$_{2}$ deduced by linearly interpolated FWHMs in Ref. \[\] is therefore overestimated.
Note that the present predicted $\kappa_{l}$ of R-VO$_{2}$ is also potentially overestimated because (1) the renormalized TA1 modes have slightly higher energies than experimental measurements, which commonly lead to larger $\kappa_{l}$ because of reduced scattering rates and increased group velocity, and (2) high-order phonon-phonon interactions beyond three phonon processes are neglected in our calculations. To illustrate the drop of $\kappa_{l}$ across the MIT upon heating, comparisons of mode-resolved phonon lifetime, group velocity and MFP between M- and R-VO$_{2}$ are shown in Fig. S8(a)-(c) [@VO2SI]. The phonon lifetimes of M-VO$_{2}$ are overall significantly longer than those of R-VO$_{2}$, while the phonon group velocities of both R- and M-VO$_{2}$ are fairly similar, resulting overall larger MFPs in M-VO$_{2}$ than in R-VO$_{2}$. As a consequence, a substantial change of $\kappa_{l}$ across the MIT is expected, suggesting that WF law may not be strongly violated as demonstrated in Ref. \[\].
In summary, we have applied a first-principles-based phonon renormalization scheme to model lattice dynamics and thermal transport properties of VO$_{2}$ across the metal-insulator transition. Using the obtained temperature-dependent 2$^{\text{nd}}$- and 3$^{\text{rd}}$-order interatomic force constants, we computed renormalized phonon dispersions of rutile VO$_{2}$ and identified its intrinsic strong anharmonicity associated with low-lying zone-center optical mode. We also computed vibrational entropy change across the MIT, based on which we found that contribution from magnetic entropy may also be important. Finally, lattice thermal transport properties were investigated for both phases and a significant change of $\kappa_{l}$ across the MIT is confirmed, suggesting that Wiedemann-Franz law might not be strongly violated in R-VO$_{2}$.
**Acknowledgements** We thank Huihuo Zheng for fruitful discussions. This material is based upon work supported by Laboratory Directed Research and Development (LDRD) funding from Argonne National Laboratory, provided by the Director, Office of Science, of the U.S. Department of Energy under Contract No. DE-AC02-06CH11357. Use of the Center for Nanoscale Materials, an Office of Science user facility, was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
**Supplementary Materials: Renormalized Lattice Dynamics and Thermal Transport in VO$_{2}$**
Yi Xia$^{1}$ and Maria K. Y. Chan$^{1}$
Methodology
===========
Compressive sensing lattice dynamics
------------------------------------
Compressive sensing lattice dynamics (CSLD) [@csld] was used to extract the 2$^\text{nd}$- and 3$^\text{rd}$-order interatomic force constants (IFCs). Within CSLD, the total energy is expressed as Taylor expansion in terms of the atomic displacements $$\label{eq:Taylor}
E=E_{0}+\Phi_{\mathbf{a}}u_{\mathbf{a}}+\frac{1}{2}\Phi_{\mathbf{ab}}u_{\mathbf{a}}u_{\mathbf{b}}+\frac{1}{3!}\Phi_{\mathbf{abc}}u_{\mathbf{a}}u_{\mathbf{b}}u_{\mathbf{c}}+\cdots,$$ where $E_{0}$ is the ground state energy, $u_{\mathbf{a}} \equiv u_{a,\alpha}$ is the displacement of atom $a$ in the Cartesian direction $\alpha$, and $\Phi_\mathbf{ab}$, $\Phi_{\mathbf{abc}}$ are the harmonic and 3$^\text{rd}$-order IFCs. The linear term with $\Phi_\mathbf{a}$ is absent if the reference lattice sites represent mechanical equilibrium, and the Einstein summation convention over repeated indices is used. CSLD belongs to the class of direct supercell methods where DFT forces are used to “invert” Eq. using the force-displacement relation $$F_{\mathbf{a}}=-\Phi_{\mathbf{a}}-\Phi_{\mathbf{ab}}u_{\mathbf{b}}-\frac{1}{2}\Phi_{\mathbf{abc}}u_{\mathbf{b}}u_{\mathbf{c}}-\cdots.$$ The determination of IFCs can be rewritten as a linear problem, $$\label{eq:Force}
\mathbf{F}=\mathbb{A} \mathbf{\Phi},$$ if the unknown IFCs are arranged in a column vector $\mathbf{\Phi}$ and $\mathbf{F}$ is a column vector of the calculated atomic forces on individual atoms in a set of training configurations with displaced atoms. $\mathbb{A}$ is a matrix formed by the products of the atomic displacements in the chosen set of training configurations and is commonly referred to as the sensing matrix $$\mathbb{A} = \begin{bmatrix} -1 & -u_{\mathbf{b}}^1 & - \frac{1}{2} u_{\mathbf{b}}^1 u_{\mathbf{c}}^1 & \cdots \\
& \cdots & & \\
-1 & -u_{\mathbf{b}}^L & - \frac{1}{2} u_{\mathbf{b}}^L u_{\mathbf{c}}^L & \cdots
\end{bmatrix}.$$ The superscripts represent a combined index labeling different atoms in the supercells of the selected training configurations; here we use one supercell with $M$ atoms and generate $N_\text{conf}$ displacement configurations, so that $L=MN_\text{conf}$. Derivative commutativity, space group symmetry and translational symmetry are used to further reduce the number of independent parameters in $\mathbf{\Phi}$. Since each symmetry operation can be written as a linear constraint on $\mathbf{\Phi}$ as $$\mathbb{B}\mathbf{\Phi}=\mathbf{0}$$ and these physical constraints can be strictly enforced by finding the basis vectors of the null-space ($\mathbb{C}$) of matrix $\mathbb{B}$. If the null-space dimension of $\mathbb{B}$ is $N_{\bm{\phi}}$, we are left with $N_{\bm{\phi}}$ independent parameters in the resultant $\bm{\phi}$. The original $\mathbf{\Phi}$ can be expressed as the product of $\mathbb{C}$ and $\bm{\phi}$ $$\mathbf{\Phi} = \mathbb{C} \bm{\phi}$$ where $ \mathbb{C}$ is a $N_{\mathbf{\Phi}} \times N_{\bm{\phi}}$ matrix.
Mathematically, the solution of Eq. is obtained from a convex optimization problem that minimizes a weighted sum of the root-mean-square fitting error and the $\ell_{1}$ norm of the IFC vector $\mathbf{\Phi}$ $$\begin{split}
\mathbf{\Phi}^\text{CS} &= {\arg \min}_\mathbf{\Phi} \; \mu \| \mathbf{\Phi} \|_1 + \frac{1}{2} \| \mathbf{F} - \mathbb{A} \mathbf{ \Phi} \|^2_2 \\
&\equiv {\arg \min}_\mathbf{\Phi} \; \mu \sum_{I} | \Phi_{I} | + \frac{1}{2} \sum_{a \alpha} \left( F_{a\alpha} - \sum_J A_{a\alpha, J} \Phi_J \right)^2,
\end{split}
\label{eq:L1min}$$ where $I$ is a composite index representing the collection of atomic sites and Cartesian indices defining the IFC tensor elements, $I \equiv (\mathbf{a},\mathbf{b}, \ldots )$ and $\mu$ is a parameter that adjusts the relative weights of the fitting error versus the absolute magnitude of the nonzero IFC components represented by the term with the $\ell_{1}$ norm; large values of $\mu$ favor solutions with very few nonzero IFCs at the expense of the accuracy of the fitted forces (“underfitting”), while very small $\mu$ will give a dense solution with many large nonzero IFCs that fits the DFT forces well, but may have poor predictive accuracy due to numerical noise, both random and systematic (“overfitting”). There is an optimal range of $\mu$ values between these two extremes where a sparse IFC vector $\mathbf{\Phi}$ can be obtained with excellent predictive accuracy. We note that the addition of the $\ell_1$ term solves both difficulties of the least-squares fitting approach described above: the linear problem Eq. can be underdetermined and the ill-conditioned nature of the sensing matrix does not present a problem because the $\ell_1$ norm suppresses the numerical noise stemming from the small eigenvalues of $\mathbb{A}^T\mathbb{A}$. As a consequence, according to tests performed on several systems [@Jiangang2016; @Jiangang2017], we find the introduction of $l_{1}$ norm term greatly improves the convergence of fitted higher-order IFCs. For example, it generally requires to calculate hundreds of supercell structures to extract 3$^{\text{rd}}$-order IFCs with a reasonable diameter cutoff, while only tens of supercell structures need to be evaluated within the CSLD framework, which significantly reduce computational cost without sacrificing accuracy. Further details of the CSLD method, numerical approaches for solving the minimization problem and its performance for calculating the lattice thermal conductivities of strongly anharmonic crystals can be found in Ref. \[,,\].
Phonon renormalization: Temperature-dependent atomic displacements and self-consistency
----------------------------------------------------------------------------------------
To generate atomic displacements $\mathbf{u} \equiv \{ u_{a,\alpha} \}$ according to a given covariance matrix $\Sigma(\mathbf{u}) $, a lower triangle matrix $L$ needs to be computed through Cholesky decomposition of $\Sigma$, i.e., $LL^{\text{T}}=\Sigma(\mathbf{u})$. Provided that a vector $\mathbf{x}$ with the same dimension of $\mathbf{u}$ has a covariance matrix of $\Sigma(\mathbf{x})$ and the variance of $\mathbf{u}\equiv L\mathbf{x}$ is $L\text{var}(\mathbf{x})$, the covariance of $L\mathbf{x}$ becomes $L\Sigma(\mathbf{x})L^{\text{T}}$. Therefore, $\Sigma(\mathbf{u}) $ can be recovered if $\mathbf{x}$ has identity covariance matrix, indicating that $\mathbf{u}$ can be generated using $\mathbf{x}$ following either continuous Gaussian distribution or discrete Rademacher distribution. According to our tests on SrTiO$_{3}$ and VO$_{2}$, these two distributions yield very similar results if atomic configurations are sufficiently sampled, and the later generally requires fewer samples (on the order of 10 for each iteration within the CSLD framework) due to a more constrained distribution. Therefore we used discrete Rademacher distribution, while Roekeghem *et. al.* used continuous Gaussian distribution [@Roekeghem2016]. Fig. \[fig:SelfCon\] shows the self-consistent loop in the iterative phonon renormalization procedure. To suppress the fluctuations of computed vibrational free energy in each iteration and achieve faster convergence, it is important to introduce a mixing parameter $\eta$, which is 0.5 in this study and used to average the phonon mode frequencies from current and previous iteration.
Computational details
=====================
Vienna $Ab\ initio$ Simulation Package (VASP) was used to perform structural relaxation and self-consistent calculations [@Vasp1; @Vasp2; @Vasp3; @Vasp4]. The projector-augmented wave (PAW) [@paw] method was used in conjunction with the generalized gradient approximation (GGA) [@gga] for the exchange-correlation (xc) functional [@dft]. Perdew-Burke-Ernzerhof (PBE) and its revised version for solids (PBEsol) [@pbe; @Perdew2008] xc functionals were used, since recent studies [@Budai2014; @Ropo2008] show that (1) PBE tends to predict lattice dynamics properties in good agreement with experiments and (2) PBEsol generally yields better lattice parameter via achieving a more balanced binding energy compared to PBE and LDA. Experimental lattice parameters were found [@Budai2014], as well as confirmed in this study (see results and discussions section), to better describe lattice dynamics properties compared to DFT-relaxed structures, therefore structures of experimental lattice parameters with relaxed internal atomic coordinates were used in subsequent calculations.
The lattice parameters of R-VO$_2$ at 425 and 600 K were obtained by interpolating experimental values [@McWhan1974], and the lattice parameters of M-VO$_{2}$ at room temperature were used [@Longo1970]. DFT calculations of primitive cells of R (M)-VO$_{2}$ were performed using a 6$\times$6$\times$10 (6$\times$6$\times$6) $\Gamma$-centered $\mathbf{k}$-point mesh and a kinetic energy cutoff of 500 eV. The force and energy convergence thresholds were set to be $10^{-3}$ eV/ and $10^{-8}$ eV respectively. 2$\times$2$\times$4 (2$\times$2$\times$2) supercell structures were constructed, which were sampled with 3$\times$3$\times$3 $\Gamma$-centered $\mathbf{k}$-point mesh, to extract 2$^{\text{nd}}$- and 3$^{\text{rd}}$-order IFCs for R (M)-VO$_{2}$. No explicit cutoff is imposed for 2$^{\text{nd}}$-order IFCs by taking into account the cumulant IFCs [@Parlinski1997]. The diameter cutoff of 3$^{\text{rd}}$-order interactions was varied from 3.0 to 4.5 for both M- and R-VO$_{2}$ (see more details in the results and discussions section). We found that 4.0 is sufficient to achieve good convergence of computed $\kappa_{l}$ well within 5%. We only performed phonon renormalization for R-VO$_{2}$ because both DFT calculation and experimental measurements have confirmed that M-VO$_{2}$ mostly exhibits harmonic behavior [@Budai2014; @Lee371]. To extract 3$^{\text{rd}}$-order IFCs of M-VO$_{2}$, 1332 2$\times$2$\times$2 supercell structures were evaluated due to low symmetry of M-VO$_{2}$ ($P2_{1}/c$). For R-VO$_{2}$, twenty 2$\times$2$\times$4 supercell structures were sampled in each iteration during phonon renormalization process, owing to its relatively high symmetry of $P4_{2}/mnm$. The ShengBTE package [@shengbte], which was modified to work with CSLD, was used to perform all phonon-related calculations including the iterative calculation of $\kappa_{l}$ with a 16$\times$16$\times$26$\ ($14$\times$14$\times$14$)\ \mathbf{q}$-point mesh for R (M)-VO$_{2}$. The $\kappa_{l}$ of polycrystalline VO$_{2}$ was computed by averaging over $\kappa_{l}$ along three Cartesian coordinates.
DFT calculations of primitive cells of SrTiO$_{3}$ were performed using a 8$\times$8$\times$8 $\Gamma$-centered $\mathbf{k}$-point mesh and a kinetic energy cutoff of 520 eV. The force and energy convergence threshold were set to be $10^{-3}$ eV/ and $10^{-8}$ eV respectively. 2$\times$2$\times$2 supercell structures were constructed, which was sampled with 4$\times$4$\times$4 $\Gamma$-centered $\mathbf{k}$-point mesh, to extract 2$^{\text{nd}}$- and 3$^{\text{rd}}$-order IFCs for SrTiO$_{3}$. In the phonon renormalization process, ten 2$\times$2$\times$2 supercell structures in each iteration were found sufficient to converge the IFCs fitting because of high symmetry of cubic SrTiO$_{3}$ ($Pm\bar{3}m$). The diameter cutoff of 3$^{\text{rd}}$-order interactions in SrTiO$_{3}$ was 5.0 , and $\kappa_{l}$ was computed using a 20$\times$20$\times$20 $\mathbf{q}$-point mesh.
Results and discussions
=======================
SrTiO$_{3}$: Validation of proposed phonon renormalization scheme
-----------------------------------------------------------------
To confirm the validity of our approach and verify the implementation, we applied our phonon renormalization scheme to a well studied system: perovskite SrTiO$_{3}$ with cubic symmetry at high-temperature, which undergoes a cubic-to-tetragonal phase transition at 105 K upon cooling [@Cowley1964]. Fig. \[fig:STO\](a) shows that the convergence of vibrational free energy within $\pm$1 meV/atom can be achieved after eleven iterations at 300 K. Additional iterations were performed to obtain forces required to converge the IFCs fitting. The phonon dispersions computed in the harmonic approximation at 0 K exhibits instability at $R$ point in the Brillouin zone of cubic cell, which corresponds to the rotation of octahedra formed by TiO$_{3}$. The renormalized phonon dispersions at 300 K, as shown in Fig. \[fig:STO\](b), display well-defined phonon frequencies without imaginary numbers, further confirming that the cubic structure can be stabilized by temperature. The computed renormalized phonon dispersions at 300 K agree well with experimental measurements [@Stirling1972; @COWLEY1969181], particularly near the $R$ point. Using the extracted renormalized IFCs at 300 K, we also computed the $\kappa_{l}$ as a function of temperature. Fig. \[fig:STO\](c) shows that the computed $\kappa_{l}$ agrees well with experiments at 300 K [@MUTA2005306; @Popuri2014], but shows significant underestimation at high temperatures, which is due to the fact that high-temperature IFCs are different from those at 300 K. Recently, we also applied our phonon renormalization scheme to other perovskites CsPbCl$_{3}$ and CsPbBr$_{3}$, the details of of which can be found in Ref. \[\]
VO$_{2}$: Crystal structures
----------------------------
R-VO$_{2}$ has a symmetry of $P4_{2}/mnm$ with one symmetrically-inequivalent V and O atom each. M-VO$_{2}$ has a symmetry of $P2_{1}/c$ with one symmetrically-inequivalent V and two symmetrically-inequivalent O atoms. As shown in Fig. \[fig:Structures\](a) and (b), V atoms in both phases are arranged in 1D chains along the rutile $c$ axis with O atoms octahedrally bonded to V atoms. Upon cooling, the tetragonal R-VO$_{2}$ transforms into monoclinic M-VO$_{2}$, which features zigzag dimerization of all V atoms, tilting of octahedra and doubling of the unit cell in the chain direction.
VO$_{2}$: Convergence of $\kappa_{l}$ with respect to the diameter cutoff of 3$^{\text{rd}}$-order interactions
---------------------------------------------------------------------------------------------------------------
Fig. \[fig:Con\_Kappa\] shows the computed $\kappa_{l}$ as a function of diameter cutoff of 3$^{\text{rd}}$-order interactions for M-VO$_{2}$ at 340 K and R-VO$_{2}$ at 425 K respectively. This plot shows that including 3$^{\text{rd}}$-order interactions up to 4.0 is sufficient to converge the $\kappa_{l}$.
VO$_{2}$: Effects of exchange correlation functional and structure relaxation on phonon dispersion
--------------------------------------------------------------------------------------------------
Fig. \[fig:PBE\_Compare\](a) shows that results obtained using PBE and PBEsol with experimental lattice parameter yield very similar phonon dispersions for M-VO$_{2}$, while PBEsol with volume-relaxed structure leads to softened phonon dispersions, which leads to values of the $\kappa_{l}$ about half of what is determined using experimental lattice parameters. Fig. \[fig:PBE\_Compare\](b) shows that results obtained using PBE and PBEsol with experimental lattice parameter display similar phonon dispersions for R-VO$_{2}$, while PBEsol with volume-relaxed structure leads to significantly hardened phonon dispersions, which gives rise to significantly enhanced $\kappa_{l}$, even higher than M-VO$_{2}$. We used PBE and PBEsol exchange correlation functionals together with experimental lattice parameters for both M- and R-VO$_{2}$.
VO$_{2}$: Renormalized phonon dispersions of R-VO$_{2}$ at 360, 425 and 600 K
-----------------------------------------------------------------------------
Fig. \[fig:Disp\_Temp\](a) shows the renormalized phonon dispersions of R-VO$_{2}$ at 360 K with small lattice instabilities near $\Gamma$ point. The instability is not associated with either the R-point transverse acoustic phonon mode or the low-lying zone-center optical mode, consistent with previous theoretical investigations [@Budai2014]. Fig. \[fig:Disp\_Temp\](b) shows the comparison of renormalized phonon dispersions of R-VO$_{2}$ at 425 and 600, which show that low-lying zone-center optical mode has a strong temperature dependence.
VO$_{2}$: Comparisons of energy cumulative $\kappa_{l}$, phonon lifetime, group velocity, and mean free path between M-VO$_{2}$ at 340 K and R-VO$_{2}$ at 425 K.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------
Fig. \[fig:Kappa\_Cum\] displays the energy cumulative lattice thermal conductivities of polycrystalline VO$_{2}$ and single-crystalline VO$_{2}$ along rutile $c$ axis, which indicates that phonon modes with energies less than 10 meV contribute only about 30% of total $\kappa_{l}$. Fig. \[fig:Kappa\_Ana\_Sup\](a)-(c) compare the phonon mode lifetime, group velocity and mean free path between monoclinic VO$_{2}$ at 340 K and rutile VO$_{2}$ at 425 K.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This is a survey on the recent theory on minimizing the normalized volume function attached to any klt singularities.'
address:
- 'Department of Mathematics, Purdue University, West Lafayette, IN 47907-2067, USA'
- 'Department of Mathematics, Yale University, New Haven, CT 06511, USA'
- 'Beijing International Center for Mathematical Research, Beijing 100871, China'
author:
- Chi Li
- Yuchen Liu
- Chenyang Xu
title: A guided tour to normalized volume
---
Introduction
============
The development of algebraic geometry and complex geometry has interwoven in the history. One recent example is the interaction between the theory of higher dimensional geometry centered around the minimal model program (MMP), and the existence of ‘good’ metrics on algebraic varieties. Both subjects have major steps forward, whose influences are beyond the subjects themselves, spurring out new progress in topics once people could not imagine. In this note, we will discuss a ‘local stability theory’ of singularities, which in our opinion provides an excellent example on the philosophy that there are many unexpected connections underlying these two different topics.
Ever since the starting of the theory of MMP in higher dimensions (that is, the dimension is at least three), people understand that a feature of such a theory is that we need to deal with singular varieties. Then it becomes very nature to investigate this class of singularities for people working on the MMP. To deal with singular varieties in complex geometry is a more recent trend, and it significantly improves people’s knowledge on the existence of interesting metrics, even in situations which people originally only want to study smooth varieties.
It becomes clear now, Kawamata log terminal (klt) singularities form an exceptionally important class of singularities for many reasons: it is the natural class of singular varieties for people to inductively prove deep results in the MMP; it is the class of singularities appearing on degenerations in many natural settings and it carries properties which globally Fano varieties have.
What we want to survey here is a rather new theory on klt singularities. The picture consists of two closely related parts: firstly, we want to establish a structure which provides a canonically determined degeneration to a stable log Fano cone from each klt singularity; secondly, to construct the degeneration, we need a valuation which minimizes the normalized volume function on the ‘non-archimedean link’, and since such minimum is a deep invariant defined for all klt singularities, we want to explore more properties of this invariant, including calculating it in many cases.
History
--------
The first prototype of the local stability theory underlies in [@MSY06; @MSY08]. They find that the existence of Ricci-flat cone metric on an affine variety with a good action by a torus group $T$ is closely related to the normalized volume minimizing problem. In our language, they concentrate on the valuations induced by the vectors in the Reeb cone provided by the torus action. Later a systematic study of K-stability in the setting of Sasaki geometry is further explored in [@CS18; @CS15].
Consider klt singularities which appear on the Gromov-Hausdorff (GH) limit of Käher-Einstein Fano manifolds. At the first sight, we do not know more algebraic structure for these singularities. Nevertheless, by looking at the metric tangent cone, it is shown in [@DS17], built on the earlier works in [@CCT02; @DS14; @Tia13], that the metric tangent cone of such singularities is an affine $T$-variety with a Ricci-flat cone metric. Furthermore, [@DS17] gives a a two-step degeneration description of the metric tangent cone. They further conjecture that this two-step degeneration should only depend on the algebraic structure of the singularity, but not the metric.
Then in [@Li15], the normalized volume function on the ‘non-archimedean link’ of a given klt singularity is defined, and a series of conjectures on normalized volume function are proposed. This attempt is not only to algebrize the work in [@DS17] without invoking the metric, but it is also of a completely local nature. Since then, the investigation on this local stability theory points to different directions.
In [@Blu18], the existence of a minimum (opposed to only infimum) which was conjectured in [@Li15] is affirmatively answered. The proof uses the properness estimates in [@Li15] and the observation in [@Liu16] that the minimizer can be computed by the minimal normalized multiplicities, and then skillfully uses the techniques from the study of asymptotic invariants (see [@Laz04]). Later in [@BL18], lower semicontinuity of the volume of singularities are also established using this circle of ideas.
In [@Li17; @LL16], the case of a cone singularity over a Fano variety is intensively studied, and it was found if we translate the minimizing question for the canonical valuation into a question on the base Fano varieties, what appears is the sign of the $\beta$-invariant developed in [@Fuj18; @Fuj16; @Li17].
Built on the previous study of cone singularities, implementing the ideas circled around the MMP in birational geometry, an effective process of degenerating a general singularity to a cone singularity is established in [@LX16], provided the minimizer is a divisorial valuation. In [@LX17], a couple of conjectural properties are added to complete the picture proposed in [@Li15], and now the package is called ‘[*stable degeneration conjecture*]{}’, see Conjecture \[conj-local\]. The investigation in [@LX16] is also extended in [@LX17] to the case when the minimizer is a quasi-monomial valuation with a possibly higher rational rank, where the study involves a considerable amount of new techniques. As a corollary, the first part of Donaldson-Sun’s conjecture in [@DS17] is answered affirmatively in [@LX17]. Later the work is extended in [@LWX18] and a complete solution of Donaldson-Sun’s conjecture is found.
Applications to global questions, especially the existence of KE metrics on Fano varieties, are also explored. In [@Liu16], built on the work of [@Fuj18], an inequality to connect the local volume and the global one is proved. Then in [@SS17; @LiuX17], via the approach of the ‘comparison of moduli’, complete moduli spaces parametrizing explicit Fano varieties with a KE metric are established by studying the local constraint posted by the lower bound of the local volumes.
Outline
-------
In the note, we will survey a large part of the results mentioned above. From the perspective of techniques, there are three closely related ways to think about the volume of a singularity: the infima of the normalized volume of valuations, of the normalized multiplicity of primary ideals or of the volume of models. The viewpoint using valuations gives the most canonical picture, e.g. the stable degeneration conjecture, but there are less techniques available to directly study the space. The viewpoint using ideals is flexible for many purposes, e.g. taking degenerations. Moreover, though usually working on a single ideal does not give too much advantage over others, working on a graded sequence of ideals really enables one to use the powerful theory on asymptotical invariants for such setting. The third viewpoint of using models allows us to apply the machinery from the MMP theory, and it is the key to degenerate the underlying singularities into cone singularities. The interplay among these three circle of techniques is fruitful, and we expect further insight can be made in the future.
In Section \[s-def\], we give the definition of the function of the normalized volumes and sketch the basic properties of its minimizer, including the existence. In Section \[s-sasaki\], we discuss the theory on searching for Sasaki-Einstein metrics on a Fano cone singularities. The algebraic side, namely the K-stability notions on a Fano cone plays an important role as we try to degenerate any klt singularity to a K-semistable Fano cone. Such an attempt is formulated in the stable degeneration conjecture, which is the focus of Section \[s-SDC\]. In Section \[s-application\], we present some applications, including the torus equivariant K-stability (Section \[ss-equiva\]), a solution of Donaldson-Sun’s conjecture (Section \[ss-dsc\]) and the K-stability of cubic threefolds (Section \[ss-cubic3\]). In the last Section \[s-ques\], we discuss many unsolved questions, which we hope will lead to some future research. Some of them give new approaches to attack existing problems.
[**Acknowledgement:**]{} We would like to use this chance to express our deep gratitude to Gang Tian, from whom we all learn a large amount of knowledge related to K-stability questions on Fano varieties in these years. We want to thank Harold Blum, Sébastien Boucksom for helpful discussions.
CL is partially supported by NSF (Grant No. DMS-1405936) and an Alfred P. Sloan research fellowship. CX is partially supported by the National Science Fund for Distinguished Young Scholars (11425101). A large part of the work is written while CX visits Institut Henri Poincaré under the program ‘Poincaré Chair’. He wants to thank the wonderful environment.
Definitions and first properties {#s-def}
================================
Definitions {#ss-firstdefinition}
-----------
In this section, we give the definition of the normalized volume ${\widehat{\rm vol}}_{(X,D),x}(v)$ (or abbreviated as ${\widehat{\rm vol}}(v)$ if there is no confusion) for a valuation $v$ centered on a klt singularity $x\in (X,D)$ as in [@Li15]. It consists of two parts: the volume ${{\rm vol}}(v)$ (see Definition \[d-vol\]) and the log discrepancy $A_{X,D}(v)$ (see Definition \[d-logd\]). Let $X$ be a reduced, irreducible variety defined over ${{\mathbb{C} }}$. A *real valuation* of its function field $K(X)$ is a non-constant map $v\colon K(X)^{\times}\to {\mathbb{R}}$, satisfying:
- $v(fg)=v(f)+v(g)$;
- $v(f+g)\geq \min\{v(f),v(g)\}$;
- $v({{\mathbb{C} }}^*)=0$.
We set $v(0)=+\infty$. A valuation $v$ gives rise to a valuation ring ${\mathcal{O}}_v:=\{f\in K(X)\mid v(f)\geq 0\}$. We say a real valuation $v$ is *centered at* a scheme-theoretic point $\xi=c_X(v)\in X$ if we have a local inclusion ${\mathcal{O}}_{\xi,X}\hookrightarrow{\mathcal{O}}_v$ of local rings. Notice that the center of a valuation, if exists, is unique since $X$ is separated. Denote by ${{\rm Val}}_X$ the set of real valuations of $K(X)$ that admits a center on $X$. For a closed point $x\in X$, we denote by ${{\rm Val}}_{X,x}$ the set of real valuations of $K(X)$ centered at $x\in X$. It’s well known that $v\in {{\rm Val}}_{X}$ is centered at $x\in X$ if $v(f)$ for any $f\in {\mathfrak{m}}_x$.
For each valuation $v\in {{\rm Val}}_{X,x}$ and any integer $m$, we define the valuation ideal ${{\mathfrak{a}}}_m(v):=\{f\in{\mathcal{O}}_{x,X}\mid v(f)\geq m\}$. Then it is clear that ${{\mathfrak{a}}}_m(v)$ is an ${\mathfrak{m}}_x$-primary ideal for each $m>0$.
Given a valuation $v\in {{\rm Val}}_X$ and a nonzero ideal ${{\mathfrak{a}}}\subset{\mathcal{O}}_X$, we may evaluate ${{\mathfrak{a}}}$ along $v$ by setting $v({{\mathfrak{a}}}) := \min\{v(f)\mid f \in {{\mathfrak{a}}}\cdot{\mathcal{O}}_{c_X(v),X} \}$. It follows from the above definition that if ${{\mathfrak{a}}}\subset {\mathfrak{b}}\subset {\mathcal{O}}_X$ are nonzero ideals, then $v({{\mathfrak{a}}}) \geq v({\mathfrak{b}})$. Additionally, $v({{\mathfrak{a}}})> 0$ if and only if $c_X(v) \in {\mathrm{Cosupp}}({{\mathfrak{a}}})$. We endow ${{\rm Val}}_X$ with the weakest topology such that, for every ideal ${{\mathfrak{a}}}$ on $X$, the map ${{\rm Val}}_X\to {\mathbb{R}}\cup\{+\infty\}$ defined by $v\mapsto v({{\mathfrak{a}}})$ is continuous. The subset ${{\rm Val}}_{X,x}\subset {{\rm Val}}_X$ is endowed with the subspace topology. In some literatures, the space ${{\rm Val}}_{X,x}$ is called the [*non-archimedean link*]{} of $x\in X$. When $X={{\mathbb{C} }}^2$, the geometry of ${{\rm Val}}_{X,x}$ is understood well (see [@FJ04]). For higher dimension, its structure is much more complicated but can be described as an inverse limit of dual complexes (see [@JM12; @BdFFU15]).
Let $Y\xrightarrow[]{\mu} X$ be a proper birational morphism with $Y$ a normal variety. For a prime divisor $E$ on $Y$, we define a valuation ${{\rm ord}}_E\in {{\rm Val}}_X$ that sends each rational function in $K(X)^{\times}=K(Y)^{\times}$ to its order of vanishing along $E$. Note that the center $c_X({{\rm ord}}_E)$ is the generic point of $\mu(E)$. We say that $v\in {{\rm Val}}_X$ is a *divisorial valuation* if there exists $E$ as above and $\lambda\in{\mathbb{R}}_{>0}$ such that $v=\lambda\cdot {{\rm ord}}_E$.
Let $\mu : Y \to X$ be a proper birational morphism and $\eta\in Y$ a point such that $Y$ is regular at $\eta$. Given a system of parameters $y_1,\cdots, y_r \in {\mathcal{O}}_{Y,\eta}$ at $\eta$ and $\alpha = (\alpha_1,\cdots,\alpha_r) \in {\mathbb{R}}_{\geq 0}^r\setminus\{0\}$, we define a valuation $v_\alpha$ as follows. For $f\in {\mathcal{O}}_{Y,\eta}$ we can write it as $f =\sum_{\beta\in{{\mathbb{Z} }}_{\geq 0}^r}c_\beta y^\beta$, with $c_\beta\in\widehat{{\mathcal{O}}_{Y,\eta}}$ either zero or unit. We set $$v_\alpha(f) = \min\{\langle\alpha,\beta\rangle\mid c_\beta\neq 0\}.$$ A *quasi-monomial valuation* is a valuation that can be written in the above form.
Let $(Y, E =\sum_{k=1}^N E_k)$ be a log smooth model of $X$, i.e. $\mu : Y \to X$ is an isomorphism outside of the support of $E$. We denote by ${\mathrm{QM}}_{\eta}(Y,E)$ the set of all quasi-monomial valuations $v$ that can be described at the point $\eta\in Y$ with respect to coordinates $(y_1,\cdots, y_r)$ such that each $y_i$ defines at $\eta$ an irreducible component of $E$ (hence $\eta$ is the generic point of a connected component of the intersection of some of the divisors $E_i$). We put ${\mathrm{QM}}(Y,E):=\bigcup_{\eta}{\mathrm{QM}}_\eta(Y,E)$ where $\eta$ runs over generic points of all irreducible components of intersections of some of the divisors $E_i$.
Given a valuation $v\in {{\rm Val}}_{X,x}$, its *rational rank* ${\mathrm{rat.rank}\,}v$ is the rank of its value group. The *transcendental degree* ${\mathrm{trans.deg}\,}v$ of $v$ is the transcendental degree of the field extension ${{\mathbb{C} }}\hookrightarrow {\mathcal{O}}_v/{\mathfrak{m}}_v$. The Zariski-Abhyankar Inequality says that $${\mathrm{trans.deg}\,}v+{\mathrm{rat.rank}\,}v\leq \dim X.$$ A valuation satisfying the equality is called an *Abhyankar valuation*. By [@ELS03], we know that a valuation $v\in{{\rm Val}}_{X}$ is Abhyankar if and only if it is quasi-monomial.
\[d-vol\] Let $X$ be an $n$-dimensional normal variety. Let $x\in X$ be a closed point. We define the *volume of a valuation* $v\in{{\rm Val}}_{X,x}$ following [@ELS03] as $${{\rm vol}}_{X,x}(v)=\limsup_{m\to\infty}\frac{\ell({\mathcal{O}}_{x,X}/{{\mathfrak{a}}}_m(v))}{m^n/n!}.$$ where $\ell$ denotes the length of the artinian module.
Thanks to the works of [@ELS03; @LM09; @Cut13] the above limsup is actually a limit.
\[d-logd\] Let $(X,D)$ be a klt log pair. We define the *log discrepancy function of valuations* $A_{(X,D)}:{{\rm Val}}_X\to (0,+\infty]$ in successive generality.
1. Let $\mu:Y\to X$ be a proper birational morphism from a normal variety $Y$. Let $E$ be a prime divisor on $Y$. Then we define $A_{(X,D)}({{\rm ord}}_E)$ as $$A_{(X,D)}({{\rm ord}}_E):=1+{{\rm ord}}_E(K_Y-\mu^*(K_X+D)).$$
2. Let $(Y,E=\sum_{k=1}^N E_k)$ be a log smooth model of $X$. Let $\eta$ be the generic point of a connected component of $E_{i_1}\cap E_{i_2}\cap\cdots\cap E_{i_r}$ of codimension $r$. Let $(y_1,\cdots,y_r)$ be a system of parameters of ${\mathcal{O}}_{Y,\eta}$ at $\eta$ such that $E_{i_j}=(y_j=0)$. Then for any $\alpha=(\alpha_1,\cdots,\alpha_r)\in{\mathbb{R}}_{\geq 0}^r\setminus\{0\}$, we define $A_{(X,D)}(v_{\alpha})$ as $$A_{(X,D)}(v_\alpha):=\sum_{j=1}^r \alpha_j A_{(X,D)}({{\rm ord}}_{E_{i_j}}).$$
3. In [@JM12], it was showed that there exists a retraction map $r_{Y,E}: {{\rm Val}}_X \to {\mathrm{QM}}(Y, E)$ for any log smooth model $(Y, E)$ over $X$, such that it induces a homeomorphism ${{\rm Val}}_X \to\varprojlim_{(Y,E)}{\mathrm{QM}}(Y, E)$. For any real valuation $v \in {{\rm Val}}_X$, we define $$A_{(X,D)}(v):=\sup_{(Y,E)} A_{(X,D)}(r_{(Y,E)}(v)).$$ where $(Y, D)$ ranges over all log smooth models over $X$. For details, see [@JM12] and [@BdFFU15 Theorem 3.1]. It is possible that $A_{(X,D)}(v) = +\infty$ for some $v\in {{\rm Val}}_X$, see e.g. [@JM12 Remark 5.12].
Then we can define the main invariant in this paper. As we mentioned in Section \[s-sasaki\], it is partially inspired the definition in [@MSY08] for a valuation coming from the Reeb vector field.
\[d-normvol\] Let $(X,D)$ be an $n$-dimensional klt log pair. Let $x\in X$ be a closed point. Then the *normalized volume function of valuations* ${\widehat{\rm vol}}_{(X,D),x}:{{\rm Val}}_{X,x}\to(0,+\infty)$ is defined as $${\widehat{\rm vol}}_{(X,D),x}(v)=\begin{cases}
A_{(X,D)}(v)^n\cdot{{\rm vol}}_{X,x}(v), & \textrm{ if }A_{(X,D)}(v)<+\infty;\\
+\infty, & \textrm{ if }A_{(X,D)}(v)=+\infty.
\end{cases}$$ The *volume of the singularity* $(x\in (X,D))$ is defined as $${\widehat{\rm vol}}(x, X,D):=\inf_{v\in{{\rm Val}}_{X,x}}{\widehat{\rm vol}}_{(X,D),x}(v).$$
Since ${\widehat{\rm vol}}(v)={\widehat{\rm vol}}(\lambda\cdot v)$ for any $\lambda \in {\mathbb{R}}_{>0}$, for any valuation $v\in {{\rm Val}}_{X,D}$ with a finite log discrepancy, we can rescale such that $\lambda\cdot v\in {{\rm Val}}^{=1}_{X,D}$ where ${{\rm Val}}^{=1}_{X,D}$ consists of all valuations $v\in {{\rm Val}}_{X,x}$ with $ A_{(X,D)}(v)=1$.
A definition of volume of singularities is also given in [@BdFF12]. Their definition is the local analogue of the volume $K_X$ whereas our definition is the one of the volume of $-K_X$. In particular, a singularity has volume 0 in the definition of [@BdFF12] if it is log canonical.
Properties
----------
In this section, we discuss some properties of ${\widehat{\rm vol}}$ on ${{\rm Val}}_{X,x}$. We start from the properness and Izumi estimates. As a corollary, we conclude that ${\widehat{\rm vol}}(x, X,D)$ is always positive for any klt singularity $x\in (X,D)$.
\[t-izumi\] Let $(x\in(X,D))$ be a klt singularity. Then there exists positive constants $C_1$, $C_2$ which only depend on $x\in (X,D)$ (but not the valuation $v$) such that the following holds.
1. (Izumi-type inequality) For any valuation $v\in{{\rm Val}}_{X,x}$, we have $$v({\mathfrak{m}}_x){{\rm ord}}_x\leq v\leq C_2\cdot A_{(X,D)}(v){{\rm ord}}_x.$$
2. (Properness) For any valuation $v\in{{\rm Val}}_{X,x}$ with $A_{(X,D)}(v)<+\infty$, we have $$C_1\frac{A_{(X,D)}(v)}{v({\mathfrak{m}}_x)} \le {\widehat{\rm vol}}(v).$$
Note that since $x\in X$ is singular, ${{\rm ord}}_x$ in the above inequality might not be a valuation. In other words, for $f,g\in {\mathcal{O}}_{X,x}$, ${{\rm ord}}_x(fg)\ge {{\rm ord}}_x(f)+{{\rm ord}}_x(g)$ may be a strict inequality.
The above Izumi type inequality is well known when $x\in X$ is a smooth point. In the case of a general klt singularity, it can be reduced to the smooth case after a log resolution and decreasing the constant. Then for the properness, it follows from a more subtle estimate that there exists a positive constant $c_2$ that $${{\rm vol}}(v)\ge c_2\left(\sup_{{\mathfrak{m}}_x} \frac{v}{{{\rm ord}}_x}\right)^{1-n} \cdot \frac{1}{v({\mathfrak{m}})}.$$
Let ${{\mathfrak{a}}}_\bullet=\{{{\mathfrak{a}}}_m\}_{m\in{{\mathbb{Z} }}}$ be a graded sequence of ${\mathfrak{m}}_x$-primary ideals. By the works in [@LM09; @Cut13], the following identities hold true: $${{\rm mult}}({{\mathfrak{a}}}_\bullet):=\lim_{m\rightarrow+\infty}\frac{\ell({\mathcal{O}}_{X,x}/{{\mathfrak{a}}}_m)}{m^n/n!}=\lim_{m\rightarrow+\infty}\frac{{{\rm mult}}({{\mathfrak{a}}}_m)}{m^n}.$$ In particular, the two limits exist. Note that, by definition, for any $v\in {{\rm Val}}_{X,x}$ and ${{\mathfrak{a}}}_\bullet(v)=\{{{\mathfrak{a}}}_m(v)\}$, we have ${{\rm vol}}(v)={{\rm mult}}({{\mathfrak{a}}}_\bullet(v))$ .
The following observation on characterizing the normalized volumes by normalized multiplicities provides lots of flexibility in the study as we will see.
\[thm:liueq\] Let $(x\in (X,D))$ be an $n$-dimensional klt singularity. Then we have $${\widehat{\rm vol}}(x,X,D)=\inf_{{{\mathfrak{a}}}\colon {\mathfrak{m}}_x\textrm{-primary}}{{\rm lct}}(X,D;{{\mathfrak{a}}})^n{{\rm mult}}({{\mathfrak{a}}})
=\inf_{{{\mathfrak{a}}}_\bullet\colon {\mathfrak{m}}_x\textrm{-primary}}{{\rm lct}}(X,D;{{\mathfrak{a}}}_\bullet)^n{{\rm mult}}({{\mathfrak{a}}}_\bullet).$$ We also set ${{\rm lct}}(X,D;{{\mathfrak{a}}}_\bullet)^n{{\rm mult}}({{\mathfrak{a}}}_\bullet)=
+\infty$ if ${{\rm lct}}(X,D;{{\mathfrak{a}}}_\bullet)=+\infty$.
Firstly, for any ${\mathfrak{m}}_x$-primary ideal ${{\mathfrak{a}}}$, we can take a divisorial valuation $v\in{{\rm Val}}_{X,x}$ computing ${{\rm lct}}({{\mathfrak{a}}})$. In other words, ${{\rm lct}}({{\mathfrak{a}}})=A_X(v)/v({{\mathfrak{a}}})$. We may rescale $v$ such that $v({{\mathfrak{a}}})=1$. Then clearly ${{\mathfrak{a}}}^m\subset{{\mathfrak{a}}}_m(v)$ for any $m\in{\mathbb{N}}$, hence ${{\rm mult}}({{\mathfrak{a}}})\geq {{\rm vol}}(v)$. Therefore, ${{\rm lct}}({{\mathfrak{a}}})^n{{\rm mult}}({{\mathfrak{a}}})\geq A_X(v)^n{{\rm vol}}(v)$ which implies $$\label{eq:liueq1}
{\widehat{\rm vol}}(x, X,D)\leq \inf_{{{\mathfrak{a}}}\colon {\mathfrak{m}}_x\textrm{-primary}}{{\rm lct}}({{\mathfrak{a}}})^n{{\rm mult}}({{\mathfrak{a}}}).$$
Secondly, for any graded sequence of ${\mathfrak{m}}_x$-primary ideals ${{\mathfrak{a}}}_\bullet$, we have $${{\rm lct}}({{\mathfrak{a}}}_\bullet)=\lim_{m\to\infty}m\cdot{{\rm lct}}({{\mathfrak{a}}}_m)$$ by [@JM12; @BdFFU15]. Hence $${{\rm lct}}({{\mathfrak{a}}}_\bullet)^n{{\rm mult}}({{\mathfrak{a}}}_\bullet)=
\lim_{m\to\infty} (m\cdot{{\rm lct}}({{\mathfrak{a}}}_m))^n\frac{{{\rm mult}}({{\mathfrak{a}}}_m)}{m^n}
=\lim_{m\to\infty}{{\rm lct}}({{\mathfrak{a}}}_m)^n{{\rm mult}}({{\mathfrak{a}}}_m).$$ As a result, $$\label{eq:liueq2}
\inf_{{{\mathfrak{a}}}\colon {\mathfrak{m}}_x\textrm{-primary}}{{\rm lct}}({{\mathfrak{a}}})^n{{\rm mult}}({{\mathfrak{a}}})
\leq \inf_{{{\mathfrak{a}}}_\bullet\colon {\mathfrak{m}}_x\textrm{-primary}}
{{\rm lct}}({{\mathfrak{a}}}_\bullet)^n{{\rm mult}}({{\mathfrak{a}}}_\bullet).$$
Lastly, for any valuation $v\in{{\rm Val}}_{X,x}$, we consider the graded sequence of its valuation ideals ${{\mathfrak{a}}}_\bullet(v)$. Since $v({{\mathfrak{a}}}_\bullet(v))=1$, we have ${{\rm lct}}({{\mathfrak{a}}}_\bullet)
\leq A_X(v)$. We also have ${{\rm mult}}({{\mathfrak{a}}}_\bullet(v))={{\rm vol}}(v)$. Hence ${{\rm lct}}({{\mathfrak{a}}}_{\bullet}(v))^n{{\rm mult}}({{\mathfrak{a}}}_\bullet(v))\leq
A_X(v)^n{{\rm vol}}(v)$, which implies $$\label{eq:liueq3}
\inf_{{{\mathfrak{a}}}_\bullet\colon {\mathfrak{m}}_x\textrm{-primary}}
{{\rm lct}}({{\mathfrak{a}}}_\bullet)^n{{\rm mult}}({{\mathfrak{a}}}_\bullet)\le {\widehat{\rm vol}}(x, X,D).$$ The proof is finished by combining , , and .
In general we have the following relation between a sequence of graded ideals and the one from a valuation: Let $\Phi^g$ be an ordered subgroup of the real numbers ${\mathbb{R}}$. Let $(R, {\mathfrak{m}})$ be the local ring at a normal singularity $o\in X$. A $\Phi^g$-graded filtration of $R$, denoted by ${{\mathcal{F}}}:=\{{{\mathfrak{a}}}^m\}_{m\in \Phi^g}$, is a decreasing family of ${\mathfrak{m}}$-primary ideals of $R$ satisfying the following conditions:
[**(i)**]{} ${{\mathfrak{a}}}^m\neq 0$ for every $m\in \Phi^g$, ${{\mathfrak{a}}}^m=R$ for $m\le 0$ and $\cap_{m\ge 0}{{\mathfrak{a}}}^m=(0)$;
[**(ii)**]{} ${{\mathfrak{a}}}^{m_1}\cdot {{\mathfrak{a}}}^{m_2}\subseteq {{\mathfrak{a}}}^{m_1+m_2}$ for every $m_1, m_2\in \Phi^g$.
Given such an ${{\mathcal{F}}}$, we get an associated order function $$v=v_{{{\mathcal{F}}}}: R\rightarrow {\mathbb{R}}_{\ge 0} \qquad v(f)=\max\{m; f\in {{\mathfrak{a}}}^m\} \mbox{\ \ for any $f\in R$}.$$ Using the above [**(i)-(ii)**]{}, it is easy to verify that $v$ satisfies $v(f+g)\ge \min\{v(f), v(g)\}$ and $v(fg)\ge v(f)+v(g)$. We also have the associated graded ring: $${{\rm gr}}_{{{\mathcal{F}}}}R=\sum_{m\in \Phi^g} {{\mathfrak{a}}}^m/{{\mathfrak{a}}}^{>m}, \text{ where } {{\mathfrak{a}}}^{>m}=\bigcup_{m'> m}{{\mathfrak{a}}}^{m'}.$$ For any real valuation $v$ with valuative group $\Phi^g$, $\{{{\mathcal{F}}}^m\}:=\{{{\mathfrak{a}}}_m(v)\}$ is a $\Phi^g$-graded filtration of $R$. We will need the following facts.
\[lem-quasi\] With the above notations, the following statements hold true:
[(1) ([@Tei14 Page 8])]{} If ${{\rm gr}}_{{{\mathcal{F}}}}R$ is an integral domain, then $v=v_{{{\mathcal{F}}}}$ is a valuation centered at $o\in X$. In particular, $v(fg)=v(f)+v(g)$ for any $f,g\in R$.
[(2) (Piltant)]{} A valuation $v$ is quasi-monomial if and only if the Krull dimension of ${{\rm gr}}_v R$ is the same as the Krull dimension of $R$.
The existence of a minimizer for ${\widehat{\rm vol}}_{(X,D),x}$ was conjectured in the first version of [@Li15] and then proved in [@Blu18].
\[t-existence\] For any klt singularity $x\in(X,D)$, there exists a valuation $v_{\min}\in{{\rm Val}}_{X,x}$ that minimizes the function ${\widehat{\rm vol}}_{(X,D),x}$.
Let us sketch the idea of proving the existence of ${\widehat{\rm vol}}$-minimizer. We first take a sequence of valuations $(v_i)_{i\in{\mathbb{N}}}$ such that $$\lim_{i\to\infty}{\widehat{\rm vol}}(v_i)={\widehat{\rm vol}}(x, X,D).$$ Then we would like to find a valuation $v^*$ that is a limit point of the sequence $(v_i)_{i\in{\mathbb{N}}}$ and then show that $v^*$ is a minimizer of ${\widehat{\rm vol}}$. Instead of seeking a limit point $v^*$ of $(v_i)_{i\in{\mathbb{N}}}$ in the space of valuations, we consider graded sequences of ideals. More precisely, each valuation $v_i$ induces a graded sequence ${{\mathfrak{a}}}_\bullet(v_i)$ of ${\mathfrak{m}}_x$-primary ideals. By Theorem \[thm:liueq\], we have $${\widehat{\rm vol}}(v_i)\geq {{\rm lct}}({{\mathfrak{a}}}_\bullet(v_i))^n{{\rm mult}}({{\mathfrak{a}}}_\bullet(v_i))\geq {\widehat{\rm vol}}(x, X,D).$$ Therefore, once we find a graded sequence of ${\mathfrak{m}}_x$-primary ideals $\tilde{{{\mathfrak{a}}}}_\bullet$ that is a ‘limit point’ of the sequence $({{\mathfrak{a}}}_{\bullet}(v_i))_{i\in{\mathbb{N}}}$, a valuation $v^*$ computing ${{\rm lct}}(\tilde{{{\mathfrak{a}}}}_\bullet)$ will minimizes ${\widehat{\rm vol}}$. The existence of such ‘limits’ relies on two ingredients: the first is an asymptotic estimate to control the growth for ${{\mathfrak{a}}}_k(v_i)$ for a fixed $k$; once the growth is controlled, we can apply the generic limit construction.
For simplicity, we will assume $D=0$. More details about log pairs can be found in [@Blu18 Section 7].
Let us choose a sequence of valuations $v_i\in{{\rm Val}}_{X,x}$ such that $$\lim_{i\to\infty}{\widehat{\rm vol}}(v_i)={\widehat{\rm vol}}(x, X).$$ Since the normalized volume function is invariant after rescaling, we may assume that $v_i({\mathfrak{m}})=1$ for all $i\in{\mathbb{N}}$ where ${\mathfrak{m}}:={\mathfrak{m}}_x$. Our goal is to show that the family of graded sequences of ${\mathfrak{m}}$-primary ideals $({{\mathfrak{a}}}_\bullet(v_i))_{i\in{\mathbb{N}}}$ satisfies the following conditions:
1. For every $\epsilon>0$, there exists positive constants $M, N$ so that $${{\rm lct}}({{\mathfrak{a}}}_m(v_i))^n{{\rm mult}}({{\mathfrak{a}}}_m(v_i))\leq {\widehat{\rm vol}}(x, X)+\epsilon
\textrm{ for all }m\geq M\textrm{ and } i\geq N.$$
2. For each $m,i\in{\mathbb{N}}$, we have ${\mathfrak{m}}^m\subset{{\mathfrak{a}}}_m(v_i)$.
3. There exists $\delta>0$ such that ${{\mathfrak{a}}}_m(v_i)\subset{\mathfrak{m}}^{\lfloor m\delta\rfloor}$ for all $m,i\in{\mathbb{N}}$.
Part (b) follows easily from $v_i({\mathfrak{m}})=1$. Hence ${{\rm vol}}(v_i)\leq {{\rm mult}}({\mathfrak{m}})=:B$. For part (c), we need to use Theorem \[t-izumi\]. By Part (2), there exists a positive constant $C_1$ such that $$A_X(v)\leq C_1^{-1}\cdot v({\mathfrak{m}}){\widehat{\rm vol}}(v) \textrm{ for all }v\in{{\rm Val}}_{X,x}.$$ Let $A:=C_1^{-1}\sup_{i\in{\mathbb{N}}}{\widehat{\rm vol}}(v_i)$, then $A_X(v_i)\leq A$ for any $i\in{\mathbb{N}}$. By Theorem \[t-izumi\](1), then there exists a positive constant $C_2$ such that $$v(f)\leq C_2\cdot A_X(v){{\rm ord}}_x(f)\textrm{ for all }v\in {{\rm Val}}_{X,x}\textrm{ and }f\in{\mathcal{O}}_{X,x}.$$ In particular, $v_i(f)\leq C_2 A\cdot{{\rm ord}}_x(f)$ for all $i\in{\mathbb{N}}$ and $f\in{\mathcal{O}}_{X,x}$. Thus by letting $\delta:=(C_2 A)^{-1}$ we have ${{\mathfrak{a}}}_m(v_i)\subset{\mathfrak{m}}^{\lfloor m\delta\rfloor}$ which proves part (c).
The proof of part (a) relies on the following result on uniform convergence of multiplicities of valuation ideals.
\[p-uniform\] Let $(x\in X)$ be an $n$-dimensional klt singularity. Then for $\epsilon, A, B, r\in {\mathbb{R}}_{>0}$, there exists $M=M(\epsilon, A, B, r)$ such that for every valuation $v\in{{\rm Val}}_{X,x}$ with $A_X(v)\leq A$, ${{\rm vol}}(v)\leq B$, and $v({\mathfrak{m}})\geq 1/r$, we have $${{\rm vol}}(v)\leq \frac{{{\rm mult}}({{\mathfrak{a}}}_m(v))}{m^n}<{{\rm vol}}(v)+\epsilon\textrm{ for all } m\geq M.$$
The first inequality is straightforward. When the point is smooth, the second inequality uses the inequality that for the graded sequence of ideals $\{{{\mathfrak{a}}}_{\bullet}\}$, there exists a $k$ such that for any $m$ and $l$ $${{\mathfrak{a}}}_{ml}\subseteq {{\mathfrak{a}}}^{l}_{m-k}.$$ The proof of such result uses the multiplier ideal, see [@ELS03]. For isolated klt singularity, then an estimate of a similar form in [@Tak06] says $$\begin{aligned}
\label{e-takagi}
\mathcal{J}_X^{l-1}\cdot {{\mathfrak{a}}}_{ml}\subset {{\mathfrak{a}}}^{l}_{m-k}\end{aligned}$$ suffices, where $\mathcal{J}_X $ is the Jacobian ideal of $X$. Finally, in the general case, an argument using and interpolating $\mathcal{J}_X$ and a power of ${\mathfrak{m}}$ gives the proof. See [@Blu18 Section 3] for more details.
To continue the proof, let us fix an arbitrary $\epsilon\in{\mathbb{R}}_{>0}$. Since $A_X(v_i)\leq A$, ${{\rm vol}}(v_i)\leq B$, and $v_i({\mathfrak{m}})=1$ for all $i\in{\mathbb{N}}$, Proposition \[p-uniform\] implies that there exists $M\in{\mathbb{N}}$ such that $$\frac{{{\rm mult}}({{\mathfrak{a}}}_m(v_i))}{m^n}\leq {{\rm vol}}(v_i)+\epsilon/(2A^n)\textrm{ for all }
i\in{\mathbb{N}}.$$ We also have ${{\rm lct}}({{\mathfrak{a}}}_m(v_i))\leq A_X(v_i)/v_i({{\mathfrak{a}}}_m(v_i))\leq m\cdot A_X(v_i)$. Let us take $N\in{\mathbb{N}}$ such that ${\widehat{\rm vol}}(v_i)\leq {\widehat{\rm vol}}(x,X)+\epsilon/2$ for any $i\geq N$. Therefore, $$\begin{aligned}
{{\rm lct}}({{\mathfrak{a}}}_m(v_i))^n{{\rm mult}}({{\mathfrak{a}}}_m(v_i))&\leq A_X(v_i)^n({{\rm vol}}(v_i)+\epsilon/(2A^n))\\
& ={\widehat{\rm vol}}(v_i)+\epsilon\cdot A_X(v_i)^n/(2A^n)\\
& ={\widehat{\rm vol}}(v_i)+\epsilon/2\\&\leq {\widehat{\rm vol}}(x,X)+\epsilon.\end{aligned}$$ So part (a) is proved.
Finally, (b) and (c) guarantee that we can apply a generic limit type construction (cf. [@Blu18 Section 5]). Then (a) implies that a ‘limit point’ $\tilde{{{\mathfrak{a}}}}_\bullet$ of the sequence $({{\mathfrak{a}}}_\bullet(v_i))_{i\in{\mathbb{N}}}$ satisfies that ${{\rm lct}}(\tilde{{{\mathfrak{a}}}}_\bullet)^n{{\rm mult}}(\tilde{{{\mathfrak{a}}}}_\bullet)\leq{\widehat{\rm vol}}(x,X)$. Thus a valution $v^*$ computing the log canonical threshold of $\tilde{{{\mathfrak{a}}}}_\bullet$, whose existence follows from [@JM12], necessarily minimizes the normalized volume.
\[t-smooth\] Let $x\in (X,D)$ be an $n$-dimensional klt singularity. Then ${\widehat{\rm vol}}(x,X,D)\leq n^n$ and the equality holds if and only if $x\in X\setminus{{\rm Supp}}(D)$ is a smooth point.
Using the fact that we can specialize a graded sequence of ideals preserving the colength, and the lower semi-continuous of the log canonical thresholds, we easily get the inequality part of Theorem \[t-smooth\]. Then the equality part gives us a characterization of the smooth point using the normalized volume. The following Theorem \[t-lower\] on the semicontinuity needs a more delicate analysis. We conjecture that the normalized volume function is indeed constructible (see Conjecture \[c-constru\]).
\[t-lower\] Let $\pi:({{\mathcal{X}}},D)\to T$ together with a section $t\in T\mapsto x_t\in{{\mathcal{X}}}_t$ be a ${{\mathbb{Q} }}$-Gorenstein flat family of klt singularities. Then the function $t\mapsto {\widehat{\rm vol}}(x_t,{{\mathcal{X}}}_t,D_t)$ is lower semicontinuous with respect to the Zariski topology.
Now we introduce a key tool that the minimal model program provides to us to understand minimizing the normalized volume. For more discussions, see Section \[ss-general\].
\[d-kollar\] Let $x\in (X,D)$ be a klt singularity. We call a proper birational morphism $\mu:Y\to X$ provides a Kollár component $S$, if $\mu$ is isomorphic over $X\setminus \{x\}$, and $\mu^{-1}(x)$ is an irreducible divisor $S$, such that $(Y,S+\mu^{-1}_*D)$ is purely log terminal (plt) and $-S$ is $\mathbb{Q}$-Cartier and ample over $X$.
We have the identity: $$\begin{aligned}
{\widehat{\rm vol}}(x, X, D)=\inf_S \{ {\widehat{\rm vol}}({{\rm ord}}_S)\ | \ \mbox{for all Koll\'ar components $S$ over }x\}. \end{aligned}$$
For the explanation of proof, see the discussions for in Section \[ss-general\].
Stability in Sasaki-Einstein geometry {#s-sasaki}
=====================================
To proceed the study of normalized volumes, we will introduce the concept of K-stability. This is now a central notion in complex geometry, which serves as an algebraic characterization of the existence of some ‘canonical metrics’.
In the local setting, such problem on an affine $T$-variety $X$ with a unique fixed point $x$ was first considered in [@MSY08]. We can then varies the Reeb vector field $\xi\in {{\mathfrak{t}}}^+_{{\mathbb{R}}}$, and call such a structure $(X,\xi)$ is a Fano cone if $X$ only has klt log terminal singularities. The name is justified since if $\xi\in {{\mathfrak{t}}}^+_{{{\mathbb{Q} }}}$, let ${\langle}\xi {\rangle}$ be the ${{\mathbb{C} }}^*$ generated by $\xi$, then $X\setminus \{x\}/{\langle}\xi {\rangle}$ is a log Fano variety.
In [@MSY08], the relation between the existence of Sasaki-Einstein metric along $(X,\xi)$ and the K-stability of $(X,\xi)$, a mimic of the absolute case, was explored. A key observation in [@MSY08] is that we can define a normalized volume function ${\widehat{\rm vol}}_{X}(\xi)$ for $\xi \in {{\mathfrak{t}}}^+_{\mathbb{R}}$, and among all choices of $\xi$ the one minimizing ${\widehat{\rm vol}}_{X}(\cdot)$ gives ‘the most stable’ direction.
Then an important step to advance such a picture is made in [@CS18; @CS15] by extending the definition of K-stability notions on $(X,\xi)$ allowing degenerations, and showing that there is a Sasaki-Einstein metric along an isolated Fano cone singularity $(X,\xi)$ if and only of $(X,\xi)$ is K-polystable, extending the solution of the Yau-Tian-Donaldson’s conjecture in the Fano manifold case (see [@CDS; @Tia15]) to the cone case.
In this section, we will briefly introduce these settings.
T-varieties {#s-tvariety}
-----------
We first introduce the basic setting using $T$-varieties. For general results of $T$-varieties, see [@AIPSV12].
Assume $X={\rm Spec}_{{{\mathbb{C} }}}(R)$ is an affine variety with ${{\mathbb{Q} }}$-Gorenstein klt singularities. Denote by $T$ the complex torus $({{\mathbb{C} }}^*)^r$. Assume $X$ admits a good $T$-action in the following sense.
\[d-good\] Let $X$ be a normal affine variety. We say that a $T$-action on $X$ is [*good*]{} if it is effective and there is a unique closed point $x\in X$ that is in the orbit closure of any $T$-orbit. We shall call $x$ (sometimes also denoted by $o_X$) the vertex point of the $T$-variety $X$.
Let $N={\rm Hom}(\mathbb{C}^*, T)$ be the co-weight lattice and $M=N^*$ the weight lattice. We have a weight space decomposition of the coordinate ring of $X$: $$R=\bigoplus_{\alpha\in \Gamma} R_\alpha \mbox{ where \ } \Gamma =\{ \alpha\in M |\ R_{\alpha}\neq 0\}.$$ The action being good implies $R_0=\mathbb{C}$, which will always be assumed in the below. An ideal ${\mathfrak{a}}$ is called homogeneous if ${{\mathfrak{a}}}=\bigoplus_{\alpha\in \Gamma}{\mathfrak{a}}\cap R_\alpha$. Denote by $\sigma^{\vee}\subset M_{\mathbb{Q}}$ the cone generated by $\Gamma$ over $\mathbb{Q}$, which will be called the [*weight cone*]{} or the [*moment cone*]{}. The cone $\sigma\subset N_{{\mathbb{R}}}$, dual to $\sigma^\vee$, is the same as the following conical set $$\mathfrak{t}^+_{{\mathbb{R}}}:=\{\ \xi \in N_{\mathbb{R}}\ \ | \ \langle \alpha, \xi \rangle>0 \mbox{ for any }\alpha\in \Gamma\setminus\{0\} \}.$$
Motivated by notations from Sasaki geometry, we will introduce:
\[defn-Reeb\] With the above notations, $\frak{t}^+_{\mathbb{R}}$ will be called the Reeb cone of the $T$-action of $X$. A vector $\xi\in {{\mathfrak{t}}}^+_{\mathbb{R}}$ will be called a Reeb vector on the $T$-variety $X$.
To adapt this definition into our setting in Section \[ss-firstdefinition\], for any $\xi \in \mathfrak{t}^+_{{\mathbb{R}}}$, we can define a valuation $${{\rm wt}}_{\xi}(f) = \min_{\alpha\in \Gamma}\{\langle \alpha,\xi \rangle \ | \ f_{\alpha}\neq 0\}.$$ It is easy to verify that ${{\rm wt}}_\xi\in {{\rm Val}}_{X,o_X}$. The rank of $\xi$, denoted by ${\rm rk}(\xi)$, is the dimension of the subtorus $T_\xi$ (as a subgroup of $T$) generated by $\xi\in {{\mathfrak{t}}}$. The following lemma can be easily seen.
\[lem-Tqmv\] For any $\xi\in \mathfrak{t}^+_{{\mathbb{R}}}$, ${{\rm wt}}_\xi$ is a quasi-monomial valuation of rational rank equal to the rank of $\xi$. Moreover, the center of ${{\rm wt}}_\xi$ is $o_X$.
We recall the following structure results for any $T$-varieties.
Let $X={\rm Spec}(R)$ be a normal affine variety and suppose $T={\rm Spec}\left({{\mathbb{C} }}[M]\right)$ has a good action on $X$ with the weight cone $\sigma^{\vee} \subset M_{{{\mathbb{Q} }}}$. Then there exist a normal projective variety $Y$ and a polyhedral divisor ${\mathfrak{D}}$ such that there is an isomorphism of graded algebras: $$R\cong H^0(X, \mathcal{O}_X)\cong \bigoplus_{u\in \sigma^{\vee} \cap M} H^0 \big(Y, \mathcal{O}({\mathfrak{D}}(u))\big)=: R(Y, {\mathfrak{D}}).$$ In other words, $X$ is equal to ${\rm Spec}_{{\mathbb{C} }}\big( \bigoplus_{u\in \sigma^{\vee} \cap M} H^0(Y, \mathcal{O}({\mathfrak{D}}(u)) ) \big)$.
In the above definition, a polyhedral divisor ${\mathfrak{D}}: u\rightarrow {\mathfrak{D}}(u)$ is a map from $\sigma^\vee$ to the set of ${{\mathbb{Q} }}$-Cartier divisors that satisfies:
1. ${\mathfrak{D}}(u)+{\mathfrak{D}}(u')\le {\mathfrak{D}}(u+u')$ for any $u, u'\in \sigma^\vee$;
2. $u\mapsto{\mathfrak{D}}(u)$ is piecewisely linear;
3. ${\mathfrak{D}}(u)$ is semiample for any $u\in \sigma^\vee$, and ${\mathfrak{D}}(u)$ is big if $u$ is in the relative interior of $\sigma^\vee$.
Here $Y $ is projective since from our assumption $$H^0(Y,\mathcal{O}_Y)=R^T=R_0=\mathbb{C}$$ (see [@LS13]). We collect some basic results about valuations on $T$-varieties.
\[t-Tcano\] Assume a $T$-variety $X$ is determined by the data $(Y, \sigma, {\mathfrak{D}})$ such that $Y$ is a projective variety, where $\sigma={{\mathfrak{t}}}^+_{{\mathbb{R}}} \subset N_{{\mathbb{R}}}$ and ${\mathfrak{D}}$ is a polyhedral divisor.
1. For any $T$-invariant quasi-monomial valuation $v$, there exist a quasi-monomial valuation $v^{(0)}$ over $Y$ and $\xi\in M_{{\mathbb{R}}}$ such that for any $f\cdot \chi^u\in R_u$, we have: $$v(f\cdot \chi^u)=v^{(0)}(f)+\langle u, \xi\rangle.$$ We will use $(\xi, v^{(0)})$ to denote such a valuation.
2. $T$-invariant prime divisors on $X$ are either vertical or horizontal. Any vertical divisor is determined by a divisor $Z$ on $Y$ and a vertex $v$ of ${\mathfrak{D}}_Z$, and will be denoted by $D_{(Z,v)}$. Any horizontal divisor is determined by a ray $\rho$ of $\sigma$ and will be denoted by $E_\rho$.
3. Let $D$ be a $T$-invariant vertical effective $\mathbb{Q}$-divisor. If $K_X+D$ is ${{\mathbb{Q} }}$-Cartier, then the log canonical divisor has a representation $K_X+D=\pi^*H+{\rm div}(\chi^{-u_0})$ where $H=\sum_Z a_Z\cdot Z$ is a principal ${{\mathbb{Q} }}$-divisor on $Y$ and $u_0\in M_{{{\mathbb{Q} }}}$. Moreover, the log discrepancy of the horizontal divisor $E_\rho$ is given by: $$A_{(X,D)}(E_\rho)=\langle u_0, n_\rho\rangle,$$ where $n_\rho$ is the primitive vector along the ray $\rho$.
For the first statement, the case of divisorial valuations follows from [@AIPSV12 Section 11]. It can be extended to the case of quasi-monomial valuations by the same proof. Note also that any $T$-invariant quasimonomial valuation can be approximated by a sequence of $T$-invariant divisorial valuations. The second statement is in [@PS08 Proposition 3.13]. The absolute case (e.g. without boundary divisor $D$) for the third statement is from [@LS13 Section 4] whose proof also works for the case of log pairs..
We will specialize the study of general affine $T$-varieties to case that the log pair is klt. Assume $X$ is a normal affine variety with ${{\mathbb{Q} }}$-Gorenstein klt singularities and a good $T$-action. Let $D$ be a $T$-invariant vertical divisor. Then there is a nowhere-vanishing $T$-equivariant section $s$ of $m(K_X+D)$ where $m$ is sufficiently divisible. The following lemma says that the log discrepancy of ${{\rm wt}}_{\xi}$ can indeed be calculated in a similar way as in the toric case (the toric case is well-known). Moreover, it can be calculated by using the weight of $T$-equivariant pluri-log-canonical sections. The latter observation was first made in [@Li15].
\[lem-ldwt\] Using the same notion as in the Theorem \[t-Tcano\], the log discrepancy of ${{\rm wt}}_{\xi}$ is given by: $A_{(X,D)}({{\rm wt}}_\xi)=\langle u_0, \xi \rangle$. Moreover, let $s$ be a $T$-equivariant nowhere-vanishing holomorphic section of $|-m(K_X+D)|$, and denote ${\mathcal{L}}_\xi$ the Lie derivative with respect to the holomorphic vector field associated to $\xi$. Then $A_{(X,D)}(\xi)=\lambda$ if and only if $$\mathcal{L}_{\xi}(s)=m \lambda s \quad \text{ for } \quad \lambda>0.$$
As a consequence of the above lemma, we can formally extend $A_{(X,D)}(\xi)$ to a linear function on ${{\mathfrak{t}}}_{{\mathbb{R}}}$: $$\label{eq-linearA}
A_{(X,D)}(\eta)=\langle u_0, \eta\rangle.$$ for any $\eta\in {{\mathfrak{t}}}_{{\mathbb{R}}}$. By Lemma \[lem-ldwt\], $A_{(X,D)}(\eta)=\frac{1}{m} {\mathcal{L}}_\eta s/s$ where $s$ is a $T$-equivariant nowhere-vanishing holomorphic section of $|-m(K_X+D)|$.
\[d-logfanocone\] Let $(X, D)$ be an affine pair with a good $T$ action. Assume $(X,D)$ is a normal pair with klt singularities. Then for any $\xi\in {{\mathfrak{t}}}^+_{\mathbb{R}}$, we call the triple $(X,D,\xi)$ a [*log Fano cone*]{} structure that is polarized by $\xi$. We will denote by ${\langle}\xi{\rangle}$ the sub-torus of $T$ generated by $\xi$. If ${\langle}\xi{\rangle}\cong {{\mathbb{C} }}^*$, then we call $(X, D, \xi)$ to be quasi-regular. Otherwise, we call it irregular.
In the quasi-regular case, we can take the quotient $(X\setminus\{x\},D\setminus\{x\})$ by the $\mathbb{C^*}$-group generated by $\xi$ in the sense of Seifert ${{\mathbb{C} }}^*$-bundles (see [@Kol04]), and we will denote by $(X,D)/\langle \xi \rangle$, which is a log Fano variety, because of the assumption that $(X,D)$ is klt at $x$ (see [@Kol13 Lemma 3.1]).
K-stability
-----------
In this section, we will discuss the K-stability notion of log Fano cones, which generalizes the K-stability of log Fano varieties originally defined by Tian and Donaldson. For irregular Fano cones, such a notion was first defined in [@CS18].
\[defn-QGTC\] Let $(X, D, \xi_0)$ be a log Fano cone singularity and $T$ a torus containing ${\langle}\xi_0{\rangle}$.
A $T$-equivariant test configuration (or simply called a test configuration) of $(X, D, \xi_0)$ is a quadruple $({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)$ with a map $\pi: ({{\mathcal{X}}}, {\mathcal{D}})\rightarrow{{\mathbb{C} }}$ satisfying the following conditions:
1. $\pi:{{\mathcal{X}}}\rightarrow {{\mathbb{C} }}$ is a flat family and ${\mathcal{D}}$ is an effective $\mathbb{Q}$-divisor such that ${\mathcal{D}}$ does not contain any component $X_0$, the fibres away from $0$ are isomorphic to $(X, D)$ and ${{\mathcal{X}}}={\rm Spec}({\mathcal{R}})$ is affine, where ${\mathcal{R}}$ is a finitely generate flat ${{\mathbb{C} }}[t]$ algebra. The torus $T$ acts on ${{\mathcal{X}}}$, and we write $\mathcal{R}=\bigoplus_{\alpha}\mathcal{R}_{\alpha}$ as decomposition into weight spaces.
2. $\eta$ is a holomorphic vector field on ${{\mathcal{X}}}$ generating a ${{\mathbb{C} }}^*(=\langle \eta\rangle)$-action on $({{\mathcal{X}}}, {\mathcal{D}})$ such that $\pi$ is ${{\mathbb{C} }}^*$-equivariant where ${{\mathbb{C} }}^*$ acts on the base ${{\mathbb{C} }}$ by the multiplication (so that $\pi_*\eta=t\partial_t$ if $t$ is the affine coordinate on ${{\mathbb{C} }}$) and there is a ${{\mathbb{C} }}^*$-equivariant isomorphism $\phi: ({{\mathcal{X}}}, {\mathcal{D}})\times_{{{\mathbb{C} }}}{{\mathbb{C} }}^*\cong
(X, D)\times {{\mathbb{C} }}^*$.
3. The torus $T$-action commutes with $\eta$. The holomorphic vector field $\xi_0$ on ${{\mathcal{X}}}\times_{{{\mathbb{C} }}}{{\mathbb{C} }}^*$ (via the isomorphism $\phi$) extends to a holomorphic vector field on ${{\mathcal{X}}}$ which we still denote to be $\xi_0$.
In most our study, we only need to treat the case that test configuration $({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)$ of $(X, D, \xi_0)$ satisfies that
1. $K_{{{\mathcal{X}}}}+{\mathcal{D}}$ is ${{\mathbb{Q} }}$-Cartier and the central fibre $(X_0, D_0)$ is klt
In other words, we will mostly consider special test configurations (see [@LX14; @CS15]).
Condition (1) implies that each weight piece ${\mathcal{R}}_{\alpha}$ is a flat ${{\mathbb{C} }}[t]$-module. So $X$ and $X_0$ have the same weight cone and Reeb cone with respect to the fiberwise $T$-action.
A test configuration $({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)$ is called a product one if there is a $T$-equivariant isomorphism $({{\mathcal{X}}}, {\mathcal{D}})\cong (X, D)\times {{\mathbb{C} }}$ and $\eta=\eta_0+t\partial_t$ where $\eta_0$ is a holomorphic vector field on $X$ that preserves $D$ and commutes with $\xi_0$. In this case, we will denote $({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)$ by $$(X\times{{\mathbb{C} }}, D\times {{\mathbb{C} }}, \xi_0; \eta)=:(X_{{\mathbb{C} }}, D_{{\mathbb{C} }}, \xi_0; \eta).$$ In [@MSY08], only such test configurations are considered.
\[d-ksemiSE\]For any special test configuration $({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)$ of $(X,D,\xi_0)$ with central fibre $(X_0, D_0, \xi_0)$, its generalized Futaki invariant is defined as $$\begin{aligned}
{{\rm Fut}}({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta):=\frac{D_{-T_{\xi_0}(\eta)}{{\rm vol}}_{X_0}(\xi_0)}{{{\rm vol}}_{X_0}(\xi_0)} \end{aligned}$$ where we denote $$\label{eq-normeta}
T_{\xi_0}(\eta)=\frac{A(\xi_0)\eta-A(\eta)\xi_0}{n}.$$ Since the generalized Futaki invariant defined above only depends on the data on the central fibre, we will also denote it by ${{\rm Fut}}(X_0, D_0, \xi_0; \eta)$.
We say that $(X, D, \xi_0)$ is K-semistable, if for any special test configuration, ${{\rm Fut}}({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)$ is nonnegative.
We say that $(X, D, \xi_0)$ is K-polystable, if it is K-semistable, and any special test configuration $({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)$ with ${{\rm Fut}}({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)=0$ is a product test configuration.
In the above definition, we used the notation and the directional derivative: $$D_{-T_{\xi_0(\eta)}}{{\rm vol}}_{X_0}(\xi_0):=\left.\frac{d}{d\epsilon}\right|_{\epsilon=0}{{\rm vol}}_{X_0}(\xi_0-\epsilon T_{\xi_0(\eta)}).$$ Recall that the $\pi_*\eta=t\partial_t$. Then the negative sign in front of $T_{\xi_0}(\eta)$ in the above formula is to be compatible with our later computation. Using the rescaling invariance of the normalized volume, it is easy to verify that the following identity holds: $$\label{eq-dirhvol}
D_{-T_{\xi_0}\eta}{{\rm vol}}_{X_0}(\xi_0)=\left.\frac{d}{d\epsilon}\right|_{\epsilon=0}{\widehat{\rm vol}}_{X_0}({{\rm wt}}_{\xi_0-\epsilon\eta})\cdot \frac{1}{n A(\xi_0)^{n-1}},$$ where $A(\xi_0)=A_{(X_0,D_0)}({{\rm wt}}_{\xi_0})$. As a consequence, we can rewrite the Futaki invariant of a special test configuration as: $$\label{eq-Futhvol}
{{\rm Fut}}({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta):=D_{-\eta}{\widehat{\rm vol}}_{X_0}({{\rm wt}}_{\xi_0})\cdot \frac{1}{n A(\xi_0)^{n-1} \cdot {{\rm vol}}_{X_0}(\xi_0)}.$$
One can show that, up to a constant, the above definition of ${{\rm Fut}}({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)$ coincides with the one in [@CS18; @CS15] defined using index characters. For convenience of the reader, we recall their definition. It is enough to define the Futaki invariant for the central fibre which we just denote by $X$. For any $\xi\in \frak{t}_{\mathbb{R}}^+$, the index character $F(\xi, t)$ is defined by: $$F(\xi, t):=\sum_{\alpha\in \Gamma}e^{-t\langle \alpha, \xi\rangle}\dim_{{{\mathbb{C} }}}R_\alpha.$$ Then there is a meromorphic expansion for $F(\xi, t)$ as follows: $$F(\xi, t)=\frac{a_0(\xi)(n-1)!}{t^n}+\frac{a_1(\xi)(n-2)!}{t^{n-1}}+O(t^{2-n}).$$ One always has the identity $a_0(\xi)={{\rm vol}}(\xi)/(n-1)!$.
For any $\eta\in \frak{t}_{{\mathbb{R}}}$, define: $$\begin{aligned}
{{\rm Fut}}_{\xi_0}(X, \eta)&=&\frac{1}{n-1}D_{-\eta}(a_1(\xi_0))-\frac{1}{n}\frac{a_1(\xi_0)}{a_0(\xi_0)}D_{-\eta} a_0(\xi_0)\\
&=&\frac{a_0(\xi_0)}{n-1}D_{-\eta} \left(\frac{a_1}{a_0}\right)(\xi_0)+\frac{a_1(\xi_0)D_{-\eta} a_0(\xi_0)}{n(n-1)a_0(\xi_0)}.\end{aligned}$$
This is a complicated expression. But in [@CS15 Proposition 6.4], it was showed that, when $X$ is ${{\mathbb{Q} }}$-Gorenstein log terminal, there is an identity $a_1(\xi)/a_0(\xi)=A(\xi)(n-1)/2$ for any $\xi\in \frak{t}_{{\mathbb{R}}}^+$ (by using our notation involving log discrepancies). Note that the rescaling properties $a_0(\lambda \xi)=\lambda^{-n}a_0(\xi)$ and $a_1(\lambda \xi)=\lambda^{-(n-1)}a_1(\xi)$ which imply ${{\rm Fut}}_{\xi_0}(X, \xi_0)=0$. If we denote $\eta'=\eta-\frac{A(\eta)}{A(\xi_0)}\xi_0$, then we get: $${{\rm Fut}}_{\xi_0}(X, \eta)={{\rm Fut}}_{\xi_0}\left(X, \eta'\right)=\frac{A(\xi_0)}{2n}D_{-\eta'}a_0(\xi_0)=\frac{1}{2(n-1)!} D_{-T_{\xi_0}(\eta)}{{\rm vol}}(\xi_0).$$ So the definition in [@CS18; @CS15] differs from our notation by a constant $2(n-1)!/{{\rm vol}}_{X}(\xi_0)$.
More precisely, our notation differs from that in [@CS18] by a sign. Our choice of minus sign for $-\eta$, besides being compatible with the sign choice in Tian’s original definition of K-stability in [@Tia97], is made for least two reasons. The first is that the careful calculation in [@LX17 Section 5.2] shows that the limiting slope of the Ding energy along the geodesic ray associated to any special test configuration is indeed the directional derivative of ${{\rm vol}}(\xi)$ along $-\eta$ instead of $\eta$. For the second reason, as we stressed in [@LX17 Remark 3.4], for the special test configuration coming from a Kollár component $S$, the $-\eta$ vector corresponds to ${{\rm ord}}_S$. Since our goal is to compare ${\widehat{\rm vol}}({{\rm wt}}_{\xi_0})$ and ${\widehat{\rm vol}}({{\rm ord}}_S)$, $-\eta$ is the correct choice of sign (see [@LX17 Proof of Theorem 3.5]).
\[r-Ding\] In fact, in a calculation, instead of the generalized Futaki invariant, it is the Berman-Ding invariant, denoted by $D^{{\rm NA}}({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)$, where $$\begin{aligned}
D^{{\rm NA}}({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta):=\frac{D_{-T_{\xi_0}(\eta)}{{\rm vol}}_{X_0}(\xi_0)}{{{\rm vol}}(\xi_0)}-(1-{{\rm lct}}({{\mathcal{X}}}, D; {{\mathcal{X}}}_0)).\end{aligned}$$ appears more naturally, whenever we know
1. there exists a nowhere vanishing section $s\in |m (K_{{{\mathcal{X}}}}+{\mathcal{D}})|$ such that we can use it to define $A(\cdot)$ as in the formula in Lemma \[lem-ldwt\].
Then we can similarly define Ding semi(poly)-stable, replacing ${{\rm Fut}}({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)$ by $D^{{\rm NA}}({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)$. For a special test configuration, since $${{\rm Fut}}({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)=D^{{\rm NA}}({{\mathcal{X}}}, {\mathcal{D}}, \xi_0; \eta)$$ the two notions coincide.
If we specialize the above definitions to the case of quasi-regular log Fano cone $(X,D,\xi_0)$, then we get the corresponding more familiar notions for the log Fano projective pair $(S,B)=(X,D)/{\langle}\xi_0{\rangle}$.
Sasaki-Einstein geometry
------------------------
The introduction of normalized volumes in [@Li15] was motivated by the minimization phenomenon in the study of Sasaki-Einstein metrics. The latter was discovered in [@MSY06; @MSY08] and was motivated by the so called AdS/CFT correspondence from mathematical physics. Here we give a short account on this. For the reader who are mostly interested in the algebraic part of the theory, one can skip this section. The results will only be used in Section \[ss-dsc\].
Classically, a Sasaki manifold is defined as an odd dimensional Riemannian manifold $(M^{2n-1}, g_M)$ such that metric cone over it, defined as: $$(X, g_X):=((M\times {\mathbb{R}}_{>0})\cup \{o_X\}, dr^2+r^2 g_M)$$ is Kähler. It’s convenient to work directly on $X=X^\circ\cup o_X$ which is an affine variety with the Kähler metric ${{\sqrt{-1}\partial\bar{\partial}}}r^2$. The Reeb vector field of $(X^\circ, g_X)$ is usually defined as $J(r\partial_r)$ where $J$ is the complex structure on $X^\circ$. The corresponding holomorphic vector field $\xi=r\partial_r-iJ(r\partial_r)$, which we also call Reeb vector field, generates a $T_\xi\cong ({{\mathbb{C} }}^*)^{{\rm rk}(\xi)}$-action on $X$ where $r(\xi)\ge 1$. For simplicity, we will denote such a torus by $\langle \xi\rangle$. Moreover the corresponding element in $(\frak{t}_\xi)_{\mathbb{R}}$, also denoted by $\xi$ is in the Reeb cone: $\xi\in (\frak{t}_\xi)^+_{\mathbb{R}}$. The volume of $\xi$ is defined to be the volume density of $g_X$: $$\begin{aligned}
\label{eq-volxi}
{{\rm vol}}(\xi)&:=&{{\rm vol}}(r^2)=\frac{1}{(2\pi)^n n!}\int_X e^{-r^2}({{\sqrt{-1}\partial\bar{\partial}}}r^2)^n\nonumber \\
&=&\frac{1}{(2\pi)^n}\int_M (-Jdr)\wedge (-dJd r)^{n-1}\nonumber \\
&=&\frac{(n-1)!}{2\pi^n}{{\rm vol}}(M, g_M)=\frac{{{\rm vol}}(M, g_M)}{{{\rm vol}}({\mathbb{S}}^{2n-1})}\nonumber \\
&=&\frac{{{\rm vol}}(B_1(X), g_X)}{{{\rm vol}}(B_1(\underline{0}), g_{{{\mathbb{C} }}^n})}.\end{aligned}$$ Here $g_X=\frac{1}{2}{{\sqrt{-1}\partial\bar{\partial}}}r^2(\cdot, J\cdot)$ and $g_M=\left.g_X\right|_M$ are the Riemannian metric on $X$ and $M$ respectively, ${\mathbb{S}}^{2n-1}$ is the standard unit sphere in ${{\mathbb{C} }}^n$ with volume ${{\rm vol}}({\mathbb{S}}^{2n-1})=2\pi^n/(n-1)!$.
This is well-defined because if two Sasaki metrics have the same Reeb vector field, then their volumes are the same. Indeed, $\omega_1={{\sqrt{-1}\partial\bar{\partial}}}r_1$ and $\omega_2={{\sqrt{-1}\partial\bar{\partial}}}r_2$ have the same Reeb vector field if $r_2=r_1 e^{{{\varphi}}}$ for a function ${{\varphi}}$ satisfying ${\mathcal{L}}_{r\partial_r}{{\varphi}}={\mathcal{L}}_{\xi}{{\varphi}}=0$ (i.e. ${{\varphi}}$ is a horizontal function on $M$ with respect to the foliation defined by ${\rm Im}(\xi_0)$). Letting $r_t^2=r^2 e^{t{{\varphi}}}$ and differentiating the volume we get: $$\begin{aligned}
C\cdot \frac{d}{dt}{{\rm vol}}(r_t^2)&=&\int_X e^{-r_t^2}(-r_t^2{{\varphi}}){{\sqrt{-1}\partial\bar{\partial}}}r_t^2)^n+e^{-r_t^2}n {{\sqrt{-1}\partial\bar{\partial}}}(r_t^2{{\varphi}})\wedge ({{\sqrt{-1}\partial\bar{\partial}}}r_t^2)^{n-1}\\
&=&\int_X -e^{-r_t^2} r_t^2{{\varphi}}({{\sqrt{-1}\partial\bar{\partial}}}r_t^2)^n+e^{-r_t^2} n \sqrt{-1}\partial r_t^2\wedge ({{\varphi}}\bar{\partial} r_t^2) \wedge ({{\sqrt{-1}\partial\bar{\partial}}}r_t^2)^{n-1}\\
&&\quad +\int_{X}e^{-r_t^2} n \sqrt{-1}\partial r_t^2\wedge (r_t^2 \bar{\partial}{{\varphi}}) \wedge ({{\sqrt{-1}\partial\bar{\partial}}}r_t^2)^{n-1}\\
&=&0.\end{aligned}$$ The second equality follows from integration by parts. The last equality follows by substituting $f=r_t^2$ and $f={{\varphi}}$ in to the following identities and using the fact that ${{\varphi}}$ is horizontal (so that $\xi_t({{\varphi}})=0$): $$\begin{aligned}
n \sqrt{-1} \partial r_t^2 \wedge \bar{\partial} f\wedge ({{\sqrt{-1}\partial\bar{\partial}}}r_t^2)^{n-1}=\xi_t(f)({{\sqrt{-1}\partial\bar{\partial}}}r_t^2)^n.\end{aligned}$$
One should compare this to the fact that two Kähler metrics in the same Kähler class have the same volume.
The Reeb vector field associated to a Ricci-flat Kähler cone metric satisfies the minimization principle in [@MSY08]. To state it in general, we assume $X$ is a $T$-variety with the Reeb cone $\frak{t}^+_{\mathbb{R}}$ with respect to $T$ and recall the variation formulas of volumes of Reeb vector fields from [@MSY08]. For any $\xi\in \frak{t}^+_{\mathbb{R}}$, we can find a radius function $r: X\rightarrow {\mathbb{R}}_+$ such that ${{\rm vol}}(\xi)$ is given by the formula .
The first order derivative of ${{\rm vol}}_{X}(\xi)$ is given by: $$\label{eq-Dvol}
D{{\rm vol}}(\xi)\cdot \eta_1=\frac{1}{(2\pi)^n (n-1)!}\int_X \theta_1 e^{-r^2}({{\sqrt{-1}\partial\bar{\partial}}}r^2)^n,$$ where $\theta_i=\eta_i(\log r^2)$. The second order variation of ${{\rm vol}}_{X}(\xi)$ is given by: $$\begin{aligned}
D^2{{\rm vol}}(\xi)(\eta_1, \eta_2)&=&\frac{n+1}{(2\pi)^n(n-1)!}\int_X \theta_1\theta_2 e^{-r^2} ({{\sqrt{-1}\partial\bar{\partial}}}r^2)^n.\end{aligned}$$
Now we fix a $\xi_0\in \frak{t}^+_{\mathbb{R}}$ and a radius function $r: X\rightarrow {\mathbb{R}}_+$ (by using equivariant embedding of $X$ into ${{\mathbb{C} }}^N$ for example), we define:
$PSH(X, \xi_0)$ is the set of bounded real functions ${{\varphi}}$ on $X^\circ$ that satisfies:
1. ${{\varphi}}\circ \tau={{\varphi}}$ for any $\tau\in \langle \xi_0 \rangle$, the torus generated by $\xi_0$;
2. $r^2_{{\varphi}}:=r^2e^{{\varphi}}$ is a proper plurisubharmonic function on $X$.
To write down the equation of Ricci-flat Kähler-cone equation, we fix a $T$-equivariant no-where vanishing section $s\in H^0(X, mK_X)$ as in the last section and define an associated volume form on $X$: $$\label{eq-MARF}
dV_X:=\left((\sqrt{-1})^{mn^2} s\wedge \bar{s}\right)^{1/m}.$$
\[defn-RFKC\] We say that $r^2_{{\varphi}}:=r^2 e^{{{\varphi}}}$ where ${{\varphi}}\in PSH(X, \xi_0)$ is the radius function of a Ricci-flat Kähler cone metric on $(X, \xi_0)$ if ${{\varphi}}$ is smooth on $X^{\rm reg}$ and there exists a positive constant $C>0$ such that $$\label{eq-RFKC}
({{\sqrt{-1}\partial\bar{\partial}}}r^2_{{\varphi}})^n=C\cdot dV,$$ where the constant $C$ is equal to: $$C=\frac{\int_X e^{-r^2_{{\varphi}}}({{\sqrt{-1}\partial\bar{\partial}}}r^2_{{\varphi}})^n}{\int_X e^{-r^2_{{\varphi}}}dV_X}=\frac{(2\pi)^n n! {{\rm vol}}(\xi_0)}{\int_X e^{-r^2_{{\varphi}}}dV_X}.$$
Motivated by standard Kähler geometry, one defines the Monge-Ampère energy $E({{\varphi}})$ using either its variations or the explicit expression on the link $M:=X\cap \{r=1\}$: $$\begin{aligned}
\delta E({{\varphi}})\cdot \delta{{\varphi}}&=&-\frac{1}{(n-1)!(2\pi)^n {{\rm vol}}(\xi_0)}\int_X \delta{{\varphi}}e^{-r^2_{{\varphi}}}({{\sqrt{-1}\partial\bar{\partial}}}r^2_{{\varphi}})^n.\end{aligned}$$ Then the equation is the Euler-Lagrange equation of the following Ding-Tian-typed functional: $$\begin{aligned}
D({{\varphi}})&=&E({{\varphi}})-\log\left(\int_X e^{-r^2_{{\varphi}}}dV_X\right).\end{aligned}$$ This follows from the identity: $$\begin{aligned}
\delta D({{\varphi}})\cdot \delta{{\varphi}}&=&\frac{1}{(2\pi)^n(n-1)!{{\rm vol}}(\xi_0)}\int_X \delta{{\varphi}}e^{-r^2_{{\varphi}}}({{\sqrt{-1}\partial\bar{\partial}}}r^2_{{\varphi}})^n-\frac{\int_X r^2_{{\varphi}}\delta{{\varphi}}e^{-r^2_{{\varphi}}}dV_X}{\int_X e^{-r^2_{{\varphi}}}dV_X}\\
&=&n \int_X e^{-r^2_{{\varphi}}} \delta{{\varphi}}\left(\frac{({{\sqrt{-1}\partial\bar{\partial}}}r^2_{{\varphi}})^n}{(2\pi)^n n!{{\rm vol}}(\xi_0)}-\frac{dV_X}{\int_X e^{-r^2_{{\varphi}}}dV_X}\right).\end{aligned}$$
Compared with the weak Kähler-Einstein case, it is expected that the regularity condition in the above definition is automatically satisfied. With this regularity assumption, on the regular part $X^{\rm reg}$, both sides of are smooth volume forms and we have $r_{{{\varphi}}}\partial_{r_{{\varphi}}}=2 {\rm Re}(\xi_0)$ or, equivalently, $\xi_0=r_{{\varphi}}\partial_{r_{{\varphi}}}-i J(r_{{\varphi}}\partial_{r_{{\varphi}}})$. Moreover, taking $\mathcal{L}_{r_{{\varphi}}\partial_{r_{{\varphi}}}}$ on both sides gives us the identity $\mathcal{L}_{r_{{\varphi}}\partial_{r_{{\varphi}}}}dV=2n\; dV$. Equivalently we have: $${\mathcal{L}}_{\xi_0}s= m n\cdot s,$$ where $s\in |-mK_X|$ is the chosen $T$-equivariant non-vanishing holomorphic section. By Lemma \[lem-ldwt\], this implies $A_{X}({{\rm wt}}_{\xi_0})=n$ (see [@HS17; @LL16] for this identity in the quasi-regular case). The main result of [@MSY08] can be stated as follows.
\[thm-irSE\] If $(X, \xi_0)$ admits a Ricci-flat Kähler cone metric, then $A_X(\xi_0)=n$ and ${{\rm wt}}_{\xi_0}$ obtains the minimum of ${{\rm vol}}$ on $\frak{t}^{+}_{\mathbb{R}}$.
The following result partially generalizes Berman’s result on K-polystability of Kähler-Einstein Fano varieties to the more general case of Ricci-flat Fano cones. Together with Theorem \[t-SDChigh\], it is used to show a generalization the minimization result [@MSY08]: the valuation ${{\rm wt}}_{\xi_0}$ minimizes ${\widehat{\rm vol}}$ where $\xi_0$ is the Reeb vector field of the Ricci-flat Fano cone.
\[thm-RF2K\] Assume $(X, \xi_0)$ admits a Ricci-flat Kähler cone metric. Then $A_{X}({{\rm wt}}_{\xi_0})=n$ and $(X, \xi_0)$ is K-polystable among all special test configurations.
Fix any smooth Kähler cone metric ${{\sqrt{-1}\partial\bar{\partial}}}r^2$ on $X$. Any special test configuration determines a geodesic ray $\{r_t^2=r^2 e^{{{\varphi}}_t}\}_{t> 0}$ of Kähler cone metrics. Denote $D(t)=D({{\varphi}}_t)$. Then we have the following formula: $$\lim_{t\rightarrow 0} \frac{D(t)}{-\log|t|^2}=\frac{D_{-\eta}{{\rm vol}}(\xi_0)}{{{\rm vol}}(\xi_0)}-(1-{{\rm lct}}({{\mathcal{X}}}, {{\mathcal{X}}}_0))=D^{{\rm NA}}(\chi, \xi_0; \eta),$$ which is a combination of two ingredients:
1. The Fano cone version of an identity from Kähler geometry which combined with gives the formula: $$\lim_{t\rightarrow 0}\frac{E({{\varphi}}_t)}{-\log|t|^2}=\frac{D_{-\eta}{{\rm vol}}(\xi_0)}{{{\rm vol}}(\xi_0)}.$$
2. $G({{\varphi}}_t)$ is subharmonic in $t$ (cone version of Berndtsson’s result) and its Lelong number at $t=0$ is given by $1-{{\rm lct}}({{\mathcal{X}}}, {{\mathcal{X}}}_0)$ (cone version of Beman’s result).
The other key result is the cone version of Berndtsson’s subharmonicity and uniqueness result, which was used to characterize the case of vanishing Futaki invariant.
The argument in [@LX17] gives a slightly more general result: Assume $(X, \xi_0)$ admits a Ricci-flat Käler cone metric, then $A_{X}({{\rm wt}}_{\xi_0})=n$ and $(X, \xi_0)$ is Ding-polystable among ${{\mathbb{Q} }}$-Gorenstein test configurations (see Remark \[r-Ding\]).
Stable degeneration conjecture {#s-SDC}
==============================
In this section, we give a conjectural description of minimizers for general klt singularities, and explain various parts of the picture that we can establish.
Statement
---------
For a klt singularity $x\in (X,D)$, one main motivation to study the minimizer $v$ of ${\widehat{\rm vol}}_{(X,D),x}$ is to establish a ‘local K-stability’ theory, guided by the local-to-global philosophy mentioned in the introduction. In particular, we propose the following conjecture for all klt singularities.
\[conj-local\] Given any arbitrary klt singularity $x\in (X={\rm Spec}(R), D)$, there is a unique minimiser $v$ up to rescaling. Furthermore, $v$ is quasi-monomial, with a finitely generated associated graded ring $R_0=_{\rm defn}{\rm gr}_v(R)$, and the induced degeneration $$(X_0={\rm Spec}(R_0), D_0, \xi_v)$$ is a K-semistable Fano cone singularity. (See below for the definitions.)
Let us explain the terminology in more details: First by the grading of $R_0$, there is a $T\cong \mathbb{C}^r$-action on $X_0$ where $r$ is the rational rank of $v$, i.e. the valuative semigroup $\Phi$ of $v$ generates a group $M\cong \mathbb{Z}^r$. Moreover, since the valuation $v$ identifies $M$ to a subgroup of $\mathbb{R}$ and sends $\Phi$ into $\mathbb{R}_{\ge 0}$, it induces an element in the Reeb cone $\xi_v$. If $R_0$ is finitely generated, then [@LX17] shows that we can embed $(x\in X)\subset (0\in \mathbb{C}^N)$ and find an rational vector $\xi \in \mathfrak{t}^{+}_{\mathbb{R}}\cap N_{\mathbb{Q}}$ sufficiently close to $\xi_v$ such that the $\mathbb{C}^*$-action generated by $\xi$ degenerates $X$ to $X_0$ with a good action. We denote by $o$ (or $o_{X_0}$) the unique fixed point on $X_0$. Furthermore, the extended Rees algebra yielding the degeneration does not depend on the choice of $\xi$. So we can define $D_0$ as the degeneration of $D$.
Conjecture \[conj-local\], if true, would characterize deep properties of a klt singularity. Various parts are known, see Theorem \[t-high\]. However, the entire picture remains open in general.
Cone case {#ss-cone}
---------
The study of the case of cone is not merely verifying a special case. In fact, since the stable degeneration conjecture predicts the degeneration of any klt singularities to a cone, understanding the cone case is a necessary step to attack the conjecture. Here we divide our presentations into two case: the rank one case and the general higher rank case. Although our argument in the higher rank case covers the rank one case with various simplifications, we believe it is easier for reader to first understand the rank one case, as it is equivalent to the more standard K-semistability theory of the base which is a log Fano pair. This connection is made via the theory of $\beta$-invariant, which is first introduced in [@Fuj18] in terms of ideal sheaves and further developed in [@Li17; @Fuj16] via valuations.
### Rank one case
The rank one Fano cone is just a cone over a log Fano pair. More precisely, let $(S,B)$ be an $(n-1)$-dimensional log Fano pair, and $r$ a positive integer such that $r(K_S+B)$ is Cartier. Then we can consider the minimizing problem of the normalized volume at the vertex of the cone $$x\in (X,D)=C(S,B; -r(K_S+B)).$$ Such a question was first extensively studied in [@Li17]. More precisely, there is a canonical divisorial valuation obtained by blowing up $x$ to get a divisor $S_0$ isomorphic to $S$, which yields the degeneration of $x\in (X,D)$ to itself with $\xi$ being the natural rescaling vector field from the cone structure. Therefore, the stable degeneration conjecture predicts $v_{S_0}={{\rm ord}}_{S_0}$ is a minimizer of ${\widehat{\rm vol}}_{(X,D),x}$ if and only if $(S,B)$ is K-semistable, and this is confirmed in [@Li17; @LL16; @LX16].
\[t-rankonecone\] The valuation $v_{S_0}$ is a stabilizer of ${\widehat{\rm vol}}_{(X,D),x}$ if and only if $(S,B)$ is K-semistable. Moreover, ${\widehat{\rm vol}}(S_0)< {\widehat{\rm vol}}(E)$ for any other divisor $E$ over $x$.
In the below, we will sketch the ideas of two slightly different proofs of Theorem \[t-rankonecone\].
In the first approach, we carry out a straightforward calculation as follows: Given a compactified nontrivial special test configuration $(\mathcal{S},\mathcal{B})$ of $(S,B)$, then we obtain a valuation $v^*$ by restricting the divisorial valuation of the special fiber $S_0$ to $K(S)\subset K(S\times \mathbb{A}^1)$, which is a multiple of some divisorial valuation (cf. [@BHJ17]). Such a valuation $v^*$ pull backs a valuation $v^*_X$ on $K(X)$. Then we define a $\mathbb{C}^*$-valuation on $K(X)$ by $v_{\infty}(f_m)=v^*_X(f_m)-mra_S(v^*)$ over $X$ for any $f_m\in H^0(S,-mr(K_S+B))$. In other words, $v_{\infty}=v^*_X-ra_S(v^*)v_{S_0}$, and we know that the induced filtration on $R$ yields the Duistermaat-Heckman (DH) measure of $(\mathcal{S},\mathcal{B})$ (see [@BHJ17 Definition 3.5]). We define the ray in $$\left\{v_t= v_{S_0}+t \cdot v_{\infty}\in {{\rm Val}}_{X,x} \ | \ t\in [0,\frac{1}{ra_S(v^*)}) \right\}.$$ Then the key computation in [@Li17] is that $$\begin{aligned}
\frac{d}{dt}{\widehat{\rm vol}}(v_t) |_{t=0}&=& \frac{n}{r^n}(-K_S-B)^{n-1}\cdot{\rm Fut}(\mathcal{S},\mathcal{B}).
$$ In fact, if for any valuation $v$ over $S$, we denote by $R_m=H^0(S,-mr(-K_S-B))$ and define $$\mathcal{F}^x_{v}R_m:=\{f\in R_m |\ \ f\in H^0(S, -mr(-K_S-B)\otimes {{\mathfrak{a}}}_x) \},$$ then we easily see $${{\mathfrak{a}}}_{k}(v_t) \cap R_m=\mathcal{F}^{\frac{k-m}{t}}_{v_{\infty}}H^0(S, -mr(-K_S-B)).$$ So $$\begin{aligned}
{{\rm vol}}(v_t)&= \lim_k \frac { l_{\mathbb{C}}(R/{{\mathfrak{a}}}_{k}(v_t))}{k^{n}/n!} \nonumber \\
& =\lim_{k\to \infty\ }\frac{n!}{k^n} \sum_{m=0}\left( \dim \mathcal{F}_{v_{\infty}}^0R_m-\dim\mathcal{F}^{\frac{k-m}{t}}_{v_{\infty}}H^0(S, -mr(-K_S-B))\right) \nonumber\\
&=-\int^{\infty}_{-\infty} \frac{d{{\rm vol}}(\mathcal{F}_{v_{\infty}}R^{(x)})}{(1+t x)^{n}} \label{e-changevb},\end{aligned}$$ where $\mathcal{F}_{v_{\infty}}R^{(x)}:=\bigoplus_{m} \mathcal{F}_{v_{\infty}}^{mx}R_m$ and the last equality is obtained by a change of variables (see Lemma [@Li17 Lemma 4.5]).
Since $A(S_0)=\frac{1}{r}$ and $A(v_{\infty})=0$, $A_{v_t}=\frac{1}{r}$, so $${\widehat{\rm vol}}(v_t)=-(\frac{1}{r})^n\int^{\infty}_{-\infty} \frac{d{{\rm vol}}(\mathcal{F}_{v_{\infty}}R^{(x)})}{(1+t x)^{n}},$$ and this implies that $$\begin{aligned}
\frac{d}{dt}{\widehat{\rm vol}}(v_t) |_{t=0}&=&\frac{n}{r^n}\int^{\infty}_{-\infty} x\cdot d{{\rm vol}}(\mathcal{F}_{v_{\infty}}R^{(x)})\\
&=&\frac{n}{r^n} \lim_{k\to \infty}\frac{w_k}{kN_k}\\
&=&-\frac{1}{r^n} (-K_{\mathcal{S}}-\mathcal{B})^{n},\\
&=&\frac{n}{r^n} (-K_S-B)^{n-1}\cdot {{\rm Fut}}(\mathcal{S},\mathcal{B}).\end{aligned}$$ where for the second equality we use that $v_{\infty}$ is the DH measure for $(\mathcal{S},\mathcal{B})$.
It is not straightforward to reverse the argument to show that $(S,B)$ is K-semistable implies that ${{\rm ord}}_{S_0}$ is a minimizer of ${\widehat{\rm vol}}_{(X,D),x}$, since a priori there could be more complicated valuations than those induced by central fibres of test configurations. In particular, originally in [@Li17], the techniques of ‘taking the limit of a sequence of filtered linear systems’ developed in [@Fuj18] were used in the case when the associated bigraded ring $$\bigoplus_{m,k}H^0(S, -rm(K_S+B)\otimes {{\mathfrak{a}}}_k)$$ is not finitely generated, and this is enough to treat all $\mathbb{C}^*$-equivariant valuations.
In [@LX16], after the MMP method was systematically applied, it was shown that $$\begin{aligned}
\label{e-Ckollar}
\inf_{v\in {{\rm Val}}_{X,x}} {\widehat{\rm vol}}(v)=\{\inf {\widehat{\rm vol}}({{\rm ord}}_S)\ | \ \ \mbox{$\mathbb{C}^*$-equivariant Koll\'ar components $S$} \}
\end{aligned}$$ (see and the discussion below it). Since Kollár components yield special degenerations, therefore, the above arguments can be essentially reversed. See Section \[sss-highrankcone\].
\[r-onetoone\]In fact, we establish a one-to-one correspondence between special test configurations of $(S,B)$ (up to a base change) and rays in ${{\rm Val}}_{X,x}$ emanating from $v_{S_0}$ containing a Kollár component (different with $v_{S_0}$).
An interesting consequence is that the above argument indeed gives an alternative way to show that K-semistability implies the valuative criterion of K-semistability with $\beta$-invariant as in [@Fuj16; @Li17], but without using the arguments of ‘taking a limit of filtered linear systems’.
The second approach to treat the cone singularity is developed in [@LL16] (see also [@LX16]). It is shown that K-semistablity of $(S,B)$ is equivalent to that of $(\bar{X},\bar{D}+(1-\frac{1}{rn})S_\infty)$, where $(\bar{X},\bar{D})$ is the projective cone of $(X,D)$ with respect to $-r(K_X+D)$ and $S_{\infty}(=S)$ is the divisor at the infinity place. This follows from a straightforward Futaki invariant calculation as in [@LX16 Proposition 5.3]. Applying the inequality \[thm:liuvolcomp\] to $x\in (\bar{X},\bar{D}+(1-\frac{1}{rn})S_\infty)$, we immediately conclude that $$\begin{aligned}
\label{q-LL}
{\widehat{\rm vol}}(x, \bar{X},\bar{D})\ge\frac{(-K_S-B)^{n-1}}{r^n} ={\widehat{\rm vol}}_{(X,D),x}({{\rm ord}}_{S_0}). \end{aligned}$$
To understand better the relation between the K-semistability of $(\bar{X}, \bar{D}+(1-\frac{1}{rn})S_\infty)$ and of $(S,B)$, we want to present a direct calculation which connects the calculation on $\beta$-invariant on $(\bar{X},\bar{D}+\frac{1}{rn}S_\infty)$ and the one on $(S,B)$.
\[l-kkcone\] Assume $\beta$-invariant is nonnegative for any divisorial valuation over $S$. Denote by $\hat{L}=\mathcal{O}(1)=\mathcal{O}(S_\infty)$ and $\delta=\frac{n+1}{rn}$. For any ${{\mathbb{C} }}^*$-invariant divisorial valuation $E$. We have the following $$\label{eq-XDVstable}
\beta(E):=A_{(\bar{X},\bar{D}+(1-\frac{1}{rn})S_\infty)}(E)-\frac{\delta}{\hat{L}^n}\int_0^{+\infty} {{\rm vol}}({{\mathcal{F}}}_{{{\rm ord}}_{E}} \hat{R}^{(x)})dx\ge 0,$$ where $\hat{R}=\bigoplus_{m=0}^{+\infty} H^0(\bar{X},m\hat L)$.
The key of the proof is to relate the $\beta$-invariant for a ${{\mathbb{C} }}^*$-invariant valuation $v$ over $\bar{X}$ to the $\beta$-invariant of the restriction of $v$ over the base $S$.
We have $K_{\bar{X}}+\bar{D}+(1-\frac{1}{rn})S_\infty=-\frac{n+1}{rn}\hat{L}=-\delta\hat{L}$, and define $$\mathcal{F}^x_{v}{\hat{R}}_m:=\{f\in \hat{R}_m |\ \ f\in \hat R_m=\oplus_{0\le k \le m}H^0(S, kr(-K_S-B)) \mbox{ and } v(f)\ge x \},$$
For any ${{\mathbb{C} }}^*$-invariant divisorial valuation $v={{\rm ord}}_E$ on $\bar{X}$, there exists $c_1\in {{\mathbb{Z} }}$, $a\ge 0$ and a divisorial valuation ${{\rm ord}}_F$ over $S$ such that for any $f\in H^0(S, mr(-K_S-B))$, we have $$\begin{aligned}
v(t)=c_1; \text{ and }
v(f)=a \cdot {{\rm ord}}_{F}(f)=:\bar{v}(f). \end{aligned}$$ We estimate $\beta(E)$ in three cases depending on the signs of $a$ and $c_1$:
$(a=0):$ The valuation $v$ is associated to the canonical ${{\mathbb{C} }}^*$-action along the ruling of the cone, up to rescaling, then we easily get $\beta(E)=0$
$(a>0$ and $c_1\ge 0):$ Then the center of $v$ is contained in $S_\infty$. In this case we can easily calculate: $$\begin{aligned}
{{\rm vol}}({{\mathcal{F}}}\hat{R}^{(x)})&=&\lim_{m\rightarrow +\infty}\frac{\dim_{{{\mathbb{C} }}} {{\mathcal{F}}}^{xm}\hat{R}_m}{m^n/n!}=\lim_{m\rightarrow+\infty} \frac{1}{m^n/n!} \sum_{k=0}^m \dim_{{{\mathbb{C} }}} {{\mathcal{F}}}^{xm-c_1(m-k)}_{\bar{v}} R_k\\
&=&n \int_0^1 {{\rm vol}}({{\mathcal{F}}}_{\bar{v}} R^{(c_1+\frac{x-c_1}{\tau})})\tau^{n-1} d\tau,\end{aligned}$$ where the last identity can be proved in the same way as in . So we have: $$\begin{aligned}
\int_0^{+\infty}{{\rm vol}}({{\mathcal{F}}}\hat{R}^{(x)})dx&=&n\int_0^{+\infty}dx \int_0^1 {{\rm vol}}({{\mathcal{F}}}_{\bar{v}}R^{(c_1+\frac{x-c_1}{\tau})})\tau^{n-1}d\tau\\
&=&n\int_0^1 \tau^{n-1}d\tau \int_0^{+\infty} {{\rm vol}}({{\mathcal{F}}}_{\bar{v}}R^{(c_1+\frac{x-c_1}{\tau})})dx \\
&=&n\int_0^1 \tau^{n-1}d\tau\left[H^{n-1}c_1(1-\tau)+\tau\int_{c_1}^{+\infty}{{\rm vol}}\left({{\mathcal{F}}}_{\bar{v}}R^{(y)}\right)dy\right]\\
&=&\frac{c_1}{n+1}+n\int_0^{+\infty} {{\rm vol}}({{\mathcal{F}}}_{\bar{v}}R^{(x)})dx \int_0^1 \tau^n d\tau \\
&=&\frac{c_1}{n+1}+\frac{n}{n+1}\int_0^{+\infty}{{\rm vol}}({{\mathcal{F}}}_{\bar{v}}R^{(x)})dx.\end{aligned}$$
On the other hand, we have $H^{n-1}=\hat{L}^n$ and: $$A_{(\bar{X},\bar{D}+(1-\beta)S_\infty)}({{\rm ord}}_E)=A_{(S,B)}(\bar{v})+c_1-(1-\beta)c_1=A_{(S,B)}(\bar{v})+\frac{c_1 }{rn}$$ So we get: $$\begin{aligned}
\beta(E)&=&A_{(S,B)}(\bar{v})+\frac{c_1}{rn}-\frac{\frac{n+1}{rn}}{H^{n-1}}\frac{n}{n+1}\left(\frac{c_1}{n+1}+ \int_0^{+\infty}{{\rm vol}}({{\mathcal{F}}}_{\bar{v}}R^{(x)})dx\right)\\
&= & A_{(S,B)}(\bar{v})-\frac{1}{rH^{n-1}}\int_0^{+\infty}{{\rm vol}}({{\mathcal{F}}}_{\bar{v}}R^{(x)})dx=\beta(\bar{v}),\end{aligned}$$ which is non-negative by our assumption.
$(a>0$ and $c_1<0)$: In this case, the center of $v$ is at the vertex. As a consequence we have: $$\begin{aligned}
A_{(\bar{X}, \bar{D}+(1-\beta)S_\infty)}(v)&=&A_{(S,B)}(\bar{v})+(-c_1)+(\frac{1}{r}-1) (-c_1) \\
&=&A_{(S,B)}(\bar{v})+\frac{-c_1}{r}\ge A_{(S,B)}(\bar{v}).\end{aligned}$$ The similar calculation as in the second case shows that $\beta(E)\ge \beta(\bar{v})$.
Finally, to show ${\widehat{\rm vol}}(S_0)<{\widehat{\rm vol}}(E)$ for $E\neq S_0$, in [@LX16], it was first proved that if $E$ is a minimizer then it has to be a $\mathbb{C}^*$-equivariant Kollár component. Then a careful study of the geometry of $E$ using the equality condition in implies $E=S$. This is similar to the analysis for the equality case in [@Fuj18; @Liu16] where they showed that the K-stable $\mathbb{Q}$-Fano variety with the maximal volume $(n+1)^n$ can only be $\mathbb{CP}^n$. We will leave the discussion on this uniqueness type result to the general case of cones of higher rational ranks, where we take a somewhat different approach, using more convex geometry.
It is worthy pointing out that there is another global invariant for an $n$-dimensional log Fano pair $(S,B)$, defined as $$\delta(S,B)=\inf_{v\in {{\rm Val}}_S} \frac{A_{(S,B)}(v)\cdot(-K_S-B)^{n}}{\int^{\infty}_{0}{{\rm vol}}(-K_S-B-tv)dt}$$ (see [@FO16; @BJ17]). $\delta$-invariant shares lots of common properties with the normalized volume. For example, the existence of minimizers were proved using similar strategy. They both have differential geometric meanings. The minimizer of ${\widehat{\rm vol}}$ is related to the metric tangent cone (see section \[ss-dsc\]); while the valuation on $K(S)$ yielding $\delta(S,B)$ is related to the the existence of twisted Kähler-Einstein metrics (see [@BoJ18]).
For a log Fano pair $(S,B)$ and a cone $x\in (X,D)=C(S,B;-r(K_S+B))$, if $(S,B)$ is not K-semistable, or equivalently $\delta=\delta(S,B)<1$, then we have $${\widehat{\rm vol}}(x, X, D)\ge \frac{\delta^n\cdot (-K_S-B)^{n-1}}{r^n}.$$ This follows from our second proof by looking at $(\bar{X}, \bar{D}+(1-\beta)S_\infty)$ and applying the inequality [@BJ17 Theorem D] which can be written as $$(K_{\bar{X}}+\bar{D}+(1-\beta)S_\infty)^{n}\le\frac{(n+1)^n}{n^n}\cdot {\widehat{\rm vol}}(x, X, D)\cdot \bar{\delta}^n,$$ where $\bar{\delta}:=\delta(\bar{X},\bar{D}+(1-\beta)S_\infty)$. We claim $\min\{\bar{\delta},1\}=\delta$. In fact, by the argument in [@BJ17 Section 7], we know that $\bar{\delta}$ is computed by a $\mathbb{C}^*$-invariant valuation and the claim follows from the calculation in the proof of Lemma \[l-kkcone\].
### Log Fano cone in general {#sss-highrankcone}
We proceed to investigate a log Fano cone $o\in (X, D,\xi)$ where the torus $T$ could have dimension larger than one. However, we consider not only the valuations in $\mathfrak{t}^+_{\mathbb{R}}(X)$ coming from the torus as in [@MSY08] (see Section \[s-tvariety\]) but all valuations in ${{\rm Val}}_{X,o}$. Compared to the proof of Theorem \[t-rankonecone\], for the higher rational rank case, we rely more on the construction of Kollár components coming from the birational geometry. More explicitly, we use the relation between special test configurations and Kollár components (see [@LX16 2.3] and [@LX17 3.1]).
By the results from the MMP (see and the explanation below), to show a valuation is a minimizer in ${{\rm Val}}_{X,x}$, we only need to show its normalized volume is not greater than that of any $T$-invariant Kollár component. On the other hand, any T-equivariant Kollár component $E$ in ${{\rm Val}}_{X,o}$ yields a special test configuration of $(\mathcal{X},\mathcal{D},\xi;\eta)$ of $(X,D)$ such that $-\eta\in {{\mathfrak{t}}}^+_{\mathbb{R}}(X_0)$ and the valuation associated to $-\eta$ coincides with ${{\rm ord}}_E$. We denote by $(X_0,D_0)$ the fiber with a cone vertex $o$. Then we can compare the volumes as ${\widehat{\rm vol}}_{X}(\xi)={\widehat{\rm vol}}_{X_0}(\xi)$ and ${\widehat{\rm vol}}_{X}(E)={\widehat{\rm vol}}(-\eta)$. Since $\xi,-\eta\in {{\mathfrak{t}}}^+_{{\mathbb{R}}}(X_0)$ we reduce the question to the set up of [@MSY08] on $X_0$. Then we only need to each time treat one degeneration $X_0$ and try to understand how to pass properties between $X_0$ and $X$.
With this strategy, we can show the following generalization of Theorem \[t-rankonecone\].
\[t-SDChigh\] Let $x\in (X, D,\xi)$ be a log Fano cone singularity. Then $v_{\xi}$ is a minimizer of ${\widehat{\rm vol}}_{(X, D),x}$ if and only if $(X, D,\xi)$ is K-semistable. In such case, ${\widehat{\rm vol}}(v_{\xi})<{\widehat{\rm vol}}(v)$ for any quasi-monomial valuation $v$ if $v$ is not a rescaling of $v_{\xi}$.
If $(X, D,\xi)$ is K-semistable, then for each special test configuration $(\mathcal{X},\mathcal{D},\xi;\eta)$, on $X_0$, we can consider the ray $\xi_t=\xi-t\eta$ for $t\in [0, \infty)$. We know $$\frac{d}{dt}{\widehat{\rm vol}}_{(X_0,D_0),o}(v_{\xi_t})|_{t=0}=c\cdot {\rm Fut}(\mathcal{X},\mathcal{D},\xi;\eta)\ge 0.$$ Moreover, when $(X_0,D_0,o)=(X_0, \emptyset, o)$ is an isolated singularity, it was shown in [@MSY08] that ${\widehat{\rm vol}}(v_{\xi_t})$ is a convex function. We obtain a stronger result for any log Fano cone $(X_0,D_0,\xi_0)$ (see Section \[sss-convex\]). In particular, we conclude that ${\widehat{\rm vol}}(v_{\xi_t})$ is an increasing function of $t$, and its limit is ${\widehat{\rm vol}}(-\eta)$, thus the inequality in the following relation holds true: $${\widehat{\rm vol}}_{(X, D),x}(\xi)={\widehat{\rm vol}}_{(X_0, D_0),o}(\xi)\le {\widehat{\rm vol}}_{(X_0, D_0),o}(-\eta)={\widehat{\rm vol}}_{(X,D),x}(E).$$
The first identity consists of two identities: $A_{(X,D)}(v_\xi)=A_{(X_0, D_0)}(v_\xi)$ and ${{\rm vol}}_{X}(v_\xi)={{\rm vol}}_{X_0}(v_\xi)$, which essentially follow from the flatness of $T$-equivariant test configuration (see [@LX17 Lemma 3.2]). The last identity is because $v_{-\eta}={{\rm ord}}_E$.
This argument is reversible since we can indeed attach to any special test configuration such a set of valuations (see Remark \[r-onetoone\]): if we consider the valuation $w_{t}$ obtained by considering the vector field $\xi_t$ as a valuation on $K({{\mathcal{X}}})$ and then take its restriction on $K(X)$. The corresponding degeneration induces the test configuration. See [@LX16 6] and [@LX17 4.2] for more details.
### Uniqueness {#sss-convex}
We have seen the convexity of the normalized volume function in the Reeb cone plays a key role. In [@MSY08], the strict convexity on the normalized function is established for the valuation varying inside the Reeb cone for an isolated singularity. This is the kind of property we need for the uniqueness of the minimizer of a K-semistable Fano cone singularity $(X,D,\xi)$. However, as we do not know the associated graded ring of other minimizer is finitely generated, we can not degenerate two minimizers into the Reeb cone. Thus we need develop a technique to deal with valuations outside the Reeb cone.
The idea of the argument in [@LX17 Section 3.2] is to use the theory of Newton-Okounkov bodies which was first developed in [@LM09; @KK12]) and in the local setting in [@Cut13; @KK14]. This is a theory which realizes the volumes in algebraic geometry with an asymptotic nature to the Euclidean volumes of some convex bodies in $\mathbb{R}^n$. So our aim is to apply the Newton-Okounkov body construction to translate the normalized volume of valuations into the volume of convex bodies, and then invoke a convexity property of the volumes functions known in the latter setting.
To start, we first need to set a valuation $\mathbb{V}$ with $\mathbb{Z}^n$-valued valuation, which sends the elements in $R$ to the lattice points inside a convex region $\tilde{\sigma}$, so that later we can realize the normalized volumes of valuations as the volume of subsets in $\tilde{\sigma}$.
For any fixed $T\cong ({{\mathbb{C} }}^*)^r$-equivariant quasi-monomial valuation $\mu$, we know it is of the form $( \xi_{\mu},v^{(0)})$ where $\xi_{\mu}\in M_{\mathbb{R}}$ and $v^{(0)}$ is a quasi-monomial valuation over $K(Y)$, such that for any function $f \in R_u$, $$\mu(f)=\langle \xi_{\mu},u\rangle+v^{(0)}(f)$$ (see Theorem \[t-Tcano\](1)). We fix a lexicographic order on ${{\mathbb{Z} }}^{r}$ and define for any $f\in R$, $${{\mathbb{V}}}_1(f)= \min\{u; f=\sum_u f_u \text{ with } f_u \neq 0\}={{\mathbb{V}}}_1(f),$$ i.e., the first factor ${{\mathbb{V}}}_1$ comes from the toric part of $\mu$.
We extend this ${{\mathbb{Z} }}^{r}$-valuation ${{\mathbb{V}}}_1$ to become a ${{\mathbb{Z} }}^n$-valued valuation in the following way: Denote $u_f={{\mathbb{V}}}_1(f)\in \sigma^{\vee}$ and $f_{u_f}$ the corresponding nonzero component. Define ${{\mathbb{V}}}_2(f)=v^{(0)}(f_{u_f})$. Because $\{\beta_i\}$ are $\mathbb{Q}$-linearly independent, we can write ${{\mathbb{V}}}_2(f)=\sum_{i=1}^{s} m^{*}_i \beta_i$ for a uniquely determined $m^*:=m^*(f_{u_f})=\{m^*_i:=m^*_i(f_{u_f})\}$. Moreover, the Laurent expansion of $f$ has the form: $$\label{eq-flaurent*}
f_{u_f}=z_1^{m^*_1}\dots z_{s}^{m^*_{s}} \chi_{m^*}(z'')+\sum_{m\neq m^*} z_1^{m_1}\dots z_{s}^{m_{s}} \chi_m(z'').$$ Then $\chi_{m^*}(z'')$ in the expansion of is contained in ${{\mathbb{C} }}(Z)$, where on some model of $Y$, we have $Z=\{z_1=0\}\cap \dots \{z_s=0\}=D_1\cap \dots\cap D_{s}$ is the center of $v^{(0)}$.
Extend the set $\{\beta_1, \dots, \beta_{s}\}$ to $d=n-r$ ${{\mathbb{Q} }}$-linearly independent positive real numbers $\{\beta_1, \dots, \beta_{s}; \gamma_1, \dots, \gamma_{d-s}\}$. Define ${{\mathbb{V}}}_3(f)=w_{\gamma}(\chi_{m^*}(z''))$ where $w_{\gamma}$ is the quasi-monomial valuation with respect to the coordinates $z''$ and the $(d-s)$ tuple $\{\beta_1, \dots, \beta_{s}; \gamma_1, \dots, \gamma_{d-s}\}$.
Now we assign the lexicographic order on $$\mathbb{G}:={{\mathbb{Z} }}^{r}\times G_2\times G_3\cong {{\mathbb{Z} }}^{r}\times {{\mathbb{Z} }}^{s}\times {{\mathbb{Z} }}^{n-r-s}$$ and define $\mathbb{G}$-valued valuation: $${{\mathbb{V}}}(f)=({{\mathbb{V}}}_1(f), {{\mathbb{V}}}_2(f_{u_f}), {{\mathbb{V}}}_3(\chi_{m^*})).$$
Let $\mathcal{S}$ be the valuative semigroup of ${{\mathbb{V}}}$. Then $\mathcal{S}$ generates a cone $\tilde{\sigma}$ which is the one we are looking for. We also let $P_1: {\mathbb{R}}^n\rightarrow {\mathbb{R}}^{r}$, $P_2: {\mathbb{R}}^n\rightarrow {\mathbb{R}}^{s}$ and $P=(P_1, P_2): {\mathbb{R}}^n\rightarrow {\mathbb{R}}^{r+s}$ be the natural projections. Then $P_1(\tilde{\sigma})=\sigma\subset {\mathbb{R}}^r$.
To continue, we consider how to construct some subsets $\Delta_{\tilde{\Xi}_t} \subset \tilde{\sigma}$ whose Euclidean volume is the same as the normalized volumes of the valuations. For any $\xi\in {\rm int}(\sigma)$, denote by ${{\rm wt}}_\xi$ the valuation associated to $\xi$. We can connect ${{\rm wt}}_\xi$ and $\mu$ by a family of quasi-monomial valuations: $\mu_t= ((1-t)\xi+t \xi_{\mu}, t v^{(0)})$ defined as $$\mu_t(f)=t v^{(0)}(f)+\langle u, (1-t)\xi+t\xi_{\mu}\rangle \mbox{\ \ \ for any $f\in R_u$}.$$ So the vertical part of $\mu_t$ corresponds to the vector $\Xi_t:=((1-t)\xi+t\xi_{\mu}, t\beta)\in {\mathbb{R}}^{r+s}$. Extend $\Xi_t$ to $\tilde{\Xi}_t:=(\Xi_t, 0)\in {\mathbb{R}}^n$ and define the following set: $$\Delta_{\tilde{\Xi}_t}=\left\{y\in \tilde{\sigma}; \langle y, \tilde{\Xi}_t\rangle \le 1 \right\}=\left\{y\in \tilde{\sigma}; \langle P(y), \Xi_t\rangle \le 1\right\}.$$ Because ${\widehat{\rm vol}}$ is rescaling invariant, we can assume $A_{(X,D)}(v)=A_{(X,D)}(\xi)=1$. Then by the $T$-invariance of $v_t$, we easily get: $$A(v_t)=t A(v^{(0)})+A_{(X,D)}((1-t)\xi+t\zeta)=t A_{(X,D)}(v)+(1-t)A_{(X,D)}(\xi)\equiv 1.$$ The Newton-Okounkov body theory implies that we have $${\widehat{\rm vol}}(v_t)={{\rm vol}}(v_t)={{\rm vol}}(\Delta_{\tilde{\Xi}_t}).$$
To finish the uniqueness argument, now we only need to look at the convex geometry of $\Delta_{\tilde{\Xi}_t}$. We note that $\tilde{\Xi}_t$ is linear with respect to $t$, and each region $\Delta_{\tilde{\Xi}_t}$ is cut out by a hyperplane $H_t$ on the convex cone $\tilde{\sigma}$. Moreover, all $H_t$ passes through a fixed point. A key result from convex geometry then shows that $\phi(t):={{\rm vol}}(\Delta_{\tilde{\Xi}_t})$ is strictly convex as a function of $t\in [0,1]$ (see [@MSY06; @Gig78]). By the assumption $\phi(0)={{\rm vol}}(v_0)={\widehat{\rm vol}}({{\rm wt}}_{\xi})$ is a minimum. So the strict convexity implies $$\phi(1)={{\rm vol}}(\Delta_{\tilde{\Xi}_1})={\widehat{\rm vol}}(v)>{\widehat{\rm vol}}({{\rm wt}}_\xi)=\phi(0).$$
Results on the general case {#ss-general}
---------------------------
To treat the general case, the key idea, suggested by the degeneration conjecture, is to understand how an arbitrary klt singularity can be degenerated to a K-semistable Fano cone singularity. In [@LX16], by localizing the setting of [@LX14], the following approach of using Kollár components is developed.
From each ideal ${{\mathfrak{a}}}$, we can take a dlt modification of $$f\colon (Y,D_Y)\to (X,D+{{\rm lct}}(X,D;{{\mathfrak{a}}})\cdot{{\mathfrak{a}}}),$$ where $D_Y=f_*^{-1}D+{\rm Ex}(f)$ and for any component $E_i\subset {\rm Ex}(f)$ we have $$A_{X,D}(E)={{\rm lct}}(X,D;{{\mathfrak{a}}})\cdot {{\rm mult}}_{E}f^*{{\mathfrak{a}}}.$$ There is a natural inclusion ${{\mathcal{D}}}(D_Y)\subset {{\rm Val}}^{=1}_{X,x}$, and using a similar argument as in [@LX14], we can show that there exists a Kollár component $S$ whose rescaling in ${{\rm Val}}^{=1}_{X,x}$ contained in ${{\mathcal{D}}}(D_Y)$ satisfies that $${\widehat{\rm vol}}({{\rm ord}}_S)= {{\rm vol}}^{\rm loc}(-A_{X,D}(S)\cdot S)\le {{\rm vol}}^{\rm loc}(-K_Y-D_Y)\le {{\rm mult}}({{\mathfrak{a}}})\cdot {{\rm lct}}^n(X,D;{{\mathfrak{a}}}).$$ Here ${{\rm vol}}^{\rm vol}(\cdot)$ is the local volume of divisors over $X$ as defined in [@Ful13]. Then Theorem \[thm:liueq\] immediately implies that $$\begin{aligned}
\label{e-kol}
{\widehat{\rm vol}}(x, X,D)=\inf\{{\widehat{\rm vol}}({{\rm ord}}_S)|\ S \mbox{ is a Koll\'ar component over $x$}\}.\end{aligned}$$ Moreover, if $x\in (X,D)$ admits a torus group $T$-action, then by degenerating to the initial ideals, as the colengths are preserved and the log canonical thresholds may only decrease, the infimum of the normalized multiplicities in Theorem \[thm:liueq\] can be only run over all $T$-equivariant ideals. Then the equivariant MMP allows us to make all the above data $Y$ and $S$ be $T$-equivariant.
In case a minimizer is divisorial, then the above discussion shows that
\[l-dmkollar\] A divisorial minimizer of ${\widehat{\rm vol}}_{X,D}$ yields a Kollár component.
In general, we know that the minimizer is a limit of a rescaling of Kollár components (see [@LX16]). So understanding the limiting process is crucial. When the minimizer is quasi-monomial $v$ of rational rank $r$, i.e., the valuation $v$ is ' etale locally a monomial valuation with respect to a log resolution $(Y,E)\to X$, then a natural candidates will be the valuations given by taking rational approximations of the monomial coordinates $\alpha \in \mathbb{R}^r_{>0}$.
Our first observation in [@LX17] is using MMP results including the ACC of log canonical thresholds, we could construct a weak log canonical model which extracts divisors whose coordinates are good linear Diophantine approximations of the coordinates of $v$.
\[p-highmodel\]For any quasi-monomial valuation $v$ computing a log canonical threshold of a graded sequence of ideals, we can find a sequence of divisors $S_1$,..., $S_r$, such that
1. there is a model $Y\to X$ which precisely extracts $S_1$,..., $S_r$ over $x$,
2. there exists a component $Z$ of $\cap^r_{i=1} S_i$ such that $(Y, E:=\sum^r_{i=1} S_i)$ is toroidal around the generic point $\eta(Z)$,
3. $v$ is étale locally a monomial valuation over $\eta(Z)$ with respect to $(Y,E)$ (see Section \[ss-firstdefinition\]),
4. $(Y,E)$ is log canonical, and $-K_Y-E$ is nef.
Fix the first model $Y_0=Y$, then one can construct a sequence of models $(Y_j, E_j)$ satisfying Proposition \[p-highmodel\] such that a suitable rescaling of the components of $E_j$ become closer and closer to $v$. To make the notation easier, we rescale $v$ into ${{\rm Val}}_{X,x}^{=1}$. Similarly, we can embed the dual complex of a dlt modification of $(Y_j, E_j)$ into ${{\rm Val}}_{X,x}^{=1}$ (see [@dFKX]). Our construction moreover satisfies that $${{\mathcal{DR}}}(Y_0,E_0)\supset {{\mathcal{DR}}}(Y_1,E_1)\supset \cdots$$ Then the above discussion indeed implies that
A quasi-monomial minimizer $v\in {{\rm Val}}_{X,x}^{=1}$ can be written as a limit of $c_j\cdot {{\rm ord}}_{S_j}\in {{\mathcal{DR}}}(Y_j, E_j)$ where $S_i$ are Kollár components.
It would be natural to expect that $c_j\cdot {{\rm ord}}_{S_j}$ is indeed contained in the simplex $\sigma_{\eta(Z)}\subset {{\rm Val}}_{X,x}^{=1}$ which corresponds to all the monomial valuations in ${{\rm Val}}_{X,x}^{=1}$ over $\eta(Z)$ with respect to $(Y,E)$. However, for now we can not show it.
If we further assume $R_0={{\rm gr}}_v(R)$ is finitely generated, then we have the following
If $R_0={{\rm gr}}_v(R)$ is finitely generated, then ${{\rm gr}}_{v}(R)\cong {{\rm gr}}_{v_i}(R)$ for any $v_i\in \sigma_{\eta(Z)}$ sufficiently close to $v$.
This immediately implies that $(X_0:={\rm Spec}(R_0),D_0)$ is semi-log-canonical (slc). The final ingredient we need is the following,
Under the above assumptions on $(X,D)$ and its quasi-monomial minimizer $v$, then $\xi_v$ is a minimizer of $(X_0,D_0)$. In particular, $${\widehat{\rm vol}}(x, X,D)={\widehat{\rm vol}}(o, X_0,D_0).$$
We claim that $\xi_v$ is indeed a minimizer of ${\widehat{\rm vol}}_{X_0,D_0}$. If not, we can find a degeneration $(Y,D_Y, \xi_{Y})$ induced by an irreducible anti-ample divisor $E$ over $o'\in X_0$ with $${\widehat{\rm vol}}_Y(\xi_E)={\widehat{\rm vol}}_{X_0}({{\rm ord}}_{E})<{\widehat{\rm vol}}_{X_0}(\xi_v)={\widehat{\rm vol}}_Y(\xi_Y).$$ This is clear by our discussion when $(X_0,D_0)$ is klt. The same thing still holds when the model extracting $S_j$ is only log canonical but not plt, which implies that $(X_0,D_0)$ is semi-log-canonical but not klt. In fact, denote by $(X^{\rm n}_0,D^{\rm n}_0)\to (X_0,D_0)$ the normalization, then Lemma \[l-volslc\] implies that $${\widehat{\rm vol}}(o', X_0,D_0):=\sum_{o_i\to o'} {\widehat{\rm vol}}(o_i, X^{\rm n}_0,D^{\rm n}_0)=0$$ in this case. The argument in [@LX17 Lemma 4.13] then says in this case, we can still extract an equivariant anti-ample irreducible divisor $E$ over $o'\in X_0$ with ${\widehat{\rm vol}}({{\rm ord}}_E)$ arbitrarily small.
Then Lemma \[l-doudeg\] shows that we can construct a degeneration from $(X,D)$ to $(Y,D_Y)$ and a family of valuations $v_t\in {{\rm Val}}_{X,x}$ for $t\in [0,\epsilon]$ (for some $0<\epsilon\ll 1$), with the property that $${\widehat{\rm vol}}_X(v_t)={\widehat{\rm vol}}_Y(\xi_Y-t \eta)<{\widehat{\rm vol}}_Y(\xi_Y)={\widehat{\rm vol}}_{X_0}(\xi_v)={\widehat{\rm vol}}_X(v),$$ where for the second inequality, we use again the fact that ${\widehat{\rm vol}}_Y(\xi_Y-t\cdot\eta)$ is a convex function in this setting as well. But this is a contradiction.
\[l-doudeg\] Let $(x\in X)\subset (0\in \mathbb C^N)$ be a closed affine variety. If $\lambda_1 \in \mathbb{N}^N$ is a coweight of $(\mathbb{C}^*)^N$ which gives an action degenerating $X$ to $X_0$ when $t\rightarrow 0$, and $\lambda_2 \in \mathbb{N}^N$ degenerates $X_0$ to $Y$ when $t\to 0$, then for $k\in \mathbb{N}$ sufficiently large, $k\lambda_1+\lambda_2$ degenerates $X$ to $Y_0$.
The proof was essentially given in [@LX16 section 6] (see also [@LWX18 Lemma 3.1]) and uses some argument in the study of toric degenerations (see e.g. [@And13 Section 5]).
\[l-volslc\] If $o\in (X,D)$ is an lc but not klt point, then $${\widehat{\rm vol}}(o, X,D):=\inf_{v\in {{\rm Val}}_{X,o}}{\widehat{\rm vol}}(v)=0.$$
Let $\pi^{\rm dlt}\colon (X^{\rm dlt},D^{\rm dlt})\to (X,D)$ be a dlt modfication and pick $o''$ a preimage of $o$ under $\pi^{\rm dlt}$, then ${\widehat{\rm vol}}(o'', X^{\rm dlt},D^{\rm dlt})\ge {\widehat{\rm vol}}(o, X,D)$, thus we can assume $(X^{\rm dlt},D^{\rm dlt})$ is dlt ${{\mathbb{Q} }}$-factorial.
By specialing a sequence of points, and applying Theorem \[t-lower\], we can assume $o\in (X,D)$ is a point on a smooth variety with a smooth reduced divisor $D$. Now we can take a weighted blow up of $(1,\epsilon,....,\epsilon)$ where the first coordinate yields $D$. Then the exceptional divisor $E$ has its normalized volume $${\widehat{\rm vol}}(E)=\frac{(n-1)^n\epsilon^n}{\epsilon^{n-1}}=(n-1)^n\epsilon\to 0 \mbox {\ as \ } \epsilon\to 0.$$
This implies that $(X_0,D_0)$ is klt and $(X_0,D_0,\xi_v)$ is a K-semistable Fano cone. To summarize, we have shown Part (a) in the following theorem which characterize what we know about the Stable Degeneration Conjecture \[conj-local\] for a general klt singularity.
\[t-high\] Let $x\in (X,D)$ be a klt singularity. Let $v$ be a quasi-monomial valuation in ${{\rm Val}}_{X,x}$ that minimises ${\widehat{\rm vol}}_{(X,D)}$ and has a finitely generated associated graded ring ${\rm gr}_v(R)$ (which is always true if the rational rank of $v$ is one by Lemma \[l-dmkollar\]). Then the following properties hold:
1. The degeneration $\big(X_0=_{\rm defn}{\rm Spec}\big({\rm gr}_v(R)\big), D_0, \xi_v \big)$ is a K-semistable Fano cone, i.e. $v$ is a K-semistable valuation;
2. Let $v'$ be another quasi-monomial valuation in ${{\rm Val}}_{X,x}$ that minimises ${\widehat{\rm vol}}_{(X,D)}$. Then $v'$ is a rescaling of $v$.
Conversely, any quasi-monomial valuation that satisfies (a) above is a minimiser.
We first show the uniqueness in general, under the assumption that it admits a degeneration $(X_0,D_0,\xi_v)$ given by a K-semistable minimiser $v$. For another quasi-monomial minimiser $v'$ of rank $r'$, by a combination of the Diophantine approximation and an MMP construction including the application of ACC of log canonical thresholds (see Proposition \[p-highmodel\]), we can obtain a model $f\colon Z\to X$ which extracts $r'$ divisors $E_i$ ($i=1,...,r'$) such that $(Z, D_Z=_{\rm defn}\sum E_i+f_*^{-1}D)$ is log canonical. Moreover, the quasi-monomial valuation $v'$ can be computed at the generic point of a component of the intersection of $E_i$, along which $(Z,D_Z)$ is toroidal. Then with the help of the MMP, one can show $Z\to X$ degenerates to a birational morphism $Z_0\to X_0$. Moreover, there exists a quasi-monomial valuation $w$ computed on $Y_0$ which can be considered as a degeneration of $v'$ with $${\widehat{\rm vol}}_{X_0}(w)={\widehat{\rm vol}}_X(v')={\widehat{\rm vol}}_X(v)={\widehat{\rm vol}}_{X_0}(\xi_v).$$ Thus $w=\xi_v$ by Section \[sss-convex\] after a rescaling. Since $w({\bf in}(f))\ge v'(f)$ and ${{\rm vol}}(w)={{\rm vol}}(v')$, we may argue this implies $$\xi_v({\bf in}(f))=v'(f)$$ (see [@LX17 Section 4.3]). Therefore, $v'$ is uniquely determined by $\xi_v$.
To show the last statement, we already know it for a cone singularity. For a valuation $v$ on a general singularity $X$ such that the degeneration $(X_0,D_0,\xi_v)$ is K-semistable, since the degeneration to the initial ideal argument implies that ${\widehat{\rm vol}}(x, X,D)\ge {\widehat{\rm vol}}(o, X_0,D_0)$, then $${\widehat{\rm vol}}_X(v)={\widehat{\rm vol}}_{X_0}(\xi_v)={\widehat{\rm vol}}(o, X_0,D_0)$$ is equal to ${\widehat{\rm vol}}(x, X,D)$.
So in other words, the stable degeneration conjecture precisely predicts the following two sets coincide: $$\left \{
\begin{tabular}{c}
Minimizers of ${\widehat{\rm vol}}$
\end{tabular}
\right \} \longleftrightarrow
\Big\{
\begin{tabular}{c}
K-semistable valuations
\end{tabular}
\Big\}.$$ Theorem \[t-existence\] and Theorem \[t-high\] together imply the existence of left hand side and the uniqueness of the right hand side, as well as the direction that any K-semistable valuation is a minimizer.
Finally, let us conclude this section with the two dimensional case.
\[thm-2dim\] Let $(X, D, x)$ be a two-dimensional log terminal singularity. The Stable Degeneration Conjecture \[conj-local\] holds for $(X, D)$. Moreover, if $D$ is a ${{\mathbb{Q} }}$-divisor, then the minimizer of ${\widehat{\rm vol}}_{(X, D)}$ is always divisorial.
We first consider the case when $X={{\mathbb{C} }}^2$. Let $v_*$ be a minimizer and denote $\frak{a}_\bullet=\{\frak{a}_m(v_*)\}_{m\in {\mathbb{N}}}$. Then it was known that $v_*$ computes the log canonical threshold of $(X, D+\frak{a}_\bullet)$. By similar argument as [@JM12], we know that $v_*$ must be quasi-monomial.
If $v_*$ is divisorial, then we know that the associated divisor is a Kollár component. Otherwise, $v_*$ satisfies ${\rm rat.rk.}(v_*)=2$ and ${\rm tr.deg.}(v_*)=0$. From the description of valuations on ${{\mathbb{C} }}^2$ using [*sequences of key polynomials*]{} (SKP), it was showed that the valuative semigroup $\Gamma$ of $v_*$ is finitely generated (see [@FJ04 Theorem 2.28]). Since the residual field of $v_*$ is ${{\mathbb{C} }}$, we know that ${\rm gr}_{v_*}R\cong {{\mathbb{C} }}[\Gamma]$, which is finitely generated. By [@LX17], we know that $v_*$ is indeed the unique minimizer of ${\widehat{\rm vol}}$ (up to scaling) which is a K-semistable valuation.
If $D$ is a ${{\mathbb{Q} }}$-divisor and $v_*$ is not divisorial, then the pair $(X_0, D_0)$ is a ${{\mathbb{Q} }}$-Gorenstein toric pair with ${{\mathbb{Q} }}$-boundary toric divisor and the associated Reeb vector field $\xi_{v_*}$ solves the convex geometric problem. But in dimension 2 case (i.e. on the plane), it is easy to see that the corresponding convex geometric problem as discussed in section \[sss-convex\] for toric valuations always has a rational solution. This is a contradiction to $v_*$ being non-divisorial.
More generally, we know that $X={{\mathbb{C} }}^2/G$ where $G$ is a finite group acting on ${{\mathbb{C} }}^2$ without pseudo-reflections. Consider the covering $({{\mathbb{C} }}^2, \tilde{D}, 0)\rightarrow (X, D, x)$. Then by the above discussion, there exists a unique minimizer $v_*$ of ${\widehat{\rm vol}}_{({{\mathbb{C} }}^2, \tilde{D}, 0)}$. In particular, $v_*$ is invariant under the $G$-action. So it descends to a minimizer of ${\widehat{\rm vol}}_{(X, D, x)}$ which is quasi-monomial and has a finitely generated associated graded ring.
Applications {#s-application}
============
In this section, we give some applications of the normalized volume. We have seen that the normalized volume question of a cone singularity is closely related the K-semistability of the base. Another situation where singularities naturally appear is on the limit of smooth Fano manifolds.
Equivariant K-semistability of Fano {#ss-equiva}
-----------------------------------
An interesting application of the minimizing theory is to treat the equivariant K-semistability.
A log Fano pair $(S,B)$ with a $G$-action is called $G$-equivariant K-semistable, if for any $G$-equivariant test configuration $ (\mathcal{S},\mathcal{B})$, the generalized Futaki invariant ${\rm Fut}(\mathcal{S},\mathcal{B})\ge 0$. We can similarly define $G$-equivariant K-polystability.
The notion of usual K-(semi,poly)stability trivially implies the equivariant one. It is a natural question to ask whether they are equivalent, and if it is confirmed it will reduce the problem of verifying K-stability into a much simpler ones if the log Fano pair carries a large symmetry. When $S$ is smooth and $B=0$, this is proved in [@DS16], using an analytic argument. Here we want to explain how our approach can give a proof of such an equivalence when $G=T$ is a torus group.
The key is the fact we obtain in and : let $x\in (X,D)$ be a klt singularity which admits a $T$ action for a torus group $T$, then $$\begin{aligned}
\label{e-Tkollar}
\ \ \ \inf_{v\in {{\rm Val}}_{X,x}} {\widehat{\rm vol}}(v)=\{\inf({\widehat{\rm vol}}({{\rm ord}}_S)) | \ \ \mbox{$T$-equivariant Koll\'ar components $S$} \}.
\end{aligned}$$
So if $(S,B)$ is not K-semistable, by Theorem \[t-rankonecone\], we know that over the cone $x\in (X,D)$, the valuation ${{\rm ord}}_{S_{\infty}}$ obtained by the canonical blow up does not give a minimizer. By , there exists a $T$-equivariant valuation $v$ such that ${\widehat{\rm vol}}(v)<{\widehat{\rm vol}}({{\rm ord}}_{S_{\infty}}) $. So we can find a $T$-equivariant Kollár component $S$ such that ${\widehat{\rm vol}}({{\rm ord}}_S)<{\widehat{\rm vol}}({{\rm ord}}_{S_{\infty}})$. Then arguing as before, we can find a $T$-equivariant test configuration $(\mathcal{S},\mathcal{B})$ with ${\rm Fut}(\mathcal{S},\mathcal{B})< 0$.
To prove a similar statement for K-polystability is more delicate. Assume a K-semistable log pair $(S,B)$ admits a test configuration $(\mathcal{S},\mathcal{B})$ with ${\rm Fut}(\mathcal{S},\mathcal{B})=0$. We still take the cone construction of a K-semistable log Fano pair as before. The special test configuration determines a ray $v_{t}$ of valuations in ${{\rm Val}}_{X,x}$, emanating from the canonical component $v_0={{\rm ord}}_{S_{\infty}}$. Using the fact that the Futaki invariant is 0, a minimal model program argument shows that this implies for $t\ll 1$, $v_{t}$ is automatically $\mathbb{C}^*$-equivariant, which immediately implies the test configuration is $\mathbb{C}^*$-equivariant. Therefore we show the following result (also see [@CS16] for an earlier attempt).
The K-semistability (resp. K-polystability) of a log Fano pair $(S,B)$ is equivalent to the $T$-equivariant K-semistablity ($T$-equivariant K-polystablity) for any torus group $T$ acting on $(S,B)$.
For other groups $G$, e.g. finite groups or general reductive groups, we haven’t proved the corresponding result as . It is a consequence of the uniqueness part of the stable degeneration conjecture. We also note that in [@LX17], it is proved that quasi-monomial minimizers over a $T$-equivariant klt singularity are automatically $T$-invariant.
Donaldson-Sun’s Conjecture {#ss-dsc}
--------------------------
One major application of what we know about the stable degeneration conjecture, formulated in Theorem \[t-high\], is the solution of [@DS17 Conjecture 3.22] (see Conjecture \[conj-DS\]), which predicts that for a singularity appearing on a Gromov-Hausdorff limit of Kähler-Einstein metrics, its metric tangent cone only depends on the algebraic structure of the singularity. In this section, we briefly explain the idea.
### K-semistable degeneration
Let $(M_k, g_k)$ be a sequence of Kähler-Einstein manifolds with positive curvature. Then possibly taking a subsequence, $(M_k, g_k)$ converges in the Gromov-Hausdorff topology to a limit metric space $(X, d_\infty)$. By the work of Donaldson-Sun and Tian, $X$ is homeomorphic to a ${{\mathbb{Q} }}$-Fano variety. For any point $x\in X$, a metric tangent cone $C_xX$ is defined as a pointed Gromov-Hausdorff limit: $$C_xX=\lim_{r_k\rightarrow 0} \left(X, x, \frac{d_\infty}{r_k}\right).$$ By Cheeger-Colding’s theory, $C_xX$ is always a metric cone. By [@CCT02], the real codimension of singularity set of $C_xX$ is at least 4 and the regular part admits a Ricci-flat Kähler cone structure. In [@DS17], it is further proved that $C_xX$ is an affine variety with an effective torus action. They proved that $C_xX$ is uniquely determined by the metric structure $d_\infty$ and can be obtained in the following steps. In the first step, they defined a filtration $\{{{\mathcal{F}}}^\lambda\}_{\lambda\in \mathcal{S}}$ of the local ring $R={\mathcal{O}}_{X,o}$ using the limiting metric structure $d_\infty$. Here $\mathcal{S}$ is a set of positive numbers that they called the holomorphic spectrum which depends on the torus action on the metric tangent cone $C$. In the second step, they proved that the associated graded ring of $\{\mathcal{F}^\lambda\}$ is finitely generated and hence defines an affine variety, denoted by $W$. In the last step, they showed that $W$ equivariantly degenerates to $C$. Notice that this process depends crucially on the limiting metric $d_\infty$ on $X$. They then made the following conjecture.
\[conj-DS\] Both $W$ and $C$ depend only on the algebraic germ structure of $X$ near $x$.
We made the following observations:
1. $\{{{\mathcal{F}}}^{\lambda}\}$ comes from a valuation $v_0$. This is due to the fact that $W$ is a normal variety. More explicitly, since the question is local, we can assume $X={\rm Spec}(R)$ with the germ of $x\in X$, by the work in [@DS17], one can embed both $X$ and $C$ into a common ambient space ${{\mathbb{C} }}^N$, and $v_0$ on $X$ is induced by the monomial valuation ${{\rm wt}}_{\xi_0}$ where $\xi_0$ is the linear holomorphic vector field with $2{\rm Im}(\xi_0)$ being the Reeb vector field of the Ricci flat Kähler cone metric on $C$. By this construction, it is clear that the induced valuation by $v_0$ on $W$ is nothing but ${{\rm wt}}_{\xi_0}$.
2. $v_0$ is a quasi-monomial valuation. This follows from Lemma \[lem-quasi\].
More importantly we conjectured in [@Li15] that $v_0$ can be characterized as the unique minimizer of ${\widehat{\rm vol}}_{X, x}$. As a corollary of the theory developed so far, we can already confirm [@DS17 Conjecture 3.22] for $W$.
The semistable cone $W$ in Donaldson-Sun’s construction depends on the algebraic structure of $(X, x)$.
The proof consists of the following steps consisting of analytic and algebraic arguments:
1. By Theorem \[thm-RF2K\], $(C, \xi_0)$ is K-polystable and in particular K-semistable. By Theorem \[t-SDChigh\], ${{\rm wt}}_{\xi_0}$ is a minimizer of ${\widehat{\rm vol}}_{C}$.
2. By Proposition \[p-semideg\], $(W, \xi_0)$ is K-semistable. By Theorem \[t-SDChigh\] again, ${{\rm wt}}_{\xi_0}$ is a minimizer of ${\widehat{\rm vol}}_W$. Moreover, by Theorem \[t-high\], $v_0$ is a minimizer of ${\widehat{\rm vol}}_X$.
3. $v_0$ is a quasi-monomial minimizer of ${\widehat{\rm vol}}_X$ with a finitely generated associated graded ring. By Theorem \[t-high\], such a $v_0$ is indeed the unique minimizer of ${\widehat{\rm vol}}$ among all quasi-monomial valuations.
The following is an immediate consequence of Theorem \[t-SDChigh\].
\[p-semideg\] Assume there is a special degeneration of a log-Fano cone $(X, D, \xi_0)$ to $(X_0, D_0, \xi_0)$. Assume that $(X_0, D_0,\xi_0)$ is K-semistable, then $(X, D, \xi_0)$ is also K-semistable, or equivalently, ${{\rm wt}}_{\xi_0}$ is the minimizer of ${\widehat{\rm vol}}_{(X,D,x)}$.
Asssume $(X, x)$ lives on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifold. Then we can define the volume density in the sense of Geometric Measure Theory as the following quantity: $$\Theta(x,X)=\lim_{r\rightarrow 0}\frac{{\rm Vol}(B_r(x))}{r^{2n} {\rm Vol}(B_1(\underline{0})}.$$ Note that $n^n={\widehat{\rm vol}}(0, {{\mathbb{C} }}^n)$. The normalized volumes of klt singularities on Gromov-Hausdorff limits have the following differential geometric meaning:
\[thm-hvol2Theta\] With the same notation as above, we have the identity: $$\frac{{\widehat{\rm vol}}(x,X)}{n^n}=\Theta(x,X).$$
From the standard metric geometry, we have $\Theta(x, X)=\Theta(o_C, C)$. Because $C$ admits a Ricci-flat Kähler cone metric, by Theorem \[thm-RF2K\], $(C, \xi_0)$ is K-semistable. ${\widehat{\rm vol}}(x, X)={\widehat{\rm vol}}(o_C, C)$.
On the other hand, since $C$ is a metric cone, from the definition of the volume of $\xi_0=\frac{1}{2}\left(r\partial_r-i J(r\partial_r)\right)$ is equal to: $$\begin{aligned}
\Theta(o_C, C)=\frac{{\rm Vol}(C\cap \{r=1\})}{{\rm Vol}(S^{2n-1})}={{\rm vol}}(\xi_0).\end{aligned}$$ By Theorem \[thm-irSE\], $A({{\rm wt}}_{\xi_0})=n$ and ${\widehat{\rm vol}}(o_C, C)=n^n {{\rm vol}}(\xi_0)=n^n \Theta(o_C, C)$.
### Uniqueness of polystable degeneration
To confirm Donaldson-Sun’s conjecture, we also need to prove the uniqueness of polystable degenerations for K-semistable Fano cones.
Since a Fano cone singularity $(C,\xi)$ with a Ricci-flat Kähler cone metric is aways K-polystable (see [@CS15 Theorem 7.1] and also Theorem \[thm-RF2K\]), once knowing that $W$ only depends on the algebraic structure of $o\in M_{\infty}$, an affirmative answer to Conjecture \[conj-DS\] follows from the following more general result by letting $(X,D, \xi_0)=(W, \emptyset, \xi_0)$:
\[t-uniquecone\] Given a K-semistable log Fano cone singularity $(X,D,\xi_0)$, there always exists a special test configuration $(\mathcal{X},\mathcal{D}, \xi_0; \eta)$ which degenerates $(X,D,\xi_0)$ to a K-polystable log Fano cone singularity $(X_0,D_0,\xi_0)$. Furthermore, such $(X_0,D_0,\xi_0)$ is uniquely determined by $(X, D, \xi_0)$ up to isomorphism.
For the special case of smooth (or ${{\mathbb{Q} }}$-Gorenstein smoothable) Fano varieties, this was proved in [@LWX 7.1] based on analytic results which also show the uniqueness of Gromov-Hausdorff limit for a flat family of Fano Kähler-Einstein manifolds. Our proof of Theorem \[t-uniquecone\] is however a completely new algebraic argument.
We briefly discuss the idea to prove Theorem \[t-uniquecone\] in [@LWX18], which heavily depends on the study of normalized volumes as discussed in Section \[ss-cone\].
Let $({{\mathcal{X}}}^{(i)}, D^{(i)}, \xi_0, \eta^{(i)}), (i=1,2)$, be two special test configurations of the log Fano cone $(X, D, \xi_0)$ with the central fibre $(X^{(i)}_0, D^{(i)}_0, \xi_0)$. To show Theorem \[t-uniquecone\], the main step is to show that if ${{\rm Fut}}({{\mathcal{X}}}^{(i)}, {\mathcal{D}}^{(i)}, \xi_0; \eta^{(i)})=0, (i=1,2)$, then there exists special test configurations $({{\mathcal{X}}}'^{(i)}, {\mathcal{D}}'^{(i)})$ of $(X^{(i)}_0, D^{(i)}_0)$ such that $({{\mathcal{X}}}'^{(i)}, {\mathcal{D}}'^{(i)})$ have isomorphic central fibres, which we will describe below.
We consider the normalized volume functional defined on the valuation space ${{\rm Val}}_{X,x}$ over the vertex $x$ of the cone $X$. Then $({{\mathcal{X}}}^{(1)}, {\mathcal{D}}^{(1)}, \xi_0; \eta^{(1)})$ determines a “ray" of valuations emanating from the toric valuation ${{\rm wt}}_{\xi_0}$ and the generalized Futaki invariant ${\rm Fut}({{\mathcal{X}}}^{(1)},{\mathcal{D}}^{(1)}, \xi_0; \eta^{(1)})$ is the derivative of the normalized volume at ${{\rm wt}}_{\xi_0}$ along this ray.
$$\xymatrix@1 @R=1.2pc @C=1pc
{
(X^{(2)}_0, D^{(2)}_0) \ar@{~>}_{({{\mathcal{X}}}'^{(2)}, {\mathcal{D}}'^{(2)})}[dd]
& & & &(X, D) \ar_{ ({{\mathcal{X}}}^{(2)}, {\mathcal{D}}^{(2)})\longleftarrow {\mathcal{Y}}^{(2)}_k\longleftarrow{\mathcal{E}}^{(2)}_k}@{~>}[llll] \ar^{({{\mathcal{X}}}^{(1)}, {\mathcal{D}}^{(1)})\leftarrow {\mathcal{Y}}_k\leftarrow {\mathcal{E}}_k=E_k\times{{\mathbb{C} }}^1}@{~>}[dd] & \ar[l] Y_k\leftarrow E_k
\\
& & & \hskip 2cm & &\\
(X'_{0}, D'_{0}) & & & & (X^{(1)}_0, D^{(1)}_0)\ar@{~>}_{({{\mathcal{X}}}'^{(1)}, {\mathcal{D}}'^{(1)})}[llll] & \ar[l] Y_{k,0} \leftarrow E_k
}$$
We can approximate $\xi_0$ by a sequence of integral vectors $\tilde{\xi}_{k}$ such that $|\tilde{\xi}_{k}-k \xi_0|\le C$. For $k\gg 1$, the vector $\tilde{\xi}_k-\eta$ corresponds to a Kollár component $E_k$ over $X$. Our key argument is to show that $E_k$ can be degenerated along $({{\mathcal{X}}}^{(2)}, {\mathcal{D}}^{(2)})$ to get a model ${\mathcal{Y}}_k^{(2)}\rightarrow {{\mathcal{X}}}^{(2)}$ with an exceptional divisor ${\mathcal{E}}^{(2)}_k$ such that $({\mathcal{Y}}_k^{(2)}, {\mathcal{E}}^{(2)}_k)\times_{{{\mathbb{C} }}}{{\mathbb{C} }}^*\cong (Y_k, E_k)\times {{\mathbb{C} }}^*$ where the isomorphism is compatible with the equivariant isomorphism of the [*second*]{} special test configuration. Note that $E_k\times{{\mathbb{C} }}^*$ determines a divisorial valuation over $X\times{{\mathbb{C} }}^*$ and hence over $({{\mathcal{X}}}^{(2)}, {\mathcal{D}}^{(2)})$. So the goal is to show that this divisorial valuation can be extracted as the only exceptional divisor over ${{\mathcal{X}}}^{(2)}$. By the work in the minimal model program (MMP) (see [@BCHM10]), this would be true if there is a graded sequence of ideals $\frak{A}_\bullet$ and a positive real number $c'_k$ such that two conditions are satisfied: $$({{\mathcal{X}}}^{(2)}, {\mathcal{D}}^{(2)}+c'_k \frak{A}_\bullet)\mbox{ is klt \ \ \ and\ \ \ } A(E_k\times{{\mathbb{C} }}; {{\mathcal{X}}}^{(2)}, {\mathcal{D}}^{(2)}+ c'_k \frak{A}_\bullet)<1,$$ where $A(E_k\times{{\mathbb{C} }}; {{\mathcal{X}}}^{(2)}, {\mathcal{D}}^{(2)}+c'_k \frak{A}_\bullet)$ is the log discrepancy of (the birational transform of) $E_k\times{{\mathbb{C} }}$ with respect to the triple $({{\mathcal{X}}}^{(2)}, {\mathcal{D}}^{(2)}+ c'_k \frak{A}_\bullet)$. To find such a $\frak{A}_\bullet$, we look at the graded sequence of valuative ideals $\{{{\mathfrak{a}}}_{\bullet}\}$ of ${{\rm ord}}_{E_k}$ and its equivariant degeneration along the second special test configuration $({{\mathcal{X}}}^{(2)}, {\mathcal{D}}^{(2)})$. The resulting graded sequence of ideals over ${{\mathcal{X}}}^{(2)}$ will be denoted by $\frak{A}_{\bullet}$. Using the study in Section \[ss-cone\] one can show the assumptions that $({{\mathcal{X}}}^{(1)}, {\mathcal{D}}^{(1)}; \xi_0)$ is K-semistable and ${{\rm Fut}}({{\mathcal{X}}}^{(1)}, {\mathcal{D}}^{(1)}, \xi_0; \eta)=0$ implies $$\mbox{$f(k):={\widehat{\rm vol}}(E_k)$ is of the order $f(0)+O(k^{-2})$. }$$ This in turn guarantees that we can find $c'_k$ satisfying the above two conditions.
Applying the relative Rees algebra construction to ${\mathcal{E}}^{(2)}_k\subset \mathcal{Y}^{(2)}_k/\mathbb{C}$, we get a family over $\mathbb{C}^2$, which over $\mathbb{C}\times \{t\}$ is the same as $({{\mathcal{X}}}^{(1)},{\mathcal{D}}^{(1)})$ for $t\neq 0$ and gives a degeneration of $(X^{(1)}_0, D^{(1)}_0)$ for $t=0$. On the other hand, over $\{0\}\times \mathbb{C}$, we get a degeneration of $(X^{(2)}_0, D^{(2)}_0)$. Therefore, we indeed show that the two special fibers of two special test configurations $({{\mathcal{X}}}^{(i)},{\mathcal{D}}^{(i)}, \xi_0; \eta^{(i)})\ (i=1,2)$ with ${\rm Fut}({{\mathcal{X}}}^{(i)},{\mathcal{D}}^{(i)}, \xi_0; \eta^{(i)})=0$ will have a common degeneration.
Estimates in dimension three and K-stability of threefolds {#ss-cubic3}
----------------------------------------------------------
In general, it is not so easy to find the minimizer of ${\widehat{\rm vol}}(\cdot)$ for a given singularity. A number of cases have been computed in [@Li15; @LL16; @LX16; @LX17; @LiuX17] including quotient singularities, ADE singularities in all dimensions (except 4-dimensional $D_4$) etc.
Here we study normalized volumes of threefold klt singularities, and then give a global application where we show that all GIT semi-stable (resp. polystable) cubic threefolds are also K-semi-stable (resp. K-polystable). Our main estimate is in Theorem \[thm:3dimhvol\], which heavily depends on classifications of canonical threefold singularities.
\[thm:3dimhvol\] Let $x\in X$ be a $3$-dimensional non-smooth klt singularity. Then ${\widehat{\rm vol}}(x, X)\leq 16$ and the equality holds if and only if it is an $A_1$ singularity;
The proof of Theorem \[thm:3dimhvol\] heavily relies on the classification theory of three dimensional canonical and terminal singularities, developed in the investigation of explicit three dimensional MMP.
The idea goes as follows. Firstly, we reduce to the case of Gorenstein canonical singularity. If $x\in X$ is not Gorenstein, let us take the index one cover $\tilde{x}\in\widetilde{X}$ of $x\in X$. Hence $\tilde{x}\in\widetilde{X}$ is a Gorenstein canonical singularity. If $\tilde{x}\in\widetilde{X}$ is smooth, then ${\widehat{\rm vol}}(x, X)=27/{\mathrm{ind}}(x,K_X)\leq 13.5<16$. If $\tilde{x}\in\widetilde{X}$ is not smooth, the a weak version of finite degree formula (Proposition \[prop:weakfinitedeg\]) implies that ${\widehat{\rm vol}}(x, X)<{\widehat{\rm vol}}(\tilde{x},\widetilde{X})$.
Next, let us assume that $x\in X$ is Gorenstein canonical. By [@KM98 Proposition 2.36], there exist only finitely many crepant exceptional divisors over $X$. By [@BCHM10], we can extract these divisors simultaneously on a birational model $Y_1\to X$. If none of these exceptional divisors are centered at $x$, then [@KM98 Theorem 5.34] implies that $x\in X$ is a cDV singularity, hence ${{\rm lct}}({\mathfrak{m}}_x)
\leq 4-{{\rm mult}}({\mathfrak{m}}_x)$ which implies $${\widehat{\rm vol}}(x, X)\leq {{\rm lct}}({\mathfrak{m}}_x)^3{{\rm mult}}({\mathfrak{m}}_x)\leq (4-{{\rm mult}}({\mathfrak{m}}_x))^3
{{\rm mult}}({\mathfrak{m}}_x)\leq 16.$$ The equality case can be characterized using the volume of birational models approach in [@LX16]. If some crepant exceptional divisor $E_1\subset Y_1$ is centered at $x$, then let us run $(Y_1,\epsilon E_1)$-MMP over $X$ for $0\ll \epsilon<1$. By [@Kol13 1.35], this MMP will terminate as $Y_1\dashrightarrow Y\xrightarrow[]{g}Y'$, where $Y_1\dashrightarrow Y$ is the composition of a sequence of flips, and $g : Y \to Y'$ contracts the birational transform $E$ of $E_1$. If $g(E)$ is a curve, then $Y'$ has cDV singularities along $g(E)$ by [@KM98 Theorem 5.34]. By choosing a point $y'\in g(E)$, we have $${\widehat{\rm vol}}(x, X)<{\widehat{\rm vol}}(y',Y')\leq 16.$$ If $g(E)=y'$ is a point, then we still have ${\widehat{\rm vol}}(x, X)<{\widehat{\rm vol}}(y',Y')$. Thus it suffices to show ${\widehat{\rm vol}}(y',Y')<16$.
If $Y$ has a singular point $y\in E$, then we know that $y\in Y$ is a cDV singularity. Hence $${\widehat{\rm vol}}(y',Y')<{\widehat{\rm vol}}(y,Y)\leq 16.$$ So we may assume that $Y$ is smooth along $E$. In particular, $E$ is a (possibly non-normal) reduced Gorenstein del Pezzo surface. If $E$ is normal, then classification of such surfaces show that $(-K_E)^2\leq 9$. Thus $${\widehat{\rm vol}}(y',Y')\leq A_{Y'}({{\rm ord}}_E)^3 {{\rm vol}}({{\rm ord}}_E)=(-K_E)^2\leq 9<16.$$ If $E$ is non-normal, then from Reid’s classification [@Rei94] either $(-K_E)^2\leq 4$ or the normalization of $E$ is a Hirzebruch surface. In the former case, we have ${\widehat{\rm vol}}(y',Y')\leq 4$. In the latter case, we need to take a general fiber $l$ of $E$ and argue that ${\widehat{\rm vol}}_{Y',y'}({{\rm ord}}_l)\leq 16$.
Here are some intermediate results in proving Theorem \[thm:3dimhvol\].
Let $\phi : (Y, y) \to (X, x)$ be a birational morphism of klt singularities such that $y \in \mathrm{Ex}(\phi)$. If $K_Y\leq\phi^*K_X$, then ${\widehat{\rm vol}}(x,X) < {\widehat{\rm vol}}(y, Y)$.
See Conjecture \[c-fdf\] for more discussions about the following Proposition.
\[prop:weakfinitedeg\] Let $\pi:(\widetilde{X},\tilde{x})\to (X,x)$ be a finite quasi-étale morphism of klt singularities of degree at least $2$. Then we have $${\widehat{\rm vol}}(x,X)<{\widehat{\rm vol}}(\tilde{x},\widetilde{X})\leq \deg(\pi)\cdot{\widehat{\rm vol}}(x,X).$$
As mentioned [@SS17], one main application of the local volume estimate Theorem \[thm:3dimhvol\] is to the K-stability question of cubic threefolds.
\[thm:Kcubic\] A cubic threefold is K-(poly/semi)stable if and only if it is GIT (poly/semi)stable. In particular, any smooth cubic threefold is K-stable.
The general strategy to prove Theorem \[thm:Kcubic\] is via the comparison of moduli spaces which has first appeared in [@MM93] built on the work of [@Tia92]. Later it was also applied in [@OSS; @SS17].
First, one can construct a proper algebraic space which is a good quotient moduli space with closed points parametrizing all smoothable K-polystable ${{\mathbb{Q} }}$-Fano varieties (see e.g. [@LWX; @Oda15]). Let $M$ be the closed subspace whose closed points parametrize KE cubic threefolds and their K-polystable limits. By [@Tia87], we know that at least one cubic threefold, namely the Fermat cubic threefold, admits a KE metric. Hence $M$ is non-empty. By the Zariski openness of K-(semi)stability of smoothable Fano varieties (cf. [@Oda15; @LWX]), the K-moduli space $M$ is birational to the GIT moduli space $M^{\rm GIT}$ of cubic threefolds.
Next, we will show that any K-semistable limit $X$ of a family of cubic threefolds $\{X_t\}$ over a punctured curve is necessarily a cubic threefold. The idea is to control the singularity of $X$ use an inequality from [@Liu16] (see Theorem \[thm:liuvolcomp\]) between the global volume of a K-semistable Fano variety and the local normalized volume. Since the volume of $X$ is the same as the volume of a cubic $3$-fold which is $24$, Theorem \[thm:liuvolcomp\] immediately implies that ${\widehat{\rm vol}}(x,X)\geq \frac{81}{8}$ for any closed point $x\in X$. The limit $X$ carries a ${{\mathbb{Q} }}$-Cartier Weil divisor $L$ which is the flat limit of hyperplane sections in the cubic threefolds $X_t$. It is clear that $-K_X\sim_{{{\mathbb{Q} }}}2L$ and $(L^3)=3$, thus once we show that $L$ is Cartier, we can claim that $X$ is a cubic threefold using a result of T. Fujita.
Assume to the contrary that $L$ is not Cartier at some point $x\in X$, then we may take the index $1$ cover $(\tilde{x}\in \widetilde{X})\to (x\in X)$ of $L$. From the finite degree formula Theorem \[t-fdf\], $${\widehat{\rm vol}}(\tilde{x},\widetilde{X})={\mathrm{ind}}(L)\cdot{\widehat{\rm vol}}(x,X)\geq 81/4.$$ Hence $\tilde{x}\in\widetilde{X}$ is a smooth point and ${\mathrm{ind}}(L)=2$ by Theorem \[thm:3dimhvol\]. Thus $x\in X$ is a quotient singularity of type $\frac{1}{2}(1,1,0)$ from the smoothable condition. Then using the local Grothendieck-Lefschetz theorem, we can show that $L$ is indeed Cartier at $x\in X$ which is a contradiction.
So far we have shown that any K-polystable point $X$ in $M$ is a cubic threefold. By an argument of Paul and Tian in [@Tia94], we know that any K-(poly/semi)stable hypersurface is GIT (poly/semi)stable. Thus we obtain an injective birational morphism $M\to M^{\rm GIT}$ between proper algebraic spaces. This implies that $M$ is isomorphic to $M^{\rm GIT}$ which finishes the proof.
\[thm:liuvolcomp\] Let $X$ be an $n$-dimensional K-semistable Fano variety. Then for any closed point $x\in X$, we have $$(-K_X)^n\leq \left(1+\frac{1}{n}\right)^n{\widehat{\rm vol}}(x,X).$$
When $X$ is smooth, the above result was first proved in [@Fuj18].
Questions and Future research {#s-ques}
=============================
Revisit stable degeneration conjecture
--------------------------------------
The following two parts of stable degeneration conjecture, proposed in [@Li15], are still missing.
Let $x\in (X, D)$ be a klt singularity. Any minimizer of ${\widehat{\rm vol}}_{(X,D),x}$ is quasi-monomial.
Let $x\in (X={\rm Spec}(R),D)$ be a klt singularity. Any minimizer of ${\widehat{\rm vol}}_{(X,D),x}$ has its associated graded ring ${{\rm gr}}_v(R)$ to be finitely generated.
Due to the fundamental role of the stable degeneration conjecture, it implies many other interesting properties. We discuss a number of special cases or consequences, with the hope that some of them might be solved first.
One interesting consequence of the uniqueness of the minimizer is the following
\[g-equiva\] If there is a group $G$ acting on the klt singularity $x\in (X, D)$ such that $x$ is a fixed point, then there exists a $G$-invariant minimizer.
Applying this conjecture to a cone singularity, it implies that to test the K-semistability of a log Fano $(S,B)$ with a $G$-action, we only need to test on $G$-equivariant test configurations, a fact known for a Fano manifold $X$ and $G$-reductive.
There are two special cases naturally appearing in contexts. The first one is that when $G$ is a torus group $T$. It follows the argument in [@Blu18] and the techniques of degenerating ideals to their initials, that there is a $T$-equivariant minimizer. This is the philosophy behind Section \[ss-equiva\]. It also follows from [@LX17] that any quasi-monomial minimizer is $T$-equivariant.
A more challenging case is when $G$ is a finite group. Indeed, Conjecture \[g-equiva\] for finite group $G$ implies the following finite degree formula.
\[c-fdf\] If $\pi\colon (y\in Y,D')\to (x\in X,D)$ is a dominant finite morphism between klt singularities, such that $K_Y+D'=\pi^*(K_X+D)$, then $$\deg(\pi)\cdot {\widehat{\rm vol}}(x, X,D)={\widehat{\rm vol}}(y, Y,D').$$
This is useful when we want to bound the klt singularities $x\in (X,D)$ with a large volume.
\[t-fdf\] Conjecture \[c-fdf\] is true when $(X, x)$ is on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds.
Let $\pi=\pi_X: (Y,y)\rightarrow (X,x)$ be a quasi-étale morphism, i.e. $\pi_X$ is étale in codimension one. Then $\pi_X$ induces a quasi-étale morphism along the 2-step degeneration of $X$.
$$\xymatrix@1 @R=1.2pc @C=1pc
{
Y \ar@{~>}_{}[r]
\ar^{\pi_X} [d] & W_Y \ar@{~>}_{}[r] \ar^{\pi_W}[d] & C_Y \ar^{\pi_C}[d] \\
X \ar@{~>}_{}[r] & W \ar@{~>}_{}[r] & C
}$$
We can use the above diagram to prove the degree multiplication formula. Roughly speaking, because $C$ admits a Ricci-flat Kähler cone metric $\omega_C$ with radius function $r^2$ and $\pi_C$ is quasi-étale, we can pull back it to get $\pi_C^*r^2$ which is also a potential for a weak Ricci-flat Kähler cone metric $\omega_{C_Y}$. By Theorem \[thm-RF2K\], Theorem \[t-SDChigh\] and Theorem \[t-high\], we know that the Reeb vecotor field associated to $\omega_C$ (resp. $\omega_{C_Y}$) induces minimizing valuations of ${\widehat{\rm vol}}_{X}$ (resp. ${\widehat{\rm vol}}_Y$). So we get $$\begin{aligned}
{\widehat{\rm vol}}(y, Y)&=&{\widehat{\rm vol}}(o_{C_Y}, C_Y)=\deg(\pi_C)\cdot {\widehat{\rm vol}}(o_C, C)=\deg(\pi_C) \cdot {\widehat{\rm vol}}(x, X).\end{aligned}$$
Another consequence of the stable degeneration conjecture is the following strengthening of Theorem \[t-lower\].
\[c-constru\] Let $\pi:({{\mathcal{X}}},{\mathcal{D}})\to T$ together with a section $t\in T\mapsto x_t\in{{\mathcal{X}}}_t$ be a ${{\mathbb{Q} }}$-Gorenstein flat family of klt singularities. Then the function $t\mapsto {\widehat{\rm vol}}(x_t,{{\mathcal{X}}}_t,{\mathcal{D}}_t)$ is construtible with respect to the Zariski topology.
Besides the stable degeneration conjecture, to prove Conjecture \[c-constru\], we also need to know the well expected speculation that K-semistability is an open condition. It is also natural to consider the volume of non-closed point. However, the following conjecture says after the right scaling, it does not contribute more information.
If a klt pair $(X,D)$ has a non-closed point $\eta$, and let $Z=\overline{\{\eta\}}$ has dimension $d$. Pick a general closed point $x\in Z$, then $${\widehat{\rm vol}}(x, X,D)={\widehat{\rm vol}}(\eta,X,D)\cdot \frac{n^n}{(n-d)^{n-d}}.$$
In fact, combining the argument in [@LZ18], for any valuation $v\in {{\rm Val}}_X$ such that its center $Z={\rm Center}_X(v)$ on $X$ is of dimension $d$ and $x\in Z $, denoted by $\eta$ is the generic point of $Z$, one can show that $$\frac{{\widehat{\rm vol}}_{(X,D),\eta}(v)\cdot n^n}{(n-d)^{n-d}}\ge {\widehat{\rm vol}}(x, X,D).$$ i.e., $${\widehat{\rm vol}}(x, X,D)=\inf_{v}\left\{ \frac{n^n\cdot {\widehat{\rm vol}}_{(X,D),\eta}(v)}{(n-d)^{n-d}}\ | \ x\in Z=\overline{\{\eta\}}={\rm Center}_X(v), \dim(Z)=d \right\}.$$
Birational geometry study
-------------------------
A different invariant attached to a klt singularities, called the minimal log discrepancy has been intensively studied in the minimal model program, though there are still many deep questions unanswered. We can formulate many similar questions for ${\widehat{\rm vol}}$.
### Inversion of adjunction
One could look for a theory of the change of the volumes when the klt pair is ‘close’ to a log canonical singularities, using the inversion of adjunction. We have some results along this line.
Let $x\in (X,\Delta)$ be an $n$-dimensional klt singularity. Let $D$ be a normal ${{\mathbb{Q} }}$-Cartier divisor containing $x$ such that $(X,D+\Delta)$ is plt. Denote by $\Delta_D$ the different of $\Delta$ on $D$. Then $$\lim_{\epsilon\to 0+}\frac{{\widehat{\rm vol}}(x, X,(1-\epsilon)D+\Delta)}{n^n\epsilon}
=\frac{{\widehat{\rm vol}}(x, D,\Delta_D)}{(n-1)^{n-1}}.$$
Using the degeneration argument in [@LZ18], we know that $$\epsilon^{-1} {\widehat{\rm vol}}(x, X,(1-\epsilon)D+\Delta)\geq \frac{n^n}{(n-1)^{n-1}}{\widehat{\rm vol}}(x, D,\Delta_D).$$ Hence it suffices to show the reverse inequality is true after taking limits. Let us pick an arbitrary Kollár component $S$ over $x\in (D,\Delta_D)$ with valuation ideals ${{\mathfrak{a}}}_m:={{\mathfrak{a}}}_m({{\rm ord}}_S)$. Choose $m$ sufficiently divisible so that ${{\mathfrak{a}}}_{im}={{\mathfrak{a}}}_m^i$ for any $i\in{\mathbb{N}}$. Then we know that ${{\rm lct}}(D,\Delta_D;{{\mathfrak{a}}}_m)=A_X({{\rm ord}}_S)/m=:c$. Let ${\mathfrak{b}}_m$ be the pull-back ideal of ${{\mathfrak{a}}}_m$ on $X$. By inversion of adjunction, we have ${{\rm lct}}(X,D+\Delta;{\mathfrak{b}}_m)={{\rm lct}}(D,\Delta_D;{{\mathfrak{a}}}_m)=c$.
Let $E$ be an exceptional divisor over $X$ computing ${{\rm lct}}(X,D+\Delta;{\mathfrak{b}}_m)$. Then $E$ is centered at $x\in X$ since $(X,D+\Delta)$ is plt. For $\epsilon_1>0$ sufficiently small, we have that $(X,\Delta+(1-\epsilon_1)(D+c\cdot{\mathfrak{b}}_m))$ is a klt pair over which the discrepancy of $E$ is negative. Thus [@BCHM10] implies that there exists a proper birational model $\mu:Y\to X$ which only extracts $E$. Moreover, $\mu:Y\to X$ is a log canonical modification of $(X,\Delta+D+c\cdot{\mathfrak{b}}_m)$. Let $\widetilde{D}$ be the normalization of $\mu_*^{-1}D$. Then by adjunction, the lifting morphism $\tilde{\mu}:\widetilde{D}\to D$ is a log canonical (in fact plt) modification of $(D,\Delta_D+c\cdot{{\mathfrak{a}}}_m)$. Since $\mathrm{Bl}_{{{\mathfrak{a}}}_m}D\to D$ provides a model of the Kollár component $S$, this is the only log canonical modification of $(D,\Delta_D+c\cdot{{\mathfrak{a}}}_m)$. Hence $E|_{\widetilde{D}}=S$ and $(\widetilde{D},\tilde{\mu}_*^{-1}\Delta_D+
E|_{\widetilde{D}})$ is plt. Then by inversion of adjunction, $(Y,\mu_*^{-1}\Delta+\mu_*^{-1}D+E)$ is qdlt and $\mu_*^{-1}D=\widetilde{D}$ is normal. Note that all the constructions so far are independent of the choice of $\epsilon$.
Over the qdlt model $(Y,\mu_*^{-1}\Delta+\mu_*^{-1}D+E)$, we consider a quasi-monomial valuation $v_{\lambda}$ of weights $1$ and $\lambda$ along divisors $\widetilde{D}$ and $E$ respectively. By adjunction, we know that $A_{(X,\Delta)}({{\rm ord}}_E)=
A_{(D,\Delta_D)}({{\rm ord}}_S)+{{\rm ord}}_E(D)$. Hence computation shows that $$A_{(X,\Delta+(1-\epsilon) D)}(v_{\lambda})
=\lambda A_{(D,\Delta_D)}({{\rm ord}}_S)+\lambda\epsilon \cdot {{\rm ord}}_E(D)+\epsilon.$$ Then using the Okounkov body description of the volume (see [@LM09; @KK12]), we easily see that ${{\rm vol}}(v_\lambda)\leq \lambda^{1-n}{{\rm vol}}({{\rm ord}}_S)$. Hence $$\begin{aligned}
{\widehat{\rm vol}}_{(X,\Delta+(1-\epsilon)D)}(v_\lambda)& \leq &\lambda^{1-n}((A_{(D,\Delta_D)}({{\rm ord}}_S)+\epsilon\cdot {{\rm ord}}_E(D))\lambda+\epsilon)^n{{\rm vol}}({{\rm ord}}_S)\\
&=:&\phi(\lambda).
\end{aligned}$$ It is easy to see that $\phi(\lambda)$ reaches its minimum at $$\lambda_0=\frac{(n-1)\epsilon}{A_{D,\Delta_D}({{\rm ord}}_S)+\epsilon\cdot {{\rm ord}}_E(D)}.$$ Hence computation shows $$\epsilon^{-1}{\widehat{\rm vol}}_{(X,\Delta+(1-\epsilon)D)}(v_{\lambda_0})
\leq \frac{n^n}{(n-1)^{n-1}}(A_{(D,\Delta_D)}({{\rm ord}}_S)+\epsilon\cdot {{\rm ord}}_E(D))^{n-1}{{\rm vol}}({{\rm ord}}_S).$$ Thus $$\limsup_{\epsilon\to 0}\epsilon^{-1}{\widehat{\rm vol}}(x, X,\Delta+(1-\epsilon)D)
\leq \frac{n^n}{(n-1)^{n-1}}{\widehat{\rm vol}}_{(D,\Delta_D)}({{\rm ord}}_S)$$ Since this inequality holds for any Kollár component $S$ over $x\in (D,\Delta_D)$, the proof is finished.
When the center is zero dimensional, we also have
Let $x\in (X,\Delta)$ be a klt singularity. Let $D\geq 0$ be a ${{\mathbb{Q} }}$-Cartier divisor such that $(X,\Delta+D)$ is log canonical with $\{x\}$ being the minimal non-klt center. Then there exists $\epsilon_0>0$ (depending only on the coefficient of $\Delta,D$ and $n$) and a quasi-monomial valuation $v\in{{\rm Val}}_{X,x}$ such that $v$ computes both ${{\rm lct}}(X,\Delta;D)$ and ${\widehat{\rm vol}}(x,X,\Delta+(1-\epsilon)D)$ for any $0<\epsilon<\epsilon_0$. In particular, $${\widehat{\rm vol}}(x, X,\Delta+(1-\epsilon)D)
={\widehat{\rm vol}}_{x,(X,\Delta)}(v)\cdot\epsilon^n\textrm{ for any }
0<\epsilon<\epsilon_0.$$
Let $Y^{{\operatorname{dlt}}}\to X$ be a dlt modification of $(X,\Delta+D)$. Let $K_{Y^{{\operatorname{dlt}}}}+\Delta^{{\operatorname{dlt}}}$ be the log pull back of $K_X+\Delta+D$. Then by [@dFKX], the dual complex ${{\mathcal{DR}}}(\Delta^{{\operatorname{dlt}}})$ form a natural subspace of ${{\rm Val}}_{X,x}^{=1}$. Any divisorial valuation ${{\rm ord}}_E$ computing ${{\rm lct}}(X,\Delta;D)$ corresponds to a rescaling of a valuation in ${{\mathcal{DR}}}(\Delta^{{\operatorname{dlt}}})$. Consider the function ${{\rm vol}}_X:{{\mathcal{DR}}}(\Delta^{{\operatorname{dlt}}})\to{\mathbb{R}}_{>0}
\cup\{+\infty\}$. Denote by ${{\mathcal{DR}}}^{\circ}(\Delta^{{\operatorname{dlt}}})$ the open subset of ${{\mathcal{DR}}}(\Delta^{{\operatorname{dlt}}})$ consisting of valuations centered at $x$. Since $\{x\}$ is the minimal non-klt center of $(X,\Delta+D)$, we know that ${{\mathcal{DR}}}^{\circ}(\Delta^{{\operatorname{dlt}}})$ is non-empty. By [@BFJ14] the function ${{\rm vol}}$ is continuous on ${{\mathcal{DR}}}(\Delta^{{\operatorname{dlt}}})$, so we can take a ${{\rm vol}}$-minimizing valuation $v\in{{\mathcal{DR}}}^{\circ}(\Delta^{{\operatorname{dlt}}})$. Hence $v$ is a minimizer of ${\widehat{\rm vol}}$ restricted to ${{\mathcal{DR}}}^{\circ}(\Delta^{{\operatorname{dlt}}})$.
Assume $S$ is an arbitrary Kollár component over $(X,\Delta+(1-\epsilon)D)$. Then we have a birational morphism $\mu: Y\to X$ such that $K_Y+\mu_*^{-1}(\Delta+(1-\epsilon)D)+S$ is plt, and $\mu$ is an isomorphism away from $x$ with $S=\mu^{-1}(x)$. Then by ACC of lct [@HMX14], we know that there exists $\epsilon_0$ such that $K_Y+\mu_*^{-1}(\Delta+D)+S$ is log canonical whenever $0<\epsilon<\epsilon_0$. Let $v'$ be an arbitrary divisorial valuation in ${{\mathcal{DR}}}^{\circ}(\Delta^{{\operatorname{dlt}}})$. Since $K_Y+\mu_*^{-1}(\Delta+D)+S\sim_{{{\mathbb{Q} }}}
\mu^*(K_X+\Delta+D)+A_{(X,\Delta+D)}({{\rm ord}}_S)S$, we have $$0\leq A_{(Y,\mu_*^{-1}(\Delta+D)+S)}(v')=A_{(X,\Delta+D)}(v')
-A_{(X,\Delta+D)}({{\rm ord}}_S)\cdot v'(S).$$ Since $A_{(X,\Delta+D)}(v')=0$ and $v'(S)>0$ since $\{x\}$ is the only lc center, we know that $A_{(X,\Delta+D)}({{\rm ord}}_S)=0$. Thus a rescaling of ${{\rm ord}}_S$ belongs to ${{\mathcal{DR}}}(\Delta^{{\operatorname{dlt}}})$. Then by [@LX16] we see that $$\begin{aligned}
{\widehat{\rm vol}}(x,X,\Delta+(1-\epsilon)D)&=\min_{v'\in{{\mathcal{DR}}}(\Delta^{{\operatorname{dlt}}})}{\widehat{\rm vol}}_{(X,\Delta+(1-\epsilon)D)}(v')\\
&=\epsilon^n\min_{v'\in{{\mathcal{DR}}}(\Delta^{{\operatorname{dlt}}})} {{\rm vol}}_{X}(v')=
{\widehat{\rm vol}}_{x,(X,\Delta)}(v)\cdot\epsilon^n.
\end{aligned}$$
One should be able to solve the following question using the above techniques.
Let $x\in (X,\Delta)$ be an $n$-dimensional klt singularity. Let $D$ be an effective ${{\mathbb{Q} }}$-Cartier ${{\mathbb{Q} }}$-Weil divisor through $x$. Let $c={{\rm lct}}(X,\Delta;D)$, and let $W$ be the minimal log canonical center of $(X,\Delta+cD)$ containing $x$. By Kawamata’s subadjunction, we have $(K_X+\Delta+cD)|_W=K_W+\Delta_W+J_W$, where $(W,\Delta_W+J_W)$ is a generalized klt pair. Denote by $k:={\mathrm{codim}}_{X}W$, then is it true that $$\lim_{\epsilon\to 0+} \epsilon^{-k}\frac{{\widehat{\rm vol}}(x,X,\Delta+(1-\epsilon)cD)}{n^n}\geq
\frac{{\widehat{\rm vol}}(w,X,\Delta)}{k^k}\cdot \frac{{\widehat{\rm vol}}(x,W,\Delta_W+J_W)}{(n-k)^{n-k}}$$ where $w$ is the generic point of $W$ in $X$ and ${\widehat{\rm vol}}(x,W,\Delta_W+J_W)$ is similarly defined as for the usual klt pair case in Definition \[d-normvol\]?
### Uniform bound
The following is conjectured in [@SS17] (see also [@LiuX17]).
Let $x\in X$ be an $n$-dimensional singular point, then ${\widehat{\rm vol}}(x, X)\le 2(n-1)^n$.
The constant $2(n-1)^n$ is the volume of a rational double point. When $n=3$, it is proved in Theorem \[thm:3dimhvol\]. The implication to the K-stability question of cubic hypersurfaces as in the argument of Theorem \[thm:Kcubic\] holds in any dimension.
We also ask whether the following strong property of the set of local volumes holds.
Fix the dimension $n$, and a finite set $I\subset [0,1]$. Is it true that the set ${\rm Vol}^{\rm loc}_{n,I}$ consisting of all possible local volumes of $n$-dimensional klt singularities $x\in (X,D)$ with $(\mbox{coefficients of }D)\subset I$ has the only accumulation point 0?
Next we give a comparison between local volumes and minimal log discrepancies.
\[thm:mldhvol\] Let $x\in (X,\Delta)$ be an $n$-dimensional complex klt singularity. Then there exists a neighborhood $U$ of $x\in X$ such that $(U,\Delta|_U)$ is $({\widehat{\rm vol}}(x,X,\Delta)/n^n)$-lc. Moreover, ${{\rm mld}}(x,X,\Delta)>{\widehat{\rm vol}}(x,X,\Delta)/n^n$.
If $x\in X$ is not ${{\mathbb{Q} }}$-factorial then we may replace $X$ by its ${{\mathbb{Q} }}$-factorial modification under which the local volume will increase by [@LiuX17 Corollary 2.11]. Let $\Delta_i$ be any component of $\Delta$ containing $x$. Then [@BL18 Theorem 33] implies that $A_{(X,\Delta)}(\Delta_i)\geq {\widehat{\rm vol}}(x,X,\Delta)/n^n$. Let $E$ be any exceptional divisor over $X$ such that $x$ is contained in the Zariski closure of $c_X(E)$ and $a(E;X,\Delta)<0$. Then by [@Kol13 Corollary 1.39], there exists a proper birational morphism $\mu:Y\to X$ such that $Y$ is normal, ${{\mathbb{Q} }}$-factorial and $E=\mathrm{Ex}(\mu)\supset\mu^{-1}(x)$. Since $K_Y+\mu_*^{-1}\Delta-a(E;X,\Delta)E=\mu^*(K_X+\Delta)$, we know that $(Y,\mu_*^{-1}\Delta-a(E;X,\Delta)E)$ is klt. Let $y\in \mu^{-1}(x)$ be a point, then $y$ lies on $E$. Hence by [@LiuX17 Corollary 2.11] and [@BL18 Theorem 33] we have $${\widehat{\rm vol}}(x,X,\Delta)< {\widehat{\rm vol}}(y,Y,\mu_*^{-1}\Delta-a(E;X,\Delta)E)\leq A_{(X,\Delta)}(E)n^n.$$ Thus $A_{(X,\Delta)}(E)>{\widehat{\rm vol}}(x,X,\Delta)/n^n$ which finishes the proof.
Next we will discuss application to boundedness generalizing a result by C. Jiang [@Jia17 Theorem 1.6].
\[cor:bdd\] Let $n$ be a natural number and $c$ a positive real number. Then the projective varieties $X$ satisfying the following properties:
- $(X,\Delta)$ is a klt pair of dimension $n$ for some effective ${{\mathbb{Q} }}$-divisor $\Delta$,
- $-(K_X+\Delta)$ is nef and big,
- $\alpha(X,\Delta)^n(-(K_X+\Delta))^n\geq c$,
form a bounded family.
By [@BJ17 Theorem A and D] (generalizing [@Liu16]), for any closed point $x\in X$ we have $$c\leq \alpha(X,\Delta)^n(-(K_X+\Delta))^n\leq
\delta(X,\Delta)^n(-(K_X+\Delta))^n
\leq \left(1+\frac{1}{n}\right)^n{\widehat{\rm vol}}(x,X,\Delta).$$ Hence Theorem \[thm:mldhvol\] implies $(X,\Delta)$ is $(c/(n+1)^n)$-lc. Therefore, the BAB Conjecture proved by Birkar in [@Bir16 Theorem 1.1] implies the boundedness of $X$.
In the conditions of Corollary \[cor:bdd\] if we also assume that the coefficients of $\Delta$ are at least $\epsilon$ for any fixed $\epsilon\in (0,1)$, then such pairs $(X,\Delta)$ are log bounded. This partially generalizes [@Che18 Theorem 1.4]. Besides, all results should hold for ${\mathbb{R}}$-pairs.
Is it true that for any $n$-dimensional klt singularity $x\in X$, we have ${{\rm mld}}(x,X)\geq {\widehat{\rm vol}}(x,X)/n^{n-1}$?
Miscellaneous Questions
-----------------------
### Positive characteristics
In this section, we consider a variety $X$ over an algebraically closed field ${\mathbbm{k}}$ of characteristic $p>0$. From [@Har98; @HW02], we know that klt singularities are closely related to strongly $F$-regular singularities in positive characteristic. Moreover, log canonical thresholds (${{\rm lct}}$) correspond to $F$-pure thresholds (${\mathrm{fpt}}$) in positive characteristic (see [@HW02]). In spirit of Theorem \[thm:liueq\], we define the $F$-volume of singularities in characteristic $p$ as follows.
Let $X$ be an $n$-dimensional strongly $F$-regular variety over an algebraically closed field ${\mathbbm{k}}$ of positive characteristic. Let $x\in X$ be a closed point. We define the *$F$-volume* of $(x\in X)$ as $${{\rm Fvol}}(x,X):=\inf_{{{\mathfrak{a}}}\colon{\mathfrak{m}}_x\textrm{-primary}}
{\mathrm{fpt}}(X;{{\mathfrak{a}}})^n{{\rm mult}}({{\mathfrak{a}}}).$$
Similar to [@dFEM], Takagi and Watanabe [@TW04] showed that if $x\in X$ is a smooth point, then ${{\rm Fvol}}(x,X)=n^n$.
Another interesting invariant of a strongly $F$-regular singularity $x\in X$ is its $F$-signature $s(x,X)$, see [@SVdB97; @HL02; @Tuc12]. In [@Liu18b], we estabilish the following comparison result between the $F$-volume and the $F$-signature.
\[t-com\] Let $x\in X$ be an $n$-dimensional strongly $F$-regular singularity. Then $$n!\cdot s(x,X)\leq {{\rm Fvol}}(x,X)\leq n^n\min\{1, n!\cdot s(x,X)\}.$$
It would be interesting to study the limiting behavior of $F$-volumes of mod-$p$ reductions of a klt singularity over characteristic zero when $p$ goes to infinity.
Let $x\in (X,\Delta)$ be a klt singularity over characteristic $0$. Let $x_p\in (X_p,\Delta_p)$ be its reduction mod $p\gg 0$, then $${\widehat{\rm vol}}(x, X,\Delta)=\lim_{p\to \infty}{{\rm Fvol}}(x_p, X_p,\Delta_p).$$
Together with Theorem \[t-com\], this will imply that for the reductions $(X_p,\Delta_p)$, the F-signature $s(x_p,X_p,\Delta_p)$ has a uniform lower bound as $p\to \infty$, as asked in [@CRST16 Question 5.9].
### Relation to local orbifold Euler numbers
In [@Lan03], Langer introduced local orbifold Euler numbers for general log canonical surface singularities and used it to prove a Miyaoka-Yau inequality for any log canonical surface. In an attempt to understand Langer’s inequality using the Kähler-Einstein metric on a log canonical surface, Borbon-Spotti conjectured recently in [@BS17] that the volume densities of the singular Kähler-Einstein metrics should match Langer’s local Euler numbers (at least for log terminal surface singularities). They verified this in special examples by comparing the known values of both sides. On the other hand, from Theorem \[thm-hvol2Theta\], we know that the normalized volume is equal to the volume density up to a factor $(\dim X)^{\dim X}$ for any point $(X,x)$ that lives on a Gromov-Hausdorff limit of smooth Kähler-Einstein manifolds ([@HS17; @LX17]). In view of this connection, one can formulate a purely algebraic problem about two algebraic invariants of the singularities. This problem was already posed by in [@BS17] at least in the log terminal case. We formulate the following form by including one of Langer’s expectations (see [@Lan03 p.381]):
\[conj\] Let $(X, D, x)$ be a germ of log canonical surface singularity with ${{\mathbb{Q} }}$-boundary. Then we have $$e_{\rm orb}(x, X, D)=
\left\{
\begin{array}{ll}
\frac{1}{4}{\widehat{\rm vol}}(x, X,D), & \text{ if } (X,D) \text{ is log terminal };\\
0, & \text{ if } (X, D) \text{ is not log terminal}.
\end{array}
\right.$$
In [@Li18], it was proved that the above conjecture is true when $(X, D, x)$ is a $2$-dimensional log-Fano cone or a log-CY cone. In particular, combined with Langer’s calculation, one gets the local orbifold Euler numbers of line arrangements.
Let $L_1, \dots, L_n$ be $m$ distinct lines in ${{\mathbb{C} }}^2$ passing through $0$. Let $D=\sum_{i=1}^m \delta_i L_i$, where $0\le \delta_1\le \delta_2\le \dots \le \delta_m\le 1$. Denote $\delta=\sum_{i=1}^m \delta_i$. Then we have: $$e_{\rm orb}(0, {{\mathbb{C} }}^2, D)=\left\{
\begin{array}{lcc}
0 & \text{ if} & ({{\mathbb{C} }}^2, D, 0) \text{ is not klt } ;\\
(1-\delta+\delta_m)(1-\delta_m)& \text{ if } & \delta<2\delta_m;\\
\frac{(2-\delta)^2}{4} & \text{ if } & 2\delta_m\le \delta\le 2.
\end{array}
\right.$$
Here we point out a possible application of Theorem \[thm-2dim\] (i.e. 2-dimensional case conjecture \[conj-local\]) for studying Conjecture \[conj\] for any log terminal singularity $(x, X, D)$. First, by Theorem \[thm-2dim\] there exists a unique Kollár component $S\cong {\mathbb{P}}^1$ which minimizes the normalized volume. Let $\mu: Y\rightarrow X$ be the extraction of $S$ and $\Delta={\rm Diff}_S(D)$. By Theorem \[t-high\] we know that $(S, \Delta)\cong ({\mathbb{P}}^1, \sum_i \delta_i p_i)$ is indeed K-semistable (see [@LX16 section 6]). Then $\mathscr{F}:=\Omega^1(\log (S+D))$ (defined using ramified coverings as in [@Lan03]) restricted to $S$ fits into an exact sequence of orbifold sheaves: $$0\longrightarrow \Omega^1_S(\log(\Delta))\rightarrow \mathscr{F}|_S\rightarrow {\mathcal{O}}_S\rightarrow 0.$$ By [@Li18 Theorem 1.3], we know that $\mathscr{E}:=\mathscr{F}|_S$ is slope semistable. Then the generalization of [@Wah93 Proposition 3.16] to the logarithmic/orbifold setting combined together with Langer’s work should imply that $e_{\rm orb}(x, X, D)=\frac{c_1(\mathscr{E})^2}{4(-S\cdot S)_Y}$ which is indeed equal to $\frac{{\widehat{\rm vol}}({{\rm ord}}_S)}{4}$.
### Normalized volume function
We have mainly concentrated on the minimizer of the normalized volume function. We can also ask questions on the general behavior of the normalized volume function. For example:
Let $\sigma\subset {{\rm Val}}_{X,x}$ be a simplex of quasi-monomial valuations. Is it true that ${\widehat{\rm vol}}(\cdot)$ is always convex on $\sigma$? Is there a more general convexity property for ${\widehat{\rm vol}}$ on ${{\rm Val}}_{X,x}$?
Is the normalized volume a lower semicontinuous function on ${{\rm Val}}_{X,x}$? If this is true, then it would directly imply the existence of minimizer of ${\widehat{\rm vol}}$ using the properness estimate in Theorem \[t-izumi\].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study non-local correlations in a three-orbital Hubbard model defined on an extended one-dimensional chain using determinant quantum Monte Carlo and density matrix renormalization group methods. We focus on a parameter regime with robust Hund’s coupling, which produces an orbital selective Mott phase (OSMP) at intermediate values of the Hubbard $U$, as well as an orbitally ordered ferromagnetic insulating state at stronger coupling. An examination of the orbital- and spin-correlation functions indicates that the orbital ordering occurs before the onset of magnetic correlations in this parameter regime as a function of temperature. In the OSMP, we find that the self-energy for the itinerant electrons is momentum dependent, indicating a degree of non-local correlations while the localized electrons have largely momentum independent self-energies. These non-local correlations also produce relative shifts of the hole-like and electron-like bands within our model. The overall momentum dependence of these quantities is strongly suppressed in the orbitally-ordered insulating phase.'
author:
- 'S. Li'
- 'N. Kaushal'
- 'Y. Wang'
- 'Y. Tang'
- 'G. Alvarez'
- 'A. Nocera'
- 'T. A. Maier'
- 'E. Dagotto'
- 'S. Johnston'
title: 'Non-local correlations in the orbital selective Mott phase of a one dimensional multi-orbital Hubbard model'
---
Introduction
============
In recent years the scientific community renewed its interest in understanding the properties of multi-orbital Hubbard models, and this has been intensified by the discovery of the iron-based superconductors.[@JohnstonAP2010; @StewartRMP2011; @DaiNaturePhys2012; @vanRoekeghem] On a theoretical front, this is a challenging problem due to a lack of non-perturbative methods for treating multi-orbital Hubbard models at intermediate or strong couplings and on extended systems. Nevertheless, considerable progress has been made using mean-field-based approaches, [@vanRoekeghem; @YuPRL2013; @FanfarilloPRB2015; @GeorgesReview; @Zhang; @Knecht; @WernerPRL2007; @LiebschPRL2005; @deMediciPRL2009; @deMediciPRB2005; @MukherjeePRB2016; @SemonPreprint; @Ferrero] resulting in new concepts such as that of a Hund’s metal [@Zhang; @YinNatureMaterials2011; @GeorgesReview; @LanataPRB2013] and the orbital-selective Mott phase (OSMP).[@Anisimov2002; @GeorgesReview] These concepts are central to our understanding the paradoxical appearance of both localized and itinerant characteristics in many multi-orbital systems [@deMediciPRL2014; @MannellaReview] and bad metallic behavior in the presence of sizable electronic correlations.[@deMediciPRL2014]
The most widely used numerical approach in this context is single-site multi-orbital dynamical mean-field theory (DMFT).[@Footnote1; @Georges; @vanRoekeghem] Generally speaking, DMFT maps the full lattice problem onto an impurity problem embedded in an effective medium, which approximates the electron dynamics on a larger length scale as a local renormalization. [@Georges] While this technique has had considerable success in addressing many aspects of the OSMP and other physics related to the multi-orbital problem,[@Knecht; @LiebschPRL2005; @ChanPRB2009; @LiebschPRB2004; @BiermannPRL2005; @deMediciPRL2009; @deMediciPRB2005; @WernerPRL2007; @LiebschPRB2010; @GregerPRL2013] it is unable to capture spatial fluctuations and non-local correlations encoded in the $k$-dependent self-energy $\Sigma({\bf k},\omega)$. This is a potential short coming as non-local correlations are known to have an impact in the case of the single-band Hubbard model.[@GullEPL2008; @ParkPRL2008] It is therefore important to assess the importance of such non-local effects on multi-orbital properties such as the OSMP.
To date, most non-perturbative studies have used cluster DMFT or the dynamical cluster approximation (DCA); [@NomuraPRB2015; @LeePRB2011; @LeePRL2010; @BeachPRB2011; @LeoPRL2008; @SemonPreprint] however, these techniques are typically limited to a handful of sites when multiple orbitals are included in the basis. This is due to technical issues related to each choice in impurity solver, such as the Fermion sign problem in the case of quantum Monte Carlo or the exponential growth of the Hilbert space in the case of exact diagonalization. As a result, these studies have only addressed short-range spatial fluctuations. One study of the OSMP has been carried out on a larger two-dimensional cluster using determinant quantum Monte Carlo (DQMC). In that case, however, the OSMP was imposed by the model by assuming that electrons in a subset of orbitals were localized as Ising spins.[@BouadimPRL2009] In light of these limitations it is desirable to find situations where multi-orbital physics can be modeled on extended clusters that support long-range spatial fluctuations and where the properties under study emerge from the underlying many-body physics of the model.
In this regard, one dimensional (1D) models are quite promising. For example, two recent density matrix renormalization group (DMRG) studies have been carried out for an effective 1D three-orbital model representative of the iron-based superconductors. [@RinconPRL2014; @RinconPRB2014] More recently, it was demonstrated that DQMC simulations for a simplified version of the same model can also be carried out to low temperatures due to a surprisingly mild Fermion sign problem.[@LiuPreprint2016] These observations open the doorway to non-perturbative studies of this model on extended clusters, thus granting access to the momentum-resolved self-energies and non-local correlations. 1D studies along these lines are also directly relevant for the recently-discovered quasi-1D selenide Ba$_{1-x}$K$_x$Fe$_2$Se$_3$.[@CaronPRB2011; @CaronPRB2012; @Patel; @Shaui; @LuoPRB2013; @LuoPRB2014] In this context, it is important to note that DMFT becomes more accurate in higher dimensions and therefore one expects its ability to describe multi-orbital Mott physics in 1D to be diminished.
Motivated by these considerations, we examine the properties of a three-orbital Hubbard Hamiltonian on an extended 1D cluster using DQMC and DMRG, with a particular focus on its $k$-resolved self-energies and spectral properties. We thus gain explicit access to non-local correlations occurring on longer length scales than those addressed in previous non-perturbative studies. In general, we find that the OSMP leads to a mixture of localized and itinerant bands, where the former are characterized by a localized (momentum-independent) self-energy while the latter exhibits significant non-local (momentum-dependent) correlations. This leads to a band-dependent shift of the underlying band structures. We also identify an insulating state driven by orbital ordering in a region of parameter space previously associated with an OSMP.[@RinconPRL2014; @RinconPRB2014]
Methods {#Sec:Methods}
=======
Model Hamiltonian
-----------------
We study a simplified three-orbital model defined on a 1D chain as introduced in Ref. . This model displays a rich variety of phases including block ferromagnetism, antiferromagnetism, Mott insulting phases, metallic and band insulating phases, and several distinct OSMPs.[@RinconPRL2014; @RinconPRB2014; @LiuPreprint2016] The Hamiltonian is $H = H_\mathrm{0} + H_\mathrm{int}$, where $$\label{Eq:H0}
H_\mathrm{0} = - \sum_{\substack{\langle i,j \rangle\\\sigma,\gamma,\gamma^\prime}}
t^{{\phantom{\dagger}}}_{\gamma\gamma^\prime} c^\dagger_{i,\gamma,\sigma}c^{{\phantom{\dagger}}}_{j,\gamma^\prime,\sigma}
+ \sum_{i,\sigma,\gamma} (\Delta_\gamma-\mu) \hat{n}_{{i},\gamma,\sigma}$$ contains the non-interacting terms of $H$, and $$\nonumber
\begin{split}
H_\mathrm{int}&=U\sum_{i,\gamma} \hat{n}_{i,\gamma,\uparrow}\hat{n}_{i,\gamma,\downarrow} +
\left(U^\prime - \frac{J}{2}\right) \sum_{\substack{i,\sigma,\sigma^\prime\\
\gamma < \gamma^\prime}} \hat{n}_{i,\gamma,\sigma}\hat{n}_{i,\gamma,\sigma^\prime}\\
&+ J \sum_{i,\gamma<\gamma^\prime} S^\mathrm{z}_{i,\gamma} S^\mathrm{z}_{i,\gamma^\prime}
\end{split}$$ contains the on-site Hubbard and Hund’s interaction terms. Here, $\langle \dots \rangle$ denotes a sum over nearest-neighbors, $c^\dagger_{i,\gamma,\sigma}$ $(c^{{\phantom{\dagger}}}_{i,\gamma,\sigma})$ creates (annihilates) a spin $\sigma$ electron in orbital $\gamma = 1,2,3$ on site $i$, $\Delta_\gamma$ are the on-site energies for each orbital, $S^\mathrm{z}_{i,\gamma}$ is the z-component of the spin operator ${\bf S}_{i,\gamma}$, and $\hat{n}^{{\phantom{\dagger}}}_{i,\gamma,\sigma} = c^\dagger_{i,\gamma,\sigma}c^{{\phantom{\dagger}}}_{i,\gamma,\sigma}$ is the particle number operator. The pair-hopping and spin-flip terms of the interaction have been neglected in order to manage the sign problem in the DQMC calculations.
Following Ref. , we set $t_{11} = t_{22} = -0.5$, $t_{33} = -0.15$, $t_{13} = t_{23} = 0.1$, $t_{12} = 0$, $\Delta_1 = -0.1$, $\Delta_2 = 0$, and $\Delta_3 = 0.8$ in units of eV while the chemical potential $\mu$ is adjusted to obtain the desired filling. These parameters produce a non-interacting band structure analogous to the iron-based superconductors, with two hole-like bands centered at $k = 0$ and an electron-like band centered at $k = \pi/a$, as shown in Fig. \[Fig:Bands\]. Due to the weak inter-orbital hopping, each of the bands is primarily derived from a single orbital, as indicated by the line thickness and colors in Fig. \[Fig:Bands\]. One can therefore (loosely) regard the orbital character as an indicator of the band in this model. For example, the top most band is primarily composed of orbital $\gamma = 3$. The total bandwidth of the non-interacting model is $W = 4.9|t_{11}| = 2.45$ eV. This will serve as our unit of energy. We further set $a = 1$ as our unit of length. The interaction parameters are fixed to $U^\prime = U - 2J$, $J = U/4$, while $U$ is varied. This parameter regime results in a robust OSMP for intermediate values of $U$, which is our focus here.
![\[Fig:Bands\] (color online) A fat band plot of the non-interacting band structure at a total filling of $\langle \hat{n} \rangle = 4$, where the thickness of the lines indicates the majority orbital content of the band. The top most band has the narrowest bandwidth and is primarily of orbital 3 character. The lower two bands disperse over a much larger energy range and are primarily composed of orbitals 1 and 2, respectively. ](./noninteractingbands.pdf){width="0.75\columnwidth"}
DQMC and DMRG Calculations
--------------------------
The model is studied using non-perturbative DQMC and DMRG methods. The details of these techniques can be found in Refs. (DQMC) and Refs. and (DMRG). These approaches are complementary to one another; DMRG works in the canonical ensemble and provides access to the ground state properties of the system while DQMC works in the grand canonical ensemble and provides access to finite temperatures and fluctuations in particle number. Both methods are capable of treating large cluster sizes such that non-local correlations can be captured without approximation for the specified Hamiltonian.
The primary drawback to DQMC is the Fermion sign problem,[@Sign1; @Sign2] which typically limits the range of accessible temperatures for many models. Indeed, when the spin-flip and pair hopping terms of the Hund’s interaction are included in the Hamiltonian, we find that the model has a prohibitive sign problem. But when these terms are neglected the corresponding sign problem becomes very mild,[@LiuPreprint2016] even in comparison to similar simplified multi-orbital models in 2D.[@RademakerPRB2013; @BouadimPRB2008] Given that these terms do not qualitatively affect the phase diagram [@LiuPreprint2016] for the current model, we have neglected them here. This has allowed us to study clusters of up to $L = 24$ sites in length ($3L$ orbitals in total) down to temperatures as low as $\beta = 74/W$.[@LiuPreprint2016] At this low of a temperature we begin to see the onset of magnetic correlations in our cluster, however, as we will show, the OSMP forms at a much higher temperature. Since the latter phase is our focus here, we primarily show DQMC results for $\beta \le 19.6/W$ throughout. In all cases shown here, the average value of the Fermion sign is greater than $0.87 \pm 0.01$. Unless otherwise stated, all of our DQMC results were obtained on an $L =24$ site cluster with periodic boundary conditions and for an average filling of $\langle n\rangle = 4$ electrons, which corresponds to $2/3$ filling.
DQMC provides direct access to various quantities defined in the imaginary time $\tau$ or Matsubara frequency ${\mathrm{i}}\omega_n$ axes. In Sec. \[Sec:Spectral\] we will examine the spectral properties of our model, which requires an analytic continuation to the real frequency axis. This was accomplished using the method of Maximum Entropy,[@MaxEnt] as implemented in Ref. .
Our DMRG results were obtained on variable length chains with open boundary conditions. The chemical potential term in Eq. (\[Eq:H0\]) is dropped for these calculations. In all of the DMRG calculations the truncation tolerance is between $10^{-5}$ – $10^{-7}$. We performed three to five full sweeps of finite DMRG algorithm and used 300 states for calculating both the ground state and the spectral function. Once the ground state is obtained using the standard DMRG algorithm, we computed the spectral function using the correction vector targeting in Krylov space [@DMRG_Method1; @DMRG_Method2], with an broadening of $\eta = 0.001$ eV.
Results {#Sec:Results}
=======
Self-energies in the OSMP
-------------------------
We begin by examining some of the standard metrics for the formation of an OSMP, namely the average filling per orbital and the quasiparticle residue $Z_\gamma(k,{\mathrm{i}}\omega_n)$. DQMC results for $\langle n \rangle = 4$ and $U/W = 0.8$ are summarized in Fig. \[Fig:n\_beta\]. The temperature dependence of the individual orbital occupations $\langle n_\gamma \rangle$, plotted in Fig. \[Fig:n\_beta\]a, has the standard indications of the formation of an OSMP: At high temperature (small $\beta$) we see noninteger fillings for all three orbitals. As the temperature is lowered (large $\beta$), however, orbitals one and two smoothly approach fillings of $\sim 1.53$ and $\sim 1.47$, respectively, while orbital three locks into an integer value of exactly 1. In some studies these values for the average occupation are often taken as an indication of an OSMP,[@RinconPRL2014; @RinconPRB2014] where orbital three has undergone a transition to a Mott insulating state while orbitals one and two host itinerant electrons. However, as we will show, this is not always the case. For $U/W =
0.8$ the two fractionally filled orbitals are in fact itinerant, but for larger values of $U/W$ these same orbitals retain a fractional filling but are driven into an insulating state by the onset of orbital ordering in these two orbitals.
![\[Fig:n\_beta\] (color online) Orbitally resolved electronic properties for $U/W = 0.8$ ($W = 2.45$ eV) at different temperatures. (a) The temperature dependence of orbital occupations. (b) The orbital resolved quasiparticle residue $Z_\gamma(k,{\mathrm{i}}\pi/\beta)$ at an inverse temperature $\beta = 19.6/W$. (c) The normalized electron self energies Im$\Sigma_\gamma(k,{\mathrm{i}}\pi/\beta)$ at $\omega_n=\pi/\beta$ as a function of momentum. Each curve is normalized by its $k = 0$ value to highlight the overall momentum dependence. The scale is determined by $\mathrm{Im}\Sigma_\gamma(0,{\mathrm{i}}\pi/\beta) = -0.53$, $-0.57$, and $-2.53$ for $\gamma = 1,2,3$, respectively, and in units of the bandwidth $W$. The blue, red, and green dash lines in (b) and (c) correspond to the bare Fermi momentum of the non-interacting bands. Panel (d) shows orbitally resolved quasiparticle residues $Z_\gamma(k_\mathrm{F}^0,{\mathrm{i}}\pi/\beta)$ and self energies Im$\Sigma_\gamma(k_\mathrm{F}^0,{\mathrm{i}}\pi/\beta)$ at Fermi momentum as a function of temperature. In each panel, error bars smaller than the marker size have been suppressed for clarity. ](./n_beta.pdf){width="0.66\columnwidth"}
The mixed itinerant/localized nature of the OSMP at $U/W = 0.8$ is reflected in the momentum dependence of quasi-particle residue $Z_\gamma(k,{\mathrm{i}}\pi/\beta)$ and the orbitally resolved normalized self-energies $R(k) = {\mathrm{Im}}\Sigma_\gamma(k,{\mathrm{i}}\pi/\beta)/{\mathrm{Im}}\Sigma_\gamma(0,{\mathrm{i}}\pi/\beta)$, plotted in Figs. \[Fig:n\_beta\]c and \[Fig:n\_beta\]d, respectively, for $\omega_n = \pi/\beta$. The self-energy is extracted from the dressed Green’s function using Dyson’s equation $$\hat{G}^{-1}(k,{\mathrm{i}}\omega_n) = \hat{G}^{-1}_0(k,{\mathrm{i}}\omega_n) - \hat{\Sigma}(k,{\mathrm{i}}\omega_n),$$ where the $\hat{G}$ notation denotes a matrix in orbital space, $\hat{G}_0(k,{\mathrm{i}}\omega_n) = [{\mathrm{i}}\omega_n\hat{I} - \hat{H}_0(k)]^{-1}$ is the non-interacting Green’s function, and $\hat{H}_0(k)$ is the Fourier transform of the non-interacting Hamiltonian defined in orbital space. The quasi-particle residue is obtained from the diagonal part of the self-energy using the identity $$\hat{Z}(k,{\mathrm{i}}\pi/\beta) = \left(\hat{I} - \frac{\mathrm{Im}\hat{\Sigma}
(k,{\mathrm{i}}\pi/\beta)}{\pi/\beta}\right)^{-1},$$ where $\hat{I}$ is a $3\times 3$ unit matrix.
As can be seen from Fig. \[Fig:n\_beta\]c, the self-energies for each orbital have a sizable $k$-dependence at this temperature. (In this case we have normalized the self-energy by its value at $k = 0$ in order to highlight the overall momentum dependence. The magnitude of $\mathrm{Im}\Sigma_\gamma(0,{\mathrm{i}}\pi/\beta)$ is given in the figure caption.) In the case of orbitals one and two, the magnitude of the self-energy varies by nearly 50% throughout the Brillouin zone. In contrast, the momentum dependence of $\Sigma_3(k,{\mathrm{i}}\pi/\beta)$ for orbital three is much weaker, varying by only 5-10% and reflecting the localized nature of the carriers in these orbitals. Similarly, the quasi-particle residue for the orbital three is essentially momentum independent, while it increases for the two itinerant orbitals as $k$ tracks towards the zone boundary. The $k$ dependence at the remaining Matsubara frequencies accessible to our simulations (not shown) exhibits a similar trend, with orbitals one and two having a strong $k$-dependence while orbital three is nearly momentum independent at each $\omega_n$.
The momentum dependence shown in Fig. \[Fig:n\_beta\] indicates that the local self-energy approximation introduced by DMFT may miss quantitative aspects of the electronic correlations in the OSMP with mixed itinerant and local characteristics. It should be noted that our results have been obtained in 1D, which is the worst case situation for DMFT. [@Akerlund_PRD] It is expected that the local approximation will perform better in higher dimensions, since DMFT becomes exact in the limit of infinite dimensions; however, it is unclear how well the method will capture similar non-local correlations in two dimensions relevant for the Fe-based superconductors. A recent study [@SemonPreprint] has argued that the local approximation is quite accurate for parameters relevant to the iron-based superconductors, however, it remains to be seen if this will remain true for all parameter regimes or when longer range fluctuations are included. Our results further highlight the need for the continued development of numerical methods capable of handling the strong Hubbard and Hund’s interactions in intermediate dimensions and on extended clusters.
Figure \[Fig:n\_beta\]d examines the temperature dependence of $Z(k^0_\mathrm{F},\frac{{\mathrm{i}}\pi}{\beta})$ and Im$\Sigma(k^0_\mathrm{F},\frac{{\mathrm{i}}\pi}{\beta})$ at the Fermi momenta $k^0_\mathrm{F}$ of the non-interacting system. (These are indicated by the dashed lines in \[Fig:n\_beta\]b and \[Fig:n\_beta\]c.) Here, we find indications of anomalous behavior for the itinerant electrons, where the quasiparticle residues of all three orbitals decrease with temperature. This is accompanied by an increase in Im$\Sigma(k_\mathrm{F},\frac{{\mathrm{i}}\pi}{\beta})$ as $T$ is lowered. This is perhaps expected for orbital three, as $Z$ (Im$\Sigma$) for the localized orbitals should decrease (increase) as this orbital becomes more localized. For the itinerant orbitals, however, one would naively expect the self-energy to decrease as temperature is lowered, which is opposite to what is observed. We believe that this is due to the Hund’s interaction between the itinerant electrons and the localized spins on orbital three. At this temperature we find no evidence of a magnetic ordering in our model,[@LiuPreprint2016] despite the fact that a local moment has clearly formed in the OSMP. This means that the orientation of the local moment is random and fluctuating at these temperatures. This produces a fluctuating potential acting on the itinerant electrons via the Hund’s coupling, thus generating a residual scattering mechanism at low temperatures that reduces the quasiparticle residue and increases the self-energy.
Momentum and Temperature Dependence of the Spectral Weight
----------------------------------------------------------
Next, we turn to the momentum dependence of the spectral weight for the three orbitals in the vicinity of the Fermi level. This can be estimated directly from the imaginary time Green’s function, where the spectral weight at momentum $k$ is proportional to $\beta G(k,\tau=\beta/2)$.[@spectral] Using this relationship we do not have to perform the extra step of analytically continuing the data to the real frequency axis.
Figures \[Fig:Gbeta\]a-\[Fig:Gbeta\]c summarize $\beta G(k,\beta/2)$ for $U/W=0.1$, $U/W=0.8$, and $U/W=2$, respectively. The results in the weak coupling limit ($U/W = 0.1$, Fig. \[Fig:Gbeta\]a) are consistent with that of a fully itinerant system: all three orbitals have a maximal spectral weight at a momentum point very close to the Fermi momenta of the non-interacting system (indicated by the dashed lines). This is exactly the behavior one expects for a well-defined quasi-particle band dispersing through $E_\mathrm{F}$, where the peak in the spectral weight occurs at $k_\mathrm{F}$. The proximity of the peaks in $\beta G(k,\beta/2)$ to the non-interacting values of $k_\mathrm{F}$ indicates that the Fermi surface is only weakly shifted for this value of the interaction parameters. However, as we will show in Sec. \[Sec:BandShifts\], these shifts are band dependent.
![\[Fig:Gbeta\] (color online) The momentum dependence of Green functions $G(k,\tau=\beta/2)$ for a) $U/W=0.1$, b) $0.8$, and c) $2.0$. The inverse temperature in all three cases is $\beta=19.6/W$. The blue, red, and green dash lines in each panel indicate the Fermi momentum of the three non-interacting bands. (d) $G(k_\mathrm{F},\tau=\beta/2)$ as a function of inverse temperatures $\beta$ for the OSMP $U/W=0.8$. Error bars smaller than the marker size have been suppressed for clarity. ](./Gbeta.pdf){width="0.7\columnwidth"}
In the intermediate coupling regime ($U/W = 0.8$, Fig. \[Fig:Gbeta\]b), where the OSMP has formed, we again see both localized and itinerant characteristics. The spectral weight of the localized orbital is small and independent of momentum, as expected for the formation of a localized Mott state. Conversely, the spectral weight of the remaining orbitals still exhibits a momentum dependence characteristic of dispersive bands. Despite this, the total spectral weight is decreased, indicating that spectral weight has been transferred to higher binding energies by the Hubbard and Hund’s interactions. This is also reflected in the position of the maximum spectral weight, which has shifted to a slightly larger $k$ value due to a renormalization of the Fermi surface by the interactions. We also observe that the spectral weight at the zone boundary increases relative to the zone center, consistent with a flattening of the bands and a broadening of the spectral function with increasing $U$. (This will be confirmed shortly when we examine the spectral functions directly.) A similar transfer of spectral weight was observed in a two-dimensional cluster DMFT study.[@NomuraPRB2015]
The temperature evolution of spectral weight $\beta G(k_\mathrm{F},\beta/2)$ at the Fermi momentum for the OSMP ($U/W = 0.8$) is shown in Figure \[Fig:Gbeta\]d. In a metallic system one generally expects the spectral weight at the Fermi level to increase as the temperature is decreased. Initially, this is what is observed for all three orbitals, however, the spectral weight for orbital three reaches a maximum around $\beta = 7.5/W$ before decreasing as the temperature is lowered further and the OSMP gap forms on this orbital. Conversely, the spectral weight of the itinerant orbitals continues to rise until saturating at $\beta/W \approx 15$. This saturation is again due to the presence of a residual scattering channel, which we associate with the fluctuating localized spins present on the localized orbital three.
The $U/W = 0.8$ results confirm the mixed itinerant/local character of the model at intermediate coupling. When the value of $U$ is further increased, we find that all three bands become localized while maintaining partial occupancies for each band. To demonstrate this, Fig. \[Fig:Gbeta\]c shows results for $U/W = 2$. In this case, the orbital occupations for the three orbitals are $\langle n_1 \rangle = 1.55$, $\langle n_2 \rangle = 1.44$, $\langle n_3 \rangle = 1$, which are similar to those obtained at $U/W = 0.8$. At face value one might therefore conclude that the system is in an OSMP,[@RinconPRL2014; @LiuPreprint2016] however, an examination of the spectral weight reveals that the system is in fact insulating. As can be seen in Fig. \[Fig:Gbeta\]c, at $U/W = 2$ and $\beta = 19.6/W$, $\beta G(k,\beta/2)$ is nearly momentum independent and the total spectral weight of all three orbitals has significantly decreased (note the change in scale of the y-axis). This behavior is indicative of the formation of a charge gap throughout the Brillouin zone. The ultimate origin of this insulating behavior is the formation of a long-range orbital ordering, as we will show in Sec. \[Sec:Spectral\].
Band-dependent Fermi surface renormalization {#Sec:BandShifts}
--------------------------------------------
![\[Fig:n\_k\] (color online) The momentum dependence of the number operator $n_\gamma(k) = \frac{1}{2}\sum_\sigma \langle c^\dagger_{{{\bf k}},\gamma,\sigma} c_{{{\bf k}},\gamma,\sigma}\rangle$ for each band. Results are shown for the non-interacting case $U = 0$ (black dashed, $\square$), $U/W = 0.1$ (blue solid, $\bigtriangleup$), $U/W = 0.8$ (red solid $\circ$), and $U/W = 2$ (green solid $\diamond$) and at an inverse temperature of $\beta = 19.6/W$.](./n_k.pdf){width="0.7\columnwidth"}
It is now well known that [*ab initio*]{} band structure calculations based on density functional theory (DFT) do not describe the electronic structure of the iron based superconductors as measured in ARPES experiments. (For a recent review, see Ref. .) Generally speaking, the calculated band structure usually needs to be rescaled by an overall factor, which is attributed to reduction in bandwidth driven by electronic correlations. In addition, the size of the Fermi surfaces is often overestimated by DFT in comparison to measurements. A prominent example of this is LiFeAs,[@Chi] where the inner most hole pocket realized in nature is substantially smaller than the one predicted by DFT [@Klaus; @Klaus2]. In order to correct this, the electron- and hole-bands need to be shifted apart,[@vanRoekeghem] which requires a momentum-dependent self-energy correction.
We examine this issue within our model in Fig. \[Fig:n\_k\], which plots the expectation value of the orbitally-resolved number operator in momentum space $n^{{\phantom{\dagger}}}_\gamma(k) = \frac{1}{2}\sum_\sigma \langle c^\dagger_{{{\bf k}},\gamma,\sigma} c^{{\phantom{\dagger}}}_{{{\bf k}},\gamma,\sigma}\rangle$ for various values of the interaction strength. In the non-interacting limit, and in a single-band case, this quantity is equal to the Fermi-Dirac distribution and the location of the leading edge corresponds to $k_\mathrm{F}$. In a multi-band system the mixing of the orbital character complicates this picture; however, in our model the leading edge still corresponds to $k_\mathrm{F}$ due to the weak hybridization between orbitals. In the weak coupling case ($U/W = 0.1$) we observe a small shift in the position of the leading edge. Within error bars, the curve $n_1(k)$ and $n_2(k)$ shift to slightly larger momenta while $n_3(k)$ shifts towards smaller momenta. This indicates that the size of the Fermi surfaces are increasing and the electron-like and hole-like bands are shifted towards one another by the interactions. This trend continues as $U/W$ is increased to $0.8$; however, in this case the electron-like band is significantly smeared out due to the formation of the OSMP.
We note that the direction of the band shifts is reversed from what is generally required for the two-dimensional iron-based superconductors, where the calculated hole-like Fermi surfaces generally need to be shrunk relative to the electron-like Fermi surfaces. We attribute this to differences in the underlying tight-binding model and differences in dimensionality. In this light, it would be interesting to compare the ARPES observed band structures in the quasi-one-dimensional pnictides against the predictions of our model and DFT calculations.[@Patel] Nevertheless, our results do show that non-local correlations arising from a local interaction can produce relative shifts of the electron-like and hole-like bands in a multi-orbital system.
Spectral Properties {#Sec:Spectral}
-------------------
### Intermediate Coupling $U/W = 0.8$
We now examine the spectral properties of the model, beginning with the OSMP. Figure \[Fig:dos\]a shows the temperature evolution of the total density of states (DOS) at $U/W = 0.8$, which is obtained from the trace of the orbital-resolved spectral function $N(\omega) = \sum_{k,\gamma}
-\frac{1}{\pi}{\mathrm{Im}}\hat{G}_{\gamma\gamma}(k,\omega+i\delta)$. In the non-interacting limit (the long-dashed (blue) curve), the DOS has a double peak structure, where the lower (upper) peak corresponds to the bands derived from orbitals one and two (orbital three). The overall structure of the DOS in the interacting case is similar at high temperatures, but some spectral weight is transferred to a broad incoherent tail extending to lower energies. As the temperature is decreased, the peak on the occupied side shifts towards the Fermi level and sharpens. At the same time, a small amount of spectral weight is transferred from the vicinity of the Fermi level into this peak. The appearance of this apparent “pseudogap" is a direct consequence of the OSMP forming on orbital three, which is easily confirmed by examining the orbital-resolved DOS $N_\gamma(\omega) = -\frac{1}{\pi}\sum_k
\mathrm{Im}\hat{G}_{\gamma,\gamma}(k,\omega)$ shown in Fig. \[Fig:dos\]b. As can be clearly seen, orbitals one and two have a finite DOS at $\omega = 0$, while orbital three is fully gapped at low-temperature.
![\[Fig:dos\] (color online) (a) The density of states at different temperatures. (b) The orbitally-resolved density of states for each orbital at an inverse temperature $\beta=19.6/W$. (c) The density of states at the Fermi surface of the orbital 3 as a function of inverse temperatures $\beta$. The Coulomb interaction strength is $U/W=0.8$ in all three graphs. ](./dos.pdf){width="0.75\columnwidth"}
We also begin to see the formation of an additional peak near the Fermi level at the lowest temperature we examined ($\beta = 19.6/W$). This feature is more clearly seen in the orbital-resolved DOS (Fig. \[Fig:dos\]b), where it is found to originate from the itinerant orbitals. This peak is due to a hybridization between the itinerant and localized orbitals, which is observable in the $k$-resolved spectral functions (see Fig. \[Fig:spectral\]).
The relevant temperature scale for the formation of the OSMP can be estimated by tracking $N_{3}(0)$ as a function of temperature, as shown in Fig. \[Fig:dos\]c. Here, a continuous suppression of $N_3(0)$ is observed, with the value reaching zero at $\beta \approx 20/W$. The rate at which $N_3(0)$ decreases also undergoes a distinct change at $\beta
\approx 10/W$, which coincides with the temperature at which the spectral weight for this orbital at $k_\mathrm{F}$ is largest (see Fig. \[Fig:Gbeta\]d). We interpret this to mean that the Mott gap on orbital three begins to form at $\beta W \approx 10$ (on the $L = 24$ site lattice), growing continuously from zero as the temperature is lowered. In this case, the finite spectral weight between $\beta W = 10$ – $20$ is due to thermal broadening across this gap. Since we have observed similar behavior on smaller clusters with DQMC and at zero temperature using DMRG, we believe that the transition to the OSMP will survive in the thermodynamic limit, however, the gap magnitude has some finite size dependence.
{width="80.00000%"}
The extended length of our 1D cluster grants us access to the momentum dependence of the spectral function, which is shown in Fig. \[Fig:spectral\]. The top row of Fig. \[Fig:spectral\] shows the results in the OSMP with $U/W = 0.8$ and $\beta = 19.6/W$, which is the same parameter set used in Fig. \[Fig:dos\]. The total spectral function $A(k,\omega) =-\frac{1}{\pi}\mathrm{Tr}\left[\mathrm{Im}\hat{G}(k,\omega)\right]$ is shown in Fig. \[Fig:spectral\]a and the orbital-resolved components $A_\gamma(k,\omega)=-\frac{1}{\pi}\mathrm{Im}\hat{G}_{\gamma\gamma}(k,\omega)$ are shown in Figs. \[Fig:spectral\]b-d, as indicated. The lower row of Fig. \[Fig:spectral\] shows similar results obtained for $U/W = 2$ and $L = 8$. (In this case a smaller cluster is sufficient due to the non-dispersing nature of the band dispersions.)
The results in the OSMP with $U/W = 0.8$ reveal localized and itinerant characteristics that are consistent with the spectral weight analysis presented earlier. The itinerant orbitals primarily contribute to dispersing bands that track through the $E_\mathrm{F}$ ($\omega = 0$), while orbital three has split into two relatively dispersionless upper and lower Hubbard bands above and below $E_\mathrm{F}$. At first glance, these Hubbard bands appear to be sharper than the corresponding Hubbard bands in the single-band Hubbard model; however, an examination of the DOS (Fig. \[Fig:dos\]b) reveals that they are spread out over an energy interval that is larger than the non-interacting bandwidth of the top most band ($W_3 \sim 0.3W \sim 0.735$ eV). In addition to the formation of the Hubbard bands for orbital 3, we also observe two additional effects. The first is an expected narrowing of the bandwidth of the itinerant bands. For this parameter set we obtain $W_1 \sim 1.7$ and $W_2 \sim 1.65$ eV for orbitals one and two, respectively, which should be compared to the non-interacting values of 1.88 and 1.97 eV. The second is the aforementioned hybridization and level repulsion between the itinerant and localized orbitals. This is manifest in the spectral function as a slight “buckling" of orbital three’s upper Hubbard band near $k = 0$, and the tracking orbital one’s spectral weight along $E_\mathrm{F}$ near $k = \pm\pi/2a$. It is this trailing intensity that forms the peak observed in the DOS just above the Fermi level at low temperatures.
### Strong Coupling $U/W = 2$
The spectral properties of the model are very different when the Hubbard interaction is increased to $U/W = 2$. In this case, the total spectral function (Fig. \[Fig:spectral\]e) and its orbitally-resolved components (Fig. \[Fig:spectral\]f-\[Fig:spectral\]h) all split into relatively flat Hubbard-like bands above and below $E_\mathrm{F}$. (In the case of orbital three, the lower band below $E_\mathrm{F}$ has been pushed outside of the energy range shown in the figure.) For this value of the interaction strength there is no spectral weight at the Fermi level, and the system is insulating even though orbitals one and two have on average 1.55 and 1.44 electrons/orbital, respectively. (These values are obtained both from the measured equal time orbital occupancies, and from integrating the total spectral weight above and below $E_\mathrm{F}$.)
The imaginary axis spectral weight analysis (Fig. \[Fig:Gbeta\]c) and the spectral function analysis (Fig. \[Fig:spectral\]) both indicate that for $U/W = 2$ the model is an insulator. The origin of this behavior is the combined action of the Hund’s coupling and the onset of an orbital ordering of the itinerant orbitals. All indications show that orbital three has already undergone an orbital selective Mott phase transition (OSMT) when $U/W = 2$. This has the effect of localizing one electron per site within this subset of orbitals while leaving three additional electrons to be distributed among the remaining two itinerant orbitals. A sizable Hund’s coupling will decouple the individual orbitals when the crystal field splittings are smaller than the bandwidth of the material.[@deMediciPRL2014] This is precisely the situation at hand, and thus the remaining nominally itinerant orbitals are decoupled from the localized orbital by the large $J = U/4$. This results in an effective nearly-degenerate two-band system with (nearly) three-quarters filling. This is special case for the two-orbital Hubbard model, which is prone to orbital ordering in one and two-dimensions.[@HeldEPJB; @ChanPRB2009; @KuboPRB2002]
The situation is sketched in Fig. \[Fig:OrbitalOrdering\]. Assuming ferromagnetic nearest neighbor correlations for orbital three, we have a low-energy ground state configuration as shown in the left side of \[Fig:OrbitalOrdering\]a. Here, orbitals one and two adopt alternating double occupations in order to maximize their delocalization energy through virtual hopping processes. This results in near-neighbor orbital correlations. Subsequent charge fluctuations such as the one shown in the right side of the Fig. \[Fig:OrbitalOrdering\]a cost a potential energy $PE \sim U^\prime - J = W/2$. This is compensated for by a kinetic energy gain $KE \sim 4t_{11} \sim 4W/4.9$. The ratio between these competing energy scales is $\sim 5/8$, suggesting that charge fluctuations are strongly suppressed by the strong electronic correlations in this subsystem. Note that the situation is worse for antiferromagnetic nearest neighbor correlations in orbital three. The energy cost in this case increases to $\sim U^\prime$, as shown in Fig. \[Fig:OrbitalOrdering\]b. Thus both ferro- and antiferromagnetic correlations in orbital three will suppress charge fluctuations and promote orbital ordering. Since the type of magnetic correlations does not matter, such orbital ordering tendencies can be expected in the paramagnetic phases, provided the localized moments have formed in orbital three. This picture is then consistent with insulating behavior (and short-range orbital ordering tendencies, see below) at high temperatures, where no magnetic correlations are observed.
![\[Fig:OrbitalOrdering\] A cartoon sketch of the relevant charge fluctuation processes leading to the insulating state when $U/W = 2$ assuming (a) ferromagnetic and (b) antiferromagnetic nearest neighbor correlations within the orbital that has undergone the orbital selective Mott transition (orbital three). ](./Orbital_Ordering.pdf){width="0.9\columnwidth"}
We verify this picture explicitly in Fig. \[Fig:Orbital\_correlation\], which plots the equal-time orbital correlation function $\langle \hat{\tau}_{i+d}\hat{\tau}_i\rangle$, with $\hat{\tau}_{i} = (\hat{n}_{i,2}-\hat{n}_{i,1})$. Here, results are shown for finite temperature DQMC calculations (Fig. \[Fig:Orbital\_correlation\]a) and zero temperature DMRG calculations (Fig. \[Fig:Orbital\_correlation\]b) and with $U/W=2$ in both cases. The “long-range" (with respect to the cluster size) anti-ferro-orbital correlation is clear in the zero temperature results obtained on $L = 8$ and $L=16$ chains. At finite temperatures ($\beta = 19.6/W$) we find that the orbital correlations are suppressed at long distances, but local anti-ferro-orbital correlation remains on shorter length scales. These combined results demonstrate the presence of short-range orbital correlations at higher temperatures, which grow in length as the temperature is decreased. The corresponding orbitally resolved DOS are plotted in Fig. \[Fig:Ueq2\] for both cases. Both methods predict that the system is insulating, with a charge gap width on orbitals one and two of about 0.5 eV. The presence of a gap at finite temperature also confirms that the short range orbital correlations are sufficient to open a gap in the spectral function. Finally, we stress these results will survive in the thermodynamic limit $L \rightarrow \infty$. This is confirmed in the inset in Fig. \[Fig:Ueq2\]b, which shows the evolution of the $T = 0$ gap $\Delta$ as a function of $L$, as obtained from DMRG. Here, the gap size decreases with increasing chain lengths, however, it saturates to 0.2 eV for an infinite length chain.
![\[Fig:Orbital\_correlation\] Results for the orbital correlation function for the system in the strong coupling case $U/W = 2$. Results are obtained at (a) finite temperature using DQMC and (b) $T = 0$ ($\beta = \infty$) using DMRG. In both cases, results are shown on $L = 8$ (red dots) and $L = 16$ (blue triangles) chains. The DQMC results were obtained on a chain with periodic boundary conditions. The DMRG results were obtained on a chain with open boundary conditions. ](./Orbital_Correlation.pdf){width="0.8\columnwidth"}
Discussion and Summary {#Sec:Discussion}
======================
![\[Fig:Ueq2\] Results for the orbitally-resolved density of states for each orbital obtained for $U/W = 2$ and on $L = 8$ site chains. Panel (a) shows DQMC results at $\beta = 19.6/W$ and the inset zooms in to energy around Fermi surface. Panel (b) shows DMRG results for the same conditons but at zero temperature ($\beta = \infty$). The inset plots a finite size scaling analysis of the charge gap obtained within DMRG (see text). The dash line in both panels indicates the Fermi energy. ](./DOS_DQMCvsDMRG.pdf){width="0.8\columnwidth"}
We have performed a momentum-resolved study of a multi-orbital model defined on extended 1D chains using non-perturbative DQMC and DMRG. This has allowed us to compute the several properties of an OSMP in a momentum resolved manner without resorting to approximate methods. We find that several properties do indeed exhibit significant momentum dependencies, not be captured by local approximations introduced by DMFT; however, the 1D case we have considered represents the worst case for DMFT. In that sense our results complement existing DMFT efforts by providing analysis in a region where the method is expected to perform badly.
Our results establish the hierarchy of charge and magnetic orderings in this model. At low temperatures, our DMRG calculations (as well as those in Ref. ) demonstrate that orbital three is ferromagnetically ordered at $T = 0$. Contrary to this, our finite temperature DQMC calculations find no indications of any magnetic order for $\beta < 19.6/W$; the magnetic structure factor $S(q)$ is completely featureless as a function of $q$ at these temperatures. Despite this, our finite $T$ calculations find an orbital-selective Mott phase, as well as a fully insulating phase arising due to short-range orbital ordering, depending on the strength of the Hubbard interaction $U$. We therefore conclude that the charge ordering occurs before any magnetic ordering in this model.
The results shown in Fig. \[Fig:n\_beta\]d and \[Fig:Gbeta\]d show that orbital three in our model, which has the narrowest band width, undergoes a transition to a Mott phase at $\beta W \sim 10-15$. This in combination with the lack of magnetic signal means that OSMP in this parameter regime is a true Mott phase as opposed to a Slater insulator where the insulating behavior is driven by magnetism. Our results also demonstrate that it is insufficient to identify an OSMP using the orbital occupations only in some instances. One should be particularly careful in regions of parameter space where the itinerant bands have average occupations close to special cases known for one and two-orbital Hubbard models. In our case, the average fillings of the itinerant orbitals are $\langle n_1\rangle \sim 1.53$ and $\langle n_2 \rangle \sim 1.47$, values very close to the special case of $3/4$ filling in a degenerate two-band Hubbard model. At zero temperature, our DMRG results obtain fillings of 1.5 for each orbital.
Finally, we discuss our results in the context of recent experimental work. ARPES results for AFe$_2$As$_2$ have found evidence that the OSMP in these materials disappears as the temperature is lowered.[@YiPRL2013] This behavior was explained using a slave-boson approach and attributed to an increase in entropy associated with the OSMP. Our results do not show this behavior, and the OSMP is found at low temperature as one might naively expect. This difference may be related to the differences in the dimensionality (one vs. two) or number of orbitals (three vs. five) between the models or the differences between our non-perturbative approach and other mean-field methods. However, one would expect the entropy to be more important in one dimension. This highlights the need for continued application of non-perturbative methods to tractable multi-orbital Hubbard models.
[*Acknowledgements*]{} — The authors thank G. Liu for useful discussions. S. L., Y. W., and S. J. are supported by the University of Tennessee’s Science Alliance Joint Directed Research and Development (JDRD) program, a collaboration with Oak Ridge National Laboratory. N. K. and E. D. were supported by the National Science Foundation (NSF) under Grant No. DMR-1404375. Y. T. and T. A. M. acknowledge support by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy. Part of this work was conducted at the Center for Nanophase Materials Sciences, sponsored by the Scientic User Facilities Division (SUFD), BES, DOE, under contract with UT-Battelle. A.N. and G.A. acknowledge support by the Early Career Research program, SUFD, BES, DOE. CPU time was provided in part by resources supported by the University of Tennessee and Oak Ridge National Laboratory Joint Institute for Computational Sciences (http://www.jics.utk.edu).
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|
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---
abstract: 'We introduce more properties of forcing notions which imply that their $\lambda$–support iterations are $\lambda$–proper, where $\lambda$ is an inaccessible cardinal. This paper is a direct continuation of Ros[ł]{}anowski and Shelah [@RoSh:777 §A.2]. As an application of our iteration result we show that it is consistent that dominating numbers associated with two normal filters on $\lambda$ are distinct.'
address:
- |
Department of Mathematics\
University of Nebraska at Omaha\
Omaha, NE 68182-0243, USA
- |
Einstein Institute of Mathematics\
Edmond J. Safra Campus, Givat Ram\
The Hebrew University of Jerusalem\
Jerusalem, 91904, Israel\
and Department of Mathematics\
Rutgers University\
New Brunswick, NJ 08854, USA
author:
- 'Andrzej Ros[ł]{}anowski'
- Saharon Shelah
date: June 2005
title: Reasonably complete forcing notions
---
[^1]
Introduction
============
There are serious ZFC obstacles to easy generalizations of properness to the case of iterations with uncountable supports (see, e.g., Shelah [@Sh:f Appendix 3.6(2)]). This paper belongs to the series of works aiming at localizing “good properness conditions” for such iterations and including Shelah [@Sh:587], [@Sh:667], Ros[ł]{}anowski and Shelah [@RoSh:655], [@RoSh:777] and Eisworth [@Ei03]. This paper is a direct continuation of Ros[ł]{}anowski and Shelah [@RoSh:777 §A.2], though no familiarity with the previous paper is assumed and the current work is fully self-contained.
In Section 2 we introduced 3 bounding–type properties (A, B, C) and we essentially show that the first two are almost preserved in $\lambda$–support iterations (Theorems \[verB\], \[verA\]). “Almost” as the limit of the iteration occur to have somewhat weaker property, but equally applicable. In the following section we show that the [ *reasonable A–bounding*]{} property is equivalent to the one introduced in [@RoSh:777 §A.2] thus showing that \[verA\] improves [@RoSh:777 Theorem A.2.4]. Finally, in the fourth section of the paper, we give an example of an interesting [*reasonable $B$–bounding*]{} forcing notion and we use it to show that it is consistent that dominating numbers associated with two normal filters on $\lambda$ are distinct (Conclusion \[conc\]).
Like in [@RoSh:777], we assume here that our cardinal $\lambda$ is inaccessible. We do not know at the moment if any parallel work can be done for a successor cardinal.
[**Notation:**]{}Our notation is rather standard and compatible with that of classical textbooks (like Jech [@J]). In forcing we keep the older convention that [*a stronger condition is the larger one*]{}.
1. Ordinal numbers will be denoted be the lower case initial letters of the Greek alphabet ($\alpha,\beta,\gamma,\delta\ldots$) and also by $i,j$ (with possible sub- and superscripts).
Cardinal numbers will be called $\kappa,\lambda,\mu$; [*$\lambda$ will be always assumed to be inaccessible*]{} (we may forget to mention it).
By $\chi$ we will denote a [*sufficiently large*]{} regular cardinal; ${{\mathcal H}}(\chi)$ is the family of all sets hereditarily of size less than $\chi$. Moreover, we fix a well ordering $<^*_\chi$ of ${{\mathcal H}}(\chi)$.
2. For two sequences $\eta,\nu$ we write $\nu\vartriangleleft\eta$ whenever $\nu$ is a proper initial segment of $\eta$, and $\nu
\trianglelefteq\eta$ when either $\nu\vartriangleleft\eta$ or $\nu=\eta$. The length of a sequence $\eta$ is denoted by ${{\rm lh}\/}(\eta)$.
3. We will consider several games of two players. One player will be called [*Generic*]{} or [*Complete*]{} or just [*COM*]{}, and we will refer to this player as “she”. Her opponent will be called [*Antigeneric*]{} or [*Incomplete*]{} or just [*INC*]{} and will be referred to as “he”.
4. For a forcing notion ${{\mathbb P}}$, $\Gamma_{{\mathbb P}}$ stands for the canonical ${{\mathbb P}}$–name for the generic filter in ${{\mathbb P}}$. With this one exception, all ${{\mathbb P}}$–names for objects in the extension via ${{\mathbb P}}$ will be denoted with a tilde below (e.g., $\name{\tau}$, $\name{X}$). The weakest element of ${{\mathbb P}}$ will be denoted by $\emptyset_{{\mathbb P}}$ (and we will always assume that there is one, and that there is no other condition equivalent to it). We will also assume that all forcing notions under considerations are atomless.
By “$\lambda$–support iterations” we mean iterations in which domains of conditions are of size $\leq\lambda$. However, we will pretend that conditions in a $\lambda$–support iteration $\bar{{{\mathbb Q}}}=\langle{{\mathbb P}}_\zeta,
\name{{{\mathbb Q}}}_\zeta:\zeta<\zeta^*\rangle$ are total functions on $\zeta^*$ and for $p\in\lim(\bar{{{\mathbb Q}}})$ and $\alpha\in\zeta^*\setminus{{\rm Dom}}(p)$ we will let $p(\alpha)=\name{\emptyset}_{\name{{{\mathbb Q}}}_\alpha}$.
5. For a filter $D$ on $\lambda$, the family of all $D$–positive subsets of $\lambda$ is called $D^+$. (So $A\in D^+$ if and only if $A\subseteq
\lambda$ and $A\cap B\neq\emptyset$ for all $B\in D$.)
The club filter of $\lambda$ is denoted by ${{\mathcal D}}_\lambda$.
\[incon\]
1. $\lambda$ is a strongly inaccessible cardinal,
2. $\bar{\mu}=\langle\mu_\alpha:\alpha<\lambda\rangle$, each $\mu_\alpha$ is a regular cardinal satisfying (for $\alpha<\lambda$) $$\aleph_0\leq\mu_\alpha\leq\lambda\qquad\mbox{ and }\qquad \big(\forall f\in
{}^\alpha \mu_\alpha\big)\big(\big|\prod_{\xi<\alpha} f(\xi)\big|<
\mu_\alpha\big),$$
3. ${{\mathcal U}}$ is a normal filter on $\lambda$.
Preliminaries on $\lambda$–support iterations
=============================================
\[strcom\] Let ${{\mathbb P}}$ be a forcing notion.
1. For a condition $r\in{{\mathbb P}}$ let $\Game_0^\lambda({{\mathbb P}},r)$ be the following game of two players, [*Complete*]{} and [*Incomplete*]{}:
> the game lasts $\lambda$ moves and during a play the players construct a sequence $\langle (p_i,q_i): i<\lambda\rangle$ of pairs of conditions from ${{\mathbb P}}$ in such a way that $(\forall j<i<\lambda)(r\leq p_j
> \leq q_j\leq p_i)$ and at the stage $i<\lambda$ of the game, first Incomplete chooses $p_i$ and then Complete chooses $q_i$.
Complete wins if and only if for every $i<\lambda$ there are legal moves for both players.
2. We say that the forcing notion ${{\mathbb P}}$ is [*strategically $({<}\lambda)$–complete*]{} if Complete has a winning strategy in the game $\Game_0^\lambda({{\mathbb P}},r)$ for each condition $r\in{{\mathbb P}}$.
3. Let $N\prec ({{\mathcal H}}(\chi),\in,<^*_\chi)$ be a model such that ${}^{<\lambda} N\subseteq N$, $|N|=\lambda$ and ${{\mathbb P}}\in N$. We say that a condition $p\in{{\mathbb P}}$ is [*$(N,{{\mathbb P}})$–generic in the standard sense*]{} (or just: [*$(N,{{\mathbb P}})$–generic*]{}) if for every ${{\mathbb P}}$–name $\name{\tau}\in N$ for an ordinal we have $p{\Vdash}$“ $\name{\tau}\in N$ ”.
4. ${{\mathbb P}}$ is [*$\lambda$–proper in the standard sense*]{} (or just: [*$\lambda$–proper*]{}) if there is $x\in {{\mathcal H}}(\chi)$ such that for every model $N\prec ({{\mathcal H}}(\chi),\in,<^*_\chi)$ satisfying $${}^{<\lambda} N\subseteq N,\quad |N|=\lambda\quad\mbox{ and }\quad{{\mathbb P}},x
\in N,$$ and every condition $q\in N\cap{{\mathbb P}}$ there is an $(N,{{\mathbb P}})$–generic condition $p\in{{\mathbb P}}$ stronger than $q$.
\[[[@RoSh:777 Prop. A.1.4]]{}\] \[obsA.4\] Suppose that ${{\mathbb P}}$ is a $({<}\lambda)$–strategically complete (atomless) forcing notion, $\alpha^*<\lambda$ and $q_\alpha\in{{\mathbb P}}$ (for $\alpha<
\alpha^*$). Then there are conditions $p_\alpha\in{{\mathbb P}}$ (for $\alpha<
\alpha^*$) such that $q_\alpha\leq p_\alpha$ and for distinct $\alpha,\alpha'<\alpha^*$ the conditions $p_\alpha, p_{\alpha'}$ are incompatible.
\[[[@RoSh:777 Prop. A.1.6]]{}\] \[pA.6\] Suppose $\bar{{{\mathbb Q}}}=\langle{{\mathbb P}}_i,\name{{{\mathbb Q}}}_i: i<\gamma\rangle$ is a $\lambda$–support iteration and, for each $i<\gamma$, $${\Vdash}_{{{\mathbb P}}_i}\mbox{`` $\name{{{\mathbb Q}}}_i$ is strategically
$({<}\lambda)$--complete ''.}$$ Then, for each ${\varepsilon}\leq\gamma$ and $r\in{{\mathbb P}}_{\varepsilon}$, there is a winning strategy ${{\bf st}}({\varepsilon},r)$ of Complete in the game $\Game_0^\lambda({{\mathbb P}}_{\varepsilon},r)$ such that, whenever ${\varepsilon}_0<{\varepsilon}_1\leq
\gamma$ and $r\in{{\mathbb P}}_{{\varepsilon}_1}$, we have:
1. if $\langle (p_i,q_i):i<\lambda\rangle$ is a play of $\Game_0^\lambda({{\mathbb P}}_{{\varepsilon}_0},r{{\restriction}}{\varepsilon}_0)$ in which Complete follows the strategy ${{\bf st}}({\varepsilon}_0,r{{\restriction}}{\varepsilon}_0)$, then $\langle (p_i{{}^\frown\!}r{{\restriction}}[{\varepsilon}_0,{\varepsilon}_1),q_i{{}^\frown\!}r{{\restriction}}[{\varepsilon}_0,{\varepsilon}_1)):i<\lambda\rangle$ is a play of $\Game_0^\lambda({{\mathbb P}}_{{\varepsilon}_1},r)$ in which Complete uses ${{\bf st}}({\varepsilon}_1,r)$;
2. if $\langle (p_i,q_i):i<\lambda\rangle$ is a play of $\Game_0^\lambda({{\mathbb P}}_{{\varepsilon}_1},r)$ in which Complete plays according to the strategy ${{\bf st}}({\varepsilon}_1,r)$, then $\langle (p_i{{\restriction}}{\varepsilon}_0,q_i{{\restriction}}{\varepsilon}_0):i
<\lambda\rangle$ is a play of $\Game_0^\lambda({{\mathbb P}}_{{\varepsilon}_0},r{{\restriction}}{\varepsilon}_0)$ in which Complete uses ${{\bf st}}({\varepsilon}_0,r{{\restriction}}{\varepsilon}_0)$;
3. is ${\varepsilon}_1$ is limit and a sequence $\langle (p_i,q_i):
i<\lambda\rangle\subseteq{{\mathbb P}}_{{\varepsilon}_1}$ is such that for each $\xi<{\varepsilon}_1$, $\langle (p_i{{\restriction}}\xi,q_i{{\restriction}}\xi):i<\lambda\rangle$ is a play of $\Game_0^\lambda({{\mathbb P}}_\xi,r{{\restriction}}\xi)$ in which Complete uses the strategy ${{\bf st}}(\xi,r{{\restriction}}\xi)$, then $\langle (p_i,q_i):i<\lambda\rangle$ is a play of $\Game_0^\lambda({{\mathbb P}}_{{\varepsilon}_1},r)$ in which Complete plays according to ${{\bf st}}({\varepsilon}_1,r)$;
4. if $\langle (p_i,q_i):i<i^*\rangle$ is a partial play of $\Game_0^\lambda({{\mathbb P}}_{{\varepsilon}_1},r)$ in which Complete uses ${{\bf st}}({\varepsilon}_1,r)$ and $p'\in{{\mathbb P}}_{{\varepsilon}_0}$ is stronger than all $p_i{{\restriction}}{\varepsilon}_0$ (for $i<i^*$), then there is $p^*\in {{\mathbb P}}_{{\varepsilon}_1}$ such that $p'=p^*{{\restriction}}{\varepsilon}_0$ and $p^*\geq p_i$ for $i<i^*$.
\[[[@RoSh:777 Def. A.1.7]]{}, see also [[@Sh:587 A.3.3, A.3.2]]{}\] \[dA.5\]
1. Let $\alpha,\gamma$ be ordinals, $\emptyset\neq w\subseteq \gamma$. [*A standard $(w,\alpha)^\gamma$–tree*]{} is a pair ${{\mathcal T}}=(T,{{\rm rk}})$ such that
- ${{\rm rk}}:T\longrightarrow w\cup\{\gamma\}$,
- if $t\in T$ and ${{\rm rk}}(t)={\varepsilon}$, then $t$ is a sequence $\langle
(t)_\zeta: \zeta\in w\cap{\varepsilon}\rangle$, where each $(t)_\zeta$ is a sequence of length $\alpha$,
- $(T,{\vartriangleleft})$ is a tree with root $\langle\rangle$ and such that every chain in $T$ has a $\vartriangleleft$–upper bound it $T$.
We will keep the convention that ${{\mathcal T}}^x_y$ is $(T^x_y,{{\rm rk}}^x_y)$.
2. Let $\bar{{{\mathbb Q}}}=\langle{{\mathbb P}}_i,\name{{{\mathbb Q}}}_i:i<\gamma\rangle$ be a $\lambda$–support iteration. [*A standard tree of conditions in $\bar{{{\mathbb Q}}}$*]{} is a system $\bar{p}=\langle p_t:t\in T\rangle$ such that
- $(T,{{\rm rk}})$ is a standard $(w,\alpha)^\gamma$–tree for some $w\subseteq
\gamma$ and an ordinal $\alpha$,
- $p_t\in{{\mathbb P}}_{{{\rm rk}}(t)}$ for $t\in T$, and
- if $s,t\in T$, $s{\vartriangleleft}t$, then $p_s=p_t{{\restriction}}{{\rm rk}}(s)$.
3. Let $\bar{p}^0,\bar{p}^1$ be standard trees of conditions in $\bar{{{\mathbb Q}}}$, $\bar{p}^i=\langle p^i_t:t\in T\rangle$. We write $\bar{p}^0\leq \bar{p}^1$ whenever for each $t\in T$ we have $p^0_t\leq
p^1_t$.
\[pA.7\] Assume that $\bar{{{\mathbb Q}}}=\langle{{\mathbb P}}_i,\name{{{\mathbb Q}}}_i:i<\gamma\rangle$ is a $\lambda$–support iteration such that for all $i<\gamma$ we have $${\Vdash}_{{{\mathbb P}}_i}\mbox{`` $\name{{{\mathbb Q}}}_i$ is strategically
$({<}\lambda)$--complete ''.}$$
1. [@RoSh:777 Prop. A.1.9]Suppose that $\bar{p}=\langle
p_t:t\in T\rangle$ is a standard tree of conditions in $\bar{{{\mathbb Q}}}$, $|T|<\lambda$, and ${{\mathcal I}}\subseteq{{\mathbb P}}_\gamma$ is open dense. Then there is a standard tree of conditions $\bar{q}=\langle q_t:t\in T\rangle$ such that $\bar{p}\leq \bar{q}$ and $(\forall t\in T)({{\rm rk}}(t)=\gamma\
\Rightarrow\ q_t\in{{\mathcal I}})$.
2. If $\bar{p}=\langle p_t:t\in T\rangle$ is a standard tree of conditions in $\bar{{{\mathbb Q}}}$ and $|T|<\lambda$, then there is a standard tree of conditions $\bar{q}=\langle q_t:t\in T\rangle$ such that $\bar{p}\leq
\bar{q}$ and
- if $t_0,t_1\in T$, ${{\rm rk}}(t_0)={{\rm rk}}(t_1)$, $\xi\in{{\rm Dom}}(t_0)$ and $(t_0)_\xi\neq (t_1)_\xi$, $t_0{{\restriction}}\xi=t_1{{\restriction}}\xi$ then $$q_{t_0}{{\restriction}}\xi{\Vdash}_{{{\mathbb P}}_\xi}\mbox{`` the conditions }q_{t_0}(\xi),
q_{t_1}(\xi) \mbox{ are incompatible in }\name{{{\mathbb Q}}}_\xi\mbox{ ''.}$$
3. Suppose that
- $w\subseteq\lambda$, $|w|<\lambda$, $1<\mu\leq\lambda$ and $T=\bigcup\limits_{\xi\leq\gamma}\prod\limits_{\zeta\in w\cap \xi}\mu$ (so ${{\mathcal T}}=(T,{{\rm rk}})$ is a standard $(w,1)^\gamma$–tree),
- $\bar{p}=\langle p_t:t\in T\rangle$ is a standard tree of conditions in $\bar{{{\mathbb Q}}}$,
- for $\xi\in w$, $\name{{\varepsilon}}_\xi$ is a ${{\mathbb P}}_\xi$–name for a non-zero ordinal below $\mu$.
Then there are a standard $(w,1)^\gamma$–tree ${{\mathcal T}}'=(T',{{\rm rk}}')$ and a tree of conditions $\bar{q}=\langle q_t:t\in T'\rangle$ such that
- $T'\subseteq T$ and for every $t\in T'$ such that ${{\rm rk}}'(t)=\xi\in w$ the condition $q_t$ decides the value of $\name{{\varepsilon}}_\xi$, say $q_t
{\Vdash}\name{{\varepsilon}}_\xi={\varepsilon}^t_\xi$, and
- $p_t\leq q_t$ for $t\in T'$, and
- if $t\in T'$, ${{\rm rk}}(t)=\xi\in w$, then $$\big\{\alpha<\mu_\xi: t\cup\{\langle\xi,\alpha\rangle\}\in T'\big\}=
{\varepsilon}^t_\xi.$$
(2)Straightforward application of \[obsA.4\].
(3)Note that we cannot apply the first part directly, as the tree $T$ may be of size $\lambda$. So we will proceed inductively constructing initial levels of $T'$ of size $<\lambda$ and applying (1) to them.
For ${\varepsilon}\leq\gamma$ and $r\in{{\mathbb P}}_{\varepsilon}$ let ${{\bf st}}({\varepsilon},r)$ be the winning strategy of Complete in $\Game^\lambda_0({{\mathbb P}}_{\varepsilon},r)$ given by \[pA.6\] (so these strategies have the coherence properties listed there). Let $\langle\xi_\beta:\beta\leq\beta^*\rangle$ be the increasing enumeration of $w\cup\{\gamma\}$, $\beta^*<\lambda$. By induction on $\beta\leq\beta^*$ we will pick ${{\mathcal T}}_\beta,\bar{q}^\beta,\bar{r}^\beta$ and $\bar{{\varepsilon}}^\beta$ such that
1. ${{\mathcal T}}_\beta=(T_\beta,{{\rm rk}}_\beta)$ is a standard $(w\cap
\xi_\beta)^\gamma$–tree, $|T_\beta|<\lambda$, and $\bar{q}^\beta=\langle
q^\beta_t:t\in T_\beta\rangle$, $\bar{r}^\beta=\langle r^\beta_t:t\in
T_\beta\rangle$ are tree of conditions, $\bar{q}^\beta\leq\bar{r}^\beta$;
2. if $\beta_0<\beta_1\leq\beta^*$, then $T_{\beta_0}=\{t{{\restriction}}\xi_{\beta_0}:t\in T_{\beta_1}\}$ and $r^{\beta_0}_{t{{\restriction}}\xi_{\beta_0}}\leq
q^{\beta_1}_t{{\restriction}}\xi_{\beta_0}$ for $t\in T_{\beta_1}$;
3. if $\beta<\beta^*$, $t\in T_\beta$ and ${{\rm rk}}_\beta(t)=\gamma$ (so ${{\rm rk}}(t)=\xi_\beta$), then $$\langle \big(q^\alpha_{t{{\restriction}}\xi_\alpha}\!{{}^\frown\!}p_t{{\restriction}}[\xi_\alpha,
\xi_\beta),r^\alpha_{t{{\restriction}}\xi_\alpha}\!{{}^\frown\!}p_t{{\restriction}}[\xi_\alpha,\xi_\beta)
\big):\alpha<\beta\rangle^{\textstyle\frown}\langle \big(q^\alpha_t,
r^\alpha_t\big):\beta\leq\alpha<\beta^*\rangle$$ is a partial play of $\Game^\lambda_0({{\mathbb P}}_{\xi_\beta},p_t)$ in which Complete uses her winning strategy ${{\bf st}}(\xi_\beta,p_t)$;
4. $\bar{{\varepsilon}}^\beta=\langle{\varepsilon}^\beta_t:t\in T_\beta,\
{{\rm rk}}_\beta(t)=\gamma\rangle\subseteq\lambda$;
5. if $\beta<\beta^*$, $t\in T_\beta$ and ${{\rm rk}}_\beta(t)=\gamma$ (so ${{\rm rk}}(t)=\xi_\beta$), then $p_t\leq q^\beta_t\in{{\mathbb P}}_{\xi_\beta}$ and $q^\beta_t{\Vdash}_{{{\mathbb P}}_{\xi_\beta}}\name{{\varepsilon}}_{\xi_\beta}=
{\varepsilon}^\beta_t$;
6. if $\beta<\beta^*$, $t\in T_\beta$ and ${{\rm rk}}_\beta(t)=\gamma$, then $\big\{\alpha<\lambda:t\cup\{\langle\xi_\beta,\alpha\rangle\big\}\in
T_{\beta+1}\}={\varepsilon}_t^\beta$.
We let $T_0=\{\langle\rangle\}$ and we choose $q^0_{\langle\rangle}\in
{{\mathbb P}}_{\xi_0}$ and ${\varepsilon}^0_{\langle\rangle}$ so that $p_{\langle\rangle}\leq
q^0_{\langle\rangle}$ and $q^0_{\langle\rangle}{\Vdash}_{{{\mathbb P}}_{\xi_0}}
\name{{\varepsilon}}_{\xi_0}={\varepsilon}^0_{\langle\rangle}$. Then we let $r^0_{\langle
\rangle}$ be the answer given by ${{\bf st}}(\xi_0,p_{\langle\rangle})$. Now suppose that we have defined ${{\mathcal T}}_\alpha,\bar{q}^\alpha,\bar{r}^\alpha$ and $\bar{{\varepsilon}}^\alpha$ for $\alpha<\beta\leq\beta^*$.
If $\beta$ is a limit ordinal then the demands (a) and (b) uniquely define the standard tree ${{\mathcal T}}_\beta$. It follows from the choice of ${{\bf st}}({\varepsilon},r)$ (see clause \[pA.6\](iii)) and demand (c) at previous stages that
1. if $t\in T_\beta$, ${{\rm rk}}_\beta(t)=\gamma$ (so ${{\rm rk}}(t)=\xi_\beta$), then the sequence $$\big\langle \big(q^\alpha_{t{{\restriction}}\xi_\alpha}{{}^\frown\!}p_t{{\restriction}}[\xi_\alpha,
\xi_\beta),r^\alpha_{t{{\restriction}}\xi_\alpha}{{}^\frown\!}p_t{{\restriction}}[\xi_\alpha,\xi_\beta)
\big):\alpha<\beta\big\rangle$$ is a partial play of $\Game^\lambda_0({{\mathbb P}}_{\xi_\beta},p_t)$ in which Complete uses her winning strategy ${{\bf st}}(\xi_\beta,p_t)$.
For $t\in T_\beta$ we define a condition $q_t\in {{\mathbb P}}_{\xi_\beta}$ as follows:
- ${{\rm Dom}}(q_t)=\bigcup\limits_{\alpha<\beta}{{\rm Dom}}(r^\alpha_{t{{\restriction}}\xi_\alpha})\cup{{\rm Dom}}(p_t)\subseteq{{\rm rk}}(t)$,
- if $\zeta\in{{\rm Dom}}(q_t)$, then $q_t(\zeta)$ is the $<^*_\chi$–first ${{\mathbb P}}_\zeta$–name for an element of $\name{{{\mathbb Q}}}_\zeta$ such that $$\begin{array}{r}
q_t{{\restriction}}\zeta{\Vdash}_{{{\mathbb P}}_\zeta}\mbox{`` if the set }\{r^\alpha_{t{{\restriction}}\xi_\alpha}(\zeta):\zeta<\xi_\alpha\ \&\ \alpha<\beta\}\cup\{p_t(\zeta)\}
\mbox{ has an upper bound, }\\
\mbox{\ \ then $q_t(\zeta)$ is such an upper bound ''.}
\end{array}$$
It follows from $(\oplus)_\beta$ (and \[pA.6\](iv)) that $p_t\leq q_t$ and $r^\alpha_{t{{\restriction}}\xi_\alpha}\leq q_t{{\restriction}}\xi_{\alpha+1}$ for $\alpha<\beta$. Now, by “the $<^*_\chi$–first”, clearly $\bar{q}=\langle
q_t:t\in T_\beta\rangle$ is a tree of conditions. Applying \[pA.7\](1) we may choose a tree of conditions $\bar{q}^\beta=\langle q^\beta_t:t\in
T_\beta\rangle$ such that $\bar{q}\leq\bar{q}^\beta$ and
- if $\beta<\beta^*$, $t\in T_\beta$ and ${{\rm rk}}_\beta(t)=\gamma$, then the condition $q^\beta_t$ decides the value of $\name{{\varepsilon}}_{\xi_\beta}$ (and let $q^\beta_t{\Vdash}\name{{\varepsilon}}_{\xi_\beta}={\varepsilon}^\beta_t$) and $q^\beta_t
\in {{\mathbb P}}_{\xi_\beta}$.
Then, for $t\in T_\beta$, we let $r^\beta_t$ be the answer given to Complete by ${{\bf st}}({{\rm rk}}(t),p_t)$ in the appropriate partial play of $\Game^\lambda_0(
{{\mathbb P}}_{{{\rm rk}}(t)},p_t)$, where at stage $\beta$ Incomplete put $q^\beta_t$ (see (c), $(\oplus)_\beta$). It follows from \[pA.6\](ii) that $\bar{r}^\beta=
\langle r^\beta_t:t\in T_\beta\rangle$ is a tree of conditions. Plainly, ${{\mathcal T}}_\beta,\bar{q}^\beta,\bar{r}^\beta$ and $\bar{{\varepsilon}}^\beta$ satisfy all relevant (restrictions of the) demands (a)–(f).
Now suppose that $\beta$ is a successor ordinal, say $\beta=\beta_0+1$. Let $$T_\beta=T_{\beta_0}\cup\big\{t\cup\{\langle\xi_{\beta_0},{\varepsilon}\rangle\}:
t\in T_{\beta_0}\ \&\ {{\rm rk}}_{\beta_0}(t)=\gamma\ \&\ {\varepsilon}<{\varepsilon}^{\beta_0}_t
\big\}$$ and for $t\in T_\beta$ define $q_t$ as follows:
- if $t\in T_{\beta_0}$, then $q_t=r^{\beta_0}_t$,
- if $t\in T_\beta\setminus T_{\beta_0}$, then $q_t= r^{\beta_0}_{t{{\restriction}}\xi_{\beta_0}}{{}^\frown\!}p_t{{\restriction}}[\xi_{\beta_0},\xi_\beta)$.
Then $\bar{q}=\langle q_t:t\in T_\beta\rangle$ is a tree of conditions, $r^{\beta_0}_t\leq q_t$ for $t\in T_{\beta_0}$. It follows from \[pA.7\](1) that we may choose a tree of conditions $\bar{q}^\beta=\langle
q^\beta_t:t\in T_\beta\rangle$ such that $\bar{q}\leq\bar{q}^\beta$ and
- if $\beta<\beta^*$, $t\in T_\beta$ and ${{\rm rk}}_\beta(t)=\gamma$, then the condition $q^\beta_t$ decides $\name{{\varepsilon}}_{\xi_\beta}$ and, say, $q^\beta_t{\Vdash}\name{{\varepsilon}}_{\xi_\beta}={\varepsilon}^\beta_t$.
Next, like in the limit case, $\bar{r}^\beta=\langle r^\beta_t:t\in
T_\beta\rangle$ is obtained by applying the strategies ${{\bf st}}({{\rm rk}}(t),p_t)$ suitably. Easily, ${{\mathcal T}}_\beta,\bar{q}^\beta,\bar{r}^\beta$ and $\bar{{\varepsilon}}^\beta$ satisfy the demands (a)–(f).
After the inductive construction is carried out look at $T_{\beta^*}$, $\bar{q}^{\beta^*}$ and $\langle\bar{{\varepsilon}}^\beta:\beta<\beta^*\rangle$.
ABC of reasonable completeness
==============================
Note that if ${{\mathbb Q}}$ is strategically $({<}\lambda)$–complete and ${{\mathcal U}}$ is a normal filter on $\lambda$, then the normal filter generated by ${{\mathcal U}}$ in ${{\bf V}}^{{{\mathbb Q}}}$ is proper. Abusing notation, we may denote the normal filter generated by ${{\mathcal U}}$ in ${{\bf V}}^{{\mathbb Q}}$ also by ${{\mathcal U}}$ or by ${{\mathcal U}}^{{\mathbb Q}}$. Thus if $\name{A}$ is a ${{\mathbb Q}}$–name for a subset of $\lambda$, then $p{\Vdash}_{{\mathbb Q}}\name{A}\in{{\mathcal U}}^{{\mathbb Q}}$ if and only if for some ${{\mathbb Q}}$–names $\name{A}_\alpha$ for elements of ${{\mathcal U}}^{{\bf V}}$ we have that $p{\Vdash}_{{\mathbb Q}}\mathop{\triangle}\limits_{\alpha<\lambda}\name{A}_\alpha\subseteq\name{A}$.
\[p.1A\] Let ${{\mathbb Q}}$ be a strategically $({<}\lambda)$–complete forcing notion.
1. For a condition $p\in{{\mathbb Q}}$ we define a game ${{\Game^{\rm rcA}_{\bar{\mu}}}}(p,{{\mathbb Q}})$ between two players, Generic and Antigeneric, as follows. A play of ${{\Game^{\rm rcA}_{\bar{\mu}}}}(p,{{\mathbb Q}})$ lasts $\lambda$ steps and during a play a sequence $$\Big\langle I_\alpha,\langle p^\alpha_t,q^\alpha_t:t\in I_\alpha\rangle:
\alpha<\lambda\Big\rangle$$ is constructed. Suppose that the players have arrived to a stage $\alpha<
\lambda$ of the game. Now,
1. first Generic chooses a non-empty set $I_\alpha$ of cardinality $<\mu_\alpha$ and a system $\langle p^\alpha_t:t\in I_\alpha
\rangle$ of conditions from ${{\mathbb Q}}$,
2. then Antigeneric answers by picking a system $\langle q^\alpha_t:t\in I_\alpha\rangle$ of conditions from ${{\mathbb Q}}$ such that $(\forall t\in I_\alpha)(p^\alpha_t\leq q^\alpha_t)$.
At the end, Generic wins the play $$\Big\langle I_\alpha,\langle p^\alpha_t,q^\alpha_t:t\in I_\alpha\rangle:
\alpha<\lambda\Big\rangle$$ of ${{\Game^{\rm rcA}_{\bar{\mu}}}}(p,{{\mathbb Q}})$ if and only if
1. there is a condition $p^*\in{{\mathbb Q}}$ stronger than $p$ and such that[^2] $$p^*{\Vdash}_{{{\mathbb Q}}}\mbox{`` }\big\{\alpha<\lambda:\big(\exists t\in I_\alpha
\big)\big(q^\alpha_t\in\Gamma_{{{\mathbb Q}}}\big)\big\}=\lambda\mbox{ ''}.$$
2. Games ${{\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb Q}}),{{\Game^{\rm rcC}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb Q}})$ are defined similarly, except that the winning criterion $(\circledast)^{\rm rc}_{\rm A}$ is replaced by
1. there is a condition $p^*\in{{\mathbb Q}}$ stronger than $p$ and such that $$p^*{\Vdash}_{{{\mathbb Q}}}\mbox{`` }\big\{\alpha<\lambda:\big(\exists t\in I_\alpha
\big)\big(q^\alpha_t\in\Gamma_{{{\mathbb Q}}}\big)\big\}\in {{\mathcal U}}^{{\mathbb Q}}\mbox{ ''},$$
2. there is a condition $p^*\in{{\mathbb Q}}$ stronger than $p$ and such that $$p^*{\Vdash}_{{{\mathbb Q}}}\mbox{`` }\big\{\alpha<\lambda:\big(\exists t\in I_\alpha
\big)\big(q^\alpha_t\in\Gamma_{{{\mathbb Q}}}\big)\big\}\in \big({{\mathcal U}}^{{\mathbb Q}}\big)^+
\mbox{ ''},$$
respectively.
3. For a condition $p\in{{\mathbb Q}}$ we define a game ${{\Game^{{\rm rc}{\bf b}}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb Q}})$ between Generic and Antigeneric as follows. A play of ${{\Game^{{\rm rc}{\bf b}}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb Q}})$ lasts $\lambda$ steps and during a play a sequence $$\Big\langle \zeta_\alpha,\langle p^\alpha_\xi,q^\alpha_\xi:\xi<
\zeta_\alpha\rangle:\alpha<\lambda\Big\rangle$$ is constructed. Suppose that the players have arrived to a stage $\alpha<
\lambda$ of the game. Now, Generic chooses a non-zero ordinal $\zeta_\alpha<\mu_\alpha$ and then the two players play a subgame of length $\zeta_\alpha$ alternatively choosing successive terms of a sequence $\langle p^\alpha_\xi,q^\alpha_\xi:\xi<\zeta_\alpha\rangle$. At a stage $\xi<\zeta_\alpha$ of the subgame, first Generic picks a condition $p^\alpha_\xi\in{{\mathbb Q}}$ and then Antigeneric answers with a condition $q^\alpha_\xi$ stronger than $p^\alpha_\xi$.
At the end, Generic wins the play $$\Big\langle \zeta_\alpha,\langle p^\alpha_\xi,q^\alpha_\xi:\xi<
\zeta_\alpha\rangle:\alpha<\lambda\Big\rangle$$ of ${{\Game^{{\rm rc}{\bf b}}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb Q}})$if and only if
1. there is a condition $p^*\in{{\mathbb Q}}$ stronger than $p$ and such that $$p^*{\Vdash}_{{{\mathbb Q}}}\mbox{`` }\big\{\alpha<\lambda:\big(\exists
\xi<\zeta_\alpha\big)\big(q^\alpha_\xi\in\Gamma_{{{\mathbb Q}}}\big)\big\}\in
{{\mathcal U}}^{{\mathbb Q}}\mbox{ ''}.$$
4. Games ${{\Game^{{\rm rc}{\bf a}}_{\bar{\mu}}}}(p,{{\mathbb Q}})$ and ${{\Game^{{\rm rc}{\bf c}}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb Q}})$ are defined similarly except that the winning criterion $(\circledast)^{\rm rc}_{\bf b}$ is changed so that “$\in {{\mathcal U}}^{{\mathbb Q}}$” is replaced by “$=\lambda$” or “$\in
\big({{\mathcal U}}^{{\mathbb Q}}\big)^+$”, respectively.
5. We say that a forcing notion ${{\mathbb Q}}$ is [*reasonably A–bounding over $\bar{\mu}$*]{} if
1. ${{\mathbb Q}}$ is strategically $({<}\lambda)$–complete, and
2. for any $p\in{{\mathbb Q}}$, Generic has a winning strategy in the game ${{\Game^{\rm rcA}_{\bar{\mu}}}}(p,{{\mathbb Q}})$.
In an analogous manner we define when the forcing notion ${{\mathbb Q}}$ is [*reasonably X–bounding over ${{\mathcal U}},\bar{\mu}$*]{} (for ${\rm X}\in\{{\rm B},
{\rm C}, {\bf a}, {\bf b}, {\bf c}\}$) — just using the game $\Game^{\rm
rcX}_{{{\mathcal U}},\bar{\mu}}(p,{{\mathbb Q}})$ appropriately.
If $\mu_\alpha=\lambda$ for each $\alpha<\lambda$, then we may omit $\bar{\mu}$ and say [*reasonably B–bounding over ${{\mathcal U}}$*]{} etc. If ${{\mathcal U}}$ is the filter generated by club subsets of $\lambda$, we may omit it as well.
6. Let ${{\bf st}}$ be a strategy for Generic in the game ${{\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb Q}})$. We will say that a sequence $\Big\langle I_\alpha,\langle p^\alpha_t,
q^\alpha_t:t\in I_\alpha\rangle:\delta<\alpha<\lambda\Big\rangle$ is [ *a $\delta$–delayed play according to ${{\bf st}}$*]{} if it has an extension $\Big
\langle I_\alpha,\langle p^\alpha_t,q^\alpha_t:t\in I_\alpha\rangle:\alpha<
\lambda\Big\rangle$ which is a play agreeing with ${{\bf st}}$ and such that $p^\alpha_t=q^\alpha_t$ for $\alpha\leq\delta$, $t\in I_\alpha$.
If ${{\bf st}}$ is a winning strategy for Generic in the game ${{\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb Q}})$, and $\bar{\sigma}=\Big\langle I_\alpha,\langle p^\alpha_t, q^\alpha_t:t
\in I_\alpha\rangle:\delta\leq\alpha<\lambda\Big\rangle$ is a $\delta$–delayed play according to ${{\bf st}}$, then $\bar{\sigma}$ satisfies the condition $(\circledast)^{\rm rc}_{\rm B}$.
For ${{\mathcal U}},\bar{\mu}$ as in \[incon\], $X\in\{A,B,C,{\bf a},{\bf b},{\bf
c}\}$ and a forcing notion ${{\mathbb Q}}$, let $\Phi({{\mathbb Q}},X,{{\mathcal U}},\bar{\mu})$ be the statement
“${{\mathbb Q}}$ is reasonably $X$–bounding over ${{\mathcal U}},\bar{\mu}$”.
Then the following implications hold
$$\begin{array}{ccccccc}
\Phi({{\mathbb Q}},A,\bar{\mu})& \Rightarrow & \Phi({{\mathbb Q}},B,{{\mathcal U}},\bar{\mu}) &
\Rightarrow & \Phi({{\mathbb Q}},C,{{\mathcal U}},\bar{\mu}) & & \\
\Downarrow & & \Downarrow & & \Downarrow & & \\
\Phi({{\mathbb Q}},{\bf a},\bar{\mu})& \Rightarrow & \Phi({{\mathbb Q}},{\bf b},{{\mathcal U}},
\bar{\mu}) & \Rightarrow & \Phi({{\mathbb Q}},{\bf c},{{\mathcal U}},\bar{\mu})& \Rightarrow &
{{\mathbb Q}}\mbox{ is $\lambda$--proper.}
\end{array}$$
\[verB\] Assume that $\lambda,{{\mathcal U}},\bar{\mu}$ are as in \[incon\] and $\bar{{{\mathbb Q}}}=
\langle{{\mathbb P}}_\xi,\name{{{\mathbb Q}}}_\xi:\xi<\gamma\rangle$ is a $\lambda$–support iteration such that for every $\xi<\gamma$, $${\Vdash}_{{{\mathbb P}}_\xi}\mbox{`` $\name{{{\mathbb Q}}}_\xi$ is reasonably B--bounding
over ${{\mathcal U}},\bar{\mu}$ ''.}$$ Then ${{\mathbb P}}_{\gamma}=\lim(\bar{{{\mathbb Q}}})$ is reasonably ${\bf b}$–bounding over ${{\mathcal U}},\bar{\mu}$ (and so also $\lambda$–proper).
For each $\xi<\gamma$ pick a ${{\mathbb P}}_\xi$–name $\name{{{\bf st}}}^0_\xi\in N$ such that $$\begin{array}{r}
{\Vdash}_{{{\mathbb P}}_\xi}\mbox{`` }\name{{{\bf st}}}^0_\xi\mbox{ is a winning strategy
for Complete in }\Game_0^\lambda\big(\name{{{\mathbb Q}}}_\xi,
\name{\emptyset}_{\name{{{\mathbb Q}}}_\xi}\big)\mbox{ such that }\ \\
\mbox{ if Incomplete plays $\name{\emptyset}_{\name{{{\mathbb Q}}}_\xi}$ then
Complete answers with $\name{\emptyset}_{\name{{{\mathbb Q}}}_\xi}$ as well ''.}
\end{array}$$ Also, for $\xi\leq\gamma$ and $r\in{{\mathbb P}}_\xi$, let ${{\bf st}}(\xi,r)$ be a winning strategy of Complete in $\Game_0^\lambda({{\mathbb P}}_\xi,r)$ with the coherence properties given in \[pA.6\].
We are going to describe a strategy ${{\bf st}}$ for Generic in the game ${{\Game^{{\rm rc}{\bf b}}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb P}}_\gamma)$. In the course of the play, at a stage $\delta<\lambda$, Generic will be instructed to construct aside
1. ${{\mathcal T}}_\delta,\bar{p}^\delta_*,
\bar{q}^\delta_*,r^-_\delta, r_\delta,w_\delta$, $\langle
\name{{\varepsilon}}_{\delta,\xi},\name{\bar{p}}_{\delta,\xi},
\name{\bar{q}}_{\delta,\xi}:\xi\in w_\delta\rangle$, and $\name{{{\bf st}}}_\xi$ for $\xi\in w_{\delta+1}\setminus w_\delta$.
These objects will be chosen so that if $$\big\langle\zeta_\delta,\langle p^\delta_\zeta,q^\delta_\zeta:\zeta<
\zeta_\delta\rangle:\delta<\lambda\big\rangle$$ is a play of ${{\Game^{{\rm rc}{\bf b}}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb P}}_\gamma)$ in which Generic follows ${{\bf st}}$, and the objects constructed at stage $\delta<\lambda$ are listed in $(\otimes)_\delta$, then the following conditions are satisfied (for each $\delta<\lambda$).
1. $r^-_\delta,r_\delta\in {{\mathbb P}}_\gamma$, $r_0(0)=p(0)$, $w_\delta\subseteq\gamma$, $|w_\delta|=|\delta|+1$, $\bigcup\limits_{\alpha<
\lambda}{{\rm Dom}}(r_\alpha)=\bigcup\limits_{\alpha<\lambda} w_\alpha$, $w_0=\{0\}$, $w_\delta\subseteq w_{\delta+1}$ and if $\delta$ is limit then $w_\delta=\bigcup\limits_{\alpha<\delta} w_\alpha$.
2. For each $\alpha<\delta<\lambda$ we have $(\forall\xi\in
w_{\alpha+1})(r_\alpha(\xi)=r_\delta(\xi))$ and $p\leq r_\alpha^-\leq
r_\alpha\leq r^-_\delta\leq r_\delta$.
3. If $\xi\in\gamma\setminus w_\delta$, then $$\begin{array}{ll}
r_\delta{{\restriction}}\xi{\Vdash}&\mbox{`` the sequence }\langle r^-_\alpha(\xi),
r_\alpha(\xi):\alpha\leq\delta\rangle\mbox{ is a legal partial play of }\\
&\quad\Game_0^\lambda\big(\name{{{\mathbb Q}}}_\xi,\name{\emptyset}_{\name{{{\mathbb Q}}}_\xi}
\big)\mbox{ in which Complete follows }\name{{{\bf st}}}^0_\xi\mbox{ ''}
\end{array}$$ and if $\xi\in w_{\delta+1}\setminus w_\delta$, then $\name{{{\bf st}}}_\xi$ is a ${{\mathbb P}}_\xi$–name for a winning strategy of Generic in ${{\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}}}}(r_\delta(\xi),\name{{{\mathbb Q}}}_\xi)$ such that if $\langle p^\alpha_t:
t\in I_\alpha\rangle$ is given by that strategy to Generic at stage $\alpha$, then $I_\alpha$ is an ordinal below $\mu_\alpha$. (And ${{\bf st}}_0$ is a suitable winning strategy of Generic in ${{\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}}}}(p(0),{{\mathbb Q}}_0)$.)
4. ${{\mathcal T}}_\delta=(T_\delta,{{\rm rk}}_\delta)$ is a standard $(w_\delta,
1)^\gamma$–tree, $|T_\delta|<\mu_\delta$.
5. $\bar{p}^\delta_*=\langle p^\delta_{*,t}:t\in T_\delta
\rangle$ and $\bar{q}^\delta_*=\langle q^\delta_{*,t}:t\in T_\delta\rangle$ are standard trees of conditions, $\bar{p}^\delta_*\leq\bar{q}^\delta_*$.
6. For $t\in T_\delta$ we have $\big(\bigcup\limits_{\alpha<
\delta}{{\rm Dom}}(r_\alpha)\cup w_\delta\big)\cap{{\rm rk}}_\delta(t)\subseteq
{{\rm Dom}}(p^\delta_{*,t})$ and for each $\xi\in {{\rm Dom}}(p^\delta_{*,t})\setminus
w_\delta$: $$\begin{array}{ll}
p^\delta_{*,t}{{\restriction}}\xi{\Vdash}&\mbox{`` if the set }\{r_\alpha(\xi):\alpha<
\delta\}\mbox{ has an upper bound in }\name{{{\mathbb Q}}}_\xi,\\
&\mbox{\quad then $p^\delta_{*,t}(\xi)$ is such an upper bound ''.}
\end{array}$$
7. $\zeta_\delta=|\{t\in T_\delta:{{\rm rk}}_\delta(t)=\gamma\}|$ and for some enumeration $\{t\in T_\delta:{{\rm rk}}_\delta(t)=\gamma\}=\{t_\zeta:
\zeta<\zeta_\delta\}$, for each $\zeta<\zeta_\delta$ we have $$p^\delta_{*,t_\zeta}\leq p^\delta_\zeta\leq q^\delta_\zeta\leq
q^\delta_{*,t_\zeta}.$$
8. If $\xi\in w_\delta$, then $\name{{\varepsilon}}_{\delta,\xi}$ is a ${{\mathbb P}}_\xi$–name for an ordinal below $\mu_\delta$, $\name{\bar{p}}_{\delta,\xi},\name{\bar{q}}_{\delta,\xi}$ are ${{\mathbb P}}_\xi$–names for sequences of conditions in $\name{{{\mathbb Q}}}_\xi$ of length $\name{{\varepsilon}}_{\delta,\xi}$.
9. If $\xi\in w_{\beta+1}\setminus w_\beta$, $\beta<\lambda$, then $$\begin{array}{r}
{\Vdash}_{{{\mathbb P}}_\xi}\mbox{`` }\langle\name{{\varepsilon}}_{\alpha,\xi},
\name{\bar{p}}_{\alpha,\xi},\name{\bar{q}}_{\alpha,\xi}:\beta<\alpha<
\lambda\rangle\mbox{ is a delayed play of }{{\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}}}}(r_\beta(\xi),
\name{{{\mathbb Q}}}_\xi)\\
\mbox{ in which Generic uses $\name{{{\bf st}}}_\xi$ ''.}
\end{array}$$
10. If $t\in T_\delta$, ${{\rm rk}}_\delta(t)=\xi<\gamma$, then the condition $p^\delta_{*,t}$ decides the value of $\name{{\varepsilon}}_{\delta,
\xi}$, say $p^\delta_{*,t}{\Vdash}$“$\name{{\varepsilon}}_{\delta,\xi}=
{\varepsilon}^t_{\delta,\xi}$”, and $\{(s)_\xi:t\vartriangleleft s\in T_\delta\}=
{\varepsilon}^t_{\delta,\xi}$ and $$q^\delta_{*,t}{\Vdash}_{{{\mathbb P}}_\xi}\mbox{`` } \name{\bar{p}}_{\delta,\xi}
({\varepsilon})\leq p^\delta_{*,t{{}^\frown\!}\langle{\varepsilon}\rangle}(\xi)\mbox{ for }{\varepsilon}<
{\varepsilon}^t_{\delta,\xi}\mbox{ and }\name{\bar{q}}_{\delta,\xi}=\langle
q^\delta_{*,s}(\xi):t\vartriangleleft s\in T_\delta\rangle\mbox{ ''.}$$
11. If $t_0,t_1\in T_\delta$, ${{\rm rk}}_\delta(t_0)=
{{\rm rk}}_\delta(t_1)$ and $\xi\in w_\delta\cap{{\rm rk}}_\delta(t_0)$, $t_0{{\restriction}}\xi=t_1{{\restriction}}\xi$ but $\big(t_0\big)_\xi\neq \big(t_1\big)_\xi$, then $$q^\delta_{*,t_0{{\restriction}}\xi}{\Vdash}_{{{\mathbb P}}_\xi}\mbox{`` the conditions
$q^\delta_{*,t_0}(\xi),q^\delta_{*,t_1}(\xi)$ are incompatible ''.}$$
12. ${{\rm Dom}}(r_\delta)=\bigcup\limits_{t\in T_\delta}
{{\rm Dom}}(q^\delta_{*,t})\cup{{\rm Dom}}(p)$ and if $t\in T_\delta$, $\xi\in{{\rm Dom}}(r_\delta)\cap {{\rm rk}}_\delta(t)\setminus w_\delta$, and $q^\delta_{*,t}{{\restriction}}\xi\leq q\in{{\mathbb P}}_\xi$, $r_\delta{{\restriction}}\xi\leq q$, then $$\begin{array}{ll}
q{\Vdash}_{{{\mathbb P}}_\xi}&\mbox{`` if the set }\{r_\alpha(\xi):\alpha<\delta\}
\cup\{q^\delta_{*,t}(\xi), p(\xi)\}\mbox{ has an upper bound in }
\name{{{\mathbb Q}}}_\xi,\\
&\mbox{\quad then $r_\delta(\xi)$ is such an upper bound ''.}
\end{array}$$
To describe the instructions given by ${{\bf st}}$ at stage $\delta<\lambda$ of a play of ${{\Game^{{\rm rc}{\bf b}}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb P}}_\gamma)$ let us assume that $$\big\langle\zeta_\alpha,\langle p^\alpha_\zeta,q^\alpha_\zeta:\zeta<
\zeta_\alpha\rangle:\alpha<\delta\big\rangle$$ is the result of the play so far and that Generic constructed objects listed in $(\otimes)_\alpha$ (for $\alpha<\delta$) with properties $(*)_1$–$(*)_{12}$.
First, Generic uses her favourite bookkeeping device to determine $w_\delta$ such that the demands in $(*)_1$ are satisfied (and that at the end we will have $\bigcup\limits_{\alpha<\lambda}{{\rm Dom}}(r_\alpha)=\bigcup\limits_{\alpha<
\lambda}w_\alpha$). Now Generic lets ${{\mathcal T}}_\delta'$ be a standard $(w_\delta,1)^\gamma$–tree such that for each $\xi\in w_\delta\cup
\{\gamma\}$ we have $\{t\in T_\delta':{{\rm rk}}_\delta'(t)=\xi\}=
\prod\limits_{{\varepsilon}\in w_\delta\cap\xi}\mu_\delta$. Then for $\xi\in
w_\delta$ she chooses ${{\mathbb P}}_\xi$–names $\name{{\varepsilon}}_{\delta,\xi},
\name{\bar{p}}_{\delta,\xi}$ such that $\name{{\varepsilon}}_{\delta,\xi}$ is a name for an ordinal below $\mu_\delta$ and $\name{\bar{p}}_{\delta,\xi}$ is a name for a sequence of conditions in $\name{{{\mathbb Q}}}_\xi$ of length $\name{{\varepsilon}}_{\delta,\xi}$ and $$\begin{array}{ll}
{\Vdash}_{{{\mathbb P}}_\xi}&\mbox{`` }\name{{\varepsilon}}_{\delta,\xi},
\name{\bar{p}}_{\delta,\xi} \mbox{ is the answer to the delayed play }\\
&\quad\langle\name{{\varepsilon}}_{\alpha,\xi},\name{\bar{p}}_{\alpha,\xi},
\name{\bar{q}}_{\alpha,\xi}:\xi\in w_\alpha\ \&\ \alpha<\delta\rangle
\mbox{ given to Complete by }\name{{{\bf st}}}_\xi\mbox{ ''}.
\end{array}$$ She lets $\bar{p}^{\delta,0}_*=\langle p^{\delta,0}_{*,t}:t\in T'_\delta
\rangle$ be a tree of conditions defined so that ${{\rm Dom}}(p^{\delta,0}_{*,t})=
\big(\bigcup\limits_{\alpha<\delta}{{\rm Dom}}(r_\alpha)\cup w_\delta\big)\cap
{{\rm rk}}_\delta'(t)$ and for each $\xi\in {{\rm Dom}}(p^{\delta,0}_{*,t})$
1. $p^{\delta,0}_{*,t}(\xi)$ is the $<^*_\chi$–first ${{\mathbb P}}_\xi$–name for an element of $\name{{{\mathbb Q}}}_\xi$ such that
- if $\xi\in w_\delta$, then $${\Vdash}_{{{\mathbb P}}_\xi}\mbox{`` if }(t)_\xi<\name{{\varepsilon}}_{\delta,\xi}\mbox{
then }p^{\delta,0}_{*,t}(\xi)=\name{\bar{p}}_{\delta,\xi}\big((t)_\xi\big),
\mbox{ otherwise }p^{\delta,0}_{*,t}(\xi)=\name{\emptyset}_{\name{{{\mathbb Q}}}_\xi}
\mbox{ '',}$$
- if $\xi\notin w_\delta$, then $$\begin{array}{ll}
{\Vdash}_{{{\mathbb P}}_\xi}&\mbox{`` if the set }\{r_\alpha(\xi):\alpha<\delta\}
\mbox{ has an upper bound in }\name{{{\mathbb Q}}}_\xi,\\
&\mbox{\quad then $p^{\delta,0}_{*,t}(\xi)$ is such an upper bound ''.}
\end{array}$$
Now Generic uses \[pA.7\](3) and then \[pA.7\](2) to choose a standard tree $(w_\delta,1)^\gamma$–tree ${{\mathcal T}}_\delta=(T_\delta,{{\rm rk}}_\delta)$ and a tree of conditions $\bar{p}^\delta_*=\langle p^\delta_{*,t}:t\in T_\delta
\rangle$ such that
1. $T_\delta\subseteq T_\delta'$ and for every $t\in
T_\delta$ such that ${{\rm rk}}_\delta(t)=\xi\in w_\delta$ the condition $p^\delta_{*,t}$ decides the value of $\name{{\varepsilon}}_{\delta,\xi}$, say $p^\delta_{*,t}{\Vdash}\name{{\varepsilon}}_{\delta,\xi}={\varepsilon}^t_{\delta,\xi}$, and
2. if $t\in T_\delta$, ${{\rm rk}}_\delta(t)=\xi\in
w_\delta$, then $\{\alpha<\lambda:t\cup\{\langle\xi,\alpha\rangle\}\in
T_\delta\}={\varepsilon}^t_{\delta,\xi}$, and
3. $p^{\delta,0}_{*,t}\leq p^\delta_{*,t}$ for all $t\in T_\delta$, and if $t_0,t_1\in T_\delta$, ${{\rm rk}}_\delta(t_0)=
{{\rm rk}}_\delta(t_1)$, $\xi\in{{\rm Dom}}(t_0)$, and $t_0{{\restriction}}\xi=t_1{{\restriction}}\xi$ but $(t_0)_\xi\neq (t_1)_\xi$, then $$p^\delta_{*,t_0{{\restriction}}\xi}{\Vdash}_{{{\mathbb P}}_\xi}\mbox{`` the conditions }
p^\delta_{*,t_0}(\xi),p^\delta_{*,t_1}(\xi)\mbox{ are incompatible in
$\name{{{\mathbb Q}}}_\xi$ '',}$$
Thus Generic has written aside ${{\mathcal T}}_\delta$, $\bar{p}^\delta_*$, $w_\delta$ and $\langle \name{{\varepsilon}}_{\delta,\xi},\name{\bar{p}}_{\delta,\xi}:\xi\in
w_\delta\rangle$. (It should be clear that they satisfy the demands in $(*)_1$, $(*)_4$–$(*)_6$, $(*)_8$ and $(*)_9,(*)_{10}$.) Now she turns to the play of ${{\Game^{{\rm rc}{\bf b}}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb P}}_\gamma)$ and she puts $$\zeta_\delta=|\{t\in T_\delta:{{\rm rk}}_\delta(t)=\gamma\}|$$ and she also picks an enumeration $\langle t_\zeta:\zeta<\zeta_\delta
\rangle$ of $\{t\in T_\delta:{{\rm rk}}_\delta(t)=\gamma\}$. The two players start playing the subgame of level $\delta$ of length $\zeta_\delta$. During the subgame Generic constructs partial plays $\langle(r^\zeta_i,s^\zeta_i):i
\leq\zeta_\delta\rangle$ of $\Game_0^\lambda({{\mathbb P}}_\gamma,p^\delta_{*,
t_\zeta})$ (for $\zeta<\zeta_\delta$) in which Complete uses the strategy ${{\bf st}}(\gamma,p^\delta_{*,t_\zeta})$ and such that
1. if $\zeta,\xi<\zeta_\delta$, $t\in T_\delta$, $t{\vartriangleleft}t_\zeta$, $t{\vartriangleleft}t_\xi$, $i\leq\zeta_\delta$, then $r^\zeta_i{{\restriction}}{{\rm rk}}_\delta(t)=r^\xi_i{{\restriction}}{{\rm rk}}_\delta(t)$ and $s^\zeta_i{{\restriction}}{{\rm rk}}_\delta(t)=
s^\xi_i{{\restriction}}{{\rm rk}}_\delta(t)$;
2. if $p^\delta_\zeta,q^\delta_\zeta$ are the conditions played at stage $\zeta$ of the subgame, then $p^\delta_{*,t_\zeta}\leq r^\zeta_i\leq p^\delta_\zeta\leq q^\delta_\zeta
=r^\zeta_\zeta$ for all $i<\zeta$.
So suppose that the two players have arrived at a stage $\zeta<\zeta_\delta$ of the subgame and $\big\langle\langle (r^\xi_i,
s^\xi_i):i<\zeta\rangle: \xi<\zeta_\delta\big\rangle$ has been defined. Generic looks at $\langle (r^\zeta_i,s^\zeta_i):i<\zeta\rangle$ – it is a play of $\Game_0^\lambda({{\mathbb P}}_\gamma,p^\delta_{*,t_\zeta})$ in which Complete uses ${{\bf st}}(\gamma,p^\delta_{*,t_\zeta})$, so we may find a condition $p^\delta_\zeta\in{{\mathbb P}}_\gamma$ stronger than all $r^\zeta_i,s^\zeta_i$ for $i<\zeta$ (and $p^\delta_\zeta\geq p^\delta_{*,t_\zeta}$). She plays this condition as her move at stage $\zeta$ of the subgame and Antigeneric answers with $q^\delta_\zeta\geq p^\delta_\zeta$. Generic lets $r^\zeta_\zeta=q^\delta_\zeta$ and she defines $r^\xi_\zeta$ for $\xi<
\zeta_\delta$, $\xi\neq \zeta$, as follows. Let $t\in T_\delta$ be such that $t{\vartriangleleft}t_\zeta$, $t{\vartriangleleft}t_\xi$ and ${{\rm rk}}_\delta(t)$ is the largest possible. Generic declares that $${{\rm Dom}}(r^\xi_\zeta)=\big({{\rm Dom}}(r^\zeta_\zeta)\cap{{\rm rk}}_\delta(t)\big)\cup
\bigcup\limits_{i<\zeta}{{\rm Dom}}(s_i^\xi)\cup{{\rm Dom}}(p^\delta_{*,t_\xi}),$$ and $r^\xi_\zeta{{\restriction}}{{\rm rk}}_\delta(t)=r^\zeta_\zeta{{\restriction}}{{\rm rk}}_\delta(t)$, and for ${\varepsilon}\in {{\rm Dom}}(r^\xi_\zeta)\setminus{{\rm rk}}_\delta(t)$ she lets $r^\xi_\zeta({\varepsilon})$ be the $<^*_\chi$–first ${{\mathbb P}}_{\varepsilon}$–name for a member of $\name{{{\mathbb Q}}}_{\varepsilon}$ such that $$r^\xi_\zeta{{\restriction}}{\varepsilon}{\Vdash}_{{{\mathbb P}}_{\varepsilon}}\mbox{`` $r^\xi_\zeta({\varepsilon})$ is
an upper bound to }\{p^\delta_{*,t_\xi}({\varepsilon})\}\cup \{s^\xi_i({\varepsilon}):
i<\zeta\}\mbox{ ''}$$ (remember \[pA.6\]). Finally, $s^\xi_\zeta$ (for $\xi<\zeta_\delta$) is defined as the condition given to Complete by ${{\bf st}}(\gamma,p^\delta_{*,
t_\zeta})$ in answer to $\langle(r^\xi_i,s^\xi_i):i<\zeta\rangle{{}^\frown\!}\langle r^\xi_\zeta\rangle$.
After the subgame is completed and both $p^\delta_\zeta,q^\delta_\zeta$ and $\big\langle\langle (r^\xi_i,s^\xi_i):i<\zeta_\delta\rangle:\xi<
\zeta_\delta \big\rangle$ has been determined, Generic chooses $r^0_{\zeta_\delta}$ as any upper bound to $\langle s^0_i:i<\zeta_\delta
\rangle$ and then defines $r^\xi_{\zeta_\delta}$ for $\xi\in\zeta_\delta\setminus 1$ like $r^\xi_\zeta$ for $\xi\neq \zeta$ above. Also $s^\xi_{\zeta_\delta}$ (for $\xi<\zeta_\delta$) are chosen like earlier (as results of applying ${{\bf st}}(\gamma,p^\delta_{*,t_\xi})$). Finally, Generic picks a standard tree of conditions $\bar{q}^\delta_*=\langle
q^\delta_{*,t}:t\in T_\delta\rangle$ such that $(\forall\zeta<\zeta_\delta)
(q^\delta_{*,t_\zeta}= s^\zeta_{\zeta_\delta})$. (Note that $(*)_5$, $(*)_7$ hold.)
Now Generic defines $r^-_\delta,r_\delta\in{{\mathbb P}}_\gamma$ so that $${{\rm Dom}}(r^-_\delta)={{\rm Dom}}(r_\delta)=\bigcup\limits_{t\in T_\delta}{{\rm Dom}}(
q^\delta_{*,t})\cup {{\rm Dom}}(p)$$ and
1. if $\xi\in{{\rm Dom}}(r^-_\delta)\setminus w_\delta$, then:\
$r^-_\delta(\xi)$ is the $<^*_\chi$–first ${{\mathbb P}}_\xi$–name for an element of $\name{{{\mathbb Q}}}_\xi$ such that $$\begin{array}{ll}
r^-_\delta{{\restriction}}\xi{\Vdash}_{{{\mathbb P}}_\xi}&\mbox{`` }r^-_\delta(\xi)\mbox{ is an
upper bound of }\{r_\alpha(\xi):\alpha<\delta\}\cup\{p(\xi)\}\mbox{ and }\\
&\mbox{\quad if }t\in T_\delta,\ \ {{\rm rk}}_\delta(t)>\xi,\mbox{ and }
q^\delta_{*,t}{{\restriction}}\xi\in\Gamma_{{{\mathbb P}}_\xi}\mbox{ and the set}\\
&\quad\{r_\alpha(\xi):\alpha<\delta\}\cup\{q^\delta_{*,t}(\xi),
p(\xi)\}\mbox{ has an upper bound in }\name{{{\mathbb Q}}}_\xi,\\
&\mbox{\quad then $r^-_\delta(\xi)$ is such an upper bound '',}
\end{array}$$ and $r_\delta(\xi)$ is the $<^*_\chi$–first ${{\mathbb P}}_\xi$–name for an element of $\name{{{\mathbb Q}}}_\xi$ such that $$\begin{array}{ll}
r_\delta{{\restriction}}\xi{\Vdash}_{{{\mathbb P}}_\xi}&\mbox{`` }r_\delta(\xi)\mbox{ is given to
Complete by }\name{{{\bf st}}}^0_\xi\mbox{ as the answer to }\ \\
&\quad\langle r^-_\alpha(\xi),r_\alpha(\xi):\alpha<\delta\rangle{{}^\frown\!}\langle r^-_\delta(\xi)\rangle \mbox{ ''}
\end{array}$$
2. if $\xi\in w_{\alpha+1}$, $\alpha<\delta$, then $r^-_\delta(\xi)=r_\delta(\xi)=r_\alpha(\xi)$.
(Note that by a straightforward induction on $\xi\in{{\rm Dom}}(r_\delta)$ one easily applies $(*)_3$ from previous stages to show that $r_\delta^-,r_\delta$ are well defined and $r_\delta\geq r^-_\delta\geq
r_\alpha,p$ for $\alpha<\delta$. Remember also $(*)_{11}$ and/or $(*)^{\rm
c}_{14}$.) If $\delta=0$ we also stipulate $r^-_0(0)=r_0(0)=p(0)$.
Finally, for each $\xi\in w_\delta$, Generic chooses a ${{\mathbb P}}_\xi$–name $\name{\bar{q}}_{\delta,\xi}$ for a sequence of conditions in $\name{{{\mathbb Q}}}_\xi$ of length $\name{{\varepsilon}}_{\delta,\xi}$ such that $$\begin{array}{ll}
{\Vdash}_{{{\mathbb P}}_\xi}&\mbox{`` }(\forall{\varepsilon}<\name{{\varepsilon}}_{\delta,\xi})(
\name{\bar{p}}_{\delta,\xi}({\varepsilon})\leq\name{\bar{q}}_{\delta,\xi}({\varepsilon}))
\mbox{ and }\\
&\mbox{\quad if }t\in T_\delta,\ {{\rm rk}}_\delta(t)>\xi,\mbox{ and }
q^\delta_{*,t}{{\restriction}}\xi\in\Gamma_{{{\mathbb P}}_\xi}\mbox{ then }
\name{\bar{q}}_{\delta,\xi}\big((t)_\xi\big)=q^\delta_{*,t}(\xi)
\mbox{ ''.}
\end{array}$$ Generic also picks $w_{\delta+1}$ by the bookkeeping device mentioned at the beginning and for $\xi\in w_{\delta+1}\setminus w_\delta$ she fixes $\name{{{\bf st}}}_\xi$ as in $(*)_3$.
This completes the description of the side objects constructed by Generic at stage $\delta$. Verification that they satisfy our demands $(*)_1$–$(*)_{12}$ is straightforward, and thus the description of the strategy ${{\bf st}}$ is complete.
We are going to argue now that ${{\bf st}}$ is a winning strategy for Generic. To this end suppose that $$\big\langle\zeta_\delta,\langle p^\delta_\zeta,q^\delta_\zeta:\zeta<
\zeta_\delta\rangle:\delta<\lambda\big\rangle$$ is the result of a play of ${{\Game^{{\rm rc}{\bf b}}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb P}}_\gamma)$ in which Generic followed ${{\bf st}}$ and constructed aside objects listed in $(\otimes)_\delta$ (for $\delta<\lambda$) so that $(*)_1$–$(*)_{12}$ hold.
We define a condition $r\in{{\mathbb P}}_\gamma$ as follows. Let ${{\rm Dom}}(r)=
\bigcup\limits_{\delta<\lambda}{{\rm Dom}}(r_\delta)$ and for $\xi\in{{\rm Dom}}(r)$ let $r(\xi)$ be a ${{\mathbb P}}_\xi$–name for a condition in $\name{{{\mathbb Q}}}_\xi$ such that if $\xi\in w_{\alpha+1}\setminus w_\alpha$, $\alpha<\lambda$ (or $\xi=0=\alpha$), then $${\Vdash}_{{{\mathbb P}}_\xi}\mbox{`` }r(\xi)\geq r_{\alpha}(\xi)\mbox{ and }
r(\xi) {\Vdash}_{\name{{{\mathbb Q}}}_\xi}\big\{\delta\!<\!\lambda\!:\big(\exists
{\varepsilon}\!<\!\name{{\varepsilon}}_{\delta,\xi}\big)\big(\bar{q}_{\delta,\xi}({\varepsilon})\in
\Gamma_{\name{{{\mathbb Q}}}_\xi}\big)\big\}\in ({{\mathcal U}}^{{{\mathbb P}}_\xi})^{\name{{{\mathbb Q}}_\xi}}
\mbox{ ''.}$$ Clearly $r$ is well defined (remember $(*)_9$) and $(\forall\delta<
\lambda)(r_\delta\leq r)$ and $r\geq p$. For each $\xi\in{{\rm Dom}}(r)$ choose a sequence $\langle\name{A}^\xi_i:i<\lambda\rangle$ of ${{\mathbb P}}_{\xi+1}$–names for elements of ${{\mathcal U}}\cap{{\bf V}}$ such that
1. $\displaystyle r{{\restriction}}(\xi+1){\Vdash}_{{{\mathbb P}}_{\xi+
1}}\big(\forall\delta\in \mathop{\triangle}\limits_{i<\lambda}
\name{A}^\xi_i\big)\big(\exists{\varepsilon}<\name{{\varepsilon}}_{\delta,\xi}\big)\big(
\name{\bar{q}}_{\delta,\xi}({\varepsilon})\in\Gamma_{\name{{{\mathbb Q}}}_\xi}\big)$.
\[cl1\] For each limit ordinal $\delta<\lambda$, $$r{\Vdash}_{{{\mathbb P}}_\gamma}\mbox{`` }\big(\forall\xi\in w_\delta\big)\big(
\delta\in\mathop{\triangle}\limits_{i<\lambda}\name{A}^\xi_i\big)\
\Rightarrow\ \big(\exists t\in T_\delta\big)\big({{\rm rk}}_\delta(t)=\gamma\ \&\
q^\delta_{*,t}\in \Gamma_{{{\mathbb P}}_\gamma}\big)\mbox{ ''.}$$
Suppose that $r'\geq r$ and a limit ordinal $\delta<\lambda$ are such that
1. $r'{\Vdash}_{{{\mathbb P}}_\gamma}$“ $\big(\forall\xi\in
w_\delta\big)\big( \delta\in\mathop{\triangle}\limits_{i<\lambda}
\name{A}^\xi_i\big)$ ”.
We are going to show that there is $t\in T_\delta$ such that ${{\rm rk}}_\delta(t)
=\gamma$ and the conditions $q^\delta_{*,t}$ and $r'$ are compatible (and then the claim will readily follow). To this end let $\langle{\varepsilon}_\alpha:
\alpha\leq\alpha^*\rangle=w_\delta\cup\{\gamma\}$ be the increasing enumeration. By induction on $\alpha\leq\alpha^*$ we will choose conditions $r^*_\alpha,r^{**}_\alpha\in{{\mathbb P}}_{{\varepsilon}_\alpha}$ and $t=\langle
(t)_{{\varepsilon}_\alpha}:\alpha<\alpha^*\rangle\in T_\delta$ such that letting $t^\alpha_\circ=\langle (t)_{{\varepsilon}_\beta}:\beta<\alpha\rangle\in T_\delta$ we have
1. $q^\delta_{*,t^\alpha_\circ}\leq r^*_\alpha$ and $r'{{\restriction}}{\varepsilon}_\alpha\leq r^*_\alpha$,
2. $\langle r^*_\beta{{}^\frown\!}r'{{\restriction}}[{\varepsilon}_\beta,
\gamma),r^{**}_\beta{{}^\frown\!}r'{{\restriction}}[{\varepsilon}_\beta,\gamma):\beta<\alpha
\rangle$ is a partial legal play of $\Game^\lambda_0({{\mathbb P}}_\gamma,r')$ in which Complete uses her winning strategy ${{\bf st}}(\gamma,r')$.
Suppose that $\alpha\leq\alpha^*$ is a limit ordinal and we have already defined $t^\alpha_\circ=\langle (t)_{{\varepsilon}_\beta}: \beta<\alpha\rangle$ and $\langle r^*_\beta, r^{**}_\beta: \beta<\alpha\rangle$. Let $\xi=\sup(
{\varepsilon}_\beta:\beta<\alpha)$. It follows from $(*)_{20}^\beta$ (for $\beta<
\alpha$) that we may find a condition $s\in {{\mathbb P}}_\xi$ stronger than all $r^{**}_\beta$ (for $\beta<\alpha$). Let $r^*_\alpha\in{{\mathbb P}}_{{\varepsilon}_\alpha}$ be such that $r^*_\alpha{{\restriction}}\xi=s$ and $r^*_\alpha{{\restriction}}[\xi,{\varepsilon}_\alpha)
=r'{{\restriction}}[\xi,{\varepsilon}_\alpha)$. It follows from $(*)_{19}^\beta$ that $q^\delta_{*,t^\alpha_\circ}{{\restriction}}\xi\leq s=r^*_\alpha{{\restriction}}\xi$ and $r'{{\restriction}}\xi\leq s=r^*_\alpha{{\restriction}}\xi$. Note also that $(\forall\beta<\alpha)(
r^{**}_\beta\leq s{{\restriction}}{\varepsilon}_\beta=r^*_\alpha{{\restriction}}{\varepsilon}_\beta)$, so $(\forall\beta<\alpha)(r^{**}_\beta{{}^\frown\!}r'{{\restriction}}[{\varepsilon}_\beta,\gamma)\leq
r^*_\alpha{{}^\frown\!}r'{{\restriction}}[{\varepsilon}_\alpha,\gamma))$. Now by induction on $\zeta
\leq {\varepsilon}_\alpha$ we show that $q^\delta_{*,t^\alpha_\circ}{{\restriction}}\zeta\leq
r^*_\alpha{{\restriction}}\zeta$ and $r'{{\restriction}}\zeta\leq r^*_\alpha{{\restriction}}\zeta$. For $\zeta\leq\xi$ we are already done, so assume that $\zeta\in [\xi,
{\varepsilon}_\alpha)$ and we have shown $q^\delta_{*,t^\alpha_\circ}{{\restriction}}\zeta \leq
r^*_\alpha{{\restriction}}\zeta$ and $r'{{\restriction}}\zeta\leq r^*_\alpha{{\restriction}}\zeta$. It follows from $(*)_6+(*)_3$ that $r^*_\alpha{{\restriction}}\zeta{\Vdash}(\forall
i<\delta)(r_i(\zeta)\leq p^\delta_{*,t^\alpha_\circ}(\zeta))$ and therefore we may use $(*)_{12}$ to conclude that $$r^*_\alpha{{\restriction}}\zeta{\Vdash}_{{{\mathbb P}}_\zeta} q^\delta_{*,t^\alpha_\circ}
(\zeta)\leq r_\delta(\zeta)\leq r(\zeta)\leq r'(\zeta)=r^*_\alpha(\zeta).$$ The limit stages are trivial and we see that $(*)_{19}^\alpha$ and (a part of) $(*)_{20}^\alpha$ hold. Finally we let $r^{**}_\alpha\in
{{\mathbb P}}_{{\varepsilon}_\alpha}$ be the condition given to Complete by ${{\bf st}}(\gamma,r')$ as the response to $\langle r^*_\beta{{}^\frown\!}r'{{\restriction}}[{\varepsilon}_\beta,\gamma),
r^{**}_\beta{{}^\frown\!}r'{{\restriction}}[{\varepsilon}_\beta,\gamma):\beta<\alpha\rangle{{}^\frown\!}\langle r^*_\alpha\rangle$.
Now suppose that $\alpha=\beta+1\leq\alpha^*$ and we have already defined $r^*_\beta, r^{**}_\beta\in{{\mathbb P}}_{{\varepsilon}_\beta}$ and $t^\beta_\circ\in
T_\delta$. It follows from $(*)_{17}^{{\varepsilon}_\beta}+(*)_{18}+
(*)_{19}^\beta+(*)_{10}$ that $$r^{**}_\beta{\Vdash}_{{{\mathbb P}}_{{\varepsilon}_\beta}}\mbox{`` }r'({\varepsilon}_\beta)
{\Vdash}_{\name{{{\mathbb Q}}}_{{\varepsilon}_\beta}}\big(\exists{\varepsilon}<
{\varepsilon}^{t^\beta_\circ}_{\delta,{\varepsilon}_\beta}\big)\big(q^\delta_{*,
t^\beta_\circ{{}^\frown\!}\langle{\varepsilon}\rangle}({\varepsilon}_\beta)\in
\Gamma_{\name{{{\mathbb Q}}}_{{\varepsilon}_\beta}}\big)\mbox{ ''.}$$ Therefore we may choose ${\varepsilon}=(t)_{{\varepsilon}_\beta}<
{\varepsilon}^{t^\beta_\circ}_{\delta,{\varepsilon}_\beta}$ (thus defining $t^\alpha_\circ$) and a condition $s\in{{\mathbb P}}_{{\varepsilon}_\beta+1}$ such that $s{{\restriction}}{\varepsilon}_\beta\geq
r^{**}_\beta\geq q^\delta_{*, t^\beta_\circ}$ and $$s{{\restriction}}{\varepsilon}_\beta{\Vdash}\mbox{`` }s({\varepsilon}_\beta)\geq
r'({\varepsilon}_\beta)\ \&\ s({\varepsilon}_\beta)\geq q^\delta_{*,t^\alpha_\circ}(
{\varepsilon}_\beta)\mbox{ ''.}$$ We let $r^*_\alpha\in{{\mathbb P}}_{{\varepsilon}_\alpha}$ be such that $r^*_\alpha{{\restriction}}({\varepsilon}_\beta+1)=s$ and $r^*_\alpha{{\restriction}}({\varepsilon}_\beta,{\varepsilon}_\alpha)=
r'{{\restriction}}[{\varepsilon}_\beta,{\varepsilon}_\alpha)$. Exactly like in the limit case we argue that $(*)_{19}^\alpha$ and (a part of) $(*)_{20}^\alpha$ hold and then in the same manner as there we define $r^{**}_\alpha$.
Finally note that $t\in T_\delta$, ${{\rm rk}}_\delta(t)=\gamma$, and the condition $r^*_{\alpha^*}$ witnesses that $r'$ and $q^\delta_{*,t}$ are compatible.
Now note that $${\Vdash}_{{{\mathbb P}}_\gamma}\mbox{`` }\big\{\delta<\lambda:\big(\forall\xi\in
w_\delta\big)\big(\delta\in\mathop{\triangle}\limits_{i<\lambda}
\name{A}^\xi_i\big)\big\}\in{{\mathcal U}}^{{{\mathbb P}}_\gamma}\mbox{ '',}$$ and hence by \[cl1\] we have $$r{\Vdash}_{{{\mathbb P}}_\gamma}\mbox{`` }\big\{\delta<\lambda:\big(\exists t\in
T_\delta\big)\big({{\rm rk}}_\delta(t)=\gamma\ \&\ q^\delta_{*,t}\in
\Gamma_{{{\mathbb P}}_\gamma}\big)\big\}\in{{\mathcal U}}^{{{\mathbb P}}_\gamma}\mbox{ ''.}$$ Therefore, by $(*)_7$, $$r{\Vdash}_{{{\mathbb P}}_\gamma}\mbox{`` }\big\{\delta<\lambda: \big(\exists
\zeta<\zeta_\delta\big)\big(q^\delta_\zeta\in\Gamma_{{{\mathbb P}}_\gamma}\big)
\big\}\in {{\mathcal U}}^{{{\mathbb P}}_\gamma}\mbox{ ''}$$ and the proof of the theorem is complete.
The reason for the weaker “[**b**]{}–bounding” in the conclusion of \[verB\] (and not “B–bounding”) is that in our description of the strategy ${{\bf st}}$, we would have to make sure that the conditions played by Antigeneric form a tree of conditions. Playing a subgame and keeping the demands of $(*)_{15}$ are a convenient way to deal with this issue. Note that (at a stage $\delta$) after playing $\zeta_\delta$ steps of the subgame, the players may start over and play another $\zeta_\delta$ steps. This small modification can be used to strengthen \[verB\] to \[verBbis\] below.
\[doubleres\] Let ${{\mathbb Q}}$ be a strategically $({<}\lambda)$–complete forcing notion.
1. For a condition $p\in{{\mathbb Q}}$ we define a game ${{\Game^{{\rm rc}{\bf 2b}}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb Q}})$ between Generic and Antigeneric as follows. A play of ${{\Game^{{\rm rc}{\bf 2b}}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb Q}})$ lasts $\lambda$ steps and during a play a sequence
1. $\Big\langle \zeta_\alpha,i_\alpha,\langle
p^\alpha_\xi,q^\alpha_\xi:\xi<\zeta_\alpha\cdot (1+i_\alpha)\rangle:\alpha<
\lambda\Big\rangle$
is constructed. Suppose that the players have arrived to a stage $\alpha<
\lambda$ of the game. First, Generic chooses a non-zero ordinal $\zeta_\alpha<\mu_\alpha$ and then the two players play the first $\zeta_\alpha$ steps of a subgame in which they alternatively choose successive terms of a sequence $\langle p^\alpha_\xi,q^\alpha_\xi:\xi<
\zeta_\alpha\rangle$. At a stage $\xi<\zeta_\alpha$ of the subgame, first Generic picks a condition $p^\alpha_\xi\in{{\mathbb Q}}$ and then Antigeneric answers with a condition $q^\alpha_\xi$ stronger than $p^\alpha_\xi$. After this part of the subgame Antigeneric picks a non-zero ordinal $i_\alpha<\lambda$ and the two players continue playing the subgame up to the total length of $\zeta_\alpha\cdot (1+i_\alpha)$ alternatively choosing successive terms of a sequence $\langle p^\alpha_\xi,q^\alpha_\xi:\xi<\zeta_\alpha\cdot
(1+i_\alpha)\rangle$. At a stage $\xi=\zeta_\alpha\cdot i+j$ (where $j<\zeta_\alpha$, $0<i<1+i_\alpha$) of the subgame, first Generic picks a $\leq_{{\mathbb Q}}$–upper bound $p^\alpha_\xi\in{{\mathbb Q}}$ to $\{q^\alpha_\xi:\xi=
\zeta_\alpha\cdot i'+j\ \&\ i'<i\}$, and then Antigeneric answers with a condition $q^\alpha_\xi$ stronger than $p^\alpha_\xi$. At the end, Generic wins the play $(\boxdot)$ of ${{\Game^{{\rm rc}{\bf 2b}}_{{{\mathcal U}},\bar{\mu}}}}(p,{{\mathbb Q}})$ if and only if both players had always legal moves and
1. there is a condition $p^*\in{{\mathbb Q}}$ stronger than $p$ and such that $$p^*{\Vdash}_{{{\mathbb Q}}}\mbox{`` }\big\{\alpha<\lambda:\big(\exists j<
\zeta_\alpha\big)\big(\forall i<1+i_\alpha\big)\big(q^\alpha_{\zeta_\alpha
\cdot i+j}\in\Gamma_{{{\mathbb Q}}}\big)\big\} \in {{\mathcal U}}^{{\mathbb Q}}\mbox{ ''}.$$
2. The game ${{\Game^{{\rm rc}{\bf 2a}}_{\bar{\mu}}}}(p,{{\mathbb Q}})$ is defined similarly except that the winning criterion $(\circledast)^{\rm rc}_{\bf 2b}$ is changed so that “$\in {{\mathcal U}}^{{\mathbb Q}}$” is replaced by “$=\lambda$”.
3. We say that a forcing notion ${{\mathbb Q}}$ is [*reasonably double [ **a**]{}–bounding over $\bar{\mu}$*]{} if
1. ${{\mathbb Q}}$ is strategically $({<}\lambda)$–complete, and
2. for any $p\in{{\mathbb Q}}$, Generic has a winning strategy in the game ${{\Game^{{\rm rc}{\bf 2a}}_{\bar{\mu}}}}(p,{{\mathbb Q}})$.
In an analogous manner we define when the forcing notion ${{\mathbb Q}}$ is [*reasonably double [**b**]{}–bounding over ${{\mathcal U}},\bar{\mu}$*]{}.
\[doubleobs\] Let ${{\mathcal U}},\bar{\mu}$ be as in \[incon\], $(X,x)\in\{(A,{\bf a}),(B,{\bf
b})\}$. Then the following implications hold for a forcing notion ${{\mathbb Q}}$:
-------------------------------------------------------------------------------------
“${{\mathbb Q}}$ is reasonably $X$–bounding over ${{\mathcal U}},\bar{\mu}$”
$\Downarrow$
“${{\mathbb Q}}$ is reasonably double $x$–bounding over ${{\mathcal U}},\bar{\mu}$”
$\Downarrow$
${{\mathbb Q}}$ is reasonably $x$–bounding over ${{\mathcal U}},\bar{\mu}$” .
-------------------------------------------------------------------------------------
\[verBbis\] Assume that $\lambda,\bar{\mu}$ are as in \[incon\] and $\bar{{{\mathbb Q}}}=
\langle{{\mathbb P}}_\xi,\name{{{\mathbb Q}}}_\xi:\xi<\gamma\rangle$ is a $\lambda$–support iteration such that for every $\xi<\gamma$, $${\Vdash}_{{{\mathbb P}}_\xi}\mbox{`` $\name{{{\mathbb Q}}}_\xi$ is reasonably B--bounding
over $\bar{\mu}$ ''.}$$ Then ${{\mathbb P}}_{\gamma}=\lim(\bar{{{\mathbb Q}}})$ is reasonably double ${\bf
b}$–bounding over $\bar{\mu}$.
\[verA\] Assume that $\lambda,\bar{\mu}$ are as in \[incon\] and $\bar{{{\mathbb Q}}}=
\langle{{\mathbb P}}_\xi,\name{{{\mathbb Q}}}_\xi:\xi<\gamma\rangle$ is a $\lambda$–support iteration such that for every $\xi<\gamma$, $${\Vdash}_{{{\mathbb P}}_\xi}\mbox{`` $\name{{{\mathbb Q}}}_\xi$ is reasonably A--bounding
over $\bar{\mu}$ ''.}$$ Then ${{\mathbb P}}_{\gamma}=\lim(\bar{{{\mathbb Q}}})$ is reasonably double ${\bf
a}$–bounding over $\bar{\mu}$ (and thus also reasonably ${\bf
a}$–bounding over $\bar{\mu}$.
The proof of Theorem \[verB\] (changed so that it works for \[verBbis\]) can be easily modified to fit the current purpose, just replace each occurrence of $\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}}, \Game^{{\rm rc}{\bf
2b}}_{{{\mathcal U}},\bar{\mu}}$ by $\Game^{\rm rcA}_{\bar{\mu}}, \Game^{{\rm
rc}{\bf 2a}}_{\bar{\mu}}$, respectively, and in the end think that ${\Vdash}_{{{\mathbb P}}_{\xi+1}}\name{A}^\xi_i=\lambda$ (for $\xi\in{{\rm Dom}}(r)$, $i<\lambda$). Then one uses the proof of \[cl1\] to argue that $$r{\Vdash}_{{{\mathbb P}}_\gamma}\mbox{`` }\big(\forall\delta<\lambda\big)
\big(\exists t\in T_\delta\big)\big({{\rm rk}}_\delta(t)=\gamma\ \&\
q^\delta_{*,t}\in \Gamma_{{{\mathbb P}}_\gamma}\big)\mbox{ ''.}$$
\[prob\]
1. Do we have a result parallel to \[verB\] and/or \[verA\] for reasonably C–bounding forcings?
2. Can the implications in \[doubleobs\] be reversed in the sense that we allow passing to an equivalent forcing notion?
Consequences of reasonable ABC
==============================
Let us note that Theorem \[verA\] improves [@RoSh:777 Theorem A.2.4]. Before we explain why, we should recall the following definition.
\[[[@RoSh:777 Def. A.2.1]]{}\] \[da2\] Let ${{\mathbb P}}$ be a forcing notion.
1. A complete $\lambda$–tree of height $\alpha<\lambda$ is a set of sequences $s\subseteq {}^{\leq\alpha}\lambda$ such that
- $s$ has the ${\vartriangleleft}$–smallest element denoted ${{\rm root}}(s)$,
- $s$ is closed under initial segments longer than ${{\rm lh}\/}({{\rm root}}(s))$, and
- the union of any ${\vartriangleleft}$–increasing sequence of members of $s$ is in $s$, and
- $\big(\forall\eta\in s\big)\big({{\rm lh}\/}(\eta)\leq\alpha\big)$ and $\big(\forall\eta\in s\big)\big(\exists\nu\in s\big)\big(\eta{\vartriangleleft}\nu\ \&\
{{\rm lh}\/}(\nu)=\alpha\big)$.
2. For a condition $p\in{{\mathbb P}}$ and an ordinal $i_0<\lambda$ we define a game ${{\Game^{\rm Sacks}_{\bar{\mu}}}}(i_0,p,{{\mathbb P}})$ of two players, [*the Generic player*]{} and [*the Antigeneric player*]{}. A play lasts $\lambda$ moves indexed by ordinals from the interval $[i_0,\lambda)$, and during it the players construct a sequence $\langle (s_i,\bar{p}^i,\bar{q}^i):i_0\leq i<\lambda\rangle$ as follows. At stage $i$ of the play (where $i_0\leq i<\lambda$), first Generic chooses $s_i\subseteq {}^{\leq i+1}\lambda$ and a system $\bar{p}^i=\langle p^i_\eta:\eta\in s_i\cap {}^{ i+1}\lambda
\rangle$ such that
1. $s_i$ is a complete $\lambda$–tree of height $i+1$ and ${{\rm lh}\/}({{\rm root}}(s_i))=i_0$,
2. for all $j$ such that $i_0\leq j<i$ we have $s_j=s_i\cap
{}^{\leq j+1}\lambda$,
3. $p^i_\eta\in{{\mathbb P}}$ for all $\eta\in s_i\cap
{}^{ i+1}\lambda$, and
4. if $i_0\leq j<i$, $\nu\in s_i\cap {}^{ j+1}
\lambda$ and $\nu{\vartriangleleft}\eta\in s_i\cap {}^{ i+1}\lambda$, then $q^j_\nu\leq p^i_\eta$ and $p\leq p^i_\eta$,
5. $|s_i\cap {}^{ i+1}\lambda|<\mu_i$.
Then Antigeneric answers choosing a system $\bar{q}^i=\langle q^i_\eta:
\eta\in s_i\cap {}^{ i+1}\lambda\rangle$ of conditions in ${{\mathbb P}}$ such that $p^i_\eta\leq q^i_\eta$ for each $\eta\in s_i\cap {}^{
i+1}\lambda$.
The Generic player wins a play if she always has legal moves (so the play really lasts $\lambda$ steps) and there are a condition $q\geq p$ and a ${{\mathbb P}}$–name $\name{\rho}$ such that
1. $q{\Vdash}_{{{\mathbb P}}}\mbox{`` }\name{\rho}\in
{{}^{\lambda}\lambda}\ \&\ \big(\forall i\in [i_0,\lambda\big))\big(\name{\rho}{{\restriction}}(i+1)
\in s_i\ \&\ q^i_{\name{\rho}{{\restriction}}(i+1)}\in\Gamma_{{{\mathbb P}}}\big)\mbox{ ''.}$
3. We say that ${{\mathbb P}}$ has the [*strong $\bar{\mu}$–Sacks property*]{} whenever
1. ${{\mathbb P}}$ is strategically $(<\lambda)$–complete, and
2. the Generic player has a winning strategy in the game ${{\Game^{\rm Sacks}_{\bar{\mu}}}}(i_0,p,{{\mathbb P}})$ for any $i_0<\lambda$ and $p\in{{\mathbb P}}$.
The following proposition explains why \[verA\] is stronger than [@RoSh:777 Theorem A.2.4].
\[thesame\] Assume that $\lambda,\bar{\mu}$ are as in Context \[incon\] and that additionally $(\forall i<j<\lambda)(\mu_i\leq \mu_j)$. Let ${{\mathbb Q}}$ be a forcing notion. Then
${{\mathbb Q}}$ is reasonably A–bounding over $\bar{\mu}$
if and only if
${{\mathbb Q}}$ has the strong $\bar{\mu}$–Sacks property.
Suppose that ${{\mathbb Q}}$ is reasonably A–bounding over $\bar{\mu}$. Since the sequence $\bar{\mu}$ is non-decreasing, it is enough to show that Generic has a winning strategy in ${{\Game^{\rm Sacks}_{\bar{\mu}}}}(0,p,{{\mathbb Q}})$ for each $p\in{{\mathbb Q}}$ (as then almost the same strategy will be good in ${{\Game^{\rm Sacks}_{\bar{\mu}}}}(i,p,{{\mathbb Q}})$ for any $i<\lambda$).
Let $p\in{{\mathbb Q}}$. We are going to define a strategy ${{\bf st}}$ for Generic in the game ${{\Game^{\rm Sacks}_{\bar{\mu}}}}(0,p,{{\mathbb Q}})$. To describe it, let us fix a winning strategy ${{\bf st}}_0$ of Complete in $\Game^\lambda_0({{\mathbb Q}},p)$ and a winning strategy ${{\bf st}}_1$ of Generic in ${{\Game^{\rm rcA}_{\bar{\mu}}}}(p,{{\mathbb Q}})$. Now, at a stage $\delta<\lambda$ of the play the strategy ${{\bf st}}$ will tell Generic to write aside
1. $I_\delta$ and $\langle r^{0,\delta}_t,
r^{1,\delta}_t:t\in I_\delta\rangle$ and $\langle r^\delta_\eta: \eta\in
s_\delta\cap {}^{\delta+1}\lambda\rangle$
so that if $\langle (s_\delta,\bar{p}^\delta,\bar{q}^\delta):\delta<\lambda
\rangle$ is a play of ${{\Game^{\rm Sacks}_{\bar{\mu}}}}(0,p,{{\mathbb Q}})$ in which Generic follows ${{\bf st}}$, then the following conditions $(\odot)_1$–$(\odot)_4$ are satisfied (for each $\delta<\lambda$).
1. $\big\langle I_\alpha,\langle r^{0,\alpha}_t,
r^{1,\alpha}_t: t\in I_\alpha\rangle:\alpha\leq\delta\big\rangle$ is a partial legal play of ${{\Game^{\rm rcA}_{\bar{\mu}}}}(p,{{\mathbb Q}})$ in which Generic uses ${{\bf st}}_1$.
2. For each $\eta\in s_\delta\cap {}^{\delta+1}\lambda$ the sequence $\langle q^\alpha_{\eta{{\restriction}}(\alpha+1)},r^\alpha_{\eta{{\restriction}}(\alpha+1)}:\alpha\leq\delta\rangle$ is a partial legal play of $\Game^\lambda_0 ({{\mathbb Q}},p)$ in which Complete uses ${{\bf st}}_0$.
3. If $t\in I_\delta$, $\alpha<\delta$, $\nu\in s_\alpha\cap
{}^{\alpha+1}\lambda$, then either $r^\alpha_\nu,r^{1,\delta}_t$ are incompatible or $r^\alpha_\nu\leq r^{1,\delta}_t$.
4. $\langle p^\delta_\nu:\nu\in s_\delta\cap{}^{\delta+1}
\lambda \rangle$ is an antichain in ${{\mathbb Q}}$.
So suppose that the two players arrived to a stage $\delta<\lambda$ of the game ${{\Game^{\rm Sacks}_{\bar{\mu}}}}(0,p,{{\mathbb Q}})$ and the objects listed in $(\boxtimes)_\alpha$ (for $\alpha<\delta$) as well as $\langle (s_\alpha,\bar{p}^\alpha,
\bar{q}^\alpha):\alpha<\delta\rangle$ have been constructed. First Generic uses ${{\bf st}}_1$ to pick the answer $\big(I_\delta,\langle r^{0,\delta}_t:t\in
I_\delta\rangle\big)$ to $\big\langle I_\alpha,\langle
r^{0,\alpha}_t,r^{1,\alpha}_t:t\in I_\alpha\rangle:\alpha<\delta\big\rangle$ in ${{\Game^{\rm rcA}_{\bar{\mu}}}}(p,{{\mathbb Q}})$. Then she uses the strategic completeness of ${{\mathbb Q}}$ and \[obsA.4\] to choose a system $\langle r^*_t:t\in I_\delta\rangle$ of conditions in ${{\mathbb Q}}$ such that
1. if $t\in I_\delta$, then $r^{0,\delta}_t\leq r^*_t$ and for every $\alpha<\delta$ and $\nu\in s_\alpha\cap {}^{\alpha+1}\lambda$, either $r^\alpha_\nu,r^*_t$ are incompatible or $r^\alpha_\nu\leq r^*_t$, and also either $p,r^*_t$ are incompatible or $p\leq r^*_t$,
2. if $t_0,t_1\in I_\delta$, $t_0\neq t_1$, then the conditions $r^*_{t_0},r^*_{t_1}$ are incompatible in ${{\mathbb Q}}$.
Now she lets $s^*=\{\eta\in {}^\delta\lambda:(\forall\alpha<\delta)(\eta
{{\restriction}}(\alpha+1)\in s_\alpha)\}$ and $$s^-=\{\eta\in s^*: (\exists t\in I_\delta)(\forall\alpha<\delta)(
r^\alpha_{\eta{{\restriction}}(\alpha+1)}\leq r^*_t\ \&\ p\leq r^*_t)\},$$ and for each $\eta\in s^-$ she fixes an enumeration $\langle t^\eta_\xi:
\xi<\xi_\eta\rangle$ of the set $$\big\{t\in I_\delta:\big(\forall\alpha<\delta\big)\big(r^\alpha_{\eta
{{\restriction}}(\alpha+1)}\leq r^*_t\ \&\ p\leq r^*_t\big)\big\}.$$ Now Generic defines $$s_\delta^+=\big\{\nu\in {}^{\delta+1}\lambda:\big(\nu{{\restriction}}\delta\in s^*
\setminus s^-\ \&\ \nu(\delta)=0\big)\mbox{ or }\big(\nu{{\restriction}}\delta\in s^-\
\&\ \nu(\delta)<\xi_{\nu{{\restriction}}\delta}\big)\big\}$$ and she lets $s_\delta$ be a $\lambda$–tree of height $\delta+1$ such that $s_\delta\cap {}^{\delta+1}\lambda=s^+_\delta$. For $\nu\in s^+_\delta$ she also chooses $p^\delta_\nu$ so that
- if $\nu{{\restriction}}\delta\notin s^-$, then $p^\delta_\nu\in{{\mathbb Q}}$ is an upper bound to $\{r^\alpha_{\nu{{\restriction}}(\alpha+1)}:\alpha<\delta\}\cup\{p\}$ (remember $(\odot)_2$),
- if $\nu{{\restriction}}\delta\in s^-$, then $p^\delta_\nu= r^*_{t^{\nu{{\restriction}}\delta}_{\nu(\delta)}}$.
And now, in the play of ${{\Game^{\rm Sacks}_{\bar{\mu}}}}(0,p,{{\mathbb Q}})$, Generic puts $$s_\delta\quad\mbox{ and }\quad \langle p^\delta_\nu:\nu\in s^+_\delta
\rangle$$ and Antigeneric answers with $\langle q^\delta_\nu:\nu\in s^+_\delta\rangle$ (so that $q^\delta_\nu\geq p^\delta_\nu$). Conditions $r^\delta_\nu$ (for $\nu\in s^+_\delta$) are determined using ${{\bf st}}_0$ (so that the demand in $(\odot)_2$ is satisfied). Finally, Generic defines also $r^{1,\delta}_t$ for $t\in I_\delta$ so that
- if $t=t^\eta_\xi$ for some $\eta\in s^-$ and $\xi<\xi_\eta$, then $r^{1,\delta}_t=r^\delta_{\eta{{}^\frown\!}\langle\xi\rangle}$,
- otherwise $r^{1,\delta}_t=r^*_t$.
This completes the description of what Generic plays and what she writes aside — it should be clear that the requirements of $(\odot)_1$–$(\odot)_4$ are satisfied. Now, why is ${{\bf st}}$ a winning strategy? So suppose that $\langle (s_\delta,\bar{p}^\delta,\bar{q}^\delta):
\delta<\lambda\rangle$ is a play of ${{\Game^{\rm Sacks}_{\bar{\mu}}}}(0,p,{{\mathbb Q}})$ in which Generic follows ${{\bf st}}$, and $I_\delta$, $\langle r^{0,\delta}_t,r^{1,\delta}_t:t\in
I_\delta\rangle$ and $\langle r^\delta_\eta: \eta\in s_\delta\cap
{}^{\delta+1}\lambda\rangle$ (for $\delta<\lambda$) are the objects constructed by Generic aside, so they satisfy $(\odot)_1$–$(\odot)_4$. It follows from $(\odot)_1$ and the choice of ${{\bf st}}_1$ that there is a condition $p^*\geq p$ such that
1. for every $\delta<\lambda$ the set $\big\{r^{1,\delta}_t:
t\in I_\delta\big\}$ is pre-dense above $p^*$.
We claim that then also
1. for every $\delta<\lambda$ the set $\big\{r^\delta_\eta:
\eta\in s_\delta\cap {}^{\delta+1}\lambda\big\}$ is pre-dense above $p^*$
(and this clearly implies that Generic won the play, remember $(\odot)_4$). Assume towards contradiction that $(\odot)_8$ fails and let $\delta<\lambda$ be the smallest ordinal for which we may find a condition $q\geq p^*$ such that $q$ is incompatible with every $r^\delta_\eta$ for $\eta\in s_\delta\cap {}^{\delta+1}\lambda$. It follows from $(\odot)_7$ that we may pick $t\in I_\delta$ such that the conditions $r^{1,\delta}_t,q$ are compatible. By the previous sentence and by the definition of $r^{1,\delta}_t$ we get that $t\neq t^\eta_\xi$ for all $\xi<\xi_\eta$, $\eta\in s^-$ and thus $r^{1,\delta}_t=r^*_t$. Look at the condition $r^*_t$ (satisfying $(\odot)_5+(\odot)_6$) — it must be stronger than $p$ and by the minimality of $\delta$ we have that $\big(\forall\alpha<\delta\big)
\big(\exists\nu\in s_\alpha\cap {}^{\alpha+1}\lambda\big)\big(r^\alpha_\nu
\leq r^*_t\big)$. It follows from $(\odot)_4$ from stages $\alpha<\delta$ that there is $\eta\in s^*$ such that $\big(\forall\alpha<\delta\big)
\big(r^\alpha_{\eta{{\restriction}}(\alpha+1)}\leq r^*_t\big)$. Then $t\in s^-$ and hence $t=t^\eta_\xi$ for some $\xi<\xi_\eta$, contradicting what we already got.
The converse implication should be clear.
The following easy proposition explains why the names of the properties defined in \[p.1A\] include the adjective “bounding”.
\[bound\] Let $\lambda,{{\mathcal U}}$ and $\bar{\mu}$ be as in \[incon\]. Assume that ${{\mathbb Q}}$ is a forcing notion, $p\in {{\mathbb Q}}$ and $\name{\tau}$ is a ${{\mathbb Q}}$–name for an element of ${{}^{\lambda}\lambda}$.
1. If ${{\mathbb Q}}$ is reasonably [**a**]{}–bounding over $\bar{\mu}$, then there are a condition $q\geq p$ and a sequence $\bar{a}=\langle a_\alpha:\alpha<
\lambda\rangle$ such that
- $a_\alpha\subseteq\lambda$, $|a_\alpha|<\mu_\alpha$ for all $\alpha<
\lambda$,
- $q{\Vdash}_{{{\mathbb Q}}}$“ $(\forall\alpha<\lambda)(\name{\tau}(\alpha)\in
a_\alpha)$ ”.
2. If ${{\mathbb Q}}$ is reasonably [**b**]{}–bounding over ${{\mathcal U}},\bar{\mu}$, then there are a condition $q\geq p$ and a sequence $\bar{a}=\langle
a_\alpha:\alpha<\lambda\rangle$ such that
- $a_\alpha\subseteq\lambda$, $|a_\alpha|<\mu_\alpha$ for all $\alpha<
\lambda$,
- $q{\Vdash}_{{{\mathbb Q}}}$“ $\{\alpha<\lambda:\name{\tau}(\alpha)\in
a_\alpha\}\in{{\mathcal U}}^{{{\mathbb Q}}}$ ”.
3. If ${{\mathbb Q}}$ is reasonably [**c**]{}–bounding over ${{\mathcal U}},\bar{\mu}$, then there are a condition $q\geq p$ and a sequence $\bar{a}=\langle
a_\alpha:\alpha<\lambda\rangle$ such that
- $a_\alpha\subseteq\lambda$, $|a_\alpha|<\mu_\alpha$ for all $\alpha<
\lambda$,
- $q{\Vdash}_{{{\mathbb Q}}}$“ $\{\alpha<\lambda:\name{\tau}(\alpha)\in
a_\alpha\}\in\big({{\mathcal U}}^{{{\mathbb Q}}}\big)^+$ ”.
Forcing notions and models
==========================
In this section, in addition to the assumptions stated in \[incon\] we will also assume that
\[extraassum\]
1. $S\subseteq \lambda$ is stationary and co-stationary, $S\in{{\mathcal U}}$,
2. ${{\mathcal V}}$ is a normal filter on $\lambda$, $\lambda\setminus
S\in{{\mathcal V}}$.
\[ext\]
1. Let $\alpha<\beta<\lambda$. [*An $(\alpha,\beta)$–extending function*]{} is a mapping $c:{{\mathcal P}}(\alpha)\longrightarrow{{\mathcal P}}(\beta)\setminus
{{\mathcal P}}(\alpha)$ such that $c(u)\cap\alpha=u$ for all $u\in{{\mathcal P}}(\alpha)$.
2. Let $C$ be an unbounded subset of $\lambda$. [*A $C$–extending sequence*]{} is a sequence ${{\mathfrak c}}=\langle c_\alpha:\alpha\in C\rangle$ such that each $c_\alpha$ is an $(\alpha,\min(C\setminus(\alpha+1)))$–extending function.
3. Let $C\subseteq\lambda$, $\|C\|=\lambda$, $\beta\in C$, $w\subseteq
\beta$ and let ${{\mathfrak c}}=\langle c_\alpha:\alpha\in C\rangle$ be a $C$–extending sequence. We define ${{\rm pos}}^+(w,{{\mathfrak c}},\beta)$ as the family of all subsets $u$ of $\beta$ such that
1. if $\alpha_0=\min\big(\{\alpha\in C:(\forall\xi\in w)(\xi<
\alpha)\}\big)$, then $u\cap\alpha_0=w$ (so if $\alpha_0=\beta$, then $u=w$), and
2. if $\alpha_0,\alpha_1\in C$, $w\subseteq\alpha_0<\alpha_1=
\min(C\setminus(\alpha_0+1))\leq\beta$, then either $c_{\alpha_0}(u\cap
\alpha_0)=u\cap\alpha_1$ or $u\cap\alpha_0=u\cap\alpha_1$,
3. if $\sup(w)<\alpha_0=\sup(C\cap\alpha_0)\notin C$, $\alpha_1=\min\big(C\setminus(\alpha_0+1)\big)\leq\beta$, then $u\cap\alpha_1=u\cap\alpha_0$.
The family ${{\rm pos}}(w,{{\mathfrak c}},\beta)$ consists of all elements $u$ of ${{\rm pos}}^+(w,{{\mathfrak c}},\beta)$ which satisfy also the following condition:
1. if $\alpha_0=\min\big(\{\alpha\in C:w\subseteq\alpha\}\big)\leq
\beta$, $\alpha_1=\min\big(C\setminus(\alpha_0+1)\big)\leq\beta$, then $u
\cap\alpha_1=c_{\alpha_0}(w)$.
4. A $C$–extending sequence ${{\mathfrak c}}=\langle c_\alpha:\alpha\in C\rangle$ is [*$S$–closed*]{} provided that
1. $C$ is a club of $\lambda$, and
2. if $\alpha\in C$ and $u\subseteq\alpha$, then $\alpha\in
c_\alpha(u)$, and
3. if $\xi\in S\setminus C$, $\alpha\in C\cap\xi$, $u\subseteq
\alpha$ and $\xi=\sup\big(c_\alpha(u)\cap\xi\big)$, then $\xi\in
c_\alpha(u)$.
5. A set $w\subseteq\lambda$ is $S$–closed if $\xi=\sup\big(w\cap\xi
\big)\in S$ implies $\xi\in w$.
6. Let ${{\mathfrak c}}=\langle c_\alpha:\alpha\in C\rangle$ be an $S$–closed $C$–extending sequence, $\beta\in C$, $w\subseteq\beta$ and $\alpha=\min
\big(C\setminus\sup(w)\big)$. Assume also that $w\cup\{\alpha\}$ is $S$–closed. Then we let $$\begin{array}{ll}
{{\rm pos}}^+_S(w,{{\mathfrak c}},\beta)=&\big\{u\in{{\rm pos}}^+(w,{{\mathfrak c}},\beta): u\cup\{\beta\}\mbox{
is $S$--closed }\big\}\\
{{\rm pos}}_S(w,{{\mathfrak c}},\beta)=&\big\{u\in{{\rm pos}}(w,{{\mathfrak c}},\beta): u\cup\{\beta\}\mbox{ is
$S$--closed }\big\}
\end{array}$$
\[easyobs\]
1. Assume that ${{\mathfrak c}}$ is a $C$–extending sequence, $\alpha,\beta\in C$, $\alpha<\beta$ and $w\subseteq\alpha$.
1. If $u\in{{\rm pos}}(w,{{\mathfrak c}},\alpha)$ and $v\in{{\rm pos}}(u,{{\mathfrak c}},\beta)$, then $v\in
{{\rm pos}}(w,{{\mathfrak c}},\beta)$.
2. If $v\in{{\rm pos}}(w,{{\mathfrak c}},\beta)$, then $v\cap\alpha\in{{\rm pos}}(w,{{\mathfrak c}},\alpha)$ and $v\in{{\rm pos}}^+(v\cap\alpha,{{\mathfrak c}},\beta)$.
3. Similarly for ${{\rm pos}}^+$.
2. Assume that ${{\mathfrak c}}$ is an $S$–closed $C$–extending sequence, $\alpha,\beta\in C$, $\alpha<\beta$, $w\subseteq\alpha$ and $w\cup\big\{
\min\big(C\setminus\sup(w)\big)\big\}$ is $S$–closed.
1. If $u\in{{\rm pos}}_S(w,{{\mathfrak c}},\alpha)$ and $v\in{{\rm pos}}_S(u,{{\mathfrak c}},\beta)$, then $v\in {{\rm pos}}_S(w,{{\mathfrak c}},\beta)$.
2. If $v\in{{\rm pos}}_S(w,{{\mathfrak c}},\beta)$, then $v\cap\alpha\in{{\rm pos}}_S(w,{{\mathfrak c}},\alpha)$ and $v\in{{\rm pos}}^+_S(v\cap\alpha,{{\mathfrak c}},\beta)$.
3. Similarly for ${{\rm pos}}^+_S$.
4. $\emptyset\neq {{\rm pos}}_S(w,{{\mathfrak c}},\beta)\subseteq{{\rm pos}}^+_S(w,{{\mathfrak c}},\beta)$.
\[clfor\] We define a forcing notion ${{{{\mathbb Q}}^1_S}}$ as follows.\
[**A condition in ${{{{\mathbb Q}}^1_S}}$**]{} is a triple $p=(w^p,C^p,{{\mathfrak c}}^p)$ such that
1. $C^p\subseteq\lambda$ is a club of $\lambda$ and $w^p\subseteq
\min(C^p)$ is such that the set $w^p\cup\{\min(C^p)\}$ is $S$–closed,
2. ${{\mathfrak c}}^p=\langle c^p_\alpha:\alpha\in C^p\rangle$ is an $S$–closed $C^p$–extending sequence.
[**The order $\leq_{{{{{\mathbb Q}}^1_S}}}=\leq$ of ${{{{\mathbb Q}}^1_S}}$**]{} is given by\
$p\leq_{{{{{\mathbb Q}}^1_S}}} q$if and only if\
1. $C^q\subseteq C^p$ and $w^q\in{{\rm pos}}^+_S(w^p,{{\mathfrak c}}^p,\min(C^q))$ and
2. if $\alpha_0<\alpha_1$ are two successive members of $C^q$, $u\in
{{\rm pos}}^+_S(w^q,{{\mathfrak c}}^q,\alpha_0)$, then $c^q_{\alpha_0}(u)\in{{\rm pos}}_S(u,{{\mathfrak c}}^p,
\alpha_1)$.
For $p\in{{{{\mathbb Q}}^1_S}}$, $\alpha\in C^p$ and $u\in{{\rm pos}}^+_S(w^p,{{\mathfrak c}}^p,\alpha)$ we let $p{{\restriction}}_\alpha u\stackrel{\rm def}{=}(u,C^p\setminus\alpha,{{\mathfrak c}}^p{{\restriction}}(C^p
\setminus\alpha))$.
\[clbas\]
1. ${{{{\mathbb Q}}^1_S}}$ is a $({<}\lambda)$–complete forcing notion of cardinality $2^\lambda$.
2. If $p\in{{{{\mathbb Q}}^1_S}}$ and $\alpha\in C^p$, then
- for each $u\in{{\rm pos}}^+_S(w^p,{{\mathfrak c}}^p,\alpha)$, $p{{\restriction}}_\alpha u\in{{{{\mathbb Q}}^1_S}}$ is a condition stronger than $p$, and
- the family $\{p{{\restriction}}_\alpha u: u\in{{\rm pos}}^+_S(w^p,{{\mathfrak c}}^p,\alpha)\}$ is pre-dense above $p$.
3. Let $p\in{{{{\mathbb Q}}^1_S}}$ and $\alpha<\beta$ be two successive members of $C^p$. Suppose that for each $u\in{{\rm pos}}^+_S(w^p,{{\mathfrak c}}^p,\alpha)$ we are given a condition $q_u\in{{{{\mathbb Q}}^1_S}}$ such that $p{{\restriction}}_\beta c^p_\alpha(u)\leq q_u$. Then there is a condition $q\in{{{{\mathbb Q}}^1_S}}$ such that letting $\alpha'=\min(
C^q\setminus\beta)$ we have
1. $p\leq q$, $w^q=w^p$, $C^q\cap\beta=C^p\cap\beta$ and $c^q_\delta=c^p_\delta$ for $\delta\in C^q\cap\alpha$, and
2. $\bigcup\big\{w^{q_u}:u\in{{\rm pos}}^+_S(w^p,{{\mathfrak c}}^p,\alpha)\big\}
\subseteq\alpha'$, and
3. $q_u\leq q{{\restriction}}_{\alpha'} c^q_\alpha(u)$ for every $u\in{{\rm pos}}^+_S(
w^p,{{\mathfrak c}}^p,\alpha)$.
4. Assume that $p\in{{{{\mathbb Q}}^1_S}}$, $\alpha\in C^p$ and $\name{\tau}$ is a ${{{{\mathbb Q}}^1_S}}$–name such that $p{\Vdash}$“$\name{\tau}\in{{\bf V}}$”. Then there is a condition $q\in{{{{\mathbb Q}}^1_S}}$ stronger than $p$ and such that
1. $w^q=w^p$, $\alpha\in C^q$ and $C^q\cap\alpha=C^p\cap\alpha$, and
2. if $u\in{{\rm pos}}^+_S(w^q,{{\mathfrak c}}^q,\alpha)$ and $\gamma=\min(C^q\setminus
(\alpha+1))$, then the condition $q{{\restriction}}_\gamma c^q(u)$ forces a value to $\name{\tau}$.
(1)It should be clear that ${{{{\mathbb Q}}^1_S}}$ is a forcing notion of size $2^\lambda$. To show that it is $({<}\lambda)$–complete suppose that $\gamma<\lambda$ is a limit ordinal and $\bar{p}=\langle p_\xi:\xi<
\gamma\rangle\subseteq{{{{\mathbb Q}}^1_S}}$ is $\leq_{{{{{\mathbb Q}}^1_S}}}$–increasing. We put $w^q=\bigcup\limits_{\xi<\gamma}w^{p_\xi}$, $C^q=\bigcap\limits_{\xi<
\gamma} C^{p_\xi}$ and for $\delta\in C^q$ we define $c^q_\delta:
{{\mathcal P}}(\delta)\longrightarrow{{\mathcal P}}\big(\min(C^q\setminus(\delta+1))\big)$ so that
- if $u\in\bigcap\limits_{\xi<\gamma}{{\rm pos}}^+_S(w^{p_\xi},{{\mathfrak c}}^{p_\xi},
\delta)$, then $c^q_\delta(u)=\bigcup\limits_{\xi<\gamma}c^{p_\xi}_\delta(
u)$,
- if $u\subseteq\delta$ but it is not in $\bigcap\limits_{\xi<\gamma}
{{\rm pos}}^+_S(w^{p_\xi},{{\mathfrak c}}^{p_\xi},\delta)$, then $c^q_\delta(u)=u\cup
\{\delta\}$.
Finally we put ${{\mathfrak c}}^q=\langle c^q_\delta:\delta\in
C^q\rangle$ and $q=(w^q,C^q,{{\mathfrak c}}^q)$. One easily checks that $q\in{{{{\mathbb Q}}^1_S}}$ is a condition stronger than all $p_\xi$’s.
(2)Straightforward (remember \[easyobs\](2)).
(3)We let $w^q=w^p$ and $C^q=(C^p\cap\beta)\cup\bigcap\big\{
C^{q_u}:u\in{{\rm pos}}^+_S(w^p,{{\mathfrak c}}^p,\alpha)\big\}$ (plainly, $C^q$ is a club of $\lambda$). Let $\alpha'=\min(C^q\setminus(\alpha+1))=\min(C^q\setminus
\beta)$. For $\delta\in C^q\cap\alpha=C^p\cap\alpha$ put $c^q_\delta=
c^p_\delta$. Next, choose an $(\alpha,\alpha')$–extending function $c^q_\alpha:{{\mathcal P}}(\alpha)\longrightarrow{{\mathcal P}}(\alpha')$ such that $(\forall u\in
{{\rm pos}}^+_S(w^p,{{\mathfrak c}},\alpha))(c^q_\alpha(u)\in{{\rm pos}}_S(w^{q_u},{{\mathfrak c}}^{q_u},\alpha'))$ and $(c^q_\alpha(u)\setminus\alpha)\cup\{\alpha'\}$ is $S$–closed for each $u\subseteq\alpha$. (Remember \[easyobs\](2d); note that, by the definition of $C^q$, $w^{q_u}\subseteq\alpha'$ for each $u\in{{\rm pos}}^+_S(w^p,
{{\mathfrak c}}^p,\alpha)$.) Finally, if $\delta_0<\delta_1$ are two successive members of $C^q\setminus\alpha'$, then choose a $(\delta_0,\delta_1)$–extending function $c^q_{\delta_0}:{{\mathcal P}}(\delta_0)\longrightarrow{{\mathcal P}}(\delta_1)$ so that
1. if $v\subseteq\delta_0$, $u=v\cap\alpha\in{{\rm pos}}^+_S(w^p,{{\mathfrak c}}^p,
\alpha)$ and $v\in{{\rm pos}}^+_S(w^{q_u},{{\mathfrak c}}^{q_u},\delta_0)$, then $c^q_{\delta_0}(v)\in{{\rm pos}}_S(v,{{\mathfrak c}}^{q_u},\delta_1)$;
2. if $v\subseteq\delta_0$ but we are not in a case covered by (i), then $c^q_{\delta_0}(v)\in{{\rm pos}}_S(v,{{\mathfrak c}}^p,\delta_1)$.
Let ${{\mathfrak c}}^q=\langle c^q_\delta:\delta\in C^q\rangle$ and $q=(w^q,C^q,{{\mathfrak c}}^q)$. It should be clear that $q\in{{{{\mathbb Q}}^1_S}}$ is a condition as required.
(4)Easily follows from (3).
\[natural\] Suppose that $\gamma<\lambda$ is a limit ordinal and $\bar{p}=\langle
p_\xi:\xi<\lambda\rangle\subseteq{{{{\mathbb Q}}^1_S}}$ is $\leq_{{{{{\mathbb Q}}^1_S}}}$–increasing. The condition $q$ constructed as in the proof of \[clbas\](1) for $\bar{p}$ will be called [*the natural limit of $\bar{p}$*]{}.
\[cllimit\]
1. Suppose $\bar{p}=\langle p_\xi:\xi<\lambda\rangle$ is a $\leq_{{{{{\mathbb Q}}^1_S}}}$–increasing sequence of conditions from ${{{{\mathbb Q}}^1_S}}$ such that
1. $w^{p_\xi}=w^{p_0}$ for all $\xi<\lambda$, and
2. if $\gamma<\lambda$ is limit, then $p_\gamma$ is the natural limit of $\bar{p}{{\restriction}}\gamma$, and
3. for each $\xi<\lambda$, if $\delta\in C^{p_\xi}$, ${{\rm otp}\/}(C^{p_\xi}
\cap\delta)=\xi$, then $C^{p_{\xi+1}}\cap (\delta+1)=C^{p_\xi}\cap
(\delta+1)$ and for every $\alpha\in C^{p_{\xi+1}}\cap\delta$ we have $c^{p_{\xi+1}}_\alpha=c^{p_\xi}_\alpha$.
Then the sequence $\bar{p}$ has an upper bound in ${{{{\mathbb Q}}^1_S}}$.
2. Suppose that $p\in{{{{\mathbb Q}}^1_S}}$ and $\name{h}$ is a ${{{{\mathbb Q}}^1_S}}$–name such that $p{\Vdash}$“$\name{h}:\lambda\longrightarrow{{\bf V}}$”. Then there is a condition $q\in{{{{\mathbb Q}}^1_S}}$ stronger than $p$ and such that
1. if $\delta<\delta'$ are two successive points of $C^q$, $u\in{{\rm pos}}^+_S(w^q,{{\mathfrak c}}^q,\delta)$, then the condition $q{{\restriction}}_{\delta'}
c^q_\delta(u)$ decides the value of $\name{h}{{\restriction}}(\delta+1)$.
(1)First let us note that if $\delta\in\mathop{\triangle}\limits_{
\xi<\lambda} C^{p_\xi}$ is a limit ordinal, then $\delta\in\bigcap\limits_{
\xi<\lambda} C^{p_\xi}$ and $c^{p_{\delta+1}}_\delta=c^{p_\xi}_\delta$ for all $\xi\geq\delta+2$ (by assumptions (b) and (c)). Now, we put $w^q=
w^{p_0}$ and $C^q=\{\delta\in\mathop{\triangle}\limits_{\xi<\lambda}
C^{p_\xi}:\delta\mbox{ is limit }\}$, and for $\delta\in C^q$ we let $c^q_\delta=c^{p_{\delta+1}}_\delta$ (thus defining ${{\mathfrak c}}^q=\langle
c^q_\delta:\delta\in C^q\rangle$). It should be clear that $q=(w^q,C^q,
{{\mathfrak c}}^q)\in{{{{\mathbb Q}}^1_S}}$ is an upper bound to $\bar{p}$.
(2)Follows from (1) above and \[clbas\](4).
\[clnames\] We let $\name{W}$ and $\name{\eta},\name{\nu}$ be ${{{{\mathbb Q}}^1_S}}$–names such that $${\Vdash}_{{{{{\mathbb Q}}^1_S}}}\name{W}=\bigcup\big\{w^p:p\in\Gamma_{{{{{\mathbb Q}}^1_S}}}\big\}$$ and $$\begin{array}{ll}
{\Vdash}_{{{{{\mathbb Q}}^1_S}}}&\mbox{`` }\name{\eta},\name{\nu}\in{{}^{\lambda}\lambda}\mbox{ and if
}\langle\delta_\xi:\xi<\lambda\rangle\mbox{ is the increasing enumeration of
}{{\rm cl}\/}(\name{W}),\\
&\ \mbox{ and }\delta_\xi\leq\alpha<\delta_{\xi+1},\ \xi<\lambda,\mbox{ then
} \name{\eta}(\alpha)=\xi\mbox{ and }\name{\nu}(\alpha)=\delta_{\xi+4}
\mbox{ ''.}\end{array}$$
\[easyW\]
1. ${\Vdash}_{{{{\mathbb Q}}^1_S}}$“ $\name{W}$ is an unbounded $S$–closed subset of $\lambda$ ”. Consequently ${\Vdash}_{{{{\mathbb Q}}^1_S}}$“ $\name{W}\in
{{\mathcal U}}^{{{{{\mathbb Q}}^1_S}}}$ ”.
2. ${\Vdash}_{{{{\mathbb Q}}^1_S}}\mbox{`` }\name{W},\lambda\setminus\name{W}\in
\big({{\mathcal V}}^{{{{\mathbb Q}}^1_S}}\big)^+\mbox{ ''}$.
3. ${\Vdash}_{{{{{\mathbb Q}}^1_S}}}\big(\forall f\in{{}^{\lambda}\lambda}{\cap}{{\bf V}}\big)
\big(\forall A\in{{\mathcal V}}^{{{{{\mathbb Q}}^1_S}}}\big)\big(\exists\alpha\in A\big)\big(
f(\alpha)<\name{\nu}(\alpha)\big)$.
(2)Suppose that $p\in{{{{\mathbb Q}}^1_S}}$ and $\name{A}_i$ (for $i<\lambda$) are ${{{{\mathbb Q}}^1_S}}$–names for elements of ${{\mathcal V}}\cap{{\bf V}}$. Build inductively sequences $\langle p_i:i\leq\lambda\rangle\subseteq{{{{\mathbb Q}}^1_S}}$ and $\langle A_i:i\leq
\lambda\rangle\subseteq{{\mathcal V}}$ such that
1. $\big(\forall i<j<\lambda\big)\big(p\leq p_i\leq p_j\big)$,
2. $p_{i+1}{\Vdash}_{{{{{\mathbb Q}}^1_S}}}\name{A}_i=A_i$ and $i\leq\sup(w^{p_i})$ for all $i<\lambda$,
3. if $\gamma<\lambda$ is limit, then $p_\gamma$ is the natural limit of $\langle p_i:i<\gamma\rangle$.
Pick $\delta\in\mathop{\triangle}\limits_{i<\lambda} A_i\setminus S$ such that $\delta=\sup\big(\bigcup\limits_{i<\delta} w^{p_i}\big)\in
C^{p_\delta}$ (possible by the normality of ${{\mathcal V}}$; remember (b,c) above). Then $p_\delta{\Vdash}\delta\in\mathop{\triangle}\limits_{i<\lambda}
\name{A}_i$. Put $\beta=\min\big(C^{p_\delta}\setminus (\delta+1)\big)$.
Let $w=c^{p_\delta}_\delta(w^{p_{\delta}})$ and $p^*=p{{\restriction}}_\beta w$. Then $p^*\geq p$ and $p{\Vdash}\delta\in\name{W}$.
On the other hand, since $\delta=\sup(w^{p_\delta})\notin S$, we have $w^{p_\delta}\in{{\rm pos}}_S^+(w^{p_\delta},{{\mathfrak c}}^{p_\delta},\beta)$ so we may let $p^{**}=p{{\restriction}}_\beta w^{p_\delta}$. Then $p^{**}\geq p$ and $p{\Vdash}\delta\notin\name{W}$.
(3)Suppose that $p\in{{{{\mathbb Q}}^1_S}}$, $f\in{{}^{\lambda}\lambda}$ and $\langle
\name{A}_\alpha:\alpha<\lambda\rangle$ is a sequence of ${{{{\mathbb Q}}^1_S}}$–names for members of ${{\mathcal V}}\cap{{\bf V}}$. By induction on $\alpha<\lambda$ construct a sequence $\langle p_\alpha,A_\alpha:\alpha<\lambda\rangle$ such that for each $\alpha$:
1. $p_\alpha\in{{{{\mathbb Q}}^1_S}}$, $A_\alpha\subseteq\lambda$, $A_\alpha\in{{\mathcal V}}$, $p_0=p$, $p_\alpha\leq_{{{{{\mathbb Q}}^1_S}}} p_{\alpha+1}$, and
2. if $\alpha$ is a limit ordinal, then $p_\alpha$ is the natural limit of $\langle p_\beta:\beta<\alpha\rangle$, and
3. $p_{\alpha+1}{\Vdash}_{{{{{\mathbb Q}}^1_S}}} \name{A}_\alpha\cap (\lambda
\setminus S)=A_\alpha$.
Next pick a limit ordinal $\delta\in\mathop{\triangle}\limits_{\alpha<
\lambda} A_\alpha\cap (\lambda\setminus S)$ such that $(\forall\alpha<
\delta)(w^{p_\alpha}\subseteq\delta)$. Then $p_\delta{\Vdash}\delta\in
\mathop{\triangle}\limits_{\alpha<\lambda}\name{A}_\alpha$ and $w^{p_\delta}\subseteq\delta$ is $S$–closed, so we may let $w^q=w^{p_\delta}$, $C^q=C^{p_\delta}\setminus\big(f(\delta)+1\big)$ and ${{\mathfrak c}}^q={{\mathfrak c}}^{p_\delta}{{\restriction}}C^q$ to get a condition $q\in{{{{\mathbb Q}}^1_S}}$ stronger than $p$ and such that $$q{\Vdash}_{{{{{\mathbb Q}}^1_S}}}\mbox{`` }\delta\in\mathop{\triangle}\limits_{\alpha<
\lambda}\name{A}_\alpha\mbox{ and }f(\delta)<\name{\nu}(\delta)\mbox{ ''.}$$
\[clfoOK\] The forcing notion ${{{{\mathbb Q}}^1_S}}$ is reasonably B–bounding over ${{\mathcal U}}$.
By \[clbas\](1), ${{{{\mathbb Q}}^1_S}}$ is $({<}\lambda)$–complete, so we have to verify \[p.1A\](5b) only. Let $p\in{{{{\mathbb Q}}^1_S}}$ and let $\bar{\mu}=\langle
\mu_\alpha':\alpha<\lambda\rangle$, $\mu_\alpha'=\lambda$ for each $\alpha<\lambda$. We are going to describe a strategy ${{\bf st}}$ for Generic in ${{\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}'}}}(p,{{{{\mathbb Q}}^1_S}})$.
In the course of a play the strategy ${{\bf st}}$ instructs Generic to build aside an increasing sequence of conditions $\bar{p}^*=\langle p^*_\alpha:\alpha<
\lambda\rangle\subseteq{{{{\mathbb Q}}^1_S}}$ such that
1. $p_0^*=p$ and $w^{p^*_\alpha}=w^p$ for all $\alpha<\lambda$, and
2. if $\gamma<\lambda$ is limit, then $p^*_\gamma$ is the natural limit of $\bar{p}^*{{\restriction}}\gamma$, and
3. for each $\alpha<\lambda$, if $\delta\in C^{p^*_\alpha}$, ${{\rm otp}\/}(C^{p^*_\alpha}\cap\delta)=\alpha$, then $C^{p^*_{\alpha+1}}\cap
(\delta+1)=C^{p^*_\alpha}\cap (\delta+1)$ and for every $\xi\in
C^{p^*_{\alpha+1}}\cap\delta$ we have $c^{p^*_{\alpha+1}}_\xi=
c^{p^*_\alpha}_\xi$, and
4. after stage $\alpha<\lambda$ of the play of ${{\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}'}}}(p,{{{{\mathbb Q}}^1_S}})$, the condition $p_{\alpha+1}$ is determined (conditions $p_\alpha$ for non-successor $\alpha<\lambda$ are determined by (a),(b) above).
So suppose that the players arrived to a stage $\alpha<\lambda$ of ${{\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}'}}}(p,{{{{\mathbb Q}}^1_S}})$, and Generic (playing according to ${{\bf st}}$ so far) has constructed aside an increasing sequence $\langle p^*_\xi:\xi\leq p^*_\alpha
\rangle$ of conditions (satisfying (a)–(d)). Let $\delta\in C^{p^*_\alpha}$ be such that ${{\rm otp}\/}(C^{p^*_\alpha}\cap\delta)=\alpha$ and let $\gamma=\min(
C^{p^*_\alpha}\setminus(\delta+1))$. Now Generic makes her move in ${{\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}'}}}(p,{{{{\mathbb Q}}^1_S}})$:
- $I_\alpha={{\rm pos}}^+_S(w^{p^*_\alpha},{{\mathfrak c}}^{p^*_\alpha},\delta)$, and
- $p^\alpha_u=p^*_\alpha{{\restriction}}_\gamma c^{p^*_\alpha}_\delta(u)$ for $u\in
I_\alpha$.
Let $\langle q^\alpha_u:u\in I_\alpha\rangle\subseteq{{{{\mathbb Q}}^1_S}}$ be the answer of Antigeneric, so $p^*_\alpha{{\restriction}}_\gamma c^{p^*_\alpha}_\delta(u)\leq
q^\alpha_u$ for each $u\in{{\rm pos}}^+(w^{p^*_\alpha},{{\mathfrak c}}^{p^*_\alpha},\delta)$. Now Generic uses \[clbas\](3) (with $\delta,\gamma,p^*_\alpha,q^\alpha_u$ here standing for $\alpha,\beta,p,q_u$ there) to pick a condition $p^*_{\alpha+1}$ such that, letting $\alpha'=\min(C^{p^*_{\alpha+1}}
\setminus\gamma)$, we have
1. $p^*_\alpha\leq p^*_{\alpha+1}$, $w^{p^*_{\alpha+1}}=w^p$, $C^{
p^*_{\alpha+1}}\cap\gamma=C^{p^*_\alpha}\cap\gamma$ and $c^{p^*_{\alpha+
1}}_\xi=c^{p^*_\alpha}_\xi$ for $\xi\in C^{p^*_{\alpha+1}}\cap\delta$, and
2. $\bigcup\big\{w^{q^\alpha_u}:u\in I_\alpha\big\}\subseteq
\alpha'$, and
3. $q^\alpha_u\leq p^*_{\alpha+1}{{\restriction}}_{\alpha'}
c^{p^*_{\alpha+1}}_\delta(u)$ for every $u\in I_\alpha$.
We claim that ${{\bf st}}$ is a winning strategy for Generic in ${{\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}'}}}(p,{{{{\mathbb Q}}^1_S}})$. So suppose that $$\Big\langle I_\alpha,\langle p^\alpha_u,q^\alpha_u:u\in I_\alpha\rangle:
\alpha<\lambda\Big\rangle$$ is a play of ${{\Game^{\rm rcB}_{{{\mathcal U}},\bar{\mu}'}}}(p,{{{{\mathbb Q}}^1_S}})$ in which Generic uses ${{\bf st}}$, and let $\bar{p}^*=\langle p^*_\alpha:\alpha<\lambda\rangle\subseteq{{{{\mathbb Q}}^1_S}}$ be the sequence constructed aside by Generic, so it satisfies (a)–(c) above, and thus also the assumptions of \[cllimit\](1). Let $p^*$ be an upper bound to $\bar{p}$ (which exists by \[cllimit\](1)). Now note that $$p^*{\Vdash}_{{{{\mathbb Q}}^1_S}}\mbox{`` if }\alpha\in C^{p^*}\cap\name{W}\mbox{ and }u=
\name{W}\cap\alpha,\mbox{ then }q^\alpha_u\in\Gamma_{{{{{\mathbb Q}}^1_S}}}\mbox{ ''}$$ and therefore $$p^*{\Vdash}_{{{{\mathbb Q}}^1_S}}\mbox{`` }\big(\forall\alpha\in C^{p^*}\cap \name{W}\big)
\big(\exists u\in I_\alpha\big)\big(q^\alpha_u\in\Gamma_{{{{{\mathbb Q}}^1_S}}}\big)
\mbox{ ''.}$$ Since $p^*{\Vdash}C^{p^*}\cap\name{W}\in{{\mathcal U}}^{{{{\mathbb Q}}^1_S}}$ (by \[easyW\]) we may conclude that the condition $p^*$ witnesses that Generic won the play.
Let ${{\mathcal F}}$ be a filter on $\lambda$ including all co=bounded subsets of $\lambda$, $\emptyset\notin{{\mathcal F}}$.
1. We say that a family $F\subseteq{{}^{\lambda}\lambda}$ is [*${{\mathcal F}}$–dominating*]{} whenever $$\big(\forall g\in {{}^{\lambda}\lambda}\big)\big(\exists f\in F\big)
\big(\{\alpha<\lambda: g(\alpha)<f(\alpha)\}\in{{\mathcal F}}\big).$$
2. The ${{\mathcal F}}$–dominating number ${{\mathfrak d}}_{{\mathcal F}}$ is the minimal size of an ${{\mathcal F}}$–dominating family in ${{}^{\lambda}\lambda}$.
3. If ${{\mathcal F}}$ is the filter of co-bounded subsets of $\lambda$, then the corresponding dominating number is also denoted by ${{\mathfrak d}}_\lambda$. If ${{\mathcal F}}$ is the filter generated by club subsets of $\lambda$, then the corresponding dominating number is called ${{\mathfrak d}}_{\rm cl}$.
It was shown in Cummings and Shelah [@CuSh:541] that ${{\mathfrak d}}_\lambda={{\mathfrak d}}_{\rm cl}$ (whenever $\lambda>\beth_\omega$ is regular). The following conclusion is an interesting addition to that result.
\[conc\] It is consistent that $\lambda$ is an inaccessible cardinal and there are two normal filters ${{\mathcal U}}',{{\mathcal U}}''$ on $\lambda$ such that ${{\mathfrak d}}_{{{\mathcal U}}'}\neq{{\mathfrak d}}_{{{\mathcal U}}''}$.
Start with the universe where $\lambda,{{\mathcal U}},{{\mathcal V}},S$ are as in \[incon\]+ \[extraassum\] and $2^\lambda=\lambda^+$. Let $\bar{{{\mathbb Q}}}=\langle{{\mathbb P}}_\xi,
\name{{{\mathbb Q}}}_\xi:\xi<\lambda^{++}\rangle$ be a $\lambda$–support iteration such that for every $\xi<\lambda^{++}$, ${\Vdash}_{{{\mathbb P}}_\xi}\mbox{``
}\name{{{\mathbb Q}}}_\xi={{{{\mathbb Q}}^1_S}}$ ”.
It follows from \[verB\] that ${{\mathbb P}}_{\lambda^{++}}$ is reasonably [**b**]{}–bounding over ${{\mathcal U}}$, and hence also $\lambda$–proper. Therefore using \[clbas\](1) and [@RoSh:777 Theorem A.1.10] (see also Eisworth [@Ei03 §3]) one can easily argue that the limit ${{\mathbb P}}_{\lambda^{++}}$ of the iteration satisfies the $\lambda^{++}$–cc, has a dense subset of size $\lambda^{++}$, is strategically $({<}\lambda)$–complete and $\lambda$–proper. Consequently, the forcing with ${{\mathbb P}}_{\lambda^{++}}$ does not collapse cardinal. Also it follows from \[bound\] that $${\Vdash}_{{{\mathbb P}}_{\lambda^{++}}}\mbox{`` ${{}^{\lambda}\lambda}\cap{{\bf V}}$ is
$\big({{\mathcal U}}\big)^{{{\mathbb P}}_{\lambda^{++}}}$--dominating in ${{}^{\lambda}\lambda}$ ''}$$ and it follows from \[easyW\](3) that for each $\xi<\lambda^{++}$ $${\Vdash}_{{{\mathbb P}}_{\lambda^{++}}}\mbox{`` ${{}^{\lambda}\lambda}\cap{{\bf V}}^{{{\mathbb P}}_\xi}$ is not
$\big({{\mathcal V}}\big)^{{{\mathbb P}}_{\lambda^{++}}}$--dominating in ${{}^{\lambda}\lambda}$ ''}$$ Therefore we may easily conclude that $$\begin{array}{ll}
{\Vdash}_{{{\mathbb P}}_{\lambda^{++}}}&\mbox{`` if }{{\mathcal U}}'=\big({{\mathcal U}}\big)^{{{\mathbb P}}_{\lambda^{++}}},\ {{\mathcal U}}''=\big({{\mathcal V}}\big)^{{{\mathbb P}}_{\lambda^{++}}}
\mbox{ then}\\
&\quad{{\mathfrak b}}_\lambda={{\mathfrak d}}_{{{\mathcal U}}'}=\lambda^+<2^\lambda=\lambda^{++}={{\mathfrak d}}_{{{\mathcal U}}''}=
{{\mathfrak d}}_{\rm cl}={{\mathfrak d}}_\lambda\mbox{ ''.}
\end{array}$$
$$\qquad *\qquad *$$
\[3.2\] We define a forcing notion ${{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ as follows.\
[**A condition in ${{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$**]{} is a triple $p=(w^p,C^p,{{\mathfrak c}}^p)$ such that
1. $C^p\in {{\mathcal U}}$, $w^p\subseteq\min(C^p)$,
2. ${{\mathfrak c}}^p=\langle c^p_\alpha:\alpha\in C^p\rangle$ is a $C^p$–extending sequence.
[**The order $\leq_{{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}}=\leq$ of ${{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$**]{} is given by\
$p\leq_{{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}} q$if and only if\
1. $C^q\subseteq C^p$ and $w^q\in{{\rm pos}}^+(w^p,{{\mathfrak c}}^p,\min(C^q))$ and
2. if $\alpha_0,\alpha_1\in C^q$, $\alpha_0<\alpha_1=\min(C^q
\setminus(\alpha_0+1))$ and $u\in{{\rm pos}}^+(w^q,{{\mathfrak c}}^q,\alpha_0)$, then $c^q_{\alpha_0}(u)\in {{\rm pos}}(u,{{\mathfrak c}}^p,\alpha_1)$.
For $p\in{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$, $\alpha\in C^p$ and $u\in{{\rm pos}}^+(w^p,{{\mathfrak c}}^p,\alpha)$ we let $p{{\restriction}}_\alpha u\stackrel{\rm def}{=}(u,C^p\setminus\alpha,{{\mathfrak c}}^p{{\restriction}}(C^p
\setminus\alpha))$.
\[3.4\]
1. ${{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ is a $\lambda$–complete forcing notion of cardinality $2^\lambda$.
2. If $p\in{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ and $\alpha\in C^p$, then
- for each $u\in{{\rm pos}}^+(w^p,{{\mathfrak c}}^p,\alpha)$, $p{{\restriction}}_\alpha u\in{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ is a condition stronger than $p$, and
- the family $\{p{{\restriction}}_\alpha u: u\in{{\rm pos}}^+(w^p,{{\mathfrak c}}^p,\alpha)\}$ is pre-dense above $p$.
3. Let $p\in{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ and $\alpha<\beta$ be two successive members of $C^p$. Suppose that for each $u\in{{\rm pos}}^+(w^p,{{\mathfrak c}}^p,\alpha)$ we are given a condition $q_u\in{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ such that $p{{\restriction}}_\beta c^p_\alpha(u)\leq q_u$. Then there is a condition $q\in{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ such that letting $\alpha'=\min(
C^q\setminus\beta)$ we have
1. $p\leq q$, $w^q=w^p$, $C^q\cap\beta=C^p\cap\beta$ and $c^q_\delta=c^p_\delta$ for $\delta\in C^q\cap\alpha$, and
2. $\bigcup\big\{w^{q_u}:u\in{{\rm pos}}^+(w^p,{{\mathfrak c}}^p,\alpha)\big\}\subseteq
\alpha'$, and
3. $q_u\leq q{{\restriction}}_{\alpha'} c^q_\alpha(u)$ for every $u\in{{\rm pos}}^+(w^p,{{\mathfrak c}}^p,\alpha)$.
4. Assume that $p\in{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$, $\alpha\in C^p$ and $\name{\tau}$ is a ${{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$–name such that $p{\Vdash}$“$\name{\tau}\in{{\bf V}}$”. Then there is a condition $q\in{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ stronger than $p$ and such that
1. $w^q=w^p$, $\alpha\in C^q$ and $C^q\cap\alpha=C^p\cap\alpha$, and
2. if $u\in{{\rm pos}}^+(w^q,{{\mathfrak c}}^q,\alpha)$ and $\gamma=\min(C^q\setminus
(\alpha+1))$, then the condition $q{{\restriction}}_\gamma c^q(u)$ forces a value to $\name{\tau}$.
Fully parallel to \[clbas\].
[*The natural limit*]{} of an $\leq_{{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}}$–increasing sequence $\bar{p}=\langle p_\xi:\xi<\lambda\rangle\subseteq{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ (where $\gamma<\lambda$ is a limit ordinal) is the condition $q=(w^q,C^q,{{\mathfrak c}}^q)$ defined as follows:
- $w^q=\bigcup\limits_{\xi<\gamma}w^{p_\xi}$, $C^q=\bigcap\limits_{\xi<
\gamma} C^{p_\xi}$ and
- ${{\mathfrak c}}^q=\langle c^q_\delta:\delta\in C^q\rangle$ is such that for $\delta\in C^q$ and $u\subseteq\delta$ we have $c^q_\delta(u)=
\bigcup\limits_{\xi<\gamma}c^{p_\xi}_\delta(u)$.
\[limit\]
1. Suppose $\bar{p}=\langle p_\xi:\xi<\lambda\rangle$ is a $\leq_{{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}}$–increasing sequence of conditions from ${{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ such that
1. $w^{p_\xi}=w^{p_0}$ for all $\xi<\lambda$, and
2. if $\gamma<\lambda$ is limit, then $p_\gamma$ is the natural limit of $\bar{p}{{\restriction}}\gamma$, and
3. for each $\xi<\lambda$, if $\delta\in C^{p_\xi}$, ${{\rm otp}\/}(C^{p_\xi}
\cap\delta)=\xi$, then $C^{p_{\xi+1}}\cap (\delta+1)=C^{p_\xi}\cap
(\delta+1)$ and for every $\alpha\in C^{p_{\xi+1}}\cap\delta$ we have $c^{p_{\xi+1}}_\alpha=c^{p_\xi}_\alpha$.
Then the sequence $\bar{p}$ has an upper bound in ${{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$.
2. Suppose that $p\in{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ and $\name{h}$ is a ${{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$–name such that $p{\Vdash}$“$\name{h}:\lambda\longrightarrow{{\bf V}}$”. Then there is a condition $q\in{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ stronger than $p$ and such that
1. if $\delta<\delta'$ are two successive points of $C^q$, $u\in{{\rm pos}}(w^q,{{\mathfrak c}}^q,\delta)$, then the condition $q{{\restriction}}_{\delta'}
c^q_\delta(u)$ decides the value of $\name{h}{{\restriction}}(\delta+1)$.
Fully parallel to \[cllimit\].
\[names\] We let $\name{W}$ and $\name{\eta},\name{\nu}$ be ${{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$–names such that $${\Vdash}_{{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}}\name{W}=\bigcup\big\{w^p:p\in\Gamma_{{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}}\big\}$$ and $$\begin{array}{ll}
{\Vdash}_{{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}}&\mbox{`` }\name{\eta},\name{\nu}\in{{}^{\lambda}\lambda}\mbox{ and if
}\langle\delta_\xi:\xi<\lambda\rangle\mbox{ is the increasing enumeration of
}{{\rm cl}\/}(\name{W}),\\
&\ \mbox{ and }\delta_\xi\leq\alpha<\delta_{\xi+1},\ \xi<\lambda,\mbox{ then
} \name{\eta}(\alpha)=\xi\mbox{ and }\name{\nu}(\alpha)=\delta_{\xi+4}
\mbox{ ''.}\end{array}$$
Note that if $p\in{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$, then $$p{\Vdash}_{{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}}\mbox{`` }\name{W}\subseteq\bigcup\big\{[\alpha_0,
\alpha_1):\alpha_0,\alpha_1\in C^p\ \&\ \alpha_1=\min\big(C^p\setminus(
\alpha_0+1)\big)\big\}\mbox{ ''}$$ and $$\begin{array}{ll}
p{\Vdash}_{{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}}&\mbox{`` }\big\{\alpha\in C^p:\big[\alpha,\min(C^p
\setminus(\alpha+1))\big)\cap\name{W}\neq\emptyset\big\}, \\
&\ \ \big\{\alpha\in C^p:\big[\alpha,\min(C^p\setminus
(\alpha+1))\big)\cap\name{W}=\emptyset\big\}\in\big({{\mathcal U}}^{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}\big)^+
\mbox{ ''}.\end{array}$$
\[3.1\] ${\Vdash}_{{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}}\big(\forall f\!\in\!{{}^{\lambda}\lambda}{\cap}{{\bf V}}\big)
\big(\forall A\!\in\!{{\mathcal U}}^{{{{{\mathbb Q}}^2_{{{\mathcal U}}}}}}\big)\big(\exists\alpha\!\in\! A\big)
\big(f(\alpha)<\name{\nu}(\alpha)\big)$.
Fully parallel to \[easyW\].
\[tefoOK\] The forcing notion ${{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ is reasonably C–bounding over ${{\mathcal U}}$.
Fully parallel to \[clfoOK\].
The following problem is a particular case of \[prob\](1).
Are $\lambda$–support iterations of ${{{{\mathbb Q}}^2_{{{\mathcal U}}}}}$ $\lambda$–proper?
[1]{}
James Cummings and Saharon Shelah. . , 75:251–268, 1995. math.LO/9509228.
Todd Eisworth. . , 179:249–266, 2003, math.LO/0210162.
Thomas Jech. . Academic Press, New York, 1978.
Andrzej Roslanowski and Saharon Shelah. . , submitted. math.LO/0210205.
Andrzej Roslanowski and Saharon Shelah. , 28:63–82, 2001. math.LO/9906024.
Saharon Shelah. . . [Springer]{}, 1998.
Saharon Shelah. . , 136:29–115, 2003. math.LO/9707225.
Saharon Shelah. . , [134]{}:127–155, 2003. math.LO/9808140.
[^1]: Both authors acknowledge support from the United States-Israel Binational Science Foundation (Grant no. 2002323). This is publication 860 of the second author.
[^2]: equivalently, for every $\alpha<\lambda$ the set $\big\{q^\alpha_t:t\in
I_\alpha\big\}$ is pre-dense above $p^*$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The abundance of brown dwarfs (BDs) in young clusters is a diagnostic of star formation theory. Here we revisit the issue of determining the substellar initial mass function (IMF), based on a comparison between NGC1333 and IC348, two clusters in the Perseus star-forming region. We derive their mass distributions for a range of model isochrones, varying distances, extinction laws and ages, with comprehensive assessments of the uncertainties. We find that the choice of isochrone and other parameters have significant effects on the results, thus we caution against comparing IMFs obtained using different approaches. For NGC1333, we find that the star/BD ratio $R$ is between 1.9 and 2.4, for all plausible scenarios, consistent with our previous work. For IC348, $R$ is found to be between 2.9 and 4.0, suggesting that previous studies have overestimated this value. Thus, the star forming process generates about 2.5-5 substellar objects per 10 stars. The derived star/BD ratios correspond to a slope of the power-law mass function of $\alpha = 0.7-1.0$ for the 0.03-1.0$\,M_{\odot}$ mass range. The median mass in these clusters – the typical stellar mass – is between 0.13-0.30$\,M_{\odot}$. Assuming that NGC1333 is at a shorter distance than IC348, we find a significant difference in the cumulative distribution of masses between the two clusters, resulting from an overabundance of very low mass objects in NGC1333. Gaia astrometry will constrain the cluster distances better and will lead to a more definitive conclusion. Furthermore, the star/BD ratio is somewhat larger in IC348 compared with NGC1333, although this difference is still within the margins of error. Our results indicate that environments with higher object density may produce a larger fraction of very low mass objects, in line with predictions for brown dwarf formation through gravitational fragmentation of filaments falling into a cluster potential.'
author:
- 'Alexander Scholz, Vincent Geers, Paul Clark, Ray Jayawardhana, Koraljka Muzic'
title: |
Substellar Objects in Nearby Young Clusters VII:\
The substellar mass function revisited
---
Introduction {#s1}
============
Brown dwarfs (BDs) are an ubiquitous outcome of the star formation process. All young regions investigated so far with sufficient depth host a population of BDs with masses down to 0.01$\,M_{\odot}$ or even below. The mechanism that governs their formation, however, remains unknown. It is clear that additional physics needs to be included in the models for the cloud fragmentation and subsequent evolution, to allow for the formation of a sizable number of BDs [@2007prpl.conf..149B]. Plausible options for these processes include fragmentation driven by turbulence, dynamical ejection of embryonic BDs from multiple systems, fragmentation of filaments falling into a cluster potential, or fragmentation of protoplanetary disks, again combined with ejection [@2007prpl.conf..459W; @2008MNRAS.389.1556B; @2009MNRAS.392..413S]. Young brown dwarfs are a critical population to test the relevance of these processes.
The standard diagnostic to distinguish between theoretical scenarios is the distribution of stellar and substellar masses after star formation is finished, or the initial mass function (IMF). In the literature several parameterisations for the IMF are used, for example a series of power laws [@2001MNRAS.322..231K] or a lognormal form [@2003PASP..115..763C]. For our goal of determining the abundance of brown dwarfs, an often used parameterisation of the IMF is the star/BD ratio $R$, the ratio of the number of objects in the two mass bins from 0.08 to 1.0$\,M_{\odot}$ and from 0.03 to 0.08$\,M_{\odot}$, where the low mass cutoff as 0.03$\,M_{\odot}$ is chosen to assure completeness. The upper mass limit for the stars of 1.0$\,M_{\odot}$ is to some extent an arbitrary definition, but the relatively small number of higher-mass stars in the nearby star forming region assures that this particular choice does not affect the result much. The star/BD ratio as a metric has the advantage of maximising the sample size in the substellar regime and thus minimizing the statistical errors. Because the stellar side of the IMF is well-determined for the nearby star forming regions and shows, in the overwhelming majority of regions, no evidence for environmental differences , any variation in the star/BD ratio from one region to another would indicate a change in the BD abundance.
Measuring the substellar mass function and the star/BD ratio is a challenging task. It needs a consistent survey procedure and a careful analysis of possible incompleteness, but the core problem is to estimate the masses. This requires one to make assumptions about the distance to and age of the region, as well as the extinction law used to deredden the photometry or spectra. Furthermore, the conversion from observed quantities to masses can only be done in the framework of a given theoretical isochrone and thus depends on the status of the evolutionary models for young stars and brown dwarfs.
In previous work within the project SONYC (’Substellar Objects in Nearby Young Clusters’) we have presented tentative evidence for regional differences in the star/BD ratio. In particular, for the young cluster NGC1333 in the Perseus star forming complex we find $R \sim 2-3$ based on a very deep survey with comprehensive spectroscopic follow-up (@2009ApJ...702..805S, hereafter SONYC-I; @2012ApJ...744....6S, hereafter SONYC-IV). For other regions we and other groups have published R-values ranging from 2 to 8 (see SONYC-IV). One of the most extreme cases in the literature is the cluster IC348, the second embedded cluster in Perseus and a slightly older sibling to NGC1333 [@2008hsf1.book..308B], with a star/BD ratio of $R \sim 8$ [@2003ApJ...593.1093L; @2008ApJ...683L.183A]. At face value, this indicates changes in the BD abundance by a factor of 4 occuring within the same star forming association. So far, however, the uncertainties for the star/BD ratio have not been assessed accurately. The goal of this paper is to carry out a benchmark test for the two extreme cases NGC1333 and IC348 to verify their discrepant star/BD ratios.
The approach
============
The core idea of this paper is to determine the mass distribution and the star/BD ratio for two young clusters in Perseus and to assess the associated uncertainties. We will start with consistently selected samples for the two clusters, i.e. samples which have been put together in a homogeneous way, to minimize the influence of selection biases and incompleteness. For these samples, we will then define a consistent way of estimating object masses by comparing photometry with a given model isochrone. In addition to the choice of the isochrone, the distance to the cluster, the extinction law, and the age of the region enter as free parameters into this procedure. We will define a set of scenarios with plausible choices for these parameters and different isochrones. We will then estimate object masses and calculate the star/BD ratio and other indicators of the mass distribution for each of these scenarios. This will yield a useful dataset to discuss the uncertainties in these indicators and assess whether there is evidence for regional differences in the substellar IMF between NGC1333 and IC348.
The samples {#s21}
-----------
We use the Spitzer-selected sample of young stellar/substellar objects presented by @2009ApJS..184...18G which includes our two target regions. For each of the regions, a region of $25'\times 25'$ centered on the core of the cluster was observed. The selection in @2009ApJS..184...18G is based on colours and magnitudes in Spitzer and 2MASS bands from 1 to 24$\,\mu m$. Their multi-colour selection process uses a series of criteria designed to exclude background stars, non-stellar emission features and extragalactic objects. According to their analysis, this process yields only minimal contamination (a few percent). An earlier version of this process was used in @2008ApJ...674..336G in a survey of NGC1333. Assuming that the distances to the clusters are similar, the identical selection procedure also ensures that the depth and completeness in terms of magnitudes in the two samples is comparable. This removes a major obstacle for an accurate comparison of the BD abundance.
Since the primary selection criterion is excess emission in the infrared, this sample only contains objects with emission from circumstellar material, i.e. either disks or envelopes. It does not include disk-less young stars and brown dwarfs (Class III objects). Therefore, to infer star/BD ratios, we have to make the assumption that the fraction of objects with disks does not change with object mass in the low-mass regime. As recently shown in @2013MNRAS.429..903D, this is a plausible assumption for many star forming regions, including IC348. For NGC1333, there might be a slight mass dependence, as the disk fraction in the total Spitzer-selected sample is found to be $83\pm 11$% [@2008ApJ...674..336G], whereas the value for the very low mass objects is only 55-66% (SONYC-IV). This indicates that the star/BD ratio in this cluster could be slightly overestimated.
The entire Perseus cloud, including the two target clusters have also been observed as part of the Spitzer Legacy program ’From Cores to Disks’ (C2D, PI: N. Evans). Their Perseus YSO catalogue derived from IRAC photometry is discussed in detail in @2006ApJ...645.1246J. We prefer to use the @2009ApJS..184...18G selection, because it is slightly deeper, which is beneficial for our purposes. The downside of the @2009ApJS..184...18G sample is the limited spatial coverage. Here the C2D catalogue is useful to check the spatial completeness of our samples.
\
In Fig. \[f0\] we show the spatial distribution of the selected objects in NGC1333 and IC348. With red squares we plot the samples from @2009ApJS..184...18G, with black crosses the C2D sample that covers the entire Perseus region. The figure shows that the number of YSO candidates outside the region covered by @2009ApJS..184...18G is small compared with the total population. This is particularly true for NGC1333 which shows a very compact profile and can be considered to be spatially complete (see also SONYC-IV). For IC348, there is a tail of a YSO population towards the south-west, which represents the transition region to another densely populated area in the Perseus star forming region . In addition, there are about 10 objects outside the coverage of the @2009ApJS..184...18G survey. However, there is also a large number of additional YSO candidates only contained in the C2D sample within the cluster core. We checked the objects only in C2D and found that they do not show an obvious magnitude or extinction bias with respect to the sample we are using, thus, even in case we are missing members in the outskirts of the cluster, this is not going to affect our analysis in any significant way. While @2003AJ....125.2029M do find a difference in the IMF between the core and the halo in IC348, both regions (in their definitions) are within the survey area of @2009ApJS..184...18G. We conclude that the samples we are using are not affected by a spatial bias.
Fig. \[f0\] also illustrates one major difference between IC348 and NGC1333. The core of IC348 has about twice the diameter of the core of NGC1333 [2.1 vs. 1.2pc, @2009ApJS..184...18G], i.e. the cluster volume in IC348 is about 8 times larger than in NGC1333. On the other hand, IC348 has only about 20% more YSOs than NGC1333, according to the Spitzer surveys [@2008ApJ...683..822J; @2009ApJS..184...18G]. The fraction of diskless Class III objects is higher in IC348, taken that into account the total the YSO population in IC348 could be up to twice as large as in NGC1333. This still implies that the object density in NGC1333 is 4-7 times higher than in NGC1333. Given the age, size, and number of members in these clusters, this difference is likely to be primordial, and not caused by dynamical evolution [see Fig. 1 in @2012MNRAS.426L..11G]. Hence, these two clusters constitute an excellent test case to probe the effects of dynamical interactions and cluster potential on the formation of BDs.
Estimating masses {#s22}
-----------------
For the overwhelming majority of young objects, masses can only be estimated indirectly by comparing an observed quantity with predictions from theoretical isochrones for a given age. For the observed quantity, there are two options, either the effective temperature or the luminosity (or a photometric proxy). The luminosity has the problem that model derivations are sensitive to the age for pre-main sequence objects that are still contracting. In addition, measurements can be affected by extinction as well as excess emission from disk and/or accretion. The effective temperature is problematic for other reasons; it depends on atmosphere models and can be altered by magnetic activity [@2012ApJ...756...47S]. This can lead us to underestimate object masses by up to a factor of two.
For our chosen samples, accurate multi-band photometry is available, while the spectroscopic follow-up is not complete. Therefore, we will rely on photometry in the optical and near-infrared to estimate masses. We complement the 2MASS photometry provided by @2009ApJS..184...18G with optical photometry from @1999ApJ...525..466L and @2003ApJ...593.1093L for IC348 (Landolt R- and I-band) and from SONYC-I for NGC1333 (Sloan i- and z-band)[^1]. For some objects without 2MASS near-infrared magnitudes, we were able to complement the dataset using the photometry from @2003AJ....125.2029M for IC348[^2] and SONYC-I for NGC1333. In total, the samples contain 142 (for IC348) and 95 (for NGC1333) objects with photometry in JHK, with smaller subsets of 86 (IC348) and 23 (NGC1333) with additional optical magnitudes available.
For the objects with 2MASS photometry, we also obtain the error as listed in the database (mostly between 0.03 and 0.05mag). Similarly, the photometry from @2003AJ....125.2029M provides errors for all measurements. For the remaining photometry, errors for individual objects are not reported in the literature. For the optical magnitudes in IC348, we adopt a generic and conservative uncertainty of 0.1mag. For the SONYC magnitudes in NGC1333, we adopt errors of 0.1mag for the optical bands and 0.05mag for the near-infrared bands.
Based on the information given in the papers listed above, the error values adopted for the optical photometry should be typical for the samples. However, some objects might be affected by additional uncertainties introduced to calibration imperfections. Since the z- and I-bands are located at the long-wavelength edge of the sensitivity of the optical CCDs, they are highly susceptible to colour terms in the calibration, which are difficult to measure with the usual photometric standard stars. This can introduce errors larger than 0.1mag in individual sources, which cannot be quantified accurately. This issue is a particular problem for very red sources, since most of their optical flux is emitted in the part of the spectrum where the CCD sensitivity declines.
We derive masses using three different sets of isochrones from the Lyon group, BT-Settl (with AGSS2009 opacities), BT-Dusty (with AGSS2009 opacities), and BT-Nextgen (with GNC93 opacities).[^3] The latter two are updated versions from the standard AMES-Dusty and Nextgen models. The main difference between the three sets is the treatment of dust. In contrast to Nextgen, Dusty includes dust opacities. Settl includes a full dust cloud model. For more information on the isochrones, see @2011ASPC..448...91A [@2001ApJ...556..357A]. Note that atmospheric dust becomes a major source of opacity for $T_{\mathrm{eff}} \lesssim 2500$K [@2008MNRAS.391.1854H], corresponding to $M \lesssim 0.02\,M_{\odot}$ for young brown dwarfs, which is the low-mass limit in our analysis, thus, the treatment of dust should not have a major effect on our results. The isochrones predict absolute magnitudes as a function of object mass in all photometric bands for which observations are available. They cover the range from 0.02$\,M_{\odot}$ or below to 1.4$M_{\odot}$ and are available for ages starting from 1Myr, which is adequate for our purposes.
To compare the observed with the predicted magnitudes, we first shift the isochrones from absolute magnitude to the distance of our target regions, which enters here as a free parameter (see Sect. \[s23\]). We also re-bin the isochrones to a uniform stepsize of 0.01$\,M_{\odot}$, using a linear interpolation over a small portion of the isochrone. We then calculate a series of reddened isochrones for $A_V = 1-20$mag in steps of 1mag. For this step, the choice of the extinction law is important (see Sect. \[s23\]). The upper limit is chosen to be 20mag, for two reasons. First, independent studies indicate that only a small fraction in the Perseus star forming complex exceeds this extinction value . Second, beyond this value the samples are biased towards bright sources, thus, incompleteness becomes an issue.
After these preparations, the best fit for mass and $A_V$ is determined with a $\chi^2$ minimization. The number of degrees of freedom in this process is the number of photometric bands for which data is available ($N=3$ to $5$) minus the number of free parameters (mass and $A_V$, i.e. 2). For each object, we saved the combination of mass and $A_V$ that results in the minimum value for $\chi^2$, the corresponding reduced $\chi^2$ ($\chi_r^2$, i.e. $\chi^2$ divided by the number of degrees of freedom) and the number of available bands $N$. Objects with best fit value of $A_V =20$ are discarded from the analysis – since this is the upper limit in our grid of isochrones, their mass estimate is not reliable. A typical example of the resulting $\chi_r^2$ plotted vs. the best mass estimate is shown in Fig. \[f2\], left panel. This figure reveals that the procedure produces similar fitting results for high- and low-mass objects and for objects with and without optical photometry. We also show a typical $A_V$ vs. mass plot from this procedure (Fig. \[f2\], right panel). The upper limit in $A_V$ is not changing significantly with mass, i.e. there is no evidence for an extinction bias in these samples (apart from the $A_V<20$ cutoff).
The distribution of reduced $\chi^2$ can in principle be used to assess the goodness-of fit for our mass estimates. For a good fit, we expect $\chi_r^2$ to have an average of 1.0 and a standard deviation of $\sim \sqrt{2/N}$. However, as shown in Fig. \[f2\], our procedure yields significantly higher values in $\chi_r^2$. This indicates that either the model does not reflect the data well or that the errors are underestimated. In our case, the high values for $\chi_r^2$ are mostly explained by the fact that our model is discretely sampled in mass-$A_V$ space. The stepsize in mass and $A_V$ results in magnitude steps that are often larger than the typical photometric error. In addition, in some cases the errors of the optical photometry may be underestimated, see above. Therefore, we only use the procedure to select the best fit solution.
The scenarios {#s23}
-------------
In our estimation of masses from photometry, the distance, age, as well as the extinction law, expressed in the quantity $R_V = A_V / E_{(B-V)}$, are considered free parameters. In addition, we have to choose the theoretical isochrone. What we call ’scenarios’ in the following are combinations of isochrone, distance, age, and extinction law for which we estimate masses. These scenarios have been chosen to cover the plausible range of these parameters and to give insight into the impact of the specific choice of a parameter or an isochrone on the mass estimates. In the following, we justify the choice of the range of the parameters.
For the extinction, we use the parameterised law by @1989ApJ...345..245C, with $R_V = 3.1$, the canonical value used for the ISM. This law yields extinction offsets that are consistent with the often-used extinction values published by @1998ApJ...500..525S. In reality, $R_V$ depends on the grain properties and is not the same for every line of sight; @1989ApJ...345..245C report values ranging from 2.6 to 5.6, with the overwhelming majority (22 out of 27 cases) below 4.5. We therefore use $R_V = 4.5$ as an alternative value to be able to assess the impact of the choice of $R_V$ on the mass estimates.
The distances to the two clusters are not well constrained. The entire Perseus cloud is usually assumed to have an average distance of $\sim 300$pc, which we use as a default value. Based on the Hipparcos parallaxes for the early-type stars @1999AJ....117..354D estimate $318 \pm 27$pc for the cloud. Based on a kinematical analysis of a much larger sample of A stars, infer 300pc (270-330pc). However, there are indications that NGC1333 is located at a shorter distance. @2011PASJ...63....1H report a distance of 235pc for NGC1333 based on interferometry of the maser emission from a source that may be associated with the cluster. This is also consistent with an earlier photometric estimate of the distance of NGC1333 (220pc) by . In addition, suggest, based on extinction map analysis, that the northern part of the Perseus region (where IC348 is located) is slightly more distant than the southern part (where NGC1333 is located). To take this into account, we use a distance of 230pc as an alternative value, noting that this is only a viable option for NGC1333.
The ages that are typically quoted are 2-4Myr for IC348 and 1-3Myr for NGC1333 [see @2008hsf1.book..308B and references therein]. Judging from model-independent indicators of evolutionary state (fraction of objects with disks, fraction of objects in Class I stage, luminosity function), NGC1333 is definitely younger than IC348, and both are clearly younger than star forming regions with established ages of 5-10Myr like Upper Scorpius and the TW Hydrae Association [e.g. @1996AJ....111.1964L; @2001ApJ...553L.153H; @2008ApJ...674..336G]. In the context of our study, the relevant quantity is not the age of the cluster, but the average age of the objects contained in our samples, which may not include the youngest, embedded population because we require a near-infrared detection. In fact, we showed in SONYC-IV that most of the very low mass objects in NGC1333 are consistent with an age of 1-5Myr, based on their position in the Hertzsprung-Russell diagram. Therefore, we use a default value of 3Myr, which is plausible for both clusters. To assess the influence of the age on the mass estimates, we additionally estimate masses for an age of 1Myr.
To evaluate the impact of the choice of the parameters, we define 6 scenarios for which we estimate object masses. These scenarios are listed in Table \[t1\]. The default scenario \#1 uses a distance of 300pc, an age of 3Myr, $R_V$ of 3.1, and the BT-Settl model, which has the most recent opacities and the most advanced treatment of dust. In scenarios \#2 to \#4 we vary the cluster parameters. In scenario \#2 we use the younger age of 1Myr, in \#3 the alternative distance of 230pc, and in \#4 the alternative value for $R_V$ of 4.5. In scenarios \#5 and \#6 we switch to the Nextgen and DUSTY isochrones.
[lccclclcccc]{} IC348 & 1 & 300 & 3 & BT-Settl & 3.1 & 3.6 (96/27) & 2.8-4.4 & 2.6-4.1 & 0.72 & 0.27\
IC348 & 2 & 300 & 1 & BT-Settl & 3.1 & 2.1 (88/42) & 1.7-2.5 & 1.5-2.9 & 0.95 & 0.13\
IC348 & 3 & 230 & 3 & BT-Settl & 3.1 & 1.9 (84/45) & 1.5-2.2 & 1.3-2.1 & 1.00 & 0.16\
IC348 & 4 & 300 & 3 & BT-Settl & 4.5 & 3.2 (94/29) & 2.6-4.0 & 2.7-4.3 & 0.76 & 0.26\
IC348 & 5 & 300 & 3 & BT-Nextgen & 3.1 & 4.0 (99/25) & 3.1-4.9 & 3.4-5.5 & 0.67 & 0.19\
IC348 & 6 & 300 & 3 & BT-Dusty & 3.1 & 2.9 (92/32) & 2.3-3.5 & 2.4-3.4 & 0.81 & 0.22\
N1333 & 1 & 300 & 3 & BT-Settl & 3.1 & 2.2 (43/20) & 1.6-2.8 & 1.9-2.5 & 0.94 & 0.27\
N1333 & 2 & 300 & 1 & BT-Settl & 3.1 & 2.1 (47/22) & 1.6-2.7 & 2.0-2.5 & 0.94 & 0.13\
N1333 & 3 & 230 & 3 & BT-Settl & 3.1 & 2.4 (47/20) & 1.8-3.0 & 2.2-2.4 & 0.90 & 0.18\
N1333 & 4 & 300 & 3 & BT-Settl & 4.5 & 2.0 (43/21) & 1.6-2.6 & 1.9-2.5 & 0.96 & 0.30\
N1333 & 5 & 300 & 3 & BT-Nextgen & 3.1 & 2.2 (44/20) & 1.7-2.8 & 1.9-2.6 & 0.93 & 0.18\
N1333 & 6 & 300 & 3 & BT-Dusty & 3.1 & 1.9 (42/22) & 1.5-2.5 & 1.8-2.0 & 0.99 & 0.22\
From the resulting mass distribution in a given scenario, we derive the cumulative distribution of object masses, i.e. the fraction of objects below a given mass, as a function of mass. These plots are shown in Fig. \[f1\]. For each scenario, we determine the number of objects with $0.03\le M \le 0.08\,M_{\odot}$ (BDs) and with $0.08< M <1.0\,M_{\odot}$ (stars) and calculate the star/BD ratio $R$. To be able to compare with the literature, we also calculate the slope of the mass function $\alpha$ for the power-law parameterisation $dN/dM \propto M^{-\alpha}$, directly from the star/BD ratios, i.e. using two bins in mass, one for BDs and one for stars. In addition, we derive the median mass $\overline{M}$ for each scenario. The resulting parameters for the 6 scenarios are given in Table \[t1\].
\
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Error budget {#s33}
------------
An important part of this work is to evaluate the errors in the derived parameters. In our chosen experiment, five factors contribute to the uncertainties:
[**1) Sample size:**]{} The part that is easiest to quantify is the statistical uncertainty, which is purely determined by the sample size. For this paper, we use the same approach as in SONYC-IV, which is based on the IDL scripts presented in @2011PASA...28..128C. In short, we calculate Bayesian confidence intervals from the beta distribution. We note that for large samples this procedure gives results that are very similar to binomial confidence intervals. The resulting values are listed in Table \[t1\], column 8. Typically, the sample size introduces an error in $R$ of about $\pm$0.3-0.9 for IC348 and $\pm$0.5-0.6 for NGC1333.
[**2) Models:**]{} Masses are only defined in relation to evolutionary models, and at young ages the deficiencies of the available tracks are well documented . However, at the moment no tracks with self-consistent treatment of the collapse and infall are available to the community.[^4] Based on a dynamical mass estimate for the very low mass pre-main sequence object AB Dor C, which is older than our target regions, @2005Natur.433..286C claim that the existing mass-luminosity relations underestimate masses by a factor of about two for young objects, but this claim has been questioned [@2006ApJ...638..887L]. At young ages and very low masses, the only direct benchmark test for these tracks is the eclipsing brown dwarf binary 2M J05352184-0546085 [@2007ApJ...664.1154S]. The Baraffe et al. isochrones fail to reproduce the surprising temperature reversal in this object (i.e. the more massive object is cooler than the secondary), an effect that is likely related to the presence of strong magnetic fields on the primary [@2012ApJ...756...47S]. The luminosities of the two components, however, are consistent with the isochrones. @2007MNRAS.380..541I have discovered and analysed a very young eclipsing binary with component masses around $\sim 0.2\,M_{\odot}$; that system confirms the isochrones within the errorbars as well. Thus, while more work is required to calibrate the isochrones, some preliminary trust in their validity seems warranted.
[**3) Cluster parameters:**]{} As discussed in Sect. \[s22\], several properties of the clusters affect the mass estimates, in particular the distance, the age, and the extinction law. From our set of scenarios documented in Table \[t1\] (scenarios 1-4) we can assess how the uncertainties in these parameters propagate through the procedure. In general, changes in age and distance cause significant changes in the estimated mass distribution, while a change in the extinction does not. For NGC1333, the induced variations in the star/BD ratios are small; $R$ varies from 2.0 to 2.4, smaller than the statistical uncertainties. For IC348, the scatter is larger, from 1.9 to 3.6, but excluding the implausible scenarios with age of 1Myr and distance of 230pc this range shrinks to 3.2-3.6. We note that the star/BD ratio increases somewhat with assumed age. For example, for IC348 and an (unlikely) age of 5Myr we obtain $R = 6.7$. This is easy to understand – as the objects evolve, they become fainter i.e. the same magnitude corresponds to a larger mass. As a result, objects move from the substellar to the stellar domain, and the star/BD ratio increases.
[**4) Completeness:**]{} The samples we are using have the same depth and completeness, in terms of magnitudes, which eliminates one major source of uncertainty. However, in terms of object masses the depth of the survey depends on the assumed distance and age. Assuming a shorter distance, as well as a younger age, will produce lower masses for the same magnitudes, i.e. the entire mass distribution would be shifted to lower masses, including the limits for completeness. For our default scenario with age of 3Myr and distance of 300pc, the magnitude limit of the samples correponds to $\sim 0.02\,M_{\odot}$. The alternative distance of 230pc (scenario \#3) implies a magnitude shift by 0.6mag. According to BT-Settl isochrones, this translates to a mass limit of 0.015$\,M_{\odot}$. The alternative age of 1Myr (scenario \#2) yields a new mass limit of 0.012-0.015$\,M_{\odot}$. Thus, in these scenarios we would be sensitive to slightly lower mass objects, but since the object density is very low in this mass domain, this has only a minuscule effect on the mass distibution. Furthermore, it is not going to affect the star/BD ratios.
[**5) Degeneracy:**]{} In many cases there are multiple mass/$A_V$ combinations that fit the data with a similar $\chi^2$. In particular, for an object with a best mass estimate around the Hydrogen burning limit (0.07-0.09$\,M_{\odot}$) this method is unable to distinguish between a star and a BD. As an estimate of the introduced uncertainty in $R$ we selected all objects in this mass regime and included them first in the BD count (for the lower limit of the star/BD ratio) and then in the star count (for the upper limit). The resulting ranges in $R$ are listed in Table \[t1\], column 9. These intervals are $\pm$0.1-0.3 for NGC1333 and $\pm$0.5-1.0 for IC348. We note that the degeneracy is less of a problem for the two other mass thresholds involved in the calculation of $R$ (0.03 and 1.0$\,M_{\odot}$), simply because the number of objects around these limits is low.
Combining these error sources, the values of $R$ are affected by an uncertainty of approximately $\pm 1$ for the two samples studied here. Currently this is about the best that can be done in terms of estimating this indicator for young star forming regions. This translates to an uncertainty of about $\pm 0.1$ in the power-law slope $\alpha$. The median of the mass function can be estimated with an accuracy of $\pm 0.1\,M_{\odot}$ for nearby star forming regions.
Looking ahead, there are obvious ways to lower these uncertainties. First, future evolutionary tracks need to include realistic initial conditions and require more detailed calibration (e.g., with eclipsing binaries). Second, the extinction parameters need to be studied in more detail for individual regions. Third, independent estimates for the (relative or absolute) ages of young clusters are needed. Fourth, the accuracy in the distances of these clusters (and many other well-studied star forming regions) should be improved. In this respect, the Gaia satellite will be a major opportunity, as it is anticipated to be able to measure distances with 1% accuracy for open clusters within 1kpc [@2011sca..conf...11P], a huge improvement over the current estimates. Fifth, it is a worthwhile goal to obtain comprehensive sets of multi-filter photometry for star forming regions. And sixth, additional survey work in rich star forming regions with significantly more members (factor 10 or more) than the well-studied nearby regions can be used to minimize the statistical uncertainties. However, since these regions are only found at large distances of $>2$kpc, such studies have to be postponed until larger facilities, namely JWST or ELTs, are available.
Results {#s3}
=======
The cumulative distribution {#s35}
---------------------------
Our procedure yields for each scenario and for each cluster a distribution of masses. Prior to calculating parameters for the IMF, we examine these distributions directly. In Fig. \[f1\] we show for all 6 scenarios the cumulative distribution of object masses, i.e. the fraction of objects below a given mass, as a function of mass. We compare the 36 combinations of the functions shown in Fig. \[f1\] (6 for IC348 and 6 for NGC1333) with a Kolmogorov-Smirnov (KS) test to search for differences in the mass distribution. By doing that, we test the null hypothesis that two given distributions are drawn from the same parent distribution. In Table \[t2\] we list the probabilities that this hypothesis is valid.
[lccccccc]{} IC348-1: default & 0.26 & [**0.02**]{} & [**0.04**]{} & 0.50 & [**0.001**]{} & 0.17\
IC348-2: $t=1$Myr & [**0.002**]{} & 0.49 & 0.10 & [**0.003**]{} & [**0.01**]{} & [**0.002**]{}\
IC348-3: $D=230$pc & 0.32 & 0.33 & 0.32 & 0.39 & [**0.03**]{} & 0.32\
IC348-4: $R_V = 4.5$ & 0.26 & [**0.003**]{} & 0.05 & 0.13 & [**0.001**]{} & 0.17\
IC348-5: Nextgen & 0.80 & [**0.016**]{} & 0.30 & 0.76 & [**0.02**]{} & 0.54\
IC348-6: Dusty & 0.47 & [**0.02**]{} & 0.08 & 0.74 & [**0.004**]{} & 0.32\
For 14 out of 36 combinations, this probability is $<5$%, i.e. the null hypothesis should be rejected. 6 of them are combinations that include scenario \#5 for NGC1333, i.e. the one that uses the Nextgen isochrone. This scenario produces an unusual large number of low-mass BDs (0.02-0.05$\,M_{\odot}$) for NGC1333, which causes the discrepancy with other distributions. This shows that apparent differences in the mass distribution of young clusters can be introduced simply by the choice of the isochrone. We caution against comparing mass distributions derived with inconsistent isochrones.
7 further combinations of scenarios with significant differences in the mass distribution include scenario \#2 for one of the clusters, i.e. the scenario with an age of 1Myr. As an example, we show in Fig. \[f3\] (left panel) a comparison between the default scenario for IC348 and scenario \#2 for NGC1333. From this figure the origin of the difference is clear – the mass distribution for NGC1333 shows a pronounced ’knee’ at 0.15$\,M_{\odot}$. However, the mass distribution in IC348 gives almost exactly the same ’knee’ when assuming an age of 1Myr (upper left panel in Fig. \[f1\]). This effect is best explained by the ’knee’ in the mass-luminosity relation at this age, which is predicted to become weaker with age. The resulting differences in the mass distribution cannot be attributed to environmental differences.
The combination of the default scenario for IC348 and scenario \#3 for NGC1333 produces a significant difference as well (see Fig. \[f3\], right panel). This combination assumes the shorter distance of 230pc for NGC1333. We note that three further combinations with this assumption give marginally significant differences with probabilities between 5-10%. As explained in Sect. \[s23\], multiple independent studies indicate that NGC1333 is located at a shorter distance than IC348, thus, this scenario is plausible.
\
In summary, his comparison shows that the choice of the cluster parameters and the choice of the isochrone can have noticable effects on the estimated distribution of object masses. We find that there could be significant differences in the mass distributions between the two clusters, if NGC1333 is at a shorter distance than IC348. In this case, our analysis indicates a larger proportion of very low mass objects with masses $<0.3\,M_{\odot}$ in NGC1333.
The star/BD ratio {#s34}
-----------------
From the mass distributions in the 6 scenarios we calculated the star/BD ratio, see Table \[t1\] for the results. For the cluster NGC1333, our 6 scenarios give star/BD ratios of 1.9-2.4, which is a range comparable to the statistical uncertainties (see Sect. \[s33\]). This means that our previous estimate for this cluster from SONYC-IV ($R\sim 2.3$) is confirmed.
For IC348, we find a wider range of values between 1.9 and 4.0. These values imply that star/BD ratios reported in the literature for IC348 of $\sim 8$ [@2003ApJ...593.1093L; @2008ApJ...683L.183A] are overestimated. A possible reason for these large values is survey incompleteness or low number statistics. As explained in Sect. \[s33\], the star/BD ratio depends on age, therefore these large star/BD ratios in the literature would become viable if IC348 is in fact significantly older than 3Myr. This is unlikely, as age-dependent observable quantities like the disk fraction are comparable to other 2-3Myr old clusters [@2013MNRAS.429..903D].
For IC348 an age of 1Myr and a distance of 230pc are not plausible (Sect. \[s23\]). Excluding the scenarios using these parameters gives a range for $R$ between 2.9 and 4.0, which is our best estimate for this cluster. These values are somewhat larger than in NGC1333, although still within the margin of error. For example, for the default case the star/BD ratio is 3.6 for IC348 with a lower limit of 2.6, and 2.2 for NGC1333 with an upper limit of 2.5. This is before taking into account the statistical uncertainties. Thus, based on the star/BD ratio the evidence for regional differences in the mass distribution of IC348 and NGC1333 is tentative.
Other parameters
----------------
In Table \[t1\] we also report two other quantities that are used in the literature to describe the IMF. The power-law slope of the mass function $\alpha$ is directly determined from the star/BD ratio and thus reflects the same trends reported in Sect. \[s34\]. For NGC1333, $\alpha$ is 0.9-1.0; for IC349 0.7-1.0, or 0.7-0.8 after excluding the implausible scenarios. Note that the slope is an average value for the mass range 0.03 to 1.0$\,M_{\odot}$; therefore it is not unexpected to find values somewhere between the tyipcal slope of 1.3 in the regime of low-mass stars [@2001MNRAS.322..231K] and the typical value of $\sim 0.6$ in the very low mass regime (see SONYC-IV).
Independent from the star/BD ratio, we determine the median mass for each scenario. These values vary between 0.13 and 0.30$\,M_{\odot}$, again indicating that the choice of the cluster parameters and the choice of the isochrone affect the results considerably. For a given scenario, the two clusters have very similar median masses (maximum difference is 0.04$\,M_{\odot}$ for scenario 4). Increasing age and distance will also increase the median mass. Note that have recently determined the mass function for IC348 from a different survey and find a characteristic mass (in the lognormal mass function) of 0.21-0.22$\,M_{\odot}$ consistent with the values derived here.
Implications for brown dwarf formation
======================================
In Sect. \[s3\] we established that differences in the mass distributions of the two clusters are significant, if NGC1333 is closer than IC348. In addition, there is tentative evidence that the star/BD ratio in IC348 is slightly larger than in NGC1333. Given that our two target regions differ in object density by a factor of 4-7 (Sect. \[s22\]), this finding can in principle be used to put constraints on theories for BD formation in which the stellar density is a critical parameter for the yield.
One popular scenario to form BDs is as part of dynamical cluster formation. Here, very low mass objects are removed from their accretion reservoir by dynamical ejections and thus stop their growth; the final mass is set by the competition between accretion and ejection [@2012MNRAS.419.3115B]. In this model the efficiency of BD formation is partly controlled by the likelihood for dynamical encounters which is related to the object density. The most recent radiation-hydrodynamical simulations by Bate (see their Table 1) are comparable to the clusters studied here in terms of initial cloud mass and number of objects produced. The simulations yield star/BD ratios ($>2.6$, $>4.1$) and median masses (0.21, 0.24$\,M_{\odot}$) that are consistent with our empirical results. However, the impact of object density is difficult to judge, since the simulations have only been carried out for a very limited set of initial conditions.
Another way to form BDs that has been suggested in the literature is gravitational fragmentation of infalling gas into a stellar cluster [@2008MNRAS.389.1556B]. Here the potential well, and thus the object density in the cluster, is a critical parameter for the efficiency of BD formation. The BDs and very low mass objects are expected to be formed preferentially in regions with high stellar density. Qualitatively the predictions from this scenario are confirmed by our analysis: Under plausible assumptions, the denser cluster NGC1333 has indeed a larger fraction of very low mass stars and brown dwarfs (see Sect. \[s34\]).
Their figure 7 shows the BD fraction as a function of object density from their simulations. The two regions investigated in the current paper are both at the low end of the considered densities (1-100pc$^{-3}$). For these densities, the predicted BD fractions are between 7 and 13%. In their paper the BD fraction is calculated as the number of BDs divided by the total number of objects. From our mass distributions, this quantity is $\sim 20$% in IC348 and $\sim 30$% in NGC1333, i.e. the predicted values are lower than the observed one. If this formation mechanism plays a role and the predictions are realistic, it could only contribute about one third to half of the BDs in the clusters. Other mechanisms, for example disk fragmentation followed by the ejection of embryonic or proto-brown dwarfs [@2009MNRAS.392..413S; @2012ApJ...750...30B], could contribute to the final tally of substellar objects in the young clusters.
Judged by their figure 7, an increase in the object density by one order of magnitudes would result in an increase in the BD fraction by a factor of about 2. This is consistent with the observed difference between IC348 and NGC1333, although such a difference is, as explained in Sect. \[s34\], still within the uncertainties. Therefore, the scenario remains viable, but cannot be rigorously verified with the current surveys. An important test for the theory would be the measurement of the BD fraction in a cluster that is significantly denser than NGC1333, such as RCW38 [@2008hsf2.book..124W] or the Orion Nebula Cluster. So far, the survey results in the ONC give inconsistent answers regarding the frequency of very low mass objects .
The aforementioned scenario for brown dwarf formation via disk fragmentation could also result in a star/BD ratio that depends on stellar density, if some of the fragmentation processes are driven by stellar encounters [@2010ApJ...717..577T] or disk-disk collisions [@2010MNRAS.401..727S]. Stellar encounters could also facilitate the ejection of bound brown dwarf companions from their host stars . With these additional mechanisms, disk fragmentation models would again produce more brown dwarfs in a region with higher stellar density, which is qualitatively what we find to be the case. However, the expected magnitude of this effect has not been estimated yet.
As pointed out in Sect. \[s21\], the current stellar densities in IC348 and NGC1333 are probably representative of their primordial densities [@2012MNRAS.426L..11G; @2012MNRAS.425..450M]. With constant star formation rate, these should scale with the gas density in the original cloud. Under these assumptions, we can also put limits on scenarios for brown dwarf formation through turbulent fragmentation. According to the model presented by @2002ApJ...576..870P, a factor of 5 in density enhancement should amount to a very large increase (about an order of magnitude, see their Fig. 1) in the number of brown dwarfs. In the gravoturbulent picture [@2009ApJ...702.1428H] the effect seems to be similar. Qualitatively the result is as seen in the Perseus clusters (i.e. the denser cluster produces more brown dwarfs), but the magnitude of the effect is much larger than what we derive. However, differences in other cluster parameters, for example in the Mach number, could partially erase the predicted effect. Since their predictions depend heavily on initial conditions, it is doubtful whether empirically derived IMFs can provide a meaningful test for these models.
An important caveat in our analysis is the fact that what we derive is a snapshot of the mass distribution, which may not necessarily represent the IMF. This is particularly relevant because NGC1333 is at an earlier evolutionary state than IC348 and might become as rich as its sibling at the other side of the Perseus star forming complex [@1996AJ....111.1964L]. In the typical picture of cluster formation, however, lower mass objects form later [e.g. @2012MNRAS.419.3115B their Fig. 8], thus, if additional formation processes in NGC1333 have any effect on the mass distribution, they are expected to amplify the observed discrepancy with IC348. For the comparison with the models quoted above, which typically only predict a mass distribution of cores, not an IMF, this issue is not of practical relevance.
Summary
=======
We present a systematic study of the mass distribution in the two young open clusters IC348 and NGC1333, with specific emphasis on the substellar regime. These two regions are of specific interest because NGC1333 has a higher spatial density (by a factor of 4-7). In the following we list our most important findings.
1. [The mass distribution as well as the parameters derived from it, e.g., the star/BD ratio $R$ or the median mass, is significantly affected by the choice of the isochrone used to estimated masses and the choice of the cluster parameters. Therefore, we caution against comparing IMF parameters derived using different assumptions.]{}
2. [If NGC1333 is in fact closer to the Sun than IC348, as indicated by several independent studies, there is a significant difference in the mass distributions of these two clusters, in the sense that NGC1333 harbours a larger fraction of very low mass stars and brown dwarfs.]{}
3. [The star/BD ratio in NGC1333 is 1.9-2.4 in NGC1333, consistent with previous estimates, and 2.9-4.0 in IC348, significantly lower than in previous estimates. The combined uncertainty in these values is approximately $\pm 1$, but can be lowered with more accurate distance estimates and age estimates. If confirmed, these values would point to a larger fraction of brown dwarfs in NGC1333.]{}
4. [These results (2 and 3) indicate that the relative number of very low mass objects in a star forming regions may depend on the stellar density, in the sense that regions with higher density (such as NGC1333) produce more very low mass objects. At this point, this conclusion is only based on two clusters and needs to be verified in other regions.]{}
We thank Nickolas Moeckel, Gilles Chabrier, and Francesco Palla for helpful suggestions regarding topics discussed in this paper. AS acknowledges financial support through the grant 10/RFP/AST2780 from the Science Foundation Ireland. Additional support for this work came from grants to RJ from the Natural Sciences and Engineering Research Council of Canada.
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[^1]: available from [http://browndwarfs.org/sonyc]{}
[^2]: downloaded from\
[http://flamingos.astro.ufl.edu/sfsurvey/datarelease.html]{}
[^3]: downloaded from\
[^4]: If, in a hypothetical future, evolutionary tracks are linked with the initial conditions via realistic models for collapse, infall, and dynamical evolution in clusters, one would directly compare the predicted with the observed luminosity functions or HRDs. The semi-empirical estimate of IMFs for star forming regions, as it is done in this paper and in many others in the literature, would become obsolete.
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{
"pile_set_name": "ArXiv"
}
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---
author:
- |
Saumya Jetley$^\ast$[^^]{} Nicholas A. Lord[^1] [^^]{} Philip H.S. Torr[^^]{}\
[[^^]{}Department of Engineering Science, University of Oxford]{}\
[[^^]{}Oxford Research Group, FiveAI Ltd.]{}\
[`{sjetley,nicklord,phst}@robots.ox.ac.uk`]{}\
bibliography:
- 'input/references.bib'
title: 'With Friends Like These, Who Needs Adversaries?'
---
#### Acknowledgements.
This work was supported by the ERC grant ERC-2012-AdG 321162-HELIOS, EPSRC grant Seebibyte EP/M013774/1 and EPSRC/MURI grant EP/N019474/1. We would also like to acknowledge the Royal Academy of Engineering, FiveAI, and extend our thanks to Seyed-Mohsen Moosavi-Dezfooli for providing his research code for curvature analysis of decision boundaries of DCNs.
[^1]: S. Jetley and N.A. Lord have contributed equally and assert joint first authorship.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'This work studies properties of the conditional mean estimator in vector Poisson noise. The main emphasis is to study conditions on prior distributions that induce linearity of the conditional mean estimator. The paper consists of two main results. The first result shows that the only distribution that induces the linearity of the conditional mean estimator is a product gamma distribution. Moreover, it is shown that the conditional mean estimator cannot be linear when the dark current parameter of the Poisson noise is non-zero. The second result produces a quantitative refinement of the first result. Specifically, it is shown that if the conditional mean estimator is close to linear in a mean squared error sense, then the prior distribution must be close to a product gamma distribution in terms of their characteristic functions. Finally, the results are compared to their Gaussian counterparts.'
author:
- 'Alex Dytso, Michael Fauß, and H. Vincent Poor, [^1][^2]'
bibliography:
- 'refs.bib'
title: 'The Vector Poisson Channel: On the Linearity of the Conditional Mean Estimator '
---
Introduction
============
This work considers a problem of estimating a random vector ${\mathbf{X}}$ from a noisy observation ${\mathbf{Y}}$ where ${\mathbf{Y}}$ given ${\mathbf{X}}=\boldsymbol{x}$ (denoted by ${\mathbf{Y}}|{\mathbf{X}}=\boldsymbol{x}$) follows a *vector Poisson distribution*. The objective is to characterize conditions under which the *conditional mean estimator* (i.e., ${\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}] $) is a linear estimator. Conditional mean estimators are an important class of estimators that are optimal under a large family loss functions, namely Bregman divergences [@banerjee2005optimality]. For example, we are interested in characterizing the set of prior distributions on ${\mathbf{X}}$ that induce linearity of ${\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}] $. Also, we are interested in which linear estimators are realizable from ${\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}] $. That is, given that ${\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}] ={ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}$, what values of a matrix ${ \boldsymbol{\mathsf{H}}}$ and vector $\boldsymbol{c}$ are permitted? Finally, we are interested in the question of the stability of linear estimators. In other words, suppose that ${\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}]$ is ‘close’ to a linear function, can we make statements about the distribution of ${\mathbf{X}}$?
Note that the aforementioned questions have been answered for the Gaussian noise model and are part of standard tools of statistical signal processing. Despite the wide use of the Poisson noise model in statistical science, such questions have not been fully addressed in the vector Poisson case. The aim of this work is to fill this gap.
The linearity of the conditional expectation is intimately connected with a notation of *conjugate priors*, which is an important element of Bayesian statistics. In its original definition in [@raiffa1961applied Ch. 3], the family of prior distributions is said to be conjugate if it is closed under sampling – the prior is said to be closed under sampling when both prior and posterior belong to the same family of distributions. In other words, the distribution of ${\mathbf{X}}$ and the distribution of ${\mathbf{X}}|{\mathbf{Y}}=\boldsymbol{y}$ are in the same family.
The structure of the conjugate prior is highly dependent on the nature of the distribution of ${\mathbf{Y}}|{\mathbf{X}}=\boldsymbol{x}$ (often termed likelihood distribution or noise distribution). For example, in [@diaconis1979conjugate], authors have made considerable progress in characterizing conjugate priors for the case when the likelihood distribution belongs to the *exponential family*. In particular, in [@diaconis1979conjugate], it has been shown that a subset of the exponential family, characterized by certain regularity conditions, has a corresponding set of conjugate priors. Moreover, this set of conjugate priors is completely characterized by the linearity of the posterior expectation: $$\begin{aligned}
{\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}] = { \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+{\bf b}, \end{aligned}$$ where ${ \boldsymbol{\mathsf{H}}}=a {\boldsymbol{\mathsf{I}}}$ for some constant $a$ and ${\bf b}$ is some constant vector.
We note that the case when ${ \boldsymbol{\mathsf{H}}}$ is a general matrix was not considered in [@diaconis1979conjugate]. Moreover, even the case when ${\mathbf{Y}}|{\mathbf{X}}=\boldsymbol{x}$ follows a Poisson distribution is not covered by the regularity conditions found in [@diaconis1979conjugate]. However, it was shown earlier in [@johnson1957uniqueness] that the conjugate prior for the scalar Poisson distribution is a gamma distribution, and that the linearity of the posterior expectation holds and is a characterizing property. The proof in [@johnson1957uniqueness] was generalized in [@chou2001characterization] to include several families of discrete distributions not covered by the regularity conditions of [@diaconis1979conjugate]. This work considers an arbitrary matrix ${ \boldsymbol{\mathsf{H}}}$ and characterizes the sufficient and necessary conditions for the existence of the conjugate prior.
The literature on the Poisson distribution is considerable, and the interested reader is referred to [@CompresseSensingPoisson; @wang2015signal] and [@wang2013designed] for applications of the Poisson model in compressed sensing; [@verdu1999poisson] and [@lapidoth2009capacity] for a summary of communication theoretic applications; [@guo2008mutualPoisson; @atar2012mutual] and [@wang2014bregman] for applications in information theory; and [@grandell1997mixed], [@StochApproxPoissonWang] and [@dytso2019estimation] for applications of the Poisson distributions in signal processing and other fields.
The paper is organized as follows. Section \[sec:Model\] presents the Poisson noise model. Section \[sec:MainResults\] presents and discusses our main results, which are described in Theorem \[thm:Main1\] and Theorem \[thm:QuantitativeRefinement\]. Section \[sec:thm:Main1\] and Section \[proof:thm:QuantitativeRefinement\] are dedicated to the proofs of Theorem \[thm:Main1\] and Theorem \[thm:QuantitativeRefinement\], respectively. Finally, Section \[sec:Applications\] concludes the paper and discusses implications of our results by reflecting on the following: a practically relevant parametrization of a Poisson noise model, which, for example, explicitly incorporates the dark current parameter; and Gaussian noise counterparts of our results.
#### Notation {#notation .unnumbered}
Throughout the paper we adopt the following notation. $\mathbb{R}^n$ denotes the space of all $n$-dimensional vectors, $\mathbb{R}^k_{+}$ the space of all $n$-dimensional vectors with non-negative components, and $\mathbb{Z}^n_{+} $ the $n$-dimension non-negative integer lattice. Vectors are denoted by bold lowercase letters, random vectors by bold uppercase letters, and matrices by bold uppercase sans serif letters (e.g., $\boldsymbol{x}, {\mathbf{X}}, \boldsymbol{\mathsf{X}}$). All vectors are are assumed to be column vectors. For $\boldsymbol{x} \in \mathbb{R}^n$, ${ \boldsymbol{\mathsf{diag}}}(\boldsymbol{x}) \in \mathbb{R}^{n \times n}$ denotes the diagonal matrix with the main diagonal given by $\boldsymbol{x}$. The vector with one at position $i$ and zero otherwise is denote by $\boldsymbol{1}_{i}$. In this paper, the gamma distribution has a probability density function (pdf) given by $$\begin{aligned}
f(x)= \frac{\alpha^\theta}{ \Gamma(\theta) } x^{\theta-1} {\mathrm{e}}^{-\alpha x},\, x \ge 0, \label{eq:pdfGamma}\end{aligned}$$ where $\theta>0$ is the shape parameter and $\alpha>0$ is the rate parameter. We denote the distribution with the pdf in by $\mathsf{Gam}(\alpha, \theta)$.
Poisson Noise Model {#sec:Model}
===================
Let ${\mathbf{Y}}\in \mathbb{Z}^k_{+}$ and ${\mathbf{X}}\in \mathbb{R}^n$. We say that ${\mathbf{Y}}$ is an output of a system with Poisson noise, if ${\mathbf{Y}}| {\mathbf{X}}={\boldsymbol{x}}$ follows a Poisson distribution, that is, $$\begin{aligned}
P_{{\mathbf{Y}}|{\mathbf{X}}}({\mathbf{y}}|{\boldsymbol{x}})= \prod_{i=1}^k P_{Y_i|{\mathbf{X}}}(y_i|{\boldsymbol{x}}) \label{eq:PoissonChannel}\end{aligned}$$ where $$\begin{aligned}
P_{Y_i|{\mathbf{X}}}(y_i|{\boldsymbol{x}})= \frac{1}{y_i!} ( [{ \boldsymbol{\mathsf{A}}}{\boldsymbol{x}}]_i +\lambda_i )^{ y_i } {\mathrm{e}}^{-( [{ \boldsymbol{\mathsf{A}}}{\boldsymbol{x}}]_i +\lambda_i )}, \label{eq:SclarNoise}\end{aligned}$$ ${ \boldsymbol{\mathsf{A}}}\in \mathbb{R}^{k \times n}$ and ${ \boldsymbol{\lambda}}=[\lambda_1, \ldots, \lambda_k]^T \in \mathbb{R}^k_{+} $. In we use the convention that $0^0=1$.
Using the terminology of laser communications, we refer to ${ \boldsymbol{\mathsf{A}}}$ as the *intensity matrix* and ${ \boldsymbol{\lambda}}$ as the *dark current* vector. Moreover, we assume that the matrix ${ \boldsymbol{\mathsf{A}}}$ must satisfy the following non-negativity preserving constraint: $$\begin{aligned}
{ \boldsymbol{\mathsf{A}}}{\boldsymbol{x}}\in \mathbb{R}^k_{+} , \, \forall {\boldsymbol{x}}\in \mathbb{R}^n_{+} . \end{aligned}$$
The random transformation of the input random variable ${\mathbf{X}}$ to an output random variable ${\mathbf{Y}}$ by the channel in is denoted by $$\begin{aligned}
{\mathbf{Y}}= \mathcal{P}( { \boldsymbol{\mathsf{A}}}{\mathbf{X}}+{ \boldsymbol{\lambda}}). \label{Eq:PoissonTransformationDefinition}\end{aligned}$$
Main Results {#sec:MainResults}
============
This section presents our main result pertaining to the linearity properties of the conditional expectation $ {\mathbb{E}}[ {\mathbf{X}}| {\mathbf{Y}}=\boldsymbol{y}]$. Specifically, our interest lies in answer various questions of optimality of linear estimators such as:
1. Under what prior distribution on ${\mathbf{X}}$ are linear estimators optimal for squared error loss and Bregman divergence[^3] loss? Since the conditional expectation is an optimal estimator for the aforementioned loss functions, this is equivalent to asking when the conditional expectation is a linear function of $\boldsymbol{y}$.
2. Which linear estimators are realizable from ${\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}] $? That is, given that ${\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}] ={ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}$, what values of the matrix ${ \boldsymbol{\mathsf{H}}}$ and vector $\boldsymbol{c}$ are permitted?
3. If the linear estimators are approximately optimal, can we say something about the prior distribution of ${\mathbf{X}}$? In other words, we are looking for a quantitative refinement of 1).
Questions 1) and 2) are answered in Theorem \[thm:Main1\] and Corollary \[cor:InputXThm1\], and question 3) is is addressed in Theorem \[thm:QuantitativeRefinement\].
Necessary and Sufficient Conditions for Linearity
-------------------------------------------------
Our first result is the following theorem, the proof of which can be found in Section \[sec:thm:Main1\].
\[thm:Main1\] Suppose that ${\mathbf{Y}}=\mathcal{P}({\mathbf{U}})$ where ${\mathbf{U}}$ is a non-degenerate[^4] random vector. Then, $$\begin{aligned}
{\mathbb{E}}[{\mathbf{U}}| {\mathbf{Y}}=\boldsymbol{y}]={ \boldsymbol{\mathsf{H}}}\boldsymbol{y}+\boldsymbol{c}, \forall \boldsymbol{y} \in \mathbb{Z}^n_{+} \label{eq:LinearityAssumption}\end{aligned}$$ if and only if $$\begin{aligned}
P_{{\mathbf{U}}}= \prod_{i=1}^n \mathsf{Gam} \left( \theta_i, \alpha_i \right). \label{eq:ProductGammaDistribuiton}\end{aligned}$$ In this case
- ${ \boldsymbol{\mathsf{H}}}$ is diagonal with entries $\displaystyle h_{ii} = \frac{1}{1+\theta_i}$
- $\displaystyle c_i = \alpha_i h_{ii} = \frac{\alpha_i}{1+\theta_i}$
Note that $0 < h_{ii} < 1$ and $c_i > 0$ for all $i \in [1:n]$.
Quantitative Refinement of Theorem \[thm:Main1\]
-------------------------------------------------
In this section, a quantitative refined of Theorem \[thm:Main1\] is shown. Namely, it is shown that if the conditional mean estimator is close to a linear function in a mean squared error sense, then the prior distribution must be close to a product gamma distribution in terms of their characteristic functions.
\[thm:QuantitativeRefinement\] Let ${ \boldsymbol{\mathsf{H}}}$ and $\boldsymbol{c}$ be as in Theorem \[thm:Main1\] and let $\phi_{\mathsf{G}}$ denote the characteristic function of the product gamma distribution in . Assume that ${\mathbf{Y}}=\mathcal{P}({\mathbf{U}})$ for some ${\mathbf{U}}\in \mathbb{R}^n_{+}$ and that $$\begin{aligned}
{\mathbb{E}}\left[ \left \|{\mathbb{E}}[{\mathbf{U}}|{\mathbf{Y}}] -({ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}) \right \|^2 \right] \le \epsilon\end{aligned}$$ for some $\epsilon \ge 0$. Then, $$\begin{aligned}
\sup_{\boldsymbol{t} \in \mathbb{R}^n} \frac{| \phi_{{\mathbf{U}}}(\boldsymbol{t} ) - \phi_{\mathsf{G}}(\boldsymbol{t} ) | }{ \|\boldsymbol{t} \| } \le \frac{\sqrt{ \epsilon}}{ 1- \max_{k} h_{kk}} , \label{eq:ControllOfCharPoisson}\end{aligned}$$ where $\phi_{{\mathbf{U}}}(\boldsymbol{t} )$ is the characteristic function of ${\mathbf{U}}$.
The proof of Theorem \[thm:QuantitativeRefinement\] is presented in Section \[proof:thm:QuantitativeRefinement\].
Proof of Theorem \[thm:Main1\] {#sec:thm:Main1}
==============================
We first establish conditions on $\boldsymbol{c}$ and ${ \boldsymbol{\mathsf{H}}}$ under which the equality is possible.
Conditions $\boldsymbol{c}$
---------------------------
To establish such conditions we need the following representation of the conditional expectation.
\[lem:EmpericalBayes\] Let $P_{\mathbf{Y}}$ denote the probability mass function of ${\mathbf{Y}}$. Then, for $\boldsymbol{y} \in \mathbb{Z}^n_{+}$ $$\begin{aligned}
{\mathbb{E}}[{\mathbf{U}}|{\mathbf{Y}}=\boldsymbol{y} ] = \left({ \boldsymbol{\mathsf{diag}}}(\boldsymbol{y})+{\boldsymbol{\mathsf{I}}}\right) \frac{ \Delta P_{\mathbf{Y}}(\boldsymbol{y})}{ P_{\mathbf{Y}}(\boldsymbol{y})}, \label{eq:TRGformula}\end{aligned}$$ where $$\begin{aligned}
[\Delta P_{\mathbf{Y}}(\boldsymbol{y})]_i= P_{\mathbf{Y}}(\boldsymbol{y}+\boldsymbol{1}_{i} ), \, i \in [1:n].\end{aligned}$$
The scalar version of Lemma \[lem:EmpericalBayes\] has been shown in [@good1953population] and in [@robbins1956empirical] and the vector version has been shown in [@palomar2007representation Lemma 3] and [@wang2014bregman Lemma 3].
We proceed to show that every element of $\boldsymbol{c}$ must be strictly positive. Choosing $\boldsymbol{y}=\boldsymbol{0}$ and combining with implies that $$\begin{aligned}
\boldsymbol{c} = \frac{ \Delta P_{\mathbf{Y}}(\boldsymbol{0} )}{ P_{\mathbf{Y}}(\boldsymbol{0} )} ,\end{aligned}$$ or equivalently for all $i$ $$\begin{aligned}
c_i =\frac{ P_{\mathbf{Y}}(\boldsymbol{0} + \boldsymbol{1}_{i} )}{ P_{\mathbf{Y}}(\boldsymbol{0} )}= \frac{ {\mathbb{E}}\left[ U_i {\mathrm{e}}^{ - \sum_{ i=1}^n U_i} \right] }{ {\mathbb{E}}\left[ {\mathrm{e}}^{ - \sum_{ i=1}^n U_i} \right] }. \end{aligned}$$ The above is zero if and only if $U_i=0$ and is positive otherwise.
Conditions on ${ \boldsymbol{\mathsf{H}}}$
------------------------------------------
We now proceed to study properties of ${ \boldsymbol{\mathsf{H}}}$. First, by combining with , we have $$\begin{aligned}
\frac{ \Delta P_{\mathbf{Y}}(\boldsymbol{y})}{ P_{\mathbf{Y}}(\boldsymbol{y})} &= \left({ \boldsymbol{\mathsf{diag}}}(\boldsymbol{y})+{\boldsymbol{\mathsf{I}}}\right)^{-1} ( { \boldsymbol{\mathsf{H}}}\boldsymbol{y}+\boldsymbol{c}) \\
&= \left({ \boldsymbol{\mathsf{diag}}}(\boldsymbol{y})+{\boldsymbol{\mathsf{I}}}\right)^{-1} { \boldsymbol{\mathsf{H}}}\boldsymbol{y}+ \left({ \boldsymbol{\mathsf{diag}}}(\boldsymbol{y})+{\boldsymbol{\mathsf{I}}}\right)^{-1} \boldsymbol{c},\end{aligned}$$ which equivalently can be written as $$\begin{aligned}
\frac{P_{\mathbf{Y}}(\boldsymbol{y} + \boldsymbol{1}_{i} )}{P_{\mathbf{Y}}(\boldsymbol{y})} = \frac{1}{y_i +1 } \sum_{j=1} h_{ij} y_j +\frac{c_i}{y_i+1}, \forall i \in [1:n]. \label{eq:IdenityFOrEqch Term}\end{aligned}$$ Observe that every entry of $ \frac{ \Delta P_{\mathbf{Y}}(\boldsymbol{y})}{ P_{\mathbf{Y}}(\boldsymbol{y})} $ is non-negative. Therefore, for every $i$ we have the following inequality: $$\begin{aligned}
0 \le \frac{1}{y_i +1 } \sum_{j=1} h_{ij} y_j +\frac{c_i}{y_i+1}, \forall \boldsymbol{y} \in \mathbb{Z}^n_{+}, \label{eq:InequalitiesOnH}\end{aligned}$$ where $h_{ij}$ is the $(i,j)$ element of ${ \boldsymbol{\mathsf{H}}}$. Since $\boldsymbol{y}$ can be chosen arbitrary in , taking limits along all possible paths as $y_i$’s go to infinity we arrive at $$\begin{aligned}
0 \le h_{ii} + \sum_{j \in S } h_{ij}, \forall i \, \text{ and } \forall S \subset[1:n]\setminus i .\end{aligned}$$ In particular, by selecting $S$ to be an empty set we arrive at the conclusion that $0 \le h_{ii}, \forall i$. To see that $h_{ii} \neq 0$, consider $$\begin{aligned}
{\mathbb{E}}[U_i | {\mathbf{Y}}=\boldsymbol{0}+ y_i\boldsymbol{1}_{i} ] = h_{ii} y_i +c_i, \forall y \in \mathbb{Z}_{+}. \end{aligned}$$ Therefore, $ h_{ii}$ can only be zero if $U_i$ is a constant.
Next, using and summing over $y_i$ we have that $$\begin{aligned}
\sum_{y_i=0}^{k} (y_i+1) P_{{\mathbf{Y}}}( \boldsymbol{y}+\boldsymbol{1}_{i})
= \sum_{y_i=0}^{k} \left( \sum_{j=1} h_{ij} y_j +c_i \right) P_{{\mathbf{Y}}}(\boldsymbol{y}), \label{eq:TransitionEquation}\end{aligned}$$ or, equivalently, by doing a change of variable on the left side of , $$\begin{aligned}
&{\mathbb{E}}[ Y_i 1_{ \{ Y_i \le k+1\}} | {\mathbf{Y}}_{-i}=\boldsymbol{y}_{-i} ] \notag\\
&= {\mathbb{E}}\left[ \left( \sum_{j=1} h_{ij} Y_j +c_i \right) 1_{ \{ Y_i \le k\}} | {\mathbf{Y}}_{-i}=\boldsymbol{y}_{-i} \right] ,\end{aligned}$$ where ${\mathbf{Y}}_{-i}$ is ${\mathbf{Y}}$ with the $i$-th element removed. Now by choosing $\boldsymbol{y}_{-i} =\boldsymbol{0} $ and re-arranging the terms we have that $$\begin{aligned}
h_{ii} &= \frac{ {\mathbb{E}}[ Y_i 1_{ \{ Y_i \le k+1\}} | {\mathbf{Y}}_{-i}=\boldsymbol{0} ] - c_i {\mathbb{E}}\left[ 1_{ \{ Y_i \le k\}} | {\mathbf{Y}}_{-i}=\boldsymbol{0} \right] }{{\mathbb{E}}[ Y_i 1_{ \{ Y_i \le k\}} | {\mathbf{Y}}_{-i}=\boldsymbol{0} ] } ,\end{aligned}$$ for all $k$. Now taking $k$ to infinity and using the fact that $c_i>0$, it immediately follows that $h_{ii} <1$.
The above discussion shows that $ 0< h_{ii} <1, \forall i$. We now proceed to show that ${ \boldsymbol{\mathsf{H}}}$ is invertible. To that end, we need the following lemma shown in Appendix \[app:lem:ConditionalCovMatrix\].
\[lem:ConditionalCovMatrix\] For $\boldsymbol{y} \in \mathbb{Z}^n_{+}$ $$\begin{aligned}
&[ \mathsf{\boldsymbol{Var}}({\mathbf{U}}|{\mathbf{Y}}=\boldsymbol{y}) ]_{ij} \notag\\
&= {\mathbb{E}}[ U_i | {\mathbf{Y}}= \boldsymbol{y} ] \left( {\mathbb{E}}[ U_j | {\mathbf{Y}}= \boldsymbol{y}+\boldsymbol{1}_{i} ] - {\mathbb{E}}[ U_j | {\mathbf{Y}}= \boldsymbol{y} ] \right) .\end{aligned}$$
Now by using Lemma \[lem:ConditionalCovMatrix\] and taking $ {\mathbb{E}}[ {\mathbf{U}}| {\mathbf{Y}}= \boldsymbol{y} ] ={ \boldsymbol{\mathsf{H}}}\boldsymbol{y} +\boldsymbol{c}$ we have that $$\begin{aligned}
\mathsf{\boldsymbol{Var}}({\mathbf{U}}|{\mathbf{Y}}=\boldsymbol{0})= \boldsymbol{c} \mathsf{1}^T \odot { \boldsymbol{\mathsf{H}}}^T \end{aligned}$$ where $\odot$ denotes the element-wise product (i.e., Hadamard product). Now using an elementary rank bound for the element-wise product, and the fact that for non-degenerate random vectors $\mathsf{\boldsymbol{Var}}({\mathbf{U}}|{\mathbf{Y}}=\boldsymbol{0})$ is a positive definite matrix, we have that $$\begin{aligned}
n& =\mathsf{Rank} \left( \mathsf{\boldsymbol{Var}}({\mathbf{U}}|{\mathbf{Y}}=\boldsymbol{0}) \right) \\
&= \mathsf{Rank} \left( \boldsymbol{c} \mathsf{1}^T \odot { \boldsymbol{\mathsf{H}}}^T \right) \\
& \le \mathsf{Rank} \left( \boldsymbol{c} \mathsf{1}^T \right) \mathsf{Rank} \left( { \boldsymbol{\mathsf{H}}}^T \right) \\
&= \mathsf{Rank} \left( { \boldsymbol{\mathsf{H}}}^T \right) \\
& \le n.\end{aligned}$$ Therefore, ${ \boldsymbol{\mathsf{H}}}$ has full rank and is invertible.
We now proceed to show that ${ \boldsymbol{\mathsf{H}}}$ must be a diagonal matrix. In order to that, we need the following definition.
The Laplace transform of the distribution of a random vector ${\mathbf{U}}\in \mathbb{R}^n$ is denoted by $$\begin{aligned}
\mathcal{L}_{\mathbf{U}}(\boldsymbol{t})= {\mathbb{E}}\left[ {\mathrm{e}}^{-\boldsymbol{t}^T {\mathbf{U}}} \right], \boldsymbol{t} \in \mathbb{R}^n_+. \end{aligned}$$
The following lemma is extensively used in this proof and the proof of Theorem \[thm:QuantitativeRefinement\].
\[lem:Gradient\] Let ${\mathbf{Y}}=\mathcal{P}({\mathbf{U}})$ and suppose that holds. Then, $$\begin{aligned}
{\mathbb{E}}\left[ \left({\mathbf{U}}-({ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}) \right) {\mathrm{e}}^{-\boldsymbol{t} ^T{\mathbf{Y}}} \right] =\boldsymbol{0} \label{eq:Consequence:Of:Orthogonality}\end{aligned}$$ for all $\boldsymbol{t} \in \mathbb{R}^n_{+}$. Moreover, for any $\boldsymbol{t} \in \mathbb{R}^n_{+}$ $$\begin{aligned}
&{\mathbb{E}}\left[ \left({\mathbf{U}}-({ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}) \right) {\mathrm{e}}^{-\boldsymbol{t} ^T{\mathbf{Y}}} \right] \notag\\
& = - ( { \boldsymbol{\mathsf{H}}}( { \boldsymbol{\mathsf{diag}}}( {\bf s} )- {\boldsymbol{\mathsf{I}}}) + {\boldsymbol{\mathsf{I}}}) \nabla_{\bf s} \mathcal{L}_U({\bf s})-\boldsymbol{c}\mathcal{L}_U({\bf s}),
\label{eq:OrthgonalityIdenity}\end{aligned}$$ where $ s_m=1-{\mathrm{e}}^{-t_m}, m=1,\ldots,n$.
The proof of follows from the orthogonality principle. To show we need to compute the following terms: $$\begin{aligned}
{\mathbb{E}}\left[ {\mathbf{U}}{\mathrm{e}}^{-\boldsymbol{t}^T{\mathbf{Y}}} \right], {\mathbb{E}}\left[ {\mathrm{e}}^{-\boldsymbol{t} ^T{\mathbf{Y}}} \right] \text{ and }{\mathbb{E}}\left[ {\mathbf{Y}}{\mathrm{e}}^{-\boldsymbol{t} ^T{\mathbf{Y}}} \right]. \end{aligned}$$ Also, recall that the Laplace transform of a distribution of a scalar Poisson random variable $W$ with the parameter $\lambda$ is given by $$\begin{aligned}
\mathcal{L}_W(t)={\mathrm{e}}^{\lambda v(t)},\end{aligned}$$ where $v(t)=({\mathrm{e}}^{-t}-1)$.
Now, first, $$\begin{aligned}
{\mathbb{E}}\left[ {\mathbf{U}}{\mathrm{e}}^{\boldsymbol{t}^T{\mathbf{Y}}} \right]&={\mathbb{E}}\left[ {\mathbf{U}}{\mathbb{E}}\left[ {\mathrm{e}}^{\boldsymbol{t} ^T{\mathbf{Y}}} \mid {\mathbf{U}}\right ] \right]\\
&={\mathbb{E}}\left[ {\mathbf{U}}\prod_{m=1}^n {\mathbb{E}}\left[ {\mathrm{e}}^{ t_m Y_m} \mid U_m \right ] \right]\\
&={\mathbb{E}}\left[ {\mathbf{U}}\prod_{m=1}^n {\mathrm{e}}^{ v(t_m) U_m } \right] \label{eq:PoissonChar}\\
&={\mathbb{E}}\left[ {\mathbf{U}}{\mathrm{e}}^{- {\bf s}^T {\mathbf{U}}} \right]\\
&= \nabla_{\bf s} \mathcal{L}_U({\bf s}) , \label{eq:CharOfU}\end{aligned}$$ where follows by using the Laplace transform of a scalar Poisson distribution. Second, using similar steps, we have $$\begin{aligned}
{\mathbb{E}}\left[ {\mathrm{e}}^{-\boldsymbol{t} ^T{\mathbf{Y}}} \right]&= {\mathbb{E}}\left[ {\mathbb{E}}\left[ {\mathrm{e}}^{-\boldsymbol{t} ^T{\mathbf{Y}}} \mid {\mathbf{U}}\right ] \right]\\
&={\mathbb{E}}\left[ \prod_{i=m}^n {\mathrm{e}}^{ v(t_m) U_i } \right]\\
&={\mathbb{E}}\left[ {\mathrm{e}}^{ -{\bf s}^T {\mathbf{U}}} \right]\\
&= \mathcal{L}_U({\bf s}). \label{eq:CharY}\end{aligned}$$ Third, $$\begin{aligned}
{\mathbb{E}}\left[ {\mathbf{Y}}{\mathrm{e}}^{-\boldsymbol{t} ^T{\mathbf{Y}}} \right]&= -\nabla_{\boldsymbol{t} } {\mathbb{E}}\left[ {\mathrm{e}}^{-\boldsymbol{t} ^T{\mathbf{Y}}} \right]\\
&= -\nabla_{\boldsymbol{t} } {\mathbb{E}}\left[ \prod_{m=1}^n {\mathrm{e}}^{ v(t_m) U_m } \right]\\
&= -\nabla_{\boldsymbol{t} } {\mathbb{E}}\left[ {\mathrm{e}}^{ - {\bf s}^T {\mathbf{U}}} \right]\\
&= {\mathbb{E}}\left[ \nabla_{\boldsymbol{t} } {\bf s}^T {\mathbf{U}}{\mathrm{e}}^{ -{\bf s}^T {\mathbf{U}}} \right]\\
&= {\mathbb{E}}\left[ ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{diag}}}( {\bf s} ) ) {\mathbf{U}}{\mathrm{e}}^{ - {\bf s}^T {\mathbf{U}}} \right]\\
&= ( { \boldsymbol{\mathsf{diag}}}( {\bf s} )- {\boldsymbol{\mathsf{I}}}) \nabla_{\bf s} \mathcal{L}_U({\bf s}), \label{eq:GradOfYchar}\end{aligned}$$ where we have used that $$\begin{aligned}
\frac{{\rm d}}{ {\rm d} t_m } s_m U_m&=\frac{{\rm d}}{ {\rm d} t_m } (1- {\mathrm{e}}^{-t_m}) U_m\\
&= {\mathrm{e}}^{-t_m} U_m\\
&= (1-s) U_m.\end{aligned}$$
Combining , and we arrive at $$\begin{aligned}
&{\mathbb{E}}\left[ \left({\mathbf{U}}-({ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}) \right) {\mathrm{e}}^{-\boldsymbol{t} ^T{\mathbf{Y}}} \right] \\
&= \nabla_{\bf s} \mathcal{L}_U({\bf s})- { \boldsymbol{\mathsf{H}}}( { \boldsymbol{\mathsf{diag}}}( {\bf s} )- {\boldsymbol{\mathsf{I}}}) \nabla_{\bf s} \mathcal{L}_U({\bf s})-\boldsymbol{c}\mathcal{L}_U({\bf s})\\
&= - ( { \boldsymbol{\mathsf{H}}}{ \boldsymbol{\mathsf{diag}}}( {\bf s} ) + ({\boldsymbol{\mathsf{I}}}-{ \boldsymbol{\mathsf{H}}}) ) \nabla_{\bf s} \mathcal{L}_U({\bf s})-\boldsymbol{c}\mathcal{L}_U({\bf s}). \end{aligned}$$ This concludes the proof.
To present the solution to the differential equation in we need the following lemma.
First using that ${ \boldsymbol{\mathsf{H}}}$ is invertible it follows that $$\begin{aligned}
\frac{ \nabla_{\bf s} \mathcal{L}_{\bf U}({\bf s}) }{ \mathcal{L}_{\bf U}({\bf s})} = - \left( { \boldsymbol{\mathsf{H}}}^{-1} ({\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{H}}}) + { \boldsymbol{\mathsf{diag}}}( {\bf s}) \right)^{-1} { \boldsymbol{\mathsf{H}}}^{-1} \boldsymbol{c} ,\end{aligned}$$ which can further be simplified to $$\begin{aligned}
\nabla g( {\bf s})= \left( { \boldsymbol{\mathsf{H}}}^{-1} ({\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{H}}}) + { \boldsymbol{\mathsf{diag}}}( {\bf s}) \right)^{-1} { \boldsymbol{\mathsf{H}}}^{-1} \boldsymbol{c} , \label{eq:SimplificationOfmatrixPDF}\end{aligned}$$ where $g( {\bf s})= \log (\mathcal{L}_{\bf U}({\bf s}) )$.
Next it is shown that has a solution only if ${ \boldsymbol{\mathsf{H}}}$ is a diagonal matrix and the solution is characterized.
\[lem:SolutionToDifferentialEquation\] For $ \boldsymbol{0} \prec { \boldsymbol{\mathsf{A}}}\in \mathbb{R}^{n \times n}$ and ${\bf b} \in \mathbb{R}^n$ where ${\bf b}$ is assumed to have all positive entries. The system $$\begin{aligned}
\nabla g( {\bf s})=- ( { \boldsymbol{\mathsf{A}}}+ { \boldsymbol{\mathsf{diag}}}( {\bf s}) )^{-1} {\bf b}, \, g( \boldsymbol{0} )=\boldsymbol{0} , \label{eq:SystemWeHaveTOSolve}\end{aligned}$$ has a solution only if ${ \boldsymbol{\mathsf{A}}}$ is a diagonal matrix with a solution given by $$\begin{aligned}
g( {\bf s})& = \sum_{i=1}^n b_i \log \left( 1+ \frac{s_i}{A_{ii}} \right). \end{aligned}$$
We first find the Hessian matrix of $f({\bf s})=\nabla g( {\bf s})$. Let $$\begin{aligned}
{ \boldsymbol{\mathsf{C}}}&= { \boldsymbol{\mathsf{A}}}+ { \boldsymbol{\mathsf{diag}}}( {\bf s}),\\
\boldsymbol{\mathsf{S}}&= { \boldsymbol{\mathsf{diag}}}( {\bf s}), \\
\boldsymbol{\mathsf{S}} f&= { \boldsymbol{\mathsf{diag}}}(f) {\bf s}.\end{aligned}$$ Then, the differential is given by $$\begin{aligned}
\partial f &=\partial { \boldsymbol{\mathsf{C}}}^{-1} {\bf b}\\
&=- { \boldsymbol{\mathsf{C}}}^{-1} (\partial { \boldsymbol{\mathsf{C}}}) { \boldsymbol{\mathsf{C}}}^{-1} {\bf b}\\
&=- { \boldsymbol{\mathsf{C}}}^{-1} (\partial { \boldsymbol{\mathsf{C}}}) f\\
&=- { \boldsymbol{\mathsf{C}}}^{-1} (\partial \boldsymbol{\mathsf{S}} ) f\\
&=- { \boldsymbol{\mathsf{C}}}^{-1} { \boldsymbol{\mathsf{diag}}}(f) \partial {\bf s}. \end{aligned}$$ Hence, $$\begin{aligned}
\frac{\partial f}{ \partial {\bf s}}= - { \boldsymbol{\mathsf{C}}}^{-1} { \boldsymbol{\mathsf{diag}}}(f)= - { \boldsymbol{\mathsf{C}}}^{-1} { \boldsymbol{\mathsf{diag}}}( ( { \boldsymbol{\mathsf{A}}}+ { \boldsymbol{\mathsf{diag}}}( {\bf s}) )^{-1} {\bf b} ). \end{aligned}$$ Therefore, the Hessian matrix of $g$ is given by $$\begin{aligned}
\nabla^2 g( {\bf s}) &= - ( { \boldsymbol{\mathsf{A}}}+ { \boldsymbol{\mathsf{diag}}}( {\bf s}) )^{-1} { \boldsymbol{\mathsf{diag}}}\left( ( { \boldsymbol{\mathsf{A}}}+ { \boldsymbol{\mathsf{diag}}}( {\bf s}) )^{-1} {\bf b} \right). \label{eq:HessianMatrix}\end{aligned}$$ Note that the Hessian matrix must be symmetric. Next, it is shown that in order for the Hessian to be symmetric ${ \boldsymbol{\mathsf{A}}}$ must be a diagonal matrix.
Let $\tilde{{ \boldsymbol{\mathsf{A}}}} = ( { \boldsymbol{\mathsf{A}}}+ { \boldsymbol{\mathsf{diag}}}( {\bf s}) )^{-1} $ and choose ${\bf s}$ such that $$\begin{aligned}
\tilde{{\bf b}}= \tilde{{ \boldsymbol{\mathsf{A}}}} {\bf b}= ( { \boldsymbol{\mathsf{A}}}+ { \boldsymbol{\mathsf{diag}}}( {\bf s}) )^{-1} {\bf b}\end{aligned}$$ has distinct elements all of which are non-zero. Note that this is possible in view of the assumption that ${\bf b}$ has non-zero entries.
Next, observe that if $\tilde{{ \boldsymbol{\mathsf{A}}}} { \boldsymbol{\mathsf{diag}}}(\tilde{{\bf b}})$ is symmetric, then $\tilde{{ \boldsymbol{\mathsf{A}}}} $ must be symmetric. This follows by letting $ \tilde{{ \boldsymbol{\mathsf{C}}}}= \tilde{{ \boldsymbol{\mathsf{A}}}} { \boldsymbol{\mathsf{diag}}}(\tilde{{\bf b}})$ and observing that $\tilde{{ \boldsymbol{\mathsf{A}}}}= \tilde{{ \boldsymbol{\mathsf{C}}}}{ \boldsymbol{\mathsf{diag}}}(\tilde{{\bf b}})^{-1}$ is symmetric. The symmetry of $\tilde{{ \boldsymbol{\mathsf{A}}}}$ implies that $$\begin{aligned}
\tilde{{ \boldsymbol{\mathsf{A}}}} { \boldsymbol{\mathsf{diag}}}(\tilde{{\bf b}})= { \boldsymbol{\mathsf{diag}}}(\tilde{{\bf b}}) \tilde{{ \boldsymbol{\mathsf{A}}}}^{T}= { \boldsymbol{\mathsf{diag}}}(\tilde{{\bf b}}) \tilde{{ \boldsymbol{\mathsf{A}}}}.\end{aligned}$$ In other words, $\tilde{{ \boldsymbol{\mathsf{A}}}}$ and $ { \boldsymbol{\mathsf{diag}}}(\tilde{{\bf b}})$ commute. However, if all elements of a diagonal matrix are distinct, then it commutes only with a diagonal matrix. Therefore, $\tilde{{ \boldsymbol{\mathsf{A}}}}$ is a diagonal matrix. This implies that for the Hessian to be symmetric ${ \boldsymbol{\mathsf{A}}}$ must be a diagonal matrix.
Since ${ \boldsymbol{\mathsf{A}}}$ is diagonal, the solution is obtained by an application of the fundamental theorem of calculus for line integrals: for a function $f$ and a smooth curve $\boldsymbol{r}(t)$ we have $$\begin{aligned}
\int_{a}^b \nabla f(\boldsymbol{r}(t)) \boldsymbol{\cdot} \dot{\boldsymbol{r}}(t) {\rm d} t= f (\boldsymbol{r}(b))- f (\boldsymbol{r}(a)). \label{eq:FTCm}\end{aligned}$$ Applying to with a choice of $\boldsymbol{r}(t)= (1-t) \boldsymbol{0} + t {\bf s}, \, t\in (0,1)$, we have that $$\begin{aligned}
g({\bf s}) &=- \int_0^1 \left( { \boldsymbol{\mathsf{A}}}+{ \boldsymbol{\mathsf{diag}}}( {\bf s})t \right)^{-1} {\bf b} \cdot {\bf s} {\rm d} t\\
&=- {\bf s}^T \int_0^1 \left( { \boldsymbol{\mathsf{A}}}+{ \boldsymbol{\mathsf{diag}}}( {\bf s})t \right)^{-1} {\rm d} t {\bf b}\\
&=- {\bf s}^T { \boldsymbol{\mathsf{diag}}}\left( \left[ \frac{\log( 1+ \frac{s_k}{A_{kk}} )}{s_{k} } \right]_k \right) {\bf b}\\
&= - \sum_{k=1}^n b_k \log \left( 1+ \frac{s_k}{A_{kk}} \right). \end{aligned}$$
Setting ${ \boldsymbol{\mathsf{A}}}= { \boldsymbol{\mathsf{H}}}^{-1} ({\boldsymbol{\mathsf{I}}}-{ \boldsymbol{\mathsf{H}}})$ and $ {\bf b} ={ \boldsymbol{\mathsf{H}}}^{-1} \boldsymbol{c}$ in Lemma \[lem:SolutionToDifferentialEquation\] and using that $g( {\bf s})= \log (\mathcal{L}_{\bf U}({\bf s}) )$ we arrive at the following form for the Laplace transform of the distribution of ${\mathbf{U}}$: $$\begin{aligned}
\mathcal{L}_{\bf U}({\bf s}) = \prod_{k=1}^n \frac{1}{\left( 1+ \frac{ h_{kk}s_k}{1- h_{kk}} \right)^{ \frac{h_{kk}}{c_k} }},\end{aligned}$$ which is the Laplace transform of a product of Gamma distributions.
Proof Theorem \[thm:QuantitativeRefinement\] {#proof:thm:QuantitativeRefinement}
============================================
Let the characteristic function of the product gamma distribution be denoted by $$\begin{aligned}
\phi_{\mathsf{G}}(\boldsymbol{t} )= \prod_{k=1}^n \left(1 -\frac{i t_k}{\alpha_k} \right)^{-\theta_k}. \end{aligned}$$ The following result, which is a generalization of the scalar result in [@dytso2019estimation], will be useful.
\[lem:BoundOnCharFunc\] Let $ \phi_{{\mathbf{U}}}(\boldsymbol{t} ) $ be a characteristic function of a distribution of a non-negative random vector ${\mathbf{U}}$ and let $$\begin{aligned}
{ \boldsymbol{\mathsf{A}}}&= { \boldsymbol{\mathsf{diag}}}^{-1} \left ( [\alpha_1, \ldots, \alpha_n]^T \right), \\
\tilde{\boldsymbol{c}}&= \left [ \frac{\theta_1}{\alpha_1}, \ldots, \frac{\theta_k}{\alpha_k} \right]^T .\end{aligned}$$ Then, for every $ \boldsymbol{t} \in \mathbb{R}^n$ $$\begin{aligned}
& | \phi_{{\mathbf{U}}}(\boldsymbol{t} ) - \phi_{\mathsf{G}}(\boldsymbol{t} ) | \notag\\
& \le \|\boldsymbol{t} \| \sup_{ \boldsymbol{t} \in\mathbb{R}^n} \left \| \left(i{\boldsymbol{\mathsf{I}}}+ { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{diag}}}( \boldsymbol{t}) \right) \nabla \phi_{{\mathbf{U}}}( \boldsymbol{t}) + \tilde{\boldsymbol{c}} \phi_{{\mathbf{U}}}( \boldsymbol{t}) \right\| . \end{aligned}$$
First, note that $$\begin{aligned}
\frac{\partial}{\partial t_k} \frac{1}{\phi_{\mathsf{G}}(\boldsymbol{t} ) }= - \frac{i \theta_k}{\alpha_k} \frac{1 }{ \left(1 -\frac{i t_k}{\alpha_k} \right) \phi_{\mathsf{G}}(\boldsymbol{t} )} , \end{aligned}$$ and hence $$\begin{aligned}
&\frac{\partial}{\partial t_k} \phi_{{\mathbf{U}}}(\boldsymbol{t} ) \frac{1}{\phi_{\mathsf{G}}(\boldsymbol{t})} \notag\\
&= \frac{\partial}{\partial t_k} \phi_{{\mathbf{U}}}(\boldsymbol{t} ) \frac{1}{\phi_{\mathsf{G}}(\boldsymbol{t})} + \phi_{{\mathbf{U}}}(\boldsymbol{t} ) \frac{\partial}{\partial t_k} \frac{1}{\phi_{\mathsf{G}}(\boldsymbol{t})}\\
&= \frac{1 }{ \left(1 -\frac{i t_k}{\alpha_k} \right) \phi_{\mathsf{G}}(\boldsymbol{t} )} \left( \left(1 -\frac{i t_k}{\alpha_k} \right) \frac{\partial}{\partial t_k} \phi_{{\mathbf{U}}}(\boldsymbol{t} ) - \frac{i \theta_k}{\alpha_k} \phi_{{\mathbf{U}}}(\boldsymbol{t} ) \right) \\
&= \frac{-i }{ \left(1 -\frac{i t_k}{\alpha_k} \right) \phi_{\mathsf{G}}(\boldsymbol{t} )} \left( \left(i +\frac{ t_k}{\alpha_k} \right) \frac{\partial}{\partial t_k} \phi_{{\mathbf{U}}}(\boldsymbol{t} ) + \frac{ \theta_k}{\alpha_k} \phi_{{\mathbf{U}}}(\boldsymbol{t} ) \right) . \end{aligned}$$
Therefore, the gradient can be upper bounded as $$\begin{aligned}
& \left \| \nabla\left( \phi_{{\mathbf{U}}}(\boldsymbol{t} ) \frac{1}{\phi_{\mathsf{G}}(\boldsymbol{t})} \right) \right\| \notag\\
& = \frac{ \left \| \left( {\boldsymbol{\mathsf{I}}}- i { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{diag}}}(\boldsymbol{t} ) \right)^{-1} \left( \left(i{\boldsymbol{\mathsf{I}}}+ { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{diag}}}(\boldsymbol{t} ) \right) \nabla \phi_{{\mathbf{U}}}(\boldsymbol{t} ) + \tilde{\boldsymbol{c}} \phi_{{\mathbf{U}}}(\boldsymbol{t} ) \right) \right\|}{ |\phi_{\mathsf{G}}(\boldsymbol{t} ) | } \\
& \le \hspace{-0.05cm} \frac{ \left \| \left( {\boldsymbol{\mathsf{I}}}- i { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{diag}}}(\boldsymbol{t} ) \right)^{-1} \right \|_{*} \hspace{-0.05cm} \left \| \hspace{-0.03cm} \left(i{\boldsymbol{\mathsf{I}}}+ { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{diag}}}(\boldsymbol{t} ) \right) \hspace{-0.05cm} \nabla \phi_{{\mathbf{U}}}(\boldsymbol{t} ) \hspace{-0.05cm} + \hspace{-0.05cm} \tilde{\boldsymbol{c}} \phi_{{\mathbf{U}}}(\boldsymbol{t} ) \hspace{-0.03cm} \right\|}{ |\phi_{\mathsf{G}}(\boldsymbol{t} ) | } \label{eq:BoundOnGradient}\end{aligned}$$ where $\| \cdot \|_{\star}$ denotes the operator norm.
Next, recall that the operator norm of a diagonal matrix is given by the maximal element and $$\begin{aligned}
\left \| \left( {\boldsymbol{\mathsf{I}}}- i { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{diag}}}(\boldsymbol{t} ) \right)^{-1} \right \|_{*} &= \max_{k \in [1:n] } \left| 1 - i \frac{t_k}{\alpha_k} \right|^{-1}\\
&= \frac{1}{\sqrt{ 1 + \min_{k \in [1:n] } \frac{t_k^2}{\alpha_k^2} }}. \end{aligned}$$
Moreover, note that $$\begin{aligned}
|\phi_{\mathsf{G}}(\boldsymbol{t} ) | = \left| \prod_{i=1}^n \left(1 -\frac{i t_i}{\alpha_i} \right)^{-\theta_i} \right| = \prod_{i=1}^n \left ( 1 +\frac{ t_i^2}{\alpha_i^2} \right) ^{- \frac{\theta_i}{2}} . \end{aligned}$$
Now let $r(\tau)= \tau \boldsymbol{t}$ and observe the following sequence of steps: $$\begin{aligned}
& | \phi_{{\mathbf{U}}}(\boldsymbol{t} ) - \phi_{\mathsf{G}}(\boldsymbol{t} ) | \notag\\
&= | \phi_{\mathsf{G}}(\boldsymbol{t} ) | \left| \frac{ \phi_{{\mathbf{U}}}(\boldsymbol{t} ) }{ \phi_{\mathsf{G}}(\boldsymbol{t} ) } - 1 \right | \\
&= | \phi_{\mathsf{G}}(\boldsymbol{t} ) | \left | \int_0^1 \nabla \frac{ \phi_{{\mathbf{U}}}( \boldsymbol{r}(\tau) ) }{ \phi_{\mathsf{G}}( \boldsymbol{r}(\tau)) } \boldsymbol{\cdot} \dot{\boldsymbol{r}}(\tau) {\rm d} \tau \right | \label{eq:FTCapp:secThm} \\
& \le | \phi_{\mathsf{G}}(\boldsymbol{t} ) | \int_0^1 \left\| \nabla \frac{ \phi_{{\mathbf{U}}}( \boldsymbol{r}(\tau) ) }{ \phi_{\mathsf{G}}( \boldsymbol{r}(\tau)) } \right\| \left \|\dot{\boldsymbol{r}}(\tau) \right\| {\rm d} \tau \label{eq:CSapplic:secThm} \\
& \le \|\boldsymbol{t} \| \sup_{ \boldsymbol{t} \in\mathbb{R}^n} \left \| \left(i{\boldsymbol{\mathsf{I}}}+ { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{diag}}}( \boldsymbol{t}) \right) \nabla \phi_{{\mathbf{U}}}( \boldsymbol{t}) + \tilde{\boldsymbol{c}} \phi_{{\mathbf{U}}}( \boldsymbol{t}) \right\|, \label{eq:MaximumElementBound}\end{aligned}$$ where follows from the fundamental theorem of calculus for line integrals; follows by using the Cauchy-Schwarz inequality; and follows by using the bound in , the fact that $ | \phi_{\mathsf{G}}(\tau\boldsymbol{t} ) | $ is an increasing function of $\tau$, and $$\begin{aligned}
& \int_0^1 \frac{ \left \| \left(i{\boldsymbol{\mathsf{I}}}+ { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{diag}}}( \tau\boldsymbol{t}) \right) \nabla \phi_{{\mathbf{U}}}( \tau\boldsymbol{t}) + \tilde{\boldsymbol{c}} \phi_{{\mathbf{U}}}( \tau\boldsymbol{t}) \right\| }{ |\phi_{\mathsf{G}}(\tau \boldsymbol{t} ) | \sqrt{ 1 + \min_{k \in [1:n] } \frac{ \tau^2 t_k^2}{\alpha_k^2} }} {\rm d} \tau \notag\\
& \le \frac{1}{ |\phi_{\mathsf{G}}( \boldsymbol{t} ) | } \int_0^1 \frac{ \left \| \left(i{\boldsymbol{\mathsf{I}}}+ { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{diag}}}( \tau\boldsymbol{t}) \right) \nabla \phi_{{\mathbf{U}}}( \tau\boldsymbol{t}) + \tilde{\boldsymbol{c}} \phi_{{\mathbf{U}}}( \tau\boldsymbol{t}) \right\| }{ \sqrt{ 1 + \min_{k \in [1:n] } \frac{ \tau^2 t_k^2}{\alpha_k^2} }} {\rm d} \tau \notag\\
& \le \sup_{ \boldsymbol{t} \in\mathbb{R}^n} \left \| \left(i{\boldsymbol{\mathsf{I}}}+ { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{diag}}}( \boldsymbol{t}) \right) \nabla \phi_{{\mathbf{U}}}( \boldsymbol{t}) + \tilde{\boldsymbol{c}} \phi_{{\mathbf{U}}}( \boldsymbol{t}) \right\| . \end{aligned}$$ This concludes the proof.
With Lemma \[lem:BoundOnCharFunc\] at our disposal we are now ready to proof the main result. First, note that by using a simple transformation from the Laplace transform to the characteristic function, the result in Lemma \[lem:Gradient\] can be re-written as $$\begin{aligned}
&{\mathbb{E}}\left[ \left({\mathbf{U}}-({ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}) \right) {\mathrm{e}}^{ i\boldsymbol{t} ^T{\mathbf{Y}}} \right] \notag\\
&=- ( i({\boldsymbol{\mathsf{I}}}-{ \boldsymbol{\mathsf{H}}})+{ \boldsymbol{\mathsf{H}}}{ \boldsymbol{\mathsf{diag}}}( {\bf s} ) ) \nabla_{\bf s} \phi_{\mathbf{U}}({\bf s})-\boldsymbol{c}\phi_{\mathbf{U}}({\bf s}). \end{aligned}$$ Moreover, $$\begin{aligned}
& \left \| ( i({\boldsymbol{\mathsf{I}}}-{ \boldsymbol{\mathsf{H}}})+{ \boldsymbol{\mathsf{H}}}{ \boldsymbol{\mathsf{diag}}}( {\bf s} ) ) \nabla_{\bf s} \phi_{\mathbf{U}}({\bf s})+\boldsymbol{c}\phi_{\mathbf{U}}({\bf s}) \right\| \notag\\
&= \left \| {\mathbb{E}}\left[ \left({\mathbf{U}}-({ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}) \right) {\mathrm{e}}^{ i\boldsymbol{t} ^T{\mathbf{Y}}} \right] \right \|\\
&= \left \| {\mathbb{E}}\left[ \left({\mathbb{E}}[{\mathbf{U}}|{\mathbf{Y}}] -({ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}) \right) {\mathrm{e}}^{ i\boldsymbol{t} ^T{\mathbf{Y}}} \right] \right. \notag\\
& \left. \quad + {\mathbb{E}}\left[ \left({\mathbf{U}}-({\mathbb{E}}[{\mathbf{U}}|{\mathbf{Y}}] \right) {\mathrm{e}}^{ i\boldsymbol{t} ^T{\mathbf{Y}}} \right] \right \| \label{eq:ApplicationOfOrthgonalityPrinciple}\\
&= \left \| {\mathbb{E}}\left[ \left({\mathbb{E}}[{\mathbf{U}}|{\mathbf{Y}}] -({ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}) \right) {\mathrm{e}}^{ i\boldsymbol{t} ^T{\mathbf{Y}}} \right] \right \|\\
&\le {\mathbb{E}}\left[ \left \|{\mathbb{E}}[{\mathbf{U}}|{\mathbf{Y}}] -({ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}) \right \| \right] \label{eq:ModulusInequalityAppliation} \\
&\le \sqrt{ {\mathbb{E}}\left[ \left \|{\mathbb{E}}[{\mathbf{U}}|{\mathbf{Y}}] -({ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}) \right \|^2 \right]}, \label{eq:J-app}
\end{aligned}$$ where follows by the orthogonality principle; follows by using the modulus inequality; and follows by using Jensen’s inequality.
Now by setting $\tilde{\boldsymbol{c}} = ({\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{H}}})^{-1} \boldsymbol{c}$ and ${ \boldsymbol{\mathsf{A}}}= ({\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{H}}})^{-1} { \boldsymbol{\mathsf{H}}}$ in Lemma \[lem:BoundOnCharFunc\] we have that $$\begin{aligned}
& | \phi_{{\mathbf{U}}}(\boldsymbol{t} ) - \phi_{\mathsf{G}}(\boldsymbol{t} ) | \notag\\
& \le \|\boldsymbol{t} \| \sup_{ \boldsymbol{t} \in\mathbb{R}^n} \left \| \left(i{\boldsymbol{\mathsf{I}}}+ { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{diag}}}( \boldsymbol{t}) \right) \nabla \phi_{{\mathbf{U}}}( \boldsymbol{t}) + \tilde{\boldsymbol{c}} \phi_{{\mathbf{U}}}( \boldsymbol{t}) \right\| \\
& \le \|\boldsymbol{t} \| \| ({\boldsymbol{\mathsf{I}}}-{ \boldsymbol{\mathsf{H}}})^{-1} \|_{\star} \notag\\
&\quad \cdot \sup_{ \boldsymbol{t} \in\mathbb{R}^n} \left \| ( i({\boldsymbol{\mathsf{I}}}-{ \boldsymbol{\mathsf{H}}})+{ \boldsymbol{\mathsf{H}}}{ \boldsymbol{\mathsf{diag}}}( {\bf s} ) ) \nabla_{\bf s} \phi_{\mathbf{U}}({\bf s})+\boldsymbol{c}\phi_{\mathbf{U}}({\bf s}) \right\| \\
& \le \|\boldsymbol{t} \| \| ({\boldsymbol{\mathsf{I}}}-{ \boldsymbol{\mathsf{H}}})^{-1} \|_{\star} \sqrt{ {\mathbb{E}}\left[ \left \|{\mathbb{E}}[{\mathbf{U}}|{\mathbf{Y}}] -({ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}) \right \|^2 \right]} \\
& = \|\boldsymbol{t} \| \frac{\sqrt{ \epsilon}}{ 1- \max_{k} h_{kk}} .\end{aligned}$$ This concludes the proof.
Concluding Remarks {#sec:Applications}
==================
This section discusses implications of our results for the practically relevant model $ {\mathbf{Y}}= \mathcal{P}( { \boldsymbol{\mathsf{A}}}{\mathbf{X}}+{ \boldsymbol{\lambda}})$, which explicitly takes into account the intensity matrix ${ \boldsymbol{\mathsf{A}}}$ and the dark current parameter ${ \boldsymbol{\lambda}}$. In addition, we also compare the Poisson result obtained in this work to their Gaussian counterparts.
We begin by adopting Theorem \[thm:Main1\] to the parametrization $ {\mathbf{Y}}= \mathcal{P}( { \boldsymbol{\mathsf{A}}}{\mathbf{X}}+{ \boldsymbol{\lambda}})$. This is done by setting ${\mathbf{U}}={ \boldsymbol{\mathsf{A}}}{\mathbf{X}}+{ \boldsymbol{\lambda}}$ in Theorem \[thm:Main1\].
\[cor:InputXThm1\] Suppose that ${\mathbf{Y}}= \mathcal{P}( { \boldsymbol{\mathsf{A}}}{\mathbf{X}}+{ \boldsymbol{\lambda}})$. Then, $$\begin{aligned}
{\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}=\boldsymbol{y}]={ \boldsymbol{\mathsf{C}}}\boldsymbol{y}+ {\bf b}, \forall \boldsymbol{y} \in \mathbb{Z}^n_{+} \label{eq:cor:linearAssumption}\end{aligned}$$ if and only if all of the following conditions hold:
- ${ \boldsymbol{\lambda}}=\boldsymbol{0} $;
- ${ \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{C}}}$ is a diagonal matrix with $ 0< [{ \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{C}}}]_{ii}<1, \forall i \in [1:n]$;
- ${ \boldsymbol{\mathsf{A}}}{\bf b} $ is a vector of positive elements; and
- $P_{{ \boldsymbol{\mathsf{A}}}{\mathbf{X}}}= \prod_{i=1}^n \mathsf{Gam} \left( \frac{1-[{ \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{C}}}]_{ii}}{[{ \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{C}}}]_{ii}}, \frac{[ { \boldsymbol{\mathsf{A}}}{\bf b} ]_{i}}{ [{ \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{C}}}]_{ii} }\right)$
Let ${\mathbf{U}}={ \boldsymbol{\mathsf{A}}}{\mathbf{X}}+{ \boldsymbol{\lambda}}$. By multiplying by ${ \boldsymbol{\mathsf{A}}}$ and adding ${ \boldsymbol{\lambda}}$ we have that $$\begin{aligned}
{\mathbb{E}}[{\mathbf{U}}| {\mathbf{Y}}=\boldsymbol{y}]= { \boldsymbol{\mathsf{A}}}{\mathbb{E}}[ {\mathbf{X}}| {\mathbf{Y}}=\boldsymbol{y}] +{ \boldsymbol{\lambda}}= { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{C}}}\boldsymbol{y}+ { \boldsymbol{\mathsf{A}}}{\bf b} +{ \boldsymbol{\lambda}}. \end{aligned}$$ Next, note that the linearity of the conditional expectation implies that ${\mathbf{U}}={ \boldsymbol{\mathsf{A}}}{\mathbf{X}}+{ \boldsymbol{\lambda}}$ is according with a product gamma distribution which has non-negative support. However, if ${ \boldsymbol{\lambda}}$ has positive components, this would imply that ${ \boldsymbol{\mathsf{A}}}{\mathbf{X}}={\mathbf{U}}-{ \boldsymbol{\lambda}}$ has negative components, which is not allowed under the Poisson model. Therefore, ${ \boldsymbol{\lambda}}$ must be zero.
The rest of the argument follows from Theorem \[thm:Main1\] by mapping ${ \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{C}}}$ to ${ \boldsymbol{\mathsf{H}}}$ and $ { \boldsymbol{\mathsf{A}}}{\bf b} $ to $\boldsymbol{c}$.
A few comments are now in order.
The case of a non-zero dark current
-----------------------------------
Somewhat regrettably Corollary \[cor:InputXThm1\] shows that the conditional expectation can only be linear if the dark current parameter is zero. To demonstrate the effect of the dark current we investigate a scalar case with an exponential distribution as a prior (i.e., a gamma distribution with $\theta=1$).
\[lem:OutputExponential\] Let $Y=\mathcal{P}(aX+\lambda)$ and take $X$ to be an exponential random variable of rate $\alpha$. Then, for every $a>0$ and $\lambda \ge 0$ $$\begin{aligned}
{\mathbb{E}}[X|Y=k]= \frac{1}{a} \frac{ (k+1) P_Y(k+1)}{P_Y(k)}-\frac{\lambda}{a}, \label{eq:expressionForConditional}\end{aligned}$$ where $$\begin{aligned}
&P_Y(0)= \frac{\alpha {\mathrm{e}}^{-\lambda}}{ \alpha+a}, \label{eq:OutputPMF0} \\
&P_Y(k)=\frac{\Gamma(k+1,\lambda)}{\Gamma(k+1)}-\frac{\Gamma(k,\lambda)}{\Gamma(k)} \notag\\
&+\frac{{\mathrm{e}}^{\frac{\alpha}{a} \lambda}}{ \left( 1+\frac{\alpha}{a} \right)^{k}} \left( \frac{\Gamma \left( k, \lambda \left(\frac{\alpha}{a} +1 \right) \right) }{\Gamma \left( k \right) } - \frac{\Gamma \left( k+1, \lambda \left(\frac{\alpha}{a} +1 \right) \right) }{\Gamma \left( k+1 \right) \left( 1+\frac{\alpha}{a} \right) } \right), \label{eq:OutputPMF1} \end{aligned}$$
where $\Gamma(\cdot, \cdot)$ is the upper incomplete gamma function.
is a scalar version of Lemma \[lem:EmpericalBayes\]. The proof of and follows by invoking standard integration techniques for exponential functions.
The effect of the dark current parameter on the conditional expectation in the scalar case for an exponential random variable is shown in Fig. \[fig:ConditionalExpectationExponential\]. Observe that the larger the dark current, the smaller the conditional expectation is. The interpretation here is that large values of dark current inflate the observed count at $Y$, and the estimator compensates by producing smaller estimates of $X$.
It is also interesting to compare the optimal linear estimator under the squared error loss to the conditional expectation. The former is given by $$\begin{aligned}
\widehat{X}(y)&= c y +b,\\
c&= \frac{a \mathbb{V}(X)}{a^2 \mathbb{V}(X) + a{\mathbb{E}}[X] +\lambda },\\
b&= {\mathbb{E}}[X] -c (a {\mathbb{E}}[X]+\lambda).\end{aligned}$$
Fig. \[fig:CEandLinear\] compares the conditional expectation to the optimal linear estimator for an exponential random variable and shows that the conditional expectation can be approximated by a piece-wise linear function. More specifically, Fig. \[fig:CEandLinear\] shows that the optimal linear estimator is a good approximation of the conditional expectation for small values of count, and the optimal zero dark current linear estimator shifted by the value of the dark current is a good approximation for large values of count.
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52 11.25\
53 11.5\
54 11.75\
55 12\
56 12.25\
57 12.5\
58 12.75\
59 13\
60 13.25\
61 13.5\
62 13.75\
63 14\
64 14.25\
65 14.5\
66 14.75\
67 15\
68 15.25\
69 15.5\
70 15.75\
71 16\
72 16.25\
73 16.5\
74 16.75\
75 17\
76 17.25\
77 17.5\
78 17.75\
79 18\
80 18.25\
81 18.5\
82 18.75\
83 19\
84 19.25\
85 19.5\
86 19.75\
87 20\
88 20.25\
89 20.5\
90 20.75\
91 21\
92 21.25\
93 21.5\
94 21.75\
95 22\
96 22.25\
97 22.5\
98 22.75\
99 23\
100 23.25\
]{};
table\[row sep=crcr\][0 0.250000000000003\
1 0.261904761904759\
2 0.27488687782805\
3 0.289084280506111\
4 0.30465677330415\
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27 2.13408253318659\
28 2.34389671062399\
29 2.56399018767538\
30 2.79335879425582\
31 3.02357447335541\
32 3.27463310512646\
33 3.49129143357618\
34 3.76203119847275\
35 4.00709438150826\
36 4.25406525801466\
37 4.50226511117587\
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39 5.0006481823217\
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41 5.50016711088253\
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43 6.00003901212864\
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53 8.50000000709334\
54 8.75000000268078\
55 9.00000000099433\
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57 9.50000000012949\
58 9.75000000004549\
59 10.0000000000157\
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table\[row sep=crcr\][0 0.224489795918367\
1 0.244897959183673\
2 0.26530612244898\
3 0.285714285714286\
4 0.306122448979592\
5 0.326530612244898\
6 0.346938775510204\
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13 0.489795918367347\
14 0.510204081632653\
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17 0.571428571428571\
18 0.591836734693878\
19 0.612244897959184\
20 0.63265306122449\
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22 0.673469387755102\
23 0.693877551020408\
24 0.714285714285714\
25 0.73469387755102\
26 0.755102040816326\
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28 0.795918367346939\
29 0.816326530612245\
30 0.836734693877551\
31 0.857142857142857\
32 0.877551020408163\
33 0.897959183673469\
34 0.918367346938776\
35 0.938775510204082\
36 0.959183673469388\
37 0.979591836734694\
38 1\
39 1.02040816326531\
40 1.04081632653061\
41 1.06122448979592\
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43 1.10204081632653\
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45 1.14285714285714\
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51 1.26530612244898\
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53 1.30612244897959\
54 1.3265306122449\
55 1.3469387755102\
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57 1.38775510204082\
58 1.40816326530612\
59 1.42857142857143\
60 1.44897959183673\
61 1.46938775510204\
62 1.48979591836735\
63 1.51020408163265\
64 1.53061224489796\
65 1.55102040816327\
66 1.57142857142857\
67 1.59183673469388\
68 1.61224489795918\
69 1.63265306122449\
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72 1.69387755102041\
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80 1.85714285714286\
81 1.87755102040816\
82 1.89795918367347\
83 1.91836734693878\
84 1.93877551020408\
85 1.95918367346939\
86 1.97959183673469\
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88 2.02040816326531\
89 2.04081632653061\
90 2.06122448979592\
91 2.08163265306122\
92 2.10204081632653\
93 2.12244897959184\
94 2.14285714285714\
95 2.16326530612245\
96 2.18367346938776\
97 2.20408163265306\
98 2.22448979591837\
99 2.24489795918367\
100 2.26530612244898\
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table\[row sep=crcr\][0 -4.75\
1 -4.5\
2 -4.25\
3 -4\
4 -3.75\
5 -3.5\
6 -3.25\
7 -3\
8 -2.75\
9 -2.5\
10 -2.25\
11 -2\
12 -1.75\
13 -1.5\
14 -1.25\
15 -1\
16 -0.75\
17 -0.5\
18 -0.25\
19 0\
20 0.25\
21 0.5\
22 0.75\
23 1\
24 1.25\
25 1.5\
26 1.75\
27 2\
28 2.25\
29 2.5\
30 2.75\
31 3\
32 3.25\
33 3.5\
34 3.75\
35 4\
36 4.25\
37 4.5\
38 4.75\
39 5\
40 5.25\
41 5.5\
42 5.75\
43 6\
44 6.25\
45 6.5\
46 6.75\
47 7\
48 7.25\
49 7.5\
50 7.75\
51 8\
52 8.25\
53 8.5\
54 8.75\
55 9\
56 9.25\
57 9.5\
58 9.75\
59 10\
60 10.25\
61 10.5\
62 10.75\
63 11\
64 11.25\
65 11.5\
66 11.75\
67 12\
68 12.25\
69 12.5\
70 12.75\
71 13\
72 13.25\
73 13.5\
74 13.75\
75 14\
76 14.25\
77 14.5\
78 14.75\
79 15\
80 15.25\
81 15.5\
82 15.75\
83 16\
84 16.25\
85 16.5\
86 16.75\
87 17\
88 17.25\
89 17.5\
90 17.75\
91 18\
92 18.25\
93 18.5\
94 18.75\
95 19\
96 19.25\
97 19.5\
98 19.75\
99 20\
100 20.25\
]{};
On the size of ${ \boldsymbol{\mathsf{A}}}$
-------------------------------------------
Observe that according to Corollary \[cor:InputXThm1\] ${ \boldsymbol{\mathsf{A}}}{\mathbf{X}}$ must have a product gamma distribution. The following scenarios can be encountered:
- ${ \boldsymbol{\mathsf{A}}}$ is full rank. In this case, the pdf of ${\mathbf{X}}$ is given by $$\begin{aligned}
f_{{\mathbf{X}}}(\boldsymbol{x})= |{\rm det}({ \boldsymbol{\mathsf{A}}})| f_{{\mathbf{U}}}( { \boldsymbol{\mathsf{A}}}\boldsymbol{x} ) \end{aligned}$$ where $ f_{{\mathbf{U}}}( \cdot ) $ is the pdf of the product gamma distribution in Corollary \[cor:InputXThm1\].
- ${ \boldsymbol{\mathsf{A}}}$ is a ‘fat’ matrix (i.e., $k<n$). In this case, there are several distributions on ${\mathbf{X}}$ that result in a product gamma distribution; and
- ${ \boldsymbol{\mathsf{A}}}$ is a ‘thin’ matrix (i.e., $k>n$). In this case, in general, it is not possible to generate a product distribution.
Comparison to the Gaussian Noise Case
-------------------------------------
It is of some value to compare the result in the Poisson case to the Gaussian noise case. The Gaussian counterpart of Theorem \[thm:Main1\] , which is a well-known result (see for example [@EPIallerton Lemma 5]), is given next.
\[thm:LinearConditionGaussian\] Suppose that ${ \boldsymbol{\mathsf{A}}}\in \mathbb{R}^{k \times n} $. Let ${\mathbf{Y}}= { \boldsymbol{\mathsf{A}}}{\mathbf{X}}+{\mathbf{Z}}$ where ${\mathbf{X}}\in \mathbb{R}^n$ and ${\mathbf{Z}}\sim \mathcal{N}(\boldsymbol{0}, {\boldsymbol{\mathsf{I}}})$ are independent. Then, $$\begin{aligned}
{\mathbb{E}}[{\mathbf{X}}|{\mathbf{Y}}={\bf y}]= { \boldsymbol{\mathsf{H}}}\boldsymbol{y}+\boldsymbol{c}, \forall \boldsymbol{y} \in \mathbb{R}^n\end{aligned}$$ if and only if ${\mathbf{X}}\sim \mathcal{N}( \boldsymbol{\mu}, \boldsymbol{ \mathsf{K}})$ such that $$\begin{aligned}
{ \boldsymbol{\mathsf{H}}}&= \boldsymbol{ \mathsf{K}} { \boldsymbol{\mathsf{A}}}^T \left( { \boldsymbol{\mathsf{A}}}\boldsymbol{ \mathsf{K}} { \boldsymbol{\mathsf{A}}}^T +{\boldsymbol{\mathsf{I}}}\right)^{-1},\\
\boldsymbol{c}&= \boldsymbol{\mu}- { \boldsymbol{\mathsf{H}}}{ \boldsymbol{\mathsf{A}}}\boldsymbol{\mu}. \end{aligned}$$ In particular, ${ \boldsymbol{\mathsf{A}}}{\mathbf{X}}\sim \mathcal{N}( { \boldsymbol{\mathsf{A}}}\mu, \boldsymbol{\Sigma}) $ where $ \boldsymbol{\Sigma}= ({\boldsymbol{\mathsf{I}}}-{ \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}})^{-1} { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}$.
The key difference is that unlike in the Poisson noise case, in the Gaussian noise case the prior does not to have to be a product distribution. In fact, in the Gaussian noise case, an arbitrary covariance matrix on ${\mathbf{X}}$ results in a linear estimator. Note that, while the distribution on ${ \boldsymbol{\mathsf{A}}}{\mathbf{X}}$ is unique, the distribution on ${\mathbf{X}}$ may not be unique and depends on the dimensionality of ${ \boldsymbol{\mathsf{A}}}$.
To the best of our knowledge, for the Gaussian noise case there exists only a scalar counterpart of Theorem \[thm:QuantitativeRefinement\], which was shown in [@du2018strong Lemma 4]. In order to make a proper comparison, the following result provides a vector Gaussian generalization.
\[thm:GaussianStabilityResult\] Let ${ \boldsymbol{\mathsf{H}}}$ and $\boldsymbol{c}$ be as in Theorem \[thm:LinearConditionGaussian\]. Denote by $ \phi_{{ \boldsymbol{\mathsf{A}}}{\mathbf{X}}}(\boldsymbol{t})$, $ \phi_{{\mathbf{Z}}}(\boldsymbol{t})$ and $\phi_{{\mathbf{Y}}}(\boldsymbol{t})$ the characteristic functions of ${ \boldsymbol{\mathsf{A}}}{\mathbf{X}}, {\mathbf{Z}}$ and ${\mathbf{Y}}$, respectively. Assume that $$\begin{aligned}
{\mathbb{E}}\left[ \left \|{\mathbb{E}}[{\mathbf{X}}|{\mathbf{Y}}] -({ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+\boldsymbol{c}) \right \|^2 \right] \le \epsilon,\end{aligned}$$ for some $\epsilon \ge 0$. Then, for all $\boldsymbol{t} \in \mathbb{R}^k $ $$\begin{aligned}
\frac{ \left| \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}}(\boldsymbol{t} ) - {\mathrm{e}}^{ -\frac{ \boldsymbol{t}^T \boldsymbol{\Sigma} \boldsymbol{t} }{2}} \right| }{ \|\boldsymbol{t} \| } \le \frac{ \sqrt{\epsilon} \|{ \boldsymbol{\mathsf{A}}}\|_{\star} }{ \sigma_{\min} \left ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\right) \phi_{{\mathbf{Z}}} \left( \boldsymbol{t} \right)} \label{eq:ControlOfCharInputGaussian}
\end{aligned}$$ where $ \boldsymbol{\Sigma}= ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}})^{-1} { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}$, $\| { \boldsymbol{\mathsf{A}}}\|_{\star}$ is the operator norm of ${ \boldsymbol{\mathsf{A}}}$, and $ \sigma_{\min} \left ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\right)$ is the smallest singular value of ${\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}$. Consequently, $$\begin{aligned}
\sup_{ \boldsymbol{t} \in \mathbb{R}^k } \frac{ \left| \phi_{ {\mathbf{Y}}}(\boldsymbol{t} ) - {\mathrm{e}}^{ -\frac{ \boldsymbol{t}^T (\boldsymbol{\Sigma} + {\boldsymbol{\mathsf{I}}}) \boldsymbol{t} }{2}} \right| }{ \|\boldsymbol{t} \| } \le \frac{ \sqrt{\epsilon} \|{ \boldsymbol{\mathsf{A}}}\|_{\star} }{ \sigma_{\min} \left ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\right) }. \label{eq:ControlOfCharOutputGaussian}
\end{aligned}$$
See Appendix \[app:thm:GaussianStabilityResult\].
It is interesting to compare the Poisson result in to the Gaussian results in and . In particular, the Poisson result appears to be stronger than the Gaussian result. In the Poisson case the control over the characteristic functions of the input in is uniform over all $\boldsymbol{t}$ (i.e., the domain of characteristic functions), but in the Gaussian counterpart in such a bound is not uniform over all $\boldsymbol{t}$. In the Gaussian case, we do get a uniform bound, but only for the characteristic functions of the output as shown in .
Proof of Lemma \[lem:ConditionalCovMatrix\] {#app:lem:ConditionalCovMatrix}
===========================================
First, compute the cross-correlation term $$\begin{aligned}
{\mathbb{E}}[U_i U_j|{\mathbf{Y}}= \boldsymbol{y} ]
&= \frac{{\mathbb{E}}[ U_i U_j P_{{\mathbf{Y}}|{\mathbf{U}}}(\boldsymbol{y} |{\mathbf{U}})]}{ P_{\mathbf{Y}}(\boldsymbol{y}) }\\
&=\frac{ (y_i+1) (y_j+1) P_{\mathbf{Y}}(\boldsymbol{y} + \boldsymbol{1}_{i} + \boldsymbol{1}_{j})}{ P_{\mathbf{Y}}(\boldsymbol{y}) }. \end{aligned}$$ Therefore, by using Lemma \[lem:EmpericalBayes\] $$\begin{aligned}
&[ \mathsf{\boldsymbol{Var}}({\mathbf{U}}|{\mathbf{Y}}=\boldsymbol{y}) ]_{ij} \notag\\
& = \frac{ (y_i+1) (y_j+1) P_{\mathbf{Y}}(\boldsymbol{y} + \boldsymbol{1}_{i} + \boldsymbol{1}_{j})}{ P_{\mathbf{Y}}(\boldsymbol{y}) } \notag\\
& - \frac{ (y_i+1) (y_j+1) P_{\mathbf{Y}}(\boldsymbol{y} + \boldsymbol{1}_{i}) P_{\mathbf{Y}}(\boldsymbol{y} + \boldsymbol{1}_{j})}{ P_{\mathbf{Y}}(\boldsymbol{y}) P_{\mathbf{Y}}(\boldsymbol{y}) } \\
&= \hspace{-0.05cm}{\mathbb{E}}[ U_i | {\mathbf{Y}}= \boldsymbol{y} ] \left(\hspace{-0.05cm} \frac{ (y_j+1) P_{\mathbf{Y}}(\boldsymbol{y} + \boldsymbol{1}_{i} + \boldsymbol{1}_{j})}{ P_{\mathbf{Y}}(\boldsymbol{y} +\boldsymbol{1}_{i}) } \hspace{-0.05cm} - {\mathbb{E}}[ U_j | {\mathbf{Y}}= \boldsymbol{y} ] \hspace{-0.05cm} \right) \\
&= {\mathbb{E}}[ U_i | {\mathbf{Y}}= \boldsymbol{y} ] \left( {\mathbb{E}}[ U_j | {\mathbf{Y}}= \boldsymbol{y}+\boldsymbol{1}_{i} ] - {\mathbb{E}}[ U_j | {\mathbf{Y}}= \boldsymbol{y} ] \right). \end{aligned}$$ This concludes the proof.
Proof of Theorem \[thm:GaussianStabilityResult\] {#app:thm:GaussianStabilityResult}
================================================
The proof for the Gaussian case is very similar to the Poisson case. We start with the following lemma.
Let $\boldsymbol{\Sigma}$ be some covariance matrix and $\phi_{{\mathbf{X}}} \left( \boldsymbol{t} \right)$ be the characteristic function of random vector ${\mathbf{X}}\in \mathbb{R}^n$. Then, for every $ \boldsymbol{t}\in \mathbb{R}^n$ $$\begin{aligned}
\left| \phi_{{\mathbf{X}}} \left( \boldsymbol{t} \right) - {\mathrm{e}}^{ -\frac{ \boldsymbol{t}^T\boldsymbol{\Sigma} \boldsymbol{t} }{2}} \right| \le \|\boldsymbol{t} \| \max_{\tau \in [0,1]} \| \nabla \phi_{{\mathbf{X}}} \left( \tau \boldsymbol{t} \right) + \boldsymbol{\Sigma} \tau \boldsymbol{t} \phi_{{\mathbf{X}}} \left( \tau \boldsymbol{t} \right) \|. \label{eq:DerivativeBoundOnDiffChar}
\end{aligned}$$
Let $\boldsymbol{r}(\tau)= \tau \boldsymbol{t} $ $$\begin{aligned}
& \left| \phi_{{\mathbf{X}}} \left( \boldsymbol{t} \right) {\mathrm{e}}^{ \frac{ \boldsymbol{t}^T \boldsymbol{\Sigma} \boldsymbol{t} }{2}} - 1 \right| \\
&= \left| \int_0^1 \nabla \phi_{{\mathbf{X}}} \left(\boldsymbol{r}(\tau) \right) {\mathrm{e}}^{ \frac{\boldsymbol{r}(\tau)^T \boldsymbol{\Sigma} \boldsymbol{r}(\tau) }{2}} \boldsymbol{\cdot} \dot{\boldsymbol{r}}(\tau) {\rm d} \tau \right| \label{eq:FTCgaussianApplication} \\
&\le \int_0^1 \left| \nabla \phi_{{\mathbf{X}}} \left(\boldsymbol{r}(\tau) \right) {\mathrm{e}}^{ \frac{\boldsymbol{r}(\tau)^T \boldsymbol{\Sigma} \boldsymbol{r}(\tau) }{2}} \boldsymbol{\cdot} \dot{\boldsymbol{r}}(\tau) \right| {\rm d} \tau \label{eq:MOdulusInequalityAppplication} \\
& \le \|\boldsymbol{t} \| \int_0^1 {\mathrm{e}}^{ \tau^2 \frac{ \boldsymbol{t}^T \boldsymbol{\Sigma} \boldsymbol{t} }{2}} \| \nabla \phi_{{\mathbf{X}}} \left( \tau \boldsymbol{t} \right) + \boldsymbol{\Sigma} \tau \boldsymbol{t} \phi_{{\mathbf{X}}} \left( \tau \boldsymbol{t} \right) \| {\rm d} \tau \label{eq:NormBoundsCauchySchwarz}\\
& \le \|\boldsymbol{t} \| \max_{\tau \in [0,1]} \| \nabla \phi_{{\mathbf{X}}} \left( \tau \boldsymbol{t} \right) + \tau \boldsymbol{\Sigma} \boldsymbol{t} \phi_{{\mathbf{X}}} \left( \tau \boldsymbol{t} \right) \|
\int_0^1 {\mathrm{e}}^{ \tau^2 \frac{ \boldsymbol{t}^T \boldsymbol{\Sigma} \boldsymbol{t} }{2}} {\rm d} \tau \\
& \le \|\boldsymbol{t} \| \max_{\tau \in [0,1]} \| \nabla \phi_{{\mathbf{X}}} \left( \tau \boldsymbol{t} \right) + \tau \boldsymbol{\Sigma} \boldsymbol{t} \phi_{{\mathbf{X}}} \left( \tau \boldsymbol{t} \right) \|
{\mathrm{e}}^{ \frac{ \boldsymbol{t}^T \boldsymbol{\Sigma} \boldsymbol{t} }{2}} , \label{eq:BoundOnRationMinus1}
\end{aligned}$$ where follows from the fundamental theorem of calculus for line integrals; follows from modulus inequality; and is a consequence of using $\dot{\boldsymbol{r}}(\tau)= \boldsymbol{t} $, $\nabla {\mathrm{e}}^{ \frac{ \boldsymbol{t}^T \boldsymbol{\Sigma} \boldsymbol{t} }{2}} = \boldsymbol{\Sigma} \boldsymbol{t} {\mathrm{e}}^{ \frac{ \boldsymbol{t}^T \boldsymbol{\Sigma} \boldsymbol{t} }{2}}$ and the Cauchy-Schwarz inequality to produces the following sequence of bounds: $$\begin{aligned}
& \left| \nabla \phi_{{\mathbf{X}}} \left(\boldsymbol{r}(\tau) \right) {\mathrm{e}}^{ \frac{\boldsymbol{r}(\tau)^T \boldsymbol{\Sigma} \boldsymbol{r}(\tau) }{2}} \boldsymbol{\cdot}\dot{\boldsymbol{r}}(\tau) \right| \notag\\
&= {\mathrm{e}}^{ \frac{ \tau^2 \boldsymbol{t}^T \boldsymbol{\Sigma} \boldsymbol{t} }{2}} \left| \left( \nabla \phi_{{\mathbf{X}}} \left( \tau \boldsymbol{t} \right) + \tau \boldsymbol{\Sigma} \boldsymbol{t} \phi_{{\mathbf{X}}} \left( \tau \boldsymbol{t} \right) \right) \boldsymbol{\cdot} \boldsymbol{t} \right| \\
& \le {\mathrm{e}}^{ \frac{ \tau^2 \boldsymbol{t}^T \boldsymbol{\Sigma} \boldsymbol{t} }{2}} \left \| \nabla \phi_{{\mathbf{X}}} \left( \tau \boldsymbol{t} \right) + \tau \boldsymbol{\Sigma} \boldsymbol{t} \phi_{{\mathbf{X}}} \left( \tau \boldsymbol{t} \right) \right \| \| \boldsymbol{t} \|.
\end{aligned}$$ This concludes the proof.
Now, using the orthogonality principle observe that $$\begin{aligned}
\boldsymbol{0}&={\mathbb{E}}\left[ ( { \boldsymbol{\mathsf{A}}}{\mathbf{X}}-{\mathbb{E}}[ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}| {\mathbf{Y}}]) {\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Y}}} \right]\\
&= {\mathbb{E}}\left[ ( { \boldsymbol{\mathsf{A}}}{\mathbf{X}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}+ { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}-{\mathbb{E}}[ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}| {\mathbf{Y}}]) {\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Y}}} \right]\\
&= {\mathbb{E}}\left[ ( { \boldsymbol{\mathsf{A}}}{\mathbf{X}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}){\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Y}}} \right ] \notag\\
&+ {\mathbb{E}}\left[ ( { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}-{\mathbb{E}}[ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}| {\mathbf{Y}}]) {\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Y}}} \right]. \label{eq:EquaivalentBetweenLinearAndNonLinear}\end{aligned}$$
Moreover, the first term in can be computed in terms of characteristic functions as follows: $$\begin{aligned}
&{\mathbb{E}}\left[ ( { \boldsymbol{\mathsf{A}}}{\mathbf{X}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}{\mathbf{Y}}){\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Y}}} \right ] \notag \\
&= {\mathbb{E}}\left[ ( { \boldsymbol{\mathsf{A}}}{\mathbf{X}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}{ \boldsymbol{\mathsf{A}}}{\mathbf{X}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}{\mathbf{Z}}){\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Y}}} \right ] \\
& ={\mathbb{E}}\left[ \left( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\right) { \boldsymbol{\mathsf{A}}}{\mathbf{X}}{\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Y}}} - { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}{\mathbf{Z}}{\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Y}}} \right ] \\
&= {\mathbb{E}}\left[ \left ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\right) { \boldsymbol{\mathsf{A}}}{\mathbf{X}}{\mathrm{e}}^{ i \boldsymbol{t}^T { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} \right] {\mathbb{E}}\left[ {\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Z}}} \right] \\
& - {\mathbb{E}}\left[ { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}{\mathbf{Z}}{\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Z}}} \right ] {\mathbb{E}}\left[{\mathrm{e}}^{ i \boldsymbol{t}^T { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} \right] \label{eq:UsingIndependence} \\
&= ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}) {\mathbb{E}}\left [ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}{\mathrm{e}}^{ i \boldsymbol{t}^T { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} \right] \phi_{{\mathbf{Z}}}(\boldsymbol{t}) \notag\\
&- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}{\mathbb{E}}\left[ {\mathbf{Z}}{\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Z}}} \right] \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} \left( \boldsymbol{t} \right) \\
&= ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}) \frac{1}{i} \nabla \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}}( \boldsymbol{t}) \phi_{{\mathbf{Z}}}(\boldsymbol{t}) - { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\frac{1}{i} \nabla \phi_{{\mathbf{Z}}}(\boldsymbol{t}) \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} \left( \boldsymbol{t} \right) \label{eq:GradientResults} \\
& = ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}) (-i) \nabla\phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}}( \boldsymbol{t}) \phi_{{\mathbf{Z}}}(\boldsymbol{t}) + { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\boldsymbol{t} (-i) \phi_{{\mathbf{Z}}}(\boldsymbol{t}) \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} \left( \boldsymbol{t} \right) \label{eq:GradientOfPhiZ}\\
&= (-i) \left( ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}) \nabla\phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}}( \boldsymbol{t}) + { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\boldsymbol{t} \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} \left( \boldsymbol{t} \right) \right) \phi_{{\mathbf{Z}}}(\boldsymbol{t}),\label{eq:ExpressionForTheLinearDiff}\end{aligned}$$ where follows from the independence of ${\mathbf{X}}$ and ${\mathbf{Z}}$; follows by observing that $\nabla \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}}( \boldsymbol{t})= {\mathbb{E}}[ i { \boldsymbol{\mathsf{A}}}{\mathbf{X}}{\mathrm{e}}^{ i \boldsymbol{t}^T { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} ]$ and $\nabla \phi_{{\mathbf{Z}}}( \boldsymbol{t})= {\mathbb{E}}[ i {\mathbf{Z}}{\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Z}}} ]$; and follows by using that $\nabla\phi_{{\mathbf{Z}}}( \boldsymbol{t})= - \boldsymbol{t} \phi_{{\mathbf{Z}}}( \boldsymbol{t}) $.
Next, by using and , and applying the norm on both sides we get that $$\begin{aligned}
&\left \| {\mathbb{E}}\left[ { \boldsymbol{\mathsf{A}}}( { \boldsymbol{\mathsf{H}}}{\mathbf{Y}}-{\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}])^T {\mathrm{e}}^{ i \boldsymbol{t}^T {\mathbf{Y}}} \right] \right\| \notag\\
&= \left\| \phi_{{\mathbf{Z}}}(\boldsymbol{t}) \left( ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}) \nabla \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}}( \boldsymbol{t}) + { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\boldsymbol{t} \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} \left( \boldsymbol{t} \right) \right) \right\|\\
&= \phi_{{\mathbf{Z}}}(\boldsymbol{t}) \left\| ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}) \nabla \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}}( \boldsymbol{t}) + { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\boldsymbol{t} \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} \left( \boldsymbol{t} \right) \right\|. \label{eq:AfterTheNorm}\end{aligned}$$ Furthermore, by using the Cauchy-Schwarz inequality in $$\begin{aligned}
& \sqrt{{\mathbb{E}}\left[ \left \| { \boldsymbol{\mathsf{A}}}( { \boldsymbol{\mathsf{H}}}{\mathbf{Y}}-{\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}]) \right\|^2 \right]} \notag\\
& \ge \phi_{{\mathbf{Z}}}(\boldsymbol{t}) \left\| ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}) \nabla \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}}( \boldsymbol{t}) + { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\boldsymbol{t} \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} \left( \boldsymbol{t} \right) \right\|,\\
& \ge \phi_{{\mathbf{Z}}}(\boldsymbol{t}) \sigma_{\min}( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}) \notag\\
& \quad \cdot \left\| \nabla \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}}( \boldsymbol{t}) + ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}})^{-1} { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\boldsymbol{t} \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} \left( \boldsymbol{t} \right) \right\|, \label{eq:SpectralBound} \end{aligned}$$ where follows by using the fact that $({\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}})$ is invertible and the inequality $\| { \boldsymbol{\mathsf{A}}}\boldsymbol{x} \| \ge \sigma_{\min}({ \boldsymbol{\mathsf{A}}}) \| \boldsymbol{x} \|, \forall \boldsymbol{x}$ where $ \sigma_{\min}({ \boldsymbol{\mathsf{A}}})$ is the small singular value of ${ \boldsymbol{\mathsf{A}}}$.
Combining bounds in and and using the bound $\| { \boldsymbol{\mathsf{A}}}\boldsymbol{x} \| \le \|{ \boldsymbol{\mathsf{A}}}\|_{\star} \| \boldsymbol{x} \|, \forall \boldsymbol{x}$ we have that $$\begin{aligned}
\frac{ \left| \phi_{ { \boldsymbol{\mathsf{A}}}{\mathbf{X}}} \left( \boldsymbol{t} \right) - {\mathrm{e}}^{ -\frac{ \boldsymbol{t}^T \boldsymbol{\Sigma} \boldsymbol{t} }{2}} \right| }{ \|\boldsymbol{t} \|} & \le \frac{ \sqrt{{\mathbb{E}}\left[ \left \| { \boldsymbol{\mathsf{A}}}( { \boldsymbol{\mathsf{H}}}{\mathbf{Y}}-{\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}]) \right\|^2 \right]} }{ \sigma_{\min} \left ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\right) \phi_{{\mathbf{Z}}} \left( \boldsymbol{t} \right)}\\
& \le \frac{ \|{ \boldsymbol{\mathsf{A}}}\|_{\star} \sqrt{{\mathbb{E}}\left[ \left \| { \boldsymbol{\mathsf{H}}}{\mathbf{Y}}-{\mathbb{E}}[{\mathbf{X}}| {\mathbf{Y}}] \right\|^2 \right]} }{ \sigma_{\min} \left ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}\right) \phi_{{\mathbf{Z}}} \left( \boldsymbol{t} \right)},
\end{aligned}$$ where $ \boldsymbol{\Sigma}= ( {\boldsymbol{\mathsf{I}}}- { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}})^{-1} { \boldsymbol{\mathsf{A}}}{ \boldsymbol{\mathsf{H}}}$. This concludes the proof.
[^1]: A. Dytso, M. Fauß, and H. V. Poor are with the Department of Electrical Engineering, Princeton University, Princeton, NY 08544, USA. E-mail: [email protected], [email protected], [email protected].
[^2]: This work was supported by the U.S. National Science Foundation under Grant CCF-1908308.
[^3]: Let $\phi: \Omega \to \mathbb{R}$ be a *continuously-differentiable* and *a strictly convex* function defined on a *closed convex* set $\Omega \subseteq \mathbb{R}^n$. The Bregman divergence between $u$ and $v$, associated with the function $\phi$, is defined as $
\ell_\phi(u,v) =\phi(u)-\phi(v)-\langle u-v, \nabla \phi(v) \rangle$.
[^4]: A random vector is said to be degenerate of its covariance of matrix is not full rank.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the textbook proofs of Lorentz covariance of the Dirac equation, one treats the wave function as a spinor and gamma matrices as scalars, leading to a quite complicated formalism with several pedagogic drawbacks. As an alternative, I propose to teach Dirac equation and its Lorentz covariance by using a much simpler, but physically equivalent formalism, in which these drawbacks do not appear. In this alternative formalism, the wave function transforms as a scalar and gamma matrices as components of a vector, such that the standard physically relevant bilinear combinations do not change their transformation properties. The alternative formalism allows also a natural construction of some additional non-standard bilinear combinations with well-defined transformation properties.'
author:
- |
Hrvoje Nikolić\
Theoretical Physics Division, Rudjer Bošković Institute,\
P.O.B. 180, HR-10002 Zagreb, Croatia\
[e-mail: [email protected]]{}\
\
title: 'How (not) to teach Lorentz covariance of the Dirac equation'
---
PACS: 03.65.Pm
Introduction
============
I like to ask tricky questions. For a warm up, here is a simple one appropriate to undergraduate students. Let $x$ be the position operator and $|\psi\rangle$ the quantum state. Which one of the two changes with time? No doubt, many students will recall how these quantities appear in the Schrödinger equation, which will lead them to the answer that $|\psi\rangle$ changes with time, while $x$ does not. Certainly not a wrong answer, but there is a much better one: it depends on the [*picture*]{}. In the Schrödinger picture $|\psi\rangle$ changes with time and $x$ does not, while in the Heisenberg picture $x$ changes with time and $|\psi\rangle$ does not. This is consistent because neither $x$ nor $|\psi\rangle$ is a physical quantity by itself, while physical quantities do not depend on the picture.
Good! Now after this warm up, here is a tricky question that I really wanted to ask. The question is appropriate to graduate students, their teachers, and even experienced experts in quantum field theory and particle physics. In the Dirac equation $$(i\gamma^{\mu}\partial_{\mu} -m)\psi =0 ,$$ which of the two quantities, $\gamma^{\mu}$ and $\psi$, changes under a Lorentz transformation? With very rare exceptions, almost everybody (assuming that they know what they are talking about) will answer that $\psi$ changes and $\gamma^{\mu}$ does not. Yet, quite analogously to the warm-up question above, that is not the best answer. A much better answer is that it depends on the picture too. In the standard picture known to everybody $\psi$ transforms and $\gamma^{\mu}$ does not, but there is also an alternative picture in which $\gamma^{\mu}$ transforms and $\psi$ does not. Neither $\gamma^{\mu}$ nor $\psi$ is a physical quantity by itself, while physical quantities do not depend on the picture.
Nevertheless, almost nobody will tell you about this alternative picture of $\gamma^{\mu}$ and $\psi$. Why? Because that is not how they are taught. The purpose of the present paper is to teach you that. And not only because the alternative picture exists, but also because it is much simpler.
Fortunately, there is a relatively small community of physicists who are more likely to tell you about the alternative picture, or at least recognize immediately that it exists if you point it out to them. These are people who work with spinors in [*curved*]{} spacetime. They know that the picture in which $\psi$ transforms and $\gamma^{\mu}$ doesn’t is not appropriate for [*general*]{} coordinate transformations, of which Lorentz transformations are nothing but a special case. Therefore, to deal with spinors in curved spacetime, they must first “unlearn” what they have learned about spinors in flat spacetime, and then learn again how to think about them in a different way. In this new picture, it turns out [@birdel; @GSW; @parker] that $\gamma^{\mu}$ transforms as a vector (which should not be surprising given the index ${\mu}$ it carries), while $\psi$ does not transform because it is a scalar (which should also not be surprising given that it does not carry any vector index at all).
Unfortunately, the treatment of spinors in curved spacetime requires some advanced concepts such as tetrads (called also [*vierbeins*]{}), with which people working in flat spacetime are usually not familiar. Hence, it is not so simple to convey the idea to the flat-spacetime people by using the techniques developed by the curved-spacetime people. Therefore, in this paper I develop a much simpler way to explain the alternative picture of spinors in flat spacetime. (A simple remark that $\gamma^{\mu}$ and $\psi$ in flat spacetime may transform as a vector and a scalar, respectively, can also be found in [@hill].) After the readers see this alternative picture, it is my hope that at least some of them will say: Wow, that’s so simple, why didn’t they taught us spinors that way from the start?
The paper is organized as follows. In Sec. \[SEC2\], I review the standard way of teaching Lorentz covariance of the Dirac equation and discuss some pedagogical drawbacks of such teaching. In Sec. \[SEC3\], I formulate the alternative picture for the Dirac equation which avoids these pedagogical drawbacks, and show that this alternative picture is much simpler and yet physically equivalent to the standard one. This alternative picture can be taught even without ever referring to the standard one, as I outline in Sec. \[SEC4\]. The conclusions are drawn in Sec. \[SEC5\].
Standard teaching and its drawbacks {#SEC2}
===================================
Elements of standard teaching of Lorentz covariance of the Dirac equation {#SEC2.1}
-------------------------------------------------------------------------
Let $$\label{e1}
(i\gamma^{\mu}\partial_{\mu} -m)\psi =0$$ be the Dirac equation in the Lorentz system ${\cal S}$, where $\gamma^{\mu}$ are the standard Dirac matrices [@bd1] obeying $$\label{anticom}
\{ \gamma^{\mu},\gamma^{\nu} \}=2\eta^{\mu\nu} ,$$ and $\eta^{\mu\nu}={\rm diag}(1,-1,-1,-1)$ is the Minkowski metric. Let $$\label{e2}
(i\gamma^{\mu}\partial'_{\mu} -m)\psi' =0$$ be the Dirac equation in another Lorentz system ${\cal S}'$. (Here $\psi=\psi(x)$ and $\psi'=\psi'(x')$, but I do not write the $x$-dependence explicitly.) The Lorentz transformation of spacetime coordinates can be written in the form $$\label{e3}
x^{\mu}=\Lambda^{\mu}_{\;\nu}x'^{\nu} ,$$ or more compactly in the matrix form $x=\Lambda x'$. The inverse of it is $x'=\Lambda^{-1}x$, which in the component form reads $$\label{vector}
x'^{\mu}=(\Lambda^{-1})^{\mu}_{\;\nu}x^{\nu} .$$ Eq. (\[e3\]) implies $$\label{e4}
\frac{\partial x^{\mu}}{\partial x'^{\nu}}=\Lambda^{\mu}_{\;\nu} .$$ Since $$\frac{\partial }{\partial x'^{\nu}}=\frac{\partial x^{\mu}}{\partial x'^{\nu}} \frac{\partial }{\partial x^{\mu}} ,$$ (\[e4\]) implies $\partial'_{\nu}=\Lambda^{\mu}_{\;\nu}\partial_{\mu}$, which we write as $$\label{e6}
\partial'_{\mu}=\Lambda^{\alpha}_{\;\mu}\partial_{\alpha} .$$ Therefore (\[e2\]) can be written as $$\label{e7}
(i\gamma^{\mu}\Lambda^{\alpha}_{\;\mu}\partial_{\alpha} -m)\psi' =0 .$$
Next write $$\label{e8}
\psi'=S\psi ,$$ where $S$ is some $x$-independent matrix the properties of which need to be determined. For that purpose one multiplies (\[e1\]) (with $\mu\rightarrow\alpha$) with $S$ from the left and inserts $1=S^{-1}S$, so $$\label{e9}
iS\gamma^{\alpha}S^{-1}\partial_{\alpha}S\psi -mS\psi =0 .$$ Using (\[e8\]), this can be written as $$\label{e10}
(iS\gamma^{\alpha}S^{-1}\partial_{\alpha} -m)\psi' =0 .$$ Comparing (\[e10\]) with (\[e7\]), one obtains $$\label{e11}
S\gamma^{\alpha}S^{-1}=\gamma^{\mu}\Lambda^{\alpha}_{\;\mu} .$$ From this equation one can determine $S$ as a function of $\Lambda^{\mu}_{\;\nu}$, but the procedure is quite complicated (see e.g. [@bd1]), so I omit it. But even without the explicit expression for $S$ as a function of $\Lambda^{\mu}_{\;\nu}$, it is clear that the inverse Lorentz transformation must correspond to the inverse $S$. Therefore (\[e11\]) can also be written as $$\label{e11'}
S^{-1}\gamma^{\alpha}S=\gamma^{\mu}(\Lambda^{-1})^{\alpha}_{\;\mu} .$$
Using (\[e11\]), it is possible to prove that $$\label{inverse}
S^{-1}=\gamma^0 S^{\dagger} \gamma^0.$$ (Unfortunately, I am not aware of any simple proof of (\[inverse\]). The simplest proof I am aware, but still quite involved, is presented in [@schweber].) By multiplying (\[inverse\]) with $\gamma^0$ from the left and using $\gamma^0\gamma^0=1$ (which follows directly from (\[anticom\])), one gets a very useful form of (\[inverse\]) $$\label{inverse2}
\gamma^0 S^{-1}=S^{\dagger} \gamma^0.$$
An important consequence of (\[inverse\]) is that $S^{-1} \neq S^{\dagger}$, i.e. $S^{\dagger}S\neq 1$. Therefore $$\label{psidpsi}
(\psi^{\dagger}\psi)'=\psi^{\dagger}S^{\dagger}S \psi \neq \psi^{\dagger} \psi ,$$ which shows that $\psi^{\dagger}\psi$ does not transform as a scalar. On the other hand, defining $$\bar{\psi}=\psi^{\dagger} \gamma^0$$ and using (\[inverse2\]) one obtains $$(\bar{\psi}\psi)'=\psi^{\dagger}S^{\dagger}\gamma^0 S \psi = \psi^{\dagger}\gamma^0 S^{-1}S \psi
= \psi^{\dagger}\gamma^0 \psi = \bar{\psi}\psi ,$$ which shows that $\bar{\psi}\psi=\psi^{\dagger}\gamma^0\psi$ transforms as a scalar.
In a similar way one finds $$\label{dircur}
(\bar{\psi}\gamma^{\mu}\psi)'=\psi^{\dagger}S^{\dagger}\gamma^0\gamma^{\mu}S \psi =
\psi^{\dagger}\gamma^0 S^{-1}\gamma^{\mu}S \psi .$$ Using (\[e11’\]), it can be written as $$\label{dircur2}
(\bar{\psi}\gamma^{\mu}\psi)'= (\Lambda^{-1})^{\mu}_{\;\nu} \; \psi^{\dagger}\gamma^0\gamma^{\nu}\psi
=(\Lambda^{-1})^{\mu}_{\;\nu} \; \bar{\psi}\gamma^{\nu}\psi ,$$ so comparing it with (\[vector\]) one concludes that $\bar{\psi}\gamma^{\mu}\psi$ transforms as a vector.
The drawbacks of standard teaching
----------------------------------
From Sec. \[SEC2.1\] one can see that the Lorentz covariance of the Dirac equation is quite complicated. For comparison, Lorentz covariance of the Maxwell equations is much simpler. If possible, it would certainly be desirable to have a simpler formulation of the Lorentz covariance for the Dirac equation.
Moreover, there is something potentially confusing about the standard teaching outlined in Sec. \[SEC2.1\]. The notation $\gamma^{\mu}$ suggests that this object also might transform as a vector. Why then $\gamma^{\mu}$ in (\[e2\]) is not replaced by $\gamma'^{\mu}$? Most textbooks which discuss Lorentz covariance of the Dirac equation, including those by Schweber [@schweber], Sakurai [@sakurai], Itzykson and Zuber [@zuber], and Zee [@zee], do not attempt to answer that question. From the pedagogical point of view, this is certainly not the best way to teach Lorentz covariance of the Dirac equation.
In some textbooks, including those by Bjorken and Drell [@bd1], Messiah [@messiah], Jauch and Rohrlich [@jauch], and Greiner [@greiner], a somewhat better approach is exploited. They note that $\gamma'^{\mu}$ is not equal to $\gamma^{\mu}$, but explain that they are related by a unitary transformation. Consequently, their argument goes, the $\gamma^{\mu}$ can be fixed and viewed as objects that do not transform under Lorentz transformations. Nevertheless, from the pedagogical point of view, such an approach is also not completely satisfying. A unitary transformation which transforms $\gamma'^{\mu}$ back to $\gamma^{\mu}$ should affect also the spinor $\psi$. So, how would $\psi$ transform if one did [*not*]{} choose to transform $\gamma'^{\mu}$ back to $\gamma^{\mu}$? In that case, would $\psi$ still transform as a spinor? Or would it perhaps become a scalar? The textbooks above say nothing about that, so it is also not the perfect way to teach Lorentz covariance of the Dirac equation.
The two approaches above have in common that they first introduce the Dirac equation, and then show that $\psi$ transforms in a specific way, known as transformation of spinors. As an alternative, some textbooks, including those by “Landau and Lifshitz” [@landau], Ryder [@ryder], and Weinberg [@weinberg], choose a reversed pedagogy. They first introduce the concept of spinors as abstract algebraic objects (that even do not need to depend on $x$), and then introduce the Dirac equation as an application of spinor mathematics to physics. No doubt, such an approach offers a much deeper mathematical understanding of spinors. In particular, their transformation properties are obtained without referring to the Dirac equation. In addition, the mathematical origin of $\gamma^{\mu}$ matrices is explained, from which it becomes clear why they are fixed matrices which do not transform. Nevertheless, even that mathematically more sophisticated approach is not perfect from the pedagogical point of view, precisely because it is mathematically sophisticated. Namely, the mathematical sophistication makes the theory even more complicated, which many practically oriented physicists view as an unnecessary distraction from their true goal – learning [*physics*]{}.
Finally, there are many textbooks which study Dirac equation but do not really attempt to prove its Lorentz covariance. Such books may be excellent for teaching what they really want to teach (e.g. how to calculate the scattering amplitude for elementary particles), but in the context of teaching Lorentz covariance of the Dirac equation they do not deserve to be mentioned.
Let me end this section with an exercise for the reader. Take three books which study the Dirac equation, not all of which are mentioned in the list of references for this paper. For each of the three books, answer the following questions: Are spinors introduced before or after introducing the Dirac equation? Is Lorentz covariance of the Dirac equation proved? Is it explicitly stated that $\gamma^{\mu}$ does not transform under Lorentz transformations? If yes, is it explained why? Is it written down explicitly how $\psi$ transforms under Lorentz transformations? If yes, is that transformation law derived?
Two pictures for the Dirac equation {#SEC3}
===================================
In Sec. \[SEC2\], I have studied the standard picture for the Dirac equation, in which the wave function $\psi$ transforms as a spinor under Lorentz transformations of spacetime coordinates, while the gamma matrices $\gamma^{\mu}$ do not transform at all. Since the transforming quantity transforms as a spinor, I refer to this picture as [*spinor*]{} picture.
Here I introduce a different picture for the Dirac equation, in which the wave function does not transform under Lorentz transformations of spacetime coordinates, while the gamma matrices transform as components of a vector. Since the transforming quantity transforms as a vector, I refer to this picture as [*vector*]{} picture. To distinguish the wave function and gamma matrices in the vector picture from those in the spinor picture, those in the vector picture are denoted by $\Psi$ and $\Gamma^{\mu}$ respectively.
The Dirac equation (\[e1\]) in the vector picture reads $$\label{e12}
(i\Gamma^{\mu}\partial_{\mu} -m)\Psi =0 .$$ Since I postulate that $\Gamma^{\mu}$ transforms as a vector, (\[e3\]) implies that it transforms according to $$\label{e13}
\Gamma^{\mu}=\Lambda^{\mu}_{\;\nu} \Gamma'^{\nu} .$$ Likewise, postulating that $\Psi$ is a scalar means $$\label{e14}
\Psi'=\Psi .$$
Since $\Gamma^{\mu}$ is a vector and $\Psi$ is a scalar, the Lorentz covariance of (\[e12\]) is quite trivial. Nevertheless, for the sake of completeness, let me present the proof explicitly. Eq. (\[e13\]) can be inverted as $$\label{e13inv}
\Gamma'^{\mu}=(\Lambda^{-1})^{\mu}_{\;\nu} \Gamma^{\nu} .$$ This together with (\[e14\]) and (\[e6\]) gives $$\begin{aligned}
\label{proof_covar}
\Gamma'^{\mu}\partial'_{\mu} \Psi'
&=& (\Lambda^{-1})^{\mu}_{\;\nu} \Gamma^{\nu} \Lambda^{\alpha}_{\;\mu}\partial_{\alpha} \Psi
= \Lambda^{\alpha}_{\;\mu} (\Lambda^{-1})^{\mu}_{\;\nu} \Gamma^{\nu} \partial_{\alpha} \Psi
\nonumber \\
&=& (\Lambda \Lambda^{-1})^{\alpha}_{\;\nu} \Gamma^{\nu} \partial_{\alpha} \Psi =
\delta^{\alpha}_{\nu} \Gamma^{\nu} \partial_{\alpha} \Psi = \Gamma^{\alpha}\partial_{\alpha} \Psi.\end{aligned}$$ This means that $\Gamma'^{\mu}\partial'_{\mu} \Psi'=\Gamma^{\mu}\partial_{\mu} \Psi$, which shows that (\[e12\]) is Lorentz covariant. Note that this simple proof does not depend on Eq. (\[e11\]).
The non-trivial aspects of (\[e12\]), however, are (i) to find out how $\Psi$ and $\Gamma^{\mu}$ are related to $\psi$ and $\gamma^{\mu}$, and (ii) to prove that (\[e12\]) is equivalent to (\[e1\]). This is what I do next. (In particular, unlike the proof of Lorentz covariance in (\[proof\_covar\]), the proof of equivalence of the two pictures will depend on (\[e11\]).) As the transformation properties of $\Psi$, $\Gamma^{\mu}$, $\psi$ and $\gamma^{\mu}$ are defined, to establish the general relation between them it is sufficient to specify the relation in one particular Lorentz system of coordinates. For convenience it can be chosen to be the laboratory system ${\cal S}_{\rm lab}$, in which I choose $$\label{e15}
\Gamma^{\mu}_{\rm lab}=\gamma^{\mu} ,\;\;\; \Psi=\psi_{\rm lab} .$$ In this sense the laboratory system can be thought of as a “preferred” system of coordinates, but it does not ruin the Lorentz covariance of the vector picture, because the only purpose of the “preferred” system is to establish the relation between the two pictures. Indeed, to use the analogy from the Introduction, this is very much analogous to the fact that the operators and states in the Heisenberg picture coincide with those in the Schrödinger picture at one particular “initial” value of time $t_0$, but it does not ruin the fact that each of the pictures by itself is invariant under time translations. Moreover, I show in the Appendix how $\Psi$ and $\Gamma^{\mu}$ can be defined in a mathematically more elegant way, without explicitly referring to any particular system of coordinates.
Now, the rest of analysis is straightforward. Eq. (\[e12\]) in the system ${\cal S}_{\rm lab}$ is $$\label{e16}
(i\Gamma^{\mu}_{\rm lab}\partial_{\mu}^{\rm lab} -m)\Psi =0 .$$ Using (\[e15\]), this can be written as $$\label{e19}
(i\gamma^{\mu}\partial_{\mu}^{\rm lab} -m)\psi_{\rm lab} =0 .$$ Now take $\Lambda$ to be the Lorentz transformation that connects the system ${\cal S}$ with the system ${\cal S}_{\rm lab}$, so that (\[e8\]) and (\[e6\]) become $$\label{e20}
\psi_{\rm lab}=S\psi ,$$ $$\label{e21}
\partial^{\rm lab}_{\mu}=\Lambda^{\alpha}_{\;\mu}\partial_{\alpha} .$$ In this way (\[e19\]) can be written as $$\label{e22}
(i\gamma^{\mu} \Lambda^{\alpha}_{\;\mu} \partial_{\alpha} -m)S\psi =0 ,$$ which after using (\[e11\]) becomes $$\label{e23}
iS\gamma^{\alpha}S^{-1}S \partial_{\alpha}\psi -mS\psi =0 .$$ Hence, by multiplying with $S^{-1}$ from the left one finally obtains $$\label{e25}
(i\gamma^{\alpha} \partial_{\alpha} -m)\psi =0 .$$ In this way, from the Dirac equation in the vector picture (\[e12\]) I have derived the Dirac equation in the spinor picture (\[e25\]). The derivation can also be inverted step by step, implying that starting from the Dirac equation in the spinor picture (\[e25\]) one can derive the Dirac equation in the vector picture (\[e12\]). This proves that [*the two pictures are equivalent*]{}.
It is also instructive to see how some bilinear combinations of $\Psi$ are related to those of $\psi$. I first define $$\bar{\Psi}=\Psi^{\dagger} \gamma^0 ,$$ which is clearly a scalar. Therefore, it is obvious that $\bar{\Psi}\Psi$ is a scalar and $\bar{\Psi}\Gamma^{\mu}\Psi$ a vector, so it does not need to be proved. What needs to be proved is that they are equal to $\bar{\psi}\psi$ and $\bar{\psi}\gamma^{\mu}\psi$, respectively. For $\bar{\Psi}\Psi$, I perform a straightforward proof $$\label{scalarbar}
\bar{\Psi}\Psi=\Psi^{\dagger}\gamma^0\Psi=\psi^{\dagger}_{\rm lab}\gamma^0\psi_{\rm lab}=
\bar{\psi}_{\rm lab}\psi_{\rm lab}=(\bar{\psi}\psi)_{\rm lab}=\bar{\psi}\psi ,$$ where in the last equality I have used the fact that $\bar{\psi}\psi$ is a scalar. For $\bar{\Psi}\Gamma^{\mu}\Psi$ the simplest way is to use use a trick. Since it is known that both $\bar{\Psi}\Gamma^{\mu}\Psi$ and $\bar{\psi}\gamma^{\mu}\psi$ transform as vectors, it is sufficient to show that they are equal in one particular Lorentz system. But they are obviously equal in the laboratory system, because in that system $\bar{\Psi}=\bar{\psi}$, $\Gamma^{\mu}=\gamma^{\mu}$, and $\Psi=\psi$. Therefore $\bar{\Psi}\Gamma^{\mu}\Psi$ is equal to $\bar{\psi}\gamma^{\mu}\psi$ in all Lorentz systems, which finishes the proof.
I have shown above that $$\bar{\Psi}\Psi=\bar{\psi}\psi , \;\;\; \bar{\Psi}\Gamma^{\mu}\Psi=\bar{\psi}\gamma^{\mu}\psi .$$ The same can be shown for other similar bilinear combinations of $\psi$ . Since physical quantities are expressed in terms of such bilinear combinations, it shows that the two pictures for the Dirac equation are [*physically equivalent*]{}.
The advantage of the vector picture is that the proof of its Lorentz covariance is much simpler. However, that is not the only advantage. The vector picture allows a simple and natural construction of some additional bilinear combinations with well-defined transformation properties, which in the spinor picture cannot be constructed so naturally. The two most interesting combinations are $$\label{rho}
\rho = \Psi^{\dagger}\Psi ,$$ $$\label{KG}
j_{\mu} = \frac{i}{2} \, \Psi^{\dagger} \stackrel{\leftrightarrow\;}{\partial_{\mu}} \Psi ,$$ where $A\!\stackrel{\leftrightarrow\;}{\partial_{\mu}}\!B \equiv A (\partial_{\mu} B)-(\partial_{\mu} A) B$. Clearly, (\[rho\]) transforms as a scalar and (\[KG\]) transforms as a vector. Their possible physical interpretation is discussed in [@nikIJMPA10; @nikBOOK12; @nik_timeprob].
For someone who never heard about the vector picture (which refers to the large majority of physicists at the time of writing this paper), it may be hard to believe that (\[rho\]) and (\[KG\]) transform as a scalar and a vector, respectively. In particular, isn’t the claim that (\[rho\]) transforms as a scalar in contradiction with the fact that (\[psidpsi\]) does not transform as a scalar? Let me show that there is no contradiction, by attempting to find a contradiction and seeing how exactly the attempt fails. Similarly to (\[scalarbar\]), one obtains $$\label{e28}
\Psi^{\dagger}\Psi=\Psi^{\dagger}\gamma^0\gamma^0\Psi=\psi^{\dagger}_{\rm lab}\gamma^0\gamma^0\psi_{\rm lab}=
\bar{\psi}_{\rm lab}\gamma^0\psi_{\rm lab} .$$ Naively one might think that the last quantity $\bar{\psi}_{\rm lab}\gamma^0\psi_{\rm lab}$ transforms as a time-component of a vector, which would contradict the claim that $\Psi^{\dagger}\Psi$ transforms as a scalar. But in fact $\bar{\psi}_{\rm lab}\gamma^0\psi_{\rm lab}$ does [*not*]{} transform as a time-component of a vector, because $$\label{e29}
\bar{\psi}_{\rm lab}\gamma^0\psi_{\rm lab} \neq \bar{\psi}\gamma^0\psi .$$ The two quantities in (\[e29\]) are equal only if $\psi$ is evaluated in the laboratory system, but in general they are different. Therefore there is no contradiction between the facts that $\Psi^{\dagger}\Psi$ and $\psi^{\dagger}\psi=\bar{\psi}\gamma^0\psi$ transform as a scalar and a time-component of a vector, respectively. These two quantities coincide in one Lorentz system (chosen to be the laboratory one), but in other Lorentz systems they are different quantities. A more formal demonstration of this fact is given also in the Appendix.
Teaching only the vector picture {#SEC4}
================================
We have seen that there are two equivalent pictures for the Dirac equation: the standard spinor picture and the alternative vector picture. We have also seen that the alternative vector picture is much simpler. Therefore, in this section I propose an alternative way to teach the Dirac equation, by teaching [*only*]{} the vector picture. Of course, a drawback of such teaching would be a clash with most of the existing literature, which could create confusion. Nevertheless, given the advantages of such teaching, I believe it is worthwhile at least to outline how such teaching might look like. So this is what I do in what follows.
In an attempt to linearize the Klein-Gordon equation $$\label{eKG}
(\partial^{\mu}\partial_{\mu} +m^2)\Psi=0 ,$$ one obtains the Dirac equation $$\label{e12v}
(i\Gamma^{\mu}\partial_{\mu} -m)\Psi =0 ,$$ where $$\label{anticomv}
\{ \Gamma^{\mu},\Gamma^{\nu} \}=2\eta^{\mu\nu} .$$ Here $\Gamma^{\mu}$ transforms as a vector and $\Psi$ as a scalar under Lorentz transformations of spacetime coordinates, so the Lorentz covariance of (\[e12v\]) is obvious. However, (\[anticomv\]) suggests that $\Gamma^{\mu}$ should be $n\times n$ matrices, so $\Psi$ should be an $n$-component column. It turns out that the smallest possible value of $n$ is 4, so one fixes $n=4$. One special choice for $\Gamma^{\mu}$ satisfying (\[anticomv\]) are the standard Dirac matrices $\gamma^{\bar{\mu}}$. (In the rest of the paper they are denoted by $\gamma^{\mu}$, but here I modify the notation by putting the bar over $\mu$ which reminds us that $\bar{\mu}$ is not a vector index. Namely, the $\gamma^{\bar{\mu}}$ are fixed matrices which do not transform under Lorentz transformations, which is why the notation $\gamma^{\bar{\mu}}$ is better than $\gamma^{\mu}$.) Thus one may determine the vector $\Gamma^{\mu}$ in any Lorentz system by choosing one particular Lorentz system, say the laboratory one, in which $$\Gamma^{\mu}_{\rm lab}=\gamma^{\bar{\mu}} .$$
Defining $$\bar{\Psi}=\Psi\gamma^{\bar{0}} ,$$ the Dirac equation (\[e12v\]) implies that the Dirac vector current $$j^{\mu}_{\rm Dirac}=\bar{\Psi}\Gamma^{\mu}\Psi$$ is conserved $$\partial_{\mu}j^{\mu}_{\rm Dirac}=0.$$ Similarly, the Klein-Gordon equation (\[eKG\]) implies that the Klein-Gordon vector current $$\label{KGv}
j_{\mu} = \frac{i}{2} \, \Psi^{\dagger} \stackrel{\leftrightarrow\;}{\partial_{\mu}} \Psi$$ is conserved too, i.e. $$\partial_{\mu}j^{\mu}=0.$$ In most physical applications only the Dirac current is relevant, but in some applications the Klein-Gordon current may be relevant as well. Similarly, one can construct two bilinear scalars $\bar{\Psi}\Psi$ and $\Psi^{\dagger}\Psi$. In most physical applications only $\bar{\Psi}\Psi$ is relevant, but in some applications $\Psi^{\dagger}\Psi$ may be of interest as well.
Conclusion {#SEC5}
==========
In this paper, I have identified some pedagogical drawbacks in the standard approaches to teaching Lorentz covariance of the Dirac equation, including the fact that the proof of Lorentz covariance is quite complicated. To avoid these drawbacks, I have proposed an alternative way to teach Lorentz covariance of the Dirac equation, by introducing a new formalism. The proposed formalism is inspired by the treatment of spinors in curved spacetime, but is in fact much simpler than that (because it is formulated in flat spacetime) and logically independent of it. The main idea of the formalism is to perform a transformation from the standard $\psi$ and $\gamma^{\mu}$, which transform as a spinor and a scalar, respectively, to new quantities $\Psi$ and $\Gamma^{\mu}$, which transform as a scalar and a vector, respectively. I have shown that the two formalisms are physically equivalent, but that the new formalism is much simpler and can be taught even without referring to the standard formalism. In addition, the new formalism allows a natural construction of some non-standard bilinear combinations with well-defined transformation properties, such as the vector Klein-Gordon current and the scalar $\Psi^{\dagger}\Psi$.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author is grateful to B. Klajn and B. Nižić for useful and encouraging comments on the manuscript, and to S. Dowker for drawing attention to some relevant references. This work was supported by the Ministry of Science of the Republic of Croatia under Contract No. 098-0982930-2864.
Formal transformation theory between spinor and vector picture of the Dirac equation
====================================================================================
By starting from the standard spinor picture described in Sec. \[SEC2.1\], in this section I rederive the main results of Sec. \[SEC3\] by developing the formal transformation theory between spinor and vector picture without explicitly referring to any particular system of coordinates.
Starting from (\[e8\]), consider the transformation $$(S^{-1}\psi)'=SS^{-1}\psi=\psi .$$ This shows that $S^{-1}\psi$ transforms as a scalar, so I define the scalar $$\label{app2}
\Psi=S^{-1}\psi .$$ Since $\Psi$ is a scalar, the quantity $$\label{app3}
\bar{\Psi}=\Psi^{\dagger} \gamma^0$$ is also a scalar. Using (\[app2\]), (\[app3\]) can also be written as $$\label{app4}
\bar{\Psi}=\psi^{\dagger} (S^{-1})^{\dagger} \gamma^0.$$ I want to define a quantity $\Gamma^{\mu}$ by requiring that $$\label{app5}
\bar{\Psi}\Gamma^{\mu}\Psi = \bar{\psi}\gamma^{\mu}\psi .$$ The right-hand side of (\[app5\]) is a vector while on the left-hand side $\bar{\Psi}$ and $\Psi$ are scalars, which implies that $\Gamma^{\mu}$ is a vector. What I need is a relation between $\Gamma^{\mu}$ and $\gamma^{\mu}$. For that purpose, I write $$\begin{aligned}
\label{app6}
\bar{\Psi}\Gamma^{\mu}\Psi & = & \psi^{\dagger} (S^{-1})^{\dagger} \gamma^0 \Gamma^{\mu} S^{-1}\psi
\nonumber \\
& = & \psi^{\dagger} (S^{-1})^{\dagger} \gamma^0 S^{-1}S \Gamma^{\mu} S^{-1}\psi
\nonumber \\
& = & \psi^{\dagger} (S^{-1})^{\dagger} S^{\dagger} \gamma^0 S \Gamma^{\mu} S^{-1}\psi
\nonumber \\
& = & \psi^{\dagger} (SS^{-1})^{\dagger} \gamma^0 S \Gamma^{\mu} S^{-1}\psi
\nonumber \\
& = & \psi^{\dagger} \gamma^0 S \Gamma^{\mu} S^{-1}\psi = \bar{\psi} S \Gamma^{\mu} S^{-1}\psi ,\end{aligned}$$ where in the third line I used (\[inverse2\]). The comparison with (\[app5\]) shows that $$\gamma^{\mu}=S \Gamma^{\mu} S^{-1} ,$$ which is equivalent to $$\label{app7}
\Gamma^{\mu} = S^{-1} \gamma^{\mu} S .$$
Now multiply the Dirac equation $(i\gamma^{\mu}\partial_{\mu} -m)\psi =0$ with $S^{-1}$ from the left and insert $1=SS^{-1}$ to obtain $$\label{app8}
iS^{-1}\gamma^{\mu} S \partial_{\mu}S^{-1} \psi -m S^{-1} \psi =0.$$ Using (\[app7\]) and (\[app2\]), this can be written as $$\label{app9}
(i\Gamma^{\mu}\partial_{\mu} -m)\Psi =0 ,$$ which is manifestly Lorentz covariant.
Now let me check that $\bar{\Psi}\Psi=\bar{\psi}\psi$. Similarly to (\[app6\]), one obtains $$\begin{aligned}
\label{app10}
\bar{\Psi}\Psi & = & \psi^{\dagger} (S^{-1})^{\dagger} \gamma^0 S^{-1}\psi
= \psi^{\dagger} (S^{-1})^{\dagger} S^{\dagger} \gamma^0 \psi
\nonumber \\
& = & \psi^{\dagger} (SS^{-1})^{\dagger} \gamma^0 \psi
= \psi^{\dagger} \gamma^0 \psi = \bar{\psi}\psi.\end{aligned}$$
Finally, let me demonstrate that the scalar nature of $\Psi^{\dagger}\Psi$ is not in contradiction with the fact that $\psi^{\dagger}\psi$ is not a scalar. This is seen from $$\label{app11}
\Psi^{\dagger}\Psi = \psi^{\dagger} (S^{-1})^{\dagger} S^{-1}\psi \neq \psi^{\dagger} \psi ,$$ which is a consequence of the fact that $S$ is not unitary due to (\[inverse\]).
[99]{}
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J. J. Sakurai, [*Advanced Quantum Mechanics*]{} (Addison-Wesley Publishing Company, Massachusetts, 1967). C. Itzykson and J.-B. Zuber, [*Quantum Field Theory*]{} (McGraw-Hill, New York, 1980). A. Zee, [*Quantum Field Theory in a Nutshell*]{} (Princeton University Press, Princeton, 2010).
A. Messiah, [*Quantum Mechanics II*]{} (North-Holland Publishing Company, Amsterdam, 1962). J. M. Jauch and F. Rohrlich, [*The Theory of Photons and Electrons*]{} (Springer-Verlag, Berlin, 1976). W. Greiner, [*Relativistic Quantum Mechanics*]{} (Springer, Berlin, 1990).
V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, [*Quantum Electrodynamics*]{} (Pergamon Press, Oxford, 1982). L. H. Ryder, [*Quantum Field Theory*]{} (Cambridge University Press, Cambridge, 1985). S. Weinberg, [*The Quantum Theory of Fields I*]{} (Cambridge University Press, 1995).
H. Nikolić, “QFT as pilot-wave theory of particle creation and destruction,” Int. J. Mod. Phys. A [**25**]{}, 1477-1505 (2010); arXiv:0904.2287. H. Nikolić, “Relativistic quantum mechanics and quantum field theory,” in X. Oriols and J. Mompart (eds.), [*Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology*]{} (Pan Stanford Publishing, 2012); arXiv:1205.1992. H. Nikolić, “Time and probability: from classical mechanics to relativistic Bohmian mechanics,” arXiv:1309.0400.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The state diagrams of a class of singular linear feedback shift registers (LFSR) are discussed. It is shown that the state diagrams of the given LFSR have special structures. An algorithm is presented to construct a new class of de Bruijn cycles from the state diagrams of these singular LFSR.'
author:
- |
XiaoFang Wang, YuJuan Sun and WeiGuo Zhang\
ISN Laboratory, Xidian University, Xi’an 710071, China\
e-mail: [email protected]
title: State Diagrams of a Class of Singular LFSR and Their Applications to the Construction of de Bruijn Cycles
---
**Keywords:** singular LFSR, state diagram, perfect binary directed tree, de Bruijn cycle.
Introduction {#sec1-1}
============
Pseudo-random sequences that come from feedback shift registers (FSR) are basic components for stream ciphers in cryptography, and FSR are also important components in modern communications due to their ease and efficiency in implementation. The problems on FSR have been important challenges in mathematics and they were extensively studied (see [@Golomb67; @GolombGong05; @Alhakim2012Spans], and the references therein).
As a kind of special FSR sequences, de Bruijn sequences have attracted much attentions due to their good randomness properties [@Fredricksen1978Necklaces; @Hauge1996De; @Hauge1996On; @Alhakim2012Spans; @Mayhew2002Extreme; @MandalGong131; @ChangEzerman161; @ChangEzerman162; @Sawada2016A]. A comprehensive survey of previous works on this subject can be found in [@Fredricksen82].
The sequences generated by nonsingular linear feedback shift registers (LFSR) have been analyzed using some methods from finite fields [@Golomb67; @GolombGong05]. Their properties can be explained by the description of their geometric structures. For examples, maximum-length LFSR, pure cycling registers and pure summing registers have special cycle structures [@EtzionLempel84]. Nonsingular LFSR can be used to generate de Bruijn cycles by the cycle-joining method [@Golomb67]. The applications of this method requires the full knowledge of the cycle structures and adjacency graphs of the original LFSR [@HemmatiSchilling84; @LiHelleseth141; @LiHelleseth142; @LiHelleseth16; @LiLin16]. However, the cycle-joining method does not work to construct de Bruijn cycles from singular LFSR since the state diagrams don’t only contains disjoint cycles. The number of singular LFSR is the same as that of nonsingular LFSR [@Golomb67]. But unfortunately, for a fixed singular LFSR, the geometry structure of its state diagram is far less developed, let alone its other properties.
The purpose of this paper is to characterize the state diagrams of a class of singular LFSR and to construct some new de Bruijn sequences. The remainder of this paper is organized as follows. In Section \[sec1-2\], we introduce some basic conceptions and related results. In Section \[sec1-3\], the state diagrams of a class of singular LFSR are completely determined, and the adjacency graphs of the LFSR are also given. In Section \[sec1-4\], we construct a new family of de Bruijn cycles. In Section \[sec1-5\], we provide an example to illuminate the algorithms given in the previous sections. Section \[sec1-6\] concludes the paper.
Preliminaries {#sec1-2}
=============
For a positive integer $n$, let $GF(2)^n$ be the $n$-dimensional vector space over $GF(2)$, where $GF(2)$ is the finite field with two elements.
FSR Sequences {#subsec1-1}
-------------
An *$n$-stage FSR* is a circuit arrangement consisting of $n$ binary storage elements (called stages) regulated by a clock. The stages are labeled from $1$ to $n$. A *state of an FSR* is an $n$-tuples vector $(x_1, x_2, \ldots , x_{n})$, where $x_i$ indicates the content of stage $i$. At each clock pulse, for some integer $i \geq 1$, the state $A_i = (a_i , a_{i+1}, \ldots , a_{i+n-1})$ is updated to $$\label{equ1-1}
A_{i+1 }= (a_{i+1}, \ldots , a_{i+n-1}, F(a_i , a_{i+1}, \ldots , a_{i+n-1})),$$ where $F: GF(2)^n\mapsto GF(2)$ is the feedback function of the FSR. The initially loaded content $A_1 = (a_1, a_2, \ldots , a_{n})$ is called an *initial state*. The feedback function $F$ can induce a *state operation* $\Gamma: GF(2)^n\mapsto GF(2)^n$ with $\Gamma(A_{i})=A_{i+1}$. After consecutive clock pulses, the FSR outputs a sequence $a = (a_1, a_2, a_3, \ldots )$ called an output sequence of the FSR. Therefore, the sequence $a = (a_1, a_2, a_3, \ldots )$ satisfies the recursive relationship $$\label{equ1-2}
a_{i+n} =F(a_i , a_{i+1}, \ldots , a_{i+n-1})$$ for all $i \geq 1$ and a given initial state $A_1 = (a_1, a_2, \ldots , a_{n})$. We denote by $\Omega( F )$ the set of all $2^n$ sequences generated by $F(x_1, x_2, \ldots , x_{n})$.
An FSR is called an *LFSR* if its feedback function $F(x_1, x_2, \ldots , x_{n})$ is linear. The output sequences of an LFSR are called LFSR sequences. A *de Bruijn sequence of order $n$* is a binary sequence $a = (a_1, a_2, a_3,\ldots)$ of period $2^n$ such that all $n$-tuples $(a_i, a_{i+1}, \ldots,a_{i+n-1}), i=1,2,\ldots,2^n$, are pairwise distinct, where $n$ is a positive integer.
Let $\mathbb{V}(GF(2))$ be the set of all binary infinite sequences. For $a =(a_1, a_2, a_3, \ldots) \in \mathbb{V}(GF(2))$, we define the *shift operator* $L$ on $\mathbb{V}(GF(2))$ by $$La =L(a_1, a_2, a_3, \ldots) = (a_2, a_3, a_4, \ldots) \in \mathbb{V}(GF(2)).$$ Generally, we have $L^ma = (a_{m+1}, a_{m+2},a_{m+3}, \ldots)$ for any integer $m$ with $m \geq 1$, and denote $L^0a = a$. A sequence $a =(a_1, a_2, a_3, \ldots) \in \mathbb{V}(GF(2))$ is called periodic if there exists a positive integer $r$ such that $L^r a = a$, i.e., $a_{i+r} = a_i$ for each integer $i \geq 1$. The smallest positive integer $r_0$ satisfying this property is called the period of $a$, denoted by $per(a)$. Specially, we can define the shift operator $L$ on $GF(2)^n$. For an $n$-stage FSR, let $A_1 =(a_1, a_2, \ldots,a_{n}) \in GF(2)^n$, we have $$\label{equ1-3}
L^iA_1=A_{i+1}=(a_{i+1}, a_{i+2},\ldots , a_{i+n}),~~ where~~ i\geq 0.$$
An $n$-stage FSR and its feedback function $F(x_1, x_2, \ldots , x_{n})$ are said to be *nonsingular* if the state operation $\Gamma$ is one-to-one, i.e., for any two states $A$ and $B$, $\Gamma A = \Gamma B$ implies $A = B$. It is well known that all sequences in $\Omega( F )$ are periodic if and only if the feedback function $F$ is nonsingular, i.e., $F$ can be written as $F(x_1 , x_2, \ldots , x_{n})=x_1+F_1(x_{2}, x_{3},\ldots , x_{n})$, where $F_1$ is a function from $GF(2)^{n-1}$ to $GF(2)$. If the state operation $\Gamma$ is not one-to-one, then the FSR and its feedback function $F$ are said to be *singular*, i.e., at least there exist two different states $A^{\prime}$ and $B^{\prime}$ satisfying $\Gamma A ^{\prime}= \Gamma B^{\prime}$. When $F$ is singular, i.e., $F$ can not be written as $F(x_1 , x_{2}, \ldots , x_{n})=x_1+F_1(x_{2}, x_{3},\ldots , x_{n})$, and all the sequences in $\Omega( F )$ are ultimately periodic. The feedback function $F(x_{1},x_{2},\ldots, x_{n})=x_{n-1}+x_n \ (n\geq3)$ discussed in this paper is obviously singular.
State diagrams {#subsec1-2}
--------------
A directed graph $D$ is an ordered pair $(V(D),A(D))$ consisting of a set $V(D)$ of vertices and a set $A(D)$ of arcs together with an incidence function $\psi_D$, such that each arc of $A(D)$ is an ordered pair of vertices of $V(D)$. If $\mathfrak{a}~:u\rightarrow v$ is an arc, i.e., $\psi_D(\mathfrak{a}) = (u, v)$, then $\mathfrak{a}$ is said to join $u$ to $v$, and we also say that $u$ dominates $v$. The vertex $u$ is the tail of $\mathfrak{a}$, and the vertex $v$ is its head. They are the two ends of $\mathfrak{a}$.
For an $n$-stage FSR with the feedback function $F(x_1, x_2, \ldots , x_{n})$, its *state diagram* $G_F$ is a directed graph with $2^n$ vertices, each vertex labeled with a unique binary $n$-length vector. For two vertices $Y^n =(y_1, y_{2}, \ldots, y_{n})$ and $Z^n =(z_1, \ldots, z_{n-1}, z_{n})$ in $G_F$, an arc is drawn from $Y^n$ to $Z^n$ if and only if $( y_2, y_3, \ldots, y_{n})=(z_1, z_2, \ldots, z_{n-1})$ and $z_n =F(y_1, y_2, \ldots, y_{n})$. In other words, an $n$-stage FSR state diagram $G_F$ is a directed graph with $2^n$ vertices such that there is an arc $Y^n\rightarrow Z^n$ if and only if $L(Y^n) = Z^n$, where $L$ is the shift operator. In this case, we say that $Y^n$ leads to $Z^n$, $Y^n$ is a *predecessor* of $Z^n$ and $Z^n$ is the *successor* of $Y^n$. We also say that the vertex $Y^n$ is the tail of the arc $Y^n\rightarrow Z^n$, and the vertex $Z^n$ is its head. In particular, if $Y^n$ is a predecessor of $Y^n$, then the state diagram has a *loop* at $Y^n$. For any given $n$-length vector $Y^n$, there are two possible predecessors for $Y^n$, and $Y^n$ also has two possible successors. Given an $n$-stage FSR, each state has only one successor.
A *cycle* $C = (Y^n_{1}, Y^n_{2}, \ldots ,Y^n_{k})$ of length $k$ in the state diagram of an $n$-stage FSR is a cyclic sequence of $k$ distinct states $Y^n_{1}, Y^n_{2}, \ldots ,Y^n_{k}$ such that $Y^n_{t}$ leads to $Y^n_{t+1}$ for each $1\leq t \leq k-1$ and $Y^n_{k}$ leads to $Y^n_{1}$, where $k\geq1$. A $k$-length cycle is denoted also by $C(k)$. A convenient representation of a cycle of length $k$ is a ring sequence $[y_1,y_2,\ldots,y_{k}]$, where $y_i$ is the first component of $Y^n_{i}$ for $1\leq i\leq k$. From the definition of a cycle, all the cyclic shifts of $[y_1,y_2,\ldots,y_{k}]$ denote the same cycle.
For a fixed singular FSR, the state diagram $G_F$ can be denoted by $G_F=G_1\cup G_2\cup\ldots\cup G_t$, where $G_i$, $i=1,2,\ldots,t$, is a connected component.
The maximum length of cycles in the state diagram of an $n$-stage FSR is $2^n$. In this case, the FSR is said to be a maximum-length FSR and the cycle is an $n$-order de Bruijn cycle (full cycle).
For a state $Y^n =(y_1,y_2,\ldots, y_{n})$ of an $n$-stage FSR, its *conjugate* $\widehat{Y^n}$ and *companion* $\widetilde{Y^n}$ are defined as $$\widehat{Y^n}=(y_1+1, y_2, \ldots, y_{n})$$ and $$\widetilde{Y^n}=(y_1, y_2, \ldots, y_{n}+1).$$ Two states $Y^n$ and $\widehat{Y^n}$ form a conjugate pair, and a conjugate pair is denoted by $conj(Y^n,\widehat{Y^n})$. Two states $Y^n$ and $\widetilde{Y^n}$ constitute a companion pair, and a conjugate pair is denoted by $comp(Y^n,\widetilde{Y^n})$. Given an $n$-length vector $Y^n =(y_1, y_2, \ldots, y_{n})$, its two possible predecessors form a conjugate pair, and its two possible successors constitute a companion pair.
For a nonsingular FSR, the problem of determining conjugate pairs among cycles in a state diagram $G_F$ leads to the definition of adjacency graph. For a fixed nonsingular FSR with feedback function $F(x_1, x_2, \ldots , x_{n})$, its adjacency graph is an undirected weighted graph where the vertices correspond to the cycles $C_1,C_2,\ldots,C_s$ in $G_F$. Two cycles $C_i$ and $C_j$ are *adjacent* if they are disjoint and there exists a state $Y^n$ on cycle $C_i$ whose conjugate $\widehat{Y^n}$ is on cycle $C_j$, where $1\leq i,j\leq s$. Then we also say that cycle $C_j$ is the adjacent cycle of cycle $C_i$. The classic idea of the cycle-joining method is that two adjacent cycles $C_i$ and $C_i$ are joined into a single cycle when the successors of $Y^n$ and $\widehat{Y^n}$ are interchanged. When two cycles can be connected by exactly $m$ edges, it is convenient to denote the $m$ edges by an edge labeled with an integer $m$, where $m\geq 2$. The weighted graph obtained in this way is called the adjacency graph of the nonsingular FSR with the feedback function $F(x_1, x_2, \ldots , x_{n})$ and denoted by $\mathcal {G}(F)$.
For any singular FSR, we can obtain its adjacency graph $\mathcal {G}(F)$ by making the connected components $G_1,G_2,\ldots,G_t$ corresponding to vertices similarly.
Perfect binary directed trees {#subsec1-3}
-----------------------------
Each acyclic connected graph is a tree. The top vertex in a tree is its root. A leaf (or terminal vertex) is a vertex with only one adjacent vertex. The depth of a vertex is the number of edges from this vertex to the tree’s root vertex. The binary tree is a connected acyclic graph, and each vertex has at most two adjacent vertices. If a directed graph is an acyclic connected graph when the direction is ignored, then we call this directed graph as a directed tree. If the tree is a binary tree when there is no consideration to the direction, then we call this directed graph as a binary directed tree.
\[def1-1\] A perfect binary directed tree is a binary directed tree in which all interior vertices have two adjacent vertices and all leaves have the same depth or same level. The depth of a perfect binary directed tree is the length of path from one leaf to the root. A perfect binary directed tree with depth $k$ is denoted by $T_k$. Obviously, there are $(2^{k+1}-1)$ vertices in $T_k$.
Given a digraph $D$, denote all vertices (or arcs) of $D$ as $V(D)$ (or $A(D)$). If $v\in V(D)$ is a head or a tail of $a\in A(D)$, then we call $a$ to be incident with $v$. Let all arcs incident with $v$ be denoted by $A(v)$, then $D\setminus v$ can be induced from $D$ such that $V(D\setminus v)=V(D)\setminus\{v\}$ and $A(D\setminus v)=A(D)\setminus A(v)$. If $C = (v_1,v_2,\ldots,v_k)$ is a $k$-length cycle and $V(C)=\{v_1,v_2,\ldots,v_k\}$, $D\setminus C$ can be induced from $D$ such that $V(D\setminus C)=V(D)\setminus\{v_1,v_2,\ldots,v_k\}$ and $A(D\setminus C)=A(D)\setminus(A(v_1)\cup A(v_2)\cup \ldots \cup A(v_k))$.
State diagrams of LFSR with the feedback function $F(x_{1},x_{2},\ldots, x_{n})=x_{n-1}+x_n $ {#sec1-3}
=============================================================================================
We next consider the singular $n$-stage LFSR with the feedback function of the form $$\label{1-4}
F(x_{1},x_{2},\ldots, x_{n})=x_{n-1}+x_n,~(n\geq3).$$
[@Golomb67] \[state-diagram1-1\] Given an FSR with feedback function $F(x_{1},x_{2},\ldots, x_{n})$, the state diagram $G_F$ only has finite connected components $G_1,G_2,\ldots,G_t$. Every connected component only contains unique cycle. In other words, every connected component is a cycle or a cycle with some branches.
[@Golomb67] \[state-diagram1-2\] Let $F(x_{1},x_{2},\ldots, x_{n})$ be the feedback function of a fixed $n$-stage FSR, and $a= (a_1, a_2, a_3, \ldots )\in \Omega( F )$ be the output sequence of a given initial state $A_1^n = (a_1, a_2, \ldots , a_{n})$. Then $A_1^n$ belongs to unique connected component $G_i$ in the state diagram $G_F$. Furthermore, if we delete some initial terms of $a$, a periodic sequence whose period is the length of the cycle $C_i$ can be obtained, where $C_i$ is the unique cycle of $G_i$.
For $(a,b,c)\in GF(2)^3$, let $$S_{abc}^n=\{Y^n=(y_1,y_2, \ldots, y_{n})\mid (y_{n-2}, y_{n-1}, y_{n})=(a,b,c)\}.$$ Let $n$-length vectors of zeros and ones be denoted by $\mathbf{0}^n=(0,0,\ldots,0)$ and $\mathbf{1}^n=(1,1,\ldots,1)$.
\[state-diagram1-3\] Given an $n$-stage singular LFSR with feedback function $ F(x_{1},x_{2},\ldots, x_{n})=x_{n-1}+x_n$, the state diagram $G_F$ only has two connected components $G_1$ and $G_2$, where $G_1$ contains the loop $[0]$ and $G_2$ contains a $3$-length cycle $[0,1,1]$.
For any state $Y^n=(y_1,y_2,\ldots,y_n)$ of $$S_{100}^n=\{Y^n=(y_1,y_2, \ldots,y_{n-2}, y_{n-1}, y_{n}) \mid (y_{n-2}, y_{n-1}, y_{n})=(1,0,0)\},$$ we have $$\begin{aligned}
LY^n &=& (a_2,~\ldots,~a_{n-3},~1,0,0,0) \\
L^2Y^n &=& (a_3,\ldots,a_{n-3}, 1, 0, 0, 0, 0) \\
&\ldots& \\
L^{n-2}Y^n &=& (0,\ldots,0, 0, 0, 0, 0, 0)= \mathbf{0}^n.\end{aligned}$$ Similarly, for any state $Y^n=(y_1,y_2,\ldots,y_n)$ belonging to $S_{000}^n$, we have $$LY^n=(y_2,y_3,\ldots,0,0,0,0),~\ldots,~L^iY^n=\mathbf{0}^n,~0\leq i\leq n-3.$$ Obviously, when $\mathbf{0}^n$ is the initial state, its successor is $\mathbf{0}^n$ itself. That is, there is a loop at $\mathbf{0}^n$. Then every state $Y^n \in S_{100}^n\bigcup S_{000}^n $ always leads to $\mathbf{0}^n$ and belongs to the same connected component $G_1$, and this connected component contains the loop $[0]$.
Let $v_1=(0,1,1,0,1,1,\ldots)$, $v_2=(1,1,0,1,1,0,\ldots)$ and $v_3=(1,0,1,1,0,1,\ldots)\in GF(2)^n$. Since the feedback function is $F(x_{1},x_{2},\ldots, x_{n})=x_{n-1}+x_n $, we can get the following relations through calculation: $Lv_1= v_2$, $Lv_2= v_3$ and $Lv_3= v_1$. In other words, $v_1\rightarrow v_2 \rightarrow v_3\rightarrow v_1$, i.e., $[0,1,1]$ is a $3$-length cycle. It is not difficult to verify that for any state $Y^n\not\in S_{100}^n\bigcup S_{000}^n$, $Y^n$ will ultimately go into the cycle $[0,1,1]$, which implies $Y^n$ belongs to the other connected component $G_2$ and $G_2$ contains the $3$-length cycle $[0,1,1]$.
\[state-diagram1-4\] With the same notations as above. $G_1\setminus[0]$ is a perfect binary directed tree $T_{n-3}^{(1)}$, and $G_2\setminus[0,1,1]$ contains three perfect binary directed trees $T_{n-3}^{(2)}$, $T_{n-3}^{(3)}$ and $T_{n-3}^{(4)}$.
Note that the $n$-stage LFSR is singular, we have that a pair conjugate states have the same successor. By Lemma \[state-diagram1-1\] and Theorem \[state-diagram1-3\], $G_1\setminus[0]$ and $G_2\setminus[0,1,1]$ are four binary directed trees. For each $Y^n\in S_{100}$, we have $L^{n-3}Y^n=\widehat{\mathbf{0}^n}$. Since the length of the directed path from any leaf $Y^n\in S_{100}$ to $\widehat{\mathbf{0}^n}$ is $n-3$ ($\widehat{\mathbf{0}^n}$ is the root of this tree), this binary directed tree $G_1\setminus[0]$ is a perfect binary directed tree $T_{n-3}^{(1)}$. Similarly, $G_2\setminus[0,1,1]$ contains three perfect binary directed trees $T_{n-3}^{(2)}$, $T_{n-3}^{(3)}$ and $T_{n-3}^{(4)}$, whose roots are $\widehat{v_1}$, $\widehat{v_2}$ and $\widehat{v_3}$, respectively.
\[state-diagram1-5\] Let $F(x_{1},x_{2},\ldots, x_{n})=x_{n-1}+x_n \ (n\geq3)$ be the feedback function of an $n$-stage singular LFSR. Then the adjacency graph $\mathcal {G}(F)$ only has two isolated vertices $G_1$ and $G_2$.
A new class of de Bruijn cycles {#sec1-4}
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In this section, we introduce a new method to obtain a new class of de Bruijn cycles by modifying the state diagrams of LFSR with the feedback function $F(x_{1},x_{2},\ldots, x_{n})=x_{n-1}+x_n \ (n\geq3)$. Note that by Corollary \[state-diagram1-5\] the cycle-joining method does not work in this case.
Given an $n$-stage singular LFSR with the feedback function $F(x_{1},x_{2},\ldots, x_{n})=x_{n-1}+x_n \ (n\geq3)$, we next present an algorithm to construct the cycles without branches from its state diagram $G_F$.
Firstly, all states on the state diagram $G_F$ are classified. The set consisting of all states on the two cycles $[0]$ and $[0,1,1]$ is denoted as $S_0^n$, i.e., $S_0^n=\{(0,1,1,0,1,1,\ldots),$ $ (1,1,0,1,1,0,\ldots),(1,0,1,1,0,1,\ldots),\mathbf{0}^n\}$. The set of all leaves is denoted as $S^n$ with $S^n=S_{100}^n\cup S_{001}^n\cup S_{010}^n\cup S_{111}^n$. An $l$-length directed path $[Y_{s,t}^n\rightarrow LY_{s,t}^n\rightarrow\ldots\rightarrow L^{l}Y_{s,t}^n]$ is written as $p_{s,t}$. For a directed path $p_{s,t}$, the tail of $p_{s,t}$ is denoted by $T(p_{s,t})=Y_{s,t}^n$, then the set of states remaining on the path except the tail $Y_{s,t}^n$ is denoted as $U_{s,t}$, i.e., $U_{s,t}=\{LY_{s,t}^n,\ldots,L^{l}Y_{s,t}^n\}$. In the state diagram $G_F$, if a state has two predecessors and one successor, then we call this state a trigeminal vertex. Obviously, $S_0^n$ and $U_{s,t}$ are all sets of trigeminal vertices.
The feedback function $F(x_{1},x_{2},\ldots, x_{n})=x_{n-1}+x_n $, and the state $Y_{i,j}^n\in S^n=S_{100}^n\cup S_{001}^n\cup S_{010}^n\cup S_{111}^n$ The set of paths $P$ and the set of cycles $C$ Set $i=1$, $S_0^n=\{(0,1,1,0,1,1,\ldots), (1,1,0,1,1,0,\ldots), (1,0,1,1,0,1,\ldots), \mathbf{0}^n\}$, $T(P_{0})=\emptyset$, $U_{0,0}=\emptyset$ $j=1$ $l_{i,j}=n-2$ $p_{i,j} \gets [Y_{i,j}^n\rightarrow LY_{i,j}^n\rightarrow\ldots\rightarrow L^{l_{i,j}}Y_{i,j}^n]$ $Y_{i,j+1}^n \gets \widetilde{L^{l_{i,j}+1}Y_{i,j}^n}$ $l_{i,j}$ be the least integer such that $L^{l_{i,j}}Y_{i,j}^n\in \bigcup_{k_1=0}^{i-1}\bigcup_{k_1=0}^{j-1}U_{k_1,k_2}$ $p_{i,j} \gets [Y_{i,j}^n\rightarrow LY_{i,j}^n\rightarrow\ldots\rightarrow L^{l_{i,j}-1}Y_{i,j}^n]$ $Y_{i,j+1}^n \gets \widetilde{L^{l_{i,j}}Y_{i,j}^n}$ $p_{i,j} \gets [Y_{i,j}^n]$, i.e., the single state $Y_{i,j}^n$ is considered as a path of length $0$ $Y_{i,j+1}^n \gets \widetilde{L^{l_{i,j}}Y_{i,j}^n}$ $j \gets j+1$ $P_i=\{p_{i,1},p_{i,2},\ldots,p_{i,j-1}\}$ $C_i: [p_{i,1}\rightarrow p_{i,2}\rightarrow\cdots\rightarrow p_{i,j-1}\rightarrow p_{i,1}]$ $T(P_{i})=\{T(p_{i,k})|1\leq k\leq j-1, T(p_{i,k})=Y_{i,k}^n\}$ $i \gets i+1$
Choose a state $Y_{1,1}^n$ from $S^n$ randomly. If $l_{1,1}$ is the least integer such that $L^{l_{1,1}}Y_{1,1}^n\in S_0^n$, then the directed path $[Y_{1,1}^n\rightarrow LY_{1,1}^n\rightarrow\ldots\rightarrow L^{l_{1,1}}Y_{1,1}^n]$ is denoted by $p_{1,1}$.
Based on $Y_{1,1}^n$, $l_{1,1}$ and $p_{1,1}$, select $Y_{1,2}^n$, and search for $l_{1,2}$ and $p_{1,2}$. In general, based on $Y_{1,j}^n$, $l_{1,j}$ and $p_{1,j}$, select $Y_{1,j+1}^n$ using the following method, and search for $l_{1,j+1}$ and $p_{1,j+1}$. If $L^{l_{1,j}}Y_{1,j}^n\in S_0^n$, then set $Y_{1,j+1}^n=\widetilde{L^{l_{1,j}+1}Y_{1,j}^n}$; If $L^{l_{1,j}}Y_{1,j}^n\in \bigcup_{k=0}^{j-1}U_{1,k}$, then set $Y_{1,j+1}^n=\widetilde{L^{l_{1,j}}Y_{1,j}^n}$. Let $l_{1,j+1}$ be the least integer such that $L^{l_{1,j+1}}Y_{1,j+1}^n\in S_0^n\bigcup_{k=0}^{j}U_{1,k}$. If $L^{l_{1,j+1}}Y_{1,j+1}^n\in S_0^n$, then the directed path $[Y_{1,j+1}^n\rightarrow LY_{1,j+1}^n\rightarrow\ldots\rightarrow L^{l_{1,j+1}}Y_{1,j+1}^n]$ is denoted by $p_{1,j+1}$; otherwise, $L^{l_{1,j+1}}Y_{1,j+1}^n\in \bigcup_{k=0}^{j}U_{1,k}$, then the directed path $[Y_{1,j+1}^n \rightarrow LY_{1,j+1}^n \rightarrow\ldots\rightarrow L^{l_{1,j+1}-1}Y_{1,j+1}^n]$ is denoted by $p_{1,j+1}$. If $l_{1,j}=1$, then the single state $Y_{1,j}^n$ is considered as a path of length $0$, i.e., $p_{1,j}: [Y_{1,j}^n]$. Repeat the directed path searching in the state diagram until the state $Y_{1,s_1+1}^n$ returns to $Y_{1,1}^n$. Set $P_1=\{p_{1,1},p_{1,2},\ldots,p_{1,s_1}\}$ and $T(P_{1})=\{T(p_{1,j})|1\leq j\leq s_1\}$. Connect the head of $p_{1,v}$ with the tail of $p_{1,v+1}$ in turn for $1\leq v\leq s_1-1$, and connect the head of $p_{1,s_1}$ with the tail of $p_{1,1}$, then the directed cycle $C_1$ is obtained, i.e. $C_1: [p_{1,1}\rightarrow p_{1,2}\rightarrow\ldots\rightarrow p_{1,s_1}\rightarrow p_{1,1}]$.
Based on the set of directed paths $P_1$ and the cycle $C_1$, the new directed path sets $P_2$ and the new cycle $C_2$ can be searched. In general, based on $P_i$ and $C_i$, search for $P_{i+1}$ and $C_{i+1}$ in the following way. Choose an $n$-stage state $Y_{i+1,1}^n$ from $S^n\setminus \bigcup_{1\leq m\leq i}T(P_{m})$ randomly. Let $l_{i+1,1}$ be the least integer such that $L^{l_{i+1,1}}Y_{i+1,1}^n\in S_0^n\bigcup_{m\leq i}U_{m,k}$. If $L^{l_{i+1,1}}Y_{i+1,1}^n\in S_0^n$, then the directed path $[Y_{i+1,1}^n\rightarrow LY_{i+1,1}^n\rightarrow\ldots\rightarrow L^{l_{i+1,1}}Y_{i+1,1}^n]$ is denoted by $p_{i+1,1}$; If $L^{l_{i+1,1}}Y_{i+1,1}^n\in \bigcup_{m\leq i}U_{m,k}$, then the directed path $[Y_{i+1,1}^n \rightarrow LY_{i+1,1}^n \rightarrow\ldots\rightarrow L^{l_{i+1,1}-1}Y_{i+1,1}^n]$ is denoted by $p_{i+1,1}$. Based on $Y_{i+1,1}^n$, $l_{i+1,1}$ and $p_{i+1,1}$, select $Y_{i+1,2}^n$ recursively, and search for $l_{i+1,2}$ and $p_{i+1,2}$. Repeat the directed path searching in the state diagram until the state $Y_{i+1,s_{i+1}+1}^n$ returns to $Y_{i+1,1}^n$. If $l_{i+1,j}=1$, then the single state $Y_{i+1,j}^n$ is considered as a path of length $0$, i.e., $p_{i+1,j}: [Y_{i+1,j}]^n$. Set $P_{i+1}=\{p_{i+1,1},p_{i+1,2},\ldots,p_{i+1,s_{i+1}}\}$ and $T(P_{i+1})=\{T(p_{i+1,j})|1\leq j\leq s_{i+1}\}$. Connect the head of $p_{i+1,v}$ with the tail of $p_{i+1,v+1}$ in turn for $1\leq v\leq s_{i+1}-1$, and connect the head of $p_{i+1,s_{i+1}}$ with the tail of $p_{i+1,1}$. Then the directed cycle $C_{i+1}$ is obtained, i.e. $C_{i+1}: [p_{i+1,1}\rightarrow p_{i+1,2}\rightarrow\ldots\rightarrow p_{i+1,s_{i+1}}\rightarrow p_{i+1,1}]$.
Repeat the directed cycle searching in the state diagram until the state set $$S^n\setminus\bigcup_{1\leq m\leq t}T(P_{m})=\emptyset.$$ Then output the set of paths $P=\bigcup_{1\leq m\leq t}P_m$ and the set of cycles $C=\{C_1,C_2,\ldots,C_t\}$.
\[state-diagram1-6\] Given an LFSR with feedback function $F(x_{1},x_{2},\ldots, x_{n})=x_{n-1}+x_n \ (n\geq3)$, we repeat the directed path searching in the state diagram as in Algorithm \[alg1-1\]. Then the original state diagram turns into a branchless one.
Noticing the $n$-stage LFSR with the feedback function $F(x_{1},x_{2},\ldots, x_{n})=x_{n-1}+x_n $, all leaves and all trigeminal vertices are completely determined.
Note that each directed path found by Algorithm \[alg1-1\] is a directed path with a leaf as its tail and a trigeminal vertex as its head. We also notice that any two paths found by Algorithm \[alg1-1\] are different in both tails and heads. The process of finding directed paths and connecting them as cycles in Algorithm \[alg1-1\], in fact, modifies the successor of one of the two predecessor states of a given trigeminal vertex to its companion state. We also note that in this state diagram, any pair of companion states must have one as a leaf and the other as a trigeminal vertex. For the head of each directed path, you can modify its successor to the companion state of the original successor, that is, the next path can be found such that its tail is connected to the head of the previous path. At the same time, the tail of each directed path can be used as the modified successor of one of the two predecessor states of its companion state, i.e., there must be a path such that its head is connected to the tail of this path. So, all the leaves and trigeminal vertices in the original state diagram have disappeared. Then the original state diagram turns into a branchless one.
Algorithm \[alg1-1\] gives the steps to generate the required two sets $P$ and $C$. The implementation of Algorithm \[alg1-1\] requires the adjacency relation of state diagram. When the feedback function is given, each state has a unique successor. The time complexity of this algorithm is $O(2^{n+1})$.
[@Golomb67] \[state-diagram1-7\] Let $F(x_{1},x_{2},\ldots, x_{n})$ be the feedback function of an $n$-stage FSR. If its state diagram only consists of disjoint cycles, then this FSR is a nonsingular FSR. Any two cycles $C_i$ and $C_j$ can be joined into a single cycle when the successors of $Y^n\in C_i$ and its conjugate $\widehat{Y^n}\in C_j$ are interchanged, and a de Bruijn cycle be obtained finally.
[@Golomb67] \[state-diagram1-8\] Let $F(x_{1},x_{2},\ldots, x_{n})$ be the feedback function of an $n$-stage nonsingular FSR. If any state $Y^n$ and its conjugate $\widehat{Y^n}$ belong to the same cycle, then the state diagram of this FSR is a de Bruijn cycle.
$P=P_1\cup P_1\cup\ldots\cup P_t$ and $C=\{C_1,C_2,\ldots,C_t\}$ determined by Algorithm \[alg1-1\]. The conjugate pairs $conj(Z^n,\widehat{Z^n})$ between any two distinct cycles $C_i,C_j\in C$. For $P_i=\{p_{i,1},p_{i,2},\ldots,p_{i,s_i}\}$ and $P_j=\{p_{j,1},p_{j,2},\ldots,p_{j,s_j}\}$, there is only one conjugate pair $conj(Z^n,\widehat{Z^n})$ between $p_{i,d}$ and $p_{j,e}$, where $Y^n_{i,d}=t(p_{i,d}),Y^n_{j,e}=t(p_{j,e})$. For any $p_{i,w}\in P_i$ and $p_{j,z}\in P_j$, there is at most one conjugate pair $conj(Z^n,\widehat{Z^n})$ between $p_{i,w}$ and $p_{j,z}$, where $Y^n_{i,w}=t(p_{i,w}),Y^n_{j,z}=t(p_{j,z})$. $C_i$ and $C_j$ are not adjacent.
According to Lemma \[state-diagram1-7\] and \[state-diagram1-8\], the conjugate pairs between any two cycles are needed in the construction of de Bruijn sequences by the cycle joining method. The method to find conjugate pairs $conj(Z^n,\widehat{Z^n})$ between two any cycles in $C$ is given in Algorithm \[alg1-2\]. Since each trigeminal vertex has two conjugate predecessor states, the conjugate pairs $conj(Z^n,\widehat{Z^n})$ appear at one trigeminal vertex nearby. Through the characteristics of state diagram, we notice that a conjugate pair consists of either one state of one cycle and one state adjacent to the cycle, or two states of the same perfect binary directed tree.
Let $P=P_1\cup P_1\cup\ldots\cup P_t$ and $C=\{C_1,C_2,\ldots,C_t\}$ be the set of directed paths and the set of cycles obtained by Algorithm \[alg1-1\], respectively, where $C_{i}: [p_{i,1}\rightarrow p_{i,2}\rightarrow\ldots\rightarrow p_{i,s_{i}}\rightarrow p_{i,1}]$ and $P_{i}=\{p_{i,1},p_{i,2},\ldots,p_{i,s_{i}}\}(1\leq i\leq t)$.
\[state-diagram1-9\] Given a directed path searching in the state diagram as in Algorithm \[alg1-1\] with feedback function $F(x_{1},x_{2},\ldots, x_{n})=x_{n-1}+x_n \ (n\geq3)$, if we repeat the searching conjugate pair $conj(Z^n,\widehat{Z^n})$ between two any cycles in $C$ as in Algorithm \[alg1-2\], then conjugate pairs $conj(Z^n,\widehat{Z^n})$ between two any cycles will be obtained.
Since the cycle obtained by the Algorithm \[alg1-1\] is a union of some directed paths, the problem of considering the conjugate pair $conj(Z^n,\widehat{Z^n})$ between the two cycles is equivalent to the problem of considering the conjugate pair $conj(Z^n,\widehat{Z^n})$ between their directed paths. Note that a conjugate pair $conj(Z^n,\widehat{Z^n})$ consists of either one state of one cycle and one state adjacent to the cycle, or two states of the same perfect binary directed tree.
Case 1: When the last states of two directed paths belong to the same cycle of the original state diagram, that is, when two $(n-2)$-length directed paths have their last states in the same cycle, they can only share one conjugate pair $conj(Z^n,\widehat{Z^n})$. In fact, the last state of one path and the last but one state of the other path are a pair of conjugate states. Since the state diagram only contains two cycles $[0]$ and $[0,1,1]$, we only consider the $(n-2)$-length paths whose tails are in the set $S_0^n\setminus\{\mathbf{0}^n\}$.
Case 2: There are only conjugate pairs $conj(Z^n,\widehat{Z^n})$ between the two directed paths of the two cycles whose tails belong to the same set of $S_{100}^n$, $S_{001}^n$, $S_{010}^n$ and $S_{111}^n$.
Based on the set of directed paths and the set of cycles obtained by Algorithm \[alg1-1\], conjugate pairs $conj(Z^n,\widehat{Z^n})$ between two any cycles be obtained.
The number of directed paths obtained by Algorithm \[alg1-1\] is $2^{n-1}$. The Algorithm \[alg1-2\] is to find conjugate pairs $conj(Z^n,\widehat{Z^n})$ by comparing the directed paths between different cycles. The time complexity of Algorithm \[alg1-2\] is $O(2^{n-1})$. For the set of cycles obtained by Algorithm \[alg1-1\] and conjugate pairs $conj(Z^n,\widehat{Z^n})$ obtained by Algorithm \[alg1-2\], a new class of de Bruijn cycles be obtained with the cycle-joining method.
Example {#sec1-5}
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Let $F(x_1,x_2,\ldots,x_6)=x_5+x_6$, the state diagram of $F(x_{1},x_{2},\ldots, x_6)=x_5+x_6 $ is given in Figure \[Fig.lable1-1\].
$Y_{1,1}^6=(1,1,1,1,1,1)$ $l_{1,1}=4$ $p_{1,1}:[Y_{1,1}^6\rightarrow LY_{1,1}^6\rightarrow\ldots\rightarrow L^4Y_{1,1}^6]$
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$Y_{1,2}^6=(1,0,1,1,0,0)$ $l_{1,2}=4$ $p_{1,2}:[Y_{1,2}^6\rightarrow LY_{1,2}^6\rightarrow\ldots\rightarrow L^4Y_{1,2}^6]$
$Y_{1,3}^6=(0,0,0,0,0,1)$ $l_{1,3}=4$ $p_{1,3}:[Y_{1,3}^6\rightarrow LY_{1,3}^6\rightarrow\ldots\rightarrow L^4Y_{1,3}^6]$
$Y_{1,4}^6=(1,1,0,1,1,1)$ $l_{1,4}=3$ $p_{1,4}:[Y_{1,4}^6\rightarrow LY_{1,4}^6\rightarrow L^2Y_{1,4}^6]$
$Y_{1,5}^6=(1,1,1,0,1,0)$ $l_{1,5}=4$ $p_{1,5}:[Y_{1,5}^6\rightarrow LY_{1,5}^6\rightarrow\ldots\rightarrow L^4Y_{1,5}^6]$
$Y_{1,6}^6=(0,1,1,0,1,0)$ $l_{1,6}=1$ $p_{1,6}:[Y_{1,6}^6]$
$Y_{1,7}^6=(1,1,0,1,0,0)$ $l_{1,7}=3$ $p_{1,7}:[Y_{1,7}^6\rightarrow LY_{1,7}^6\rightarrow L^2Y_{1,7}^6]$
$Y_{1,8}^6=(1,0,0,0,0,1)$ $l_{1,8}=1$ $p_{1,8}:[Y_{1,8}^6]$
$Y_{1,9}^6=(0,0,0,0,1,0)$ $l_{1,9}=3$ $p_{1,1}:[Y_{1,9}^6\rightarrow LY_{1,9}^6\rightarrow L^2Y_{1,9}^6]$
$Y_{1,10}^6=(0,1,0,1,1,1)$ $l_{1,10}=1$ $p_{1,2}:[Y_{1,10}^6]$
$Y_{1,11}^6=(1,0,1,1,1,1)$ $l_{1,11}=2$ $p_{1,3}:[Y_{1,11}^6\rightarrow LY_{1,11}^6]$
$Y_{1,12}^6=(1,1,1,1,0,0)$ $l_{1,12}=2$ $p_{1,4}:[Y_{1,12}^6\rightarrow LY_{1,12}^6]$
$Y_{1,13}^6=(1,1,0,0,0,1)$ $l_{1,13}=2$ $p_{1,13}:[Y_{1,13}^6\rightarrow LY_{1,13}^6]$
$Y_{1,14}^6=(0,0,0,1,1,1)$ $l_{1,14}=2$ $p_{1,14}:[Y_{1,14}^6\rightarrow LY_{1,14}^6]$
$Y_{1,15}^6=(0,1,1,1,0,0)$ $l_{1,15}=1$ $p_{1,15}:[Y_{1,15}^6]$
$Y_{1,16}^6=(1,1,1,0,0,1)$ $l_{1,16}=3$ $p_{1,16}:[Y_{1,16}^6\rightarrow LY_{1,16}^6\rightarrow L^2Y_{1,16}^6]$
$Y_{1,17}^6=(0,0,1,1,0,0)$ $l_{1,17}=1$ $p_{1,17}:[Y_{1,17}^6]$
$Y_{1,18}^6=(0,1,1,0,0,1)$ $l_{1,18}=1$ $p_{1,18}:[Y_{1,18}^6]$
$Y_{1,19}^6=(1,1,0,0,1,0)$ $l_{1,19}=2$ $p_{1,19}:[Y_{1,19}^6\rightarrow LY_{1,19}^6]$
$Y_{1,20}^6=(0,0,1,0,1,0)$ $l_{1,20}=2$ $p_{1,20}:[Y_{1,20}^6\rightarrow LY_{1,20}^6]$
$Y_{1,21}^6=(1,0,1,0,1,0)$ $l_{1,21}=1$ $p_{1,21}:[Y_{1,21}^6]$
$Y_{1,22}^6=(0,1,0,1,0,0)$ $l_{1,22}=1$ $p_{1,22}:[Y_{1,22}^6]$
$Y_{1,23}^6=(1,0,1,0,0,1)$ $l_{1,23}=2$ $p_{1,23}:[Y_{1,23}^6\rightarrow LY_{1,23}^6]$
$Y_{1,24}^6=(1,0,0,1,1,1)$ $l_{1,24}=1$ $p_{1,24}:[Y_{1,24}^6]$
$Y_{1,25}^6=(0,0,1,1,1,1)$ $l_{1,25}=1$ $p_{1,25}:[Y_{1,25}^6]$
$Y_{1,26}^6=(0,1,1,1,1,1)$ $l_{1,26}=1$ $p_{1,26}:[Y_{1,26}^6]$
$Y_{2,1}^6=(1,0,0,1,0,0)$ $l_{2,1}=2$ $p_{2,1}:[Y_{2,1}^6\rightarrow LY_{2,1}^6]$
$Y_{2,2}^6=(0,1,0,0,0,1)$ $l_{2,2}=1$ $p_{2,2}:[Y_{2,2}^6]$
$Y_{2,3}^6=(1,0,0,0,1,0)$ $l_{2,3}=1$ $p_{2,3}:[Y_{2,3}^6]$
$Y_{2,4}^6=(0,0,0,1,0,0)$ $l_{2,4}=1$ $p_{2,4}:[Y_{2,4}^6]$
$Y_{2,5}^6=(0,0,1,0,0,1)$ $l_{2,5}=1$ $p_{2,5}:[Y_{2,5}^6]$
$Y_{2,6}^6=(0,1,0,0,1,0)$ $l_{2,6}=1$ $p_{2,6}:[Y_{2,6}^6]$
$C_{1}$ $ [p_{1,1}\rightarrow p_{1,2}\rightarrow\ldots\rightarrow p_{1,26}\rightarrow p_{1,1}]$
$C_{2}$ $ [p_{2,1}\rightarrow p_{2,2}\rightarrow\ldots\rightarrow p_{2,6}\rightarrow p_{2,1}]$
Step 1: Start a search at one of the leaves $Y_{1,1}^6=(1,1,1,1,1,1)$, and trace a directed path downward in the tree. Record the path and all vertices of the path from the tree. Then repeat the directed path search from the state diagram until the forest is empty. Note that each isolated vertex is considered as a path of length $0$, and we finally get $32$ different directed paths and $2$ cycles. In Table \[tab1-1\], we give the searching way.
Step 2: In Table \[tab1-2\], the classification of the tails of paths of Algorithm \[alg1-1\] is given.
Step 3: In Table \[tab1-3\], $5$ conjugate pairs $conj(Z^n,\widehat{Z^n})$ between two cycles $C_1$ and $C_2$ are found. Then interchange their successors, and $5$ de Bruijn cycles can be obtained.
$C_1$ $C_2$
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$S_{111}^6$ $Y_{1,1}^6,Y_{1,4}^6,Y_{1,10}^6,Y_{1,11}^6,Y_{1,14}^6,Y_{1,24}^6,Y_{1,25}^6,Y_{1,26}^6$ —–
$S_{100}^6$ $Y_{1,2}^6,Y_{1,7}^6,Y_{1,12}^6,Y_{1,15}^6,Y_{1,17}^6,Y_{1,22}^6$ $Y_{2,3}^6,Y_{2,6}^6$
$S_{001}^6$ $Y_{1,3}^6,Y_{1,8}^6,Y_{1,13}^6,Y_{1,16}^6,Y_{1,18}^6,Y_{1,23}^6$ $Y_{2,2}^6,Y_{2,5}^6$
$S_{010}^6$ $Y_{1,5}^6,Y_{1,6}^6,Y_{1,9}^6,Y_{1,19}^6,Y_{1,20}^6,Y_{1,21}^6$ $Y_{2,1}^6,Y_{2,4}^6$
[|c|c|]{}\
$(1,0,1,0,0,0)$& $(0,0,1,0,0,0)$\
$(1,1,0,0,0,1)$& $(0,1,0,0,0,1)$\
$(0,0,0,0,1,0)$& $(1,0,0,0,1,0)$\
$(1,0,1,0,0,1)$& $(0,0,1,0,0,1)$\
$(1,1,0,0,1,0)$& $(0,1,0,0,1,0)$\
Conclusion {#sec1-6}
==========
The state diagrams of a class of singular LFSR are discussed. Some properties of these singular linear feedback shift registers are also given. An algorithm is presented to construct a new class of de Bruijn cycles from the state diagrams of these singular LFSR. This is the first time to construct de Bruijn cycles based on singular linear shift registers. In this method, cycle structures are obtained by modifying the state diagrams firstly, then the conjugate pairs between cycles are searched in the directed paths set. The de Bruijn cycles are realized by using the cycle-joining method finally.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by the National Natural Science Foundation of China under grants 61672414, and the National Cryptography Development Fund under Grant MMJJ20170113.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Stephen Semmes\
Rice University
title: |
Some remarks about Cauchy integrals\
and fractal sets
---
If $\mu$ is a finite Borel measure on the complex plane, then the Cauchy integral $$C(\mu)(z) = \int_{\bf C} \frac{1}{z - \zeta} \, d\mu(\zeta)$$ defines a holomorphic function of $z$ on the complement of the support of $\mu$. For simplicity, let us restrict our attention to measures with compact support in ${\bf C}$, although one can also make sense of Cauchy integrals of measures with noncompact support and infinite mass under suitable conditions. In the classical situation where $\mu$ is supported on a nice curve, the Cauchy integral of $\mu$ has a jump discontinuity across the curve. For more regular measures, the Cauchy integral converges absolutely for every $z \in {\bf C}$ and defines a continuous function on the plane. Of course, this function is not holomorphic in any neighborhood of the support of $\mu$. For any finite measure $\mu$ on ${\bf C}$, the Cauchy integral $C(\mu)(z)$ makes sense as a locally integrable function on ${\bf C}$, whose $\overline{\partial}$ derivative is a constant multiple of $\mu$ in the sense of distributions.
On nice regions in ${\bf C}$, the Cauchy integral formula can be used to recover arbitrary holomorphic functions from their boundary values, under suitable conditions. The Cauchy integral leads to a projection from general functions on the boundary to boundary values of holomorphic functions.
In more fractal situations, one might prefer to think of Cauchy integrals as a way of solving $\overline{\partial}$ problems, to make corrections to get holomorphic functions instead of producing them directly. One might view this as being more like several complex variables. A basic scenario would be to multiply a holomorphic function by a non-holomorphic function with some regularity, and to try to make some relatively small corrections to the product to get a holomorphic function. Perhaps the holomorphic function has nice boundary values and the other function is defined on the boundary, and the correction is intended to yield the boundary values of a holomorphic function. For example, the product of a holomorphic function and a rational function with poles in the interior may not be holomorphic, and this can be corrected with a finite-rank operator to get rid of the singularities.
Bergman spaces and projections could also be considered on the interior. It would still be regularity of the non-holomorphic functions up to the boundary that matters, though.
In ${\bf R}^n$, one can use Cauchy integrals associated to Clifford analysis. However, remember that the product of Clifford holomorphic functions is not necessarily holomorphic, because of noncommutativity of the Clifford algebra,
In any dimension, it is already somewhat interesting to look at Bergman spaces of locally constant functions on regions with infinitely many connected components.
There seem to be a lot of choices involved with fractals. This may be a bit disconcerting, but perhaps it is also just as well.
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C}^n$*]{}, Springer-Verlag, 1980.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'K. W. Murch'
- 'U. Vool'
- 'D. Zhou'
- 'S. J. Weber'
- 'S. M. Girvin'
- 'I. Siddiqi'
bibliography:
- 'supplementaryref.bib'
title: 'Supplementary information for “Cavity-assisted quantum bath engineering”'
---
Sample fabrication and parameters
=================================
The qubit sample was fabricated from aluminum deposited on high resistivity silicon in a two-step, double-angle evaporation process with an intervening oxidation step. The qubit consisted of a single Josephson junction with critical current $I_0 = 46$ nA shunting two paddles that set the charging energy ($E_\mathrm{C}/h = 145$ MHz) of the qubit and provide coupling to the cavity. The coupling rate of the qubit to the cavity, $g = 70$ MHz was set by placing the sample away from the cavity electric field antinode.
The cavity was machined from aluminum (alloy 6061) and had dimensions of $35.6 \times 5.2\times28.1$ mm$^3$. The cavity decay rate, $\kappa/2\pi = 4.3$ MHz, was set by adjusting the length of the center pin of a SMA coaxial connector protruding into the cavity volume. The cavity was also addressed by a weakly coupled port which was used for the qubit manipulation pulses.
The dispersive coupling rate $\chi/2\pi = - 0.55 (.11)$ MHz was determined by measuring both the AC stark shift and measurement induced dephasing rate of a drive tone at the cavity resonance frequency as in our previous work[@vija12feedback]. We use $\chi/2\pi = -0.66$ MHz for comparison to theory and simulation because this value gave the best quantitative agreement.
Experiment setup and state measurement
======================================
Drive signals were combined at room temperature and were sent to the sample via heavily attenuated coaxial lines. The reflected signal from the strongly coupled cavity port was amplified by a near quantum limited lumped-element Josephson parametric amplifier (LJPA) operating in phase sensitive mode. The LJPA was separated from the cavity by 4 cryogenic circulators and isolated from the HEMT amplifier at 2.7 K by three more isolators. The LJPA was biased with a tone at 6.823 GHz and exhibited a gain of 19 dB with a 3 dB (instantaneous) bandwidth of 30 MHz. The phase of the LJPA pump was adjusted to amplify the quadrature of the reflected signal that contained information about the qubit states.
State measurement was performed by pulsing the readout and LJPA pump tones on. After allowing 200 ns for transients to decay, a 200 ns section of data was integrated. After repeating the experiment many times, a histogram of the readout voltage revealed three well-separated distributions corresponding to the qubit ground, excited, and higher excited transmon states. From these measurements we determined that the equilibrium population of the qubit states was 77%, 14%, and 9% for the ground, excited and higher excited states respectively. During cooling, the population of the higher excited states increased to $12\%$, owing to the the increase in the average excited state population. In the experiment we use post-selection to remove the higher excited state population from our tomography measurements, but we note that coupling between the excited states results in a significant reduction of the measurement contrast. We measured a $2 \ \mu$s timescale for equilibration between the excited states of the transmon, which results in a $\sim20\%$ reduction in the state purity determined by tomography.
We note that it is in principle possible to reduce the second excited state population using sideband cooling as demonstrated in Figure 3d simultaneously with cavity cooling to prepare pure states without the need to post select away population in higher qubit states.
In a typical experimental sequence the qubit and cavity drives were suddenly turned on for a 50 $\mu$s duration, followed by state tomography and readout. The experiment was repeated at 6 kHz. Each data point in the tomography plots was the result of $10^5$ experimental sequences.
Derivation of the system Hamiltonian & Master Equation
======================================================
We begin our treatment with the Jaynes-Cummings Hamiltonian and two external drives on the cavity, one with amplitude $\epsilon_\mathrm{d}$ and frequency $\omega_\mathrm{d}$ near the cavity resonance frequency $\omega_\mathrm{c}$, and the other with amplitude $\epsilon_\mathrm{r}$ and frequency $\omega_\mathrm{r}$ near the frequency of the qubit $\omega_\mathrm{q}$: $$H=\omega_\mathrm{c}a^{\dagger}a-\frac{\omega_\mathrm{q}}{2}\sigma_{z}+g(a^{\dagger}\sigma_{-}+a\sigma_{+})+\left[\epsilon_\mathrm{d}e^{-i\omega_\mathrm{d}t}a^{\dagger}
+\epsilon_\mathrm{r}e^{-i\omega_\mathrm{r}t}a^{\dagger}+\mathrm{h.c}\right] ,$$ where $\sigma_{+}$ is the operator exciting the qubit and $\sigma_{-}$ de-exciting it. By including qubit and cavity decay terms we can write the master equation[@blai07proces]: $$\dot{\rho}=-i\left[H,\rho\right]+\kappa\mathcal{D}[a]\rho+\frac{\Gamma_{\varphi}}{2}\mathcal{D}[\sigma_{z}]\rho
+\Gamma_{-}\mathcal{D}[\sigma_{-}]\rho +\Gamma_{+}\mathcal{D}[\sigma_{+}]\rho,$$ where $\mathcal{D}[L]\rho=(2L\rho L^{\dagger}-L^{\dagger}L\rho-\rho
L^{\dagger}L)/2$, $\Gamma_{\varphi}=\frac{1}{T_{\varphi}}$ is the phase relaxation rate and the population relaxation rate is given by $\frac{1}{T_{1}}=\Gamma_{-}+\Gamma_{+}$.
By applying the dispersive shift, $U=e^{\frac{g}{\Delta}(a\sigma_{+}-a^{\dagger}\sigma_{-})}$, where $\Delta=\omega_\mathrm{q}-\omega_\mathrm{c}$, and moving to the rotating frames for both the qubit and cavity, $U_\mathrm{c}=e^{ia^{\dagger}a\omega_\mathrm{d}t},\; U_\mathrm{q}=e^{-i\sigma_{z}\frac{\omega_\mathrm{r}}{2}t}$, we can simplify the Hamiltonian. Taking terms only up to second order in $\frac{g}{\Delta}$ we are left with (assuming $\epsilon_\mathrm{r}$ and $\epsilon_\mathrm{d}$ are real): $$H=-\Delta_\mathrm{c}a^{\dagger}a-\frac{\Delta_\mathrm{q}+\chi}{2}\sigma_{z}-\frac{\Omega_\mathrm{R}}{2}\sigma_{x}-\chi a^{\dagger}a\sigma_{z}+\epsilon_\mathrm{d}(a^{\dagger}+a),$$ where, $\Omega_\mathrm{R}=-\frac{2\epsilon_\mathrm{r}g}{\Delta}$, $\chi=\frac{g^2}{\Delta}$, $\Delta_\mathrm{c}=\omega_\mathrm{d}-\omega_\mathrm{c}$, $\Delta_\mathrm{q}=\omega_\mathrm{q}-\omega_\mathrm{r}$, and we have ignored terms rotating at $\omega_\mathrm{r}-\omega_\mathrm{d}$ or higher frequencies. Note that the second-order correction also includes the well-known Purcell effect[@purc46effect] which adds a term proportional to $\kappa$ to the qubit relaxation rate due to the coherent mixing of qubit and cavity excitations.
We eliminate the drive term by displacing the field operator $a=\bar{a}+d$. Note that this displacement acting on the cavity dissipator term introduces $$\kappa \mathcal{D}[a]\rho = \kappa \mathcal{D}[d]\rho
+ \frac{\kappa}{2} \left[\left( \bar{a}^{\star} d - \bar{a} d^\dagger\right),\rho\right].$$ Because the last term has the form of a commutator, it is equivalent to a shift in the Hamiltonian of $$\Delta H = \frac{i\kappa}{2}\left({\bar a}^{\star}d - {\bar a}d^\dagger\right).$$ A convenient displacement choice will be: $$\bar{a} = \frac{\epsilon_\mathrm{d}}{\Delta_\mathrm{c}+i\kappa/2}.$$ With this choice we eliminate the drive and all of the terms linear in $d$ in $H+\Delta H$ except for terms of the form $d\sigma_{z}$. Our Hamiltonian can now be written as: $$H=-\Delta_\mathrm{c}d^{\dagger}d-\frac{\Delta_\mathrm{q}+\chi(2\bar{n})}{2}\sigma_{z}-\frac{\Omega_\mathrm{R}}{2}\sigma_{x}-\chi(\bar{a}^{\star}d+\bar{a}d^{\dagger}+d^{\dagger}d)\sigma_{z},
\label{eq:Hamil}$$ where $\bar{n}=|\bar{a}|^{2}$, and we have absorbed the Lamb shift into the definition of the qubit frequency \[via the replacement $(2\bar n + 1) \rightarrow (2\bar n)$\]. The Hamiltonian above is the one used in our simulations.
To better understand this Hamiltonian and present a more intuitive understanding of the cooling process we choose the Rabi drive frequency such that $\Delta_\mathrm{q}'=\Delta_\mathrm{q}+\chi(2\bar{n})=0$, meaning the qubit drive is exactly on resonance if there are $\bar{n}$ photons in the cavity. By performing a Hadamard rotation to interchange $\sigma_{z}$ and $\sigma_{x}$ and writing the new $\sigma_{x}$ as $\sigma_{+}+\sigma_{-}$, we arrive at an effective Hamiltonian (ignoring rapidly rotating terms as we choose the cavity drive (red) detuning to match the Rabi frequency, $\Delta_\mathrm{c}=-\Omega_\mathrm{R}$), $$H=-\Delta_\mathrm{c}d^{\dagger}d-\frac{\Omega_\mathrm{R}}{2}\sigma_{z}-\chi(\bar{a}^{\star}d\sigma_{+}+\bar{a}d^{\dagger}\sigma_{-}). \label{eq:effectiveJC}$$ The new Hamiltonian is an effective Jaynes-Cummings model in which the effective qubit ground state is the lower energy eigenstate of $\sigma_{z}$ \[$\sigma_{x}$ in the original (rotating) spin frame before the Hadamard transformation\]. With the Hamiltonian in this form, one understands that the cooling scheme is simply the dissipation of the dressed qubit to its effective ground state via photon emission into the cavity, just as in the ordinary (un-driven) Jaynes-Cummings model. \[Here however the photon emission is actually Raman scattering of the cavity pump photons.\] Using the fact that the peak value of the resonator density of states is $\rho=\frac{2}{\pi\kappa}$ Fermi’s Golden Rule yields for this particular resonant case, the following simple expression for the cooling rate $$\Gamma = 2\pi |\chi\bar{a}|^2 \rho = \frac{4\chi^2}{\kappa}\bar{n},$$ which is valid in the limit of weak coupling ($|\chi|\sqrt{\bar n}\ll\kappa)$.
General Heating and cooling rates
=================================
Before extending the above results to the more general case of arbitrary detunings for the cavity and qubit drives, it is convenient to reformulate the above derivation. Without making the Hadamard transformation and the cavity displacement transformation mentioned above, the Hamiltonian for the qubit alone can be written in the frame rotating at the Stark-shifted qubit frequency as: $$\begin{aligned}
H_\mathrm{q} = -\frac{\Omega_\mathrm{R}}{2}\sigma_x - \frac{2\chi [a^\dagger a-\bar n]}{2} \sigma_z, $$ in which the dispersive coupling term can be thought of as adding photon shot noise that causes dephasing of the qubit [@cler10noise; @blai04archit]. Following the derivation of Fermi’s Golden Rule in terms of noise spectral densities presented in[@cler10noise], the qubit excitation and de-excitation rates can be calculated from the spectral density of the noise perturbing the qubit in directions orthogonal to the $x$ quantization direction. Spectral density at negative (positive) frequencies correspond[@cler10noise] to energy emitted (absorbed) by the bath: $$\begin{aligned}
\Gamma_\pm &= \frac{1}{4}\left\{\tilde S_{zz}(\mp\Omega_\mathrm{R}) +\tilde S_{yy}(\mp\Omega_\mathrm{R})\right\}, \\
&= \frac{1}{4}\left\{4\chi^2{S}_{nn}(\mp\Omega_\mathrm{R}) +
\tilde{S}^{'}_{zz}(\mp\Omega_\mathrm{R}) + \tilde S_{yy}(\mp\Omega_\mathrm{R})\right\},\end{aligned}$$ where $\tilde S_{zz}$ is the noise power spectral density related to the autocorrelation function of the noisy coefficient of $\sigma_z$ in the Hamiltonian. It can be broken into two parts: one due to the photon number fluctuations, i.e., $S_{nn}$ and the other one due to all other processes $\tilde{S}^{'}_{zz}$. $\Gamma_-$ refers to the transition rate from the high-energy eigenstate of $\sigma_x$ with $-1$ eigenvalue to the low-energy state with $+1$ eigenvalue. This cooling transition is the dominant one for our assumption of a red-detuned cavity drive, i.e., $\Delta_\mathrm{c} = -\Omega_\mathrm{R}$.
Being cognizant[@ithi05decoher] that $S_{yy}$ is evaluated in the rotating frame and with further approximations that: $$\begin{aligned}
&\tilde{S}_{yy}(\mp\Omega_\mathrm{R}) \simeq \tilde S_{yy}(0)= \Gamma_1 ,\\ &\tilde{S}^{'}_{zz}(\mp\Omega_\mathrm{R}) \simeq \tilde{S}^{'}_{zz}(0)=2\Gamma_\varphi ,\\ &S_{nn}[\omega] = \frac{{\bar n}\cdot\kappa}{(\kappa/2)^2+(\omega+\Delta_\mathrm{c})^2} ,\end{aligned}$$ where $\Gamma_\varphi$ is the pure-dephasing rate due to other sources, and we take $S_{nn}$ to be the result for an uncoupled driven cavity. Note that noise along the $z$ direction is not affected by the transformation to the rotating frame. Using these terms we obtain[@ithi05decoher]: $$\begin{aligned}
\Gamma_- = & \frac{4\chi^2}{\kappa}{\bar n} + \frac{\Gamma_\varphi}{2} +
\frac{\Gamma_1}{4}, \\
\Gamma_+ = & \frac{\kappa\chi^2}{(\kappa/2)^2+4\Omega_\mathrm{R}^2}{\bar n} +
\frac{\Gamma_\varphi}{2} + \frac{\Gamma_1}{4} .\end{aligned}$$ For the cooling to be effective, we need: $$\begin{aligned}
\frac{4\chi^2}{\kappa}{\bar n} \gg \frac{\Gamma_\varphi}{2} +
\frac{\Gamma_1}{4} . \label{eq:rabi_cooling}\end{aligned}$$ i.e., the shot-noise term in $\Gamma_-$ should dominate so that the asymmetry in $S_{nn}(\pm\Omega_\mathrm{R})$ strongly affects the transition rates. If the qubit is driven off-resonance by a detuning $\Delta_\mathrm{q}'$ from its (Stark-shifted) frequency, the Hamiltonian takes the form: $$\begin{aligned}
H_\mathrm{q} = -\frac{\Delta_\mathrm{q}'}{2}\sigma_z- \frac{\Omega_\mathrm{R}}{2}\sigma_x
- \frac{2\chi [a^\dagger a-{\bar{n}}]}{2} \sigma_z .\end{aligned}$$ If the cavity detuning is set to the new Rabi frequency $\Delta_\mathrm{c} = \pm\sqrt{\Omega_\mathrm{R}^2+\Delta_\mathrm{q}'^2}$, it is possible to cool the qubit to an arbitrary position on the Bloch sphere using the same mechanism. The cooling and heating will occur between the two states ${|\pm\rangle}$, which are the eigenstates of $(\Delta_\mathrm{q}'\sigma_z + \Omega_\mathrm{R}\sigma_x)/\sqrt{\Omega_\mathrm{R}^2+\Delta_\mathrm{q}'^2}$ with eigenvalue $\pm1$.\
The transition rates can be calculated in a manner identical to the on-resonance case and are found to be: $$\begin{aligned}
\Gamma_\pm = &\frac{1}{4}\left\{
\tilde{S}_{zz}\left(\mp\sqrt{\Omega_\mathrm{R}^2+\Delta_\mathrm{q}'^2}\right)\sin^2\theta
+\tilde S_{xx}\left(\mp\sqrt{\Omega_\mathrm{R}^2+\Delta_\mathrm{q}'^2}\right)\cos^2\theta
+ \tilde S_{yy}\left(\mp\sqrt{\Omega_\mathrm{R}^2+\Delta_\mathrm{q}'^2}\right)
\right\}, \\
\simeq& \chi^2S_{nn}(\mp\Omega_\mathrm{R}/\sin\theta)\sin^2\theta
+ \frac{1}{2}\Gamma_\varphi\sin^2\theta
+ \frac{1}{4}\Gamma_1\left(1+\cos^2\theta\right), \label{eq:g_pm}\end{aligned}$$ where we define $\tan\theta = \Omega_\mathrm{R}/\Delta_\mathrm{q}'$.
With weak qubit drive at large detuning $\Delta_\mathrm{q}'\gg\Omega_\mathrm{R}$, the qubit is no longer dressed and cooling is along the $z$ direction. In this limit, the physics devolves into the ordinary sideband cooling process in which both the qubit and cavity drives are detuned by $\sim\Delta_\mathrm{q}'$. The rates are given by: $$\begin{aligned}
\Gamma_- &\simeq \frac{4\chi^2{\bar n}}{\kappa}
\left(\frac{\Omega_\mathrm{R}}{\Delta_\mathrm{q}'}\right)^2
+ \frac{\Gamma_\varphi}{2}\left(\frac{\Omega_\mathrm{R}}{\Delta_\mathrm{q}'}\right)^2
+ \frac{\Gamma_1}{2}, \label{eq:g+}\\
\Gamma_+ &\simeq\frac{\kappa\chi^2{\bar n}}{(\kappa/2)^2+4\Delta_\mathrm{q}'^2}
\left(\frac{\Omega_\mathrm{R}}{\Delta_\mathrm{q}'}\right)^2
+ \frac{\Gamma_\varphi}{2}\left(\frac{\Omega_\mathrm{R}}{\Delta_\mathrm{q}'}\right)^2
+ \frac{\Gamma_1}{2} \label{eq:g-}.\end{aligned}$$ Note that the cooling is less effective than the resonant case, as we have an additional factor of $\left(\frac{\Omega_\mathrm{R}}{\Delta_\mathrm{q}'}\right)^2$ which suppresses the rate.
Raman transition rates
======================
To complete the picture, we provide a derivation of the transition rates for Raman sideband cooling to show that they indeed coincide with the Rabi cooling for $\Delta_\mathrm{q}'\gg\Omega_\mathrm{R}$. The Hamiltonian can be written as: $$\begin{aligned}
H =& H_0 + V, \\
H_0 =& -\frac{\Delta_\mathrm{q}'}{2}\sigma_z - \Delta_\mathrm{c} d^\dagger d
- \frac{\Omega_\mathrm{R}}{2} \sigma_x, \\
V =& -\chi \left({\bar a}d^\dagger+ {\bar a}^{\star}d + d^\dagger d\right)\sigma_z.\end{aligned}$$ Here we have displaced the cavity field $a = {\bar a} + d$, with ${\bar a} = \epsilon_\mathrm{d}/(\Delta_\mathrm{c}+i\kappa/2)$. The cooling/heating processes rely on the fact that the dispersive shift term couples the qubit and cavity. We thus treat the dispersive term as a perturbation and use Fermi’s golden rule to calculate the transition rates.
The qubit part of $H_0$ can be further diagonalized and we get: $$\begin{aligned}
{{|\tilde e\rangle}} \simeq {|e\rangle} + \frac{\Omega_\mathrm{R}}{2\Delta_\mathrm{q}' } {|g\rangle}, \\
{{|\tilde g\rangle}} \simeq {|g\rangle} - \frac{\Omega_\mathrm{R}}{2\Delta_\mathrm{q}' } {|e\rangle},\end{aligned}$$ where we have used the fact that $\Omega_\mathrm{R}\ll\Delta_\mathrm{q}$. Note that, $$\begin{aligned}
\left|{\langle \tilde{e},1|}V{|\tilde{g},0\rangle}\right|^2 =
\left|{\langle \tilde{e},0|}V{|\tilde{g},1\rangle}\right|^2 = \chi^2{\bar n}
\left(\frac{\Omega_\mathrm{R}}{\Delta_\mathrm{q}' }\right)^2 .\end{aligned}$$ We also make use of the effective cavity density of states $\rho(\omega)=-\frac{1}{\pi}\operatorname{Im}\frac{1}{\omega-\Delta_\mathrm{q}'+i\frac{\kappa}{2}}$. Applying Fermi’s golden rule, we get the same cooling/heating rates as in the noise spectral density calculation, i.e., $$\begin{aligned}
\Gamma_- = & \frac{4\chi^2}{\kappa}{\bar n}
\left(\frac{\Omega_\mathrm{R}}{\Delta_\mathrm{q}'}\right)^2, \\
\Gamma_+ = & \frac{\kappa\chi^2}{(\kappa/2)^2+4\Delta_\mathrm{q}'^2}{\bar n}
\left(\frac{\Omega_\mathrm{R}}{\Delta_\mathrm{q}'}\right)^2.\end{aligned}$$
Simulations of the master equation
==================================
A set of numerical simulations were performed to better understand our experimental results. All the simulations involved numerically solving the Master equation presented in Section III and especially the Hamiltonian in Eq. (\[eq:Hamil\]). The simulations were written in python and make significant use of the QuTip toolbox[@joha12qutip]. Figure 1 shows a comparison between the tomography results produced by the simulation and the experiment for $\langle \sigma_x\rangle$ in Figure 1a,b and $\langle \sigma_z\rangle$ in Figure 1c,d. The experimental results are seen to be consistent with the theoretical predictions of our model.
Strong coupling regime
======================
The theoretical calculations in the sections above are all based on the weak coupling assumption $\sqrt{{\bar{n}}}|\chi|\ll\kappa$, namely that the coupling constant in the effective Hamiltonian \[e.g. in Eq. (\[eq:effectiveJC\])\] is much smaller than the decay rate. When this assumption starts to fail the calculated decay rates are no longer a good model for the behavior of the system, which instead of exponential decay to the effective ground state shows oscillations[@brun96vacuum] in the time domain. Intuitively this can be described as a coherent oscillation of the system (following a sudden switch of the coupling) between two states. The first state has the qubit in the high-energy eigenstate of $\sigma_{x}$ and no Raman photon in the cavity. The second has the qubit in the low-energy dressed state and one Raman photon in the cavity. The coupling is strong enough that the system coherently oscillates back and forth between these states multiple times before the cavity decay eventually brings the system to equilibrium. This regime can be reached experimentally with currently available parameters (strong drives and a low $\kappa$ cavity). Figure 2 shows a simulation of the system in the strong coupling regime in which this effect can be observed. One sees excellent agreement of the oscillation rate with the predicted effective ‘vacuum Rabi frequency’ $2|\chi|\sqrt{{\bar{n}}}$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In [@ST1] the authors introduced a parabolic flow for pluriclosed metrics, referred to as pluriclosed flow. We also demonstrated in [@ST2] that this flow, after certain gauge transformations, gives a class of solutions to the renormalization group flow of the nonlinear sigma model with $B$-field. Using these transformations, we show that our pluriclosed flow preserves generalized Kähler structures in a natural way. Equivalently, when coupled with a nontrivial evolution equation for the two complex structures, the $B$-field renormalization group flow also preserves generalized Kähler structure. We emphasize that it is crucial to evolve the complex structures in the right way to establish this fact.'
address:
- |
Rowland Hall\
University of California, Irvine\
Irvine, CA 92617
- 'Beijing University, China and Princeton University, Princeton, NJ 08544'
author:
- Jeffrey Streets
- Gang Tian
title: Generalized Kähler geometry and the pluriclosed flow
---
Introduction
============
The purpose of this note is to show that the pluriclosed flow introduced in [@ST1] preserves generalized Kähler geometry. This introductory section introduces the main results in a primarily mathematical context, while a physical discussion of the results appears in section \[phys\]. First we recall the concept of a generalized Kähler manifold.
A *generalized Kähler manifold* is a Riemannian manifold $(M^{2n},
g)$ together with two complex structures $J_+, J_-$, each compatible with $g$, further satisfying $$\begin{gathered}
\label{GK}
\begin{split}
d_+^c \omega_+ = - d_-^c \omega_- =&\ H,\\
d H =&\ 0.
\end{split}\end{gathered}$$
This concept first arose in the work of Gates, Hull, and Roček [@GHR], in their study of $N = (2,2)$ supersymmetric sigma models. Later these structures were put into the rich context of Hitchin’s generalized geometric structures [@Hitchin] in the thesis of Gualtieri [@Gualtieri] (see also [@Gualt2]).
Recall that a Hermitian manifold $(M^{2n}, \omega, J)$ is *pluriclosed* if the Kähler form $\omega$ satisfies $i {\partial}{{\overline{\partial}}}\omega = d d^c \omega = 0$. Note that a generalized Kähler manifold $(M,g,J_+,J_-)$ consists of a pair of pluriclosed structures $(M,\omega_+, J_+)$ and $(M, \omega_-, J_-)$ whose associated metrics are equal and furthermore satisfy the first equation of (\[GK\]), where $\omega_{\pm} (\cdot,\cdot) = g
(J_{\pm}\cdot,\cdot)$. The pluriclosed flow is the time evolution equation $$\begin{aligned}
\label{PCF}
{\frac{\partial}{\partial t}}\omega =&\ {\partial}{\partial}^*_{\omega} \omega + {{\overline{\partial}}}{{\overline{\partial}}}^*_{\omega} \omega +
\frac{\sqrt{-1}}{2} {\partial}{{\overline{\partial}}}\log \det g.\end{aligned}$$ It follows from Theorem 1.2 in [@ST1] that with $\omega_{\pm}$ as initial metrics, we get solutions $\omega_{\pm}(t)$ of on $M\times [0,T_{\pm})$. Let $g_{\pm}(t)$ be the Hermitian metric on $M$ whose Kähler form is $ \omega_{\pm}(t)$. As in Theorem 6.5 of [@ST2], we set $$\label{eq:cfield1}
X_{\pm} \,=\, \left (- J_{\pm} d^*_{g_{\pm}(t)} \omega_{\pm}
(t)\right)^{\sharp_{\pm}},$$ where $\sharp_{\pm}$ denotes the natural isomorphism from $T^* M$ onto $TM$ defined using $g_{\pm}(t)$. Further, let $\phi_{\pm}(t)$ denote the one-parameter family of diffeomorphisms generated by $X_{\pm}$, with $\phi_{\pm}(0) = \operatorname{Id}$. Then our main theorem can be stated as follows:
\[PCF2BEFD\] Let $(M, g, J_+, J_-)$ be a generalized Kähler manifold. With notations as above, one has that $\phi_+(t)^*(g_+(t)) \,=\, \phi_-(t)^*(g_-(t))$, and we denote this metric $g(t)$. Furthermore, $T_+ = T_- =: T$, and $(M, g(t), \phi_+(t)^*J_+,
\phi_-(t)^*J_-)$ is a family of generalized Kähler manifolds on $[0, T)$ with initial value $(M,g,J_+,J_-)$.
Hence, preserves generalized Kähler structures. In [@ST2], we found a striking relationship between solutions to and the B-field renormalization group flow, and the proof of Theorem \[PCF2BEFD\] makes essential use of this. The B-field renormalization group flow arises from physical considerations. Consider a pair $(g, H)$ of a Riemannian metric $g$ and closed three-form $H$ on a manifold $M$. The form $H$ is thought of as the field strength of a locally defined $2$-form $B$ (i.e. $H = dB$). Given this data, and a dilaton $\Phi$ on $M$, one can associate a Lagrangian of maps of Riemann surfaces $f: (\Sigma, h) \to M$, called the worldsheet nonlinear sigma model action, given by $$\begin{gathered}
\label{action}
S = - \frac{1}{2} \int_{\Sigma} \left[ {\left| {\nabla}f \right|}^2 +
\frac{{\epsilon}^{{\alpha}{\beta}}}{\sqrt{h}} B_{ij} {\partial}_{{\alpha}} f^i {\partial}_{{\beta}} f^j - 2 \Phi
R(h) \right] dV_h.\end{gathered}$$ We have suppressed a scaling parameter ${\alpha}'$ which is often included in this definition, see ([@Polchinski] p. 111) for more detail on this action and what follows. Imposing cutoff independence of the associated quantum theory leads to first order renormalization group flow equations $$\begin{gathered}
\begin{split} \label{flow}
{\frac{\partial}{\partial t}}g_{ij} =&\ - 2 \operatorname{Rc}_{ij} + \frac{1}{2} H_{ipq} H_j^{\ pq}\\
{\frac{\partial}{\partial t}}H =&\ {\Delta}_{d} H,
\end{split}\end{gathered}$$ where ${\Delta}_{d} = - \left( d d^* + d^* d \right)$ is the Laplace-Beltrami operator. In general there is a dilaton evolution as well, but this decouples from the above system after applying a diffeomorphism gauge transformation ([@Woolgar] pg. 6), and so is not directly relevant to the discussion here.
In view of results in [@ST2], one can ask: [*Does (\[flow\]) preserve generalized Kähler geometry*]{}? As it turns out, in the naive sense in which this question is usually asked, the answer is no. Specifically, given $(M^{2n}, g, J_{\pm})$ a generalized Kähler manifold, one may ask whether the solution to (\[flow\]) with initial condition $(g, d^c_+ \omega_+)$ remains generalized Kähler in the sense that $g$ remains compatible with $J_{\pm}$ and the equations (\[GK\]) hold. This is false in general. One has to evolve the complex structures appropriately so that they are compatible with $(M,g(t))$ and consequently, give rise to generalized Kähler structures. The corresponding evolution equation is not obvious at all, and would be quite difficult to guess directly from . The key insight comes from the pluriclosed flow and its relation to established in [@ST2].
The next theorem is a reformulation of Theorem \[PCF2BEFD\] in terms of the B-field flow.
\[mainthm\] Let $(M^{2n}, g, J_+, J_-)$ be a generalized Kähler structure. The solution to (\[flow\]) with initial condition $(g,
d^c_+ \omega_+)$ remains a generalized Kähler structure in the following sense: There exists a parabolic flow of complex structures such that if $J_{\pm}(t)$ are its solutions with initial value $J_{\pm}$, then the triple $(g(t), J_+(t), J_-(t))$ satisfies the conditions of (\[GK\]).
In fact, $J_{\pm}(t) = (\phi_t^{\pm})^* J_{\pm}$ for the one-parameter families of diffeomorphisms $\phi_t^{\pm}$ which relate to , as described in Theorem \[PCF2BEFD\]. However, it is unclear yet how to construct $\phi_{\pm}(t)$ from since does not tell how to get $J_{\pm}(t)$. A more precise statement of Theorem \[mainthm\] is given below as Corollary \[maincor\].
We end the paper in section \[class\] with some structural results that must be satisfied for pluriclosed structures which evolve under (\[flow\]) by homotheties, which we call *static* structures. In particular, we exhibit some properties showing that a static structure is automatically Kähler-Einstein, and give a complete classification in the case of non-Kähler complex surfaces.
\[classthm\] Let $(M^4, g, J)$ be a static pluriclosed structure and suppose $b_1(M)$ is odd. Then $(M^4, J)$ is locally isometric to $\mathbb R \times S^3$ with the standard product metric. The universal cover of $(M, J)$ is biholomorphic to $\mathbb C^2 \setminus \{(0, 0) \}$, and $M$ admits a finite sheeted cover ${\widetilde{M}}$ with fundamental group $\mathbb Z$, specifically $$\begin{aligned}
\pi_1({\widetilde{M}}) \cong \mathbb Z = \left< (z_1, z_2) \to ({\alpha}z_1, {\beta}z_2)
\right>\end{aligned}$$ where ${\alpha}, {\beta}\in \mathbb C$, $1 < {\left| {\alpha}\right|} = {\left| {\beta}\right|}$.
**Acknowledgements:** The authors would like to thank Sergey Cherkis for his comments, and the referee for a careful reading and some helpful suggestions.
Physical Interpretation {#phys}
=======================
The first order RG flow equations (\[flow\]) are derived by imposing cutoff independence for the quantum field theory associated to a nonlinear sigma model. For the pure gravity model these equations were first derived by Friedan [@Friedan], yielding the Ricci flow for the order ${\alpha}'$ approximation, while for the model including a skew-symmetric background field, these equations were derived in [@Friedan2] (see also [@Friedan3]). Recently, due partly to the mathematical breakthroughs of Perelman [@P1], this flow has garnered more interest in the mathematics and physics communities. In particular, in [@Woolgar] the authors generalized Perelman’s $\mathcal F$ functional to show that (\[flow\]) is in fact the gradient flow of the lowest eigenvalue of a certain Schrödinger operator. This property is suggested by Zamolodchikov’s $c$-theorem [@Zal], which implies the irreversibility of some RG flows. Furthermore, the first author showed in [@Str] that a certain generalization of Perelman’s entropy functional is monotone for (\[flow\]).
In this paper we address a different issue related to the RG flow. Recall that, as exhibited in [@GHR], when imposing $N = (2,2)$ supersymmetry, the equations (\[GK\]) are induced on the target space of a $2$-dimensional nonlinear sigma model, whose underlying sigma model action (\[action\]). A very natural question in this context is whether one can expect the supersymmetry equations to be preserved along the solution to the RG flow, when away from a fixed point. Our results show that the system of equations (\[flow\]), will *not* in general preserve the $N = (2,2)$ supersymmetry equations (\[GK\]). However, if one adds an evolution equation for the complex structures $J_{\pm}$, specified in (\[GKflow\]), then the renormalized coupling constants $(g(t), H(t),
J_{\pm}(t))$, *will* define a supersymmetric model, for all cutoff scales. In fact it is clear from our proofs that the entire discussion is true for the weaker $N = 2$ supersymmetry equations.
Our derivation of the evolution equation for $J_{\pm}$ comes from recognizing special diffeomorphism gauges relating solutions to (\[flow\]) to solutions of (\[PCF\]), where half of the supersymmetry equations (\[GK\]) are clearly preserved with respect to a fixed complex structure. Thus the evolution equations for $J_{\pm}$ come from the action of the gauge group, hence it is unlikely one could modify the sigma model action (\[action\]) to derive these equations. This makes equation for $J_{\pm}$ in (\[GKflow\]) all the more surprising and mysterious. Understanding the physical meaning of the evolution for $J_{\pm}$ therefore remains an interesting open problem. We also remark that our results only apply to the order ${\alpha}'$ approximation of the renormalization group flow. It remains an interesting open problem to ask whether higher order approximations, or even the full RG flow, preserve $N = (2,2)$ supersymmetry in the sense we have described here.
Finally, Theorem \[classthm\] can be thought of as a “No-Go” theorem for certain string vacua. In particular, we have given a complete classification of supersymmetric solutions to (\[flow\]) which evolve purely by homothety on non-Kähler surfaces. In the end only a restricted class of Hopf surfaces can possibly admit solutions to these equations. Other structural results on these vacua in arbitrary dimension appear in section \[class\]. An interesting further problem is to classify solutions to the RG flow which evolve entirely by the action of the diffeomorphism group.
Proof of main theorems {#mainsec}
======================
Consider the Hermitian manifold $(M^{2n}, g, J_+)$. By (\[GK\]), this is a pluriclosed structure, i.e. $$\begin{aligned}
d d^c_+ \omega_+ = 0.\end{aligned}$$ By ([@ST1] Theorem 1.2), there exists a solution to (\[PCF\]) with initial condition $\omega_+$ on $[0, T)$ for some maximal $T \leq \infty$. Call this one-parameter family of Kähler forms $\omega_+(t)$, and define $\omega_-(t)$ analogously as the solution to (\[PCF\]) on the complex manifold $(M, J_-)$ with initial condition $\omega_-$. Next consider the time-dependent vector fields $$\begin{aligned}
X^{\pm} = \left(- J_{\pm} d^*_{g_{\pm}} \omega_{\pm} \right)^{\sharp_{\pm}},\end{aligned}$$ and let $\phi_{\pm}(t)$ denote the one-parameter family of diffeomorphisms of $M$ generated by $X^{\pm}$, with $\phi^{\pm}_0 = \operatorname{Id}$. Theorem 1.2 in [@ST2] implies that $(\phi_+(t)^*g_+(t), \phi_+(t)^*(d^c_+
\omega_+(t)))$ is a solution to (\[flow\]) with initial condition $(g, d^c_+
\omega_+)$. Likewise, we have a solution $(\phi_-(t)^*g_-(t), \phi_-(t)^*(d^c_-
\omega_-(t)))$ to (\[flow\]) with initial condition $(g, d^c_- \omega_-)$. However, if we let $({\widetilde{g}}(t), {\widetilde{H}}(t))$ denote this latter solution, we observe that $$\begin{gathered}
\begin{split}
{\frac{\partial}{\partial t}}{\widetilde{g}}_{ij} =&\ - 2 {\widetilde{\operatorname{Rc}}}_{ij} + \frac{1}{2} {\widetilde{H}}_{ipq}
{\widetilde{H}}_j^{\ pq} = - 2 {\widetilde{\operatorname{Rc}}}_{ij} + \frac{1}{2} \left( - {\widetilde{H}}_{ipq}
\right)
\left(- {\widetilde{H}}_j^{\ pq} \right)\\
{\frac{\partial}{\partial t}}\left( - {\widetilde{H}} \right) =&\ {\Delta}_{d} \left( - {\widetilde{H}} \right),
\end{split}\end{gathered}$$ i.e. $({\widetilde{g}}(t), - {\widetilde{H}}(t))$ is a solution to (\[flow\]) with initial condition $(g, - d^c_- \omega_-)$. By (\[GK\]), we see that $(\phi_+(t)^*g_+(t), \phi_+(t)^*(d^c_+
\omega_+(t)))$ and $(\phi_-(t)^*g_-(t), -\phi_-(t)^*(d^c_-
\omega_-(t)))$ are two solutions of (\[flow\]) with the same initial condition. Using the uniqueness of solutions of (\[flow\]) ([@Str] Proposition 3.3), we conclude that these two solutions coincide, and call the resulting one-parameter family $(g(t), H(t))$.
Next we want to identify the two complex structures with which $g$ remains compatible. We observe by that for arbitrary vector fields $X$, $Y$, $$\begin{gathered}
\begin{split}
g\left( \phi_{\pm}(t)^* J_{\pm} X, \phi_{\pm}(t)^* J Y \right) =&\ g\left(
\phi_{\pm}(t)^{-1}_*\cdot J_{\pm} \cdot\phi_{\pm} (t)_* X, \phi_{\pm}(t)^{-1}
_* \cdot J_{\pm} \cdot \phi_{\pm} (t)_* Y \right)\\
=&\ \left[\phi_{\pm}(t)^{-1,*} g \right] \left(J_{\pm} \cdot
\phi_{\pm}(t)_* X, J_{\pm} \cdot \phi_{\pm}(t)_* Y \right)\\
=&\ g_{\pm} \left(J_{\pm} \cdot \phi_{\pm} (t)_*
X, J_{\pm} \cdot \phi_{\pm} (t)_* Y \right)\\
=&\ g_{\pm} \left(\phi_{\pm} (t)_*
X, \phi_{\pm} (t)_* Y \right)\\
=&\ \left[\phi_{\pm}(t)^* g_{\pm} \right] (X, Y)\\
=&\ g(X, Y).
\end{split}\end{gathered}$$ Therefore $g(t)$ is compatible with $\phi_{\pm}(t)^* J_{\pm}(t)$. Denote these two time dependent complex structures by ${\widetilde{J}}_{\pm}$. It follows that ${\widetilde{\omega_{\pm}}} = \phi_{\pm} (t)^* \omega_{\pm}$. Next we note by naturality of $d$ that $$\begin{gathered}
\begin{split}
{\widetilde{d^c_{\pm}}} {\widetilde{\omega_{\pm}}} (X, Y, Z) =&\ - \left[d
{\widetilde{\omega_{\pm}}}\right] \left({\widetilde{J}}_{\pm} X, {\widetilde{J}}_{\pm} Y,
{\widetilde{J}}_{\pm}
Z \right)\\
=&\ - \left[ d \phi_{\pm}(t)^* \omega_{\pm} \right]
\left(\phi_{\pm}(t)^{-1}_*\cdot J_{\pm} \cdot \phi_{\pm} (t)_* X,
\cdots
\right) \\
=&\ \left[ \phi_{\pm} (t)^* \left( - d \omega_{\pm} \right)
\right]
\left( \phi_{\pm}(t)^{-1}_* \cdot J_{\pm} \cdot \phi_{\pm} (t)_*X, \cdots
\right)\\
=&\ - d \omega_{\pm} \left( J_{\pm} \cdot \phi_{\pm} (t)_* X, \cdots
\right)\\
=&\ d^c_{\pm} \omega_{\pm} \left( \phi_{\pm}(t) X, \cdots
\right)\\
=&\ \phi_{\pm} (t)^* \left( d^c_{\pm} \omega_{\pm} \right)(X, Y,
Z)\\
=&\ \pm H(X, Y, Z).
\end{split}\end{gathered}$$ It follows that $$\begin{aligned}
{\widetilde{d^c_{+}}} {\widetilde{\omega_{+}}} =&\ - {\widetilde{d^c_{-}}} {\widetilde{\omega_-}} = H, \qquad d
H = 0,\end{aligned}$$ showing that the triple $(g(t), {\widetilde{J}}_+(t), {\widetilde{J}}_-(t))$ is generalized Kähler for all time. This finishes the proof of Theorem \[PCF2BEFD\].
To prove Theorem \[mainthm\], we need to find the evolution equation for $J_{\pm}(t)$. Note that our curvature convention is that $({\nabla}^2_{e_i, e_j} - {\nabla}^2_{e_j, e_i} )e_k = R_{ijk}^{\hskip 0.18in l} e_l$.
\[Jev\] Let $(M^{2n}, {\widetilde{g}}(t), J)$ be a solution to the pluriclosed flow. Let $\phi_t$ be the one parameter family of diffeomorphisms generated by $\left(- J d^*_{{\widetilde{g}}} {\widetilde{\omega}} \right)^{\sharp}$ with $\phi_0 = \operatorname{Id}$, and let $g(t) = \phi_t^*({\widetilde{g}}(t)), J(t) = \phi_t^*(J)$. Then $$\begin{gathered}
\label{Jflow}
\begin{split}
{\frac{\partial}{\partial t}}J_k^l =&\ \left({\Delta}J\right)_k^l - [J, g^{-1} \operatorname{Rc}]_k^l\\
&\ - J_k^p D^s J_i^l D_p J_s^i - J_i^l D^s J_k^p D_p J_s^i + J_s^p D^s J_i^l
D_p J_k^i + J_i^l D^s J_s^p D_p J_k^i\\
&\ - J_p^l D_k J_t^p D^s J_s^t + J_k^p D_p J_t^l D^s J_s^t - J_t^p D^s J_s^t
D_p J_k^l.
\end{split}\end{gathered}$$
It suffices to compute the time derivative of $J$ at $t = 0$. First we note $$\begin{aligned}
\left({\frac{\partial}{\partial t}}J(t) \right)_{|t = 0} =&\ {\frac{\partial}{\partial t}}\left( \phi_t^* J \right)_{|t = 0} =
\mathcal L_{X(0)} J.\end{aligned}$$ One may compute $$\begin{aligned}
( \mathcal L_X J)(Y) = [X, JY] - J[X, Y].\end{aligned}$$ In coordinates this reads $$\begin{aligned}
( \mathcal L_X J)_k^l = J_p^l {\partial}_k X^p - J_k^p {\partial}_p X^l + X^p {\partial}_p
J_k^l.\end{aligned}$$ Furthermore, since the Levi-Civita connection $D$ is torsion-free we have $$\begin{aligned}
( \mathcal L_X J)_k^l = J_p^l D_k X^p - J_k^p D_p X^l + X^p D_p J_k^l.\end{aligned}$$ We next observe a formula for the vector field $X$. $$\begin{aligned}
X^p =&\ - g^{pq} J_q^r (d^* \omega)_r = - J_t^p D^s J_s^t.\end{aligned}$$ Thus $$\begin{gathered}
\label{Jev10}
\begin{split}
\left(\mathcal L_X J \right)_k^l=&\ - J_p^l D_k \left( J_t^p D^s J_s^t
\right)+ J_k^p D_p \left( J_t^l D^s J_s^t \right) - J_t^p D^s J_s^t D_p
J_k^l\\
=&\ D_k D^s J_s^l - J_p^l D_k J_t^p D^s J_s^t + J_k^p J_t^l D_p D^s J_s^t +
J_k^p D_p J_t^l D^s J_s^t\\
&\ - J_t^p D^s J_s^t D_p J_k^l\\
=&\ D^s D_k J_s^l + g^{uv} \left(R_{u k v}^{\hskip 0.18in p} J_p^l - R_{u k
p}^{\hskip 0.18in l} J_v^p
\right)\\
&\ + J_k^p J_t^l D^s D_p J_s^t + J_k^p J_t^l g^{uv} \left( R_{u p v}^{\hskip
0.18in q} J_q^t
-R_{u p q}^{\hskip 0.18in t} J_v^q \right)\\
&\ - J_p^l D_k J_t^p D^s J_s^t + J_k^p D_p J_t^l D^s J_s^t - J_t^p D^s J_s^t
D_p J_k^l.
\end{split}\end{gathered}$$ We will simplify this expression using the vanishing of the Nijenhuis tensor of $J$. Recall $$\begin{aligned}
N(X, Y) =&\ [JX, JY] - [X, Y] - J[JX, Y] - J[X, JY].\end{aligned}$$ In coordinates we may express $$\begin{aligned}
N_{jk}^i = J_j^p {\partial}_p J_k^i - J_k^p {\partial}_p J_j^i - J_p^i {\partial}_j J_k^p +
J_p^i {\partial}_k J_j^p.\end{aligned}$$ Again since $D$ is torsion-free we may express $$\begin{aligned}
N_{jk}^i = J_j^p D_p J_k^i - J_k^p D_p J_j^i - J_p^i D_j J_k^p + J_p^i D_k
J_j^p.\end{aligned}$$ Thus since $J$ is integrable we may conclude $$\begin{gathered}
\begin{split}
0 =&\ D^k \left( J_i^l N_{jk}^i \right)\\
=&\ D^k \left[J_i^l \left( J_j^p D_p J_k^i - J_k^p D_p J_j^i - J_p^i D_j
J_k^p+ J_p^i D_k J_j^p \right) \right]\\
=&\ D^k D_j J_k^l - D^k D_k J_j^l + D^k \left[J_i^l J_j^p D_p J_k^i - J_i^l
J_k^p D_p J_j^i \right].
\end{split}\end{gathered}$$ Plugging this into the first term of (\[Jev10\]) we conclude $$\begin{gathered}
\label{Jev20}
\begin{split}
\left( \mathcal L_X J \right)_k^l =&\ D^s D_s J_k^l - D^s \left[ J_i^l J_k^p
D_p J_s^i - J_i^l J_s^p D_p J_k^i \right] + J_k^p J_t^l D^s D_p J_s^t\\
&\ + g^{uv} \left(R_{u k v}^{\hskip 0.18in p} J_p^l - R_{u k p}^{\hskip 0.18in
l} J_v^p - J_k^p R_{u p v}^{\hskip 0.18in l} -
J_k^p J_t^l J_v^q R_{u p q}^{\hskip 0.18in t} \right)\\
&\ - J_p^l D_k J_t^p D^s J_s^t + J_k^p D_p J_t^l D^s J_s^t - J_t^p D^s J_s^t
D_p J_k^l\\
=&\ \left({\Delta}J\right)_k^l + J_i^l J_s^p D^s D_p J_k^i\\
&\ + g^{uv} \left(R_{u k v}^{\hskip 0.18in p} J_p^l - R_{u k p}^{\hskip 0.18in
l} J_v^p - J_k^p R_{u p v}^{\hskip 0.18in l} -
J_k^p J_t^l J_v^q R_{u p q}^{\hskip 0.18in t} \right)\\
&\ - J_k^p D^s J_i^l D_p J_s^i - J_i^l D^s J_k^p D_p J_s^i + J_s^p D^s J_i^l
D_p J_k^i + J_i^l D^s J_s^p D_p J_k^i\\
&\ - J_p^l D_k J_t^p D^s J_s^t + J_k^p D_p J_t^l D^s J_s^t - J_t^p D^s J_s^t
D_p J_k^l.
\end{split}\end{gathered}$$ Using the skew-symmetry of $J$ one has $$\begin{gathered}
\label{simp1}
\begin{split}
J_i^l J_s^p D^s D_p J_k^i =&\ \frac{1}{2} J_i^l J_s^p \left( D^s D_p - D_p
D^s\right) J_k^i\\
=&\ \frac{1}{2} J_i^l J_s^p g^{st} \left( R_{p t k}^{\hskip 0.18in m} J_m^i -
R_{p t m}^{\hskip 0.18in i}
J_k^m \right)\\
=&\ - \frac{1}{2} g^{st} \left( R_{p t k}^{\hskip 0.18in l} J_s^p + J_i^l J_s^p
J_k^m R_{p t
m}^{\hskip 0.18in i} \right).
\end{split}\end{gathered}$$ Plugging this into (\[Jev20\]) yields $$\begin{gathered}
\begin{split}
{\frac{\partial}{\partial t}}J_k^l =&\ \left({\Delta}J\right)_k^l - \frac{1}{2} g^{uv} \left( R_{p v
k}^{\hskip 0.18in l}
J_u^p
+ J_i^l J_u^p J_k^m R_{p v m}^{\hskip 0.18in i} \right)\\
&\ + g^{uv} \left(R_{u k v}^{\hskip 0.18in p} J_p^l - R_{u k p}^{\hskip 0.18in
l} J_v^p - J_k^p R_{u p v}^{\hskip 0.18in l} -
J_k^p J_t^l J_v^q R_{u p q}^{\hskip 0.18in t} \right)\\
&\ - J_k^p D^s J_i^l D_p J_s^i - J_i^l D^s J_k^p D_p J_s^i + J_s^p D^s J_i^l
D_p J_k^i + J_i^l D^s J_s^p D_p J_k^i\\
&\ - J_p^l D_k J_t^p D^s J_s^t + J_k^p D_p J_t^l D^s J_s^t - J_t^p D^s J_s^t
D_p J_k^l.
\end{split}\end{gathered}$$ Next we observe the simplification $$\begin{gathered}
\begin{split}
g^{uv} \left( J_i^l J_u^p J_k^m R_{pvm}^{\hskip 0.18in i} + 2 J_k^p J_t^l J_v^q
R_{upq}^{\hskip 0.18in t} \right) =&\ g^{vu} J_u^p J_i^l J_k^m \left( R_{p v m}^{\hskip 0.18in i}
+ 2 R_{v m p}^{\hskip 0.18in i} \right)\\
=&\ g^{vu} J_u^p J_i^l J_k^m \left( R_{pvm}^{\hskip 0.18in i} +
R_{v m p}^{\hskip 0.18in i} + R_{m p v}^{\hskip 0.18in i} \right)\\
=&\ 0.
\end{split}\end{gathered}$$ Likewise $$\begin{gathered}
\begin{split}
g^{uv} \left( R_{pvk}^{\hskip 0.18in l} J_u^p + 2 R_{ukp}^{\hskip 0.18in l} J_v^p
\right) =&\ g^{uv} J_u^p \left( R_{pvk}^{\hskip 0.18in l} + 2 R_{vkp}^{\hskip 0.18in l} \right)\\
=&\ g^{uv} J_u^p \left( R_{pvk}^{\hskip 0.18in l} + R_{vkp}^{\hskip 0.18in l}
+ R_{k p v}^{\hskip 0.18in l} \right)\\
=&\ 0
\end{split}\end{gathered}$$ Finally we note that $$\begin{aligned}
g^{uv} \left( R_{u k v}^{\hskip 0.18in p} J_p^l - J_k^p R_{u p v}^{\hskip 0.18in l}
\right) =&\ J_k^p \operatorname{Rc}_p^l - \operatorname{Rc}_k^p J_p^l = [J, g^{-1} \operatorname{Rc}]_k^l\end{aligned}$$ Plugging these simplifications into (\[Jev20\]) yields the result.
With this proposition in hand we can add an equation to the $B$-field flow system to yield a new system of equations which preserves the generalized Kähler condition. Specifically, given a Riemannian manifold $(M^n, g)$ and $J \in
\operatorname{End}(TM)$, let $$\begin{gathered}
\label{RQdef}
\begin{split}
\mathcal R(J)_k^l =&\ [J, g^{-1} \operatorname{Rc}]_k^l\\
\mathcal Q(DJ)_k^l =&\ - J_k^p D^s J_i^l D_p J_s^i - J_i^l D^s J_k^p D_p J_s^i +
J_s^p D^s J_i^l
D_p J_k^i + J_i^l D^s J_s^p D_p J_k^i\\
&\ - J_p^l D_k J_t^p D^s J_s^t + J_k^p D_p J_t^l D^s J_s^t - J_t^p D^s J_s^t
D_p J_k^l.
\end{split}\end{gathered}$$ Now consider the system of equations for an a priori unrelated Riemannian metric $g$, three-form $H$, and tangent bundle endomorphisms $J_{\pm}$: $$\begin{gathered}
\label{GKflow}
\begin{split}
{\frac{\partial}{\partial t}}g_{ij} =&\ - 2 \operatorname{Rc}_{ij} + \frac{1}{2} H_{ipq} H_j^{\ pq}\\
{\frac{\partial}{\partial t}}H =&\ {\Delta}_{d} H,\\
{\frac{\partial}{\partial t}}J_{\pm} =&\ {\Delta}J_{\pm} + \mathcal R(J_{\pm}) + \mathcal Q(DJ_{\pm}).
\end{split}\end{gathered}$$
Let $M^n$ be a smooth compact manifold. Let $g_0 \in \operatorname{Sym}^2(T^*M)$ be a Riemannian metric, $H_0 \in \Lambda^3(T^*M)$, $(J_{\pm})_0 \in \operatorname{End}(TM)$. There exists $T > 0$ and a unique solution to (\[GKflow\]) on $[0,
T)$ with initial condition $(g_0, H_0, (J_{\pm})_0)$.
The proof is by now standard and we only give a sketch. For a metric $g$ consider the vector field $X_{g}^k = g^{ij} \left( {\Gamma}_{ij}^k -
{\left({\Gamma}^0\right)}_{ij}^k \right)$. Now consider the gauge-fixed system $$\begin{gathered}
\label{gfflow}
\begin{split}
{\frac{\partial}{\partial t}}g_{ij} =&\ - 2 \operatorname{Rc}_{ij} + \frac{1}{2} H_{ipq} H_j^{\ pq} + \left(L_{X_g}
g\right)_{ij},\\
{\frac{\partial}{\partial t}}H =&\ {\Delta}_{d} H + L_{X_g} H,\\
{\frac{\partial}{\partial t}}J_{\pm} =&\ {\Delta}J_{\pm} + \mathcal R(J_{\pm}) + \mathcal Q(DJ_{\pm}) +
L_{X_g} J_{\pm}.
\end{split}\end{gathered}$$ Let $\mathcal O(g, H, J_{\pm})$ denote the total differential operator representing the right hand sides of (\[gfflow\]). A calculation shows that the principal symbol of the linearized operator of $\mathcal O$ is elliptic. More specifically, $$\begin{aligned}
\left[{\sigma}D \mathcal O \right](\xi)\left(
\begin{matrix}
{\delta}g\\
{\delta}H\\
{\delta}J_{+}\\
{\delta}J_-
\end{matrix} \right) =
\left(
\begin{matrix}
{\left| \xi \right|}^2 \operatorname{Id}& 0 & 0 & 0\\
\star & {\left| \xi \right|}^2 \operatorname{Id}& 0 & 0\\
\star & 0 & {\left| \xi \right|}^2 \operatorname{Id}& 0\\
\star & 0 & 0 & {\left| \xi \right|}^2 \operatorname{Id}\end{matrix} \right)
\left(
\begin{matrix}
{\delta}g\\
{\delta}H\\
{\delta}J_{+}\\
{\delta}J_-
\end{matrix} \right).\end{aligned}$$ It follows from standard results that there is a unique solution to (\[gfflow\]) on some maximal time interval $[0, T)$. Moreover, if we let $\phi_t$ denote the one-parameter family of diffeomorphisms generated by $-X_t$ satisfying $\phi_0 = \operatorname{Id}$, it follows that $(\phi_t^*(g(t)),
\phi_t^*(H(t)), \phi_t^*(J_{\pm}(t)))$ is a solution to (\[GKflow\]). Finally, the proof of uniqueness is the same as that for Ricci flow, where one uses that in the modified gauge the diffeomorphisms $\phi_t$ satisfy the harmonic map heat flow equation. For more detail see ([@ChowLu] pg. 117).
We now give a corollary to this discussion which is a more precise statement of Theorem \[mainthm\].
\[maincor\] Let $(M^{2n}, g, J_{\pm})$ be a compact generalized Kähler manifold. The solution to (\[GKflow\]) with initial condition $(g,
d^c_{+} \omega_+, J_{\pm})$ remains a generalized Kähler structure in that $g$ is compatible with $J_{\pm}$, $H = \pm d^c_{\pm} \omega_{\pm}$, and (\[GK\]) holds at all time the solution exists.
Let $(g(t), H(t))$ be the solution to (\[flow\]) with initial condition $(g, d^c_+ \omega_+)$. We showed in Theorem \[mainthm\] that $(g(t), H(t), J_{\pm}(t))$ is a generalized Kähler structure for all times, where $J_{\pm}(t) = \phi_{\pm}(t)^* J_{\pm}$ and $\phi_t^{\pm}$ denote the one-parameter families of diffeomorphisms used above. However, from Proposition \[Jev\] we have that $J_{\pm}$ are solutions of $$\begin{aligned}
{\frac{\partial}{\partial t}}J_{\pm} =&\ {\Delta}J_{\pm} + \mathcal R(J_{\pm}) + \mathcal Q(DJ_{\pm}).\end{aligned}$$ It follows that $(g(t), H(t), J_{\pm}(t))$ is the unique solution to (\[GKflow\]) with initial condition $(g, d^c_+ \omega_+, J_{\pm})$, and the result follows.
The structure of static metrics {#class}
===============================
In this section we will collect some results on static pluriclosed solutions to (\[flow\]). We begin with some general definitions.
Let $(M^{2n}, \omega, J)$ be Hermitian manifold with pluriclosed metric, and let $H = d^c \omega$. We say that $\omega$ is a *$B$-field flow soliton* if there exists a vector field $X$ and ${\lambda}\in \mathbb R$ such that $$\begin{gathered}
\label{soliton}
\begin{split}
\operatorname{Rc}- \frac{1}{4} H^2 + L_X g =&\ {\lambda}g,\\
- \frac{1}{2} {\Delta}_{d} H + L_X H =&\ {\lambda}H.
\end{split}\end{gathered}$$ The form $\omega$ is called *$B$-field flow static*, or simply *static* for short, if (\[soliton\]) is satisfied with $X = 0$.
\[staticprop\] Let $(M^{2n}, \omega, J)$ be a static structure. Then
- [If ${\lambda}= 0$ then $d^* H = 0$ and $b_1(M) \leq {2n}$]{}
- [ If ${\lambda}< 0$ then $g$ is Kähler, i.e. $H = 0$.]{}
- [ If ${\lambda}> 0$ then ${\left| \pi_1(M) \right|} < \infty$.]{}
- [ If ${\lambda}\neq 0$ then $[H] = 0$.]{}
We note that $$\begin{aligned}
\int_M {\left| d^* H \right|}^2 = \int_M \left<d d^* H, H \right> = 2 {\lambda}\int_M
{\left| H \right|}^2.\end{aligned}$$ The first part of the first statement and second statement immediately follow. Now note that $$\begin{aligned}
\operatorname{Rc}= {\lambda}g + \frac{1}{4} H^2.\end{aligned}$$ Since $H^2$ is positive semidefinite, we conclude that if ${\lambda}> 0$ then $\operatorname{Rc}> 0$. It follows from the Bonnet-Meyers Theorem that ${\left| \pi_1(M) \right|} <
\infty$. Also, if ${\lambda}= 0$, the Bochner argument yields the bound $b_1(M)
\leq 2n$. Finally, since $H$ is closed, if ${\lambda}\neq 0$ we conclude that $H =
\frac{1}{{\lambda}} {\Delta}_{d} H = \frac{1}{{\lambda}} d d^* H$ and hence $[H] = 0$.
\[constantfix\] Let $(M^4, \omega, J)$ be a static structure, and suppose $M$ is not of Kähler type, i.e. $b_1$ is odd. Then ${\lambda}= 0$.
Since the manifold $M$ does not admit Kähler metrics, the second statement of Proposition \[staticprop\] rules out ${\lambda}< 0$. Likewise, if ${\lambda}> 0$ then by the third statement of Proposition \[staticprop\] we conclude that ${\left| \pi_1(M) \right|} < \infty$, so that $b_1(M) =
0$, contradicting that $b_1$ is odd. Thus ${\lambda}= 0$.
Let $(M^4, \omega, J)$ be a static structure and suppose $M$ is of Kähler type and ${\lambda}= 0$. Then $\omega$ is Kähler-Einstein.
It follows from the first statement of Proposition \[staticprop\] that $H$ is harmonic. It follows from ${\partial}{{\overline{\partial}}}$-lemma that $H$ is exact, and hence $H$ vanishes and so the metric is Kähler-Einstein.
\[parallelLee\] Suppose $(M^4, g, J)$ is a static structure with ${\lambda}= 0$. Then $D \theta = 0$, that is, the Lee form is parallel with respect to the Levi-Civita connection.
As noted above, if ${\lambda}= 0$ then $\operatorname{Rc}= \frac{1}{4} H^2 \geq
0$. Also, we have $d^* H = 0$. But in the case of complex surfaces, $\theta =
\star H$, therefore $\theta$ is harmonic. It then follows from the Bochner technique that $\theta$ is parallel.
Finally we give the proof of Theorem \[classthm\].
It follows from Corollary \[constantfix\] that ${\lambda}= 0$. Thus from Proposition \[parallelLee\] we conclude that $D \theta = 0$. Recalling that the pluriclosed flow equations and $B$-field flow equations differ by the Lie derivative of the vector dual to $\theta$, we conclude that in fact $\omega$ is static for the pluriclosed flow (see [@ST2] Theorem 6.5), i.e. $$\begin{aligned}
{\partial}{\partial}^*_{\omega} \omega + {{\overline{\partial}}}{{\overline{\partial}}}^*_{\omega} \omega +
\frac{\sqrt{-1}}{2} {\partial}{{\overline{\partial}}}\log \det g =&\ 0.\end{aligned}$$ One can check (see [@ST1] Proposition 3.3, [@IvPap] Proposition 3.3) that this is the same as $$\begin{aligned}
S - Q =&\ 0\end{aligned}$$ where $S = \operatorname{tr}_{\omega} \Omega$ is the curvature endomorphism associated to the Chern connection, and $$\begin{aligned}
Q_{i{{\overline{j}}}} =& g^{k {{\overline{l}}}} g^{m {{\overline{n}}}} T_{i k {{\overline{n}}}} T_{{{\overline{j}}}{{\overline{l}}}m},\end{aligned}$$ where $T$ is the torsion of the Chern connection. In the case of surfaces, one has ([@ST1] Lemma 4.4) that $Q = \frac{1}{2} {\left| T \right|}^2 \omega$. Therefore the metric defines a Hermitian-Einstein connection on $TM$. It follows from ([@GaudIv] Theorem 2) that $(M, g, J)$ is either Kähler-Einstein or $g$ is locally isometric, up to homothety, to $\mathbb R \times S^3$. It follows from [@Gauduchon] that the manifold is a Hopf surface. More specifically, it follows from ([@Gauduchon] III Lemma 11) that the fundamental group takes the form claimed in the theorem.
Indeed each Hopf surface described in Theorem \[classthm\] admits static metrics, given by the metric $\frac{1}{\rho^2} {\partial}{{\overline{\partial}}}\rho^2$, where $\rho = \sqrt{z_1 {\overline{z}}_1 + z_2 {\overline{z}}_2}$.
[10]{} C.G. Callan, D. Friedan, E.J. Martinec, M.J. Perry, *Strings in background fields*, Nuc. Phys. B262 (1985) 593-609.
B. Chow, P. Lu, L. Ni *Hamilton’s Ricci Flow*, Lectures in Contemporary Mathematics, American Mathematical Society, Providence, RI 2005.
D. Friedan, *Nonlinear models in $2 + {\epsilon}$ Dimensions* Ann. Phys. 163, 318-419 (1985).
D.Friedan, E.J. Martinec, S. Shenker, *Conformal invariance, supersymmetry and string theory* Nuc. Phys. B271 (1986), 93-165.
S.J. Gates, C.M. Hull, M. Roček, *Twisted multiplets and new supersymmetric nonlinear sigma models*, Nuc. Phys. B 248 (157-186), 1984.
P. Gauduchon, *Structures de Weyl-Einstein, espaces de twisteurs et variétés de type $S^1 \times S^3$,* J. Reine Angew. Math. 469 1-50.
P. Gauduchon, S. Ivanov, *Einstein-Hermitian surfaces and Hermitian Einstein-Weyl structures in dimension $4$*. Math. Z. 226, 317-326 (1997).
M. Gualtieri *Generalized Complex Geometry.* D. Phil.thesis, Oxford University, 2003, .
M. Gualtieri, *Generalized Kähler Geometry*,
N. Hitchin, *Generalized Calabi-Yau manifolds* Q.J. Math, 54(3), 281-308, 2003, .
S. Ivanov, G. Papadpoulos, *Vanishing theorems and string backgrounds*, Class. Quantum Grav. 18(2001) 1089-1110.
T. Oliynyk, V. Suneeta, E. Woolgar, *A gradient flow for worldsheet nonlinear sigma models*, Nucl. Phys. B739 (2006), 441-458.
G. Perelman, *The entropy formula for Ricci flow and its geometric applications*,
J.G. Polchinski, *String Theory* Vol. I (Cambridge 1998).
J. Streets, *Regularity and expanding entropy for connection [R]{}icci flow*, J. Geom. Phys. 58 (2008), 900-912.
J. Streets, G. Tian, *A parabolic flow of pluriclosed metrics*, Int. Math. Res. Not. 16 (2010), 3101-3133.
J. Streets, G. Tian, *Regularity results for pluriclosed flow*,
A.B. Zamolodchikov *“Irreversibility” of the flux of the renormalization group in a $2D$ field theory.* JETP Lett. 43, p. 730-732, 1986.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Dense optical flow ground truths of non-rigid motion for real-world images are not available due to the non-intuitive annotation. Aiming at training optical flow deep networks, we present an unsupervised algorithm to generate optical flow ground truth from real-world videos. The algorithm extracts and matches objects of interest from pairs of images in videos to find initial constraints, and applies *as-rigid-as-possible* deformation over the objects of interest to obtain dense flow fields. The ground truth correctness is enforced by warping the objects in the first frames using the flow fields. We apply the algorithm on the DAVIS dataset to obtain optical flow ground truths for non-rigid movement of real-world objects, using either ground truth or predicted segmentation. We discuss several methods to increase the optical flow variations in the dataset. Extensive experimental results show that training on non-rigid real motion is beneficial compared to training on rigid synthetic data. Moreover, we show that our pipeline generates training data suitable to train successfully FlowNet-S, PWC-Net, and LiteFlowNet deep networks.'
author:
- 'Hoang-An Le$^1$'
- 'Tushar Nimbhorkar$^{2}$'
- Thomas Mensink$^1$
- 'Anil S. Baslamisli$^1$'
- 'Sezer Karaoglu$^{1,2}$'
- 'Theo Gevers$^{1,2}$'
- '$^1$Computer Vision Lab, Informatics Institute, University of Amsterdam'
- '$^{2}$3DUniversum'
- '[{hoang-an.le, thomas.mensink, a.s.baslamisli, s.karaoglu, th.gevers}@uva.nl]{}'
bibliography:
- 'UvA-iccv19.bib'
title: Unsupervised Generation of Optical Flow Datasets
---
**Acknowledgements**: This project was funded by the EU Horizon 2020 program No. 688007 (TrimBot2020). We would like to thank Thành V. Lê for his support in making the figures.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Viviane M de Oliveira and J F Fontanari\
Instituto de Física de São Carlos\
Universidade de São Paulo\
Caixa Postal 369\
13560-970 São Carlos SP\
Brazil
title: 'Landscape statistics of the p-spin Ising model'
---
The statistical properties of the local optima (metastable states) of the infinite range Ising spin glass with $p$-spin interactions in the presence of an external magnetic field $h$ are investigated analytically. The average number of optima as well as the typical overlap between pairs of identical optima are calculated for general $p$. Similarly to the thermodynamic order parameter, for $p>2$ and small $h$ the typical overlap $q_t$ is a discontinuous function of the energy. The size of the jump in $q_t$ increases with $p$ and decreases with $h$, vanishing at finite values of the magnetic field.
[**Short Title:**]{} landscape of the $p$-spin model
[**PACS:**]{} 05.50+q, 87.10+e, 64.60Cn
Introduction {#sec:level1}
============
The emphasis professed by Kauffman on the role of the topology of the fitness landscape as a source of order in contraposition to natural selection has arisen considerable interest in the study of the statistical properties of fitness landscapes [@Kauffman]. The central issue is the limitation imposed by the structure of the fitness landscapes on adaptive evolution, viewed as a local hill-climbing procedure via fitter mutants. (See [@Dennet] for a lucid criticism of these ideas.) For sake of concreteness, let us consider a population of asexually reproducing haploid organisms whose genotypes are described by sequences of $N$ Ising spins ${\bf{s}} =
(s_1, \ldots, s_N)$ with $s_i = \pm 1$. In the discrete space of the $2^N$ possible sequences, evolution is modelled by an adaptive walk defined as a connected walk through a succession of neighboring sequences (i.e., sequences that differ by a single spin only) each of which possessing improved fitness [@Kauffman]. There are several questions of interest whose answers may shed light on the structure of the landscapes as, for instance, the number of fitness optima in the sequence space and the similarity between these optima, to mention only those we will address in this paper.
Most of the analyses have concentrated on the NK model of random epistatic interactions since it possesses a tunable control parameter that regulates the ruggedness of the fitness landscape [@Kauffman; @Levin; @Weinberger_1]. An alternative (and more appealing to the physicists) class of fitness functions was proposed by Amitrano [*et al. *]{} [@Amitrano], namely, the Ising spin glass with $p$-spin interactions defined by the random energy function [@Derrida; @Gross] $$\label{H_p}
{\cal{H}}_p \left ( {\bf s} \right ) =
- \sum_{1 \leq i_1 <i_2 \ldots < i_p \leq N}
J_{i_1 i_2 \ldots i_p} \, s_{i_1} s_{i_2} \ldots s_{i_p}
- h \, \sum_i s_i$$ where the coupling strengths are statistically independent random variables with a Gaussian distribution $$\label{prob}
{\cal{P}} \left ( J_{i_1 i_2 \ldots i_p} \right ) =
\sqrt{\frac{N^{p-1}}{\pi p!}} \exp \left [
-\frac{ \left( J_{i_1 i_2 \ldots i_p} \right)^2 N^{p-1}}{p!}
\right ] ,$$ and $h$ is the external magnetic field. In this context the fitness value ascribed to a sequence or genotype ${\bf s}$ is the reverse of its energy. Thus the fitness maxima correspond to the energy minima of (\[H\_p\]). Henceforth we will refer to the fitness maxima or energy minima as simply optima. For $p=1$ or $h \rightarrow \infty$ the energy (\[H\_p\]) gives a single-peaked, smooth correlated landscape, while the limit $p \rightarrow \infty$ corresponds to the random energy model of Derrida [@Derrida] and yields an extremely rugged, uncorrelated landscape. The case $p=2$ is the well-known SK model [@SK], which exhibits a large number of highly correlated local optima [@Binder; @Mezard].
For general $p$, little is known about the statistical features of the landscape generated by the energy function (\[H\_p\]). A result worth mentioning is that, for $h=0$, the correlation between values of ${\cal{H}}_p$ for different configurations is given by [@Amitrano; @Weinberger_2] $$\left \langle {\cal{H}}_p \left ( {\bf{s}}^a \right )
{\cal{H}}_p \left ( {\bf{s}}^b \right ) \right \rangle
= \left [ q \left ( {\bf{s}}^a, {\bf{s}}^b \right ) \right ]^p$$ where $$\label{overlap}
q \left ( {\bf s}^a, {\bf s}^b \right ) = \frac{1}{N}
\sum_{i=1}^N s_i^a s_i^b$$ is the overlap between the two arbitrary states ${\bf{s}}^a$ and $ {\bf{s}}^b$. Here the average indicated by $\langle \ldots \rangle$ is taken over the probability distribution of the couplings (\[prob\]). Thus, as mentioned before, the correlations between energy levels vanish for $p \rightarrow
\infty$.
The thermodynamics of the $p$-spin Ising model has been investigated within the replica framework [@Gross; @Gardner; @Stariolo]. In particular, for $p=2$ the order parameter function $q(x)$ tends to zero continuously as the temperature approaches a critical value at which the transition between the spin glass and the high temperature (disordered) phases takes place [@Binder; @Mezard]. For $p \rightarrow \infty$, the system has a critical temperature $T_c$ at which it freezes completely into the ground state: $q(x)$ is a step function with values zero and one, and with a break point at $x = T/T_c$ [@Gross]. The situation for finite $p > 2$ is considerably more complicated. There is a transition from the disordered phase to a partially frozen phase characterized by a step function $q(x)$ with values zero and $q_1 < 1$. As the temperature is lowered further, a second transition occurs, leading to a phase described by a continuous order parameter function [@Gardner; @Stariolo].
The goal of this paper is to investigate the statistical properties of the fixed points (local or global optima) of adaptive walks on the fitness landscape defined by equation (\[H\_p\]). The energy cost of flipping the spin $s_i$ is $\delta {\cal{H}}_p = 2 \Delta_i$ where $$\Delta_i = \sum_{i_2 <\ldots < i_p}
J_{i i_2 \ldots i_p} \, s_{i} s_{i_2} \ldots s_{i_p} + h s_i .$$ is termed the stability of $s_i$. Since in an adaptive walk only flippings or moves that decrease the energy (i.e., increase the fitness) are allowed, any state ${\bf s}$ that satisfies $$\label{const}
\Delta_i > 0 ~~~~~\forall i$$ is an optima of the fitness landscape. Clearly, counting the number of states that obey (\[const\]) is equivalent to calculating the number of solutions of the zero-temperature limit of the celebrated TAP equations [@TAP]. For non-zero temperature, the quite involved calculation of the average number of solutions of the TAP equations has been carried out for $p=2$ [@Bray] as well as for general $p$ [@Rieger]. However, systematic analyses of the typical energy of the local optima and of the effects of the external magnetic field have been undertaken for the simplest case only, namely, $p=2$ at zero temperature [@Tanaka; @Roberts; @Dean]. We note that in the statistical mechanics context the local optima are usually termed metastable states.
In this paper we study at length the effects of the magnetic field $h$ on the structure of the local optima of the $p$-spin energy landscape. More pointedly, we calculate analytically the average number of local optima with a fixed energy density $\epsilon$, denoted by $ \langle
{\cal{N}}(\epsilon) \rangle$. Although this analysis is quite straighforward, it is justified since the dependence of that quantity on $\epsilon$ and $h$ has not been investigated for general $p$. In fact, we note that results of extensive numerical simulations aimed at measuring $ \langle {\cal{N}} (\epsilon) $ have been reported recently [@Mexico]. More importantly, we calculate the average number of pairs of local optima with overlap $q$ and fixed energy density $\epsilon$. This quantity, denoted by $\langle {\cal{M}} (q,\epsilon) \rangle$, allows us to determine the typical overlap $q_t$ between pairs of local optima with energy density $\epsilon$. Since $ \langle {\cal{M}} (q_t,\epsilon) \rangle$ is directly related to the second moment of ${\cal{N}} (\epsilon)$, we can determine the regions in the space of parameters $(p,\epsilon,h)$ where this random variable is self-averaging.
The remainder of the paper is organized as follows. In Sec. \[sec:level2\] we derive the formal equation for the $n$th moment of the random variable ${\cal{N}} (\epsilon)$. Then we use that result to calculate the average number of local optima $ \langle {\cal{N}} (\epsilon) \rangle$ in Sec. \[sec:level3\], and the average number of pairs of local optima $ \langle {\cal{M}} (q,\epsilon) \rangle$ in Sec. \[sec:level4\]. Finally, some concluding remarks are presented in Sec. \[sec:level5\].
The formalism {#sec:level2}
=============
The number of local optima ${\cal{N}}(\epsilon) $ with fixed energy density $\epsilon$ can be calculated by introducing the quantity $Y_{\bf s}$ defined by $$\label{Y}
Y_{\bf s} = \left \{ \begin{array}{ll}
1 & \mbox{if
$\epsilon N = {\cal{H}}_p \left ( {\bf s} \right ) $ and
$\Delta_i > 0 ~~\forall i$} \\
0 & \mbox{otherwise}
\end{array}
\right. ,$$ so that $${\cal{N}}(\epsilon) = \mbox{Tr}_{\bf s} Y_{\bf s},$$ where $\mbox{Tr}_{\bf s}$ denotes the summation over the $2^N$ states of the system. We are interested in the evaluation of the moment $\langle \left [{\cal{N}}(\epsilon)\right]^n \rangle$ for $n =1,2$, which can be written as $$\begin{aligned}
\langle \left [{\cal{N}}(\epsilon) \right ]^n \rangle & = &
\left \langle \prod_{a=1}^n \mbox{Tr}_{{\bf s}^a} Y_{{\bf s}^a}
\right \rangle \nonumber \\
& = & \mbox{Tr}_{{\bf s}^1} \ldots \mbox{Tr}_{{\bf s}^n}
{\cal W} \left ( Y_{{\bf s}^1} = 1,\ldots, Y_{{\bf s}^n} = 1
\right )\end{aligned}$$ where ${\cal W} \left ( Y_{{\bf s}^1} = 1,\ldots, Y_{{\bf s}^n} = 1 \right )$ is the joint probability that the $n$ random variables $ Y_{{\bf s}^1},\ldots, Y_{{\bf s}^n}$ assume the value $1$. Using the definition $${\cal{W}}\left ( Y_{{\bf s}^1} = 1,\ldots, Y_{{\bf s}^n} = 1 \right ) =
\left \langle \prod_{a=1}^n
\delta \left [ \epsilon - {\cal{H}}_p \left ( {\bf s}^a \right )/N
\right] \prod_i \Theta \left ( \Delta_i^a \right ) \right \rangle ,$$ the equation for the $n$th moment becomes $$\label{N_fin}
\langle \left [{\cal{N}}(\epsilon)\right ]^n \rangle =
\left \langle \prod_{a=1}^n \mbox{Tr}_{{\bf s}^a}
\delta \left [ \epsilon N - {\cal{H}}_p \left ( {\bf s}^a \right )
\right] \prod_i \Theta \left ( \Delta_i^a \right ) \right \rangle .$$ where $\Theta (x) = 1$ if $x > 0$ and $0$ otherwise. We have presented the derivation of equation (\[N\_fin\]) in detail because some authors have written the random variable ${\cal{N}}(\epsilon)$ in terms of the delta function directly [@Roberts; @Dean]. Clearly, this procedure is correct only for the moments of ${\cal{N}}(\epsilon)$ as shown above.
In the next two sections we concentrate on the explicit evaluation of equation (\[N\_fin\]) for $n=1$ and $2$. To facilitate those calculations, we express the energy $ {\cal{H}}_p \left ( {\bf s} \right )$ in terms of the stabilities $\Delta_i$, $$\label{aid}
{\cal{H}}_p \left ( {\bf s} \right ) =
-\frac{1}{p} \sum_i \left ( \Delta_i + h(p-1) s_i \right ) ,$$ so that the dependence on the couplings in equation (\[N\_fin\]) appears only through the stabilities $\Delta_i$.
Average number of optima {#sec:level3}
========================
Using the integral representation of the delta function and the auxiliary relation (\[aid\]) we can write the first moment of ${\cal{N}}(\epsilon)$ as $$\begin{aligned}
\langle {\cal{N}}(\epsilon) \rangle & = &
\int_{-\infty}^\infty \frac{d\tilde{\epsilon}}{2 \pi}
\exp \left ({\bf i} N \epsilon \tilde{\epsilon} \right )
\prod_i \int_{-\infty}^\infty \frac{d\Delta_i d \tilde{\Delta}_i}{2 \pi}
\Theta \left ( \Delta_i \right)
\exp \left ( {\bf i} \Delta_i \tilde{\Delta}_i \right ) \nonumber \\
& & \times \mbox{Tr}_{{\bf s}}
\exp \left [-{\bf i} h \sum_i \tilde{\Delta}_i s_i
+\frac{{\bf i}}{p} \tilde{\epsilon}
\sum_i \left( \Delta_i +h(p-1) s_i \right )
\right ] \nonumber \\
& & \times \left \langle \exp \left (
- {\bf i} \sum_i \tilde{\Delta}_i
\sum_{i_2 < \ldots < i_p} J_{i i_2 \ldots i_p}
s_{i} s_{i_2} \ldots s_{i_p} \right ) \right \rangle .\end{aligned}$$ The average over the couplings can be easily carried out using the identity $$\label{id_1}
\sum_i \tilde{\Delta}_i \sum_{i_2 < \ldots < i_p} J_{i i_2 \ldots i_p}
s_{i} s_{i_2} \ldots s_{i_p} =
\sum_{i_1 < \ldots < i_p} \left ( \sum_{k=1}^p
\tilde{\Delta}_{i_k} \right )
J_{i_1 \ldots i_p} s_{i_1} \ldots s_{i_p}$$ and yields, in the limit $N \rightarrow \infty$, $$\begin{aligned}
\label{av_fin}
\left \langle \ldots \right \rangle & = &
\exp \left [ - \frac{p!}{4 N^{p-1}}
\sum_{i_1 < \ldots < i_p} \left ( \sum_{k=1}^p \tilde{\Delta}_{i_k}
\right )^2 \right ] \nonumber \\
& = & \exp \left [
-\frac{p}{4} \sum_i \left ( \tilde{\Delta}_i \right )^2
-\frac{p(p-1)}{4N} \left ( \sum_i \tilde{\Delta}_i \right )^2
\right ] .\end{aligned}$$ The remaining calculations are straightforward: a Gaussian transformation allows us to decouple the sites in (\[av\_fin\]), so that the integrals over $\Delta_i$ and $\tilde{\Delta}_i$ as well as the trace over the spins can be readily performed. As usual, we conclude the calculation by carrying out a saddle-point integration over two appropriately rescaled saddle-point parameters. The final result for the exponent $f$ in $\langle {\cal{N}}(\epsilon) \rangle = \mbox{e}^{Nf}$ is $$\begin{aligned}
\label{f}
f & = &
\frac{\epsilon \nu}{\sqrt{p}} - \frac{1}{p-1} \left( \mu^2
- \mu \nu +\frac{\nu^2}{4p} \right ) - \ln 2 \nonumber \\
& & + \ln \left [ \mbox{e}^{ \bar{h} \nu }
\mbox{erfc} \left (
-\mu - \bar{ h} \right ) + \mbox{e}^{- \bar{h} \nu}
\mbox{erfc} \left (
-\mu + \bar{h} \right ) \right ] ,\end{aligned}$$ where $$\label{h}
\bar{h} = \frac{h}{\sqrt{p}} .$$ Here the saddle-point parameters $\nu$ and $\mu$ are obtained by solving the equations $\partial f/\partial\nu = 0$ and $\partial f/\partial \mu = 0$ simultaneously. In figure 1 we present the exponent $f$ as a function of $\epsilon$ for $p=2$ and several values of $h$. For sake of clarity we present only positive values of $f$. The decrease in the number of local optima as $h$ increases indicates that the landscape becomes smoother, as expected. The results for $p > 2$ are qualitatively similar, except that the peaks are higher and slightly broader. Two values of the energy density are particularly important, namely, the value at which $f$ reaches its maximum value $f_t$, denoted by $\epsilon_t$, and the lowest value of $ \epsilon $ for which $f$ vanishes, denoted by $\epsilon_0$. While $\epsilon_t$ gives the typical value of the energy density of the local optima, $\epsilon_0$ gives a lower bound to the ground state energy density of the spin model defined by the hamiltonian (\[H\_p\]) [@Tanaka]. In figures 2 an 3 we present $\epsilon_t$ and $f_t$, respectively, as a function of $h$ for several values of $p$. These quantities are easily obtained by setting $\nu = 0$ in equation (\[f\]). The single saddle-point equation $\partial f/\partial\mu = 0$ possesses either one root (for either small or large values of $h$) or three roots (for intermediate values of $h$). The discontinuity in $\epsilon_t$ that can be observed in figure 2 for $p \geq 7$ is due to the simultaneous disappearance of two of those roots. For $p \rightarrow \infty$ and finite $h$ we find $\epsilon_t \rightarrow \langle {\cal{H}}_p \rangle = 0$, signaling thus the emergence of the so-called complexity catastrophe, i.e., the energy density of typical local optima equals the expected energy of a randomly chosen state [@Kauffman]. We note that $\langle {\cal{N}}(\epsilon_t) \rangle = \exp( f_t N)$ yields the average number of optima regardless of their energy values, i.e., the same result is obtained by dropping the energy constraint in the definition of $Y_{{\bf s}}$ given in equation (\[Y\]). In figure 4 we present $\epsilon_0$ as a function of $h$ for several values of $p$. Clearly, since in the limit $h \rightarrow \infty$ there is only one optimum, namely, ${\bf s} = {\bf 1}$, we find $\epsilon_0 \rightarrow \epsilon_t = -h$. It is important to note that for $p \rightarrow \infty$, $\epsilon_0 $ tends to a non-zero limiting value. This result illustrates the fact that the complexity catastrophe phenomenon affects the typical optima only. In fact, the increase of $p$ has little effect on the ground-state lower bound $\epsilon_0$, which for $h=0$ decreases from $-0.791$ for $p=2$ [@Bray] to $-\sqrt{\ln 2} \approx -0.832$ for $p \rightarrow \infty$ [@Derrida].
Average number of pairs of optima {#sec:level4}
=================================
We define the number of pairs of optima with overlap $q =-1, -1 + \frac{2}{N},\ldots, 1 $ and energy density $\epsilon$ as $$\label{M}
{\cal{M}}(q,\epsilon) = \frac{1}{2}
\mbox{Tr}_{{\bf s}^1} \mbox{Tr}_{{\bf s}^2} Y_{{\bf s}^1} Y_{{\bf s}^2}
\delta \left ( Nq, \sum_i s_i^1 s_i^2 \right )$$ where $\delta(m,n)$ is the Kronecker delta and $Y_{{\bf s}}$ is given by equation (\[Y\]). Following the procedure presented in Sec. \[sec:level2\], the average of ${\cal{M}}$ over the couplings is cast into the form $$\label{M_1}
\langle {\cal{M}}(q,\epsilon) \rangle = \frac{1}{2}
\left \langle \mbox{Tr}_{{\bf s}^1} \mbox{Tr}_{{\bf s}^2}
\delta \left ( Nq, \sum_i s_i^1 s_i^2 \right ) \prod_{a=1}^2
\delta \left [ \epsilon N - {\cal{H}}_p \left ( {\bf s}^a \right )
\right] \prod_i \Theta \left ( \Delta_i^a \right ) \right \rangle .$$ The integral representations of the delta function and the Kronecker delta allow us to write this equation as $$\begin{aligned}
\langle {\cal{M}}(q,\epsilon) \rangle & = & \frac{1}{2}
\int_{-\pi}^\pi \frac{d\tilde{q}}{2 \pi}
\exp \left ({\bf i} N q \tilde{q} \right )
\prod_a
\int_{-\infty}^\infty \frac{d\tilde{\epsilon}^a}{2 \pi}
\exp \left ({\bf i} N \epsilon^a \tilde{\epsilon}^a \right ) \nonumber \\
& & \times \prod_{ai} \mbox{Tr}_{{\bf s}^a}
\int_{-\infty}^\infty \frac{d\Delta_i^a d \tilde{\Delta}_i^a}{2 \pi}
\Theta \left ( \Delta_i^a \right)
\exp \left ( {\bf i} \Delta_i^a \tilde{\Delta}_i^a \right ) \nonumber \\
& & \times \exp \left [-{\bf i} \tilde{q} \sum_{i} s_i^1 s_i^2
-{\bf i} h \sum_{ai} \tilde{\Delta}_i^a s_i^a
+\frac{{\bf i}}{p} \sum_{ai} \tilde{\epsilon}^a
\left( \Delta_i^a +h(p-1) s_i^a \right )
\right ] \nonumber \\
& & \times
\left \langle \exp \left (
- {\bf i} \sum_{ai} \tilde{\Delta}_i^a
\sum_{i_2 < \ldots < i_p} J_{i i_2 \ldots i_p}
s_{i}^a s_{i_2}^a \ldots s_{i_p}^a \right ) \right \rangle .\end{aligned}$$ As in the previous section, the average can be performed with the aid of an identity analogous to (\[id\_1\]), yielding $$\label{av_2}
\left \langle \ldots \right \rangle =
\exp \left \{ - \frac{p!}{4 N^{p-1}}
\sum_{i_1 < \ldots < i_p} \left [\sum_{a=1}^2 \left
( \sum_{k=1}^p \tilde{\Delta}_{i_k}^a \right )
s_{i_1}^a \ldots s_{i_p}^a \right ]^2 \right \} .$$ After some algebra, the argument of this exponential is rewritten in the limit $N \rightarrow \infty$ as $$\begin{aligned}
\left \{ \ldots \right \} & = &
-\frac{p}{4} \sum_{a=1}^2 \left [ \sum_i \left ( \tilde{\Delta}_i^a \right )^2
+\frac{p-1}{N} \left ( \sum_i \tilde{\Delta}_i^a \right )^2 \right ]
-\frac{p \, q^{p-1}}{2} \sum_i \tilde{\Delta}_i^1 \tilde{\Delta}_i^2
s_i^1 s_i^2
\nonumber \\
& & - \frac{p(p-1)\, q^{p-2}}{2N}
\left( \sum_i \tilde{\Delta}_i^1 s_i^1 s_i^2 \right )
\left( \sum_i \tilde{\Delta}_i^2 s_i^1 s_i^2 \right ) .\end{aligned}$$ The next step is to introduce via delta functions the auxiliary parameters: $N m_1 = \sum_i
\tilde{\Delta}_i^1 $, $N m_2 = \sum_i
\tilde{\Delta}_i^2 $, $N v_{1} = \sum_i \tilde{\Delta}_i^1 s_i^1 s_i^2 $, $N v_{2} = \sum_i \tilde{\Delta}_i^2 s_i^1 s_i^2 $, and their respective Lagrange multipliers in order to decouple the variables $s_i^a$ and $\tilde{\Delta}_i^a$ for different sites $i$. Then the integrals over $\Delta_i^a$ and $\tilde{\Delta}_i^a$, and the trace over $s_i^a$ can be easily performed. As before, the auxiliary parameters as well as the Lagrange multipliers $\tilde{q}$ and $\tilde{\epsilon}$ are integrated out via a saddle-point integration. This part of the calculation is straightforward and quite unilluminating so we do not present any further detail. To proceed further we assume that the symmetry ${\bf s}^1 \leftrightarrow {\bf s}^2$ between the two replicas remains intact, i.e., $m_1 = m_2$ and $v_1 = v_2$. This is a quite sensible assumption since the breaking of the replica symmetry that pervades the thermodynamic calculations [@Gross; @Gardner; @Stariolo] is very probably a consequence of the limit where the number of replicas goes to zero. In any event we will, conservatively, restrict the forthcoming analysis to pairs of identical optima only. The final result for the exponent $g$ in $\langle {\cal{M}}(q,\epsilon) \rangle
= \frac{1}{2} \exp ( g N )$ is written more simply in terms of a new set of saddle-point parameters that are linear combinations of those introduced above. We find $$\begin{aligned}
\label{g}
g & = & \frac{\epsilon \nu}{\sqrt{p}} + q z - \frac{1}{2(p-1)} \left [
\left ( x+y \right )^2 + q^{2-p} \left ( x- y \right )^2
+ (1 + q^p) \frac{\nu^2}{4p} \right ] \nonumber \\
& & + \frac{\nu}{2(p-1)} \left [
\left ( 1 +q \right ) x + \left ( 1-q \right ) y \right ] +
\ln \Xi \left ( \nu,x,y,z \right) - \ln 2\end{aligned}$$ where $$\begin{aligned}
\Xi & = & \mbox{e}^{ \nu \bar{h} - z }
\int_{-x - \bar{h}}^\infty Dt \, \mbox{erfc}
\left [- \frac{x + \bar{h} + q^{p-1} t}{\sqrt{1 - q^{2p-2}}}
\right ] \nonumber \\
& & + \mbox{e}^{ - \nu \bar{h} - z }
\int_{-x + \bar{h}}^\infty Dt \, \mbox{erfc}
\left [- \frac{x - \bar{h} + q^{p-1} t}{\sqrt{1 - q^{2p-2}}}
\right ] \nonumber \\
& & + \mbox{e}^z
\int_{-y + \bar{h}}^\infty Dt \, \mbox{erfc}
\left [- \frac{y + \bar{h} - q^{p-1} t}{\sqrt{1 - q^{2p-2}}}
\right ] \nonumber \\
& &
+ \mbox{e}^z
\int_{-y - \bar{h}}^\infty Dt \, \mbox{erfc}
\left [- \frac{y - \bar{h} - q^{p-1} t}{\sqrt{1 - q^{2p-2}}}
\right ] .\end{aligned}$$ Here $Dt = dt \mbox{e}^{-t^2}/\sqrt{\pi}$ is the Gaussian measure and $\bar{h}$ is given by (\[h\]). The saddle-point parameters $\nu, x,y,z$ must be determined so as to maximize $g$. This is achieved by solving the four coupled saddle-point equations $\partial g/\partial \nu =0$, $\partial g/\partial x =0$, $\partial g/\partial y = 0$, and $\partial g/\partial z =0$. For $q=1$ we find $y = 0$ and hence $g = f$, as expected. Furthermore, for $q=0$ and $h=0$ we find $x=y$ and $z=0$ so that $g = 2 f$. Once $\langle {\cal{M}}(q,\epsilon) \rangle $ is known, the second moment of ${\cal{N}}(\epsilon)$ can be calculated using the identity $$\begin{aligned}
\sum_q \langle {\cal{M}}(q,\epsilon) \rangle & = & \frac{1}{2}
\langle \left [ {\cal{N}} (\epsilon) \right ]^2 \rangle
\nonumber \\
& \approx & \langle {\cal{M}}(q_t,\epsilon) \rangle ,\end{aligned}$$ since the sum is dominated by the overlap $q = q_t$ that maximizes equation (\[g\]) in the limit $N \rightarrow \infty$. Hence we have $ \langle \left [{\cal{N}} (\epsilon) \right ]^2 \rangle
= \exp \left ( f^{(2)} N \right ) $ with $f^{(2)}$ given by (\[g\]) calculated at $q_t$. Thus for $h=0$ the variance of the random variable ${\cal{N}}(\epsilon)$ vanishes in the thermodynamic limit, provided that $q_t = 0$. We note that although $q=0$ is always a point of maximum of $g$ for $h=0$, that maximum may not be the global one and, in that case, $q_t \neq 0$.
For fixed $q$, the dependence of $g$ on $\epsilon$ is similar to that shown in figure 1. Likewise, the maximum of $g$ with respect to $\epsilon$, denoted by $g_t$, is determined by setting $\nu = 0$. In figure 5 we show this maximum as a function of $q$ for $p=7$ and several values of $h$. The quantity $ \frac{1}{2}\exp \left ( g_t N \right )$ can be viewed as the number of pairs of identical optima (in the sense that their energies and saddle-point parameters are identical) with overlap $q$, regardless of the specific value of their energies. For $h$ not too large there appears a minimum for $q \approx 1$, indicating that around a typical optimum there is a region where other optima are rarer. The picture that emerges is one of clusters of many optima surrounded by comparatively smoother valleys. The typical energy $\epsilon_t$ of these optima is shown in figure 6 as a function of the overlap $q$. The typical overlap $q_t$ between the optima increases from zero at $h=0$ to one in the limit $h \rightarrow \infty$ since, as expected, the external magnetic field induces correlations between the optima. This is shown in figure 7, where we present $q_t$ as a function of $h$ for several values of $p$. The discontinuity that appears for $ p \geq 7$ is caused by the competition between the two maxima shown in figure 5.
Next we consider the dependence of the typical overlap between identical optima on their energies. This analysis is more involved since, besides the four saddle-point equations, we have to solve the equation $\partial g/\partial q =0$ too. In figures 8 and 9 we show $q_t$ as a function of $\epsilon$ for $p=2$ and $p=3$, respectively, and several values of the external magnetic field. For $h=0$, in both cases we find $q_t =0$ up to a certain value of the energy density ($\epsilon = -0.672$ for $p=2$ and $\epsilon = -0.792$ for $p=3$). Thus, as mentioned before, ${\cal{N}}(\epsilon)$ is self-averaging in this regime. Our results for $p=2$ are remarkably similar to those found in the replica calculation of the quenched average $ \langle \ln {\cal{N}}(\epsilon) \rangle$, with the typical overlap $q_t$ replaced by the saddle-point parameter $\hat{q} = \langle \langle s_i \rangle^2_\epsilon \rangle$ [@Roberts]. Here $\langle \ldots \rangle_\epsilon$ means an average over optima with energy density $\epsilon$. In particular, $\hat{q}$ vanishes for $\epsilon > - 0.672$, indicating thus that ${\cal{N}}(\epsilon)$ is self-averaging in this regime, in agreement with our results. However, while for $p=2$, $q_t$ increases continuously from zero, for $p = 3$ there is a discontinuity at $\epsilon = - 0.792$. The same phenomenon is observed for $p > 3$, with the size of the jump in $q_t$ increasing with $p$. This finding is reminiscent of the jump in the order parameter found in the thermodynamic calculations for $p >2$ [@Gardner; @Stariolo]. The discontinuity in $q_t$ can be understood by studying the dependence of the exponent $g$ on the overlap $q$ for $p=3$ and $h=0$, shown in figure 10. Since the typical overlap is associated to the global maximum of $g$, the competition between the maximum at $q=0$ and the maximum at $q > 0$ originates the jump in $q_t$, which takes place at the energy density where the two maxima have precisely the same height. The situation for non-zero $h$ is more complicated. The correlations induced by the magnetic field destroy the region of self-averageness of ${\cal{N}}(\epsilon)$. Interestingly, for a given $h > 0$ there is value of the energy density for which the typical overlap is minimal. For $p=3$ the effect of the magnetic field is to decrease the size of the jump in $q_t$ till it disappears altogether for $h \approx 0.29$. The results for $p > 3$ are qualitatively similar to those for $p=3$. We mention only that the larger $p$, the larger the value of $\epsilon$ at which the discontinuity occurs, and the larger the value of $h$ at which it disappears. Unfortunately, the enormous difficulty of solving the system of five coupled equations prevents a more systematic analysis of these discontinuities.
Conclusion {#sec:level5}
==========
The analytical investigation of the statistical structure of the energy landscape of the $p$-spin Ising model presented in this paper is of interest from the viewpoint of the traditional statistical mechanics of disordered systems [@Gross; @Tanaka; @Roberts; @Dean] as well as from the perspective of the study of adaptive walks in rugged fitness landscapes [@Amitrano; @Weinberger_2; @Mexico]. Besides extending the calculation of the average number of optima to general $p$ and non-zero magnetic field, we have focused on the characterization of the typical overlap $q_t$ between pairs of identical optima. Interestingly, the dependence of $q_t$ on the energy density $\epsilon$ is reminiscent of the dependence of the thermodynamic order parameter on the temperature $T$ [@Gardner; @Stariolo]. We must note, however, that there is no relation between $T$ and $\epsilon$ since $\ln {\cal{N}}(\epsilon)$ is not the entropy of the spin system. The quite complex effect of the magnetic field on the statistical properties of the energy optima motivates a more detailed study of the thermodynamics of the $p$-spin model for non-zero $h$. In fact, even the unambitious analysis of the first moment $\langle {\cal{N}}(\epsilon) \rangle$ has unveiled an interesting interplay between $h$ and $p$ that lead to a discontinuity in the typical energy density of the optima. Moreover, we have found that the magnetic field decreases the size of the jump in the typical overlap $q_t$ that occurs for $p>2$. It would be interesting to investigate whether a similar effect occurs for the thermodynamic order parameter as well, which might lead, eventually, to a continuous phase transition.
To conclude, we must mention that the calculations presented in this paper are free of all the mathematical subtleties that permeate the replica analyses of the infinite range Ising spin glass [@Binder; @Mezard]. Thus our results present a reliable account of the statistical properties of the $p$-spin energy landscape which, though may have little relevance to the thermodynamics of the model, are of considerable interest to the characterization of the fixed points (metastable states) of adaptive walks (zero-temperature Monte Carlo dynamics) on that landscape.
[**Acknowledgments**]{} This work was supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). VMO holds a fellowship.
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Figure captions {#figure-captions .unnumbered}
===============
[**Fig. 1**]{} The exponent $f$ in $\langle {\cal {N}} (\epsilon) \rangle
= \mbox{e}^{f N}$ as a function of the energy density $\epsilon$ for $p=2$ and $h =0$, $0.5$, $1.0$, and $1.5$.
[**Fig. 2**]{} The typical energy density $\epsilon_t$ of the local optima as a function of $h$ for (from top to bottom) $p=2$ to $p=10$. For $ p \rightarrow \infty$ we find $\epsilon_t \rightarrow 0$. The dashed straight line is $\epsilon_t = - h$.
[**Fig. 3**]{} The exponent $f_t$ in the expression for the average number of optima $\langle {\cal {N}} (\epsilon_t) \rangle
= \mbox{e}^{f_t N}$ as a function of $h$ for (from bottom to top) $p=2$ to $p=10$. For $ p \rightarrow \infty$ we find $f_t \rightarrow \ln 2$.
[**Fig. 4**]{} The lower bound $\epsilon_0$ to the ground state energy density as a function of $h$ for (from bottom to top) $p=2$, $3$, $4$, and $\infty$. The dashed straight line is $\epsilon_0 = - h$.
[**Fig. 5**]{} The exponent $g_t$ in the expression for the average number of pairs of identical optima $\langle {\cal {M}} (\epsilon_t, q) \rangle
= \mbox{e}^{g_t N}$ as a function of $q$ for $p=7$ and (from top to bottom) $h=0$, $1$, $2$, $2.5$, $3$, $3.3$, $3.6$, $3.8$, $4$ and $4.2$.
[**Fig. 6**]{} The typical value of the energy density of a pair of identical optima as a function of the overlap $q$ for $p=7$ and (from bottom to top) $h=0$, $1$, $2$, $2.5$, $3$, $3.3$, $3.6$, $3.8$, $4$ and $4.2$.
[**Fig. 7**]{} The typical value of the overlap between pair of identical optima as a function of $h$ for (from left to right) $p=2$ to $p=8$.
[**Fig. 8**]{} The typical value of the overlap between pair of identical optima as a function of their energy density for $p=2$ and $h = 0$, $0.5$, $1.0$, and $1.5$. The marked points correspond to $f^{(2)} = 0$.
[**Fig. 9**]{} Same as figure 8 but for $p=3$, and $h = 0$, $0.27$, $0.5$, $1.0$, and $1.5$.
[**Fig. 10**]{} The exponent $g$ in the expression for the average number of pairs of identical optima $\langle {\cal {M}} (\epsilon_, q) \rangle
= \mbox{e}^{g N}$ as a function of $q$ for $p=3$, $h=0$, and (from top to bottom) $\epsilon =-0.73$, $-0.75$, $-0.77$, $-0.79$, and $-0.81$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we study some cohomlogical properties of Banach algebras. For a Banach algebra $A$ and a Banach $A$-bimodule $B$, we investigate the vanishing of the first Hochschild cohomology groups $H^1(A^n,B^m)$ and $H_{w^*}^1(A^n,B^m)$, where $0\leq m,n\leq 3$. For amenable Banach algebra $A$, we show that there are Banach $A$-bimodules $C$, $D$ and elements $\mathfrak{a}, \mathfrak{b}\in A^{**}$ such that $$Z^1(A,C^*)=\{R_{D^{\prime\prime}(\mathfrak{a})}:~D\in Z^1(A,C^*)\}=\{L_{D^{\prime\prime}(\mathfrak{b})}:~D\in Z^1(A,D^*)\}.$$ where, for every $b\in B$, $L_{b}(a)=ba$ and $R_{b}(a)=a b,$ for every $a\in A$. Moreover, under a condition, we show that if the second transpose of a continuous derivation from the Banach algebra $A$ into $A^*$ i.e., a continuous linear map from $A^{**}$ into $A^{***}$, is a derivation, then $A$ is Arens regular. Finally, we show that if $A$ is a dual left strongly irregular Banach algebra such that its second dual is amenable, then $A$ is reflexive.'
address:
- ' $^1$ Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran'
- ' $^{2}$Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran .'
- ' $^{3}$Young Researchers and Elite Club, Islamic Azad University, Ardabil Branch, Ardabil, Iran.'
author:
- 'Hossein Eghbali Sarai$^1$, Kazem Haghnejad Azar$^2$$^{*}$ and Ali Jabbari$^3$'
title: Cohomological properties and Arens regularity of Banach algebras
---
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Introduction
============
A derivation from a Banach algebra $A$ into a Banach $A-$bimodule $B$ is a bounded linear mapping $D:A\longrightarrow B$ such that $$D(ab)=aD(b)+D(a)b\qquad\text{for all}\qquad a, b\in A.$$
The space of continuous derivations from $A$ into $B$ is denoted by $Z^1(A,B)$. The easiest example of derivations is the inner derivations, which are given for each $b\in B$ by $$\delta_b(a)=ab-ba\qquad\text{for all}\qquad a\in A.$$
The space of inner derivations from $A$ into $B$ is denoted by $B^1(A,B)$. The Banach algebra $A$ is said to be amenable, when for every Banach $A$-bimodule $B$, the inner derivations are only derivations existing from $A$ into $B^*$, in the other word, $H^1(A,B^*)=Z^1(A,B^*)/ B^1(A,B^*)=\{0\}$ and $A$ is said to be weakly amenable if $H^1(A,A^*)=\{0\}$.
The concept of amenability for a Banach algebra $A$, introduced by Johnson in 1972, see [@14]. For a Banach $A$-bimodule $B$, the quotient space $H^1(A,B)$ of all continuous derivations from $A$ into $B$ modulo the subspace of inner derivations is called the first cohomology group of $A$ with coefficients in $B$. Following [@ru.1] the Banach algebra $A$ is called super-amenable if $H^1(A,B)=\{0\}$ for every Banach $A$-bimodule $B$ (super-amenable Banach algebras are called contractible, too). It is clear that if $A$ is super-amenable, then $A$ is amenable.
In [@JKR], Johnson, Kadison, and Ringrose introduced the notion of amenability for von Neumann algebras. The basic concepts, however, make sense for arbitrary dual Banach algebras. But is most commonly associated with Connes, see [@A.Connes]. For this reason, this notion of amenability is called Connes-amenability (the origin of this name seems to be Helemskii, see [@A.Ya]).
Let $A$ be a Banach algebra. A Banach $A$-bimodule $X$ is called dual if there is a closed submodule $X_*$ of $X^*$ such that $X = (X_*)^*$ ($X_*$ is called the predual of $X$). A Banach algebra $A$ is called dual if it is dual as a Banach $A$-bimodule.
Let $A$ be a dual Banach algebra. A dual Banach $A$-bimodule $X$ is called normal if, for every $x\in X$, the maps $$A\longrightarrow X,\hspace{0.2cm} a\mapsto\left\{
\begin{array}{ll}
a\cdot x & \\
x\cdot a&
\end{array}
\right.$$ are weak$^*$-[continuous]{} ($w^*$-continuous). The dual Banach algebra $A$ is called Connes-amenable if, for every dual Banach $A$-bimodule $X$, every $w^*$-continuous derivation $D:A\longrightarrow X$ is inner; or equivalently, ${H}_{w^*}^1(A,X) = \{0\}$ [@ru.1].
The second dual $A^{**}$ of Banach algebra $A$ endowed with the either Arens multiplications is a Banach algebra. The constructions of the two Arens multiplications in $A^{**}$ lead us to the definition of topological centers for $A^{**}$ with respect to both Arens multiplications. The topological centers of Banach algebras, module actions and applications of them were introduced and discussed in [@1; @12a; @13]. To state our results, we need to fix some notations and recall some definitions.
Let $X,Y,Z$ be normed spaces and $m:X\times Y\rightarrow Z$ be a bounded bilinear mapping. Arens in [@1] offers two natural extensions $m^{***}$ and $m^{t***t}$ of $m$ from $X^{**}\times Y^{**}$ into $Z^{**}$, for more information see [@eshaghi; @12a; @13]
The mapping $m^{***}$ is the unique extension of $m$ such that $x^{\prime\prime}\rightarrow m^{***}(x^{\prime\prime},y^{\prime\prime})$ from $X^{**}$ into $Z^{**}$ is $weak^*-weak^*$ continuous for every $y^{\prime\prime}\in Y^{**}$, but the mapping $y^{\prime\prime}\rightarrow m^{***}(x^{\prime\prime},y^{\prime\prime})$ is not in general $weak^*-weak^*$ continuous from $Y^{**}$ into $Z^{**}$ unless $x^{\prime\prime}\in X$. Hence the first topological center of $m$ may be defined as follows $$Z_1(m)=\{x^{\prime\prime}\in X^{**}:~~y^{\prime\prime}\rightarrow m^{***}(x^{\prime\prime},y^{\prime\prime})~~\text{is weak}^*\text{-weak}^*~~\text{continuous}\}.$$
Now, let $m^t:Y\times X\rightarrow Z$ be the transpose of $m$ defined by $m^t(y,x)=m(x,y)$ for every $x\in X$ and $y\in Y$. Then $m^t$ is a continuous bilinear map from $Y\times X$ to $Z$, and so it may be extended as above to $m^{t***}:Y^{**}\times X^{**}\rightarrow Z^{**}$. The mapping $m^{t***t}:X^{**}\times Y^{**}\rightarrow Z^{**}$ in general is not equal to $m^{***}$, see [@1], if $m^{***}=m^{t***t}$, then $m$ is called Arens regular. The mapping $y^{\prime\prime}\rightarrow m^{t***t}(x^{\prime\prime},y^{\prime\prime})$ is $weak^*-weak^*$ continuous for every $x^{\prime\prime}\in X^{**}$, but the mapping $x^{\prime\prime}\rightarrow m^{t***t}(x^{\prime\prime},y^{\prime\prime})$ from $X^{**}$ into $Z^{**}$ is not in general weak$^*$-weak$^*$ continuous for every $y^{\prime\prime}\in Y^{**}$. So we define the second topological center of $m$ as $$Z_2(m)=\{y^{\prime\prime}\in Y^{**}:~~x^{\prime\prime}\rightarrow m^{t***t}(x^{\prime\prime},y^{\prime\prime})~~\text{is weak}^*\text{-weak}^*~~\text{continuous}\}.$$
It is clear that $m$ is Arens regular if and only if $Z_1(m)=X^{**}$ or $Z_2(m)=Y^{**}$. Arens regularity of $m$ is equivalent to the following $$\lim_i\lim_j\langle z^\prime,m(x_i,y_j)\rangle=\lim_j\lim_i\langle z^\prime,m(x_i,y_j)\rangle,$$ whenever both limits exist for all bounded sequences $(x_i)_i\subseteq X$ , $(y_i)_i\subseteq Y$ and $z^\prime\in Z^*$, see [@Dales].
The mapping $m$ is left strongly Arens irregular if $Z_1(m)=X$ and $m$ is right strongly Arens irregular if $Z_2(m)=Y$. The first Arens product is defined as follows in three steps. For $a,b$ in $A$, $f$ in $A^{*}$ and $m,n$ in $A^{**}$, the elements $f. a$, $m. f$ of $A^{*}$ and $m. n$ of $A^{**}$ are defined as follows: $$\langle f. a,b\rangle=\langle f,ab\rangle,\quad\langle m. f,a\rangle=\langle m,f. a\rangle,\quad\langle m. n,f\rangle=\langle m,n. f\rangle.$$
The second Arens product is defined as follows. For $a,b$ in $A$, $f$ in $A^{*}$ and $m,n$ in $A^{**}$, the elements $ a\Diamond f$ , $f\Diamond m$ of $A^{*}$ and $m\Diamond n$ of $A^{**}$ are defined by the equalities $$\langle a\Diamond f,b\rangle=\langle f,ba\rangle,\quad\langle f\Diamond m,a\rangle=\langle m,a\Diamond f\rangle,\quad\langle m\Diamond n,f\rangle=\langle n,f\Diamond m\rangle.$$
The Arens regularity of a normed algebra $A$ is defined to be the Arens regularity of its algebra multiplication when considered as a bilinear mapping $m:A\times A\rightarrow A$. Let $B$ be a Banach $A$-bimodule, and let $$\pi_\ell:~A\times B\longrightarrow B\quad\text{and}\quad\pi_r:~B\times A\longrightarrow B,$$ be the right and left module actions of $A$ on $B$. By above notation, the transpose of $\pi_r$ denoted by $\pi_r^t:A\times B\rightarrow B$. Then $$\pi_\ell^*:B^*\times~A\longrightarrow B^*\quad\text{and}\quad\pi_r^{t*t}: A\times B^*\longrightarrow B^*.$$ Thus $B^*$ is a left Banach $A$-module and a right Banach $A$-module with respect to the module actions $\pi_r^{t*t}$ and $\pi_\ell^*,$ respectively. The second dual $B^{**}$ is a Banach $A^{**}$-bimodule with the following module actions $$\pi_\ell^{***}:~A^{**}\times B^{**}\longrightarrow B^{**}\quad\text{and}\quad\pi_r^{***}:~B^{**}\times A^{**}\longrightarrow B^{**},$$ where $A^{**}$ is considered as a Banach algebra with respect to the first Arens product. Similarly, $B^{**}$ is a Banach $A^{**}$-bimodule with the module actions $$\pi_\ell^{t***t}:~A^{**}\times B^{**}\longrightarrow B^{**}\quad\text{and}\quad\pi_r^{t***t}:~B^{**}\times A^{**}\longrightarrow B^{**},$$ where $A^{**}$ is considered as a Banach algebra with respect to the second Arens product. In this way we write $Z(\pi_{\ell})=Z_{B^{**}}(A^{**})$ and $Z(\pi_{r})=Z_{A^{**}}(B^{**})$.
Let $B$ be a Banach $A$-bimodule. Then we say that $B$ factors on the left (right) with respect to $A$, if $B=BA~(B=AB)$. Thus $B$ factors on both sides, if $B=BA=AB$.
Weak$^*$-weak$^*$ continuous derivations
=========================================
Let $B$ be a Banach $A$-bimodule. In this section, we study the cohomological properties of Banach algebra $A$ whenever every derivation in $Z^1(A^{**},B^*)$ is weak$^*$-weak$^*$ continuous.
\[3.1a\] Let $B$ be a Banach $A$-bimodule and let every derivation $D:A^{**}\longrightarrow B^*$ is weak$^*$-weak$^*$ continuous. If $Z^\ell_{B^{**}}(A^{**})=A^{**}$ and $H^1(A,B^*)=\{0\}$, then $H^1(A^{**},B^*)=\{0\}$.
Let $D:A^{**}\longrightarrow B^*$ be a derivation. Then $D\mid_A:A\rightarrow B^*$ is a derivation. Since $H^1(A,B^*)=\{0\}$, there exists $b^\prime\in B^*$ such that $D\mid_A=\delta_{b^\prime}$. Suppose that $a^{\prime\prime}\in A^{**}$ and $(a_\alpha^{})_\alpha\subseteq A^{}$ such that $a_\alpha^{} \stackrel{w^*} {\longrightarrow} a^{\prime\prime}$ in $A^{**}$. Then
$$\begin{aligned}
D(a^{\prime\prime})&=w^*-\lim_\alpha D\mid_A(a_\alpha)\\
&=w^*-\lim_\alpha\delta_{b^\prime}(a_\alpha)\\
&=w^*-\lim_\alpha(a_\alpha {b^\prime}-{b^\prime}a_\alpha)\\
&=a^{\prime\prime}{b^\prime}-{b^\prime}a^{\prime\prime}.
\end{aligned}$$
We now show that $b^\prime a^{\prime\prime}\in B^*$. Assume that $(b^{\prime\prime}_{\beta})_{\beta}\in B^{**}$ such that $b^{\prime\prime}=w^*-\lim_{\beta}b^{\prime\prime}_{\beta}$. Since $Z^\ell_{B^{**}}(A^{**})=A^{**}$, we have $$\langle b^\prime a^{\prime\prime},b^{\prime\prime}_{\beta}\rangle=\langle a^{\prime\prime}.b^{\prime\prime}_{\beta},b^\prime\rangle\rightarrow
\langle a^{\prime\prime}.b^{\prime\prime},b^\prime\rangle=\langle b^\prime a^{\prime\prime},b^{\prime\prime}\rangle.$$
Thus, $b^\prime a^{\prime\prime}\in (B^{**},weak^*)^*=B^*$, and so $H^1(A^{**},B^*)=\{0\}$.
\[c1\] Let $A$ be an Arens regular Banach algebra and let every derivation $D:A^{**}\rightarrow A^*$ is weak$^*$-weak$^*$ continuous. If $A$ is weakly amenable, then $H^1(A^{**},A^*)=\{0\}$.
By the following result, we show that weak amenability of the Banach algebra $A$ is essential in vanishing of $H^1(A^{**},A^*)$.
\[p1\] Let $A$ be a Banach algebra such that is an ideal in $A^{**}$. If $A$ is not weakly amenable, then $H^1(A^{**},A^*)\neq\{0\}$.
Let $d:A\longrightarrow A^*$ be a derivation and $\pi:A^{**}\longrightarrow A$ be a bounded homomorphism. Now; define $D:=d\circ\pi:A^{**}\longrightarrow A^*$. Clearly, $D$ is a bounded derivation which it is not inner. This shows that $H^1(A^{**},A^*)\neq\{0\}$.
\[ex1\]
- Let $K$ be a compact metric space, $d$ be a metric on $K$ and $\alpha\in(0,1]$. The Lipchitz algebra $\mathrm{Lip}_\alpha K$ is the space of complex-valued functions $f$ on $K$ such that $$p_\alpha(f)=\sup\left\{\frac{|f(x)-f(y)|}{d(x,y)^\alpha}:x,y\in K, x\neq y\right\}$$ is finite. A subspace of $\mathrm{Lip}_\alpha K$ that contains $f\in\mathrm{Lip}_\alpha K$ such that $$\frac{|f(x)-f(y)|}{d(x,y)^\alpha}\to 0\quad\text{as}\quad d(x,y)\to0$$ is denoted by $\mathrm{lip}_\alpha K$. Let $\alpha\in(0,\frac{1}{2})$. Then by [@3Dales Theorem 4.4.34] or [@BCD Theorem 3.8], $\mathrm{lip}_\alpha K$ is Arens regular and by [@BCD Theorem 3.10] it is weakly amenable. Then by Corollary \[c1\], $H^1\left(\left(\mathrm{lip}_\alpha K\right)^{**},\left(\mathrm{lip}_\alpha K\right)^*\right)=\{0\}$.
- Let $\omega$ be a weight sequence on $\mathbb{Z}$ such that $$\sup\left\{\frac{\omega(m+n)}{\omega(m)\omega(n)}\left(\frac{1+|n|}{1+|m+n|}\right):m,n\in\mathbb{Z}\right\}$$ is finite. The Beurling algebra $\ell^1(\mathbb{Z},\omega)$ is not weakly amenable [@BCD Theorem 2.3]. Then by Proposition \[p1\], we have
$H^1\left(\ell^1(\mathbb{Z},\omega)^{**},\ell^\infty(\mathbb{Z},\omega)\right)\neq\{0\}$.
Let $B$ be a dual Banach algebra, with predual $X$ and suppose that $$X^\perp=\{x^{\prime\prime\prime}:~x^{\prime\prime\prime}\mid_X=0~~\text{where}~~x^{\prime\prime\prime}\in X^{***}\}=\{b^{\prime\prime}:~b^{\prime\prime}\mid_X=0~~\text{where}~~b^{\prime\prime}\in B^{**}\}.$$
Then the canonical projection $P:X^{***}\longrightarrow X^*$ gives a continuous linear map $P:B^{**}\longrightarrow B$. Thus, we can write the following equality $$B^{**}=X^{***}=X^*\oplus \ker P=B\oplus X^\perp ,$$ as a direct sum of Banach $A$-bimodules.
\[3.3aa\] Let $B$ be a Banach $A$-bimodule such that every derivation from $A^{**}$ into $ B$ is weak$^*$-weak continuous and $A^{**}B,BA^{**}\subseteq B$.
- If $H^1(A,B)=0$, then $H^1(A^{**},B)=\{0\}$.
- Suppose that $A$ has a left bounded approximate identity (=LBAI), $B$ has a predual $X$ and $AB^*,~B^*A\subseteq X$. If $H^1(A,B)=0$, then $H^1(A^{**},B^{**})=\{0\}$.
\(i) Proof is similar to the proof of Theorem \[3.1a\].
\(ii) Set $B^{**}=B\oplus X^\perp .$ Then we have $$H^1(A^{**},B^{**})=H^1(A^{**},B)\oplus H^1(A^{**},X^\perp).$$
Since $H^1(A,B)=\{0\}$, by (i), $H^1(A^{**},B)=\{0\}$. Now let $\widetilde{D}\in Z^1(A^{**},X^\perp)$ and we take $D=\widetilde{D}\mid_A$. It is clear that $D\in Z^1(A^{**},X^\perp)$. Assume that $a^{\prime\prime}, x^{\prime\prime}\in A^{**}$ and $(a_\alpha)_\alpha, (x_\beta)_\beta\subseteq A$ such that $a_\alpha^{} \stackrel{w^*} {\rightarrow} a^{\prime\prime}$ and $x_\beta \stackrel{w^*} {\rightarrow} x^{\prime\prime}$ on $A^{**}$. Since $AB^*,~B^*A\subseteq X$, for every $b^ \prime \in B^*$, by using the weak$^*$-weak continuity of $\widetilde{D}$, we have $$\begin{aligned}
\langle \widetilde{D}(a^{\prime\prime}\Diamond x^{\prime\prime}), b^\prime\rangle&=\lim_\beta\lim_\alpha \langle D(a_\alpha x_\beta), b^\prime\rangle\\
&=\lim_\beta\lim_\alpha \langle (D(a_\alpha) x_\beta +a_\alpha D(x_\beta)),b^\prime\rangle\\
&=\lim_\beta\lim_\alpha \langle D(a_\alpha) x_\beta ,b^\prime\rangle+\lim_\beta\lim_\alpha \langle a_\alpha D(x_\beta),b^\prime\rangle\\
&=\lim_\beta\lim_\alpha \langle D(a_\alpha), x_\beta b^\prime\rangle+\lim_\beta\lim_\alpha \langle D(x_\beta)),b^\prime a_\alpha \rangle\\
&=0.
\end{aligned}$$
Since $A$ has a LBAI, $A^{**}$ has a left unit $e^{\prime\prime}$ with respect to the second Arens product [@3Dales Proposition 2.9.16]. Then $D(x^{\prime\prime})= D(e^{\prime\prime}\Diamond x^{\prime\prime})=0$, and so $D=0$.
- Assume that $G$ is a compact group. Then we know that $L^1(G)$ is $M(G)$-bimodule and $L^1(G)$ is an ideal in the second dual of $M(G)$, $M(G)^{**}$. By [@12 Corollary 1.2], we have $H^1(L^1(G),M(G))=\{0\}$. Then by Theorem \[3.3aa\], every weak$^*$-weak continuous derivation from $L^1(G)^{**}$ into $M(G)$ is inner.
- We know that $c_0$ is a C$^*$-algebra and every C$^*$-algebra is weakly amenable, so $c_0$ is weakly amenable. Then by Theorem \[3.3aa\], every weak$^*$- weak continuous derivation from $\ell^\infty$ into $\ell^1$ is inner.
\[t2\] Let $B$ be a Banach $A$-bimodule and $A$ has a $LBAI$. Suppose that $AB^{**},~B^{**}A\subseteq B$ and every derivation from $A^{**}$ into $B^*$ is [weak]{}$^*$-[weak]{}$^*$ continuous. If $H^1(A,B^*)=\{0\}$, then $H^1(A^{**},B^{***})=\{0\}$.
Take $B^{***}=B^*\oplus B^\perp$, where $B^\perp=\{b^{\prime\prime\prime}\in B^{***}:~b^{\prime\prime\prime}\mid_B=0\}$. Then we have $$H^1(A^{**},B^{***})=H^1(A^{**},B^*)\oplus H^1(A^{**},B^\perp).$$
Since $H^1(A,B^*)=\{0\}$, similar to Theorem \[3.3aa\](i), we have $H^1(A^{**},B^*)=\{0\}$. It suffices to show that $H^1(A^{**},B^\perp)=0$. Let $(e_{\alpha})_{\alpha}\subseteq A$ be a LBAI for $A$ such that $e_{\alpha} \stackrel{w^*} {\rightarrow}e^{\prime\prime}$ in $A^{**}$ where $e^{\prime\prime}$ is a left unit for $A^{**}$ with respect to the second Arens product. Let $a^{\prime\prime}\in A^{**}$ and suppose that $(a_{\beta})_{\beta}\subseteq A$ such that $a_{\beta} \stackrel{w^*} {\rightarrow}a^{\prime\prime}$ in $A^{**}$. Let $D\in Z^1(A^{**},B^\perp)$. Then for every $b^{\prime\prime}\in B^{**}$, by [weak]{}$^*$-[weak]{}$^*$ continuity of $D$, we have $$\begin{aligned}
\langle D(a^{\prime\prime}), b^{\prime\prime}\rangle &= \langle D(e^{\prime\prime}\Diamond a^{\prime\prime}), b^{\prime\prime}\rangle\\
&=\lim_\beta\lim_\alpha \langle (D(e_\alpha a_\beta),b^{\prime\prime}\rangle\\
&=\lim_\beta\lim_\alpha \langle (D(e_\alpha) a_\beta +e_\alpha D(a_\beta)),b^{\prime\prime}\rangle\\
&=\lim_\beta\lim_\alpha \langle D(e_\alpha) a_\beta ,b^{\prime\prime}\rangle+\lim_\beta\lim_\alpha \langle e_\alpha D(a_\beta),b^{\prime\prime}\rangle\\
&=\lim_\beta\lim_\alpha \langle D(e_\alpha), a_\beta b^{\prime\prime}\rangle+\lim_\beta\lim_\alpha \langle D(a_\beta),b^{\prime\prime}e_\alpha \rangle\\
&=0.
\end{aligned}$$
It follows that $D=0$, and so the result holds.
It is known that neither the weak amenability of $A$ implies that of $A^{**}$, nor the weak amenability of $A^{**}$ implies that of $A$. The question “*when the weak amenability of $A^{**}$ implies that of $A$?*” is investigated in many works; see [@BHJ; @Dales; @EF; @GL; @f.ghahramani] for more details. We now by Theorem \[t2\] consider the converse of the above question, i.e., “*under which conditions the weak amenability of $A$ implies that of $A^{**}$?*”, as follows:
\[c2\] Assume that $A$ is a Banach algebra with $LBAI$ such that it is two-sided ideal in $A^{**}$ and every derivation $D:A^{**}\rightarrow A^{***}$ is weak$^*$- weak$^*$ continuous. If $A$ is weakly amenable, then $A^{**}$ is weakly amenable.
Assume that $G$ is a locally compact group. We know that $L^1(G)$ is weakly amenable Banach algebra, see [@Johnson]. Then by Corollary \[c2\], every weak$^*$- weak$^*$ continuous derivation from $L^1(G)^{**}$ into $L^1(G)^{***}$ is inner.
Let $A$ be an amenable and Arens regular Banach algebra. If for any normal Banach $A$-bimodule $B$ with predual $X$, we have $AB^*,~B^*A\subseteq X$, then $H^1_{w^*}(A^{**},B^{**})=\{0\}$.
If the Banach algebra $A$ is amenable and Arens regular, then $A^{**}$ is Connes-amenable and the converse holds whenever $A$ is an ideal in $A^{**}$, too [@ru.1 Theorem 4.4.8]. Thus $H^1_{w^*}(A^{**},B)=\{0\}$ and by the argument before Theorem \[3.3aa\], we have $B^{**}=B\oplus X^\bot$. These imply that $H^1_{w^*}(A^{**},B^{**})=H^1_{w^*}(A^{**},X^\bot)$. It is known that every amenable Banach algebra possesses a BAI, so by a similar argument in the proof of Theorem \[3.3aa\](ii), we obtain that $H^1_{w^*}(A^{**},X^\bot)=\{0\}$.
\[p2\] Suppose that $A$ is an amenable Banach algebra. If for every Banach $A$-bimodule $B$, we have $AB^{**},~B^{**}A\subseteq B$, then $$\begin{aligned}
H^1_{w^*}(A^{**},B^{***})=\{0\}.
\end{aligned}$$
By applying a similar argument in the proof of Theorem \[3.3aa\](ii), we obtain the desire.
\[c3\] Assume that $A$ is a weakly amenable Banach algebra with a LBAI. If $A$ is an ideal in $A^{**}$, it follows that $$H^1_{w^*}(A^{**},A^{***})=\{0\}.$$
Assume that $G$ is a compact group. It is known that $L^1(G)$ has a BAI and is a two-sided ideal in $L^1(G)^{**}$. We know that $L^1(G)$ is weakly amenable, hence by Corollary \[c3\], $$H^1_{w^*}(L^\infty(G)^*,L^\infty(G)^{**})=\{0\}.$$
\[3.1\] Let $A$ be a Banach algebra such that $A$ is an ideal in $A^{**}$ and $A^*$ factors. Then $A$ is amenable if and only if $A^{**}$ is Connes-amenable.
By [@BHJ Corollary 2.8](i), $A$ is Arens regular. Then by [@run.1 Theorem 4.4], the proof completes.
A Banach space $A$ is called weakly sequentially complete if every weakly Cauchy sequence in $A$ has a weak limit in $A$.
Let $A$ be an Arens regular dual Banach algebra such that $A^{*}$ is weakly sequentially complete (WSC). If $H^1_{w^*}(A^{**},A^{***})=\{0\}$, then $H^1_{w^{*}}(A,A^{*})=\{0\}$.
Let $D:A\longrightarrow A^{*}$ be a $w^*$-continuous derivation. Since $A^{*}$ is WSC, every derivation $D:A\longrightarrow A^{*}$ is weakly compact. Then by [@conway Theorem 6.5.5], we have $D''(A^{**})\subseteq A^{*}$ and hence, by Arens regularity of $A$, $A^{*}$ is an $A^{**}$-submodule of $(A^{**})^{*}$ and $D''(A^{**}).A^{**}\subseteq A^{*}.A^{**}\subseteq A^{*}$. Then by [@Dales Theorem 7.1], $D'':A^{**}\longrightarrow A^{***}$ is a $w^{*}$-continuous derivation. Thus, there exists $a^{\prime\prime\prime}\in A^{***}$ such that $D''(F)=F.a^{\prime\prime\prime}-a^{\prime\prime\prime}.F$, for each $F\in A^{**}$. Now, let $E:A\longrightarrow A^{**}$ be the canonical map and set $f=E^{*}(a^{\prime\prime\prime})$, then $D(a)=a.f-f.a$, for all $a\in A$. This means that $D$ is an inner $w^*$-continuous derivation. Thus the proof follows.
In the following, we extend the [@BHJ Corollary 2.8](i) to the general case as follows:
\[3.3a\] If $A^{(2n)}$ is a two-sided ideal in $A^{(2n+2)}$ and $A^{(2n+1)}$ factors, then $A^{(2n)}$ is Arens regular, where $n\in\mathbb{N}\cup\{0\}$.
\[3.5\] Let $A$ be a Banach algebra such that $A^{**}$ is an ideal in $A^{****}$ and $A^{***}$ factors. If $A$ is weakly amenable, then $H^1_{w^{*}}(A^{**},A^{***})=\{0\}$.
Lemma \[3.3a\] implies that $A^{**}$ is Arens regular. Now, let $D:A^{**}\longrightarrow A^{***}$ be a weak$^*$- weak$^*$-continuous derivation. First, we prove that $A^{***}$ is a normal Banach $A^{**}$-bimodule. Let $(a^{\prime\prime}_{\alpha})_{\alpha}$ be a net in $A^{**}$ and $a^{\prime\prime\prime}\in A^{***}$. Then, by Arens regularity of $A^{**}$, for every $b^{\prime\prime}\in A^{**}$ we have $$\begin{aligned}
\langle(w^*-\lim_{\alpha}a^{\prime\prime}_{\alpha}).a^{\prime\prime\prime},b^{\prime\prime}\rangle&=\langle a^{\prime\prime\prime},b^{\prime\prime} .(w^*-\lim_{\alpha}a^{\prime\prime}_{\alpha})\rangle\\&=\lim_{\alpha}\langle a^{\prime\prime\prime},b^{\prime\prime}.a^{\prime\prime}_{\alpha}\rangle\\&=\lim_{\alpha}\langle a^{\prime\prime}_{\alpha}.a^{\prime\prime\prime},b^{\prime\prime}\rangle\\&=\langle w^*-\lim_{\alpha}(a^{\prime\prime}_{\alpha}.a^{\prime\prime\prime}),b^{\prime\prime}\rangle.
\end{aligned}$$
Moreover, $$\begin{aligned}
\langle a^{\prime\prime\prime}.(w^*-\lim_{\alpha}a^{\prime\prime}_{\alpha}),b^{\prime\prime}\rangle&=\langle a^{\prime\prime\prime},w^*-\lim_{\alpha}a^{\prime\prime}_{\alpha}.b^{\prime\prime}\rangle\\&=\lim_{\alpha}\langle a^{\prime\prime\prime},a^{\prime\prime}_{\alpha},b^{\prime\prime}\rangle\\&=\lim_{\alpha}\langle a^{\prime\prime\prime}.a^{\prime\prime}_{\alpha},b^{\prime\prime}\rangle\\&=\langle w^*-\lim_{\alpha}(a^{\prime\prime\prime}.a^{\prime\prime}_{\alpha}),b^{\prime\prime}\rangle.
\end{aligned}$$
Hence, the mapping $a^{\prime\prime}\mapsto a^{\prime\prime}.a^{\prime\prime\prime}$ and $a^{\prime\prime}\mapsto a^{\prime\prime\prime}.a^{\prime\prime}$ are weak$^*$-weak$^*$-continuous from $A^{**}$ into $ A^{***}$. Thus, $A^{***}$ is a normal Banach $A^{**}$-bimodule. For each $a\in A$, we define $\bar{D}:A\longrightarrow A^*$ by $$\bar{D}(a)=D(\widehat{a})\mid _A,$$ where $\widehat{a}\in A^{**}$ with $\widehat{a}(a^\prime )=a^\prime (a)$, for all $a\in A$. As the following equalities $\bar{D}$ is a continuous derivation from $A$ into $A^*$.
$\bar{D}(ab)=D(\widehat{ab})=D(\widehat{a}.\widehat{b})=a.D(\widehat{b})+D(\widehat{a}).b=a.\bar{D}(b)+\bar{D}(a).b$,
where $a,b\in A$. By weak amenability of $A$, we have $\bar{D}$ is inner. Then there exist $a^{\prime\prime\prime}\in A^{***}$ such that
$D(\widehat a)=\bar{D}(a)=a.a^{\prime\prime\prime}\mid_A-
a^{\prime\prime\prime}\mid_A.a=\widehat{a}.a^{\prime\prime\prime}\mid_A-a^{\prime\prime\prime}\mid_A.\widehat{a}.$
We consider the canonical mapping $E:A^*\longrightarrow A^{***}$. Then there exists $b^{\prime\prime\prime}\in A^{***}$ such that $E(a^{\prime\prime\prime}\mid_A)=b^{\prime\prime\prime}$. So
$D(\widehat{a})=\widehat{a}.b^{\prime\prime\prime}-b^{\prime\prime\prime}.\widehat{a}.$
Then $D$ is inner. It follows that $H^1_{w^{*}}(A^{**},A^{***})=\{0\}$.
Let $A^{(2n+2)}$ be a two sided ideal in $A^{(2n+4)}$ and $A^{(2n+3)}$ factors. If $A^{(2n)}$ is weakly amenable, then $H^1_{w^*}(A^{(2n+2)},A^{(2n+3)})=\{0\}$.
Apply Lemma $\ref{3.3a}$ and Theorem $\ref{3.5}$.
Weak$^*$-continuous derivations from dual Banach algebras into their ideals are studied in [@EJ].
If $M$ is subspace of $A$ and $N$ is subspace of $A^*$, then $M^{\perp}=\{x^*\in X^*: \langle x^*,x\rangle=0,\quad \forall x\in M\}$ and $^{\perp}N=\{x\in A: \langle x^*,x\rangle=0,\quad\forall x^{*}\in N\}$. If $A$ is a dual Banach algebra and $I$ is $w^*-$closed ideal of $A$, then $I$ is dual with predual $I_*=\frac{A_{*}}{^{\perp}I}$ that $(I_{*})^{*}=(\frac{A_{*}}{^{\perp}{I}})^*=(^{\perp}I)^{\perp}=I$ and $I^*=\frac{A^*}{I^{\perp}}$, see [@conway].
\[pro1\] Let $A$ be a dual Banach algebra and $I$ be an arbitrary $w^*$-closed ideal of $A$ such that $H^1(A,I^{**})=\{0\}$. Then $H^1_{w^*}(A,I)=\{0\}$.
Let $D\in Z^1_{w^*}(A,I)$ and $E:I\rightarrow I^{**}$ be the natural embedding. Then $E\circ D:A\rightarrow I^{**}$ is a bounded derivation. Since $H^1(A,I^{**})=\{0\}$, there exists $a^{**}\in I^{**}$ such that $E\circ D=\delta_{a^{**}}$. Consider the decomposition $I^{**}=I\oplus I_{*}^{\perp}$ as an $A$-bimodule. If $P:I^{**}\rightarrow I$ is a projection, we have $D=\delta_{p(a^{**})}$. Then $H^1_{w^*}(A,I)=\{0\}$.
A Banach algebra $A$ is without of order if for any $a,b\in A$, $ab=0$ implies that $a=0$ or $b=0$. Semisimple and unital Banach algebras are without of order Banach algebras. Now by Proposition \[pro1\] and [@EJ Theorem 3.1], we have the following result.
\[cc1\] Let $A$ be a dual Banach algebra and $I$ be a closed two-sided ideal in $A$ such that $I$ is without order. If $H^1(A,I^{**})=\{0\}$, then $H^1_{w^*}(I,I)=\{0\}$.
$\mathrm{(i)}$ Let $G$ be a locally compact group. A linear subspace $S^1(G)$ of $L^1(G)$ is said to be a Segal algebra, if it satisfies the following conditions:
- $S^1(G)$ is a dense in $L^1(G)$;
- If $f\in S^1(G)$, then $L_xf\in S^1(G)$, i.e. $S^1(G)$ is left translation invariant;
- $S^1(G)$ is a Banach space under some norm $\|\cdot\|_S$ and $\|L_xf\|_s=\|f\|_s$, for all $f\in S^1(G)$ and $x\in G$;
- $x\mapsto L_xf$ from $G$ into $S^1(G)$ is continuous.
For more details about Segal algebras, see [@ri1; @ri]. Now, let $G$ be an abelian locally compact group. Then $H^1(L^1(G),S^1(G)^{**})=\{0\}$. Then by Proposition \[pro1\] and Corollary \[cc1\], we have $H_{w^*}^1(L^1(G),S^1(G))=\{0\}$ and $H_{w^*}^1(S^1(G),S^1(G))=\{0\}$.
$\mathrm{(ii)}$ Let $\Lambda$ be a non-empty, totally ordered set, and regard it as a semigroup by defining the product of two elements to be their maximum. The resulting semigroup, which we denote by $\Lambda_\vee$, is a semilattice. We may then form the $\ell^1$-convolution algebra $\ell^1(\Lambda_\vee)$. For every $t\in\Lambda_\vee$ we denote the point mass concentrated at $t$ by $e_t$. The definition of multiplication in $\ell^1(\Lambda_\vee)$ ensures that $e_se_t = e_{\max(s,t)}$ for all $s$ and $t$.
The semilattice $\Lambda_\vee$, is a commutative semigroup in which every element is idempotent. If we denote the set of idempotent elements of $\Lambda_\vee$ by $E(\Lambda_\vee)$, then $E(\Lambda_\vee)=\Lambda_\vee$. The $\ell^1$-convolution algebras of semilattices provide interesting examples of commutative Banach algebras. By [@J Proposition 3.3], $H^1(\ell^1(\Lambda_\vee),I^{**})=\{0\}$, for any closed two-sided $I$ of $\ell^1(\Lambda_\vee)$. Then by Corollary \[cc1\], $H^1_{w^*}(I,I)=\{0\}$, for any closed two-sided $I$ of $\ell^1(\Lambda_\vee)$.
$\mathrm{(iii)}$ Let $K$ be an infinite compact metric space, $\alpha\in(0,1)$ and $\mathrm{lip}_\alpha K$ be the small Lipchitz algebra (see Example \[ex1\]). By [@J Proposition 3.4], $H^1(\mathrm{lip}_\alpha K,I^{**})=\{0\}$, for any closed two-sided $I$ of $\mathrm{lip}_\alpha K$. Then by Corollary \[cc1\], $H^1_{w^*}(I,I)=\{0\}$, for any closed two-sided $I$ of $\mathrm{lip}_\alpha K$.
Representations of derivations and Arens regularity
====================================================
Let $A$ be a Banach algebra and $B$ be a Banach $A$-bimodule with the module action “$\bullet$”. Then for every $b\in B$, we define $$L_{b}(a)=b\bullet a\quad\text{and}\quad R_{b}(a)=a\bullet b,$$ for every $a\in A$. These are the operation of left and right multiplication by $b$ on $A$. In the following by using the super-amenability of Banach algebra $A$, we give a representation for $Z^1(A,C)$, where $C$ is a Banach $A$-bimodule.
For a Banach $A$-bimodule $B$ and for a derivation $D:A\rightarrow B^{*}$, we show that the left module action $\pi_\ell:A\times B\rightarrow B$ is Arens regular whenever $D^{\prime\prime}:A^{**}\rightarrow B^{***}$ is a derivation and $B^*\subseteq D^{\prime\prime}(A^{**})$. On the other hand, if $A$ is a left strongly Arens irregular and $A^{**}$ is amenable Banach algebra with respect to the first Arens product, then $A$ is unital. Moreover, if $A$ is a dual Banach algebra, it follows that $A$ is reflexive.
\[t3\] Assume that $A$ is an amenable Banach algebra. Then there are Banach $A$-bimodules $C$, $D$ and elements $\mathfrak{a}, \mathfrak{b}\in A^{**}$ such that $$Z^1(A,C^*)=\{R_{D^{\prime\prime}(\mathfrak{a})}:~D\in Z^1(A,C^*)\}=\{L_{D^{\prime\prime}(\mathfrak{b})}:~D\in Z^1(A,D^*)\}.$$
Suppose that $B$ is a Banach $A$-bimodule with a module action $\bullet$. Every amenable Banach algebra has a BAI [@ru.1 Proposition 2.2.1], so $A$ has a BAI such as $(e_{\alpha})_{\alpha}$. Then by Cohen factorization Theorem we have $B\bullet A=B=A\bullet B$, i.e., for every $b\in B$, there are $y,z\in B$ and $a,t\in A$ such that $y\bullet a=b=t\bullet z$. Then we have $$\label{eqeq1}
\lim_\alpha b\bullet e_{\alpha}=\lim_\alpha (y\bullet a)\bullet e_{\alpha}=\lim_\alpha y\bullet(a e_{\alpha})=y\bullet a=b$$ and $$\label{eqeq11}
\lim_\alpha e_{\alpha}\bullet b=\lim_\alpha e_{\alpha}\bullet (t\bullet z)=\lim_\alpha (te_{\alpha})\bullet z=t\bullet z=b.$$
It follows that $B$ has a BAI as $(e_{\alpha})_{\alpha}\subseteq A$. Let $e^{\prime\prime}$ and $f''$ be the right and left unit for $A^{**}$, respectively such that $e_{\alpha} \stackrel{w^*} {\rightarrow}e^{\prime\prime}$ and $e_{\alpha} \stackrel{w^*} {\rightarrow}f^{\prime\prime}$ in $A^{**}$.
Take $C=B$ and define a module action “$\cdot$” as $a\cdot x=0$ and $x\cdot a=x\bullet a$, for all $a\in A$ and $x\in C$. Clearly, $(C,\cdot)$ is a Banach $A$-bimodule. Suppose that $D\in Z^1(A,C^*)$. Then there is an element $c\in C^*$ such that $D=\delta_c$. Then for every $a\in A$, we have $$D(a)=\delta_c (a)=a\cdot c-c\cdot a=a\bullet c.$$
From and module actions of $C$, for any $x\in C$ and $x'\in C^*$, we have $$\label{eqeq2}
\lim_\alpha\langle x,x'\cdot e_\alpha\rangle= \lim_\alpha\langle e_\alpha\cdot x,x' \rangle=0$$ and $$\label{eqeq3}
\lim_\alpha\langle x,e_\alpha\cdot x' \rangle= \lim_\alpha\langle x\cdot e_\alpha,x' \rangle=\lim_\alpha\langle x\bullet e_\alpha,x' \rangle=\langle x, x' \rangle.$$
It follows that $e_{\alpha}\cdot x' \stackrel{w^*}{\rightarrow}x'$ in $C^*$. Since $D^{\prime\prime}$ is a weak$^*$-to-weak$^*$ continuous linear operator, we have $$\begin{aligned}
D^{\prime\prime}(e^{\prime\prime})&=D^{\prime\prime}(w^*-\lim_\alpha e_{\alpha})=w^*-\lim_\alpha D^{\prime\prime}(e_{\alpha})=w^*-\lim_\alpha D(e_{\alpha})\\
&=w^*-\lim_\alpha (e_{\alpha}x')=x'.
\end{aligned}$$
Thus we conclude that $D(a)=a\cdot D^{\prime\prime}(e^{\prime\prime})=a\bullet D^{\prime\prime}(e^{\prime\prime})$ for all $a\in A$. It follows that $D=R_{D^{\prime\prime}(e^{\prime\prime})}.$ On the other hand, since for every derivation $D\in Z^1(A,C^*)$, $R_{D^{\prime\prime}(e^{\prime\prime})}\in
Z^1(A,C^*)$, the result holds.
Now, again consider $B$ as a Banach $A$-bimodule with the module action “$\bullet$” and set $D=B$ with the module action $\triangleleft$ such that $a\triangleleft y=a\bullet y$ and $y\triangleleft a=0$, for all $a\in A$ and $y\in D$. By a similar argument that we have discussed above, and setting $\mathfrak{b}=f''$, the proof completes.
- Let $G$ be an amenable locally compact group. Then by Johnson Theorem $H^1(L^1(G),X^*)=\{0\}$, for every Banach $A$-bimodule $X$. Then by defining the similar module actions of $L^\infty(G)$ as a Banach $L^1(G)$-bimodule in the proof of Theorem \[t3\] and by this Theorem, we have $$\begin{aligned}
Z^1(L^1(G),L^\infty(G))&=\{R_{D(e'')}:D\in Z^1(L^1(G),L^\infty(G))\} \\
& =\{L_{D(f'')}:D\in Z^1(L^1(G),L^\infty(G))\},
\end{aligned}$$ where $e''$ and $f''$ are the left and right units of $L^1(G)^{**}$, indeed they $w^*$-accumulations of the BAI of $L^1(G)$.
- Let $G$ be locally compact group. Then by [@12 Corollary 1.2] $H^1(L^1(G),M(G))=\{0\}$. Then by applying the module actions defined in the proof of Theorem \[t3\], we can see $M(G)$ as a Banach $L^1(G)$-bimodule. Then by Theorem \[t3\], we have $$\begin{aligned}
Z^1(L^1(G),L^\infty(G))&=\{R_{D(e'')}:D\in Z^1(L^1(G),L^\infty(G))\} \\
& =\{L_{D(f'')}:D\in Z^1(L^1(G),L^\infty(G))\},
\end{aligned}$$ where $e''$ and $f''$ are the left and right units of $L^1(G)^{**}$.
\[t4\] Let $A$ be a Banach algebra, $B$ be a Banach $A$-bimodule and $D:A\longrightarrow B^*$ be a continuous derivation. If $D^{\prime\prime}:A^{**}\longrightarrow B^{***}$ is a derivation and $B^*\subseteq D^{\prime\prime}(A^{**})$, then $Z^\ell_{A^{**}}(B^{**})=B^{**}$.
Since $D^{\prime\prime}:A^{**}\rightarrow B^{***}$ is a derivation, by [@20 Theorem 4.2], $D^{\prime\prime}(A^{**})B^{**}\subseteq B^*$. Due to $B^*\subseteq D^{\prime\prime}(A^{**})$, we have $B^*B^{**}\subseteq B^*$. Let $(a_{\alpha}^{\prime\prime})_{\alpha}\subseteq A^{**}$ such that $a^{\prime\prime}_{\alpha} \stackrel{w^*} {\rightarrow}a^{\prime\prime}$ in $A^{**}$. Assume that $b^{\prime\prime}\in B^{**}$. Then for every $b^\prime\in B^*$, since $b^\prime b^{\prime\prime}\in B^*$, we have $$\langle b^{\prime\prime}.a_{\alpha}^{\prime\prime},b^\prime\rangle=\langle a_{\alpha}^{\prime\prime}b^\prime, b^{\prime\prime}\rangle\rightarrow
\langle a^{\prime\prime},b^\prime b^{\prime\prime}\rangle=\langle b^{\prime\prime}.a^{\prime\prime},b^\prime \rangle.$$
Thus $b^{\prime\prime}.a^{\prime\prime}_{\alpha} \stackrel{w^*} {\rightarrow}b^{\prime\prime}.a^{\prime\prime}$ is in $B^{**}$, and so $b^{\prime\prime}\in Z^\ell_{A^{**}}(B^{**})$.
\[c3.4\] Let $A$ be a Banach algebra and $D:A\longrightarrow A^*$ be a continuous derivation such that $A^*\subseteq D^{\prime\prime}(A^{**})$. If $D^{\prime\prime}:A^{**}\longrightarrow A^{***}$ is a derivation, then $A$ is Arens regular.
Let $G$ be an infinite locally compact group. Thus, $L^1(G)$ is not Arens regular. Then Corollary \[c3.4\] implies that there is no $D\in Z^1(L^1(G),L^1(G)^*)$ such that $L^1(G)^*\subseteq D^{\prime\prime}(L^1(G)^{**})$ and its second transpose $D''$ is a derivation.
\[lem3\] Let $B$ be a Banach left $A$-module and $B^{**}$ has a LBAI with respect to $A^{**}$. Then $B^{**}$ has a left unit with respect to $A^{**}$.
Assume that $(e^{\prime\prime}_{\alpha})_{\alpha}\subseteq A^{**}$ is a LBAI for $B^{**}$. By passing to a suitable subnet, we may suppose that there is an $e^{\prime\prime}\in A^{**}$ such that $e^{\prime\prime}_{\alpha} \stackrel{w^*} {\rightarrow}e^{\prime\prime}$ in $A^{**}$. Then for every $b^{\prime\prime}\in B^{**}$ and $b^\prime\in B^*$, we have $$\begin{aligned}
\langle \pi_\ell^{***}(e^{\prime\prime},b^{\prime\prime}),b^\prime\rangle &=
\langle e^{\prime\prime},\pi_\ell^{**}(b^{\prime\prime},b^\prime)\rangle=
\lim_\alpha \langle e_\alpha^{\prime\prime},\pi_\ell^{**}(b^{\prime\prime},b^\prime)\rangle\\
&=\lim_\alpha \langle \pi_\ell^{***}(e_\alpha^{\prime\prime},b^{\prime\prime}),b^\prime\rangle=
\langle b^{\prime\prime},b^\prime\rangle.
\end{aligned}$$
It follows that $\pi_\ell^{***}(e^{\prime\prime},b^{\prime\prime})=b^{\prime\prime}$.
\[t44\] Let $A$ be a left strongly Arens irregular and suppose that $A^{**}$ is an amenable Banach algebra. Then we have the following assertions.
- $A$ has an identity.
- If $A$ is a dual Banach algebra, then $A$ is reflexive.
\(i) Amenability of $A^{**}$ implies that it has a BAI. By using Lemma \[lem3\], $A^{**}$ has an identity say that $e^{\prime\prime}$. So, the mapping $x^{\prime\prime}\rightarrow e^{\prime\prime}. x^{\prime\prime}=x^{\prime\prime}$ is weak$^*$-to-weak$^*$ continuous from $A^{**}$ into $A^{**}$. It follows that $e^{\prime\prime}\in Z_1(A^{**})=A$. This means that $A$ has an identity.
\(ii) Assume that $E$ is a predual of $A$. Then we have $A^{**}=A\oplus E^\bot$. Since $A^{**}$ is amenable, by [@f.ghahramani Theorem 1.8] or [@G Theorem 2.3], $A$ is amenable, and so $E^\bot$ is amenable. Thus $E^\bot$ has a $BAI$ such as $(e^{\prime\prime}_\alpha)_\alpha\subseteq E^\bot$. Since $E^\bot$ is a closed and weak$^*$-closed subspace of $A^{**}$, without loss generality, there is $e^{\prime\prime}\in E^\bot$ such that $$e^{\prime\prime}_\alpha\stackrel{w^*} {\longrightarrow}e^{\prime\prime}\qquad \text{and}\qquad e^{\prime\prime}_\alpha\stackrel{\|\cdot\|} {\longrightarrow}e^{\prime\prime}.$$
Then $e^{\prime\prime}$ is a left identity for $E^\bot$. On the other hand, for every $x^{\prime\prime}\in E^\bot$, since $E^\bot$ is an ideal in $A^{**}$, we have $x^{\prime\prime}.e^{\prime\prime}\in E^\bot$. Thus, for every $a^\prime\in A^*$, $$\begin{aligned}
\langle x^{\prime\prime}.e^{\prime\prime},a^\prime\rangle &=\lim_\alpha
\langle (x^{\prime\prime}.e^{\prime\prime}).e^{\prime\prime}_\alpha,a^\prime\rangle=\lim_\alpha
\langle x^{\prime\prime}.(e^{\prime\prime}.e^{\prime\prime}_\alpha),a^\prime\rangle\\
&=\lim_\alpha
\langle x^{\prime\prime}.e^{\prime\prime}_\alpha,a^\prime\rangle= \langle x^{\prime\prime},a^\prime\rangle.
\end{aligned}$$
It follows that $x^{\prime\prime}.e^{\prime\prime}=x^{\prime\prime}$, and so $e^{\prime\prime}$ is a right identity for $E^\bot$. Consequently, $e^{\prime\prime}$ is a two-sided identity for $E^\bot$. Now, let $a^{\prime\prime}\in A^{**}$. Then $$e^{\prime\prime}.a^{\prime\prime}= (e^{\prime\prime}.a^{\prime\prime}).e^{\prime\prime}=
e^{\prime\prime}.(a^{\prime\prime}.e^{\prime\prime})=a^{\prime\prime}.e^{\prime\prime}.$$
Hence $e^{\prime\prime}\in Z_1(A^{**})=A$. It follows that $e^{\prime\prime}=0$, and so $E^\bot=0$. This implies that $A^{**}=A$.
Let $G$ be a locally compact group. If $M(G)^{**}$ is amenable, then by Theorem \[t44\](ii), because $C_0(G)^*=M(G)$, we conclude that $M(G)$ is reflexive. This means that $G$ is a finite group, moreover see [@f.ghahramani Corollary 1.4].
[**Acknowledgment.**]{} We would like to thank the referee for her/his careful reading of our paper and many valuable suggestions.
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A. Ebadian and A. Jabbari, *Weak$^*$-continuous derivations on module extension of dual Banach algebras*, South. Asian Bull. Math., **39**(3) (2015), 347–363. M. Eshaghi Gordji and M. Filali, [*Arens regularity of module actions*]{}, Studia Math., [**181**]{}(3) (2007), 237-254. M. Eshaghi Gordji and M. Filali, [*Weak amenability of the second dual of a Banach algebra*]{}, Studia Math., [**182**]{}(3) (2007), 205-213. F. Ghahramani and J. Laali, *Amenability and topological centers of the second duals of Banach algebras*, Bull. Austral. Math. Soc., **65** (2002), 191-197. F. Ghahramani, R. J. Loy and G. A. Willis, [*Amenability and weak amenability of second conjugate Banach algebras*]{}, Proc. Amer. Math. Soc., [**129**]{} (1996), 1489-1497. F. Gourdeau, *Amenability and the second dual of a Banach algebra*, Studia Math., **125**(1) (1997), 75-81. A. Jabbari, *On ideal amenability of Banach algebras*, J. Math. Physics, Analysis, Geometry, **8**(2) (2012), 135-143. B. E. Johnson, [*Cohomology in Banach algebra*]{}, Mem. Amer. Math. Soc., [**127**]{}, 1972.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We discuss techniques for probing the effects of a constant force acting on cold atoms using two configurations of a grating echo-type atom interferometer. Laser-cooled samples of $^{85}$Rb with temperatures as low as 2.4 $\mu$K have been achieved in a new experimental apparatus with a well-controlled magnetic environment. We demonstrate interferometer signal lifetimes approaching the transit time limit in this system ($\sim 270$ ms), which is comparable to the timescale achieved by Raman interferometers. Using these long timescales, we experimentally investigate the influence of a homogeneous magnetic field gradient using two- and three-pulse interferometers, which enable us to sense changes in externally applied magnetic field gradients as small as $\sim 4 \times 10^{-5}$ G/cm. We also provide an improved theoretical description of signals generated by both interferometer configurations that accurately models experimental results. With this theory, absolute measurements of $B$-gradients at the level of $3 \times 10^{-4}$ G/cm are achieved. Finally, we contrast the suitability of the two- and three-pulse interferometers for precision measurements of the gravitational acceleration, $g$.'
author:
- 'B. Barrett'
- 'I. Chan'
- 'A. Kumarakrishnan'
bibliography:
- 'MasterBib.bib'
title: |
Atom interferometric techniques for measuring uniform magnetic field gradients\
and gravitational acceleration
---
Introduction
============
Atom interferometers (AIs) have been employed to investigate a host of inertial effects over the past few decades. Such effects include the acceleration due to gravity [@Kasevich-PRL-1991; @Peters-Nature-1999; @Peters-Metrologia-2001; @Hughes-PRL-2009; @Poli-PRL-2011], gravity gradients [@Snadden-PRL-1998; @McGuirk-PRA-2002; @Yu-ApplPhysB-2006], and rotations [@Gustavson-PRL-1997; @Wu-PRL-2007; @Burke-PRA-2009]. Raman interferometric measurements of gravity [@Kasevich-PRL-1991; @Peters-Nature-1999; @Peters-Metrologia-2001] use cold atoms and transit time limited experiments in an atomic fountain to reach a precision of $\sim 3$ parts per $10^9$ (ppb) with 1 minute of interrogation time. This technique requires two phase-locked lasers to drive Raman transitions between two hyperfine ground states. It also requires state selection into the $m_F = 0$ magnetic sub-level to avoid sensitivity to $B$-fields and $B$-gradients, as well as velocity selection to guarantee that all interfering atoms have the same initial sub-recoil velocity.
In contrast to the Raman interferometer, the grating echo-type AI [@Cahn-PRL-1997; @Strekalov-PRA-2002] uses a single off-resonant excitation frequency that drives a cycling transition with the same initial and final state. This AI requires no state or velocity selection, and is insensitive to both the AC Stark effect and the Zeeman effect [^1]. Additionally, as we will show on the basis of a theoretical model, the intensity of the AI signal is insensitive to uniform $B$-gradients provided the atoms are pumped into a single magnetic sub-level—which need not be $m_F = 0$.
In this work, we use two configurations of the grating echo AI to demonstrate experiments with timescales comparable to those of Raman AIs. An improved theoretical description of the echo AI has enabled accurate modeling of experimental data from which sensitive measurements of an externally applied $B$-gradient can be extracted. This model is sufficiently general to describe all time-domain configurations of grating echo AIs, while accounting for a constant force on the atoms, as well as the sub-level structure of the atomic ground state. Recent work [@Su-PRA-2010] with a particular configuration of this AI, has shown that measurements of the phase of the electric field are less sensitive to mirror vibrations. Investigations of the influence of $B$-gradients on this AI configuration validate our predictions of gravitational effects, and indicate that this AI is particularly well-suited for precise measurements of the gravitational acceleration, $g$.
We begin with a review of the two AI configurations used in this work, which are illustrated in \[fig:1-RecoilDiagrams\]. The two-pulse echo AI [@Cahn-PRL-1997; @Strekalov-PRA-2002; @Beattie-PRA-2008; @Barrett-PRA-2010] utilizes short (Raman-Nath) standing wave (sw) pulses to diffract a sample of laser cooled atoms at $t = T_1$ into a superposition of momentum states: ${\left| n\hbar\bm{q} \right\rangle}$. Here, $n$ is an integer and $\bm{q} = \bm{k}_1 - \bm{k}_2 \approx 2\bm{k}$ is the difference between the traveling wave vectors comprising the sw. At $t = T_2$, a second sw pulse further diffracts the atomic wave packets—creating sets of center-of-mass trajectories that overlap and produce interference in the form of a density modulation in the vicinity of $t_{\rm{echo}}^{(2)} = T_1 + (\bar{N}+1) T_{21}$, where $T_{21} \equiv T_2 - T_1$ and $\bar{N} = 1, 2, \ldots$ is the order of the echo, as shown in \[fig:1a-RecoilDiagram-TwoPulse\]. The induced density modulation is coherent for $\tau_{\rm{coh}} = 2/q\sigma_v \sim 3$ $\mu$s about these “echo” times, beyond which the modulation dephases due to the distribution of velocities in the sample. Here, $\sigma_v = (2 k_B \mathcal{T}/M)^{1/2}$ characterizes the width of the velocity distribution along $\hat{\bm{z}}$. A traveling wave pulse is applied along the $\hat{\bm{z}}$-direction in the vicinity of $t_{\rm{echo}}^{(2)}$ to “read out” the amplitude of the grating by coherently Bragg scattering light along the $-\hat{\bm{z}}$-direction. The duration of this signal is limited by the coherence time, $\tau_{\rm{coh}}$. Due to the nature of Bragg diffraction, this back-scattered light is proportional to the Fourier component of the density distribution with spatial frequency $q$. This harmonic is only produced by interference of momentum states that differ by $\hbar q$ ($\Delta n = \pm 1$). As a result, the two-pulse AI exhibits a temporal modulation at the atomic recoil frequency, $\omega_q = \hbar q^2/2M$, and is therefore sensitive to recoil effects.
The three-pulse “stimulated” grating echo AI (henceforth referred to as the three-pulse AI) was first demonstrated in using a single hyperfine ground state, and was termed a “stimulated” echo due to similarities in pulse geometry with the stimulated photon echo scheme [@Mossberg-PRA-1979; @Borde-PRA-1984; @Allen-BOOK-1987; @Dubetsky-PRA(R)-1992]. Recent work involving this interferometer [@Su-PRA-2010] has shown certain advantages over the two-pulse scheme for phase measurements of the atomic grating. The three-pulse AI involves applying two sw pulses at $t = T_1$ and $t = T_2$, followed by a third pulse applied at $t = T_3 = T_2 + T_{32}$, where $T_{32} \equiv T_3 - T_2$. This pulse geometry produces an echo in the vicinity of $t_{\rm{echo}}^{(3)} = T_1 + (\bar{N}+1) T_{21} + T_{32}$, as shown in \[fig:1b-RecoilDiagram-ThreePulse\]. However, unlike the two-pulse AI where all pairs of trajectories produced by the second pulse interfere at the echo times, for the three-pulse AI only momentum states of the same order $(\Delta n = 0)$ after the second pulse produce interference at the echo times for arbitrary $T_{21}$ and $T_{32}$. For this reason, the signal produced by this interferometer as a function of $T_{32}$ (with $T_{21}$ fixed) is insensitive to atomic recoil effects (i.e. no temporal modulation) and is therefore ideal for probing other effects—such as those due to a constant force on the atoms.
Reference extensively reviews the grating echo AI and discusses applications relating to atomic recoil [@Weel-PRA(R)-2003; @Beattie-PRA-2008; @Beattie-PRA(R)-2009; @Beattie-PRA-2009; @Barrett-PRA-2010; @Barrett-SPIE-2011], gravity and magnetic gradients [@Weel-PRA-2006].
Previous experiments based on this AI [@Cahn-PRL-1997; @Strekalov-PRA-2002; @Weel-PRA(R)-2003; @Beattie-PRA-2008; @Beattie-PRA(R)-2009; @Beattie-PRA-2009; @Andersen-PRL-2009; @Barrett-PRA-2010] were typically limited to $T_{21} < 10$ ms by decoherence effects due to spatially and temporally varying $B$-fields. Additionally, the sample temperature (typically $\sim 50$ $\mu$K) and excitation beam configuration (fixed frequency sw with $\sim 0.5$ cm diameter) limited the transit time in these experiments. In this work, we have improved the level of $B$-field and $B$-gradient suppression by using a non-magnetic vacuum chamber, which has enabled the extension of AI signal lifetimes. The magnetically controlled environment allows a sample of $^{85}$Rb atoms to be cooled to temperatures as low as 2.4 $\mu$K. By expanding the excitation beam diameter to $\sim 2$ cm, and chirping the sw pulses to cancel Doppler shifts, echo AI signal lifetimes of $\sim 220$ ms and transit times of $\sim 270$ ms have been achieved. These timescales are comparable to those of fountain experiments involving Raman AIs [@Kasevich-PRL-1991; @Peters-Nature-1999; @Wicht-PhysScr-2002]. In contrast, long-lived echo AI signals have been observed only by using magnetic guides to limit transverse cloud expansion [@Su-PRA-2010].
The experimental apparatus presented here has made it possible to exploit the aforementioned advantages of the echo AI for a variety of precision measurements, such as the atomic recoil frequency [@Barrett-SPIE-2011] and the gravitational acceleration [@Barrett-Advances-2011], that are currently underway. Additionally, we recently utilized this apparatus to perform a coherent transient experiment with cold Rb atoms to achieve a precise determination of the atomic $g$-factor ratio [@Chan-PRA-2011].
In this Article, we apply long timescales to understanding and detecting the effects of $B$-gradients using the two-pulse grating echo AI, as well as a three-pulse “stimulated” grating echo AI [@Strekalov-PRA-2002; @Su-PRA-2010]. The passive detection of magnetic anomalies is of interest for various applications, such as submarine and mine detection where the ambient magnetic noise of the environment is large compared to the sensitivity of the instrument [@Davis-JModOpt-2008]. The influence of gravity and $B$-gradients on AI experiments has been considered in the past. Reference calculates how such forces affect the visibility of interference patterns in atomic diffraction experiments. In previous work [@Weel-PRA-2006], we demonstrated the effect of both gravity and $B$-gradients on the two-pulse AI. A theoretical description of these effects based on a spin-1/2 system was able to explain the basic signal dependence on the pulse separation, $T_{21}$, but was insufficient to model experimental data.
This work relies on an improved theoretical description of a generalized echo AI that includes an arbitrary number of sw excitation pulses, the effects of a constant force on the atoms, spontaneous emission and the sub-level structure of the atomic ground state (the 5S$_{1/2}$ $F = 3$ state of $^{85}$Rb is used in the experiment). Coupled with these theoretical predictions, we achieve sensitivity to *changes* in $B$-gradients at the level of $\sim 0.04$ mG/cm. In addition, absolute measurements of $B$-gradients as small as $\sim 0.3$ mG/cm, and sensitivity to the curvature of $B$-fields are demonstrated. These results are consistent with independent measurements of the spatial variation in the $B$-field using a flux-gate magnetometer. These studies help place limits on the sensitivity of a broad class of time-domain AIs to $B$-gradients.
We also consider implications for achieving precise measurements of $g$ using the two- and three-pulse echo AIs. In particular, analysis of the three-pulse AI suggests there are significant advantages for measuring $g$ over the two-pulse AI. Although the experimental apparatus used in this work is not designed to detect gravitational effects, predictions of the grating phase modulation due to gravity for both AIs have been validated by measuring the effects of externally applied $B$-gradients. Measurements of $g$ using these AIs will be presented elsewhere.
This Article is organized as follows. In \[sec:Theory\] we present theoretical predictions for the two- and three-pulse AI signals in the presence of a homogeneous $B$-gradient. In \[sec:Setup\], we describe details related to the experimental apparatus. We present the results of experiments related to long timescales in \[sec:Results\], and discuss measurements of $B$-gradients using both the two-pulse and three-pulse techniques. Section \[sec:Gravity\] discusses the feasibility of a precise measurement of $g$ using the formalism developed to describe $B$-gradients. We conclude in \[sec:Conclusions\]. The Appendix presents a calculation of the signal generated by a generalized echo AI—encompassing the two- and three-pulse AIs—in the presence of a constant force.
Theory {#sec:Theory}
======
In this section, we present the key results of calculations for both the two- and three-pulse AI signals in the presence of a homogeneous $B$-gradient. Details of the calculations—which are sufficiently general to account for any constant force on the atoms, and an arbitrary number of excitation pulses—are presented in the Appendix.
In general, the sensitivity of these interferometers can be characterized by the space-time area they enclose. Since only those states differing by $\hbar q$ at the echo time contribute to the signal, the area of both AIs is primarily controlled by $T_{21}$. In the absence of any external forces, the areas of the two- and three-pulse AIs can be calculated by inspecting their recoil diagrams \[\[fig:1a-RecoilDiagram-TwoPulse\] and \[fig:1b-RecoilDiagram-ThreePulse\], respectively\]
\[eqn:A(2)\] A\^[(2)]{} & = & |[N]{} (|[N]{} + 1) ( T\^[(2)]{}\_[21]{} )\^2,\
\[eqn:A(3)\] A\^[(3)]{} & = & ,
where $M$ is the mass of the atom. Henceforth, quantities containing superscripts $(2)$ or $(3)$ indicate the interferometer for which that quantity applies. At first glance, it might appear that the three-pulse AI encloses a larger area than the two-pulse AI due to the extra term in [(\[eqn:A(3)\])]{}. However, one must compare the enclosed areas at the same echo times, which are given by $t^{(2)}_{\rm{echo}} = T_1 + (\bar{N} + 1)T^{(2)}_{21}$ and $t^{(3)}_{\rm{echo}} = T_1 + (\bar{N} + 1) T^{(3)}_{21} + T_{32}$ for the two- and three-pulse schemes, respectively. By setting $t^{(2)}_{\rm{echo}} = t^{(3)}_{\rm{echo}}$, it can be shown that $A^{(2)} - A^{(3)} = \hbar q \bar{N} T_{32}^2 / 2M(\bar{N} + 1)$. This suggests that the two-pulse AI is always more sensitive to external forces than the three-pulse AI. Nevertheless, the three-pulse AI offers a unique feature: the spatial separation between interfering wave packets remains constant between the application of the second and third sw pulses. This is advantageous because larger spatial separations leads to increased decoherence, and therefore reduced timescale in the experiment [@Su-PRA-2010]. Since the separation can be precisely controlled by the pulse separation $T_{21}$, one can increase the signal lifetime by using smaller $T_{21}$.
Additionally, since the signal generated by the two-pulse AI is modulated at the recoil frequency, $\omega_q$, there are periodic regions where the signal-to-noise ratio is less than one and not well-suited for accurate phase measurements. However, the three-pulse technique is insensitive to atomic recoil if $T_{21}$ is fixed. Therefore, the scattered field amplitude has no additional modulation at $\omega_q$ as $T_{32}$ is varied—allowing regions of low signal-to-noise ratio to be avoided.
Both the gravitational force and a constant $B$-gradient produce a constant force on the atoms, $\bm{\mathcal{F}} = \mathcal{F} \hat{\bm{z}}$, which generates a phase shift in the atomic interference pattern. The basic physical mechanism that produces this phase shift is a difference in potential energy between the two arms of the AI. One can compute the relative phase between the two arms $\Delta \phi = (\mathcal{S}_{\rm{B}} - \mathcal{S}_{\rm{A}})/\hbar$ using the classical action [@Peters-Metrologia-2001] (t) = \_0\^[t]{} \[z(t’), (t’)\] dt’, where $\mathcal{L} = M\dot{z}^2/2 + \mathcal{F} z$ is the Lagrangian in this case. If $\mathcal{S}_{\rm{B}}$ and $\mathcal{S}_{\rm{A}}$ represent, respectively, the action along the upper and lower arms of the two-pulse AI, it can be shown that the phase shift between these arms is \[eqn:DeltaPhi(2)\] \^[(2)]{} = |[N]{}(|[N]{} + 1) + |[N]{}\^2 q v\_0 T\_[21]{}, where $v_0$ is the initial velocity of the atom along the $\hat{\bm{z}}$-direction. The term proportional to $v_0$ is due to the relative Doppler shift between the two arms of the AI. Since the atomic sample has a finite velocity distribution (characterized by a $1/e$ radius, $\sigma_v$, and temperature, $\mathcal{T}$), this term is responsible for the coherence time of the echo: $\tau_{\rm{coh}} = 2/q \sigma_v$. As expected, the contribution to the phase shift from the potential energy (the term proportional to $\mathcal{F}$) is independent of the initial velocity of the cloud.
A similar calculation for the relative phase shift between the arms of the three-pulse AI yields $$\begin{aligned}
\begin{split}
\label{eqn:DeltaPhi(3)}
& \Delta \phi^{(3)} = \bar{N}(\bar{N} + 1) \left[ \omega_q T_{21} + \frac{q \mathcal{F}}{M} T_{21}^2 \right] \\
& \;\;\;\;\; + \bar{N} \frac{q \mathcal{F}}{M} T_{32} T_{21} + \bar{N} q v_0 (T_{32} + \bar{N} T_{21}).
\end{split}\end{aligned}$$ This expression is similar to [(\[eqn:DeltaPhi(2)\])]{}, with additional terms proportional to the pulse spacing $T_{32}$. One can vary either $T_{21}$ or $T_{32}$ to detect phase modulation produced by an external force, $\mathcal{F}$. However, since there are no terms containing the phase $\omega_q T_{32}$, one can effectively turn off the sensitivity to atomic recoil by fixing $T_{21}$. This makes the three-pulse AI ideal for investigating the effects due to $\mathcal{F}$, especially when $q\mathcal{F} T_{21}/M \gg \omega_q$ since no additional modulation at $\omega_q$ is present. This is particularly advantageous for measurements of gravity, as discussed in \[sec:Gravity\].
Since the two-pulse AI is intrinsically more sensitive than the three-pulse AI, it is better suited to measurements of $\mathcal{F}$ when $q\mathcal{F} T_{21}/M < \omega_q$. In this work, we demonstrate this feature by measuring externally applied $B$-gradients. The sensitivity of the three-pulse AI to $\mathcal{F}$ can be enhanced by utilizing the additional phase proportional $T_{32} T_{21}$ in [(\[eqn:DeltaPhi(3)\])]{}. Experimentally, this can be accomplished by varying *both* pulse separations, $T_{21}$ and $T_{32}$, with $T_{21}$ varied in integer multiples of the recoil period: $\tau_q = \pi/\omega_q$.
To determine the response of the grating echo AI in the presence of a constant force, we assume a potential energy with the form \[eqn:U(r)\] (z) = - z, where $\hat{\mathcal{M}} = -\partial \hat{U}/\partial z$ is a matrix operator with units of force that commutes with both the position ($z$) and momentum ($p$) operators. In the case of a constant $B$-gradient, the potential is (z) = - (z) = - \_z z, where $g_F$ is the Landé $g$-factor, $\mu_B$ is the Bohr magneton, $\bm{B}(z) = \beta \bm{z}$ is the magnetic field vector with gradient $\beta$ along the $z$-direction (also assumed to be the quantization axis), and $\hat{F}_z$ is the projection operator for total angular momentum, $\bm{F}$. In this case, $\hat{\mathcal{M}} = \mathcal{F} \hat{F}_z/\hbar$ and the force is $\mathcal{F} = g_F \mu_B \beta$, where $\hat{F}_z$ operates on the basis states ${\left| F\,m_F \right\rangle}$ and has eigenvalues $\hbar m_F$.
In both interferometer schemes, the phase of the grating is imprinted on the electric field back-scattered from a traveling-wave read-out pulse applied in the vicinity of an echo time. For the two-pulse AI, measuring the phase of the scattered field is equivalent to measuring the relative position of the grating, since both the position and the phase scale as $T_{21}^2$. Similarly, in the three-pulse case, the phase measured as a function of $T_{32}$ is proportional to the velocity of the grating—which scales as $T_{21}$.
We first examine the effects of $B$-gradients on the two-pulse AI, followed by a comparison with the three-pulse AI.
Two-Pulse Interferometer {#sec:Theory-2Pulse}
------------------------
In general, the electric field scattered by the atoms at the time of an echo is proportional to the amplitude of the Fourier harmonic of the atomic density grating with spatial frequency $q$. For the case of an external $B$-gradient, $\beta$, the scattered field has distinct contributions from each magnetic sub-level: \[eqn:E(2)\_beta\] E\_\^[(2)]{}(t;) = \_[m\_F]{} E\^[(2)]{}\_[m\_F]{}(t;) e\^[i m\_F \^[(2)]{}\_(t;)]{}, where $E^{(2)}_{m_F}$ is the field scattered by the state ${\left| F\,m_F \right\rangle}$ \[given by [(\[eqn:E(2)\_mF\])]{}\] and $m_F \phi^{(2)}_{\beta}$ is the phase shift of the density grating produced by the same state in the presence of the $B$-gradient. For the $\bar{N}^{\rm{th}}$ order echo at $t_{\rm{echo}}^{(2)} = T_1 + (\bar{N} + 1)T_{21}$, with the set of onset times $\bm{T} = \{ T_1, T_1 + T_{21} \}$ and $\Delta t = t - t_{\rm{echo}}^{(2)}$, the phase shift of the grating $\phi^{(2)}_{\beta}(t;\bm{T})$ is given by $$\begin{aligned}
\begin{split}
\label{eqn:phi(2)_beta}
& \phi^{(2)}_{\beta}(\Delta t;T_{21}) = \frac{q g_F \mu_B \beta}{2M} \left\{ \bar{N}(\bar{N}+1)T_{21}^2 \right. \\
& \;\;\;\;\; \left. + \, 2 \big[T_1 + (\bar{N}+1) T_{21}\big] \Delta t + \Delta t^2 \right\}.
\end{split}\end{aligned}$$ The general form of this equation for a constant force, $\mathcal{F}$, is given by [(\[eqn:phi(2)\_F\])]{} in the Appendix. In the discussions that follow, we take $\Delta t = 0$ which corresponds to the echo time. Since the echo lasts for $\tau_{\rm{coh}} \sim 3$ $\mu$s about $\Delta t = 0$, the signal is obtained by integrating the back-scattered field over this time.
Equations [(\[eqn:E(2)\_beta\])]{} and [(\[eqn:phi(2)\_beta\])]{} indicate that the field amplitude scattered from state ${\left| F\,m_F \right\rangle}$ exhibits phase modulation as a function of $T_{21}$ at a frequency $m_F \omega^{(2)}_{\beta}(T_{21})$ due to the presence of the gradient, where $$\begin{aligned}
\begin{split}
\label{eqn:omega(2)_beta}
& \omega^{(2)}_{\beta}(T_{21}) = \left| \frac{\partial \phi^{(2)}_{\beta}}{\partial T_{21}} \right| \\
& \;\;\;\;\; = \frac{q g_F \mu_B \beta}{M} \left[ \bar{N}(\bar{N} + 1) T_{21} + (\bar{N} + 1) \Delta t \right].
\end{split}\end{aligned}$$ This modulation frequency depends linearly on $\beta$ and the pulse separation, $T_{21}$ (i.e. the frequency is chirped with $T_{21}$). The phase modulation of the grating produced by state ${\left| F\;m_F \right\rangle}$ also scales linearly with the magnetic quantum number, $m_F$, as shown in [(\[eqn:E(2)\_beta\])]{}. For an arbitrary set of magnetic sub-level populations, the total scattered field \[[(\[eqn:E(2)\_beta\])]{}\] contains all harmonics $m_F \omega^{(2)}_{\beta}(T_{21})$, where $m_F = -F,\ldots,F$. If more than one sub-level is populated, interference between the fields scattered off of each state produces modulation in the total scattered field. This effect can then be detected in the field amplitude, $E_{\beta}^{(2)}$, or the field intensity, $|E_{\beta}^{(2)}|^2$, by varying $\beta$ or the pulse separation, $T_{21}$. The amplitude of each harmonic comprising this modulation is determined by the sub-level populations, as well as the transition probabilities between ground and excited state sub-levels.
If the system is optically pumped into a single sub-level, such as the extreme state: ${\left| F\;F \right\rangle}$, then the phase modulation of the grating only affects the phase of the electric field—which cannot be observed using intensity detection. Instead, one can use heterodyne detection to measure the electric field amplitude and obtain the relative phase of the scattered light [@Cahn-PRL-1997; @Weel-PRA(R)-2003; @Weel-PRA-2006]. Furthermore, if the system is optically pumped into the ${\left| F\;0 \right\rangle}$ state, there is no phase modulation due to $B$-gradients since this state is insensitive to magnetic fields.
Figures \[fig:2a-GradientSignal\] and \[fig:2b-GradientSignal\] show the expected two-pulse AI signal as a function of $T_{21}$ in steps of the recoil period, $\tau_q = \pi/\omega_q$ ($\sim 32$ $\mu$s for $^{85}$Rb). Since $\omega^{(2)}_{\beta} < \omega_q$, incrementing $T_{21}$ in this fashion eliminates additional modulation due to atomic recoil. Figure \[fig:2a-GradientSignal\] shows the signal for equally distributed sub-level populations, while \[fig:2b-GradientSignal\] is for an optically pumped system in the two extreme states: $|3\;$$-3\rangle$ and ${\left| 3\;3 \right\rangle}$. Both of these figures show amplitude modulation, but in the optically pumped case there is only one frequency component present and the modulation occurs with maximum contrast—increasing the sensitivity to gradients.
Eliminating the amplitude modulation in the signal due to $B$-gradients \[shown by the dashed lines in \[fig:2a-GradientSignal\]\] is a key requirement for precision measurements of $\omega_q$. We will show in \[sec:Results\] that these conditions can be realized with sufficient suppression of ambient $B$-gradients in a glass cell. It is also possible to eliminate sensitivity to $B$-gradients using intensity detection if the atoms are pumped into a single magnetic sub-level.
Three-Pulse Interferometer {#sec:Theory-3Pulse}
--------------------------
The effects due to $B$-gradients manifest themselves differently in the three-pulse interferometer. We derive the expression for the signal in the Appendix \[see [(\[eqn:E(N)\])]{} and [(\[eqn:E(3)\_mF\])]{}\] and find that the amplitude of the scattered field does not depend on the time between the second and third sw pulses, $T_{32}$, but only on $T_{21}$—similar to the two-pulse AI. However, the phase of the grating in the three-pulse case depends on both $T_{21}$ and $T_{32}$: \[eqn:phi(3)\_beta\] & & \_\^[(3)]{}(t; ) = { |[N]{} (|[N]{} + 1) T\_[21]{}\^2 + 2|[N]{} T\_[32]{} T\_[21]{} .\
& & . + 2 t + t\^2 }. In this case, the set of pulse onset times is given by $\bm{T} = \{ T_1, T_1 + T_{21}, T_1 + T_{21} + T_{32} \}$ and $\Delta t = t - t^{(3)}_{\rm{echo}}$. This phase is identical to [(\[eqn:phi(2)\_beta\])]{} for the two-pulse interferometer with the addition of the two terms proportional to $T_{32}$. Equation [(\[eqn:phi(3)\_beta\])]{} suggests that the force can be determined by measuring the phase modulation of the grating as a function of either $T_{21}$ or $T_{32}$, or by varying both pulse separations simultaneously. Varying $T_{32}$ produces a phase modulation of the atomic grating at a frequency that is proportional to $T_{21}$: \[eqn:omega(3)\_beta\] \^[(3)]{}\_(T\_[21]{}) = | | = ( |[N]{} T\_[21]{} + t ).
Figures \[fig:2c-GradientSignal\] and \[fig:2d-GradientSignal\] show the expected three-pulse signal as a function of $T_{32}$, with $T_{21}$ fixed at a typical experimental value of 5 ms, in the presence of a $B$-gradient $\beta = 10$ mG/cm. When the sub-level populations are equally distributed \[\[fig:2c-GradientSignal\]\] the phase of the total scattered field contains multiple frequency components—one for each sub-level: $m_F \omega_{\beta}^{(3)}$. The interference between these components produces a modulation in the total scattered field amplitude. This is similar to the two-pulse case shown in \[fig:2a-GradientSignal\], except that the modulation occurs at a single frequency that is fixed by $\beta$, $T_{21}$ and $\bar{N}$. For a sample that is optically pumped equally into the two extreme states: $|3\;$$-3\rangle$ and ${\left| 3\;3 \right\rangle}$, as shown in \[fig:2d-GradientSignal\], there is only one frequency component present in the scattered field. In this case, the amplitude modulation occurs with greater contrast than for any other configuration of sub-level populations.
Experimental Setup {#sec:Setup}
==================
We now review the experimental setup that has made possible long-lived grating echo AI signals. This setup is substantially different from previous echo experiments [@Weel-PRA-2006; @Beattie-PRA-2008; @Beattie-PRA(R)-2009; @Beattie-PRA-2009] after implementing many improvements. These include suppression of stray magnetic gradients using a non-magnetic chamber, increasing the trapped atom number with large diameter beams, extending the transit time by cooling the sample to $\sim 10$ $\mu$K and implementing large diameter excitation beams, and by chirping the excitation frequencies to eliminate Doppler shifts associated with the falling cloud.
The experiment utilizes a sample of laser-cooled $^{85}$Rb atoms in a magneto-optical trap (MOT) containing approximately $10^9$ atoms in a Gaussian spatial distribution with a horizontal $e^{-1}$ radius of $\sim 1.7$ mm. The MOT is contained in a borosilicate glass cell maintained at a pressure of $\sim 1 \times 10^{-9}$ Torr. In addition to the anti-Helmholtz coils used for trapping, three pairs of square quadrupole coils are centered on the MOT, as shown in \[fig:3-ExpSetup\]. Each square frame contains two overlapping coils, one connected in the Helmholtz configuration with the coil in the opposite frame, and the other in the anti-Helmholtz configuration. These sets of coils are used to cancel ambient magnetic fields and field gradients over the volume of the MOT at the level of $\sim 1$ mG and $\sim 0.1$ mG/cm, respectively. The initial set points for the currents in the canceling coils that produced $\sim 1$ mG of $B$-field suppression were determined using an atomic magnetometer experiment [@Chan-PRA-2011] that allowed the field at the location of the MOT to be measured.
Light derived from a Ti:sapph laser with frequency $\nu_L$ and linewidth $\sim 1$ MHz is locked to the 5S$_{1/2}$ $F=3 \to F'=4$ transition ($\nu_L = \nu_0$) using saturated absorption spectroscopy. The light is then shifted 130 MHz above resonance by an acousto-optic modulator (AOM) operating in dual-pass mode such that $\nu_L = \nu_0 + 130$ MHz. A separate “trapping” AOM shifts this light by $-148$ MHz such that the detuning is $\Delta = -14$ MHz ($\nu_L = \nu_0 - 14$ MHz). Approximately 370 mW of this light is transmitted through an anti-reflection-coated, single-mode optical fiber (operating at 60% efficiency) and expanded to a diameter of $\sim 5.4$ cm for trapping atoms from background vapor.
![(Color online) Diagram of the experiment. The excitation beams are both $\sigma^+$-polarized by the $\lambda/4$-plates. The glass cell has approximate dimensions $7.6 \times 7.6 \times 84$ cm. Each pair of square quadrupole coils have a side length of $\sim 66$ cm and contain overlapped coils wired in both Helmholtz and anti-Helmholtz configurations for canceling $B$-fields and $B$-gradients, respectively.[]{data-label="fig:3-ExpSetup"}](Fig3-ExpSetup.eps){width="35.00000%"}
An external cavity diode laser is used to derive repump light for the trapping setup. It is locked to the 5S$_{1/2}$ $F = 2 \to F'=(2,3)$ crossover transition and up-shifted by $\sim 32$ MHz using an AOM. Approximately 25 mW of repump light is obtained after coupling through the same optical fiber as the trapping light. At $t = 0$, the MOT coils are pulsed off in $\sim 100$ $\mu$s, while the trapping and repump beams are left on for 6 ms of molasses cooling. For $\sim 3$ ms of this time, the detuning of the trapping light is linearly chirped from $\Delta = -14$ MHz to $-50$ MHz to further cool the atoms, and the power is simultaneously ramped down in order to reduce heating due to photon scattering. With this procedure we achieve temperatures as low as $\mathcal{T} = 2.4$ $\mu$K.
Light from the Ti:sapph laser is also used to derive the AI pulses. A “gate” AOM operating in dual-pass configuration shifts the undiffracted light from the “trapping” AOM from $\nu_L = \nu_0 + 130$ MHz to $\nu_L = \nu_0 + 290$ MHz. The gate AOM is also pulsed so as to serve as a high-speed shutter during the experiment. The light from the gate AOM is split and sent into two separate AOMs (referred to as the “$\bm{k}_1$” and “$\bm{k}_2$” AOMs) operating at $240 \mbox{ MHz} \pm \delta(t)$ that produce the sw pulses. Here, $\delta(t) = g t/\lambda$ is a time-dependent frequency shift that is added to (subtracted from) the radio frequency (rf) driving the $\bm{k}_1$ ($\bm{k}_2$) AOM using an arbitrary waveform generator, as shown in \[fig:4-RFSchematic\]. Chirping the excitation pulses in this manner cancels the Doppler shift of the atoms falling under gravity. The rf driving these AOMs is also phase locked to a 10 MHz rubidium clock to eliminate any electronically induced phase shifts. Light entering the $\bm{k}_1$ AOM is downshifted by $240 \mbox{ MHz} - \delta(t)$ and sent into an optical fiber that carries the light toward the MOT. Similarly, the $\bm{k}_2$ AOM downshifts the light by $240 \mbox{ MHz} + \delta(t)$. In this configuration, the detuning of the $\bm{k}_1$ ($\bm{k}_2$) pulse is $\Delta_1 = 50 \mbox{ MHz} - \delta(t)$ \[$\Delta_2 = 50 \mbox{ MHz} + \delta(t)$\]. This light is coupled into a separate fiber and aligned through the MOT along the vertical direction, as illustrated in \[fig:3-ExpSetup\]. The output of both fibers is expanded to a $e^{-2}$ diameter of $\sim 2$ cm. The rf pulses driving the $\bm{k}_1$ and $\bm{k}_2$ AOMs are controlled using TTL switches with an isolation ratio of 100 dB, which produces optical pulses with rise times of $\sim 20$ ns. The “gate” AOM is turned off between excitation pulses to further reduce background light from reaching the atoms.
![(Color online) Schematic of the rf chain used for chirped AI pulses. A phase-locked loop (PLL) generates a 220 MHz rf signal which is split and mixed with the output of two separate arbitrary waveform generators (AWGs). The AWGs are triggered at the start of the experiment to output a frequency sweep from 20 MHz to $20 \mbox{ MHz} \pm \delta(t)$ after a time $t$, where $\delta(t) = g t/\lambda$, $g \sim 9.8$ m/s$^2$ and $\lambda \sim 780$ nm. The sum frequency from the mixers is isolated using a band-pass filter (BPF) with a center frequency of 240 MHz and a 5 dB pass-band of 4 MHz. The outputs of the BPFs are pulsed using a set of transistor-transistor logic (TTL) switches, which are then sent to the $\bm{k}_1$ and $\bm{k}_2$ AOMs. The two-pulse AI sequence for both $\bm{k}_1$ and $\bm{k}_2$ are shown. Here, P1 and P2 refer to traveling wave components comprising the first and second sw pulses, and RO denotes the traveling wave read-out pulse sent along $\bm{k}_1$. Both AWGs and the PLL are externally referenced to a 10 MHz Rb clock.[]{data-label="fig:4-RFSchematic"}](Fig4-RFSchematic.eps){width="45.00000%"}
In the vicinity of any given echo (see \[fig:1-RecoilDiagrams\]), the read-out pulse is applied to the sample along the $\bm{k}_1$-direction and a coherent back-scattered field from the atoms occurs along the direction of $\bm{k}_2$. The power of the scattered field is recorded as a function of time using a photo-multiplier tube (PMT) that is gated on for 9 $\mu$s. The echo signal lasts $\tau_{\rm{coh}} \sim 3$ $\mu$s before coherence is lost due to Doppler dephasing. For $T_{21} \lesssim 10$ ms, the scattered field can reach powers greater than 100 $\mu$W. However, for $T_{21} > 10$ ms, the signal size decreases exponentially. The noise floor for the PMT is approximately 0.1 $\mu$W. Typically, one computes the time-integrated area of the echo signal as a measure of the signal size for a given set of parameters. Since this quantity has units of energy, it is henceforth referred to as the echo energy.
Results and Discussion {#sec:Results}
======================
We now review the main experimental results of this work relating to long-lived AI signals and sensing externally applied $B$-gradients.
Investigations of AI Timescale
------------------------------
Figure \[fig:5-SignalLifetime\](a) shows a measurement of the temperature of the laser cooled sample. At $t = T_0$, all optical and magnetic fields associated with the MOT are switched off and the atoms are allowed to thermally expand in the dark. At $t = T_0 + T_{\rm{exp}}$, the trapping and repump beams are turned back on and a calibrated charged-coupled device (CCD) is triggered to photograph the cloud with an exposure time of 100 $\mu$s. This process is repeated for various expansion times, $T_{\rm{exp}}$, and the $e^{-1}$ radius of the cloud, $R$, is measured by fitting to the Gaussian intensity profiles obtained from each image. The temperature is obtained by fitting to a hyperbola [@Weiss-JOSAB-1989; @Vorozcovs-JOSAB-2005] with the form $R = [R_0^2 + \sigma_v^2 (T_{\rm{exp}} - t_0)^2]^{1/2}$, where $R_0$ is the initial cloud radius, $\sigma_v = (2 k_B \mathcal{T}/M)^{1/2}$ is the $e^{-1}$ radius of the velocity distribution and $t_0$ is a phenomenological offset from $T_{\rm{exp}} = 0$. The data shown in \[fig:5-SignalLifetime\](a) give a temperature of $\mathcal{T} \sim 2.4$ $\mu$K in $^{85}$Rb. This relatively low MOT temperature is attributed to the well-controlled magnetic environment within the glass cell, as well as the molasses cooling procedure described above.
{width="80.00000%"}
Measurements of the AI signal lifetime under different pulse configurations are shown in \[fig:5-SignalLifetime\](b), with each configuration explained schematically in \[fig:5-SignalLifetime\](c). For the transit time measurement, the two-pulse AI configuration was used with $T_{21}$ fixed. The excitation and read-out pulses were incremented synchronously. The signal lifetime for the three-pulse AI was determined by fixing $T_{21}$ and varying the third sw pulse and read-out synchronously. For the two-pulse AI, the lifetime was measured by fixing the first sw pulse and incrementing the second sw pulse and read-out in steps $n \tau_q$ and $2 n \tau_q$, respectively, where $n = 10$.
Here, all ambient $B$-fields and $B$-gradients are canceled along all three axes at the level of $\sim 1$ mG and $\sim 0.1$ mG/cm, respectively. The transit time data was obtained by using the two-pulse AI with $T_{21}$ fixed at $\sim 1.690$ ms and varying the time of all sw pulses relative to the time of trap turn-off, $T_0$. In this measurement, the AI signal is proportional to the number of atoms that remain in the volume defined by the $\sim 2$ cm diameter excitation beams during the thermal expansion of the cloud. Although the echo energy spans almost three orders of magnitude as it decays exponentially, signals are clearly distinguishable from the noise floor ($\sim 0.1$ pJ) at times as large as $\sim 270$ ms, as shown in \[fig:5-SignalLifetime\](b). This time represents the transit time limit for the conditions of our experiment—corresponding to a drop height of $\sim 36$ cm. This distance nearly coincides with the bottom viewport of the vacuum system. We emphasize here that such lifetimes are not possible with this interferometer unless the frequencies of the $\bm{k}_1$ and $\bm{k}_2$ beams are oppositely chirped such that the Doppler shift due to gravity \[$\delta(t) = g t / \lambda$\] is canceled *or* the bandwidth of the sw pulses is large enough to account for such a shift. The frequency chirp puts the sw pulses on resonance for the two-photon transition back to the same ground state for all times during the sample’s free-fall.
The signal lifetime for the two-pulse AI configuration is shown as the red curve in \[fig:5-SignalLifetime\](b). Here, the signal lasts approximately 130 ms, corresponding to $T_{21} \sim 65$ ms. To the best of our knowledge, this is the largest timescale observed with this interferometer, corresponding to more than a factor of 6 improvement over our previous work [@Weel-PRA-2006; @Beattie-PRA-2008; @Beattie-PRA(R)-2009; @Beattie-PRA-2009]. However, the lifetime of the two-pulse echo is still limited by decoherence from a small, inhomogeneous $B$-gradient that the atoms sample over the $\sim 8$ cm they have fallen in 130 ms. A non-linear $B(z)$ produces a spatially-dependent force between interfering trajectories—resulting in a differential phase shift between paths of the interferometer that causes dephasing and, therefore, a loss of signal. Such a non-linearity in $B(z)$ has been measured to be $\partial^2 B/\partial z^2 \sim -0.4$ mG/cm$^2$ with a flux-gate magnetometer placed at different spatial locations around the glass chamber. This curvature is produced by a combination of non-ideal coil configurations and the presence of nearby ferromagnetic materials.
There are two important features that should be recognized from the data for the three-pulse AI shown in \[fig:5-SignalLifetime\](b). First, at $T_{\rm{RO}} - T_0 \approx 0$, the echo energy for the three-pulse AI is a factor of $\sim 2$ smaller than that of the two-pulse AI. This comes about because the additional Kapitza-Dirac pulse involved in the three-pulse AI produces fewer pathways that result in interference at the echo time compared to the two-pulse AI. Second, the lifetime of the three-pulse echo depends strongly on the value of $T_{21}$. As $T_{21}$ increases, the signal lifetime approaches that of the two-pulse AI. This feature comes about because, between the second and third sw pulses, the wave packets that interfere at the echo times have a constant spatial separation \[see \[fig:1b-RecoilDiagram-ThreePulse\]\], which is given by $\Delta z = \bar{N} \hbar q T_{21}/M$. From this expression, it is clear that $\Delta z$ can be controlled by $T_{21}$ and the choice of echo order, $\bar{N}$. By decreasing this separation, the interferometer becomes less sensitive to decoherence from non-linear $B$-fields since phase shifts produced by this effect become approximately common mode between interfering momentum states. Reference also used this interferometer and a magnetic guide to show that smaller spatial separations lead to increased timescales.
In general, the lifetime for the three-pulse echo can be tailored to last much longer than that of the two-pulse echo, which is advantageous for precisely measuring the effects of external forces. For example, we achieve timescales as large as $\sim 220$ ms for $T_{21}$ fixed at $\sim 1.3$ ms—which is much closer to the transit time limit than the lifetime of the two-pulse echo. To the best of our knowledge, the only experiment that has achieved longer timescales for this AI have employed magnetic guides [@Su-PRA-2010] to limit transverse expansion of the sample and thereby extending the transit time.
Investigations of External $B$-Gradients
----------------------------------------
When $T_{21}$ is large, the two-pulse AI can be used to explore the sensitivity to small external $B$-gradients. We demonstrate the detection of changes in the $B$-gradient as small as $\sim 4 \times 10^{-5}$ G/cm in \[fig:6a-EchoSignalVsGradient\]. Here, the $\bar{N} = 1$ echo signal was recorded with $T_{21}$ fixed at $\sim 40.6$ ms for various applied gradients. Changes in the gradient were facilitated by varying the current through the set of vertical quadrupole coils centered on the MOT (see \[fig:3-ExpSetup\]). The smallest controllable increment in current we could achieve was 1 mA, which corresponds to a change of $\sim 0.04$ mG/cm as estimated from an independent calibration based on a flux-gate magnetometer.
In a similar experiment, the $\bar{N} = 1$ echo energy was measured for $T_{21}$ fixed at $\sim 40.6$ ms as a function of $\beta$, as shown in \[fig:6b-EchoEnergyVsGradient\]. Here, it is clear that the echo energy has a strong periodic dependence on the applied $B$-gradient. These data provide confirmation of the theoretical prediction given by [(\[eqn:E(2)\_beta\])]{} and [(\[eqn:phi(2)\_beta\])]{}. This dependence is produced by the interference between electric fields scattered off of gratings produced by different magnetic sub-levels. For example, for a given $\beta$, gratings produced by states ${\left| F\,m_F \right\rangle}$ and ${\left| F\,m_F' \right\rangle}$ undergo phase shifts $m_F \phi^{(2)}_{\beta}$ and $m_F' \phi^{(2)}_{\beta}$, respectively, where $\phi^{(2)}_{\beta}$ is given by [(\[eqn:phi(2)\_beta\])]{}. For constructive interference between fields scattered by these states, the $B$-gradient must satisfy $(m_F - m_F') \phi^{(2)}_{\beta} / 2 = 2 n \pi$, where $n$ is an integer. Thus, as $\beta$ is varied, the phase shift induced in the ${\left| F\,m_F \right\rangle}$ and ${\left| F\,m_F' \right\rangle}$ gratings produces periodic constructive (destructive) interference in the total scattered field, and therefore, maxima (minima) in the echo energy. This process occurs simultaneously in all $2F+1$ sub-levels. As a result, the observed signal is a weighted sum of the scattered fields from all states. Here, there are $2F (2F+1) = 42$ pairs of states that produce interference—although not all pairs have unique contributions. Since the excitation beams were circularly polarized in the experiment, the fields scattered from the extreme states (${\left| 3\,3 \right\rangle}$ or $|3\,$$-3\rangle$) dominate the signal.
We use the following model, based on the squared modulus of [(\[eqn:E(2)\_beta\])]{}, to fit the data shown in \[fig:6b-EchoEnergyVsGradient\]: $$\begin{aligned}
\begin{split}
\label{eqn:ModelFit-2Pulse}
S^{(2)}(\beta, T_{21})
& = S_0 e^{-(T_{21} - t_0)^2/\tau^2} \sum_{m_F, m_F'} a_{m_F} a_{m_F'} \\
& \times e^{i A (m_F - m_F') \beta \bar{N} (\bar{N} + 1) (T_{21} - t_1)^2},
\end{split}\end{aligned}$$ where $S_0$, $t_1$ and the set of $\{a_{m_F}\}$ are free parameters, $A = q g_F \mu_B/2M$ is a constant and $t_0$ was set to $T_{21}$ for this data. In this model, the Gaussian factor outside the sum is added phenomenologically to account for signal loss due to both the transit time and any decoherence in the system. Also, each $a_{m_F}$ is proportional to the magnetic sub-level population, $|\alpha_{m_F}|^2$, through [(\[eqn:E(2)\_mF\])]{}. As a result, these parameters are constrained to be positive. All other fit parameters are unconstrained. In principle, it should be possible to obtain the sub-level populations from the set of best fit parameters $\{a_{m_F}\}$. However, determining the constant of proportionality between the $a_{m_F}$, the populations and the scattered field intensity is complicated [@Slama-PRA-2006; @Schilke-PRL-2011] and not addressed by the theory presented here. We emphasize, however, that fits to data presented in this work give similar results for the set of $\{a_{m_F}\}$, which are consistent with our expectations for circularly polarized excitation beams.
It is interesting that a measurement of the parameter $A$ from data similar to that shown in \[fig:6b-EchoEnergyVsGradient\] can be used to test the theory of magnetic interactions [@Chan-PRA-2011; @Anthony-PRA-1994].
Surveys of gradient-induced modulation on the echo signal shown in \[fig:7-Gradients-N1-N2\] provide additional confirmation of the theory outlined in \[sec:Theory\] and the Appendix. Figure \[fig:7a-2Pulse-Gradients-N1-N2\] indicates that, in the presence of a $B$-gradient, the two-pulse echo energy becomes modulated at a frequency that increases linearly with $T_{21}$ (i.e. the modulation is chirped), as predicted by [(\[eqn:omega(2)\_beta\])]{}. This figure shows gradient oscillations for both the $\bar{N} = 1$ and the $\bar{N} = 2$ orders of the two-pulse echo. Since the chirp rate increases as $\bar{N} (\bar{N} + 1)$, the second order echo is modulated at a rate three times that of the first order echo. Confirmation of this is provided by a least-squares fit to the data based on [(\[eqn:ModelFit-2Pulse\])]{}, as shown by the solid lines in \[fig:7a-2Pulse-Gradients-N1-N2\]. Since the gradient was held fixed in the experiment, the fits to the two data sets should provide similar measurements of $|\beta|$. The two measurements yield $|\beta| = 9.30(1)$ mG/cm for $\bar{N} = 1$ and $|\beta| = 9.34(1)$ mG/cm for $\bar{N} = 2$ [^2], where the quoted error is the $1\sigma$ statistical uncertainty generated by the fit. These measurements are in good agreement with each other and an independent measurement from a flux-gate magnetometer. We emphasize that accurate fits to these data and the extraction of $\beta$ were possible only through the development of the multi-level formalism presented in the Appendix. In particular, since the oscillations shown in \[fig:7a-2Pulse-Gradients-N1-N2\] do not occur with 100% contrast (i.e. each oscillation minima does not reach the level of the noise), a model including only two magnetic sub-levels with equal excitation probabilities, such as that described in , is insufficient to model the data.
Figure \[fig:7b-3Pulse-Gradients-N1-N2\] shows data similar to that shown in \[fig:7a-2Pulse-Gradients-N1-N2\], but for the first two orders of the three-pulse echo and a slightly larger $B$-gradient. This data illustrates that the three-pulse AI is less sensitive to gradients than the two-pulse AI. Since $T_{21}$ is fixed at 2.0 ms, the modulation frequency is constant and proportional to $\bar{N}$ and $T_{21}$—confirming the predictions of [(\[eqn:omega(3)\_beta\])]{}. The data is fit to the following model: \[eqn:ModelFit-3Pulse\] & & S\^[(3)]{}(, T\_[32]{}, T\_[21]{}) = S\_0 e\^[-(T\_[32]{} - t\_0)\^2/\^2]{} \_[m\_F, m\_F’]{} a\_[m\_F]{} a\_[m\_F’]{}\
& & e\^[i A (m\_F - m\_F’) ]{}, which is based on [(\[eqn:E(N)\])]{}, [(\[eqn:E(N)\_mF\])]{} and [(\[eqn:phi(N)\])]{}, with a Gaussian decay factor added phenomenologically. All other parameters in this model are similar to those discussed in reference to [(\[eqn:ModelFit-2Pulse\])]{}. Measurements of the magnitude of the gradient from fits to these data yield $|\beta| = 17.50(4)$ mG/cm and $|\beta| = 17.78(5)$ mG/cm for the $\bar{N} = 1$ and $\bar{N} = 2$ echoes, respectively. These two measurements differ by more than $5\sigma$, which deserves some explanation. By inspecting the fit to the $\bar{N} = 2$ echo, it is clear that the data is not well-modeled by a single frequency sinusoid as $T_{32}$ becomes large. This provides evidence that the atoms are sampling different gradients as they drop under gravity—an effect that is not accounted for in the theory. By analyzing different sections of this data, we estimate that the gradient varies by as much as $\sim 1.6$ mG/cm between $T_{32} \sim 40$ ms and 100 ms—during which time atoms fall $\sim 4$ cm. Independent measurements of the curvature of the $B$-field, where $|\beta|$ was found to change by $\sim 0.4$ mG/cm every centimeter, are consistent with the variation in $\beta$ detected by atoms.
Although we have demonstrated sensitivity to *changes* in the $B$-gradient as small as $\sim 4 \times 10^{-5}$ G/cm using $T_{21} \sim 40$ ms with the two-pulse AI, our ability to measure the absolute magnitude of the applied gradient is less sensitive. This is primarily because the measurement is based on fitting data to an oscillatory model and extracting the modulation rate—which cannot be done accurately without the presence of an oscillatory component in the data. To estimate the smallest measurable $B$-gradient with the two interferometers, we tuned the applied fields for each AI separately such that the first revival in the $\bar{N} = 1$ echo energy occurred at the largest time. The resulting data are shown in \[fig:8-SmallGradient\], which yielded measurements of $|\beta| = 0.26(3)$ mG/cm for the two-pulse AI \[\[fig:8a-2Pulse-SmallGradient\]\] and $|\beta| = 9.5(1)$ mG/cm for the three-pulse AI \[\[fig:8b-3Pulse-SmallGradient\]\].
Applications to Gravity {#sec:Gravity}
=======================
The apparatus shown in \[fig:3-ExpSetup\] is designed for measurements of the atomic recoil frequency [@Barrett-SPIE-2011]. As a result, it is not isolated from external vibrations and is unsuitable for measurements of the optical phase of the scattered read-out light using heterodyne detection. For this reason, a measurement of $g$ from the phase of the atomic grating [@Weel-PRA-2006; @Barrett-Advances-2011] is beyond the scope of this Article and will be presented elsewhere. However, the aforementioned results relating to $B$-gradients validate theoretical predictions that can be applied to precise measurements of gravity. In this section, we discuss the feasibility of such a measurement by applying the formalism presented in the Appendix.
The best portable gravimeter [@Niebauer-Metrologia-1995] uses an optical Mach-Zehnder interferometer where one arm contains a free-falling corner-cube for position-sensitive measurements of $g$ at the level of $\sim 1$ ppb over a few minutes. The position sensitivity in these devices comes from detecting interference fringes as a function of the drop time of the cube relative to an inertial frame defined by a stationary mirror. The frequency at which the fringes accumulate scales linearly with the drop time (i.e. the frequency is chirped). The matter-wave analog of this gravimeter is the two-pulse echo AI [@Weel-PRA-2006; @Barrett-Advances-2011], where changes in the phase of the grating due to gravity are detected relative to the nodes of a pulsed sw—which serves as the inertial reference frame. In this case, the accumulation of fringes due to matter-wave interference is also described by a chirped-frequency sinusoid.
We now review the main results of the grating echo theory that pertain to gravity. The gravitational potential can be written as (z) = M g z, where the force is $\mathcal{F} = - M g$ and $\hat{I}$ is the $(2F+1) \times (2F+1)$ identity matrix. The effect on the echo AI is similar to that of the $B$-gradient on a sample that has been optically pumped into a single state. Since gravity acts equally on all states, the phase shift of the grating produced by each state is the same. Therefore, the expression for the field scattered from the grating simplifies significantly compared to [(\[eqn:E(2)\_beta\])]{}: \[eqn:E(2)\_g\] E\_g\^[(2)]{}(t;) = e\^[i \^[(2)]{}\_g(t;)]{}, where the grating phase due to gravity is $$\begin{aligned}
\begin{split}
\label{eqn:phi(2)_g}
& \phi^{(2)}_g(\Delta t;T_{21}) = -\frac{q g}{2} \left\{ \bar{N}(\bar{N}+1)T_{21}^2 \right. \\
& \;\;\;\;\; \left. + \, 2 \big[ T_1 + (\bar{N}+1)T_{21} \big] \Delta t + \Delta t^2 \right\},
\end{split}\end{aligned}$$ as determined by [(\[eqn:phi(2)\_F\])]{}. This phase cannot be detected from the intensity of the scattered light because there is no differential phase shift between magnetic sub-levels—thus, there is no amplitude modulation of the grating [@Weel-PRA-2006]. The scaling of the grating phase with $T_{21}^2$ in [(\[eqn:phi(2)\_g\])]{} shows the similarity between the two-pulse AI and the optical Mach-Zehnder interferometer discussed in .
Figures \[fig:9a-GravitySignal\] and \[fig:9b-GravitySignal\] show the expected two-pulse AI signal in the presence of gravity as a function of $T_{21}$—illustrating that the modulation frequency is chirped linearly with $T_{21}$ ($\omega^{(2)}_g = \partial \phi^{(2)}_g/ \partial T_{21} \propto T_{21}$). As $T_{21}$ increases, $\omega^{(2)}_g$ becomes larger than the recoil frequency (for the first order echo in $^{85}$Rb, this occurs when $T_{21} \gtrsim 300$ $\mu$s), and $T_{21}$ must be incremented in steps less than $\tau_q$ to avoid undersampling the frequency. However, this effect causes reduced sensitivity to the grating phase, since modulation at the recoil frequency produces periodic regions with small signal amplitude. Additionally, as shown in , this AI is very sensitive to phase changes due to mirror vibrations, which can be detrimental to measurements of $g$ using this technique.
Figures \[fig:9c-GravitySignal\] and \[fig:9d-GravitySignal\] show the expected three-pulse signal as a function of $T_{21}$ in the presence of gravity. It is clear that the envelope of the scattered field has a complicated periodic dependence on $T_{21}$, with a zero every $\tau_q \sim 32$ $\mu$s due to the destructive interference of momentum states differing by the two-photon recoil momentum, $\hbar q$. This is similar to the two-pulse case shown in \[fig:9a-GravitySignal\] and \[fig:9b-GravitySignal\]. Here, the grating phase modulation frequency is given by | | = q g \[|[N]{} (|[N]{} + 1) T\_[21]{} + |[N]{} T\_[32]{} + (|[N]{} + 1) t\], which is identical to the two-pulse case, $\omega^{(2)}_g$, with the addition of the term proportional to $T_{32}$.
Figures \[fig:9e-GravitySignal\] and \[fig:9f-GravitySignal\] show the expected three-pulse signal as a function of $T_{32}$, with $T_{21}$ fixed at 5 ms. In this case, there is no sensitivity to atomic recoil, so the envelope remains at a constant level as $T_{32}$ is varied. The frequency of the phase modulation is also fixed by $\bar{N}$ and $T_{21}$, as given by \^[(3)]{}\_g (T\_[21]{}) = | | = q g ( |[N]{} T\_[21]{} + t ). For the conditions presented in these figures, $\omega_g^{(3)} \sim 2\pi \times 125$ kHz. The work of shows that the three-pulse AI is significantly less sensitive to mirror vibrations than the two-pulse AI if $T_{32} \gg T_{21}$. Our results have also shown that this configuration is less sensitive to $B$-gradients. For all these reasons, this AI is particularly well-suited for precise measurements of $g$.
Simulations of the two-pulse AI signal with $\bar{N} = 1$, $T_{21} \sim 150$ ms and a phase error of 1% suggest the precision of a measurement of $g$ should be $\sim 1.4$ ppb. Similarly, we estimate a precision of $\sim 0.4$ ppb for the three-pulse AI using $\bar{N} = 1$, $T_{21} = 75$ ms, $T_{32}$ varied over 150 ms and the same phase error. From these estimates, it is clear that these AIs can have greater sensitivity than the best industrial sensor [@Niebauer-Metrologia-1995]. Since the precision scales linearly with the phase error, we anticipate further improvements in sensitivity without extending the timescale. If systematic effects of such a cold atom gravimeter are characterized, it may be possible for the AI experiment to serve as a reference to calibrate other gravimeters.
Conclusions {#sec:Conclusions}
===========
Measurements of applied $B$-gradients using both the two- and three-pulse techniques are in good agreement with independent measurements of $\beta$ using a flux gate magnetometer. We have demonstrated sensitivity to changes in the $B$-gradient at the level of $\sim 4 \times 10^{-5}$ G/cm. Absolute measurements of $\beta$ as small as $\sim 3 \times 10^{-4}$ G/cm were also possible using the two-pulse AI. These measurements indicate that an accurate description of the data presented above requires the inclusion of multiple magnetic sub-levels. We have also shown sensitivity to spatial variation in the $B$-gradient using a long-lived second order ($\bar{N} = 2$) three-pulse echo. It is this non-linearity in the $B$-field that affects the timescale in echo AIs rather than the presence of a small, uniform $B$-gradient.
As tests of the theoretical results presented in \[sec:Theory\], we have separately confirmed the linear dependence of the $\beta$-induced oscillation frequencies, $\omega^{(2)}_{\beta}$ and $\omega^{(3)}_{\beta}$ \[given by [(\[eqn:omega(2)\_beta\])]{} and [(\[eqn:omega(3)\_beta\])]{}, respectively\], on the $B$-gradient. We have also verified that these frequencies both scale linearly with $T_{21}$, and, for the three-pulse AI, $\omega^{(3)}_{\beta}$ is constant as a function of $T_{32}$.
Since we have achieved signal lifetimes approaching the transit time limit, we have shown that fountain-based experiments are possible with grating echo AIs. The advantage of a fountain configuration is that the spatial extent of the AI ($\sim 11$ cm for 300 ms timescale) can be made small, which reduces the requirements for inhomogeneous $B$-field suppression. Such a configuration is ideal for precise measurements of gravity, particularly with the three-pulse AI. Passive suppression of $B$-fields with larger cancelation coils, or optically pumping into the $m_F = 0$ sub-level, represent two ways in which such a measurement can be realized.
Despite the widespread use of Raman-type AIs for inertial sensing [@Yu-ApplPhysB-2006; @LeGouet-ApplPhysB-2006; @Young-OSA-2007], grating echo-type AIs—which offer reduced experimental complexity—are also excellent candidates for precision measurements of $\omega_q$ and $g$. This work has brought about understanding of systematic effects produced by $B$-gradients on these measurements.
In summary, we have developed a complete understanding of the effects of a constant force that applies to all time-domain AIs. Although the sensitivity for AI-based gradient detection cannot compete with commercial magnetic gradiometers (which offer sensitivities of the order of $\sim 1$ pT/m), the technique is useful for absolute measurements of gradients in cold atom experiments.
This work was supported by the Canada Foundation for Innovation, Ontario Innovation Trust, Natural Sciences and Engineering Research Council of Canada, Ontario Centres of Excellence and York University. We would also like to thank Itay Yavin of McMaster University for helpful discussions and Adam Carew of York University for building phase-locked loops.
Appendix {#appendix .unnumbered}
========
In this appendix, we derive expressions for the signals generated by the two- and three-pulse interferometers in the presence of a constant external force, $\mathcal{F}$. In , a similar calculation for the two-pulse signal is given, in which only two ground state sub-levels are considered, and effects due to spontaneous emission are ignored. Here, we account for $2F+1$ magnetic sub-levels in the field scattered from the atoms, as well as spontaneous emission during the excitation pulses. Both of these effects are crucial for an accurate description of these interferometers. We also give a general expression for the signal generated by an $N$-pulse AI from which all classes of grating-echo interferometers can be realized.
The potential is assumed to have the form $\hat{U}(z) = -\hat{\mathcal{M}} z$, where $\hat{\mathcal{M}} = -\partial \hat{U}/\partial z$ is an operator that computes with $z$ and $p$, and acts on the basis states ${\left| F\;m_F \right\rangle}$ with eigenvalues $m_F \mathcal{F}$. Here, $\mathcal{F}$ is a constant with units of force. We proceed by computing the ground state wave function after the application of each sw pulse at times $t = T_1$ and $T_2$, with a period of evolution before, between and after each pulse (with durations $T_1$, $T_2 - T_1$ and $t - T_2$, respectively) in the presence of the force. During the application of each sw pulse, the kinetic and potential energy terms in the Hamiltonian are ignored by assuming the pulses are sufficiently short such that the atom does not move significantly (Raman-Nath approximation). In this manner, the sw pulses are treated as Dirac $\delta$-function excitations, even though they are given durations $\tau_j$ for the purposes of the calculation.
The interferometer signal is defined as the back-scattered electric field amplitude at the time of an echo, which is proportional to the amplitude of the $q$-Fourier harmonic of the density distribution at these times. The results for the two-pulse AI signal are then generalized for an $N$-pulse AI, from which we compute the three-pulse AI signal.
The Hamiltonian for the ground state ${\left| F\;m_F \right\rangle}$ in the presence of a sw field and an external potential, $\hat{U}(z)$, can be approximated by [@Beattie-PRA-2008; @Barrett-PRA-2010] \[eqn:Hg\] \_[m\_F]{} = + \_[m\_F]{} e\^[i]{} (q z) + (z), where $\theta$ is a phase associated with spontaneous emission during the sw pulse = \^[-1]{} ( - ), and $\chi_{m_F}$ is a two-photon Rabi frequency given by \_[m\_F]{} = ( 1 + )\^[-1/2]{} ( C\^[F1F+1]{}\_[m\_Fq\_Lm\_F + q\_L]{} )\^2. Here, $\Omega_0$ is the on-resonance Rabi frequency for a two-level atom, $\Delta = \omega_L - \omega_0$ is the atom-field detuning with atomic resonance frequency $\omega_0$ and laser frequency $\omega_L$, $\gamma$ is half of the spontaneous emission rate, and $(C^{F\;\;\;\;1\;\;\;F+1}_{m_F\;q_L\;m_F + q_L})$ is a Clebsch-Gordan coefficient for a light field with a polarization state $q_L$. We ignore the excited state in this treatment, since the field is assumed to be relatively weak and far off-resonance $(|\Delta| \gg \Omega_0,\, \gamma)$. We also neglect the Zeeman shift of magnetic sub-levels by assuming $|\Delta| \gg g_F \mu_B B/\hbar$.
The amplitude of the ground state wave function at $t = 0$ can be written as a superposition of spin states: a(z,0) = \_[m\_F]{} a\_[m\_F]{}(z,0) [| Fm\_F ]{}, where the amplitude of each spin state is
a\_[m\_F]{}(z,0) & = & e\^[i p\_0 z/]{},\
a\_[m\_F]{}(p,0) & = & \_[m\_F]{} (p - p\_0).
Here, $p_0$ is the initial momentum of the atom along the $z$-direction, $|\alpha_{m_F}|^2$ is the population of state ${\left| F\;m_F \right\rangle}$, with $\sum_{m_F} |\alpha_{m_F}|^2 = 1$, and $a_{m_F}(p,0)$ is the amplitude of the spin state in momentum space.
The main challenge in this calculation is evolving the wave function between sw pulses in the presence of the additional potential energy, $\hat{U}(z)$. In the absence of this potential, it is straightforward to integrate the Schrödinger equation in momentum space. However, with $\hat{U}(z)$ present, we have the following equation of motion: i = ( - z ) a\_[m\_F]{}(p,t). One can integrate this equation to find \[eqn:aexp(z+p2)\] a\_[m\_F]{}(p,t) = e\^[-i(- z + p\^2/2M) t/]{} a\_[m\_F]{}(p,0), but some care must be taken when evaluating the right hand side. The challenge arises from the fact that $z$ and $p = -i\hbar \partial/\partial z$ are non-commuting operators. As a result, the exponential in [(\[eqn:aexp(z+p2)\])]{} is really a matrix exponential of non-commuting matrices $\hat{A}$ and $\hat{B}$. In general $e^{\hat{A} + \hat{B}} \not= e^{\hat{A}}e^{\hat{B}}$, but one can use the Zassenhaus formula [@Suzuki-CommunMathPhys-1977] to expand the matrix exponential as $$\begin{aligned}
\begin{split}
\label{eqn:Zassenhaus}
e^{\xi(\hat{A} + \hat{B})}
& = e^{\xi\hat{A}} e^{\xi\hat{B}} e^{-\xi^2 [\hat{A},\hat{B}]/2} \\
& \times e^{\xi^3([\hat{A},[\hat{A},\hat{B}]] - 2[[\hat{A},\hat{B}],\hat{B}])/6} \cdots,
\end{split}\end{aligned}$$ where $\xi$ is an arbitrary constant. The higher order factors (represented by $\cdots$ in the above equation) vanish if $[[\hat{A},\hat{B}],\hat{B}]$ and $[\hat{A},[\hat{A},\hat{B}]]$ commute with all higher order nested commutators. Choosing $\hat{A} = -\hat{\mathcal{M}} z$ and $\hat{B} = p^2/2M$ [^3], and using the commutation relations $[z,p^2] = i 2\hbar p$, $[z, p] = i\hbar$, we find:
\[eqn:commutators\] & = & -i p,\
& = & - ,\
& = & 0.
Using [(\[eqn:Zassenhaus\])]{} with $\xi = -i t /\hbar$ and the commutators in [(\[eqn:commutators\])]{}, [(\[eqn:aexp(z+p2)\])]{} becomes \[eqn:aexp-expanded\] a\_[m\_F]{}(p,t) = e\^[i t z/]{} e\^[-i p\^2 t/2M]{} e\^[-i p t\^2/2M]{} e\^[-i\^2 t\^3/6M]{} a\_[m\_F]{}(p,0). Since $e^{\xi (\hat{\mathcal{M}})^n} {\left| F\,m_F \right\rangle} = e^{\xi (m_F \mathcal{F})^n} {\left| F\,m_F \right\rangle}$, it follows that the amplitude of the state ${\left| F\;m_F \right\rangle}$ before the onset of the first sw pulse is
a\_[m\_F]{}(p,t) & = & \_[m\_F]{} e\^[i (m\_F ) t z/]{} e\^[-i p\^2 t/2M]{} e\^[-i (m\_F ) p t\^2/2M]{} e\^[-i (m\_F )\^2 t\^3/6M]{} (p - p\_0),\
a\_[m\_F]{}(z,t) & = & e\^[i (p\_0 + m\_F t) z/]{} e\^[-i \_0 t/]{} e\^[-i (m\_F ) p\_0 t\^2/2M]{} e\^[-i (m\_F )\^2 t\^3/6M]{},
where $\epsilon_0 = p_0^2/2M$ is the initial kinetic energy of the atom.
The first sw pulse, applied at $t = T_1$, diffracts the atom into a superposition of momentum states. The wave function is computed in position space using the Raman-Nath approximation and integrating the Schrödinger equation to obtain
a\_[m\_F]{}\^[(1)]{}(z,T\_1) & = & a\_[m\_F]{}(z,T\_1) \_n (-i)\^n J\_n(\^[(1)]{}\_[m\_F]{}) e\^[inqz]{},\
\[eqn:a\_mF(1)(p,T\_1)\] a\_[m\_F]{}\^[(1)]{}(p,T\_1) & = & \_[m\_F]{} e\^[-i\_0 T\_1/]{} e\^[-i (m\_F ) p\_0 T\_1\^2 / 2 M ]{} e\^[-i(m\_F )\^2 T\_1\^3/6 M ]{}\
& & \_n (-i)\^n J\_n(\_[m\_F]{}\^[(1)]{}) (p - p\_0 - m\_F T\_1 - nq).
Here, $\Theta_{m_F}^{(1)} \equiv u_{m_F}^{(1)} e^{i\theta}$ is the (complex) area of pulse 1, $u_{m_F}^{(1)} = \chi_{m_F} \tau_1$, $\tau_1$ is the duration of the pulse, and $a_{m_F}^{(1)}(p,T_1)$ is the wave function in momentum space. The superscript $(1)$ on $a_{m_F}^{(1)}$ denotes the number of sw pulses that have been applied to the atom so far. We use the prescription of [(\[eqn:aexp-expanded\])]{} to evolve the amplitude in momentum space \[[(\[eqn:a\_mF(1)(p,T\_1)\])]{}\] until the onset of the second pulse $$\begin{aligned}
\begin{split}
a_{m_F}^{(1)}(p,t)
& = \alpha_{m_F} e^{i (m_F \mathcal{F}) (t - T_1) z/\hbar} e^{-i [p_0^2 T_1 + p^2 (t - T_1)]/2M\hbar}
e^{-i (m_F \mathcal{F}) [p_0 T_1^2 + p (t - T_1)^2]/2M\hbar} \\
& \times e^{-i(m_F \mathcal{F})^2 [T_1^3 + (t - T_1)^3]/6M\hbar}
\sum_n (-i)^n J_n(\Theta_{m_F}^{(1)}) \delta(p - p_0 - m_F \mathcal{F} T_1 - n\hbar q).
\end{split}\end{aligned}$$ To apply the next sw pulse to the wave function, it is convenient to transform back to position space: $$\begin{aligned}
\begin{split}
a_{m_F}^{(1)}(z,t)
& = \frac{\alpha_{m_F}}{\sqrt{2\pi\hbar}} e^{i(p_0 + m_F \mathcal{F} t) z /\hbar} e^{-i[p_0^2 T_1 + (p_0 - m_F \mathcal{F} T_1)^2(t - T_1)]/2M\hbar} \\
& \times e^{-i (m_F \mathcal{F}) [p_0 T_1^2 + (p_0 + m_F \mathcal{F} T_1) (t - T_1)^2]/2M\hbar} e^{-i(m_F \mathcal{F})^2 [T_1^3 + (t - T_1)^3]/6M\hbar} \\
& \times \sum_n (-i)^n J_n(\Theta_{m_F}^{(1)}) e^{i n q z} e^{-i n q v_0 (t - T_1)} e^{-i n^2 \omega_q (t - T_1)}
e^{-i n q (m_F \mathcal{F}) [(t - T_1)^2 + 2 T_1 (t - T_1)]/2M}.
\end{split}\end{aligned}$$ Here, $v_0 = p_0/M$ is the initial velocity of the atom and $\omega_q = \hbar q^2/2M$ is the two-photon recoil frequency. Applying the second pulse at $t = T_2$, the wave function becomes $$\begin{aligned}
\begin{split}
a_{m_F}^{(2)}(z,T_2)
& = \frac{\alpha_{m_F}}{\sqrt{2\pi\hbar}} e^{i(p_0 + m_F \mathcal{F} T_2) z /\hbar} e^{-i \epsilon_0^2 T_2/\hbar}
e^{-i p_0 (m_F \mathcal{F}) T_2^2/2M\hbar} e^{-i (m_F \mathcal{F})^2 T_2^3/6M\hbar} \\
& \times \sum_{n,m} (-i)^{(n+m)} J_n(\Theta_{m_F}^{(1)}) J_m(\Theta_{m_F}^{(2)}) e^{i(n+m)qz} e^{-i n q v_0 (T_2 - T_1)} e^{-i n^2 \omega_q (T_2 - T_1)} e^{-i n q (m_F \mathcal{F}) (T_2^2 - T_1^2)/2M}.
\end{split}\end{aligned}$$ To evolve the wave function in the presence of the external force until time $t$, once again we transform into $p$-space and use [(\[eqn:aexp-expanded\])]{} to obtain $$\begin{aligned}
\begin{split}
a_{m_F}^{(2)}(p,t)
& = \alpha_{m_F} e^{i(m_F \mathcal{F}) (t - T_2) z /\hbar} e^{-i [p_0^2 T_2 + p^2 (t - T_2)]/2M\hbar}
e^{-i (m_F \mathcal{F}) [p_0 T_2^2 + p (t - T_2)^2]/2M\hbar} e^{-i (m_F \mathcal{F})^2 [T_2^3 + (t - T_2)^3]/6M\hbar} \\
& \times \sum_{n,m} (-i)^{(n+m)} J_n(\Theta_{m_F}^{(1)}) J_m(\Theta_{m_F}^{(2)}) e^{-i n q v_0 (T_2 - T_1)} e^{-i n^2 \omega_q (T_2 - T_1)} e^{-i n q (m_F \mathcal{F}) (T_2^2 - T_1^2)/2M} \\
& \times \delta[p - p_0 - m_F \mathcal{F} T_2 - (n + m) \hbar q].
\end{split}\end{aligned}$$ Finally, the amplitude in position space after the second pulse can be shown to be $$\begin{aligned}
\begin{split}
a_{m_F}^{(2)}(z,t)
& = \frac{\alpha_{m_F}}{\sqrt{2\pi\hbar}} e^{i(p_0 + m_F \mathcal{F} t) z /\hbar} e^{-i \epsilon_0 t/\hbar}
e^{-i (m_F \mathcal{F}) p_0 t^2/2M\hbar} e^{-i (m_F \mathcal{F})^2 t^3/6M\hbar} \\
& \times \sum_{n,m} (-i)^{(n+m)} J_n(\Theta_{m_F}^{(1)}) J_m(\Theta_{m_F}^{(2)}) e^{i(n+m)qz} e^{-i q v_0 [n (T_2 - T_1) + (n+m) (t - T_2)]} \\
& \times e^{-i \omega_q [n^2 (T_2 - T_1) + (n+m)^2 (t - T_2)]} e^{-i q (m_F \mathcal{F})[ n (T_2^2 - T_1^2) + (n+m) (t^2 - T_2^2)]/2M}.
\end{split}\end{aligned}$$ To compute the field scattered from the atomic interference as a function of $t$, we use the $q$-Fourier component of the ground state density, $\rho_{m_Fm_F}^{(2)}(z,t) = |a_{m_F}^{(2)}(z,t)|^2$, which can be shown to be $$\begin{aligned}
\begin{split}
\label{eqn:rho(2)}
\rho_{m_F m_F}^{(2)}(z,t)
& =
\frac{|\alpha_{m_F}|^2}{2\pi\hbar} \sum_{n,m,n',m'} (-i)^{n+m-n'-m'} J_n(\Theta_{m_F}^{(1)}) J_m(\Theta_{m_F}^{(2)}) J_{n'}(\Theta_{m_F}^{(1)\,*}) J_{m'}(\Theta_{m_F}^{(2)\,*}) e^{i(n+m-n'-m')qz} \\
& \times
e^{-i q v_0 [(n-n') (T_2 - T_1) + (n+m-n'-m') (t - T_2)]} e^{-i \omega_q \{(n^2-n'^2) (T_2 - T_1) + [(n+m)^2 - (n'+m')^2](t - T_2)\}} \\
& \times
e^{-i q (m_F \mathcal{F})[(n-n') (T_2^2 - T_1^2) + (n+m-n'-m') (t^2 - T_2^2)]/2M}.
\end{split}\end{aligned}$$ Since the density distribution contains frequency components that depend only on the *difference* between interfering momentum states, we recast the sums over $n'$ and $m'$ in terms of $\nu\bar{N} = n-n'$ and $\nu = n'+m'-n-m$ (the integer difference between momentum states after the first and second pulses, respectively): $$\begin{aligned}
\begin{split}
\label{eqn:rho(2)-reduced}
\rho_{m_F m_F}^{(2)}(z,t)
& =
-\frac{|\alpha_{m_F}|^2}{2\pi\hbar} \sum_{\nu,\bar{N},n,m} i^\nu J_n(\Theta_{m_F}^{(1)}) J_{n-\nu\bar{N}}(\Theta_{m_F}^{(1)\,*}) J_m(\Theta_{m_F}^{(2)}) J_{m+\nu(\bar{N}+1)}(\Theta_{m_F}^{(2)\,*}) e^{-i\nu qz} \\
& \times
e^{i \nu q v_0 [(t - T_2) - \bar{N} (T_2 - T_1)]}
e^{i \nu \omega_q \{[2(n+m)+\nu](t - T_2) - \bar{N}(2n-\nu\bar{N}) (T_2 - T_1)\}} \\
& \times
e^{i \nu q (m_F \mathcal{F}) [(t^2-T_2^2) - \bar{N}(T_2^2 - T_1^2)]/2M}.
\end{split}\end{aligned}$$ The scattered field is proportional to the $q$-Fourier harmonic of $\rho_{m_F m_F}^{(2)}(z,t)$ \[the coefficient of the $e^{-i \nu q z}$ term in [(\[eqn:rho(2)-reduced\])]{}, with $\nu = 1$\]. Summing over all magnetic sub-levels in the ground state, one can show that E\^[(2)]{}\_(t;) = \_[m\_F]{} E\^[(2)]{}\_[m\_F]{}(t;) e\^[i m\_F \^[(2)]{}\_(t;)]{}, where $$\begin{aligned}
\begin{split}
\label{eqn:E(2)_mF}
E^{(2)}_{m_F}(t;\bm{T})
& \propto |\alpha_{m_F}|^2 \left( C^{F\;\;\;\;1\;\;\;F+1}_{m_F\;q_L\;m_F + q_L} \right)^2 \sum_{\bar{N}} (-1)^{\bar{N}+1} e^{-\left[ \left(t - t_{\rm{echo}}^{(2)}\right)/\tau_{\rm{coh}} \right]^2} e^{i q v_0 \left( t - t^{(2)}_{\rm{echo}} \right)} \\
& \times J_{\bar{N}} \left( 2 u_{m_F}^{(1)} \sqrt{\sin(\varphi_1 - \theta) \sin(\varphi_1 + \theta)} \right)
J_{\bar{N}+1} \left( 2 u_{m_F}^{(2)} \sqrt{\sin(\varphi_2 - \theta) \sin(\varphi_2 + \theta)} \right) \\
& \times \left( \frac{\sin(\varphi_1 + \theta)}{\sin(\varphi_1 - \theta)} \right)^{\bar{N}/2} \left( \frac{\sin(\varphi_2 - \theta)}{\sin(\varphi_2 + \theta)} \right)^{(\bar{N}+1)/2}
\end{split}\end{aligned}$$ is the field scattered from each magnetic sub-level, with recoil phases
\_1(t;) & = & \_q (t - t\_[[echo]{}]{}\^[(2)]{} ),\
\_2(t;) & = & \_q (t - T\_2),
and $m_F \phi^{(2)}_{\mathcal{F}}$ is the phase shift of the density grating produced in the ground state ${\left| F\,m_F \right\rangle}$ due to the presence of the external force, $\mathcal{F}$, with $\phi^{(2)}_{\mathcal{F}}$ given by \[eqn:phi(2)\_F\] \^[(2)]{}\_(t;) = . In deriving [(\[eqn:E(2)\_mF\])]{} we have made use of the Bessel function summation theorem [@Gradshteyn-Book-2007; @Beattie-PRA-2008; @Barrett-PRA-2010] \_n J\_n ( u e\^[i]{} ) J\_[n+]{} ( u e\^[-i]{} ) e\^[i(2n+)]{} = i\^ J\_ ( 2 u ) ( )\^[/2]{}, and we averaged over the velocity distribution of the sample assuming a Maxwellian distribution centered at $v_0$ with $e^{-1}$ width $\sigma_v = \sqrt{2 k_B \mathcal{T}/M}$. In this way, we account for the possibility of an initial launch of the atomic cloud and for the dephasing of the echo due to the distribution of Doppler phases in the sample. An additional factor of $\left( C^{F\;\;\;\;1\;\;\;F+1}_{m_F\;q_L\;m_F + q_L} \right)^2$ was added to the scattered field to account for the atom-field coupling by the read-out pulse. The scattered field lasts for a time $\tau_{\rm{coh}} = 2 /q \sigma_v$—called the coherence time—about each echo, which occur at times $t_{\rm{echo}}^{(2)} = \bar{N} (T_2 - T_1) + T_2$. The phase $\theta$ in [(\[eqn:E(2)\_mF\])]{}, associated with spontaneous emission during the excitation pulses, affects only the recoil-dependent component of the signal [@Barrett-PRA-2010].
These results can be generalized for the case of an $N$-pulse interferometer with a set of onset times $\bm{T} = \{ T_1, T_2, \ldots, T_N \}$ for which $T_{j+1} > T_j$. After $N$ sw pulses, each with pulse area $u_{m_F}^{(j)}$, the total scattered field at time $t$ is \[eqn:E(N)\] E\^[(N)]{}\_(t;) = \_[m\_F]{} E\^[(N)]{}\_[m\_F]{}(t;) e\^[i m\_F \^[(N)]{}\_(t;)]{} where $$\begin{aligned}
\begin{split}
\label{eqn:E(N)_mF}
E^{(N)}_{m_F}(t;\bm{T})
& \propto -|\alpha_{m_F}|^2 \left( C^{F\;\;\;\;1\;\;\;F+1}_{m_F\;q_L\;m_F + q_L} \right)^2 \sum_{l_1, l_2, \ldots, l_{N-1}} e^{-\left[ \left(t - t_{\rm{echo}}^{(N)}\right)/\tau_{\rm{coh}} \right]^2} e^{i q v_0 \left( t - t^{(N)}_{\rm{echo}} \right)} \\
& \times \prod_{j=1}^{N} J_{(l_j - l_{j-1})} \left( 2 u_{m_F}^{(j)} \sqrt{\sin(\varphi_j - \theta) \sin(\varphi_j + \theta)} \right) \left( \frac{\sin(\varphi_j - \theta)}{\sin(\varphi_j + \theta)} \right)^{(l_j - l_{j-1})/2}.
\end{split}\end{aligned}$$ Here, $\bm{l} = \{l_1, l_2, \ldots, l_N\}$ denotes the set of momentum states that interfere after the pulse sequence, where $l_j$ is the difference between interfering momentum states (in units of $\hbar q$) after pulse $j$. The echo times and the recoil phases are given by
\[eqn:techo(N)\] t\_[[echo]{}]{}\^[(N)]{}() & = & T\_N - \_[j=1]{}\^[N-1]{} l\_j (T\_[j+1]{} - T\_j),\
\[eqn:varphi(N)\] \_j(t;) & = & \_q \_[k=j]{}\^N l\_k (T\_[k+1]{} - T\_k),
and the contribution to the phase of the grating due to the force, $\mathcal{F}$, is \[eqn:phi(N)\] \_\^[(N)]{}(t;) = \_[j=1]{}\^N l\_j (T\_[j+1]{}\^2 - T\_j\^2). In [(\[eqn:E(N)\_mF\])]{}–[(\[eqn:phi(N)\])]{} $l_N = 1$, which corresponds to the scattered field from the $q$-Fourier harmonic of the density formed after the sw pulses, and it is understood that $l_0 = 0$ and $T_{N+1} = t$.
We now use the formalism for the $N$-pulse echo signal \[[(\[eqn:E(N)\_mF\])]{}\] to obtain an expression for the three-pulse interferometer signal discussed in \[sec:Theory\]. We begin by setting $N = 3$ and $\bm{T} = \{ T_1, T_1 + T_{21}, T_1 + T_{21} + T_{32} \}$. For an echo to occur at $t_{\rm{echo}}^{(3)} = T_1 + T_{32} + (\bar{N} + 1) T_{21}$ for any $T_1$, $T_{32}$ and $T_{21}$, [(\[eqn:techo(N)\])]{} dictates the set of $l_j$ to be $\bm{l} = \{-\bar{N}, 0, 1\}$. Then, it can be shown that the scattered field is given by $$\begin{aligned}
\begin{split}
\label{eqn:E(3)_mF}
E^{(3)}_{m_F}(t; \bm{T})
& \propto |\alpha_{m_F}|^2 \left( C^{F\;\;\;\;1\;\;\;F+1}_{m_F\;q_L\;m_F + q_L} \right)^2 \sum_{\bar{N}} (-1)^{\bar{N}+1} e^{-\left[ \left(t - t_{\rm{echo}}^{(3)}\right)/\tau_{\rm{coh}} \right]^2} e^{i q v_0 \left( t - t^{(3)}_{\rm{echo}} \right)} \\
& \times
J_{\bar{N}} \left( 2 u_{m_F}^{(1)} \sqrt{\sin(\varphi_1 - \theta) \sin(\varphi_1 + \theta)} \right)
J_{\bar{N}} \left( 2 u_{m_F}^{(2)} \sqrt{\sin(\varphi_2 - \theta) \sin(\varphi_2 + \theta)} \right) \\
& \times J_1 \left( 2 u_{m_F}^{(3)} \sqrt{\sin(\varphi_3 - \theta) \sin(\varphi_3 + \theta)} \right)
\left( \frac{\sin(\varphi_1 + \theta)}{\sin(\varphi_1 - \theta)} \right)^{\bar{N}/2}
\left( \frac{\sin(\varphi_2 - \theta)}{\sin(\varphi_2 + \theta)} \right)^{\bar{N}/2}
\left( \frac{\sin(\varphi_3 - \theta)}{\sin(\varphi_3 + \theta)} \right)^{1/2},
\end{split}\end{aligned}$$ where the recoil phases in this case are
\_1 & = & \_q ( t - t\_[[echo]{}]{}\^[(3)]{} ),\
\_2 & = & \_3 = \_q ( t - t\_[[echo]{}]{}\^[(3)]{} + |[N]{} T\_[21]{} ),
and the grating phase due to $\mathcal{F}$ is \_\^[(3)]{}(t;) & = &\
& = & { |[N]{} (|[N]{} + 1) T\_[21]{}\^2 + 2|[N]{} T\_[32]{} T\_[21]{} + 2 t + t\^2 }.
[^1]: This condition is true for far off-resonant excitation fields only. For fields closer to resonance, both the AC Stark effect and the Zeeman effect can induce a relative shift between the ground and excited states, thus affecting the response of the interferometer in a systematic way.
[^2]: Measurements of the $B$-gradient from the scattered field intensity are not sensitive to the sign of $\beta$. However, the sign can be determined using a heterodyne technique to measure the scattered electric field amplitude.
[^3]: This choice is not arbitrary. Since the $p$-space wave function is an eigenstate of the operator $p^2/2M$, but not $-\hat{\mathcal{M}} z$, we save ourselves some effort by choosing $\hat{B} = p^2/2M$ since $e^{\hat{B}}$ operates on the wave function before $e^{\hat{A}}$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $G$ be a compact connected Lie group acting on a stable complex manifold $M$ thtat has an equivariant vector bundle $E$ on it. In addition, suppose that $\phi$ is an equivariant map from $M$ to the Lie algebra $\mathfrak{g}$. We define an equivalence relation on the triples $(M, E, \phi)$ such that the set of equivalence classes forms an abelian group. We prove that this group is isomorphic to a completion of character ring $R(G)$ and so give a geometric proof of the Quantization Commutes with Reduction conjecture in the non-compact setting.'
author:
- Yanli Song
bibliography:
- 'mybib.bib'
title: 'Proper Maps, Bordism, and Geometric Quantization'
---
Introduction
============
This article focuses on the Quantization Commutes with Reduction conjecture [@Guillemin82] in the non-compact setting, a conjecture proved by Ma and Zhang [@Zhang09] (Paradan later gave a different proof [@Paradan09]). We provide a new approach to this conjecture, one that is closely related to noncommutative geometry. In addition, our methods lead to further extensions.
In the standard setting of the quantization commutes with reduction conjecture, $M$ is a compact symplectic manifold with symplectic form $\omega$. Assume that $E$ is a complex line bundle carrying a Hermitian metric and a Hermitian connection $\nabla^{E}$ such that
$$\frac{\sqrt{-1}}{2\pi}(\nabla^{E})^{2} = \omega.$$ Additionally, fix an almost complex structure $J$ such that
$$g^{TM}(v, w) = \omega(v, Jw), \ \ v, w \in TM$$ defines a Riemannian metric on $M$. Let $G$ be a compact connected Lie group, with Lie algebra denoted by $\mathfrak{g}$, acting on $M$ and $E$ in a Hamiltonian fashion. That is,
$$d\mu(\xi) = \iota_{\xi_{M}} \omega,$$ where, by the Kostant formula [@Kostant70], the moment map $\mu : M {\longrightarrow}\mathfrak{g}^{*}$ is defined by
$$\label{s70}
\mu(\xi) = \frac{\sqrt{-1}}{2\pi}(\nabla^{E}_{\xi_{M}} - L_{\xi} ), \ \xi \in \mathfrak{g}.$$
Here, $\xi_{M}$ is the induced infinitesimal vector field and $L_{\xi}$ is the Lie derivative. In this case, we call $(M, E, \omega, \mu)$ $\emph{pre-quantum data}$ [@Guillemin82]. Then one can canonically construct a Spin$^{c}$-Dirac operator
$$D^{E} : \Omega^{0, *}(M, E) \to \Omega^{0, *}(M, E),$$ which gives a finite dimensional virtual vector space
$$\mathrm{Index}(M, E) = \mathrm{Ker}(D^{E} ) \cap \Omega^{0, \mathrm{even}}(M, E)- \mathrm{Ker}(D^{E} ) \cap \Omega^{0, \mathrm{odd}}(M, E).$$
Given any pre-quantum data $(M, E, \omega, \mu)$, we define its geometric quantization
$$Q(M, E) = \mathrm{Index}(M, E),$$ which is a virtual representation of $G$.
Let $\hat{G}$ be the set of dominant weights in $\mathfrak{g}^{*}$ [@Humphreys78]. It is well-known that there is an one to one correspondence between $\hat{G}$ and all the irreducible G-representations [@Kirillov04]. Take any $\gamma \in \hat{G}$. When $\gamma$ is a regular value of the moment map $\mu$ (the singular case was discussed in [@Meinrenken99]) , the action of $G$ on $\mu^{-1}(G \cdot \gamma)$ is locally free. Therefore, $M_{\gamma} = \mu^{-1}(G\cdot \gamma) /G$ is an orbifold and $E_{\gamma}= (E|_{\mu^{-1}(G\cdot \gamma)})/G$ is an orbifold line bundle[@Weinstein79] (for basic definitions of orbifold, see [@Sateka57] [@Kawasaki79]). Moreover, $M_{\gamma}$ inherits symplectic structure from $M$. Hence, we can build a Spin$^{c}$-Dirac operator, with $\mathrm{Index}(M_{\gamma}, E_{\gamma})$ given by the orbifold index theorem of Kawasaki [@Kawasaki79].
\[k3\] We can define the quantization of pre-quantum data $(M, E, \omega, \mu)$ in an alternative way:
$$Q_{\mathrm{RED}}(M, E, \mu) = \sum_{\gamma \in \hat{G}} \mathrm{Index}(M_{\gamma}, E_{\gamma}) \cdot V_{\gamma} \in R(G),$$ where $V_{\gamma}$ is the irreducible representation with highest weight $\gamma$.
The celebrated quantization commutes with reduction theorem [@Guillemin82] says that the two definitions of quantization coincide.
If $(M, E, \omega, \mu)$ is pre-quantum data, then
$$Q(M, E) = Q_{\mathrm{RED}}(M, E)$$
Now, we assume that $M$ is noncompact. In order to formulate a suitable $[Q, R]=0$ theorem in noncompact case, the basic problem is “how to quantize a noncompact manifold". For the case in which $\mu$ is proper (i.e, the inverse image of a compact subset is compact), Ma and Zhang introduced a formal geometric quantization for $(M, E, \mu)$, by means of the Atiyah-Patodi-Singer-type index [@Atiyah79] for Dirac-type operators on manifolds with a boundary, denoted by $Q_{\mathrm{APS}}(M, E, \mu)$(see [@Zhang09] for details).
On the other hand, since $\mu$ is proper, the reduced orbifold $M_{\gamma} = \mu^{-1}(G\cdot \gamma) /G$ is compact. Thus, we extend Definition \[k3\] to the noncompact case[@Weitsman01] which we also denote $Q_{\mathrm{RED}}(M, E, \mu)$. It is necessary to point out that both $Q_{\mathrm{APS}}(M, E, \mu)$ and $Q_{\mathrm{RED}}(M, E, \mu)$ take values in $\hat{R}(G) = \mathrm{Hom}_{{\mathbb{Z}}}(R(G), {\mathbb{Z}})$, the completion of character ring $R(G)$.
We can now state the result of Ma-Zhang [@Zhang09; @Paradan09].
\[G-S theorem\] When $(M, E, \mu)$ is pre-quantum data and $\mu$ is proper, we have
$$Q_{\mathrm{APS}}(M, E, \mu) = Q_{\mathrm{RED}}(M, E, \mu).$$
In this paper, we take a topological account of this problem.
A $\emph{stable complex structure}$ on an orbifold $M$ is an equivalent class of complex structures on $TM \oplus {\mathbb{R}}^{k}$. We say two stable complex structures on $TM \oplus {\mathbb{R}}^{k_{1}}$ and $TM \oplus {\mathbb{R}}^{k_{2}}$ are equivalent if there exists $r_{1}$ and $r_{2}$ such that
$$TM \oplus {\mathbb{R}}^{k_{1}} \oplus {\mathbb{C}}^{r_{1}} \ \mathrm{and} \ TM \oplus {\mathbb{R}}^{k_{2}} \oplus {\mathbb{C}}^{r_{2}}$$ are isomorphic complex vector bundles.
When a group $G$ acts on M, we define an equivariant stable complex structure by requiring the complex structures to be invariant and the isomorphism to be equivariant. Here, the group acts on $TM$ by the natural lifting of its action on $M$, and it acts trivially on the trivial bundles ${\mathbb{C}}^{r}$. However, unlike the usual case, we do not require that $G$ acts trivially on ${\mathbb{R}}^{k}$.
Instead of pre-quantum data, we consider more general data $(M, E, \phi)$ as follows:
- $M$ is a stable complex $G$-orbifold, possibly noncompact.
- $E$ is a $G$-equivariant orbifold vector bundle over $M$.
- $\phi$ is a $G$-equivariant map from $M$ to $\mathfrak{g} \cong \mathfrak{g}^{*}$ (we can identify $\mathfrak{g}$ and its dual by making a choice of invariant inner product on $\mathfrak{g}$).
Given any $G$-equivariant map $\phi : M \mapsto \mathfrak{g}$, define a vector field $V^{\phi}$ by the formula:
$$V^{\phi}(m):= \frac{d}{dt} \Big|_{t=0} \mathrm{exp}(-t\phi(m)) \cdot m, \ \forall m \in M.$$ Let $M^{\phi}$ be the $\it{vanishing \ subset}$ in $M$:
$$M^{\phi} = \{ m \in M \big | V^{\phi}(m) = 0 \}.$$
We do not require $\phi$ to be moment map. Rather, we relax the moment map condition in the following way.
We say that $\phi$ is *compatible with E*, if there exists a constant $K$ such that $$\label{s71}
\| \frac{\sqrt{-1}}{2\pi}L_{\xi} + \langle \phi(m), \xi \rangle \cdot I_{m} \| \leq K \| \xi\|, \ \text{for \ all} \ m \in M^{\phi} ,$$ where $I_{m}$ is the identity map from $E|_{m}$ to itself, $L_{\xi}$ is the Lie derivative on $E|_{m}$, and $\xi$ lies in the isotropy Lie algebra $\mathfrak{g}_{m}$.
When $E$ is an actual orbifold line bundle with a moment map $\mu$, (\[s71\]) is equivalent to
$$\label{s27}
| \langle \mu(m), \xi \rangle - \langle \phi(m) , \xi \rangle | \leq K \cdot \| \xi \|.$$
Notice that (\[s27\]) does not depend on the choice of moment map $\mu$.
We say that a triple $(M, E, \phi)$ in which $M$ may have boundary is a $\emph{K-chain}$ if $\phi$ is proper over $M^{\phi}$ and compatible with $E$. When $M$ has no boundary, we say that $(M, E, \phi)$ is a $\emph{K-cycle}$.
In particular, all pre-quantum data $(M, E, \mu)$ and compact triples are $K$-cycles.
Next, we will define the equivalence relation between $K$-cycles. To begin with, a bordism between two $n$-dimensional orbifolds $M$ and $N$ is a $n+1$ dimensional orbifold $W$ with boundary, such that $$\partial W = M \sqcup (-N).$$ In ordinary bordism theory, we only consider compact orbifold (otherwise, any orbifold $M$ is bordant to the empty set via $W = [0, 1) \times M$). In order to obtain a nontrivial theory, we use the following definition.
\[s12\] Suppose that $(W, L, \psi)$ is a $K$-chain, whose boundary is divided into two parts $ M \sqcup N$. Hence, we obtain two $K$-cycles:
$$(M, L|_{M}, \psi|_{M}) \ \mathrm{and} \ (N, L|_{N}, \psi|_{N}).$$ We say that the first $K$-cycle $\it bordant$ to (the opposite of) the second.
\[d1\]
Our equivalence relation between $K$-cycles is generated by the following three elementary steps:
Disjoint Union
: $ (M, E, \phi) \bigsqcup (M, F, \phi) \sim (M, E \oplus F, \phi). $
Bundle Modification [@Baum07]
: Suppose that $P$ is a principal bundle over $M$ whose structure group is the compact Lie group $H$. Let $N$ be a compact, even dimensional, stable complex $H$-orbifold. The orbifold index of the associated Spin$^{c}$-Dirac operator gives an element $[D_{N}] \in R(H)$. And, if $[D_{H}] = [1]$, then
$$(M, E, \phi) \sim (\hat{M}, \hat{E}, \hat{\phi}),$$ where $\hat{M} = P \times_{H} N$, $\hat{E}$ is the pull back of $E$ and $\hat{\phi}$ is the composition of $\phi$ with the projection to $M$.
Bordism
: Definition \[s12\].
\[def-K\]
We denote by $\hat{K}(G)$ the set of equivalence classes of $K$-cycles.
The set $\hat{K}(G)$ is an abelian group, whose addition operation is given by disjoint union; the additive inverse of a $K$-cycle is obtained by reversing the stable complex structure.
For any $K$-cycle $(M, E, \phi)$, we use $n \cdot (M, E, \phi)$ to denote the disjoint union of $n$ copies of $(M, E, \phi)$. Suppose $\Gamma = \sum_{\gamma \in \hat{G}} n_{\gamma} V_{\gamma}$ is an arbitrary element in $\hat{R}(G)$. Define
$$\mathcal{O}_{\Gamma} = \bigsqcup_{\gamma \in \hat{G}}n_{\gamma} \cdot (\mathcal{O}_{\gamma}, E_{\gamma}, \iota_{\gamma}),$$ where $\mathcal{O}_{\gamma}$ is the coadjoint orbit through $\gamma \in \mathfrak{t}_{+}$; $E_{\gamma}$ is the natural line bundle defined using weight $\gamma$ [@Bott65] [@Kirillov04]; and $\iota_{\gamma}$ is the inclusion. It is clear that $ \mathcal{O}_{\Gamma}$ is a $K$-cycle.
The following theorems constitute the main results of this paper.
The map $P : \hat{R}(G) \to\hat{K}(G)$ \[o9\]
$$P : \Gamma \longmapsto \mathcal{O}_{\Gamma}$$ gives an isomorphism of abelian groups and $R(G)$-modules.
The idea of the definition of $\hat{K}(G)$ comes from the geometric $K$-homology defined by Baum and Douglas [@Baum81]. We can generalize $\hat{K}(G) = \hat{K}^{G}(\mathrm{pt})$ so as to obtain not just a group but a functor:
$$X \mapsto \hat{K}^{G}(X),$$ where $X$ is a paracompact Hausdorff $G$-space. To be more precise, let $(M, E, \phi, f)$ be a 4-tuple, where $M, E, \phi$ are the same as in the definition of $K$-cycles and $f$ is an equivariant map from $M$ to $X$. The equivalence relation extends naturally to this general case. In a work that is nearly finished, we show that the set of equivalence classes $\hat{K}(G)(X)$ is isomorphic to the Kasparov group $KK(C^{*}(G,X), {\mathbb{C}})$ [@Kasparov88].
\[main theorem\] The inverse map $Q_{\mathrm{TOP}} = P^{-1} : \hat{K}(G) {\longrightarrow}\hat{R}(G)$ has the following properties:
1. When $M$ is compact, $ Q_{\mathrm{TOP}}(M, E, \phi) = Q(M, E) \in R(G).$
2. When $(M,E, \phi_{1})$ and $(N, F, \phi_{2})$ are two K-cycles and $N$ is compact, we have
$$Q_{\mathrm{TOP}}(M, E, \phi_{1}) \times Q_{\mathrm{TOP}}(N, F, \phi_{2}) = Q_{\mathrm{TOP}}(M \times N, E \boxtimes F, \hat{\phi}_{1} + \hat{\phi}_{2}),$$ where $\hat{\phi}_{1}$ and $\hat{\phi}_{2}$ are the pullbacks of $\phi_{1}$ and $\phi_{2}$.
According to (1), $Q_{\mathrm{TOP}}$ can be considered as a generalization of the usual quantization for compact manifold. According to (2), $Q_{\mathrm{TOP}}$ satisfies the “multiplicative property", which is one of the main difficulties in [@Zhang09].
\[o7\] When $(M, E, \phi)$ is pre-quantum data and $\phi$ is proper, we have
$$Q_{\mathrm{TOP}}(M, E, \phi) = Q_{\mathrm{RED}}(M, E, \phi).$$
$\mathbf{Acknowledgments}$: I am grateful to my advisor, Professor N. Higson, for his kind guidance and advice. Also, I would like to thank Professor W. Zhang and Professor P. Baum for the many helpful conversations we have shared.
Basic Properties of $K$-Cycles
==============================
In this section, we will discuss some of the basic properties of $K$-cycles.
For convenience, we define some special $K$-chain ($K$-cycle):
- We say that a $K$-chain or a $K$-cycle $(M, E, \phi)$ $\it has \ compact \ vanishing \ set$ if $M^{\phi} \subseteq M$ is compact.
- We say that a $K$-cycle is $\it closed$ if $M$ is compact and has no boundary.
- We say that a $K$-cycle is $\it discrete$ if it has the following form:
$$\bigsqcup_{k=1}^{\infty}(N_{k}, E_{k}, \rho_{k}) \in\hat{K}(G),$$ where $N_{k}$ are closed orbifolds.
These lemmas follow immediately from the definition of $K$-cycles.
When $M$ is compact, we have
$$(M, E, \phi) \sim (M, E, 0).$$
Every $K$-cycle $(M, E, \phi)$ is equivalent to a finite sum of $K$-cycles
$$(M, E, \phi) \sim \sum_{i=1}^{n} (M_{i}, E_{i}, \phi_{i})$$ where $\{ E_{i}\}$ are line bundles.
By bundle modification, we have
$$(M, E, \phi) \sim (\hat{M}, \hat{E}, \hat{\phi}),$$
where $\hat{M}$ is a bundle of flag manifold over $M$ induced by $E$; $\hat{E}$ is the pullback of $E$; and $\hat{\phi}$ is the composition of $\phi$ with the projection from $\hat{M}$ to $M$. Hence, $\hat{E}$ splits into sum of line bundles.
Let us recall the definition of the geometric $K$-homology by Baum and Douglas [@Baum81] [@Baum10]. Let $X$ be a paracompact Hausdorff $G$-space. A cycle is a triple $(M, E, f)$ consisting of:
- $M$ = a weakly complex compact manifold with $G$-action.
- $E$ = a $G$-equivariant vector bundle over $M$.
- $f$ = a G-equivariant continuous map from $M$ to $X$.
The equivalence relations are generated by direct sum, bordism (in the compact sense) and bundle modification. The equivalence classes of cycles form an abelian group $K^{G}(X)$. In particular, when $X$ is a point, the map $f$ is trivial. Therefore, we have a natural map:
$$B : K^{G}(\mathrm{pt}) {\longrightarrow}\hat{K}(G) : B(M, E) = (M, E, 0).$$
\[localization theorem\] Suppose $\{ U_{\alpha} \}$ is a family of disjoint G-invariant open subsets such that
$$M^{\phi} \subseteq \bigsqcup_{\alpha} U_{\alpha}.$$ We have
$$(M, E, \phi) \sim \bigsqcup_{\alpha}(U_{\alpha}, E|_{U_{\alpha}}, \phi|_{U_{\alpha}}).$$
Let $W = M \times [0, 1]$ and $\hat{E}$ be the pullback of $E$. Define a map
$$\hat{\phi} : W {\longrightarrow}\mathfrak{g} :(m, t) \rightarrow \phi(m).$$ In addition, let $F = M \setminus (\bigcup(U_{\alpha})$ and $\hat{W} = W \setminus (F \times \{ 1 \})$. We also denote by $\hat{E}$ and $\hat{\phi}$ their restrictions to $\hat{W}$. It is easy to verify that
$$(\hat{W}, \hat{E}, \hat{\phi})$$ is a $K$-chain which gives desired bordism.
If $G$ is the circle group $S^{1}$, Theorem \[localization theorem\] is similar to the Linearization Theorem in [@Ginzburg96] in which $\phi$ is required to be an abstract moment map.
\[gluing\]
Let $(M, E, \phi)$ be any $K$-cycle. Suppose that $\Sigma \subset M$ is a smooth G-invariant hypersurface in M and $\Sigma$ cuts M into two oriented piece: $M\setminus \Sigma= M_{+}\sqcup M_{-}$. We obtain two $K$-cycles:
$$(M_{+}, E|_{M_{+}}, \phi_{+}) \ \mathrm{and} \ (M_{-}, E|_{M_{-}}, \phi_{-}).$$ When the vector field $V^{\phi}$ does not vanish over $\Sigma$, we have
$$(M, E, \phi) \sim (M_{+}, E|_{M_{+}}, \phi_{+}) + (M_{-}, E|_{M_{-}}, \phi_{-}).$$
Vanishing set of $K$-Cycles
===========================
The main goal of this section and the next one is to study $K$-cycles with compact vanishing. These two sections are the main part of this paper.
For any arbitrary $K$-cycle $(M, E, \phi)$, the vanishing set $M^{\phi}$ will play an important role in defining its quantization. In general, $M^{\phi}$ is a subset of $M$, which may be very complicated. However, the following theorem shows that $M^{\phi}$ can be separated into compact parts.
\[separate theorem\] Let $(M, E, \phi)$ be a $K$-cycle in which $E$ is a line bundle. There exists a covering of $M^{\phi}$ by $G$-invariant disjoint open subsets of $M$, $\{ U_{\alpha} \}$ such that each $F_{\alpha} = U_{\alpha} \cap M^{\phi}$ is compact.
When $(M, E, \phi)$ is pre-quantum data and $\phi$ is proper, Theorem \[separate theorem\] is trivial. In fact, let $\mathcal{H} = \| \phi \|^{2} : M {\longrightarrow}{\mathbb{R}}$. We can find a series of regular values of $\mathcal{H}$
$$c_{1}, c_{2}, \dots, c_{n}, \dots$$ such that $\lim_{n \to \infty}c_{n} = \infty$. Over $\mathcal{H}^{-1}(c_{i}) \subseteq M$, the vector field $V^{\phi}$ does not vanish. Hence,
$$\{ \ U_{i} = \mathcal{H}^{-1}((c_{i} , c_{i+1})) \}_{i=1}^{\infty} $$ give desired covering.
In general, the proof of Theorem \[separate theorem\] is based on Lemma \[y4\] and Lemma \[y8\]. To begin with, let ${\mathbb{T}}$ be a maximal torus in $G$, $\mathfrak{t}$ be the Lie algebra of ${\mathbb{T}}$, and $\mathfrak{t}_{+}$ be a chosen positive Weyl chamber. We observe that $m \in M^{\phi} $ if and only if $m \in M^{\gamma} \cap \phi^{-1}(\gamma)$, where $\gamma = \phi(m) \in \mathfrak{g}$. Thus,
$$\label{y3}
M^{\phi} = \bigcup_{\gamma \in \mathfrak{g}}(M^{\gamma} \cap \phi^{-1}(\gamma)) = \bigcup_{\gamma \in \mathfrak{t}_{+}}G.(M^{\gamma} \cap \phi^{-1}(\gamma)).$$
Consider the set of stabilizers $\{ H_{i} \}_{i=1}^{\infty}$ for the action of maximal torus ${\mathbb{T}}$ on $M$. Put an invariant connection on $E$ and denote by $\mu$ the associated moment map. Since $H_{i}$ are subgroups of ${\mathbb{T}}$, we can identify their Lie algebras $\mathfrak{h}_{i}$ as subspaces in $\mathfrak{t}$. Let
$$\mu_{\mathfrak{t}} : M \to \mathfrak{t}$$ be the moment map associated to the action of ${\mathbb{T}}$ and $\mu_{\mathfrak{h}_{i}}$ be the composition of $\mu_{\mathfrak{t}}$ with the projection from $\mathfrak{t}$ to $\mathfrak{h}_{i}$. From (\[y3\]), we have
$$\label{y9}
M^{\phi} = \bigcup_{\mathfrak{h}_{i} \in \Gamma}G\cdot (M^{H_{i}} \cap \phi^{-1}(\mathfrak{h}_{i})),$$
where $\Gamma$ is the set of Lie algebras $\mathfrak{h}_{i}$ such that $\mathfrak{h}_{i} \cap \mathfrak{t}_{+} \neq \{ 0\}$. For the decomposition in (\[y9\]), we have the following lemma.
\[y4\] For every $\mathfrak{h}_{i} \in \Gamma$, the set
$$G \cdot (M^{H_{i}} \cap \phi^{-1}(\mathfrak{h}_{i}))$$ can be separated by $G$-invariant disjoint open subsets $\{ V^{i}_{k} \}_{k=1}^{\infty}$ in $(G\cdot M^{H_{i}})$ such that $\phi$ is bounded on all $V^{i}_{k}$.
Notice that the map $\mu_{\mathfrak{h}_{i}}$ is locally constant on $M^{H_{i}}$. The set
$$\mu_{\mathfrak{h}_{i}}(M^{H_{i}}) \cap \mathfrak{t}_{+}$$ is a set of countable many points in $\mathfrak{h_{i}} \cap \mathfrak{t}_{+}$. Hence, we can find a covering of $G \cdot (M^{H_{i}} \cap \phi^{-1}(\mathfrak{h}_{i}))$, consisting of disjoint open subsets $\{ V^{i}_{k} \}_{k=1}^{\infty}$ in $(G\cdot M^{H_{i}})$ such that
$$\label{y2}
\mu_{\mathfrak{h}_{i}}(V^{i}_{k}) \cap \mathfrak{t}_{+} = \gamma_{k},$$
where $\gamma_{k} \in \mathfrak{h}_{i} \cap \mathfrak{t}_{+}$. In addition, we can choose $V^{i}_{k}$ small enough such that
$$\label{s10}
\phi(V^{i}_{k}) \cap \mathfrak{t}_{+} \subseteq \{ x \in \mathfrak{t}_{+} \big| \ \text{distance}(x, \mathfrak{h}_{i}) < \epsilon \},$$
where $\epsilon$ is arbitrary small positive number. From (\[s27\]), (\[y2\]), and (\[s10\]), we know that
$$\phi(V^{i}_{k}) \cap \mathfrak{t}_{+} \subseteq B(\gamma_{k}, K + \epsilon)= \{x \in \mathfrak{t}_{+}\big| \|x - \gamma_{k}\| < K + \epsilon \}.$$ It follows that $\phi$ is bounded on all $V_{k}^{i}$.
Obviously, the set $\bigcup_{i, k}V^{i}_{k}$ covers the vanishing set $M^{\phi}$. We define $\{U_{\alpha} \}$ to be disjoint $G$-invariant neighborhoods of connected components of $\bigcup_{i, k}V^{i}_{k}$ in $M$. Thus, the open sets $\{ U_{\alpha} \}$ give a covering of $M^{\phi}$. It remains to show that $F_{\alpha} = M^{\phi} \cap U_{\alpha}$ is compact.
\[y8\] The set $F_{\alpha} = M^{\phi} \cap U_{\alpha}$ is compact for all $\alpha$.
Because $\phi$ is proper over $M^{\phi}$, it is equivalent to prove that $\phi$ is bounded on $U_{\alpha}$.
Suppose $\phi$ is unbounded on $U_{\alpha}$. According to Lemma \[y4\], there must exist an infinite chain in $\{ V_{k}^{i} \}$,
$$\label{s5}
V_{1}, V_{2}, \dots, V_{m} \dots$$
such that
$$V_{i} \cap V_{i+1} \neq \emptyset \ \ \text{and} \ \ V_{i} \subseteq U_{\alpha}.$$ Notice that every $V_{i}$ is contained in some $(G\cdot M^{H_{k}})$. From (\[y2\]), let us denote
$$\label{y7}
\gamma_{i} = \mu_{\mathfrak{h}_{k}}(V_{i}) \cap \mathfrak{t}_{+}.$$
It is clear that
$$\label{s11}
\lim_{i \to \infty} \| \gamma_{i} \| = \infty.$$
Without loss of generality, we can assume that for any constant $T$, there exists $N_{T} \geq T$ and a chain
$$\gamma_{N_{T}}, \dots, \gamma_{N_{T}+\text{dim} ({\mathbb{T}})},$$ such that they are two by two distinct.
For $N_{T} \leq k \leq N_{T}+\text{dim} ({\mathbb{T}})-1$, pick an arbitrary point $m_{k} \in V_{k} \cap V_{k+1} \cap \phi^{-1}(\mathfrak{t}_{+})$ and denote by $H_{m_{k}}$ the isotropy group of ${\mathbb{T}}$-action. Let $\omega_{k}$ be the $H_{m_{k}}$-weight of the $E|_{m_{k}}$. Here, we can identify $\omega_{k}$ with a point in $\mathfrak{t}$. By (\[y7\]), we have that $$\label{s13}
\omega_{k} -\gamma_{k} \perp \gamma_{k} \ \text{and} \ \omega_{k} -\gamma_{k+1} \perp \gamma_{k+1}.$$ In addition, from (\[s10\]), we have
$$\label{y5}
\mathrm{distance}(\phi(m_{k}), \mathfrak{h}_{k}) < \epsilon.$$
It follows from (\[s27\]) that
$$\label{y6}
\|\omega_{k} - \phi(m_{k})\| \leq K.$$
From (\[s13\])-(\[y6\]), we get
$$\| \gamma_{k} - \omega_{k} \| \leq K+ \epsilon.$$ Similarly, we also have
$$\| \gamma_{k+1} - \omega_{k} \| \leq K+ \epsilon.$$ Therefore, these points $\{ \omega_{k}, \gamma_{k} \}_{k=N_{T}}^{N_{T} + \text{dim} ({\mathbb{T}})}$ are within finite distance from each other. Moreover, they must lie in the integer lattice of $\mathfrak{t}$ since they are all weights. By the orthogonal condition (\[s13\]), there are only finitely many possibilities of $\{ \omega_{k}, \gamma_{k} \}_{k=N_{T}}^{N_{T} + \text{dim} ({\mathbb{T}})}$. This leads to a contradiction to $(\ref{s11})$.
This complete the proof of Theorem \[separate theorem\]. Additionaly, from the proof, one can see that the assumption that $E$ is a line bundle is not necessary.
\[s30\] In Theorem \[separate theorem\], the cover $\{ U_{\alpha} \}$ has the property that $\phi$ is uniform bounded. That is, there exists a constant $R$ such that for all $\alpha$,
$$\| \phi(x) - \phi(y) \| \leq R, \ \mathrm{for} \ x, y \in U_{\alpha} \cap \phi^{-1}(\mathfrak{t}_{+}).$$
\[regular\] Suppose that $(M, E, \phi)$ is a $K$-cycle with compact vanishing set. Let $\partial M = \overline{M} \setminus M$. Given any small neighborhood $U$ of $\partial M$ in $\overline{M}$, we can identify
$$\label{s65}
U \cong \partial M \times [0,1).$$
In fact, by rescaling, we can assume that $\phi(m)$ tends to infinity as $m$ tends to $\partial M$. Take $\mathcal{H} = \| \phi \|^{2} : M \to {\mathbb{R}}$ and pick a regular value $c$. Let $M_{c}$ be a subset of $M$ defined by
$$M_{c} = \{ m \in M \big| \mathcal{H}(m) < c \}.$$ When $c$ is large enough, we have $M^{\phi} \subseteq M_{c}$. By Theorem \[localization theorem\],
$$(M, E, \phi) \sim (M_{c}, E|_{M_{c}}, \phi|_{M_{c}} ),$$ where the second $K$-cycle satisfies (\[s65\]). Unless stated otherwise, from here when we refer to the $K$-cycle with compact vanishing set, we always assume that it automatically satisfies $(\ref{s65})$. In this case, we can extend the vector bundle E and $\phi$ to $\partial M$, denoted by $\partial E$ and $\partial \phi$ respectively. In addition, we denote by $\partial(M, E, \phi)$ the $K$-cycle $(\partial M, \partial E, \partial \phi)$.
\[o1\] Suppose $(M_{1}, E_{1}, \phi_{1})$ and $(M_{2}, E_{2}, \phi_{2})$ are two $K$-cycles with compact vanishing set. Assume that there is a diffeomorphism $f : \partial M_{1} \cong \partial M_{2}$. By Remark \[regular\], the map $f$ also induces a diffeomorphism :
$$\hat{f} : U_{1} \cong U_{2},$$ where $U_{i}$ are neighborhoods of $\partial M_{i}$ as in (\[s65\]). When the map $\hat{f}$ lifts to an isomorphism between vector bundles $E_{1}|_{U_{1}}$ and $E_{2}|_{U_{2}}$, we can obtain a compact $K$-cycle by gluing the two $K$-cycles using the map $\hat{f}$. In the gluing process, we do not require that $\partial \phi_{1} = \partial \phi_{2}$ because we can alway vary the map $\phi$ without changing $K$-cycle class.
$K$-Cycles with Compact Vanishing Set
=====================================
From the previous two sections, we know that it is enough to study the $K$-cycles with compact vanishing set. Suppose that $(M, E, \phi)$ is a $K$-cycles with compact vanishing set. The general strategy is to build a “cap”, another $K$-cycle, so that we can compactify $(M, E, \phi)$ by gluing on the cap. The geometric construction of the cap is the main part of this section. To be more prices, we are going to the prove the following theorems.
\[a21\] Let $(M, E, \phi)$ be a $K$-cycle with compact vanishing set. There is a $K$-cycle with compact vanishing set $(W, L, \psi)$ such that
$$\partial(W, L, \psi) \cong \partial(M, E, \phi).$$ In addition, the $K$-cycle $(W, L, \psi)$ is bordant to a discrete $K$-cycle.
\[compact theorem\] Every $K$-cycle with compact vanishing set is bordant to a discrete $K$-cycle.
Suppose that $\Sigma$ is a closed $G$-manifold and $\phi : \Sigma \to \mathfrak{g}$ is an equivariant map. If the vector field $V^{\phi}$ induced by $\phi$ is nowhere vanishing over $\Sigma$, then $\Sigma$ is a boundary.
Circle Case
-----------
Let $(M, E, \phi)$ be a $K$-cycle with compact vanishing set. According to our discussion in Remark \[regular\], we can assume that $M$ is interior of some compact orbifold with boundary $\Sigma = \partial M$ on which $G = S^{1}$ acts locally freely.
Suppose that $D^{2}$ is the open disk with standard $S^{1}$-action. We can define an orbifold by
$$W = \Sigma\times_{S^{1}} D^{2},$$ and an orbifold vector bundle on $W$ by
$$L = \pi^{*}(\partial E)/S^{1},$$ where $\pi$ is the projection of $\Sigma \times D^{2}$ to $\Sigma$, and $S^{1}$-action is the diagonal action. In addition, the map $(\partial \phi) \circ \pi$ over $\Sigma \times D^{2}$ descends to a map over $W$ denoted by $\psi$. We can verify that $(W, L, \psi)$ indeed constitute a $K$-cycle with compact vanishing set such that
$$\partial(W, L, \psi) \cong \partial (M, E, \phi).$$ As in Remark \[o1\], we can obtain a compact $K$-cycle by gluing $(W, L, \psi)$ and $(M, E, \phi)$ together.
Next, we will show that $(W, L, \psi)$ is bordant to a discrete $K$-cycle, beginning with the following lemma.
\[lemma-2\] Consider the $K$-cycle
$$(D^{2}, {\mathbb{C}}, f(z)),$$ where ${\mathbb{C}}$ is the trivial line bundle over $D^{2}$ on which $S^{1}$ acts trivially, and $f(z)$ is a positive function on $D^{2}$. Then, we have that
$$\label{a1}
(D^{2}, {\mathbb{C}}, f) \sim \sum_{n=0}^{\infty} (S^{2}, F_{n}, f_{n}),$$
where $S^{2}$ is the sphere with standard $S^{1}$-action, $F_{n}$ are trivial line bundles over $S^{2}$ on which $S^{1}$ acts with $S^{1}$-weight $n$, and $f_{n}$ are equivariant functions on $S^{2}$ such that $f_{n}$ equals to $n$ when we restrict to the south pole and north pole.
Without losing generality, we assume that $f_{n}$ is positive along the equator. Thus, we can break the compact $K$-cycle $(S^{2}, F_{n}, f_{n})$ into two $K$-cycles with compact vanishing set:
$$(S^{2}, F_{n}, f_{n}) \sim (S_{+}, F_{n}|_{S_{+}}, f_{n}|_{S_{+}}) + (S_{-}, F_{n}|_{S_{-}}, f_{n}|_{S_{-}}),$$ where $S_{\pm}$ are the hemispheres. In particular, we have that
$$(S_{+}, F_{0}|_{S_{+}}, f_{0}) \cong (D^{2}, {\mathbb{C}}, f).$$ Next, let $(S^{2}, F_{n}^{n+1})$ be the compact $K$-cycle obtained by gluing
$$(S_{-}, F_{n}|_{S_{-}}, f_{n}) \ \mathrm{and} \ (S_{+}, F_{n+1}|_{S_{+}}, f_{n+1}).$$ Here, $F_{n}^{n+1}$ is an equivariant line bundle on $S^{2}$ with fiber weights equal to $n$ at south pole and $n+1$ at north pole. By Atiyah-Bott fixed point theorem, we have that
$$\label{a10}
\mathrm{Index}(S^{2}, F_{n}^{n+1}) = 0 \in R(S^{1}).$$
Hence, according to [@Baum07], we can conclude that the $K$-cycle $(S^{2}, F_{n}^{n+1})$ is equivalent to an empty $K$-cycle. This completes the proof.
If $f(z)$ is a negative function on $D^{2}$, then we have a similar result: $$\label{a2}
(D^{2}, {\mathbb{C}}, f) \sim -\sum_{n=1}^{\infty} (S^{2}, F_{-n}, -f_{n}).$$
\[p1\] The $K$-cycle $(W, L, \psi)$ is bordant to a discrete K-cycle.
Let $P = \Sigma \times_{S^{1}} S^{2}$. An orbifold vector bundle $L_{n}$ over $P$ is defined by
$$L_{n} = [ E \boxtimes F_{n}]/S^{1},$$ where $S^{1}$-action is the diagonal action. Additionally, the function
$$\Psi : \Sigma \times S^{2} \to {\mathbb{R}}: \Psi(m, x) = \phi(m) + f_{n}(x)$$ descends to a function on $P$, denoted by $\psi_{n}$. For each $n$, we obtain a compact $K$-cycle $(P, L_{n}, \psi_{n})$. Using Lemma \[lemma-2\], one can show that if $\psi$ is positive, then
$$(W, L, \psi) \sim \sum_{n=0}^{\infty} (P, L_{n}, \psi_{n}).$$ And if $\psi$ is negative, then we have that
$$(W, L, \psi) \sim -\sum_{n=-1}^{-\infty} (P, L_{n}, \psi_{n}).$$
Torus Case
----------
Now, let us assume that $G = {\mathbb{T}}$ is a torus and $\Sigma$ is a compact orbifold together with ${\mathbb{T}}$-action.
\[o2\] Let $\{ U_{i} \}_{i=1}^{n}$ be an open cover of $\Sigma$. We say $\{ U_{i}, S_{i} \}_{i=1}^{n}$ is a $\emph{good cover}$ if
- Every $U_{i}$ is ${\mathbb{T}}$-invariant.
- The circle action $S_{i}$ is a factor in some presentation ${\mathbb{T}}= S^{1} \times \dots \times S^{1}$ and $S_{i}$ acts locally freely on $U_{i}$.
- Let us define $$\mathcal{A} = \{ I \subseteq \{ 1, \dots, n\} \big| \ U_{I} = \bigcap_{i \in I} U_{i} \neq \emptyset \}.$$ For all $I \in \mathcal{A}$, $\{ S_{i} \}_{i \in I}$ generate an $|I|$-dimensional subgroup $S_{I}$ and it acts locally freely on $U_{I}$.
Suppose that $\Sigma$ is a compact orbifold with a good cover $\{U_{i}, S_{i}\}_{i=1}^{n}$. We want to construct a cap for $\Sigma$. If we naively build the caps locally, that is, defining $W_{i} = U_{i} \times _{S_{i}} D^{2}$ as in the circle case, then there is a problem that $\{ W_{i} \}$ may not be glued together. In order to overcome this difficulty, we define the local caps in a more subtle way using the compatibility conditions in Definition \[o2\]. This idea comes from cutting surgery by Lerman [@Lerman95].
\[a14\] For any $I \in \mathcal{A}$, we can define an orbifold $W_{I}$ with boundary isomorphic to $U_{I}$. Moreover, for any $I, J \in \mathcal{A}$ and $I \subsetneq J$, there exists a diffeomorphism $\Phi_{I}^{J}$ from an open subset $U_{I, J}$ of $W_{I}$ to an open subset $U_{J, I}$ of $W_{J}$:
$$\Phi_{I}^{J} : U_{I, J} \cong U_{J, I}$$ with the property that for any $K \in \mathcal{A}$ and $J \subsetneq K$,
$$\label{a13}
U_{I, K} = U_{I, J} \cap (\Phi_{I}^{J})^{-1}(U_{J, K}) \ \mathrm{and} \ \Phi_{I}^{J} \circ \Phi_{J}^{K} = \Phi_{I}^{K}.$$
To begin with, we choose a partition of unity $\{ \varphi_{i} \}_{i=1}^{n}$ subordinate to the open cover $\{ U_{i} \}_{i=1}^{n}$. Let us fix $n$ numbers $\{ \alpha_{1}, \dots, \alpha_{n} \}$ such that
- $\alpha_{i} > 1$ for all i, and
- For any $I \in \mathcal{A}$, $\prod_{i \in I}\alpha_{i}$ is a regular value for map $$\label{a17}
\rho_{I} : U_{I} \times [1, \infty) \to {\mathbb{R}}^{|I|} : (\rho_{I,i})_{i \in I} = ( t \cdot \varphi_{i}(m))_{i\in I}.$$
For any $I \in \mathcal{A}$, let us define
$$\label{a12}
\check{I} = \bigcup_{K \in \mathcal{A}, I \subseteq K} (K \setminus I).$$
Let $V_{I}$ be an open subset in $U_{I} \times [1, \infty)$ defined by
$$V_{I} = \{ (m, t) \in U_{I} \times [1, \infty) \big| t \cdot \varphi_{i}(m) < \alpha_{i}, \mathrm{for \ all} \ i \in \check{I} \}.$$ Apparently, $V_{I}$ has a boundary isomorphic to $U_{I}$ and $S_{I}$ acts locally freely on $V_{I}$.
Suppose that ${\mathbb{C}}^{|I|}$ is the product of $|I|$ copies of ${\mathbb{C}}$, where every copy has a individual standard circle action. This gives an $|I|$-dimensional torus $T_{I}$ action on ${\mathbb{C}}^{|I|}$. Let us consider
$$V_{I} \times {\mathbb{C}}^{|I|},$$ on which a torus action ${\mathbb{T}}_{I}$ acts diagonally. Let $\chi_{I}$ be a map defined by:
$$\chi_{I} : V_{I} \times {\mathbb{C}}^{|I|} \to {\mathbb{R}}^{|I|} : \chi_{I}(m, t, z) = (\alpha_{i} - t\varphi_{i}(m) - |z_{i}|^{2})_{i \in I}.$$ It is clear that 0 is a regular value for $\chi_{I}$ and ${\mathbb{T}}_{I}$ is locally free on $\chi_{I}^{-1}(0)$. Therefore,
$$\label{a11}
W_{I} = \chi_{I}^{-1}(0) / {\mathbb{T}}_{I}$$
defines an open orbifold with boundary isomorphic to $U_{I}$.
For the second part of the lemma, let $V_{I}^{J}$ be an orbifold defined by
$$V_{I}^{J} = \{ (m, t) \in U_{J} \times [1, \infty) \big| t \cdot \varphi_{i}(m) < \alpha_{i}, \mathrm{for \ all} \ i \in \check{I} \}.$$ As the construction of $W_{J}$, we can define a map $\tilde{\chi}_{J}$ on $V_{I}^{J} \times {\mathbb{C}}^{|J|}$ and
$$W_{I}^{J} = \tilde{\chi}_{J}^{-1}(0) / {\mathbb{T}}_{J},$$ gives an orbifold with boundary isomorphic to $U_{J}$.
For $I \subsetneq J$, we have that $U_{I} \subseteq U_{J}$ and $\check{J} \subseteq \check{I}$. Hence, $V_{I}^{J}$ is an open subset of both $V_{J}$ and $V_{I}$. And $W_{I}^{J}$ can be identified as open suborbifold of both $W_{I}$ and $W_{J}$. We define the diffeomorphism $\Phi_{I}^{J}$ to be the map between $W_{I}$ and $W_{J}$ factoring through $W_{I}^{J}$. The verification of (\[a13\]) is straightforward.
Thanks to Lemma \[a14\], we can obtain an orbifold $W$ by gluing all the $\{ W_{I} \}_{I \in \mathcal{A}}$ using $\phi_{I}^{J}$. From the construction, one can check that $W$ is a compact orbifold with boundary isomorphic to $\Sigma$. Therefore, we have the following theorem.
\[a16\] Let $\Sigma$ be a compact orbifold. If it has a good cover, then we can construct an orbifold $W$ with boundary isomorphic to $\Sigma$.
Now, let $(M, E, \phi)$ be a $K$-cycle with compact vanishing set and $\Sigma = \partial M$. We will show that $\Sigma$ has a good cover.
Let $H$ be an isotropy group of ${\mathbb{T}}$-action on $\Sigma$ and
$$\Sigma_{H} = \{x \in \Sigma \big| {\mathbb{T}}_{x} = H \},$$ where ${\mathbb{T}}_{x}$ is the isotropy group of $x$. For any connected component $F$ of $\Sigma_{H}$, we denote by $U_{F}$ a ${\mathbb{T}}$-invariant neighborhood of $F$ in $\Sigma$. Here, we can choose $U_{F}$ small enough such that
- For all $x \in U_{F}$, we have that ${\mathbb{T}}_{x} \subseteq H$.
- For any other component $F^{'}$ of $\Sigma^{H}$, we have that $U_{F} \cap U_{F^{'}} = \emptyset$.
In this case, we say that $U_{F}$ has $\it{level}$ equal to $\mathrm{dim}(H)$. As $H$ ranges over all the isotropy group, we obtain an open cover of $\Sigma$, denoted by $\{U_{i} \}$.
\[a3\] Every point $x \in \Sigma$ can be covered by at most $\mathrm{dim}({\mathbb{T}}) - \mathrm{dim}({\mathbb{T}}_{x})$ open sets in $\{ U_{i} \}$.
The lemma follows from the fact that every point $x \in \Sigma$ can only be covered by open set $U_{i}$ with level no lower than $\mathrm{dim}({\mathbb{T}}_{x})$.
\[a15\] There exists a good cover on $\Sigma$.
It is enough to associate every $U_{i}$ with a compatible circle action. We will complete the proof by induction on the level of $\{U_{i}\}$.
If $U_{i}$ has the highest level, then we can always find a circle action $S_{i}$ which acts locally freely on $U_{i}$, using the fact that $\phi$ induces a nowhere vanishing vector field on $\Sigma$.
Suppose that we have already associated the open sets whose level is greater than $K$ with compatible circle actions.
Let $U_{k}$ be an open set with level $K$. Suppose that $I$ is a subset of $\{1, \dots, n\}$ such that
$$U_{k} \cap ( \bigcap_{i \in I} U_{i}) \neq \emptyset,$$ and every $\{ U_{i} \}_{i \in I}$ has level greater than $K$. From Lemma \[a3\], we know that $|I| \leq \mathrm{dim}({\mathbb{T}}) -1- K$. Meanwhile, for any point $x \in U_{k}$, the isotropy group $T_{x}$ has dimension no greater than $K$. Hence, the compatible circle action $S_{k}$ always exists. This completes the proof.
In order to construct a cap for the $K$-cycle $(M, E, \phi)$, we will show how to define an orbifold vector bundle $L$ and an equivariant map $\psi$ on $W$.
\[a4\] Fixed any ${\mathbb{T}}$-weight $\gamma$, we can construct a $K$-cycle $(W, L, \psi)$ such that
$$\partial (W, L, \psi) \cong (\Sigma, E, \phi)$$ and $L|_{x}$ has ${\mathbb{T}}$-weight equal to $\gamma$ for any $x \in W^{{\mathbb{T}}}$.
According to the construction in Theorem \[a16\], it is enough to define the vector bundle and equivariant map on every piece $W_{I}$.
Let $\gamma_{I}$ be the restriction of $\gamma$ to $S_{I}$, which is subgroup of ${\mathbb{T}}$. Let $F_{\gamma_{I}}$ be the trivial line over ${\mathbb{C}}^{|I|}$ on which $T_{I}$ acts with weight $\gamma_{I}$. Recall that $\chi_{I}^{-1}(0)$ is an open subset of $V_{I} \times {\mathbb{C}}^{I}$ and the diagonal action ${\mathbb{T}}_{I}$ is locally free on $\chi_{I}^{-1}(0)$. Thus,
$$\label{t1}
L_{I} = ((E \boxtimes F_{\gamma_{I}})|_{\chi_{I}^{-1}(0)}) / {\mathbb{T}}_{I}$$
defines an orbifold vector bundle on $W_{I} = \chi_{I}^{-1}(0)/ {\mathbb{T}}_{I}$.
For the equivariant map, let $\hat{\phi}$ be the pullback of $\phi$ to $V_{I} \times {\mathbb{C}}^{|I|}$. After restricting to $\chi_{I}^{-1}(0)$, $\hat{\phi}$ descends to a map on $W_{I}$. Using the diffeomorphism in Lemma \[a14\], we can get an orbifold vector $L$ and an equivariant map $\psi$ by gluing. And the triple $(W, L, \psi)$ constitute a $K$-cycle satisfying all the desired properties.
Next, we are going to show that $(W, L, \psi)$ is bordant to a discrete $K$-cycle.
Let ${\mathbb{D}}^{n} = D^{2} \times \dots \times D^{2}$ be the product of $n$ copies of open disks. A $n$-dimensional torus ${\mathbb{T}}^{n} = S^{1} \times \dots \times S^{1}$ acts on ${\mathbb{D}}^{n}$ in such a way that the $i$-th factor of $S^{1}$ acts on the $i$-th Disk by rotation.
For every $I \in \mathcal{A}$, let us define $K_{I} = \rho_{I}^{-1}(\alpha_{I}) \cap V_{I}$, that is
$$K_{I} = \{ (m,t) \in U_{I} \times [1, \infty) \big| t\cdot \varphi_{i}(m) = \alpha_{i}, i \in I \, \mathrm{and} \ t\cdot \varphi_{j}(m) < \alpha_{j}, j \in \check{I} \}.$$ When $\check{I} = \emptyset$, $K_{I}$ is a compact orbifold without boundary. Otherwise, it is possible that $\partial K_{I} \neq \emptyset$. However, we have the following.
\[a24\] For every $I \in \mathcal{A}$, we can obtain a closed orbifold by gluing
$$(\bigsqcup_{J \in \mathcal{A}, I \subsetneq J} K_{J} \times_{{\mathbb{T}}^{I}_{J}} {\mathbb{D}}^{|J|-|I|}) \sqcup K_{I},$$ where ${\mathbb{T}}^{I}_{J}$ is a $(|J|-|I|)$-dimensional torus and it acts on $K_{J}$ through $S_{J}^{I} = \prod_{i \in J \cap \check{I}}S_{i}$.
It is straightforward based on the fact that $\alpha_{I}$ are regular values for
$$\rho_{I} : U_{I} \times [1, \infty) \to {\mathbb{R}}^{|I|} .$$
\[a26\] The $K$-cycle $(W, L, \psi)$ is bordant to a $K$-cycle in the following form:
$$(W, L, \psi) \sim \sum (M_{i} \times_{{\mathbb{T}}_{i}}{\mathbb{D}}^{|{\mathbb{T}}_{i}|}, E_{i}, \phi_{i}) + \sum (N_{j}, F_{j}, \psi_{j}),$$ where $\{ M_{i}, N_{j} \}$ are compact oribifolds without boundary.
If we identify $K_{I}/{\mathbb{T}}_{I}$ with a subset in $W_{I}$, then the set $$\label{a23}
Z_{I} = K_{I} \times_{{\mathbb{T}}_{I}} {\mathbb{D}}^{|I|}$$ can be identified as a neighborhood of $K_{I}/ {\mathbb{T}}_{I}$ in $W_{I}$. We define $Z$ to be the orbifold obtained by gluing $Z_{I}$ as in Lemma \[a14\]. Since $V^{\phi}$ is nowhere vanishing on $\Sigma$, the vanishing set $V^{\psi}$ in $W$ must be contained in $Z$. Therefore,
$$(W, L, \psi) \sim (Z, L|_{Z}, \psi|_{Z}).$$ Without losing generality, we can furthermore assume that
$$\label{a25}
(Z, L|_{Z}, \psi|_{Z}) \sim \sum_{I}(Z_{I}, L|_{Z_{I}}, \psi|_{Z_{I}}).$$
Therefore, the Lemma follows from (\[a23\]), (\[a25\]), and Lemma \[a24\].
By repeatedly using Lemma \[lemma-2\], the following theorem follows from Lemma \[a26\].
The $K$-cycle $(W, L, \psi)$ is bordant to a discrete $K$-cycle.
Nonabelian Case
---------------
Now, we assume that $G$ is a compact connected Lie group, ${\mathbb{T}}$ is a maximal torus, and $\mathfrak{t}_{+}$ is a fixed positive Weyl chamber. For any $x \in \mathfrak{t}_{+}$, we denote by $G_{x}$ the isotropy group of adjoint action.
Let $\Delta$ be a face of $\mathfrak{t}_{+}$. For any point in $\mathrm{int}(\Delta)$, they have the same isotropy group, denoted by $G_{\Delta}$. Moreover, we have that $G_{\Delta} \subset G_{\Delta^{'}}$ if and only if $\Delta$ is a sub-face of $\Delta^{'}$.
Let us recall symplectic cross-section theorem.
Let $(M, \omega)$ be a compact connected symplectic orbifold with a moment map $\mu : M \to \mathfrak{g}^{*}$ arising from an action of a compact Lie group G. For any face $\Delta$ in $\mathfrak{t}_{+}$, let $V_{\Delta}$ be a small neighborhood of $\mathrm{int}(\Delta)$ in $\mathfrak{t}_{+}$ such that $G_{x} \subseteq G_{\Delta}$ for any $x \in V_{\Delta}$. If we denote $U_{\Delta} = G_{\Delta} \cdot V_{\Delta}$, then the cross section $R = \mu^{-1}(U_{\Delta})$ is a $G_{\Delta}$-invariant symplectic sub-orbifold and
$$U = G\cdot R = G \times_{G_{\Delta}} R$$ is an open subset of $M$. Moreover, if $A_{\Delta}$ is the abelian part of $G_{\Delta}$, then the $A_{\Delta}$-action on $R$ extends in a unique way to an action on $U$ which commutes with the G-action.
See [@Guillemin90] [@Lerman98].
In this paper, we are considering stably complex orbifolds instead of symplectic orbifolds. Hence, the symplectic cross-section theorem does not apply. However, the idea of building the cap in nonabelian case comes from the symplectic cross-section theorem and symplectic surgery by Meinrenken [@Meinrenken98].
Let $(M, E, \phi)$ be a $K$-cycle with compact vanishing set and $\Sigma = \partial M$.
For each $m \in \Sigma$, let $\mathfrak{g}_{m} \subset \mathfrak{g}$ be the corresponding isotropy Lie algebra. It is clear that $\mathfrak{g}_{g \cdot m} = \mathrm{Ad}(g)(\mathfrak{g}_{m})$. We call the set of subalgebras
$$(\mathfrak{g}_{m}) = \{ \mathrm{Ad}(g)(\mathfrak{g}_{m}) \big| g \in G\}$$ the *[orbit type]{} of $m$. There are only finite many orbit type on $\Sigma$. Moreover, there is a unique orbit type $(\mathfrak{g}_{0})$ such that the set*
$$\Sigma_{(\mathfrak{g}_{0})} = \{ m \in \Sigma \big|(\mathfrak{g}_{m}) = (\mathfrak{g}_{0}) \}$$ is a dense, open subset in $\Sigma$ [@Guillemin90][@Lerman98].
Let $(\mathfrak{h})$ be an arbitrary orbit type of $\Sigma$. There exists a face $\Delta$ with maximum dimension such that $(\mathfrak{h})$ is subconjugated to $(\mathfrak{g}_{\Delta})$. Suppose that $F$ is a connected component of $\Sigma_{(\mathfrak{h})}$ and $U$ is a small $G$-invariant neighborhood of $F$ in $\Sigma$ such that for any $m \in U$
$$(\mathfrak{g}_{m}) \subseteq (\mathfrak{g}_{\Delta}).$$ In this case, as in cross-section theorem, we can find a $G_{\Delta}$-invariant subset $R$ of $U$ such that
$$U = G \cdot R \cong G \times_{G_{\Delta}} R,$$ where the isomorphism map is given by
$$G \times_{G_{\Delta}} R \to G \cdot R, \ [a, u] \to a \cdot u.$$ Moreover, the $A_{\Delta}$-action on $R$ extends to an action on $U$, which commutes with the $G$-action [@Woodward96] [@Meinrenken98].
\[a27\] There exists an open coving $\{ U_{i} \}$ of $\Sigma$ with circle actions $\{S_{i} \}$ such that
- Every $U_{i}$ is $G$-invariant.
- The circle action $S_{i}$ acts locally freely on $U_{i}$ and commutes with $G$-action.
- For all $I \in \mathcal{A}$, $\{ S_{i} \}_{i \in I}$ generate an $|I|$-dimensional torus and it acts locally freely on $U_{I}$.
The proof is similar to the torus case except the circle actions are induced from the locally defined action $A_{\Delta}$.
By similar argument in the torus case, we can prove the following theorem.
There exists a $K$-cycle $(W, L, \psi)$ such that
$$\partial(W, L, \psi) \cong (\Sigma, E, \phi).$$ Moreover, $(W, L, \psi)$ is bordant to a discrete $K$-cycle.
Quantization Map
================
In this section, we will first give the definition of quantization map for all $K$-cycles:
$$Q_{\mathrm{TOP}} : \{(M, E, \phi)\} {\longrightarrow}\hat{R}(G).$$ Then, we will show that $Q_{\mathrm{TOP}}$ induces an isomorphism from $\hat{K}(G)$ to $\hat{R}(G)$.
To begin with, by Theorem \[localization theorem\], we know that
$$\label{t11}
(M, E, \phi) \sim \bigsqcup_{k} (U_{k}, E|_{U_{k}}, \phi|_{U_{k}}),$$
where the right-hand side consists of $K$-cycles with compact vanishing set. For each $(U_{k}, E|_{U_{k}}, \phi|_{U_{k}})$, we can build a cap $(W_{k}, L_{k}, \psi_{k})$. Recall that the constructions of $L_{k}$ are not unique. In fact, we can build $L_{k}$ in a way such that $\{ (W_{k}, L_{k}, \psi_{k}) \}$ forms a global $K$-cycle by putting all them together.
\[t14\] We can build $(W_{k}, L_{k}, \psi_{k})$ such that
$$\bigsqcup_{k} (W_{k}, L_{k}, \psi_{k})$$ is a $K$-cycle.
Let us denote $\Sigma_{k} = \partial U_{k}$. Every $W_{k}$ is constructed as in section 4 and $\psi_{k}$ is induced from $\phi|_{\Sigma_{k}}$. Since $\phi$ is proper over $M^{\phi}$, we can choose $U_{k}$ small enough so that the maps $\{ \psi_{k} \}$ are proper over $W = \bigsqcup_{k} W_{k}$. The problem left is to construct line bundles $\{ L_{k} \}$ which are compatible with $\{ \psi_{k} \}$ globally. That is, there exists a constant $C$ such that for all $k$,
$$\label{t2}
\| \frac{\sqrt{-1}}{2\pi}L_{\xi} + \langle \psi_{k}(m) , \xi \rangle \| \leq C \cdot \| \xi \|, \ m \in W_{k}^{\psi_{k}} \ \text{and} \ \xi \in \mathfrak{g}_{m}.$$
Due to Remark \[s30\], there exists a constant $R$ and a series of dominant weights $\{ \gamma_{k} \} \in \mathfrak{t}_{+}$ such that
$$\label{t3}
\phi_{k}(\Sigma_{k}) \cap \mathfrak{t}_{+} \subseteq B(\gamma_{k}, R) = \{ x \in \mathfrak{t}_{+} \big| \| x - \gamma_{k} \| \leq R \}$$
and $\lim_{k \to \infty} \| \gamma_{k} \| = \infty$. As in Proposition \[a4\], we can construct $L_{k}$ using the fixed weights $\gamma_{k}$. These $\{ L_{k} \}$ satisfy condition (\[t2\]).
From section 4, we know that every $(W_{k}, L_{k}, \psi_{k})$ is bordant to a discrete $K$-cycle:
$$(W_{k}, L_{k}, \psi_{k}) \sim \sum_{i} (N_{k}^{i}, F_{k}^{i}, \rho_{k}^{i}).$$ It is natural to ask that if we put all them together, do they form a $K$-cycle?
\[a33\] The infinite sum
$$\sum_{k, i} (N_{k}^{i}, F_{k}^{i}, \rho_{k}^{i})$$ constitute a $K$-cycle.
Let $N = \bigsqcup_{k, i} N_{k}^{i}$, $F$ be the orbifold vector bundle such that $F|_{N_{k}^{i}} = F_{k}^{i}$, and $\rho$ be the map on $N$ such that $\rho|_{N_{k}^{i}} = \rho_{k}^{i}$. To check that $(N, F, \rho)$ is a $K$-cycle, it is enough to show that $\rho$ is proper. To be more precise, for every $R>0$, we need to show that there exists constant $T_{R}$ such that for all $i, k \geq T_{R}$,
$$\rho_{k}^{i} (N_{k}^{i}) \cap B_{R} = \emptyset,$$ where
$$B_{R} = \{ x \in \mathfrak{t}_{+} \big| \|x \| \leq R \}.$$
First, given any fixed $k$, we know that
$$\rho_{k}^{i} (N_{k}^{i}) \cap B_{R} = \emptyset,$$ when $i$ is large enough. On the other hand, there exists a constant $R^{'} \gg R$ such that if $\psi_{k}(W_{k}) \cap B_{R^{'}} = \emptyset$, then
$$\rho_{k}^{i} (N_{k}^{i}) \cap B_{R} = \emptyset \ \mathrm{for \ all} \ i.$$ By (\[t3\]), we have that $\psi_{k}(W_{k}) \cap B_{R^{'}} = \emptyset$ as $k \to \infty$. This completes the proof.
\[t8\] Every $K$-cycle $(M, E, \phi)$ is bordant to a discrete $K$-cycle:
$$\label{k4}
(M, E, \phi) \sim \sum_{k=1}^{\infty} (M_{k}, E_{k}, \phi_{k})$$
In addition, for any irreducible representation $\gamma \in \hat{G}$, there exists a constant $T_{\gamma}$ such that
$$\label{o10}
[\mathrm{Index}(M_{k}, E_{k})]^{\gamma} = 0, \ \mathrm{if} \ k \geq T_{\gamma}.$$
The first part follows from Lemma \[a33\]. For the section part, it is enough to prove the abelian case. The general case follows from the induction argument in [@Paradan01].
Suppose that $G = {\mathbb{T}}$ is abelian. By the discussion before, there exists a constant $R$ and a series of weights $\{ \gamma_{k} \} \in \mathfrak{t}$ such that
$$\label{t19}
\phi_{k}(M_{k}) \subseteq B(\gamma_{k}, R) = \{ x \in \mathfrak{t} \big| \| x - \gamma_{k} \| \leq R \}$$
and $\lim_{k \to \infty} \| \gamma_{k} \| = \infty$. Therefore, we know that when $k$ is large enough, there exists a cyclic unit vector $\xi \in \mathrm{Lie}({\mathbb{T}})$ such that
$$\langle \phi_{k}(x), \xi \rangle \gg 0, \ \mathrm{for \ all} \ x \in M_{k}^{{\mathbb{T}}}.$$ By (\[s27\]), we conclude that as $k \to \infty$
$$\langle \omega^{k}_{x}, \xi \rangle \gg 0, \ \mathrm{for \ all} \ x \in M_{k}^{{\mathbb{T}}},$$ where $\omega^{k}_{x}$ is the fiber weight of $E_{k}|_{x}$. Therefore, we can finish the proof by the Atiyah-Bott fixed point theorem (for details, one can see Theorem 5.1 in [@Meinrenken99]).
Now, we are able to define the quantization map.
\[t16\] Let $\bigsqcup_{k=1}^{\infty} (M_{k}, E_{k}, \phi_{k})$ be a discrete $K$-cycle satisfying (\[o10\]), we define the quantization map $Q_{\mathrm{TOP}}$ to be
$$Q_{\mathrm{TOP}}\big( \bigsqcup_{k=1}^{\infty} (M_{k}, E_{k}, \phi_{k})\big) = \sum_{k} \mathrm{Index}(M_{k}, E_{k}) \in \hat{R}(G).$$
Suppose that $(M, E, \phi)$ is an arbitrary $K$-cycle. As in (\[k4\]), let us assume that
$$\label{t12}
(M, E, \phi) \sim \sum_{k}(M_{k}, E_{k}, \phi_{k}).$$
If we can prove that $Q_{\mathrm{TOP}}$ is invariant under bordism, then we can get a well-defined quantization map for any $K$-cycle:
$$Q_{\mathrm{TOP}}(M, E, \phi) = Q_{\mathrm{TOP}}\big( \bigsqcup_{k=1}^{\infty} (M_{k}, E_{k}, \phi_{k})\big)$$
\[t22\] Suppose that $(W, L, \psi)$ is a $K$-chain. For any constant $R > 0$, there exists a hypersurface $\Sigma_{R}$ in W such that
- The vector field $V^{\psi}$ is nowhere vanishing over $\Sigma_{R}$;
- $\Sigma_{R}$ subdivides W into two parts: a bounded part $W_{-}$ and an unbounded part $W_{+}$, with the property that
$$\psi(W_{+}) \cap B_{R} = \emptyset .$$
Let $\mathcal{H}$ be a function defined by $\mathcal{H} = \| \psi \|^{2} : W \to {\mathbb{R}}$. By Theorem \[separate theorem\], the vanishing set $W^{\psi}$ can be covered by $\{ U_{\alpha} \}$ such that $F_{\alpha}= W^{\psi} \cap U_{\alpha}$ is compact. Further, by Remark \[s30\], there exists a constant $K$ such that for all $\alpha$,
$$\label{t21}
\| \psi(x) - \psi(y) \| \leq K, \mathrm{for \ all} \ x, y \in U_{\alpha} \cap \psi^{-1}(\mathfrak{t}_{+}).$$
By (\[t21\]) and the fact that $\psi$ is proper over the vanishing set, there are only finitely many $\alpha$ such that
$$\label{q1}
F_{\alpha} \cap \mathcal{H}^{-1}(R + K) \neq \emptyset.$$
Let us denote them by $\{ \alpha_{1}, \dots, \alpha_{n} \}$. Define a G-invariant non-negative function $\rho : W \to {\mathbb{R}}$ such that
- $\rho(x) = 0$ for all $x \notin \bigsqcup_{i=1}^{n} U_{\alpha_{i}}$
- $ \rho(x) \geq 2(R+K)$ for $x \in U_{\alpha_{i}} \cap W^{\psi}$, $i = 1, \dots, n$.
Select a regular value $c$ for $\mathcal{H} + \rho$, which is very close to $R+K$. The hypersurface $\Sigma_{R} = (\mathcal{H} + \rho)^{-1}(c)$ satisfies all the conditions.
Let $(M, E, \phi)$ and $(M^{'}, E^{'}, \phi^{'})$ be two $K$-cycles. If they are bordant, then
$$Q_{\mathrm{TOP}}(M, E, \phi) = Q_{\mathrm{TOP}}(M^{'}, E^{'}, \phi^{'}) \in \hat{R}(G).$$
Fixing an irreducible representation $\gamma \in \widehat{G}$, it is enough to show that
$$[Q_{\mathrm{TOP}}(M, E, \phi)]^{\gamma} = [Q_{\mathrm{TOP}}(M^{'}, E^{'}, \phi^{'})]^{\gamma}.$$ By Definition \[t12\], let us assume that
$$Q_{\mathrm{TOP}}(M, E, \phi) = \sum_{k=1}^{\infty} \mathrm{Index}(M_{k}, E_{k})$$ and
$$Q_{\mathrm{TOP}}(M^{'}, E^{'}, \phi^{'}) = \sum_{k=1}^{\infty} \mathrm{Index}(M_{k}^{'}, E_{k}^{'}).$$ Since $(M, E, \phi)$ is bordant to $(M^{'}, E^{'}, \phi^{'})$, we also have
$$\label{o5}
\bigsqcup_{k}(M_{k}, E_{k}, \phi_{k}) \sim \bigsqcup_{k}(M_{k}^{'}, E_{k}^{'}, \phi_{k}^{'}).$$
Suppose $(W, L, \psi)$ gives the bordism in (\[o5\]).
As in Lemma \[t22\], we can construct a hypersurface $\Sigma_{R}$ which cuts $(W, L, \psi)$ into two pieces: $(W_{-}, L|_{W_{-}}, \psi|_{W_{-}})$ and $(W_{+}, L|_{W_{+}}, \psi|_{W_{+}})$. We observe that $W_{-}$ is an orbifold with boundary consisting of
- The part which doesn’t intersect with $\Sigma_{R}$: $\bigsqcup_{k \in \mathscr{A}} M_{k} $ and $\bigsqcup_{j \in \mathscr{A}^{'}}M_{j}^{'}$.
- The part which intersects with $\Sigma_{R}$ : $\bigsqcup_{k \in \mathscr{B}} U_{k} $ and $\bigsqcup_{j \in \mathscr{B}^{'}}U_{j}^{'}$, where $U_{k}, U_{j}^{'}$ are subsets of $M_{k}$ and $M_{j}^{'}$.
For any fixed $\gamma \in \widehat{G}$, we can choose $R$ large enough such that for all $k \notin \mathscr{A}$ and $j \notin \mathscr{A}^{'}$
$$[Q(M_{k}, E_{k})]^{\gamma} = [Q(M_{j}^{'}, E_{j}^{'})]^{\gamma} = 0.$$
Notice that $(W_{-}, L|_{W_{-}}, \psi|_{W_{-}})$ is a $K$-chain with compact vanishing set. As the construction in Section 4, we can build a cap and obtain a compact $K$-chain (which gives a compact bordism) by gluing on the cap. During the gluing process, we notice that
$$\{ (M_{k}, E_{k}, \phi_{k}) \}_{k \in \mathscr{A}} \ \mathrm{and} \{ (M_{j}^{'}, E_{j}^{'}, \phi_{j}^{'}) \}_{j \in \mathscr{A}^{'}}$$ remain the same. When the constant $R$ is large enough, we can also assume that the multiplicity of $\gamma$ does not change during the gluing. Therefore, due to the fact that index map is invariant under compact bordism, we can conclude that
$$[Q_{\mathrm{TOP}}(M, E, \phi)]^{\gamma} = [Q_{\mathrm{TOP}}(M^{'}, E^{'}, \phi^{'})]^{\gamma} \in {\mathbb{Z}}.$$
Given any $K$-cycle $(M, E, \phi)$, we have
$$Q_{\mathrm{TOP}}(M, E, \phi) = Q_{\mathrm{TOP}}(\hat{M}, \hat{E}, \hat{\phi}),$$ where $(\hat{M}, \hat{E}, \hat{\phi})$ is the bundle modification in Definition \[d1\].
Assume that $M$ is modified by the principle bundle $P$ with fiber $N$. First, by Definition \[t16\], we have
$$(M, E, \phi) \sim \sum(M_{k}, E_{k}, \phi_{k}),$$ Suppose that $(W, L, \psi)$ gives the bordism above. As the construction of vector bundle over $W$, we can extend the principle bundle $P$ on $M$ to $W$, denoted by $P_{W}$. Now, we can define the bundle modification (with respect to $P_{W}$) of $W$ to be
$$\hat{W} = P_{W} \times_{H} N .$$ Let $\hat{L}$ and $\hat{\psi}$ be the pullback of $L$ and $\psi$ to $\hat{W}$. Then $(\hat{W}, \hat{L}, \hat{\psi})$ is a $K$-chain, giving a bordism between
$$(\hat{M}, \hat{E}, \hat{\phi}) \ \text{and} \ \sum_{k} (\hat{M}_{k}, \hat{E_{k}}, \hat{\phi_{k}}).$$ By the definition of quantization, we have
$$Q_{\mathrm{TOP}}(\hat{M}, \hat{E}, \hat{\phi} ) = \sum_{k} \mathrm{Index}(\hat{M}_{k}, \hat{E_{k}}).$$ On the other hand, because $(\hat{M}_{k}, \hat{E}_{k})$ is bundle modification of $(M_{k},E_{k})$, we have that
$$\mathrm{Index}(M_{k}, E_{k}) = \mathrm{Index}(\hat{M}_{k}, \hat{E_{k}}).$$ This completes the proof.
By the two propositions above, we can conclude that the quantization map $Q_{\mathrm{TOP}}$ is a well-defined map from $\hat{K}(G)$ to $\hat{R}(G)$.
The quantization map $Q_{\mathrm{TOP}}$ gives an isomorphism:
$$Q_{\mathrm{TOP}} : \hat{K}(G) \cong \hat{R}(G).$$
Let $P$ be the map defined in Theorem \[o9\]. It is clear that $Q_{\mathrm{TOP}} \circ P : \hat{R}(G) {\longrightarrow}\hat{R}(G)$ is the identity map. Therefore, $Q_{\mathrm{TOP}}$ is surjective.
Since every $K$-cycle is equivariant to a discrete $K$-cycle, it is enough to prove the injectivity of compact $K$-cycles. This immediately follows from the fact that geometric $K$-homology is isomorphic to analytic $K$-homology [@Baum10] [@Baum07].
Quantization Commutes with Reduction
====================================
In the previous section, we give the definition of quantization for general $K$-cycles. In this framework, we will provide a new approach to the Quantization Commutes with Reduction theorem in non-compact setting. Let us begin with the multiplicative property.
\[t17\] Let $(M,E, \phi_{1})$ and $(N, F, \phi_{2})$ be two K-cycles in $\hat{K}(G)$, where $N$ is compact. We have the following:
$$Q_{\mathrm{TOP}}(M, E, \phi_{1}) \times Q_{\mathrm{TOP}}(N, F, \phi_{2}) = Q_{\mathrm{TOP}}(M \times N, E \boxtimes F, \phi_{1} \circ \pi_{1} + \phi_{2} \circ \pi_{2})$$ where $\pi_{1}$ is the projection from $M \times N$ to M and where $\pi_{2}$ is the one to N.
Consider the $K$-chain $(M \times N \times [0, 1] , \hat{E} \otimes \hat{F}, \phi)$, where $\hat{E}$ and $\hat{F}$ are the pullback of line bundles $E$ and $F$, and the map $\phi$ is defined by
$$\phi(m, n, t) = \phi_{1}(m) + t\phi_{2}(n).$$ This gives a bordism between
$$\label{k2}
(M \times N, E \boxtimes F, \phi_{1} \circ \pi_{1}) \ \text{and} \ (M \times N, E \boxtimes F, \phi_{1} \circ \pi_{1} + \phi_{2} \circ \pi_{2}).$$
By Theorem \[t8\], let us assume that there is a bordism
$$\label{o6}
(M, E, \phi_{1}) \sim \bigsqcup_{k} (M_{k}, E_{k}, \rho_{k}).$$
Since $N$ is compact, (\[o6\]) induces another bordism:
$$(M \times N, E \boxtimes F, \phi_{1} \circ \pi_{1}) \sim \bigsqcup_{k} (M_{k} \times N, E_{k} \boxtimes F, \hat{\rho_{k}}),$$ where $\hat{\rho_{k}}$ is the composition of $\rho_{k}$ with the projection from $M_{k} \times N$ to $M_{k}$. By the bordism invariance of quantization, we have
$$Q_{\mathrm{TOP}}(M \times N, E \boxtimes F, \phi_{1} \circ \pi_{1}) = \sum_{k} \mathrm{Index}(M_{k} \times N, E_{k} \boxtimes F).$$ Meanwhile, we have that
$$\sum_{k} \mathrm{Index}(M_{k} \times N, E_{k} \boxtimes F) = ( \sum_{k} \mathrm{Index}(M_{k}, E_{k})) \times \mathrm{Index}(N, F).$$ Hence, we get
$$\label{k1}
Q_{\mathrm{TOP}}(M \times N, E \boxtimes F, \phi_{1} \circ \pi_{1}) = Q_{\mathrm{TOP}}(M, E, \phi_{1}) \times Q_{\mathrm{TOP}}(N, F, \phi_{2}).$$
The theorem follows from (\[k2\]) and (\[k1\]).
Let $(M, E, \mu)$ be pre-quantum data. If the moment map $\mu$ is proper and 0 is a regular value, then
$$[Q_{\mathrm{TOP}}(M, E, \mu)]^{G} = Q_{\mathrm{TOP}}(M_{0}, E_{0}),$$ where $M_{0} = \mu^{-1}(0)/G$ and $E_{0} = (E|_{\mu^{-1}(0)})/G$.
$\mathbf{Sketch \ of \ the \ proof}$: First, recall the decomposition of the vanishing set $M^{\mu}$ [@Paradan01]:
$$M^{\mu} = \sum_{\gamma \in \Gamma} G . (M^{\gamma} \cap \mu^{-1}(\gamma)),$$ where $\Gamma$ is a discrete set in $\mathfrak{t}_{+}$. For each $\gamma \in \Gamma$, let $U_{\gamma}$ be a small open G-invariant neighborhood of $G . (M^{\gamma} \cap \mu^{-1}(\gamma))$ in $M$. By Theorem \[localization theorem\], we have
$$Q_{\mathrm{TOP}}(M, E, \mu) = \sum_{\gamma \in \Gamma} Q_{\mathrm{TOP}}(U_{\gamma}, E|_{U_{\gamma}}, \mu|_{U_{\gamma}}).$$ In accord with [@Paradan01], one can show that for all $\gamma \neq 0$,
$$[Q_{\mathrm{TOP}}(U_{\gamma}, E|_{U_{\gamma}}, \mu|_{U_{\gamma}})]^{G} = 0.$$ Thus,
$$[Q_{\mathrm{TOP}}(M, E, \mu)]^{G} = [Q_{\mathrm{TOP}}(U_{0}, E|_{U_{0}}, \mu|_{U_{0}})]^{G}.$$ Since $0$ is a regular value, we have that $U_{0} \cong \mu^{-1}(0) \times \mathfrak{g}^{*}$. As in Section 4, we can build a cap $(W, L, \psi)$. By gluing on the cap, we can get a compact $K$-cycle $(N, F)$. Remember that the construction of the cap (in particular, the line bundle $L$) is not unique. In fact, we can build $(W, L, \psi)$ in a way such that
$$\mathrm{Index}(N, F) = [Q_{\mathrm{TOP}}(M, E, \mu)]^{G}.$$ In this case, we can show that $(N, F)$ is in fact a bundle modification of $(M_{0}, E_{0})$ (for example, when $G = S^{1}$, $N$ is a sphere bundle over $M_{0}$). Therefore,
$$Q_{\mathrm{TOP}}(M_{0}, E_{0}) =[Q_{\mathrm{TOP}}(M, E, \mu)]^{G} .$$
As a result, Theorem \[o7\] follows from the two theorems above.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Genome-wide eQTL mapping explores the relationship between gene expression values and DNA variants to understand genetic causes of human disease. Due to the large number of genes and DNA variants that need to be assessed simultaneously, current methods for eQTL mapping often suffer from low detection power, especially for identifying *trans*-eQTLs. In this paper, we propose a new method that utilizes advanced techniques in large-scale signal detection to pursue the structure of eQTL data and improve the power for eQTL mapping. The new method greatly reduces the burden of joint modeling by developing a new ranking and screening strategy based on the higher criticism statistic. Numerical results in simulation studies demonstrate the superior performance of our method in detecting true eQTLs with reduced computational expense. The proposed method is also evaluated in HapMap eQTL data analysis and the results are compared to a database of known eQTLs.'
author:
- |
Jacob Rhyne$^1$, Jung-Ying Tzeng$^{1,2,3,4}$, Teng Zhang$^1$, and X. Jessie Jeng$^1$\
1. Department of Statistics, North Carolina State University\
2. Bioinformatics Research Center, North Carolina State University\
3. Department of Statistics, National Cheng-Kung University, Tainan, Taiwan.\
4. Institute of Epidemiology and Preventive Medicine,\
National Taiwan University, Taipei, Taiwan\
title: |
**eQTL Mapping via Effective SNP\
Ranking and Screening [^1]**
---
0
[0]{}
1
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[**eQTL Mapping via Effective SNP Ranking and Screening\
**]{}
Key Words: Dimension reduction; HC-LORS; Hotspot; Multivariate response; Penalized regression.
Introduction {#s:intro}
============
Expression quantitative trait loci (eQTLs) are genomic regions that carry DNA sequence variants that influence gene expression [@review2015]. eQTLs act either in *cis*, if the SNP is located near the gene whose expression is influenced, or in *trans*, if the SNP is located further away from the gene [@SNP_cis_trans; @eQTL_mapping_cookson]. *trans*-eQTLs are particularly challenging to identify in human population studies because of their weak effects [@westra2013].
Genome-wide eQTL mapping, first proposed by Jansen and Nap in 2001, explores the relationship between gene expression levels and DNA variants, including single nucleotide polymorphisms (SNPs), copy number variants (CNVs), and short tandem repeats and single amino acid repeats [@eQTL_mapping_cookson]. The goal of eQTL mapping is to identify genetic variants, usually SNPs, that are significantly associated with the expression of genes [@shabalin2012matrix]. Figure \[fig:eqtl\_example\] shows an example of eQTL results from [@wolen2011identifying]. In this figure, each dot denotes an identified significant association between a gene and a SNP. The results reveal that the expression of a gene can be associated with multiple SNPs, suggesting there exist potential joint effects of multiple SNPs on the expression value of a gene. In addition, some SNPs are strongly associated with a few genes, (e.g., red dots), and in this paper, we refer to them as strong-sparse eQTLs. Some are moderately associated with a number of genes, (e.g., a SNP with multiple blue dots), and we refer to them as hotspot eQTLs or weak-dense eQTLs.
\[h\] \[fig:eqtl\_example\]
eQTL mapping plays a pivotal role in human disease research, and can help to elucidate underlying genetic mechanisms of diseases [@eQTL_mapping_cookson; @eQTL_Present_Future]. Many links between genetic markers, such as SNPs, and diseases have been found through genome-wide association studies (GWAS). eQTL mapping provides a link to understand how SNPs actually influence diseases; when SNPs are associated with the expression of a gene in eQTL mapping and with a disease in GWAS, this may imply that the expression of the gene mediates the SNP effect on the disease [@review2015]. eQTL mapping has helped to elucidate the connection between SNPs and diseases such as Type 1 diabetes, ulcerative colitis, Chron’s disease, and many autoimmune diseases, especially when SNPs are in non-coding regions [@review2015].
In traditional eQTL analysis, the SNP effect on gene expression is assessed marginally, i.e. one SNP-gene pair at a time, which leads to a large number of tests [@shabalin2012matrix]. The significant gene-SNP pairs are identified by controlling type I error rates, for example, the family-wise error rate or the false discovery rate [@benjamini1995controlling]. This marginal strategy has the advantage of being computationally efficient and easy to apply. However, marginal eQTL selection has two primary drawbacks: (1) The total number of marginal tests, i.e., the number of SNPs times the number of genes, is large and multiplicity adjustment tends to result in a conservative finding, especially for identifying *trans*-eQTLs and eQTL hotspots. (2) Marginal selection overlooks the potential joint effect of SNPs on gene expression and cannot account for the non-genetic effects. Ignoring these two features may not precisely capture the underlying SNP-expression relationship and lead to a loss of power. These flaws of marginal eQTL selection have motivated the consideration of joint modeling methods that explore the relationship of gene expression and SNPs simultaneously.
There are many joint modeling methods available for eQTL analysis [@PANAMA; @StatMethods; @LMMEH; @yang2013LORS]. However, without first performing screening to reduce the number of SNPs, joint modeling is often computationally infeasible [@yang2013LORS]. For example, the authors of LMM-EH [@LMMEH], an earlier method performing joint modeling while accounting for hidden factors, reported that an analysis of a mice dataset required 10 hours to complete even when parallelized across 1,100 processors. LORS [@yang2013LORS], another popular joint modeling method that accounts for hidden factors, requires $O(qn^2 + n^2pq)$ operations to run, where $n$ is the sample size, $q$ is the number of genes, and $p$ is the number of SNPs. Since human datasets often have $O(10^4)$ genes and $O(10^6)$ SNPs, LORS is computationally very expensive for genome-wide eQTL mapping in human datasets. In order to make joint modeling feasible for large datasets, screening methods can be used to reduce the number of SNPs.
Since only a fraction of SNPs influence gene expression, an optimal screening method will remove the unimportant SNPs, i.e., SNPs not associated with expression, while keeping the important SNPs. The results of the screening greatly impact the joint modeling. If the screening method is not powerful enough, weak eQTLs may be lost too early and not have a chance to be detected with joint modeling. On the other hand, if the number of SNPs removed by screening is too conservative, there may be too many SNPs for joint modeling to be computationally feasible. There is a great need for a screening method that is powerful enough to keep weak eQTLs but still reduce the number of SNPs enough so that the overall computation complexity is reasonable.
A classic example of screening is to use linear regression to model the marginal relationship between each SNP-gene pair, record the p-value of the regression coefficient, and select those SNPs that have at least one p-values less than a pre-specified threshold [@marginal_SNP_selection]. The SNPs selected are used in further analysis such as joint modeling. This classical method and other similar screening methods that select those SNPs associated with at least one genes have three main flaws: (1) The threshold for marginal $p$-values is often chosen arbitrarily or by convention; (2) these methods generally struggle to identify weak effects of *trans*-eQTLs; and (3) the existing screening criteria cannot directly indicate the computation complexity of the follow-up joint modeling. Additionally, these methods have a tendency to carry over a large number of SNPs that do not actually influence gene expression into joint modeling, which affects the efficiency of joint modeling.
In this paper, we propose a new ranking and screening strategy to largely reduce the number of SNPs before joint modeling while retaining both strong and weak eQTLs. Instead of ranking SNP-gene pairs, we propose to rank only SNPs utilizing the Higher Criticism (HC) statistic that summarizes the association of the SNP to all genes. The HC statistic has been developed in high-dimensional statistics for simultaneous detection of sparse and dense signals [@donoho2004higher; @HC2011]. In eQTL mapping, an important SNP may be strongly associated with a few genes (strong-sparse) or weakly associated with a number of genes (weak-dense). Ranking the SNPs by their HC statistics favors both strong-sparse and weak-dense SNPs over noise SNPs.
Consequently, a screening procedure can be built upon the HC ranking to efficiently retain strong-sparse and weak-dense SNPs for joint modeling. We suggest to restrict the number of SNPs that are allowed to be carried over to joint modeling to the sample size $n$, so that joint modeling can be performed effectively with adequate sample size. Comparing to the existing screening methods that select the top-ranked SNP-gene pairs, our method directly controls the total number of SNPs carried over to joint modeling while effectively retains strong-sparse and weak-dense eQTLs, and greatly reduces the computation complexity.
After HC ranking and screening, many joint modeling methods can be applied to gene expression data and the reduced subset of SNPs. We outline our methodology in detail in Section 2 and provide an overview of a candidate joint modeling method that can be used after screening. Following this, we perform a simulation study to demonstrate the finite-sample performance of our method, and apply our method to eQTL data from the International HapMap Project.
Methodology
===========
Initial Estimate
----------------
The first step of our method is to construct an initial estimate $\hat{\textbf{B}}_{p\times q}$, where $p$ and $q$ are the numbers of SNPs and genes, respectively. The initial estimate can be obtained by the marginal regression method introduced in @yang2013LORS that solves the optimization problem
$$\label{LORS_Screening_optimization}
min_{(\beta_i, \mu, \textbf{L})} \frac{1}{2} \lVert \textbf{Y} - \textbf{X}_i\beta_i - \textbf{1}\mu - \textbf{L} \rVert_F^2 + \lambda \lVert \textbf{L} \rVert_*,$$
where $\textbf{Y} \in \mathbb{R}^{n \times q}$ is a matrix of gene expression levels, $\textbf{X}_i \in \mathbb{R}^{n \times 1}$ is the vector of genotypes of the $i^{th}$ SNP, $\beta_i \in \mathbb{R}^{1 \times q}$ is a row vector of coefficients corresponding to the association of the $i^{th}$ SNP to all the $q$ genes, $\mu \in \mathbb{R}^{1 \times q}$ is a row vector of intercepts, $\textbf{L} \in \mathbb{R}^{n \times q}$ is a matrix of non-genetic factors that influence gene expression, and $\lVert \textbf{L} \rVert_*$ is the nuclear norm of [**L**]{}. The optimization problem in equation (\[LORS\_Screening\_optimization\]) provides a mechanism for handling hidden, non-genetic factors that influence gene expression. Accounting for these hidden, unobservable factors has been shown to lead to better eQTL mapping performance [@PANAMA; @LMMEH; @yang2013LORS]. The algorithm to solve (\[LORS\_Screening\_optimization\]) is presented in the appendix. Denote the estimate for the coefficients of the $i^{th}$ SNP as $\hat{\beta}_i \in \mathbb{R}^{1 \times q}$ for $i=1,\dots,p$. We stack these $p$ row vectors to form the initial estimate $\hat{\textbf{B}}_{p\times q}$.
Summary Statistic for each SNP using Higher Criticism
-----------------------------------------------------
The next phase of our methodology is to rank each SNP by its importance to the gene expression levels. We propose to use the higher criticism (HC) statistic to measure the importance of each SNP because the HC statistic has a desirable property of being sensitive to both sparse and dense signals [@HC2011].
In order to construct the HC statistic, the initial estimate $\hat{\textbf{B}}_{p \times q}$ must be standardized. Let $Z_{ij}$ be the standardized $\hat{\beta}_{ij}$ corresponding to the $i^{th}$ SNP and the $j^{th}$ gene. We calculate the standardized estimates $Z_{ij}$ as $$\label{std_est}
Z_{ij} = \frac{\hat{\beta}_{ij}}{\sqrt{(\textbf{X}_i^T\textbf{X}_i)^{-1}\hat{V}(\textbf{Y}_j - \textbf{X}_i\hat{\beta}_{ij})}},$$ where $\hat{V}(\textbf{Y}_j - \textbf{X}_i\hat{\beta}_{ij})$ denotes the sample variance of $\textbf{Y}_j - \textbf{X}_i\hat{\beta}_{ij}$.
For a given SNP, the null hypothesis is that the SNP is not associated with any genes. The alternative hypothesis is that the SNP is associated with at least one gene. The hypotheses for all SNPs can be formulated as follows for $i=1,\dots,p$,
$$\label{Hyp_Test}
\begin{split}
H_{0i}: Z_{ij} \sim N(0, 1), \hspace{4.3cm} j = 1, \ldots, q,
\\
H_{1i} : Z_{ij} \sim (1 - \eta_i) N(0, 1) + \eta_i N(\mu_{i}, \sigma_{i}^2), \quad j = 1, \ldots, q,
\end{split}$$
where $\eta_i \in [0,1]$ denotes the proportion of genes regulated by SNP $i$; $\mu_{i}$ $(\ne 0)$ and $\sigma_{i}^2$ $(> 0)$ are the mean and variance of $Z_{ij}$, respectively, if SNP $i$ is associated with gene $j$. All of the parameters $\eta_i$, $\mu_{i}$, and $\sigma_{i}$ are unknown and varying with $i$.
In reality, most SNPs are not truly associated with any genes; therefore most of the null hypotheses $H_{0i}, i = 1, \ldots, p$, are true. The alternative hypotheses describe the distribution of $Z_{ij}$ for important SNPs by mixture models. Such mixture models include the strong-sparse case ($\mu_i$ is relatively large and $\eta_i$ is small) and the weak-dense case ($\mu_i$ is small and $\eta_i$ is relatively large).
Donoho and Jin (2004) developed the HC statistic to test the existence of sparse signals among a large number of observations. In this paper, we propose to use the HC statistic to summarize the importance of each SNP in regulating the genes. We present an analogous version of the HC statistic based on the standardized test statistic $Z_{ij}$. Let $S_i(t) = \sum_{j = 1}^q I\{|Z_{ij}|\geq t\}$. The HC statistic of SNP $i$ is calculated as
$$\label{hc}
HC_i = \sup_{t \geq 0} \bigg\{ \sqrt{q} \frac{S_i(t)/q - \bar{\Phi}(t)}{\sqrt{\bar{\Phi}(t)(1 - \bar{\Phi}(t))}} \bigg\}. \nonumber$$
Note that under the null hypotheses in equation (\[Hyp\_Test\]), the expectation and variance of $S_i(t)/q$ are
$$\label{exp_var}
E(S_i(t)/q) = \bar{\Phi}(t); \quad V(S_i(t)/q) = \frac{\bar{\Phi}(t)(1 - \bar{\Phi}(t))}{q}, \nonumber$$
where $\bar{\Phi}(t) = P(N(0, 1) > t)$. Therefore large values of $HC_i$ suggest that SNP $i$ is important in regulating the genes. More specifically, when SNP $i$ is important ($H_{1i}$ is true), $Z_{ij}$ has non-zero mean value for some $j$, which causes elevated mean of $S_i(t)$ for some $t$. Consider the case that SNP $i$ is strong-sparse, we observe several large $Z_{ij}$, and the standardized $S_i(t)$ may reach its max at a large $t$. On the other hand, when SNP $i$ is weak-dense, the standardized $S_i(t)$ may reach its max at a moderate $t$. By taking the maximum over $t$, the HC statistic can capture the information for both strong-sparse and weak-dense SNPs.
HC Ranking and Screening
------------------------
After the HC statistics are calculated for all SNPs, we propose to rank the SNPs by their HC statistics in a decreasing order. Since ranking SNPs by their HC statistics tends to place strong-sparse and weak-dense SNPs before irrelevant SNPs, we can select only the top-ranked SNPs for joint modeling. We may select the top $n$ ranked SNPs, where $n$ is the sample size, so that the ratio of sample size to dimensionality is adequate for joint modeling. Similar screening criterion has been used in [@yang2013LORS]. However, [@yang2013LORS] does not rank SNPs by a summary statistic; they rank and keep the top $n$ SNPs by the initial estimate $\hat{\textbf{B}}$ for each gene. The SNPs entering joint modeling are the union of the SNPs selected for each gene. Compared to [@yang2013LORS], our ranking and screening strategy can be more efficient in selecting much less SNPs for joint modeling while retaining both strong-sparse and weak-dense SNPs.
Joint Modeling after SNP Screening
----------------------------------
Let $\textbf{X}_r \in \mathbb{R}^{n \times n}$ represent the reduced SNP matrix corresponding to the $n$ SNPs selected by the HC ranking and screening. A joint modeling method can be used to explore the relationship between the gene expression matrix $\textbf{Y}$ and $\textbf{X}_r$. A popular method is LOw-Rank representation to account for confounding factors and make use of Sparse regression for eQTL mapping (LORS) [@yang2013LORS], the model structure for which is
$$\label{LORS_Model}
\textbf{Y}= \textbf{1}\textbf{$\mu$} + \textbf{X}_r\textbf{B}_r + \textbf{L} + \textbf{e}, \nonumber$$
where $\textbf{Y} \in \mathbb{R}^{n \times q}$ is the matrix of gene expression levels, $\textbf{X}_r \in \mathbb{R}^{n \times n}$ is the reduced matrix of SNPs, $\textbf{B} \in \mathbb{R}^{n \times q}$ is a matrix of coefficients, $\textbf{L} \in \mathbb{R}^{n \times q}$ is a matrix of non-genetic factors, $\textbf{1} \in \mathbb{R}^{n \times 1}$ is a vector of ones, $\textbf{$\mu$} \in \mathbb{R}^{1 \times q}$ is a row vector of intercepts, and $\textbf{e}\in \mathbb{R}^{n \times q}$ is an error matrix with $e_{ij} \sim N(0,\sigma^2)$. It is assumed that there are only a few hidden, non-genetic factors that influence gene expression and that there are only a small fraction of true SNP-gene associations. In other words, $\textbf{L}$ is low-rank and $\textbf{B}_r$ is sparse.
In Lagrangian form, the LORS optimization problem is
$$\label{LORS_Opt}
min_{\textbf{B}, \mu, \textbf{L}} || \textbf{Y} - \textbf{X}_r\textbf{B}_r - \textbf{1}\textbf{$\mu$} - \textbf{L}||^2_F + \rho||\textbf{B}_r||_1 + \lambda||\textbf{L}||_*,$$
where $\rho$ enforces the sparsity constraint on $\textbf{B}_r$ and $\lambda$ enforces the low-rank constraint on $\textbf{L}$. The LORS MATLAB implementation is freely available at <http://zhaocenter.org/software/>. We also implemented LORS algorithm in R, which is available upon requests from the authors.
Besides LORS, there are other methods that can be applied at the joint modeling stage, giving the users freedom to select the method that best suits their needs; for example [@LMMEH; @PANAMA; @VR2013; @HEFT; @Confetti]; etc.
Simulation Study {#s:sim_study}
================
In this section, we perform simulation studies to demonstrate the efficiency of HC ranking and compare HC ranking with other popular ranking methods. The SNPs selected by HC ranking and screening are carried over to joint modeling. We compare the whole procedure with the marginal screening and joint modeling procedure in @yang2013LORS.
Simulation Setup
----------------
Genotype data were obtained from HapMap3, the third phase of the International HapMap Project. We focus on the data of chromosome 1, which includes $24806$ SNPs for 160 subjects after LD pruning. Details of the data are provided in Section \[sec:realdata\] (eQTL Data Analysis). We simulate expression data of 200 genes for the 160 subjects.
The setup of the simulation study is similar to the design of the simulation study in the original LORS paper [@yang2013LORS]. Let $\textbf{B} \in \mathbb{R}^{24806 \times 200}$ be the coefficient matrix of SNP effects and let $\textbf{U} \in \mathbb{R}^{160 \times 200}$ be a matrix of non-genetic factors. Since we assume that $\textbf{B}$ is sparse, only 20 SNPs were set to be active, and for each active SNP, $m=10$ or $m=50$ genes were randomly selected to be influenced. We set $\beta$ to be either 0.5 or 2. The case where $\beta=2$ and $m=10$ represents the strong-sparse scenario and the case where $\beta=0.5$ and $m=50$ represents the weak-dense scenario.
It is known that hidden, non-genetic factors influence gene expression [@PANAMA; @Confetti; @LMMEH; @yang2013LORS]. We simulate a matrix of hidden factors **U** through the following steps: (1) $\textbf{H}_{n\times k} \sim N(0, I)$ where $k$ is the number of hidden factors and is set to 10; (2) $\textbf{$\Sigma$}_{n \times n} = \textbf{H}\textbf{H}^T$; (3) $\textbf{U}_j \sim N(\textbf{0},0.1 \times \textbf{$\Sigma$})$, where $\textbf{U}_j$ is the $j^{th}$ column of **U**. Finally, $\textbf{Y}$ was simulated by
$$\label{sim_setup}
\textbf{Y} = \textbf{XB} + \textbf{U} + \textbf{e}, \nonumber$$
where $\textbf{e}$ has its $j^{th}$ column simulated by $e_j \sim N(\textbf{0},\textbf{I})$.
Effectiveness of Ranking SNPs by HC Statistic {#sec:sim_ranking}
---------------------------------------------
First, we evaluate the effectiveness of the HC ranking. It is critical that an effective ranking procedure would rank active SNPs highly. Two baseline methods are used in comparison to the HC ranking: (1) ranking SNPs by the means of each row of $\hat{\textbf{B}}$ and (2) ranking SNPs by maximum absolute value by row. We refer to these two alternative ranking procedures as ROWMEANS and EXTREMEVAL respectively. By design, ROWMEANS should be good at detecting weak-dense SNPs and EXTREMEVAL should be good at detecting strong-sparse SNPs.
To evaluate the effectiveness of the three ranking methods, we use precision-recall curves. A precision-recall curve, commonly used in information retrieval, shows the precisions of a selection rule for different values of recall. The precision and recall are defined as $$\label{prec_rec}
Precision=\frac{TP}{TP+FP} \qquad Recall=\frac{TP}{TP+FN}, \nonumber$$ where $TP$ is the number of true positive SNPs, $FP$ is the number of false positive SNPs, and $FN$ is the number of false negative SNPs. Recall takes values in $\{1/20, 2/20, \ldots, 20/20\}$ because there are 20 active SNPs. For a given recall value, we calculate the average precisions (over 100 simulations) for all three ranking methods and report the results in Figure \[fig:sim\_PR\]. A method with higher precision-recall curve ranks more active SNPs before noise SNPs, relative to the other methods.
As seen in Figure \[fig:sim\_PR\], the HC ranking performs better at detecting and prioritizing active SNPs over noise SNPs than either the ROWMEANS ranking or EXTREMEVAL ranking in both strong-sparse and weak-dense settings. The ROWMEANS method performs particularly poorly, with a very low precision for each value of recall. The EXTREVEVAL ranking is more competitive with the HC ranking in the strong-sparse case, but the HC ranking still maintains an advantage.
\[fig:sim\_PR\]
Screening and Joint Modeling
----------------------------
In this section, we compare the performance of our procedure with the marginal screening and joint modeling procedure in [@yang2013LORS]. For fairness of comparison, LORS algorithm is used for joint modeling in both procedures. We refer to the procedure in [@yang2013LORS] as MS-LORS and our multi-step procedure as HC-LORS that provides eQTL mapping by a) using the algorithm in equation (\[LORS\_Screening\_optimization\]) to provide an initial estimate; b) standardizing the estimate and calculating HC statistics; c) ranking SNPs by HC and selecting the top $n$ SNPs; and d) applying LORS algorithm for joint modeling on the selected SNPs and the expression matrix. We compare both methods by their success in detecting true eQTLs and the computation time required to run each method.
Denote the estimates of $\textbf{B}$ from HC-LORS and MS-LORS as $\hat{\textbf{B}}_{HC}$ and $\hat{\textbf{B}}_{MS}$, respectively. The entries of $\hat{\textbf{B}}_{HC}$ and $\hat{\textbf{B}}_{MS}$ are ranked by magnitude in descending order. For each $\hat{\textbf{B}}_{HC_{ij}}$ or $\hat{\textbf{B}}_{{MS}_{ij}}$ that is nonzero, if the corresponding true $\textbf{B}_{ij} \ne 0$ the association is called a true positive (TP); if $\textbf{B}_{ij} = 0$, the entry is called a false positive (FP). The precision for the largest 1,000 associations sorted by the estimates of $\textbf{B}_{ij}$ is presented in Figure 3. Higher precision means a larger proportion of true eQTLs are captured in the selected SNP-gene association pairs. As seen in Figure \[fig:sim\_precision\], HC-LORS performs better than MS-LORS in detecting true eQTLs and prioritizing them over noise in both strong-sparse and weak-dense scenarios.
\[h\]
![Average precision for the top 1,000 associations in strong-sparse (left) and weak-dense (right) scenarios. In the legend, HC-LORS refers to the proposed multi-step procedure and MS-LORS refers to the screening and modeling method in [@yang2013LORS].](Prec_Assoc_beta_2_m_10_new.png "fig:"){width="80mm" height="75mm"} ![Average precision for the top 1,000 associations in strong-sparse (left) and weak-dense (right) scenarios. In the legend, HC-LORS refers to the proposed multi-step procedure and MS-LORS refers to the screening and modeling method in [@yang2013LORS].](Prec_Assoc_beta_0_5_m_50_new.png "fig:"){width="80mm" height="75mm"} \[fig:sim\_precision\]
HC-LORS requires less computational expense than MS-LORS due to effective ranking and screening. In fact, the screening step of MS-LORS selects 3066 and 3059 SNPs on average in the strong-sparse and weak-dense scenarios, respectively, whereas our HC ranking and screening selects only $160$ SNPs in both scenarios. Consequently, the computation time of HC-LORS for joint modeling is much less than that of MS-LORS. Table \[tab:sim\_time\] summarizes the median values of the computation times for joint modeling of the two methods from 100 replications. The improvement in computational cost coupled with more accurate eQTL detection makes HC-LORS a promising method for large scale eQTL analysis.
Setting Method Computation time
--------------------------------- --------- ------------------
strong-sparse $(\beta=2, m=10)$ HC-LORS 509
MS-LORS 1423
weak-dense $(\beta=0.5, m=50)$ HC-LORS 581
MS-LORS 1223
: Median values of computation time (in seconds) for joint modeling[]{data-label="tab:sim_time"}
eQTL Data Analysis {#sec:realdata}
==================
We illustrate the utility of HC-LORS on the expression and SNP data from the third phase of International HapMap Project (HapMap3). The dataset includes 1301 samples from a variety of human populations. The genotype data are provided on <ftp://ftp.ncbi.nlm.nih.gov/hapmap/genotypes/hapmap3_r3/plink_format/> and the details of gene expression data are provided on <http://www.ebi.ac.uk/arrayexpress/experiments/E-MTAB-264/>. We focus on chromosome 1 and use PLINK to remove correlated SNPs via linkage disequilibrium pruning with a window size 50, a moving window increment of 5 SNPs, and a cutoff value of $r^2 = 0.5$. Again using PLINK, the pruned genotype data were converted into numerical SNP data. Following this, the data were set to include only Asian individuals for which there were gene expression data available. The resulting data have 2010 gene probes and 24806 SNPs for 160 subjects. Although the influence of tuning the parameters $\lambda$ and $\rho$ in equation (\[LORS\_Opt\]) should be minor, we address potential issues by using cross validation to select the parameters in (\[LORS\_Opt\]) [@monte_carlo_cross_valid; @yang2013LORS].
We use the classification method of Westra et al. (2013) to identify eQTLs as *cis* or *trans* [@westra2013]. We call an eQTL *cis* if the distance between the base pair positions of the SNP and the probe midpoint is less than 250 kilobases (kb) and call the eQTL *trans* if this distance is greater than 5 megabases (mb). The top 5 SNPs, in terms of the estimated association effects, along with the associated probes are seen in Table \[tab:data\_top\]. A complete list of identified SNP and probe associations for the two methods, ordered by the estimated association effects, can be found in the supplemental document.
**Method** **SNP (Gene)** **Probe (Gene)** **Distance** **Class.**
------------ ------------------------- ------------------------- -------------- ------------
rs12745189 (Intergenic) ILMN\_1762255 (GSTM1) 45.34 (kb) cis
rs12745189 (Intergenic) ILMN\_1668134 (GSTM1) 46.55 (kb) cis
rs4657741 (TIPRL) ILMN\_1779432 (TIPRL) 21.53 (kb) cis
rs2138686 (BMP8B) ILMN\_1653730 (OXCT2) 7.68 (kb) cis
rs3007708 (Intergenic) ILMN\_1796712 (S100A10) 12.23 (kb) cis
rs2239892 (GSTM1) ILMN\_1762255 (GSTM1) 2.38 (kb) cis
rs10802457 (ZNF695) ILMN\_1705078 (CHI3L2) 133.63 (mb) trans
rs2239892 (GSTM1) ILMN\_1668134 (GSTM1) 1.18 (kb) cis
rs10802457 (ZNF695) ILMN\_1685045 (CHI3L2) 133.64 (mb) trans
rs12121466 (Intergenic) ILMN\_1769839 (L1TD1) 44.50 (mb) trans
: SNPs with the strongest estimated associations and the distance between the SNP location and probe midpoint. HC-LORS denotes the proposed multi-step procedure; MS-LORS denotes the screening and modeling method in [@yang2013LORS].[]{data-label="tab:data_top"}
As seen in Table \[tab:data\_top\], the strongest associations detected by HC-LORS represent *cis*-eQTLs, as the distance between the probe midpoint and the SNP position is small, and the strongest associations detected by MS-LORS are *cis* and *trans*-eQTLs. Since *trans*-eQTLs tend to have very weak signals, we expect the strongest signals to correspond to *cis*-eQTLs [@westra2013]. Both methods detect a large proportion of *trans*-eQTLs; out of the 158 associations detected by HC-LORS, 113 of them are classified as potential *trans*-eQTLs and out of the 117 associations detected by MS-LORS, 106 are classified as *trans*-eQTLs. This suggests that both methods show promise in *trans*-eQTL detection.
In addition to identifying candidate *cis* and *trans*-eQTLs, both methods identify SNPs associated with multiple genes, or hotspots. Since Yang et al. (2013) define hotspot as a SNP being associated with the expression of at least 15 genes out of 7084 (or 0.21%), we define hotspots as cases where a SNP is associated with at least $\lceil 0.0021 \times 2010 \rceil = 5$ genes. It can be seen in Table \[tab:data\_hotspot\] that HC-LORS detects four hotspots and MS-LORS detects two hotspots.
**Method** **SNP (Gene)** **Associated Genes**
------------ ------------------------- ---------------------------------------
HC-LORS rs2297663 (LRP8) AIM2 GSTM1 L1TD1 CAMK1G C1orf123
PHGDH SH2D2A
rs12408890 (WDR8) LRRC8C WDR8 GSTM1 L1TD1 FAM79A
GPX7
rs12745189 (Intergenic) GSTM1 GSTM1 GSTM4 GSTM4 HSPA6
rs11204737 (ARNT) CTSS ARNT CGN MNDA PRRX1
MS-LORS rs4926440 (SCCPDH) LRRC8C ATP1B1 IL23R SFN BCAR3 MFSD2
NECAP2 WDR8 RAP1GAP EDG1 PIGR
IBRDC3 CNN3 GSTM1 GSTM1 RALGPS2
ZC3H12A L1TD1 MAP3K6 FLJ20054 IPP
WNT3A C1orf63 CGN NPL ATP1B1 DARC
PADI4 DENND2D FCGR2B VASH2 SCCPDH
LOC645436 ATP2B4 LOC644094 C1orf54
SSR2 LOC391157 HSPA6 LOC729853 NMNAT2
FLJ25476 LPGAT1 TNFSF4
rs2239892 (GSTM1) LRRC8C WDR8 GSTM1 L1TD1 FAM79A GPX7
: SNPs that are declared hotspots by HC-LORS and MS-LORS[]{data-label="tab:data_hotspot"}
To further examine the eQTL detection by HC-LORS and MS-LORS, we compare our findings with the *cis* and *trans* identification in the seeQTL database of the University of North Carolina at Chapel Hill, available at <http://www.bios.unc.edu/research/genomic_software/seeQTL/> [@seeQTL], which report eQTLs identified from a meta-analysis from HapMap human lymphoblastoid cell lines. We find that out of the 158 SNP-probe associations detected by HC-LORS, 23 are confirmed by the seeQTL database. On the other hand, out of the 117 associations detected by MS-LORS, only 3 are confirmed by the seeQTL database. Figure \[fig:data\_verify\] presents the percentage of identified eQTLs that also appear in seeQTL database for each method. It is clearly shown that HC-LORS achieves higher percentage and, therefore, better consistency with the database. This is particularly obvious when comparing the overlapping percentage of the top 25 ranked eQTLs detected by each method (right panel of Figure \[fig:data\_verify\]). This result further supports the efficiency of our method in detecting true eQTLs and prioritizing them over noise.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Proportion of identified eQTLs that also appear in seeQTL database. The left plot presents the proportion for the ranked eQTLs of each method. The right plot zooms in on the top 25 detected eQTLs of each method. In the legend, HC-LORS and MS-LORS refer to our multi-step method and the screening and modeling method presented in [@yang2013LORS], respectively. []{data-label="fig:data_verify"}](all_eQTLs_legend "fig:"){width="80mm" height="75mm"} ![Proportion of identified eQTLs that also appear in seeQTL database. The left plot presents the proportion for the ranked eQTLs of each method. The right plot zooms in on the top 25 detected eQTLs of each method. In the legend, HC-LORS and MS-LORS refer to our multi-step method and the screening and modeling method presented in [@yang2013LORS], respectively. []{data-label="fig:data_verify"}](top_eQTLs_no_legend "fig:"){width="80mm" height="75mm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Table \[tab:data\_verify\] summarizes the identified SNPs that are also in seeQTL, the genes that the SNPs are associated with, and [*cis/trans*]{} classifications. It can be seen that, although both methods detect many *trans*-eQTLs, almost all the overlapped eQTLs are [*cis*]{}. This could be due to the limited record of [*trans*]{}-eQTLs in the database. The proposed study has the potential to provide timely development for [*trans*]{} detection in eQTL mapping.
**Method** **SNP** **Associated Gene(s)** **Classification**
------------ ------------ ----------------------------------- --------------------
rs10888391 $CTSS^{*+}$ cis
rs11204737 $CTSS^{*+}$ cis
rs12137269 $ST7L^{*+}$ cis
rs12568757 $CTSK^{*+}$ cis
rs12745189 $GSTM1^{*+} GSTM4^{*+}$ cis
rs2138686 $OXCT2^{*+} PPIE^{*+} BMP8B^{*+}$ cis
rs2297663 $C1orf123^{*+}$ cis
rs234098 $FAM129A^{*+}$ cis
rs3007708 $S100A10^{*+} THEM4^{*+}$ cis
rs4654748 $NBPF3^{*+}$ cis
rs4660652 $NDUFS5^{*+}$ cis
rs518365 $IPP^{*+}$ cis
rs6684005 $EFCAB2^{*+}$ cis
rs954679 $ST7L^{*+}$ cis
rs2310752 $TGFBR3^* DPYD^+$ trans
rs2239892 $GSTM1^{*+}$ cis
rs4926440 $SCCPDH^{*+}$ cis
: SNP gene pairs identified by HC-LORS and LORS also found in the UNC seeQTL database. \* denotes that the association was found through HC-LORS or LORS and + denotes that the association was found in the seeQLT database[]{data-label="tab:data_verify"}
Discussion
==========
In this paper, we present a new ranking and screening strategy based on the higher criticism (HC) statistic for eQTL mapping. The HC ranking and screening is effective in prioritizing and selecting both strong-sparse and weak-dense SNPs over noise SNPs. Combing HC ranking and screening with joint modeling, our multi-step procedure, HC-LORS, shows higher accuracy in detecting true eQTLs with lower computation cost compared to existing methods. In an analysis of eQTL data from the International HapMap Project, the result of HC-LORS shows higher consistency with the seeQTL database than existing methods. Due to limited sample size ($n=160$), SNPs on chromosome 1 have been included in our analysis. The proposed method can be applied to genome-wide eQTL mapping with larger sample size. The emerging technological discoveries will result in more data being available for eQTL analysis and greater computational challenge. Because of this, an effective screening method to reduce the number of SNPs will be of even greater importance. Our method shows promise in detecting [*cis*]{}- and *trans*-eQTLs and dramatically reducing computational expense.
Acknowledgment {#acknowledgment .unnumbered}
==============
Dr. Jeng was partially supported by National Human Genome Research Institute of the National Institute of Health under grant R03HG008642. Dr. Tzeng was partially supported by National Institutes of Health under grant P01 CA142538.
Appendix {#appendix .unnumbered}
========
Marginal Estimate Algorithm {#marginal-estimate-algorithm .unnumbered}
---------------------------
We obtain the solution of (\[LORS\_Screening\_optimization\]) using the screening algorithm presented in the supplemental document of [@yang2013LORS] based on the following Lemma from [@Mazumder].
\[Mazudmer\_Lemma\] Suppose matrix **W** has rank $r$. The solution to the optimization problem $min_{\textbf{Z}} \lVert \textbf{W} - \textbf{Z} \rVert_F^2 + \lambda \lVert \textbf{Z} \rVert_*$ is given by $\hat{\textbf{Z}} = S_\lambda (\textbf{W})$ where $S_\lambda (\textbf{W}) = \textbf{U}\textbf{D}_\lambda \textbf{V}^T$ with $\textbf{D}_\lambda = diag[(d_1 - \lambda)_+,...,(d_r - \lambda)_+]$ and $\textbf{UDV}^T$ is the SVD of **W**
The screening algorithm iteratively solve the follows.
1. Fix $(\textbf{B}_i, \mu)$. The optimization problem becomes $min_{\textbf{L}} || \textbf{Y} - \textbf{X}_i\textbf{B}_i - \textbf{1}\textbf{$\mu$} - \textbf{L}||^2_F + \lambda||\textbf{L}||_*$
- By Lemma \[Mazudmer\_Lemma\] the solution is $\textbf{L} = S_{\lambda}(\textbf{Y} - \textbf{X}_i \textbf{B}_i - \textbf{1}\mu)$
2. Fix $\textbf{L}$. The optimization problem becomes $argmin_{(\textbf{B}_i,\mu)} \frac{1}{2} \lVert \textbf{Y} - \textbf{L} - \textbf{X}_i\textbf{B}_i - \textbf{1}\mu \rVert_2^2$
- This problem can be solved by ordinary least squares
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[^1]: Address for correspondence: X. Jessie Jeng, Department of Statistics, North Carolina State University, SAS Hall, 2311 Stinson Dr., Raleigh, NC 27695-8203, USA. E-mail: [email protected].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A group is combable if it can be represented by a language of words satisfying a fellow traveller property; an automatic group has a synchronous combing which is a regular language. This article surveys results for combable groups, in particular in the case where the combing is a formal language.'
address: 'University of Newcastle, Newcastle NE1 7RU, UK'
author:
- Sarah Rees
title: |
Hairdressing in groups: a survey of combings\
and formal languages
---
Introduction
============
The aim of this article is to survey work generalising the notion of an automatic group, in particular to classes of groups associated with various classes of formal languages in the same way that automatic groups are associated with regular languages.
The family of automatic groups, originally defined by Thurston in an attempt to abstract certain finiteness properties of the fundamental groups of hyperbolic manifolds recognised by Cannon in [@Cannon], has been of interest for some time. The defining properties of the family give a geometrical viewpoint on the groups and facilitate computation with them; to such a group is associated a set of paths in the Cayley graph of the group (a ‘language’ for the group) which both satisfies a geometrical ‘fellow traveller condition’ and, when viewed as a set of words, lies in the formal language class of regular languages. (A formal definition is given in section \[automatic\].) Epstein et al.’s book [@ECHLPT] gives a full account; the papers [@BGSS] and [@Farb] are also useful references (in particular, [@Farb] is very readable and non-technical).
The axioms of an automatic group are satisfied by all finite groups, all finitely generated free and abelian groups, word hyperbolic groups, the fundamental groups of compact Euclidean manifolds, and of compact or geometrically finite hyperbolic manifolds [@ECHLPT; @Lang], Coxeter groups , braid groups, many Artin groups [@Charney; @Charney2; @Peifer; @Juhasz], many mapping class groups [@Mosher], and groups satisfying various small cancellation conditions . However some very interesting groups are not automatic; the family of automatic groups fails to contain the fundamental groups of compact 3–manifolds based on the [*Nil*]{} or [*Sol*]{} geometries, and, more generally, fails to contain any nilpotent group (probably also any soluble group) which is not virtually abelian. This may be surprising since nilpotent groups have very natural languages, with which computation is very straightforward.
A family of groups which contains the fundamental groups of all compact, geometrisable 3–manifolds was defined by Bridson and Gilman in , through a weakening of both the fellow traveller condition and the formal language requirement of regularity for automatic groups. The fellow traveller condition was replaced by an asynchronous condition of the same type, and the regularity condition by a requirement that the language be in the wider class of ‘indexed languages’. The class of groups they defined can easily be seen to contain a range of nilpotent and soluble groups.
Bridson and Gilman’s work suggests that it is sensible to examine other families of groups, defined in a similar way to automatic groups with respect to other formal language classes. This paper surveys work on this theme. It attempts to be self contained, providing basic definitions and results, but referring the reader elsewhere for fuller details and proofs. Automatic groups are defined, and their basic properties described in section \[automatic\]; the more general notion of combings is then explained in section \[combings\]. A basic introduction to formal languages is given in section \[formal\_languages\] for the sake of the curious reader with limited experience in this area. (This section is included to set the results of the paper into context, but all or part of it could easily be omitted on a first reading.) Section 5 describes the closure properties of various classes of combable groups, and section 6 gives examples (and non-examples) of groups with combings in the classes of regular, context-free, indexed and real-time languages.
The author would like to thank the Fakultät für Mathematik of the Universität Bielefeld for its warm hospitality while this work was carried out, and the Deutscher Akademischer Austauschdienst for financial support.
Automatic groups {#automatic}
================
Let $G$ be a finitely generated group, and $X$ a finite generating set for $G$, and define $X^{-1}$ to be the set of inverses of the elements of $X$. We define a [*language*]{} for $G$ over $X$ to be a set of [*words*]{} over $X$ (that is, products in the free monoid over $X \cup X^{-1}$) which maps onto $G$ under the natural homomorphism; such a language is called [*bijective*]{} if the natural map is bijective.
The group $G$ is automatic if it possesses a language satisfying two essentially independent conditions, one a geometric ‘fellow traveller condition’, relating to the Cayley graph $\Gamma$ for $G$ over $X$, the other a restriction on the computational complexity of the language in terms of the formal language class in which the language lives. Before a precise definition of automaticity can be given, the fellow traveller condition needs to be explained.
Figure \[fellow\_travellers\] gives an informal definition of fellow travelling; we give a more formal definition below.
= 5cm
In the figure, the two pairs of paths labelled 1 and 2, and 3 and 4 synchronously fellow travel at a distance approximately equal to the length of the woman’s nose; the pair of paths labelled 2 and 3 asynchronously fellow travel at roughly the same distance. Particles moving at the same speeds along 1 and 2, or along 3 and 4, keep abreast; but a particle on 3 must move much faster than a particle on 2 to keep close to it.
More formally let $\Gamma$ be the Cayley graph for $G$ over $X$. (The vertices of $\Gamma$ correspond to the elements of $G$, and an edge labelled by $x$ leads from $g$ to $gx$, for each $g \in G, x \in X$). A word $w$ over $X$ is naturally associated with the finite path $\gamma_w$ labelled by it and starting at the identity in $\Gamma$. The path $\gamma_w$ can be parametrised by continuously extending the graph distance function $d_\Gamma$ (which gives edges length 1); where $|w|=d_\Gamma(1,w)$ is the string length of $w$, for $t\leq |w|$, we define $\gamma_w(t)$ to be a point distance $t$ along $\gamma_w$ from the identity vertex, and, for $t \geq |w|$, $\gamma_w(t)$ to be the endpoint of $\gamma_w$. Two paths $\gamma_1$ and $\gamma_2$ of $\Gamma$ are said to [*synchronously $K$–fellow travel*]{} if, for all $t\geq 0$, $d_\Gamma(\gamma_1(t), \gamma_2(t)) \leq K$, and [*asynchronously $K$–fellow travel*]{} if a strictly increasing positive valued function $h=h_{\gamma_1,\gamma_2}$ can be defined on the positive real numbers, mapping $[0,l(\gamma_1)+1]$ onto $[0,l(\gamma_2)+1]$, so that, for all $t\geq 0$, $d_\Gamma(\gamma_1(t), \gamma_2(h(t))) \leq K$.
Precisely, $G$ is [*automatic*]{} if, for some generating set $X$, $G$ has a language $L$ over $X$ satisfying the following two conditions. Firstly, for some $K$, and for any $w,v \in L$ for which $\gamma_v$ and $\gamma_w$ lead either to the same vertex or to neighbouring vertices of $\Gamma$, $\gamma_v$ and $\gamma_w$ synchronously $K$–fellow travel. Secondly $L$ is regular. A language is defined to be regular if it is the set of words accepted by a finite state automaton, that is, the most basic form of theoretical computer; the reader is referred to section \[formal\_languages\] for a crash course on automata theory and formal languages. The regularity of $L$ ensures that computation with $L$ is easy; the fellow traveller property ensures that the language behaves well under multiplication by a generator. Although this is not immediately obvious, the definition of automaticity is in fact independent of the generating set for $G$; that is, if $G$ has a regular language over some generating set satisfying the necessary fellow traveller condition, it has such a language over every generating set.
If $G$ is automatic, then $G$ is finitely presented and has quadratic isoperimetric inequality (that is, for some constant $A$, any loop of length $n$ in the Cayley graph $\Gamma$ can be divided into at most $An^2$ loops which are labelled by relators). It follows that $G$ has soluble word problem, and in fact there is a straightforward quadratic time algorithm to solve that.
If $G$ is automatic, then so is any subgroup of finite index in $G$, or quotient of $G$ by a finite normal subgroup, as well as any group in which $G$ is a subgroup of finite index, or of which $G$ is a quotient by a finite normal subgroup. The family of automatic groups is also closed under the taking of direct products, free products (with finite amalgamation), and HNN extensions (over finite subgroups), but not under passage to arbitrary subgroups, or under more general products or extensions.
Combings
========
In an attempt to find a family of groups which has many of the good properties of automatic groups, while also including the examples which are most clearly missing from that family, we define [*combable*]{} groups, using a variant of the first axiom for automatic groups.
Let $G=\langle X \rangle$ be a finitely generated group with associated Cayley graph $\Gamma$. We define an [*asynchronous combing*]{}, or [*combing*]{} for $G$ to be a language $L$ for $G$ with the property that for some $K$, and for any $w,v \in L$ for which $\gamma_v$ and $\gamma_w$ lead either to the same vertex or to neighbouring vertices of $\Gamma$, $\gamma_v$ and $\gamma_w$ asynchronously $K$–fellow travel; if $G$ has a combing, we say that $G$ is combable. Similarly, we define a [*synchronous combing*]{} to be a language for which an analogous synchronous fellow traveller condition holds; hence automatic groups have synchronous combings. Of course, every synchronous combing is also an asynchronous combing.
In the above definitions, we have no requirement of bijectivity, no condition on the length of words in $L$ relative to geodesic words, and no language theoretic restriction. In fact, the term ‘combing’ has been widely used in the literature, with various different meanings, and some definitions require some of these properties. Many authors require combings to be bijective; in [@ECHLPT] words in the language are required to be quasigeodesic, and in [@Gersten] combings are assumed to be synchronous.
The term ‘bicombing’ is also fairly widely used in the literature, and so, although we shall not be specifically interested in bicombability here, we give a definition for the sake of completeness. Briefly a bicombing is a combing for which words in the language related by left multiplication by a generator also satisfy a fellow traveller property. Specifically, a combing $L$ is a (synchronous, or asynchronous) [*bicombing*]{} if paths of the form $\gamma_v$ and $x\gamma_w$ (synchronously, or asynchronously) fellow travel, whenever $\gamma_v,\gamma_w \in L$, $x \in X$, and $v=_G xw$, and where $x\gamma_w$ is defined to be the concatenation of $x$ and a path from $x$ to $xw$ following edges labelled by the symbols of the word $\gamma_w$. A group is [*biautomatic*]{} if it has a synchronous bicombing which is a regular language.
Most known examples of combings for non-automatic groups are not known to be synchronous; certainly this is true of the combings for the non-automatic groups of compact, geometrisable 3–manifolds found by Bridson and Gilman. However, in recent and as yet unpublished work, Bestvina and N. Brady have constructed a synchronous, quasigeodesic (in fact linear) combing for a non-automatic group. By contrast, Burillo, in [@Burillo], has shown that none of the Heisenberg groups $$\begin{aligned}
H_{2n+1}&=&\langle x_1,\ldots x_n,y_1,\ldots y_n,z\,\mid [x_i,y_i]=z,\forall i,\\
& & [x_i,x_j]=[y_i,y_j]=[x_i,y_j]=1,\forall
i,j,i\neq j \rangle\end{aligned}$$ or the groups $U_n(\Z)$ of $n$ by $n$ unipotent upper-triangular integer matrices can admit synchronous combings by quasigeodesics (all of these groups are asynchronously combable). Burillo’s result was proved by consideration of higher-dimensional isoperimetric inequalities; the case of $H_3$ had been previously dealt with in [@ECHLPT].
Let $G$ be a combable group. Then, by [@Bridson] theorem 3.1, $G$ is finitely presented, and, by [@Bridson] theorems 4.1 and 4.2, $G$ has an exponential isoperimetric inequality; hence $G$ has soluble word problem (see [@ECHLPT], theorem 2.2.5). By [@Gersten], if $G$ has a synchronous, ‘prefix closed’ combing (that is, all prefixes of words in the language are in the language), then $G$ must actually have a quadratic isoperimetric inequality. Note that, by [@Kharlampovich] (or see [@BGS]), there are finitely presented class 3 soluble groups which have insoluble word problem, and so certainly cannot be combable.
For a combing to be of practical use, it must at least be recognisable. It is therefore natural to consider combings which lie in some formal language class, or rather, which can be defined by some theoretical model of computation. Automatic groups are associated with the most basic such model, that is, with finite state automata and regular languages. In general, where $\F$ is a class of formal languages we shall say that a group is $\F$–combable if it has a combing which is a language in $\F$. Relevant formal languages are discussed in section \[formal\_languages\].
An alternative generalisation of automatic groups is discussed in . This approach recognises that the fellow traveller condition for a group with language $L$ implies the regularity of the language $L'$ of pairs of words in $L$ which are equal in the group or related by right multiplication by a generator, and examines what happens when both $L$ and $L'$ are allowed to lie in a wider language class (in this particular case languages are considered which are intersections of context-free languages, and hence defined by series of pushdown automata). Some of the consequences of such a generalisation are quite different from those of the case of combings; for example, such groups need not be finitely presented.
Hierarchy of computational machines and formal languages {#formal_languages}
========================================================
Let $A$ be a finite set of symbols, which we shall call an [*alphabet*]{}. We define a [*language*]{} $L$ over $A$ to be a set of finite strings (words) over $A$, that is a subset of $A^* = \cup_{i\in \N}A^i$. We define a [*computational machine*]{} $M$ for $L$ to be a device which can be used to recognise the words in $L$, as follows. Words $w$ over $A$ can be input to $M$ one at a time for processing. If $w$ is in $L$, then the processing of $w$ terminates after some finite time, and $M$ identifies $w$ as being in $L$; if $w$ is not in $L$, then either $M$ recognises this after some time, or $M$ continues processing $w$ indefinitely. We define $L$ to be a [*formal language*]{} if it can be recognised by a computational machine; machines of varying complexity define various families of formal languages.
We shall consider various different types of computational machines. Each one can be described in terms of two basic components, namely a finite set $S$ of [*states*]{}, between which $M$ fluctuates, and (for all but the simplest machines) a possibly infinite [*memory*]{} mechanism. Of the states of $S$, one is identified as a [*start state*]{} and some are identified as [*accept states*]{}. Initially (that is, before a word is read) $M$ is always in the start state; the accept states are used by $M$ to help it in its decision process, possibly (depending on the type of the machine) in conjunction with information retrieved from the memory.
We illustrate the above description with a couple of examples of formal languages over the alphabet $A = \{-1,1\}$, and machines which recognise them.
We define $L_1$ to be the language over $A$ consisting of all strings containing an even number of $1$’s. This language is recognised by a very simple machine $M_1$ with two states and no additional memory. $S$ is the set $\{even,\ odd \}$; $even$ is the start state and only accept state. $M_1$ reads each word $w$ from left to right, and switches state each time a $1$ is read. The word $w$ is accepted if $M_1$ is in the state $even$ when it finishes reading $w$. $M_1$ is an example of a (deterministic) finite state automaton.
We define $L_2$ to be the language over $A$ consisting of all strings containing an equal number of $1$’s and $-1$’s. This language is recognised by a machine $M_2$ which reads an input word $w$ from left to right, and keeps a record at each stage of the sum of the digits so far read; $w$ is accepted if when the machine finishes reading $w$ this sum is equal to $0$. For this machine the memory is the crucial component (or rather, the start state is the only state). The language $L_2$ cannot be recognised by a machine without memory. $M_2$ is an example of a pushdown automaton.
A range of machines and formal language families, ranging from the simplest finite state automata and associated regular (sometimes known as rational) languages to the Turing machines and recursively enumerable languages, is described in ; a treatment directed towards geometrical group theorists is provided by [@Gilman]. One-way nested stack automata and real-time Turing machines (associated with indexed languages and real-time languages respectively) are also of interest to us in this article, and are discussed in [@Aho; @Aho2] and in [@Rabin; @Rosenberg]. We refer the reader to those papers for details, but below we try to give an informal overview of relevant machines and formal languages.
Figure \[hierarchy\] shows known inclusions between the formal language classes which we shall describe.
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(90.00,100.00)(30.00,00.00)[(30.00,05.00)[regular]{}]{} (30.00,05.00)[(-3,2)[15.00]{}]{} (60.00,05.00)[(3,5)[15.00]{}]{} (00.00,15.00)[(45.00,05.00)[deterministic context-free]{}]{} (15.00,20.00)[(0,1)[10.00]{}]{} (00.00,30.00)[(30.00,05.00)[context-free]{}]{} (15.00,35.00)[(0,1)[10.00]{}]{} (00.00,45.00)[(30.00,05.00)[indexed]{}]{} (15.00,50.00)[(3,2)[15.00]{}]{} (60.00,30.00)[(30.00,05.00)[real-time]{}]{} (75.00,35.00)[(-3,5)[15.00]{}]{} (30.00,60.00)[(30.00,05.00)[context sensitive]{}]{} (45.00,65.00)[(0,1)[10.00]{}]{} (30.00,75.00)[(30.00,05.00)[recursive]{}]{} (45.00,80.00)[(0,1)[10.00]{}]{} (25.00,90.00)[(40.00,05.00)[recursively enumerable]{}]{}
We continue with descriptions of various formal language classes; these might be passed over on a first reading.
Finite state automata and regular languages
-------------------------------------------
A set of words over a finite alphabet is defined to be a [*regular*]{} language precisely if it is the language defined by a finite state automaton. A [*finite state automaton*]{} is a machine without memory, which moves through the states of $S$ as it reads words over $A$ from left to right. The simplest examples are the so-called [*deterministic*]{} finite state automata. For these a transition function $\tau\co S \times A \rightarrow S$ determines passage between states; a word $w=a_1\ldots a_n$ ($a_i \in A$) is accepted if for some sequence of states $s_1,\ldots s_n$, of which $s_n$ is an accept state, for each $i$, $\tau(s_{i-1},a_i) = s_i$. Such a machine is probably best understood when viewed as a finite, directed, edge-labelled graph (possibly with loops and multiple edges), of which the states are vertices. The transition $\tau(s,a) = s'$ is then represented by an edge labelled by $a$ from the vertex $s$ to the vertex $s'$. At most one edge with any particular label leads from any given vertex (but since dead-end non-accept states can easily be ignored, there may be less that $|A|$ edges out of a vertex, and further, several edges with distinct labels might connect the same pair of vertices). A word $w$ is accepted if it labels a path through the graph from the start vertex/state $s_0$ to a vertex which is marked as an accept state. Figure \[automaton\] gives such a graphical description for the machine $M_1$ described at the beginning of section \[formal\_languages\]. In such a figure, it is customary to ring the vertices which represent accept states, and to point at the start state with a free arrow, hence the state $even$ is recognisable in this figure as the start state and sole accept state.
(90.00,30.00)(40.00,22.00)[(40.00,10.00)\[t\]]{} (40.00,27.00)[(1,0)[1]{}]{} (40.00,27.00)[(05.00,05.00)[1]{}]{} (40.00,18.00)[(40.00,10.00)\[b\]]{} (40.00,13.00)[(-1,0)[1]{}]{} (40.00,13.00)[(05.00,05.00)[1]{}]{} (10.00,20.00)[(15.00,20.00)]{} (25.00,20.00)[(05.00,05.00)[even]{}]{} (10.00,30.00)[(-1,0)[1]{}]{} (05.00,30.00)[(05.00,05.00)[-1]{}]{} (19.00,20.00) (19.00,20.00) (19.00,10.00)[(0,1)[6]{}]{} (70.00,20.00)[(15.00,20.00)]{} (50.00,20.00)[(05.00,05.00)[odd]{}]{} (70.00,30.00)[(1,0)[1]{}]{} (65.00,30.00)[(05.00,05.00)[-1]{}]{} (61.00,20.00)
-9mm
A [*non-deterministic*]{} finite state automaton is defined in the same way as a deterministic finite state automaton except that the transition function $\tau$ is allowed to be multivalued. A word $w$ is accepted if some (but not necessarily all) sequence of transitions following the symbols of $w$ leads to an accept state. The graphical representation of a non-deterministic machine may have any finite number of edges with a given label from each vertex. In addition, further edges labelled by a special symbol $\epsilon$ may allow the machine to leap, without reading from the input string, from one state to another, in a so-called $\epsilon$–move.
Given any finite state automaton, possibly with multiple edges from a vertex with the same label, possible with $\epsilon$–edges, a finite state automaton defining the same language can be constructed in which neither of these possibilities occur. Hence, at the level of finite state automata, there is no distinction between the deterministic and non-deterministic models. However, for other classes of machines (such as for pushdown automata, described below) non-determinism increases the power of a machine.
Turing machines and recursively enumerable languages
----------------------------------------------------
The [*Turing machines*]{}, associated with the [*recursively enumerable*]{} languages, lie at the other end of the computational spectrum from finite state automata, and are accepted as providing a formal definition of computability. In one of the simplest models (there are many equivalent models) of a Turing machine, we consider the input word to be written on a section of a doubly-infinite tape, which is read through a movable [*tape-head*]{}. The tape also serves as a memory device. Initially the tape contains only the input word $w$, the tape-head points at the left hand symbol of that word, and the machine is in the start state $s_0$. Subsequently, the tape-head may move both right and left along the tape (which remains stationary). At any stage, the tape-head either reads the symbol from the section of tape at which it currently points or observes that no symbol is written there. Depending on the state it is currently in, and what it observes on the tape, the machine changes state, writes a new symbol (possibly from $A$, but possibly one of finitely many other symbols, or blank) onto the tape, and either halts, or moves its tape-head right or left one position. The input word $w$ is accepted if the machine eventually halts in an accept state; it is possible that the machine may not halt on all input.
Non-deterministic models, where the machine may have a choice of moves in some situations (and accepts a word if some allowable sequence of moves from the obvious initial situation leads it to halt in an accept state), and models with any finite number of extra tapes and tape-heads, are all seen to be equivalent to the above description, in the sense that they also define the recursively enumerable languages.
Halting Turing machine and recursive languages
----------------------------------------------
A [*halting Turing machine*]{} is a Turing machine which halts on all input; thus both the language of the machine and its complement are recursively enumerable. A language accepted by such a machine is defined to be a [*recursive language*]{}.
Linear bounded automaton and context sensitive languages
--------------------------------------------------------
A [*linear bounded automaton*]{} is a non-deterministic Turing machine whose tape-head is only allowed to move through the piece of tape which initally contains the input word; special symbols, which cannot be overwritten, mark the two ends of the tape. Equivalently (and hence the name), the machine is restricted to a piece of tape whose length is a linear function of the length of the input word. A language accepted by such a machine is defined to be a [*context sensitive language*]{}.
Real-time Turing machines and real-time languages
-------------------------------------------------
A [*real-time Turing machine*]{} is most easily described as a deterministic Turing machine with any finite number of doubly-infinite tapes (one of which initially contains the input, and the others of which are initially empty), which halts as it finishes reading its input. Hence such a machine processes its input in ‘real time’.
A ‘move’ for this machine consists of an operation of each of the tape heads, together with a state change, as follows. On the input tape, the tape-head reads the symbol to which it currently points, and then moves one place to the right. On any other tape, the tape-head reads the symbol (if any) to which it currently points, prints a new symbol (or nothing), and then either moves right, or left, or stays still. The machine changes to a new state, which depends on its current state, and the symbols read from the tapes. When the tape-head on the input head has read the last symbol of the input, the whole machine halts, and the input word is accepted if the machine is in an accept state.
A language accepted by such a machine is defined to be a [*real-time language*]{}. $\{a^nb^nc^n:n \in \N\}$ is an example [@Rosenberg]. Examples are descibed in [@Rosenberg] both of real-times languages which do not lie in the class of context-free languages (described below), and of (even deterministic) context-free languages which are not real-time.
Pushdown automata and context-free languages
--------------------------------------------
A [*pushdown automaton*]{} can be described as a Turing machine with a particularly restricted operation on its tape, but it is probably easier to visualise as a machine formed by adding an infinite stack (commonly viewed as a spring-loaded pile of plates in a canteen) to a (possibly non-deterministic) finite state automaton. Initially the stack contains a single start symbol. Only the top symbol of the stack can be accessed at any time, and information can only be appended to the top of the stack. The input word $w$ is read from left to right. During each move, the top symbol of the stack is removed from the stack, and a symbol from $w$ may be read, or may not. Based on the symbols read, and the current state of the machine, the machine moves into a new state, and a string of symbols (possibly empty) from a finite alphabet is appended to the top of the stack. The word $w$ is accepted if after reading it the machine may be in an accept state. The language accepted by a pushdown automaton is defined to be a [*context-free language*]{}.
The machine $M_2$ described towards the beginning of this section can be seen to be a pushdown automaton as follows. The ‘sum so far’ is held in memory as either a sequence of $+1$’s or as a sequence of $-1$’s with the appropropriate sum. When the top symbol on the stack is $+1$ and a $-1$ is read from the input tape, the top stack symbol is removed, and nothing is added to the stack. When the top symbol on the stack is $-1$ and a $+1$ is read from the input tape, the top stack symbol is removed, and nothing is added to the stack. Otherwise, the top stack symbol is replaced, and then the input symbol is added to the stack. Hence the language $L_2$ recognised by $M_2$ is seen to be context-free. Similarly so is the language $\{a^nb^n: n \in N\}$ over the alphabet $\{a,b\}$. Neither language is regular. For symbols $a,b,c$, the language $\{a^nb^nc^n: n \in N\}$ is not context-free.
A pushdown automaton is deterministic if each input word $w$ defines a unique sequence of moves through the machine. This does not in fact mean that a symbol of $w$ must be read on each move, but rather that the decision to read a symbol from $w$ at any stage is determined by the symbol read from the stack and the current state of the machine. The class of deterministic context-free languages forms a proper subclass of the class of context-free languages, which contains both the examples of context-free languages given above. The language consisting of all words of the form $ww^R$ over some alphabet $A$ (where $w^R$ is the reverse of $w$) is non-deterministic context-free , but is not deterministic context-free.
One-way nested stack automata and indexed languages
---------------------------------------------------
A [*one-way nested stack automaton*]{} is probably most easily viewed as a generalisation of a pushdown automaton, that is, as a non-deterministic finite state automaton with an attached nest of stacks, rather than a single stack. The input word is read from left to right (as implied by the term ‘one-way’). In contrast to a pushdown automaton, the read/write tape-head of this machine is allowed some movement through the system of stacks. At any point of any stack to which the tape-head has access it can read, and a new nested stack can be created; while at the top of any stack it can also write, and delete. The tape-head can move down through any stack, but its upward movement is restricted; basically it is not allowed to move upwards out of a non-empty stack.
The language accepted by a one-way nested stack automaton is defined to be an [*indexed language*]{}. For symbols $a,b,c$, the languages $\{a^nb^nc^n:n \in \N\}$, $\{ a^{n^2}: n\geq 1\},\{a^{2^n}:n\geq 1\}$ and $\{ a^nb^{n^2}: n \geq 1\}$ are indexed , but $\{ a^{n!} : n \geq 1\}$ is not [@Hayashi], nor is $\{ (ab^n)^n: n \geq 1 \}$ [@Gilman2; @Hayashi].
From one $\F$–combing to another {#closure}
================================
Many of the closure properties of the family of automatic groups also hold for other classes of combable groups, often for synchronous as well as asynchronous combings.
In the list below we assume that $\F$ is either the set of all languages over a finite alphabet, or is one of the classes of formal languages described in section \[formal\_languages\], that is that $\F$ is one of the regular languages, context-free languages, indexed languages, context-sensitive languages, real-time languages, recursive languages, or recursively enumerable languages. (These results for all but real-time languages are proved in and [@Rees2], and for real-time languages in [@GHR].) Then just as for automatic groups, we have all the following results:
- If $G$ has a synchronous or asynchronous $\F$–combing then it has such a combing over any generating set.
- Where $N$ is a finite, normal subgroup of $G$, and $G$ is finitely generated, then $G$ is synchronously or asynchronously $\F$–combable if and only the same is true of $G/N$.
- Where $J$ is a finite index subgroup of $G$, then $G$ is synchronously or asynchronously $\F$–combable if and only if the same is true of $J$.
- If $G$ and $H$ are both asynchronously $\F$–combable then so are both $G\times H$ and $G \ast H$.
A crucial step in the construction of combings for 3–manifold groups in is a construction of Bridson in [@Bridson2]; combings for $N$ and $H$ can be put together to give an asynchronous combing for a split-extension of the form $N \rtimes H$ provided that $N$ has a combing which is particularly stable under the action of $H$. The set of all geodesics in a word hyperbolic group has that stability, and is a regular language; hence, for any of the language classes $\F$ considered in this section, any split extension of a word hyperbolic group by an $\F$–combable group is $\F$–combable. The free abelian group $\Z^n$ also possesses a combing with the necessary stability; hence all split extensions of $\Z^n$ by combable groups are asynchronously combable. It remains only to ask in which language class these combings lie.
Stable combings for $\Z^n$ are constructed by Bridson in [@Bridson2] as follows. $\Z^n$ is seen embedded as a lattice in $\R^n$, and the group element $g$ is then represented by a word which, as a path through the lattice, lies closest to the real line joining the point $0$ to the point representing $g$. For some group elements there is a selection of such paths; a systematic choice can clearly be made. It was proved in that $\Z^2$ has a combing of this type which is an indexed language; hence all split extensions of the form $\Z^2 \rtimes \Z$ were seen to be indexed combable. It followed from this that the fundamental groups of all compact, geometrisable 3–manifolds were indexed combable; for these are all commensurable with free products of groups which are either automatic or finite extensions of $\Z^2 \rtimes \Z$.
It is unclear whether or not the corresponding combing for $\Z^n$ is also an indexed language when $n>2$. Certainly it is a real-time language [@GHR]. Hence many split extensions of the form $\Z^n \rtimes H$ are seen to have asynchronous combings which are real-time languages. We give some examples in the final section.
Combing up the language hierarchy
=================================
Regular languages
-----------------
A group with a synchronous regular combing is, by definition, automatic. More generally, a group with a regular combing is called [*asynchronously automatic*]{} [@ECHLPT]. It is proved in [@ECHLPT] that the asynchronicity of an asynchronously automatic group is bounded; that is the relative speed at which particles must move along two fellow-travelling words in order to keep apace can be kept within bounds. The Baumslag–Solitar groups $$G_{p,q} = \langle a, b \mid ba^p = a^q b \rangle$$ are asynchronously automatic, but not automatic, for $p \neq \pm q$ (see [@ECHLPT; @Rees]), and automatic for $p = \pm q$.
It is proved in [@ECHLPT] that a nilpotent group which is not abelian-by-finite cannot be asynchronously automatic. From this it follows that the fundamental groups of compact manifolds based on the $Nil$ geometry cannot be asynchronously automatic; N. Brady proved that the same is true of groups of the compact manifolds based on the $Sol$ geometry [@Brady].
Context-free languages
----------------------
No examples are currently known of non-automatic groups with context-free combings. It is proved in that a nilpotent group which is not abelian-by-finite cannot have a bijective context-free combing; however it remains open whether a context-free combing with more that one representative for some group elements might be possible.
Indexed languages
-----------------
Bridson and Gilman proved that the fundamental group of every compact geometrisable 3–manifold (or orbifold) is indexed combable. By the results of described above for regular and context-free combings, this result must be close to being best possible.
It follows immediately from Bridson and Gilman’s results that a split extension of $\Z^2$ by an indexed combable (and so, certainly by an automatic) group is again indexed combable.
Real-time languages
-------------------
Since the stable combing of $\R^n$ described in section \[closure\] is a real-time language [@GHR], it follows that any split extension over $\Z^n$ of a real-time combable group is real-time combable. Hence (see [@GHR]), any finitely generated class 2 nilpotent group with cyclic commutator subgroup is real-time combable, and also any 3–generated class 2 nilpotent group. Further the free class 2 nilpotent groups, with presentation, $$\langle x_1,\ldots x_k \mid [[x_i,x_j],x_k],\,\forall i,j,k \rangle,$$ as well as the $n$–dimensional Heisenberg groups and the groups of $n$–dimensional, unipotent upper-triangular integer matrices, can all be expressed as split extensions over free abelian groups, and hence are real-time combable. It follows that any polycyclic-by-finite group (and so, in particular, any finitely generated nilpotent group) embeds as a subgroup in a real-time combable group.
Torsion-free polycyclic metabelian groups with centre disjoint from their commutator subgroup are far from being nilpotent, but are also real-time combable (see [@GHR]). Such groups split over their commutator subgroup, by a theorem of [@Robinson]. An example is provided by the group $$\langle x,y,z \mid yz = zy, y^x = yz, z^x = y^2z \rangle$$ which is certainly not automatic (it has exponential isoperimetric inequality). In fact this group is also indexed combable, since it is of the form $\Z^2 \rtimes \Z$.
1.5pt
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---
abstract: 'We consider a noncommutative scalar field with a covariantly constant noncommutative parameter in a curved space-time background. For a potential as a noncommutative polynomial it is shown that the stability conditions are unaffected by the noncommutativity, a result that is valid irrespective whether space-time has horizons or not.'
---
DF/IST–8.2008\
February 15, 2009
[**Stability Conditions For a Noncommutative Scalar Field Coupled to Gravity**]{}
Orfeu Bertolami$^{*}$ and Carlos A. D. Zarro$^{**}$\
\
\
[$^{*}$ Also at Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Lisboa]{}\
[E-mail address: [email protected]]{}\
[$^{**}$ Also at Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Lisboa]{}\
[E-mail address: [email protected]]{}
Introduction
============
Noncommutative geometry is believed to be a fundamental ingredient of quantum gravity [@Connes:1996gi] and it is shown to arise, under conditions, in String Theory [@Seiberg:1999vs]. Noncommutativity introduces a minimum length scale and can be implemented by generalizing the Heisenberg-Weyl algebra of Quantum Mechanics [@Snyder:1947]. This scale is presumably associated to the Planck length $L_{P}$, so that the structure of the space-time is assumed to be altered at this scale. Given its potentialities, noncommutative features can be implemented in Quantum Field Theories (for reviews, see $e.g.$ Refs. [@Szabo:2001kg; @Douglas:2001ba]), however, it is shown that the existence of a minimum length scale does not solve the problem of IR divergences and it actually introduces additional unitarity and causality problems. Other critical issues associated with noncommutative geometry involve the violation of translational invariance [@Bertolami:2003nm] and the question of noncommutative fields on a classical cosmological background [@Lizzi:2002ib; @Bertolami:2002eq].
Another interesting subject of research associated to noncommutative geometry concerns extensions of the quantum mechanics Heisenberg-Weyl algebra in order to generalize quantum mechanics both at configuration space level as well as at full phase space [@Gamboa:2001fg; @Zhang:2004yu; @Bertolami:2005jw; @Acatrinei:2003id; @Bertolami:2005ud; @Bastos:2006kj; @Bastos:2006ps]. Furthermore, noncommutative quantum cosmological models in the context of the minisuperspace Kantowski-Sachs metric have also been studied [@GarciaCompean:2001wy; @Barbosa:2004kp; @Bastos:2007bg]. Phase space noncommutative extensions which exhibit momenta space noncommutativity yield particularly interesting new feature in what concerns the selection of states for the early universe [@Bastos:2007bg].
In this work we examine the stability of noncommutative scalar fields with a polynomial potential in a curved space-time background. For this purpose we consider extensions of the positive energy theorem for gravity as originally deduced in Ref. [@Witten:1981mf]. The positive energy theorem states that the total gravitational energy cannot be negative if matter fields satisfy the dominant energy condition [@Witten:1981mf; @ChoquetBruhat:1985xy]. This establishes the classical and semi-classical stability of the Minkowski space-time. We consider the extension of this theorem that includes other kinds of fields such as scalar and vector fields. It is interesting that this setup allows, for instance, obtaining Bogolmony bounds for electromagnetic fields [@Gibbons:1982fy]. The stability of supergravity gauge theories was examined in Ref. [@Gibbons:1983aq], where it is shown the stability of supersymmetric theories in AdS spaces even when they exhibit negative local energy densities. This method was generalized to tackle situations where fields that do not admit a supersymmetric extension in Ref.[@Boucher:1984yx] and used for studying the stability conditions for scalar fields non-minimally coupled to gravity [@Bertolami:1987wj]. The generalization of the positive energy theorem to include black hole-type space-time was discussed in Ref. [@Gibbons:1982jg].
In our approach, we shall obtain the stability conditions for noncommutative scalar fields in the presence of gravity using the method of Refs. [@Boucher:1984yx; @Bertolami:1987wj]. This is achieved through a model in which the noncommutativity on scalar field is implemented via a Moyal product adapted to a curved space-time with a covariantly constant noncommutative parameter [@Lizzi:2002ib] and an additional condition to ensure the associativity of the noncommutative polynomial scalar field potential, as suggested in Ref. [@Bertolami:2002eq].
This Letter is organized as follows: in Section \[sec:model\] we present our noncommutative scalar field model and the conditions for consistently coupling it to gravity. In Section \[sec:gpet\] we discuss the commutative positive energy theorem and obtain its noncommutative counterpart. In Section \[sec:stability\] we obtain the stability conditions for noncommutative scalar fields with a noncommutative polynomial potential. In Section \[sec:bhspt\] we introduce space-times with horizons and show that the stability conditions previously obtained are also valid for these spaces. Finally Section \[sec:conclusions\] contains our conclusions.
The model {#sec:model}
=========
In a Minkowski space-time noncommutativity of fields is introduced via the so-called Moyal product [@Bayen:1977ha]
\[eq:notcovmoyal\] f g=\_[n=0]{}\^\^[\_[1]{}\_[1]{}]{}\^[\_[n]{}\_[n]{}]{}(\_[\_[1]{}]{}\_[\_[n]{}]{}f)(\_[\_[1]{}]{}\_[\_[n]{}]{}g),
where $\theta^{\mu\nu}$ is a constant noncommutative parameter. This parameter is related to the commutator between noncommutative coordinates in configuration space $[x^{\mu},x^{\nu}]=i\theta^{\mu\nu}$.
This product is not covariant, thus when considering a curved space-time a natural implementation for a covariant Moyal product would involve instead $\theta^{\mu\nu}$ as a tensor and covariant derivatives [@Lizzi:2002ib; @Bertolami:2002eq; @BarcelosNeto:2002bh; @Harikumar:2006xf][^1]
\[eq:covmoyal\] f g=\_[n=0]{}\^\^[\_[1]{}\_[1]{}]{}\^[\_[n]{}\_[n]{}]{}(\_[\_[1]{}]{}\_[\_[n]{}]{}f)(\_[\_[1]{}]{}\_[\_[n]{}]{}g).
However since covariant derivatives do not commute, the resulting Moyal product is not associative, $i.e.$ $(f\star g)\star h \neq f\star(g\star h)$. One could consider instead the Kontsevich product [@Kontsevich:1997vb], but its covariant version is also nonassociative [@Harikumar:2006xf]. Since one usually implements the noncommutativity through a mapping, the Seiberg-Witten map [@Seiberg:1999vs] up to some order in $\theta^{\mu\nu}$, this procedure usually maintains, up to that order, the associativity, and hence, one chooses the simplest form of covariant deformed product as defined by Eq. (\[eq:covmoyal\]).
One assumes that for curved space-times that
\[eq:covtheta\] \_\^ = 0,
$i.e.$ the noncommutative tensor is covariantly constant (see discussion below). This condition was considered in Refs. [@Lizzi:2002ib; @Harikumar:2006xf] as it generalizes the condition that $\theta^{\mu\nu}$ is constant.
Following Ref. [@Bertolami:2002eq], one consider a scalar field whose commutative analytic potential $V(\Phi)=\sum_{n=0}^{\infty}\frac{\lambda_{n}}{n!}\Phi^{n}$ is defined by substituting the product between functions by the Moyal product
\[eq:ncpot\] ()=\_[n=0]{}\^\^[ntimes]{},
where the tilde denotes a noncommutative function. Although the covariant Moyal product is nonassociative, one can choose an auxiliary condition to keep Eq. (\[eq:ncpot\]) associative [@Bertolami:2002eq] up to second order in the noncommutative parameter[^2]
\[eq:asscond\] \^\_=0,
and in this case, one can expand the noncommutative potential (\[eq:ncpot\]) up to the second order in $\theta^{\mu\nu}$ as [@Bertolami:2002eq]:
\[eq:ncpotexp\] ()= V() +(-\^[\_[1]{}\_[1]{}]{}\^[\_[2]{}\_[2]{}]{}\_[\_[1]{}]{}\_[\_[2]{}]{}\_[\_[1]{}]{}\_[\_[2]{}]{}).
Eq. (\[eq:asscond\]) admits two classes of solutions. For $\det\theta^{\mu\nu}\neq 0$, that is $\theta$ is invertible and then $\nabla_{\nu}\Phi=0$, a too strong condition for our problem. For $\det\theta^{\mu\nu}= 0$, then $\nabla_{\mu}\Phi$ can be written as powers of the noncommutative parameter, a solution that does not trivialize our problem (cf. Eq. (\[eq:condderphi\]) and ensued discussion).
One assumes that the gravity sector of the model is not affected by noncommutativity, therefore the space-time is still described by the usual Einstein equation with noncommutative sources
\[eq:eeq\] G\_ = \_,
where $\kappa =8\pi G$ and in the case under investigation the noncommutative energy-momentum tensor can be split into a scalar field and matter fields contributions: $\tilde{T}_{\mu\nu}=\tilde{T}^{\Phi}_{\mu\nu}+\tilde{T}^{\mbox{M}}_{\mu\nu}$. It is further assumed that matter fields satisfy the dominant energy condition[^3]. The noncommutative action then reads
=d\^[4]{}x.
The noncommutative generalization of the energy-momentum tensors are given by
\^\_ &=& ,\
\^\_ &=& (\_\_+ \_\_) - g\_\_\^+ g\_(). \[eq:ncemtsf\]
In order to discuss the stability conditions for the scalar field one considers the energy-momentum density for the gravitational field so that the associated four-momentum vector $p_{\mu}$ for a asymptotically flat space can be written as [@Nester:1982tr]
\[eq:wi\] 16G p\_V\^ = \_[S=]{} E\^dS\_ = \_\_E\^d\_,
where $V^{\mu}=\overline{\epsilon_{0}}\gamma^{\mu}\epsilon_{0}$, $\epsilon_{0}$ represents a constant Dirac spinor, $\Sigma$ is an arbitrary three-dimensional hypersurface and $S$ its boundary $\partial\Sigma$ at infinity. The two-form $E^{\sigma\alpha}$ is defined as[^4]
\[eq:ntf\] E\^=2(\^\_- \^),
where $\epsilon$ is a Dirac spinor that at infinity behaves as $\epsilon \rightarrow \epsilon_{0} + \mathcal{O}\left(\frac{1}{r}\right)$. The total energy-momentum can be written with the use of spinor fields. Since one assumes that gravity is not affected by noncommutativity, the product between spinor fields and gamma matrices is actually the usual one. One further assumes that spinor fields commute with the noncommutative scalar field.
Generalized positive energy theorem {#sec:gpet}
===================================
For supersymmetric theories the method used in Ref. [@Nester:1982tr] can be generalized by replacing Eq. (\[eq:ntf\]) by
\[eq:cntf\] \^=2(\^\_\^[i]{} - \^\^[i]{}),
where $\hat{\nabla}_{\mu}$ is the supercovariant derivative related to the change of the gravitino field $\psi^{i}_{\;\;\mu}$ under a supersymmetric transformation and $i=1,\ldots,N$ is the number of supersymmetries. One can show that Eq. (\[eq:wi\]) is then generalized to
\[eq:gwi\] 16G p\_ \^\_[0]{}\^[i]{} = \_d\_,
where $\delta\chi^{a}$ represents the change of spin-$\frac{1}{2}$ fields under a supersymmetric transformation. In the case of asymptotic Anti-de Sitter (AdS) space-time one requires another term on the L.H.S. of this equation in order to fix the four-momentum vector $p_{\mu}$. If $T^{\mbox{M}\sigma}_{\;\;\;\alpha}$ satisfies the dominant energy condition, then since vector $\overline{\epsilon_{0}^{i}}\gamma^{\alpha}\epsilon_{0}^{i}$ is non-space-like the first term in the integrand of Eq.(\[eq:gwi\]) is positive. Considering the time direction orthogonal to $\Sigma$, thus the last two terms of the R.H.S. of Eq. (\[eq:gwi\]) can be expressed as[^5]
$$\begin{aligned}
4\overline{\hat{\nabla}_{m}\epsilon^{i}}(\gamma^{0}\sigma^{mn}&+&\sigma^{mn}\gamma^{0})\hat{\nabla}_{n}\epsilon^{i} + \left(\delta\chi^{a}\right)^{\dagger}\delta\chi^{a} = \nonumber \\
&=& -4g^{mn}\left(\hat{\nabla}_{m}\epsilon^{i}\right)^{\dagger}\hat{\nabla}_{n}\epsilon^{i} + 4\left(\hat{\nabla}_{m}\epsilon^{i}\right)^{\dagger}\gamma^{m}\gamma^{n}\hat{\nabla}_{n}\epsilon^{i} + \left(\delta\chi^{a}\right)^{\dagger}\delta\chi^{a}.\end{aligned}$$
This term is positive definite if one chooses the Witten condition [@Gibbons:1983aq]
\[eq:witten\] \^[n]{}\_[n]{}\^[i]{} = 0.
For supersymmetric theories the values of $\hat{\nabla}_{n}\epsilon^{i}$ and $\delta\chi^{a}$ are automatically set by supersymmetry [@Gibbons:1983aq; @Boucher:1984yx]. If a theory does not admit a supersymmetric extension this setup can be used as discussed in Ref. [@Boucher:1984yx].
For the scalar field, we define, generalizing the result of Ref. [@Bertolami:1987wj],
$$\begin{aligned}
\hat{\nabla}_{\mu}\epsilon^{i} &=& \nabla_{\mu}\epsilon^{i} + \frac{i}{2}\kappa \gamma_{\mu}\tilde{f}^{ij}(\Phi)\epsilon^{j}, \label{eq:defscd} \\
\delta\chi^{a} &=& i\gamma^{\mu}\nabla_{\mu}\Phi\star\tilde{f}^{ai}_{2}(\Phi)\epsilon^{i}+\tilde{f}^{ai}_{3}(\Phi)\epsilon^{i}, \label{eq:defchi}\end{aligned}$$
where $\tilde{f}^{ij}(\Phi)$, $\tilde{f}^{ai}_{2}(\Phi)$ and $\tilde{f}^{ai}_{3}(\Phi)$ are noncommutative real scalar functions to be determined. Using the spinor identity $[\nabla_{\mu},\nabla_{\nu}]\epsilon=\frac{1}{2}R^{\alpha\beta}_{\;\;\mu\nu}\sigma_{\alpha\beta}\epsilon$ and Eqs. (\[eq:eeq\]) and (\[eq:ncemtsf\]), we can obtain $\nabla_{\alpha}\hat{E}^{\sigma\alpha}$
$$\begin{aligned}
\label{eq:ncdiv}
\nabla_{\alpha}\hat{E}^{\sigma\alpha} &=& 2\kappa\tilde{T}^{\mbox{M}\sigma}_{\;\;\;\alpha}\overline{\epsilon^{i}}\gamma^{\alpha}\epsilon^{i} + 4\overline{\hat{\nabla}_{\alpha}\epsilon^{i}}\star\Gamma^{\sigma \alpha\beta}\hat{\nabla}_{\beta}\epsilon^{i} + \overline{\delta\chi^{a}}\star\gamma^{\sigma}\delta\chi^{a} \nonumber \\
&+& \left(\tilde{f}^{ai}_{2}(\Phi)\star\nabla_{\alpha}\Phi\right)\star\left(\nabla_{\beta}\Phi\star\tilde{f}^{aj}_{2}(\Phi)\right)\overline{\epsilon}^{i}\Gamma^{\sigma \alpha\beta}\epsilon^{j} \nonumber \\
&+&\left\{2\kappa\delta^{ij}\left[\frac{\nabla^{\sigma}\Phi\star\nabla_{\alpha}\Phi + \nabla_{\alpha}\Phi\star\nabla^{\sigma}\Phi}{2} - \frac{\delta^{\sigma}_{\;\;\alpha}\nabla^{\rho}\Phi\star\nabla_{\rho}\Phi}{2} \right] \right. \nonumber \\
&-&\left[\left(\tilde{f}^{ai}_{2}(\Phi)\star\nabla_{\alpha}\Phi\right)\star\left(\nabla^{\sigma}\Phi\star\tilde{f}^{aj}_{2}(\Phi)\right) + \left(\tilde{f}^{ai}_{2}(\Phi)\star\nabla^{\sigma}\Phi\right)\star\left(\nabla_{\alpha}\Phi\star\tilde{f}^{aj}_{2}(\Phi)\right) \right. \nonumber \\
&-&\left. \left. \delta^{\sigma}_{\;\;\alpha}\left(\tilde{f}^{ai}_{2}(\Phi)\star\nabla^{\rho}\Phi\right)\star\left(\nabla_{\rho}\Phi\star\tilde{f}^{aj}_{2}(\Phi)\right)\right]\right\}\overline{\epsilon}^{i}\gamma^{\alpha}\epsilon^{j} + i\left[4\kappa\nabla_{\alpha}\tilde{f}^{ij}(\Phi) \right. \nonumber \\
&-& \left. \left(\tilde{f}^{ai}_{2}(\Phi)\star\nabla_{\alpha}\Phi\right)\star\tilde{f}^{aj}_{3}(\Phi) - \tilde{f}^{ai}_{3}(\Phi)\star\left(\nabla_{\alpha}\Phi\star\tilde{f}^{aj}_{2}(\Phi)\right)\right]\overline{\epsilon}^{i}\Gamma^{\sigma\alpha}\epsilon^{j} \nonumber \\
&+&\left[-\tilde{f}^{ai}_{3}(\Phi)\star\tilde{f}^{aj}_{3}(\Phi)+2\kappa\delta^{ij}\tilde{V}(\Phi)+ 6\kappa^{2}\tilde{f}^{il}(\Phi)\star\tilde{f}^{lj}(\Phi)\right]\overline{\epsilon}^{i}\gamma^{\sigma}\epsilon^{j} \nonumber \\
&+&i\left[\left(\tilde{f}^{ai}_{2}(\Phi)\star\nabla^{\sigma}\Phi\right)\star\tilde{f}^{aj}_{3}(\Phi) - \tilde{f}^{ai}_{3}(\Phi)\star\left(\nabla^{\sigma}\Phi\star\tilde{f}^{aj}_{2}(\Phi)\right) \right]\overline{\epsilon}^{i}\epsilon^{j}.\end{aligned}$$
One is now in conditions to examine the stability conditions for a noncommutative scalar field. Following Ref. [@Boucher:1984yx], the stability problem consists in obtaining the noncommutative functions $\tilde{f}^{ij}(\Phi)$, $\tilde{f}^{ai}_{2}(\Phi)$ and $\tilde{f}^{ai}_{3}(\Phi)$ for a given $\tilde{V}(\Phi)$ that ensure that Eq. (\[eq:ncdiv\]) is positive definite.
Stability conditions {#sec:stability}
====================
In order to obtain the stability conditions, one must identify Eq. (\[eq:ncdiv\]) with Eq. (\[eq:gwi\]). Therefore the coefficients of the last five terms in Eq. (\[eq:ncdiv\]) must vanish. One first notices that the resulting system of equations is quite difficult to solve, so one assumes that the conditions for indices $i,j,a$ are single valued. This simplifies considerably the system of equations.
One needs now to examine each term at the R. H. S. of Eq. (\[eq:ncdiv\]). The first term is positive definite given that the matter fields satisfy the dominant energy condition. Choosing “0" as the direction orthogonal to $\Sigma$, through Eq. (\[eq:witten\]) one gets that the second and the third terms can be written as
-4g\^[mn]{}(\_[m]{})\^\_[n]{}+ ()\^.
As $\theta^{\mu\nu}$ is covariantly constant, this will be positive definite if one chooses the conditions:
\^\_\_[n]{}&=&0, \[eq:condspin\]\
\^\_&=&0. \[eq:condchi\]
One considers now the expansion of a noncommutative function $\tilde{h}(\Phi)$ up to second order in the noncommutative parameter
\[eq:expncf\] ()=h+i\^h\_+\^[\_[1]{}\_[1]{}]{}\^[\_[2]{}\_[2]{}]{}h\_[\_[1]{}\_[2]{}\_[1]{}\_[2]{}]{},
where $h$ is a function of $\Phi$, $h_{\mu\nu}$ is an antisymmetric function of $\Phi$ and its derivatives, and so on. One uses this expansion to compute terms at Eq. (\[eq:ncdiv\]) that are functions of $\Phi$.
One looks now to the term proportional to $\overline{\epsilon}\Gamma^{\sigma\alpha\beta}\epsilon$. After using that $\Gamma^{\sigma\alpha\beta}$ is totally antisymmetric and Eq. (\[eq:id1\]) found in of the Appendix one obtains
f\_[2]{}\^[2]{}\_\_\_\_\^-\^[\_[1]{}\_[1]{}]{}\^[\_[2]{}\_[2]{}]{}f\_[2]{}f\_[2\_[2]{}\_[2]{}]{}\_[\_[1]{}]{}\_\_[\_[1]{}]{}\_\^,
which vanishes if one chooses that
\[eq:condderphi\] \^\_\_=0.
Using Eqs. (\[eq:id2\]) and (\[eq:id3\]) in the Appendix, the term proportional to $\overline{\epsilon}\epsilon$ can be computed:
i\^[\_[1]{}\_[1]{}]{}\^[\_[2]{}\_[2]{}]{}(f\_[2]{}\_[\_[1]{}]{}f\_[3\_[2]{}\_[2]{}]{} - f\_[3]{}\_[\_[1]{}]{}f\_[2\_[2]{}\_[2]{}]{} )\_[\_[1]{}]{}\^,
which vanishes given condition (\[eq:condderphi\]).
The term proportional to $\overline{\epsilon}\gamma^{\alpha}\epsilon$ reads, after using Eqs. (\[eq:condderphi\]) and (\[eq:id1\])
& & { (2-2f\_[2]{}\^[2]{})-i\^(4f\_[2]{}f\_[2]{})+\^[\_[1]{}\_[1]{}]{}\^[\_[2]{}\_[2]{}]{} (2f\_[2\_[1]{}\_[1]{}]{}f\_[2\_[2]{}\_[2]{}]{}-4f\_[2]{}f\_[2\_[1]{}\_[2]{}\_[1]{}\_[2]{}]{})}\
& & (\^\_- \^\_)\^.
Clearly, since coefficients of every order in the noncommutative parameter must vanish, one gets
\[eq:f2\] f\_[2]{}=f\_[2]{}=0f\_[2\_[1]{}\_[2]{}\_[1]{}\_[2]{}]{}=0,
and thus that $\tilde{f}_{2}(\Phi)=\sqrt{\kappa}$.
The term proportional to $\overline{\epsilon}\gamma^{\sigma}\epsilon$ reads after using Eqs. (\[eq:asscond\]), (\[eq:ncpotexp\]), (\[eq:condderphi\]) and (\[eq:id4\])
& &{-f\_[3]{}\^[2]{}+2V() + 6\^[2]{}f\^[2]{} + 2i\^(-f\_[3]{} f\_[3]{}+6\^[2]{}f f\_) + \^[\_[1]{}\_[1]{}]{}\^[\_[2]{}\_[2]{}]{}(-2f\_[3]{}f\_[3\_[1]{}\_[2]{}\_[1]{}\_[2]{}]{} . .\
& &. . +f\_[3\_[1]{}\_[1]{}]{}f\_[3\_[2]{}\_[2]{}]{} +12\^[2]{}ff\_[\_[1]{}\_[2]{}\_[1]{}\_[2]{}]{} -6\^[2]{}f\_[\_[1]{}\_[1]{}]{}f\_[\_[2]{}\_[2]{}]{}) } \^. \[eq:gammaalpha\]
In order to proceed one assumes that $\tilde{f}(\Phi)=a+b\Phi\star\Phi$, where constants $a$ and $b$ must be obtained by the boundary conditions of the system of equations; this condition generalizes the procedure of Ref. [@Bertolami:1987wj]. Using Eq. (\[eq:condderphi\]) one gets
-f\_[3]{}\^[2]{}+2V() + 6\^[2]{}f\^[2]{} &=&0, \[eq:f3b\]\
f\_[3]{}f\_[3]{}&=&0, \[eq:f3final\]\
f\_[3\_[1]{}\_[2]{}\_[1]{}\_[2]{}]{}&=&. \[eq:f3a\]
Eq. (\[eq:f3final\]) yields
\[eq:f3munu\] f\_[3]{}=0,
substituting this into Eq. (\[eq:f3a\]), it follows that
\[eq:f3so\] f\_[3\_[1]{}\_[2]{}\_[1]{}\_[2]{}]{}=0.
Finally, the term proportional to $\overline{\epsilon}\Gamma^{\sigma\alpha}\epsilon$ is given by
& &i{ \_-2i\^(f\_[2]{}f\_[3]{}+f\_[3]{}f\_[2]{}) .\
& & . - 2\^[\_[1]{}\_[1]{}]{}\^[\_[2]{}\_[2]{}]{} (f\_[3]{}f\_[2\_[1]{}\_[2]{}\_[1]{}\_[2]{}]{}+f\_[2]{}f\_[3\_[1]{}\_[2]{}\_[1]{}\_[2]{}]{}-f\_[2\_[1]{}\_[1]{}]{}f\_[3\_[2]{}\_[2]{}]{} )} \^.
Using Eqs. (\[eq:f2\]), (\[eq:f3munu\]) and (\[eq:f3so\]), it yields
4()-2f\_[2]{}f\_[3]{} =0. \[eq:derphif\]
Thus, the problem of stability consists in solving the system of equations
2()&=&f\_[3]{}, \[eq:stab1\]\
-f\_[3]{}\^[2]{}+2V() + 6\^[2]{}f\^[2]{} &=&0. \[eq:stab2\]
However, this is precisely the set of equations for the commutative case for a quartic potential solved in Ref. [@Bertolami:1987wj]. Our result is then that the stability conditions for a scalar with a noncommutative potential are not affected by noncommutativity.
Let us now examine the consistency of Eqs. (\[eq:condspin\]) and (\[eq:condchi\]) after solving the stability conditions (\[eq:f2\]), (\[eq:f3munu\]) and (\[eq:f3so\]). At first order in perturbation of the noncommutative parameter, one obtains
\^\_\_[n]{}=\^\_\_[n]{}+\_[n]{}f()\^\_+\_[n]{}(\^\_f())=0.
The first two terms vanish by the assumption that spinors are not affected by noncommutativity. The last term vanishes on account of Eq. (\[eq:asscond\]). Therefore, this equation is consistent with the results obtained above. One can also show that $\theta^{\mu\nu}\nabla_{\nu}\delta\chi=0$, using the assumption that spinors are not altered by noncommutativity and Eqs. (\[eq:asscond\]) and (\[eq:condderphi\]).
Space-time with horizons {#sec:bhspt}
========================
One considers now space-time configurations which admit horizons. In this situation, the divergence theorem must be modified so to include the horizon
\[eq:sggt\] \_[S]{} \^dS\_ - \_[H]{} \^dS\_ = \_\_\^d\_,
where $H$ denotes the horizon. Clearly, if the second term in the L. H. S. of Eq. (\[eq:sggt\]) vanishes the presence of horizons does not affect the stability conditions obtained in Section \[sec:stability\].
Following Ref. [@Straumann:1984xf] one introduces a orthonormal tetrad field at the horizon $\{e_{\hat{\mu}}\}$, where $e_{\hat{0}}$ is normal to the hypersurface $\Sigma$, $e_{\hat{1}}$ is normal to the two-surface $H$ and $e_{\hat{A}}$ ($A=2,3$) are tangent to $H$. Using this coordinate system then one has only to evaluate the term
\_[H]{} \^dS\_.
For simplicity one omits the hat on the indices. First one restricts the two-form to $\Sigma$, and thus through Witten’s condition $\gamma^{a}\hat{\nabla}_{a}\epsilon=0$, one finds that
.\^[0a]{}|\_ = -2\^\^[a]{}+ .
Using the definition of the supercovariant derivative and $\nabla_{b}\epsilon= ^{(3)}\!\!\nabla_{b}\epsilon+\frac{1}{2}K_{ab}\gamma^{0}\gamma^{a}\epsilon$, where $^{(3)}\!\nabla_{b}$ is the intrinsic three-dimensional covariant derivative and $K_{ab}$ is the second fundamental form of $\Sigma$, then the value of two-form on $H$ is given by
$$\begin{aligned}
\left.\hat{E}^{01}\right|_{H} &=& 2\epsilon^{\dagger}\hat{\nabla}_{1}\epsilon + \mbox{h. c.} \nonumber \\
&=& 2\epsilon^{\dagger} { }^{(3)}\!\nabla_{1}\epsilon + K_{1b}\epsilon^{\dagger}\gamma^{0}\gamma^{b}\epsilon - i\kappa\tilde{f}(\Phi)\epsilon^{\dagger}\gamma^{1}\epsilon + \mbox{h. c.} \label{eq:e01h1}\end{aligned}$$
From Witten’s condition:
\[eq:nabla1\] [ ]{}\^[(3)]{}\_[1]{}= \^[1]{}\^[A]{}[ ]{}\^[(3)]{}\_[A]{}- K\^[1]{}\^[0]{}+ i()\^[1]{},
where $K=K^{a}_{\;\;a}$. Substituting Eq. (\[eq:nabla1\]) into Eq. (\[eq:e01h1\]) and using that ${ }^{(3)}\!\nabla_{A}\epsilon={ }^{(2)}\!\nabla_{A}\epsilon - \frac{1}{2}J_{AB}\gamma^{1}\gamma^{B}\epsilon$[^6], it follows that
\[eq:e01h2\] .\^[01]{}|\_[H]{}=\^+ ,
where $\mathcal{D}_{A} \equiv \left({ }^{(2)}\!\nabla_{A} - \frac{1}{2}K_{1A}\gamma^{1}\gamma^{0} \right)$. A further condition is required to restrict the spinor field on $H$. This has been put forward in Ref. [@Gibbons:1982jg], namely: $\gamma^{1}\gamma^{0}\epsilon =\epsilon$. Now Eq. (\[eq:e01h2\]) reads
\[eq:e01h3\] .\^[01]{}|\_[H]{}=\^+ 2i()\^\^[1]{}+ .
Notice that $\left(J + K + K_{11}\right)=-\sqrt{2}\psi$, where $\psi$ is the expansion scalar [@Straumann:1984xf], which is related to the rate of increase of the absolute value of the element of area. If two neighbouring geodesics are converging, then $\psi<0$, if instead they diverge, then $\psi>0$. This quantity vanishes if $H$ is an apparent horizon. Given that $\gamma^{1}\gamma^{0}$ anticommutes with $\gamma^{1}\gamma^{A}\mathcal{D}_{A}$ and with $\gamma^{1}$, then
\[eq:ac1\] 2i()\^\^[1]{}=2i()\^\^[1]{}\^[1]{}\^[0]{}=-2i()\^\^[1]{}\^[0]{}\^[1]{}=-2i()\^\^[1]{}=0,
and
\[eq:ac2\] 2\^\^[1]{}\^[A]{}\_[A]{}=2\^\^[1]{}\^[A]{}\_[A]{}\^[1]{}\^[0]{}=-2\^\^[1]{}\^[0]{}\^[1]{}\^[A]{}\_[A]{}=-2\^\^[1]{}\^[A]{}\_[A]{}=0.
Thus, choosing the boundary $H$ to be an apparent horizon, from Eqs. (\[eq:ac1\]) and (\[eq:ac2\]) one finds that
\_[H]{} \^dS\_=0,
and therefore the presence of spaces with horizons does not affect the stability conditions found in Section \[sec:stability\].
Conclusions {#sec:conclusions}
===========
In this work the stability conditions for a noncommutative scalar field coupled to gravity have been examined. Gravity is assumed not to be affected by noncommutativity and also that in the Moyal product usual derivatives are replaced by covariant derivatives. Associativity is ensured through an auxiliary condition, namely $\theta^{\mu\nu}\nabla_{\nu}\Phi=0$. It is then found that for a scalar field with a polynomial potential, the stability conditions are the very ones for the commutative case studied in Ref. [@Bertolami:1987wj].
At first sight one might think that this result was already expected, given that no noncommutative corrections to $\tilde{V}(\Phi)$ and $\tilde{f}(\Phi)$ were considered up to the second order in $\theta$. This is not quite the case as one encounters that we obtain a nontrivial condition for the term proportional to $\overline{\epsilon}\Gamma^{\sigma\alpha\beta}\epsilon$ (Eq. (\[eq:condderphi\])), which is actually absent in the commutative case. It is interesting to point out that the obtained conditions for the stability of a noncommutative scalar field, Eqs. (\[eq:condspin\]), (\[eq:condchi\]) and (\[eq:condderphi\]), are structurally related with the associativity condition, Eq. (\[eq:asscond\]).
Finally, it has also been shown that the contribution of the surface integral $\oint_{H}\hat{E}^{\sigma\alpha}dS_{\sigma\alpha}$ on an apparent horizon vanishes. This means that stability results are not altered whether one considers space-time configurations with an apparent horizon.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of O. B. is partially supported by the Fundação para a Ciência e a Tecnologia (FCT) under the project POCI/FIS/56093/2004. The work of C. A. D. Z. is fully supported by the FCT fellowship SFRH/BD/29446/2006.
Appendix
========
In Section \[sec:stability\] after expanding terms up to second order in $\theta$, one cannot fail to see the similarity of many of the obtained terms. Here one derives all terms encountered in Eq. (\[eq:ncdiv\]). One uses the expansion of noncommutative functions in powers of the noncommutative parameter (Eq. (\[eq:expncf\])), the definition of the covariant Moyal product (Eq. (\[eq:covmoyal\])), and the associativity condition Eqs. (\[eq:covtheta\]) and (\[eq:asscond\]). Four types of noncommutative products are found:
(()\_)&&(\_())=f\^[2]{}\_\_+ i\^\
&+&\^[\_[1]{}\_[1]{}]{}\^[\_[2]{}\_[2]{}]{}, \[eq:id1\]
(()\^)()&=&fg\^+i\^(fg\_+gf\_)\^\
&+&\^[\_[1]{}\_[1]{}]{}\^[\_[2]{}\_[2]{}]{}, \[eq:id2\]
()(\^())&=&fg\^+i\^(fg\_+gf\_)\^\
&+&\^[\_[1]{}\_[1]{}]{}\^[\_[2]{}\_[2]{}]{}, \[eq:id3\]
\[eq:id4\] ()()=f\^[2]{}+2i\^ff\_+\^[\_[1]{}\_[1]{}]{}\^[\_[2]{}\_[2]{}]{}(2ff\_[\_[1]{}\_[2]{}\_[1]{}\_[2]{}]{}-f\_[\_[1]{}\_[1]{}]{}f\_[\_[2]{}\_[2]{}]{}).
[99]{}
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[^1]: Where f and g are in general tensor fields although their indices are omitted for simplicity.
[^2]: Another possible way to implement associativity would involve an $m$-dimensional Riemannian manifold with an $SO(m)$ holonomy group and condition (\[eq:covtheta\]). For a nondegenerate noncommutative parameter, this would require obtaining a “Maxwell" field without sources in a Kähler manifold, which for $m$ even, could yield an associative Moyal product. The authors thank Luis Alvarez-Gaumé for this remark.
[^3]: Physically this condition states that local energy density is positive, that is for any time-like vector $W^{\mu}$, $T_{\mu\nu}W^{\mu}W^{\nu}\geq 0$, and $T_{\mu\nu}W^{\mu}$ is not a space-like vector [@Hawking:1973uf].
[^4]: Our conventions are the following: the metric signature is $(+,-,-,-)$, $\overline{\epsilon}=\epsilon^{\dagger}\gamma^{0}$, $\{\gamma^{\mu}$,$\gamma^{\nu}\}=2g^{\mu\nu}$, $\sigma^{\mu\nu}=\frac{1}{4}[\gamma^{\mu},\gamma^{\nu}]$, $\epsilon_{0123}=+1$, $\nabla_{\alpha}\epsilon=\partial_{\alpha}\epsilon - \frac{1}{2}\omega^{\mu\nu}_{\alpha}\sigma_{\mu\nu}\epsilon$, $\Gamma^{\sigma\alpha\beta}=\gamma^{[\sigma}
\gamma^{\alpha}\gamma^{\beta]}$, $\Gamma^{\sigma\alpha}=\gamma^{[\sigma}
\gamma^{\alpha]}$.
[^5]: Latin indices span over $1,2,3$.
[^6]: ${ }^{(2)}\!\nabla_{A}$ is the intrinsic covariant derivative on $H$ and $J_{AB}$ is the second fundamental form on $H$ with $J=J^{A}_{\;\;A}$.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Despite oxides and some fluorides perovskites have emerged as prototypes of multiferroic and magnetoelectric materials, they have not impacted real devices. Unfortunately, their working temperatures are very low and the magnetoelectric coupling has been reported to be rather small. Herewith, we report from first-principles calculations an ideal magnetization reversal through polarization switching in BaCuF$_4$ which, according to our results, could be achieved close to room temperature. We also show that this ideal coupling is driven by a soft mode that combines both, polarization and octahedral rotation. The later being directly coupled to the weak ferromagnetism of BaCuF$_4$. This, added to its strong Jahn-Teller distortion and its orbital ordering, makes this material as a very appealing prototype for crystals in the $ABX_4$ family in multifunctional applications.'
author:
- 'A. C. Garcia-Castro$^{1,2}$'
- 'W. Ibarra-Hernandez$^{3,4}$'
- 'Eric Bousquet$^{1}$'
- 'Aldo H. Romero$^{3,4}$'
bibliography:
- 'library.bib'
title: 'Direct magnetization-polarization coupling in BaCuF$_4$ fluoride'
---
The search for materials that own ferroelectricity, magnetization, and orbital ordering with a large coupling between those properties, has become one of the most active research fields in condensed matter physics. Since last decade, it has received a vigorous interest by many different research groups due to their high potential in new technologies where multifunctionalities are required. Among these couplings, magnetoelectricity promises to reduce the computer memory energy consumption, to improve magnetic field sensors or to be used for spintronic applications [@Fiebig2016]. The magnetoelectric crystals, however, suffer from their scarcity, their small response and their low functioning temperature. In spite of the great improvements in identifying and understanding the underneath mechanism of magnetoelectricity, finding new room-temperature candidates has been difficult and their number is quite scarce. One of possible solution is to create those materials as composites but magnetoelectric single crystals are still very rare at the present stage of knowledge in this research field [@Fiebig2016]. Another solution to find good magnetoelectric single crystals has been to identify new ferroelectric materials, where the ferroelectric (FE) ordering can be coupled to the magnetization [@PhysRevLett.106.107204; @bousquet2008]. To that end, novel stoichiometries and compositions have been investigated and a promising approach is to look for layered perovskite materials where the octahedra rotations are linked to the polarization through improper-like couplings [@benedek2015; @young2015; @bousquet2008; @PhysRevLett.106.107204]. Indeed, Ruddlesden-Popper, Aurivillius, and Dion-Jacobson [@benedek2015] phases have been shown to be good candidates to exhibit the coupling between polarization and magnetism. The proof of concept has been shown theoretically by Benedek and Fennie in Ca$_3$Mn$_2$O$_7$ [@PhysRevLett.106.107204] where the coupling mechanism with magnetism relies on the improper origin of the polarization that indirectly couples with the magnetization. Unfortunately, the experimental efforts to verify this prediction have, so far, shown that the electric polarization in Ca$_3$Mn$_2$O$_7$ cannot be switched which seems to indicate that this crystal might not be the best candidate for magnetization reversal through an electric field [@doi:10.1063/1.4984841]. Following the same strategy, layered materials with formula $A_nB_nX_{3n+2}$ appeared to be other favorable candidates [@doi:10.1080/01411594.2014.986731; @PhysRevLett.109.217202; @Scarrozza2013]. In these crystal types, the octahedral rotations and the polar distortions are intrinsically coupled in a single mode through improper-like couplings too [@PhysRevB.84.075121]. However, here again, while the proof of concept has been shown, a specific candidate is still missing.
In this letter, we show from first-principles calculations that within the family of the Barium-based layered fluorides BaMF$_4$ [@Eibschutz1969], the case of BaCuF$_4$ has an ideal direct coupling between the polarization and the magnetization that does not rely on an improper mechanism. Combined with the rather high Néel temperature T$_N$ = 275 K (the largest over $M$ = Mn, Ni, Co, and Fe series), this makes BaCuF$_4$ an appealing new candidate for electric-field-tuned magnetization. We also show that the peculiar coupling is linked to subtle interplay between ferroelectricity, magnetism and Jahn-Teller (JT) effect.
We performed density-functional theory (DFT) [@PhysRev.136.B864; @PhysRev.140.A1133] calculations by using the <span style="font-variant:small-caps;">vasp</span> code (version 5.3.3) [@Kresse1996; @Kresse1999]. The projected-augmented waves, PAW [@Blochl1994], approach to represent the valence and core electrons was used. The electronic configurations taken into account in the pseudo-potentials as valence electrons are (5$p^6$6$s^1$, version 06Sep2000), (3$d^{10}$4$s^1$, version 06Sep2000), and (2$s^2$2$p^5$, version 08Apr2002) for Ba, Cu, and F atoms respectively. The exchange correlation was represented within the generalized gradient approximation GGA-PBEsol parametrization [@Perdew2008] and the $d$-electrons were corrected through the DFT$+U$ approximation within the Liechtenstein formalism [@Liechtenstein1995]. We used $U$ = 7.0 eV and $J$ = 0.9 eV parameter values that were optimized to reproduce the electronic band gap (3.9 eV) and magnetic moment of the HSE06 hybrid-functional calculation [@HSE; @HSE06]. These parameter values are in good agreement with the ones used in KCuF$_3$ [@0953-8984-25-11-115404; @Liechtenstein1995]. The periodic solution of the crystal was represented by using Bloch states with a Monkhorst-Pack [@PhysRevB.13.5188] *k*-point mesh of 6$\times$4$\times$6 and 600 eV energy cut-off to give forces convergence of less than 0.001 eV$\cdot$$^{-1}$. Spin-Orbit coupling (SOC) was included to consider non collinear magnetic configurations [@Hobbs2000]. Born effective charges and phonon calculations were performed within the density functional perturbation theory (DFPT) [@gonze1997] as implemented in <span style="font-variant:small-caps;">vasp</span>. The atomic structure figures were elaborated with the <span style="font-variant:small-caps;">vesta</span> code [@vesta]. Finally, the spontaneous polarization was computed by means of the Berry-phase approach [@Vanderbilt2000]. In what follows, we start by describing the characterization of the ferroic orders in this compound to understand the correlation between the ferroelectricity and the weak-ferromagnetic state of BaCuF$_4$.
![(Color online) $Cmcm$ and $Cmc2_1$ phases of the BaCuF$_4$. The phase transition responsible of the ferroelectric switching involves the octahedral (in blue with Cu at the center) rotations around the $x$-axis as well as the polar Ba-sites (in dark grey). The later rotations and displacements, denoted with arrows, belong to the $\Gamma_2^-$ mode.[]{data-label="F1"}](F1){width="8.5cm"}
*Ferroelectric ordering:* The Ba$M$F$_4$ family of compounds is structurally characterized by octahedral $M$F$_6$ layers separated by Ba sheets stacked along the $y$-axis as shown in Fig. \[F1\]. We start by analyzing the phonon modes of the hypothetical high-symmetry structure of BaCuF$_4$ fluoride ($Cmcm$ space group, No. 63), not observed experimentally before the melting temperature (around 1000 K) [@DiDomenico1969]. The computed phonons dispersion of the high-symmetry reference (Fig.S1 in Supplementary Material [@sup-mat]) reveals the presence of three unstable phonon modes, when looking at the zone center and zone boundary points: $\Gamma_2^-$ at 66 $i$cm$^{-1}$, $S_2^+$ at 57 $i$cm$^{-1}$, and $Y_2^-$ at 47 $i$cm$^{-1}$.
The $\Gamma_2^-$ mode is polar and combines polar motion of the Ba atoms along the $z$ direction with in-phase fluorine octahedra rotation around the $x$ direction (see Fig. \[F1\]a) and its condensation reduces the symmetry of the crystal to the $Cmc2_1$ (No. 36) space group. The full relaxation of the crystal within the $Cmc2_1$ lowers the energy with respect to the $Cmcm$ phase by $\Delta$E = -27 meV per formula unit (meV/f.u.) and by decomposing the final distortions into symmetry adapted mode [@Orobengoa:ks5225; @Perez-Mato:sh5107] we find a contribution of the $Cmcm$ $\Gamma_1^+$ and $\Gamma_2^-$ mode. The $\Gamma_1^+$ mode is the mode that is invariant under all the symmetry operations of the $Cmcm$ phase, which means that it is a relaxation of the initial $Cmcm$ degrees of freedom that favors the development of the polarization. A similar combination of modes has been reported in the ferroelectric LaTaO$_4$ [@LIU201631] and La$_2$Ti$_2$O$_7$ [@PhysRevB.84.075121].
The $S_2^+$ mode drives the system to a non-polar phase with $P2_1/c$ space group (No. 14) and its eigenvector is an out-of-phase octahedra rotations around the $x$-axis (See Fig.S1 in Supplementary Material [@sup-mat]). We note that it is the same type of distortion as for the $\Gamma_2^+$ mode (*i.e.* octahedral rotation around the $x$-axis) but in the case of the $S_2^+$ mode, the out-of-phase octahedral rotations do not break the space inversion symmetry. The gain of energy due to the relaxation of the $P2_1/c$ phase is $\Delta$E = -7 meV/f.u., thus about four times smaller than the $Cmc2_1$ phase.
The condensation of the $Y_2^-$ mode reduces the symmetry of the crystal into the $Pnma$ space group (No. 62) where its eigenvector involves in-phase clockwise rotation of the octahedra around the $x$-axis in one octahedral layer and an in-phase counter-clockwise in the next octahedral layer (Fig.S1 in Supplementary Material [@sup-mat]). The relaxation of the $Pnma$ phase lowers the energy by $\Delta$E = -10 meV/f.u., which is larger than the $P2_1/c$ phase but more than two times smaller than the $Cmc2_1$ phase.
We thus find that the ferroelectric $Cmc2_1$ phase is the ground state of BaCuF$_4$, which is in agreement with the experiments [@DANCE1981599; @BABEL198577]. In this compound, in contrast with the other members of the same Ba$M$F$_4$ family, we note a strong JT distortion. The latter caused by Cu:$d^9$ orbital filling, which is also present in the high-symmetry $Cmcm$ structure and induces a large octahedra elongation along the $x$-axis. Therefore, we found that the Cu–F bonding distances in the $Cmc2_1$ phase are 2.25, 1.88, and 1.91 for the bonds along the \[1,0,0\], \[0,1,1\], and \[0,-1,-1\] directions respectively. The relaxed $a$, $b$, and $c$ lattice parameters are 4.453, 13.892, and 5.502 respectively which are close to the experimental values of 4.476, 13.972, and 5.551 [@DANCE1981599; @BABEL198577]. Thus, when comparing with other members of the family, the bonding Cu–F bonding length, along the x-axis, is by far the largest with an elongation close to 0.2 . The later, induced by the strong JT-effect and having strong effects in the magnetic structure as will be discussed later.
[c c c c]{} Compound & $M^{2+}$ radii \[pm\] & P$_s$ \[$\mu$C$\cdot$cm$^{-2}$\] & $\Delta$E \[meV/f.u.\]
------------------------------------------------------------------------
\
BaNiF$_4$ & 83 & 6.8 [@PhysRevB.74.024102] & 28 [@PhysRevB.74.024102]\
BaMgF$_4$ & 86 & 9.9 & 133 [@PhysRevB.93.064112]\
BaCuF$_4$ & 87 & 10.9 & 27\
BaZnF$_4$ & 88 & 12.2 & 218 [@PhysRevB.93.064112]\
BaCoF$_4$ & 88.5 & 9.0 [@PhysRevB.74.024102] & 58 [@PhysRevB.74.024102]\
BaFeF$_4$ & 92 & 10.9 [@PhysRevB.74.024102] & 122 [@PhysRevB.74.024102]\
BaMnF$_4$ & 97 & 13.6 [@PhysRevB.74.024102] & 191 [@PhysRevB.74.024102]\
\[tab:1\]
The computed polarization in the ground state is $P_s = 10.9$ $\mu$C$\cdot$cm$^{-2}$, being in the range of amplitudes of Ba$M$F$_4$ compounds (see Table \[tab:1\]). The computed energy difference between the $Cmcm$ to the $Cmc2_1$, is $\Delta$E = 27 meV/f.u., which is lower than the one reported for other family members such as BaMgF$_4$ and BaZnF$_4$ with barriers of 133 and 218 meV/f.u. but similar to BaNiF$_4$ and BaCoF$_4$ where the ferroelectric switching has been experimentally demonstrated [@Eibschutz1969]. This low barrier value suggests that the ferroelectric switching can be easier. In order to estimate how the polarization of BaCuF$_4$ fits with respect to the members of the Ba$M$F$_4$ family, in Table \[tab:1\] we compare the trend of the polarization’s amplitude as a function of the $M^{2+}$ ionic radii [@Shannon1976]. We observe that the polarization follows the trend of the ionic radii size, which is expected from geometrically-driven polar displacements [@PhysRevB.89.104107], also concluded from their close-to-nominal Born effective charges (see Supplementary Material [@sup-mat]) similarly as theoretically predicted [@PhysRevLett.116.117202] and later experimentally demonstrated in the multiferroic NaMnF$_3$ perovskite fluoride [@Yang2017]. Interestingly, it can be also noted that BaMnF$_4$ and BaFeF$_4$, which lack of experimentally proved polarization reversal [@Eibschutz1969], are those with the largest $M^{2+}$ ionic radii. The latter suggests a delicate balance between geometric effects and the switching process that needs to be further investigated.
![(Color online) $a$) $Cmc2_1$ structure where the magnetic exchange constants of the intralayer $J_a$ and $J_c$ and the interlayer $J_b$ constant are depicted. $b$ Spin-polarized charge-density where a clear orbital ordering, induced by the strong JT distortion, is observed. Here the magnetic moment up and down are depicted in red and blue colors respectively.[]{data-label="F2"}](F2){width="7.0cm"}
*Magnetic ordering:* Our analysis of the possible main collinear magnetic orderings of BaCuF$_4$ reveals the existence of an 1D-AFM thanks to its strong JT-distortion. This ordering is confirmed by the exchange constants (computed following the procedure of Ref. ) where values of 0.04, 0.03, and -15.91 meV were obtained for $J_a$, $J_b$, and $J_c$ respectively (see notation in Fig. \[F2\]$a$). We find that $J_a$ and $J_b$ are very small which explains the 1D-AFM character at high temperatures. Moreover, the spin-polarized charge density (see Fig. \[F2\]$b$), clearly shows the ferrodistortive character of the $d_{x^2-y^2}$ orbital ordering thanks to the strong JT-distortion present in this Cu: $d^9$ compound and then leading as a result to the 1D-AFM character. More details about the magnetic orderings can be found in the Supplementary Material [@sup-mat] and we would like to focus on the non-collinear magnetism analysis instead as next.
Starting from the 1D-AFM (also known as $A$-AFM) in the non-collinear magnetic ordering regime, we observe the appearance of spin canting giving a weak-ferromagnetic (w-FM) canting along the $z$-axis, with $m_z$ = 0.059 $\mu_B$/atom. The system can be described by the modified Bertaud’s notation [@bertaut; @bousquet2016] as $A_yF_z$ where the $A$-AFM is the main ordering along the $y$-axis ($m_y$ = 0.829 $\mu_B$/atom) and canted ferromagnetism $F_z$ along the $z$-direction. We thus obtain a magnetic canting angle of about 4.12$^\circ$ with respect to the $y$-axis. Although, a non collinear ordering has been observed for $M$ = Mn, Fe, Co, and Ni too, the canted structure give rise to a weak-AFM ordering instead [@PhysRevB.74.024102; @PhysRevB.74.020401] Then, our findings in the Cu case confirm that both, the spontaneous polarization and the ferromagnetic moment, are conveniently aligned along the $c$-axis. Interestingly, this canting angle is larger than the those reported for the weak-AFM $M$ = Ni [@PhysRevB.74.020401] and Ca$_3$Mn$_3$O$_7$ [@PhysRevLett.106.107204]. Besides, even when the magnetic measurements [@DANCE1981599] show that above 275 K the material exhibit a paramagnetic behavior, the JT-distortion is expected to remain in the structure due to a survival of the orbital ordering beyond T~N~ as we observed in the relaxed $Cmcm$. This has a direct effect on the octahedral structure as observed for KCuF$_3$ [@PhysRevLett.101.266405].
*Intertwined magnetization and polarization:* Through the presence of both magnetization and polarization one can see that BaCuF$_4$ can hold up to 4 multiferroic states (*i.e.* $M^-P^+$, $M^-P^-$, $M^+P^+$, and $M^+P^-$) as shown in Fig. \[F3\]$a$). In the following, we are going to show that the the magnetization can be switched by means of an applied electric field, thanks to the spontaneous polarization reversal, and by an applied magnetic field thanks to its canted structure.
![(Color online) $a$) Schematic representation of the four multiferroic states where the ferroelectric polarization and magnetization are shown in BaCuF$_4$. It can be observed that the magnetic moments, depicted as red and blue arrows for up- and down-orientation respectively, can be reversed by reversing the magnetic ordering and/or the polarization reversal. $b$) Double-well energy profile obtained thought the ferroelectric switching between the up and down spontaneous polarization orientations. $c$) Full polarization reversal going from $-$10.9 to $+$10.9 $\mu$C$\cdot$cm$^{-2}$. $d$) Magnetic ($m_z$) and orbital ($l_z$) moment, per u.c. reversal by means of the ferroelectric switching showing the correlation between the switching, rotations, and non collinear magnetic ordering.[]{data-label="F3"}](F3){width="8.0cm"}
We performed the computed experiment where the full ground state distortion is gradually frozen into the $Cmcm$ reference phase. At each point the electronic structure is relaxed (and thus the non-collinear magnetization) with fixed structural parameters. In Fig. \[F3\]$b$ we show the energy well and the associated polarization at different amplitude of distortions in Fig. \[F3\]$c$. The ferroelectric polarization shows a full reversal from $-$10.9 to $+$10.9 $\mu$C$\cdot$cm$^{-2}$. The spin ($m_z$) and orbital ($l_z$) moments, plotted in Fig. \[F3\]$d$, shows that $m_z\gg l_z$, and most importantly that the magnetization, changes its sign when the polarization is reversed. Therefore as expected, the four multiferroic states are possible, and more importantly, their magnetization and polarization directions could be reversed. The later showing that a proof of concept of a 4-states memory is possible in this type of materials.
This link between the polarization and the magnetization can be explained in terms of the Dzyaloshinskii-Moriya (DM) interaction [@Dzyaloshinsky; @Moriya] that is related to the rotation of the octahedra. The DM interaction energy is defined by the relationship E = **D$_{ij}$**$\cdot$($s_i$ $\times$ $s_j$), where **D$_{ij}$** is the DM tensor and the $s_i$ and $s_j$ are the spins related to the ions $i$ and $j$ respectively. As demonstrated in perovskites [@Hyun2011], the DM tensor can be related to the inter ionic vectors as **D$_{ij}$** $\propto$ ($\hat{x}_i$ $\times$ $\hat{x}_j$) [@Hyun2011], in which $\hat{x}_i$ and $\hat{x}_j$ are unitary vectors along the Cu–F–Cu bonds. In BaCuF$_4$ the absence of octahedral rotation in the high-symmetry $Cmcm$ structure gives a 180$^\circ$ Cu–F–Cu bonding angle and thus forbids a weak-FM ordering. A key feature in this type of systems is that the polarization and octahedra rotations are embedded into the same unstable mode of the $Cmcm$ structure such that, reversing the polarization will systematically reverse the octahedral rotation and thus the weak-FM. It is important to mention that the non collinear ordering is also observed in all of the other magnetic phases ($G$-AFM and $C$-AFM) but never with a weak-FM moment.
BaCuF$_4$ ideally combines the desired magnetic ordering, thanks to the JT-distortion, with the appropriated structural ground state over the $AM$F$_4$ family to exhibit a perfect electric-field magnetization reversal. It has thus a large magnetoelectric effect at rather large temperature and can be used to build a 4-states memory device as discussed before.
Although at first glance the magnetic moment could be seen to be weak, it could be amplified by layer-engineering and growing a ferromagnetic layer on the top of it as demonstrated through the exchange-bias effect [@Matsukura2015] in \[Co/Pd(Pt)\]/Cr$_2$O$_3$ [@PhysRevLett.94.117203] and NiFe/h-YMnO$_3$ (LuMnO$_3$) [@PhysRevLett.97.227201] but also in the Barium-based family of $AM$F$_4$ crystals with $M$ = Ni and Mn [@Zhou2015; @Zhou2017]. Therefore, bilayered EB-effect could be combined to bring about a novel electrically-controlled magnetic system based-on BaCuF$_4$ [@Matsukura2015]. In conclusion, we believe the BaCuF$_4$ compound is an ideal candidate to show a strong multiferroic/magnetoelectric coupling close to room-temperature, being to our knowledge, the only fluoride material that exhibits such behavior close to room-temperature [@Scott2011a].
*Acknowledgements:* This work used the XSEDE which is supported by National Science Foundation grant number ACI-1053575. ACGC and EB acknowledge the ARC project AIMED and the F.R.S-FNRS PDR project MaRePeThe. The authors also acknowledge the support from the Texas Advances Computer Center (with the Stampede2 and Bridges supercomputers), the PRACE project TheDeNoMo and on the CECI facilities funded by F.R.S-FNRS (Grant No. 2.5020.1) and Tier-1 supercomputer of the Fédération Wallonie-Bruxelles funded by the Walloon Region (Grant No. 1117545). This work was supported by the DMREF-NSF 1434897, NSF OAC-1740111 and DOE DE-SC0016176 projects.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Classical real-time lattice simulations play an important role in understanding non-equilibrium phenomena in gauge theories and are used in particular to model the prethermal evolution of heavy-ion collisions. Due to instabilities, small quantum fluctuations on top of the classical background may significantly affect the dynamics of the system. In this paper we argue for the need for a numerical calculation of a system of classical gauge fields and small linearized fluctuations in a way that keeps the separation between the two manifest. We derive and test an explicit algorithm to solve these equations on the lattice, maintaining gauge invariance and Gauss’s law.'
author:
- 'A. Kurkela'
- 'T. Lappi'
- 'J. Peuron'
bibliography:
- 'spires.bib'
title: ' Time evolution of linearized gauge field fluctuations on a real-time lattice '
---
Introduction
============
Particle production at central rapidities in collisions of high energy hadrons or nuclei is dominated by the clouds of small-$x$ gluons surrounding the projectiles. The high density of these gluons has been argued to lead to “gluon saturation”, i.e., the emergence of a dominant semihard transverse momentum scale ${Q_{\mathrm{s}}}\gg {\Lambda_{\mathrm{QCD}}}$ where the physics become nonperturbative due to the nonlinear interactions of the gluons even at weak coupling [@Gelis:2010nm]. The saturation picture of a weak coupling and a nonperturbatively large phase space density of gluons $f\sim 1/g^2$ leads to a description of the initial stages of a heavy ion collision in terms of “glasma” fields [@Lappi:2006fp], strong boost invariant color fields with transverse coherence length $\sim {Q_{\mathrm{s}}}^{-1}$ . How these maximally anisotropic far-from-equilibrium gauge fields hydrodynamize, isotropize, and reach local thermal equilibrium to form quark-gluon plasma has been a central open question in understanding the spacetime evolution of the matter produced in a heavy-ion collisions.
The large phase space occupancy, or equivalently the strength of the gauge fields, at the early stages of the collision admits a classical description of the glasma fields accurate to leading order in $g$. The classical description, however, poses a problem phenomenologically as the boost invariance of the fields is not broken and the system remains anisotropic at all times, never thermalizing or reaching hydrodynamical flow.
For the process of isotropization to proceed, it is necessary (but not sufficient) that the boost invariance is broken by small rapidity-dependent fluctuations. The origin of the fluctuations may be quantum [@Fukushima:2006ax; @Dusling:2010rm; @Epelbaum:2011pc; @Dusling:2012ig; @Epelbaum:2013waa] or arise from the longitudinal structure of the colliding nuclei [@Gelfand:2016yho; @Schenke:2016ksl]. It is then expected that in the presence of the anisotropic background, some of these fluctuations are unstable and experience a period of exponential growth, playing an important role in the isotropization process [@Mrowczynski:1994xv; @Mrowczynski:1996vh; @Mrowczynski:2004kv; @Kurkela:2011ub].
Assuming a parametric scale separation between the dominant scale ${Q_{\mathrm{s}}}$ and the inverse wavelength of the unstable modes $g f^{1/2} {Q_{\mathrm{s}}}$, the growth and saturation of the plasma instabilities can be studied in a “hard loop” (HL) framework in which the modes at the scale ${Q_{\mathrm{s}}}$ are treated as quasiparticles and the unstable modes as classical fields. Many calculations have been performed in this framework both analytically [@Arnold:2003rq; @Romatschke:2003ms; @Romatschke:2004jh; @Arnold:2004ti; @Arnold:2004ih; @Kurkela:2011ub; @Rebhan:2005re; @Rebhan:2004ur; @Kurkela:2011ti] and numerically [@Nara:2005fr; @Dumitru:2005gp; @Bodeker:2007fw; @Rebhan:2008uj; @Attems:2012js]. This is indeed a valid approach when the isotropization process is already under way and the system is only moderately anisotropic and the occupation numbers $f$ of gluonic states with ${{p_T}}\sim {Q_{\mathrm{s}}}$ have decreased from their initial value $\sim g^{-2}$. The method however fails at the earliest time scale after the collision, $\tau \sim 1/{Q_{\mathrm{s}}}$, when the role of the instabilities are expected to be the most important.
The contribution of plasma instabilities to isotropization has also been studied using purely classical field simulations [@Romatschke:2005pm; @Romatschke:2006nk; @Berges:2012cj; @Berges:2013eia] without performing the Hard Loop approximation. These calculations typically proceed using the so called “classical statistical approximation” (CSA). This consists of identifying the initial field fluctuations of the fields, adding these to the classical background field, and then solving the time evolution of the system using the full classical equations of motion on a discrete lattice. Some of these calculations have pointed towards the possibility of a very rapid isotropization caused by the plasma instabilities seeded by the quantum fluctuations of the gauge fields [@Gelis:2013rba].
The treatment of quantum fluctuations in CSA however is problematic due the backreaction of the fluctuations on the background field. Including the quantum fluctuations in the equations of motion of the background is justified only for the modes that grow large and become effectively classical [@Khlebnikov:1996wr; @Khlebnikov:1996mc]; for the other modes, the time evolution of the fluctuations is mistreated. The problem is severe in the case of quantum fluctuations, which have a highly UV-divergent spectrum and occupy modes with $f\sim 1/2$ at all scales supported by the lattice. In the CSA these fluctuations are superimposed on top of the background with $f\sim 1/g^2$ [@Gelis:2013rba]. Even though the occupancy of the mistreated fluctuations is parametrically smaller than that of the background field, the phase space opens up like $\int^{1/a} {\mathrm{d}}^3 p$, with lattice spacing $a$. Therefore, on a fine enough lattice the UV tail of the fluctuation spectrum dominates the energy density, particle number, and eventually the dynamics of the system[^1] and the time evolution of the combined system cannot be reliably followed in a classical simulation [@Moore:2001zf]. No continuum limit may be taken (see also [@Epelbaum:2014yja]).
To avoid this problem, we propose to study the evolution of the fluctuations on a mode-by-mode basis in a setup where the evolution of the fluctuation is explicitly linearized. In this case one can treat the fluctuations to one-loop order, explicitly excluding interactions between the fluctuations and any backreaction to the classical field. One loses the ability to resum late-time “secular divergences” that was one of the motivations for adopting the CSA [@Dusling:2010rm; @Dusling:2012ig]. However, the later-time behavior and eventual hydrodynamization in the context of a heavy ion collision is in any case better described in terms of kinetic theory [@Epelbaum:2015vxa; @Kurkela:2015qoa; @Keegan:2016cpi]. Instead, one keeps the analytical control given by a well defined weak coupling expansion, where different orders in $g$ remain separate. The growth and evolution of the unstable modes can be followed in a clean numerical setup, and one may choose to include only the unstable modes in the simulation. One can also formulate the calculation of gluon production in a dense-dense collision system to NLO accuracy [@Gelis:2008rw; @Gelis:2008ad] analogously to the way quark pair production from the classical field is calculated by solving the Dirac equation in the classical background [@Gelis:2003vh; @Gelis:2004jp; @Gelis:2005pb; @Gelfand:2016prm; @Mueller:2016ven].
We will write down the equations of motion for the system of a classical gauge field and linearized fluctuations in Sec. \[sec:eoms\], noting in particular that maintaining Gauss’s law in a calculation with discretized time requires some care. In Sec. \[sec:num\] we will present results from simple numerical tests of our algorithm, before pointing in Sec. \[sec:conc\] towards some of its potential future applications.
Equations of motion for fluctuations {#sec:eoms}
====================================
In this Section we construct the equations of motion for the linearized fluctuations of the gauge and the chromoelectric field $\{a_i, e^i\}$ on top of the background field. On the lattice we will use the Kogut-Susskind Hamiltonian [@Kogut:1974ag] for the background field and in discretizing the equations of motion for the background field we will take special care to make sure that the discretized and linenarized equations of motion exactly conserve the Gauss’s law constraint.
In this paper, for simplicity, we will constrain the discussion to a system not undergoing longitudinal expansion (fixed box), however, the extension to a expanding coordinate system is trivial.
Small fluctuations in the continuum
-----------------------------------
In the continuum the Hamiltonian of a pure gauge theory can be written, in temporal gauge $A_0=0$ as $$H =\int {\mathrm{d}}^3{{\mathbf{x}}}\left[ {\, \mathrm{Tr} \, }E^i E^i + {\frac{1}{2}}{\, \mathrm{Tr} \, }F_{ij} F_{ij}\right],$$ with field strength tensor $F_{ij} = (ig)^{-1}[D_i,D_j] = \partial_i A_j - \partial_j A_i + i g [A_i,A_j]$, where the covariant derivative is $D_i = \partial_i +igA_i$. Here we write the gauge and chromoelectric fields in matrix form $ A_i = A_i^a t^a,$ with the fundamental representation generators $t^a$ normalized as ${\, \mathrm{Tr} \, }t^at^b = {\frac{1}{2}}\delta^{ab}$. From this Hamiltonian one derives the equations of motion $$\begin{aligned}
\dot{A}_i &= &E^i
\\
\dot{E}^i &= & \left[ D_j,F_{ji} \right].\end{aligned}$$ In order to project to the physical charge sector, also the Gauss’s law constraint must be fulfilled $$C({{\mathbf{x}}},t) \equiv \left[ D_i , E^i \right] =0,$$ which is conserved exactly by the equations of motion $\partial_t C({{\mathbf{x}}},t) = 0$.
Dividing the field into a background field and linearized fluctuations $$(E^i,A_i) \to (E^i + e^i, A_i + a_i),$$ the equations of motion and Gauss’s law for the fluctuations become $$\begin{aligned}
\label{eq:contaeom}
\dot{a}_i &= &e^i
\\
\dot{e}^i &=& \left[D_j,\left[D_j,a_i\right]\right]
- \left[D_j,\left[D_i,a_j\right]\right] + i g \left[a_j,F_{ji}\right]
\\ &=&
\label{eq:conteeom}
\left[D_j,\left[D_j,a_i\right]\right]
- \left[D_i,\left[D_j,a_j\right]\right] + 2 i g \left[a_j,F_{ji}\right]\end{aligned}$$ where the second form of the equation for $\dot{e}^i$ allows for an interpretation in the background field gauge $[D_i,a^i]=0$ in terms of an adjoint representation scalar field equation for $a_i$ supplemented with a gluon chromomagnetic moment term (see, e.g., [@Greiner:1985ce]) . Similarly, the Gauss’s law constraint for the fluctuation reads $$\label{eq:contgauss}
c({{\mathbf{x}}},t) =\left[ D_i,e^i \right] + i g \left[a_i,E^i\right] = 0.$$
Discretized equations for background
------------------------------------
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In order to conserve the gauge symmetry exactly, it is convenient to trade the gauge fields $A_i$ belonging to the Lie algebra of the group to link matrices $U_i$ which are members of the group. The Kogut-Susskind Hamiltonian [@Kogut:1974ag] in terms of the link matrices reads $$\begin{gathered}
\label{eq:ksh}
H =\frac{a^3}{g^2} \sum_{{\mathbf{x}}}\Bigg\{ {\, \mathrm{Tr} \, }\big[ a^{-2} E^i({{\mathbf{x}}}) E^i({{\mathbf{x}}}) \big]
\\
+ \frac{2}{ a^4} \sum_{i<j} {\mathrm{Re}}{\, \mathrm{Tr} \, }\big[ \mathbb{1} - \Box_{i,j}({{\mathbf{x}}}) \big]
\Bigg\},\end{gathered}$$ where the spatial coordinate ${\bf x}$ takes discrete values on a cartesian lattice ${\bf x}=a (n_i,n_j,n_k)$, with integers $n_i,n_j,n_k$ and lattice spacing $a$. Here the plaquette $\Box_{i,j}({{\mathbf{x}}})$ is written in terms of the link matrices $U_i({{\mathbf{x}}})$ $$\Box_{i,j}({{\mathbf{x}}}) = U_i({{\mathbf{x}}}) U_j({{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}) U^\dag_i({{\mathbf{x}}}+{{\boldsymbol{\hat{\j}}}})U^\dag_j({{\mathbf{x}}}),$$ where ${{\boldsymbol{\hat{\i}}}},{{\boldsymbol{\hat{\j}}}}$ are unit vectors in the $i,j$ directions; see [Fig. ]{}\[fig:clestep\] for an illustration.[^2] The lattice fields are related to continuum quantities by $U_i({{\mathbf{x}}})\approx e^{i a g A_i({{\mathbf{x}}})}$ and $E^i_\textup{lat} \approx a g E^i_\textup{cont}$.
The Kogut-Susskind Hamiltonian gives us equations of motion that are discrete in space but continuous in time: $$\begin{aligned}
\dot{U}_i({{\mathbf{x}}}) &=& i E^i({{\mathbf{x}}}) U_i({{\mathbf{x}}})
\\
a^2 \dot{E}^i({{\mathbf{x}}}) &=& - \sum_{j\neq i } \left[\Box_{i,j}({{\mathbf{x}}}) + \Box_{i,-j}({{\mathbf{x}}}) \right]_{\mathrm{ah}}, \label{eq:Edot}\end{aligned}$$ where the plaquette in the negative $j$ direction is $\Box_{i,-j}({{\mathbf{x}}}) = U_i({{\mathbf{x}}}) U^\dag_j({{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}-{{\boldsymbol{\hat{\j}}}}) U^\dag_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\j}}}})U_j({{\mathbf{x}}}-{{\boldsymbol{\hat{\j}}}})$. Here the notation $[]_{\mathrm{ah}}$ denotes the antihermitian traceless part of a matrix: $$[V]_{\mathrm{ah}}\equiv \frac{-i}{2} \left[ V - V^\dag - \frac{\mathbb{1}}{{{N_\mathrm{c}}}}{\, \mathrm{Tr} \, }(V-V^\dag)\right],$$ where ${{N_\mathrm{c}}}$ is the number of colors.
In order to perform a practical simulations, also the time direction must be discretized. To guarantee time reversal invariance and second order accuracy in the time step ${\mathrm{d}}t$, the time direction is commonly discretized with the leaprog algorithm, where the electric fields and the links live on alternate timesteps $$\begin{aligned}
U(t+{\mathrm{d}}t) &= e^{i E^i(t+{\mathrm{d}}t/2) {\mathrm{d}}t} U_i(t)
\label{eq:cllinkstep}
\\
\label{eq:clestep}
a^2 E^i(t+{\mathrm{d}}t) &=a^2 E^i(t) \\ -{\mathrm{d}}t \sum_{j\neq i } & \left[\Box_{i,j}\left(t+\frac{{\mathrm{d}}t}{2}\right) + \Box_{i,-j}\left(t+\frac{{\mathrm{d}}t}{2} \right) \right]_{\mathrm{ah}}, \nonumber\end{aligned}$$ where we have dropped the explicit position arguments for brevity. It is a straightforward exercise to show that both the link and electric field timesteps [(\[eq:cllinkstep\])]{} and [(\[eq:clestep\])]{} separately conserve the discretized version of Gauss’s law constraint $$\begin{aligned}
\label{eq:clgauss}
&C({{\mathbf{x}}},t) = \sum_i \frac{1}{a^2}\left\{E^i({{\mathbf{x}}}) - U^\dag_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})E^i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})U_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})\right\}, \nonumber\\
&C({{\mathbf{x}}},t+{\mathrm{d}}t)= C(t).\end{aligned}$$ Finally, let us recall that under a lattice gauge transformation $V({{\mathbf{x}}})$ (which must be time-independent in order to conserve the temporal gauge condition) the links and electric fields transform as $$\begin{aligned}
\label{eq:gaugetrlink}
U_i({{\mathbf{x}}}) &\to& V({{\mathbf{x}}}) U_i({{\mathbf{x}}})V^\dag({{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}})
\\
\label{eq:gaugetre}
E^i({{\mathbf{x}}}) &\to& V({{\mathbf{x}}}) E^i({{\mathbf{x}}})V^\dag({{\mathbf{x}}}).\end{aligned}$$ It is easy to see that the Hamiltonian [(\[eq:ksh\])]{} is gauge invariant and the equations of motion [(\[eq:cllinkstep\])]{}, [(\[eq:clestep\])]{} and Gauss’s law [(\[eq:clgauss\])]{} gauge covariant under these transformations.
(-1.4,0) – node\[left\] [$j$]{}(-1.4,0.5); (-1.2,-0.2) – node\[below\] [$i$]{} (-0.7,-0.2); (b1) \[line width=1pt,circle,draw,inner sep=0.2pt\] at (0,0)[$\boldsymbol{\rightarrow}$]{}; (b1) – ++(1,0) – ++(0,1) – ++ (-1,0) – ++(0,-0.6); at (1.5,0.5) [$+$]{}; (b2) at (2,0); (b2b) \[line width=1pt,circle,draw,inner sep=1pt\] at (3,0)[$\boldsymbol{\uparrow}$]{}; (b2) – (b2b) – ++(0,1) – ++ (-1,0) – ++(0,-0.8); at (3.5,0.5) [$-$]{}; (b3) at (4,0); (b3b) \[line width=1pt,circle,draw,inner sep=0.2pt\] at (4,1)[$\boldsymbol{\rightarrow}$]{}; (b3) – ++(1,0) – ++(0,1) – (b3b) – ++(0,-0.8); at (5.5,0.5) [$-$]{}; (b4) \[line width=1pt,circle,draw,inner sep=1pt\] at (6,0)[$\boldsymbol{\uparrow}$]{}; (6.4,0) – ++(0.6,0) – ++(0,1) – ++ (-1,0) – ++(b4); at (-0.5,-1.5) [$+$]{}; (b5) \[line width=1pt,circle,draw,inner sep=0.2pt\] at (0,-1)[$\boldsymbol{\rightarrow}$]{}; (b5) – ++(1,0) – ++(0,-1) – ++ (-1,0) – ++(0,0.6); at (1.5,-1.5) [$-$]{}; (b6) at (2,-1); (b6b) \[line width=1pt,circle,draw,inner sep=1pt\] at (3,-2)[$\boldsymbol{\uparrow}$]{}; (b6) – ++(1,0) – (b6b) – ++ (-1,0) – ++(0,0.8); at (3.5,-1.5) [$-$]{}; (b7) at (4,-1); (b7b) \[line width=1pt,circle,draw,inner sep=0.2pt\] at (4,-2)[$\boldsymbol{\rightarrow}$]{}; (b7) – ++(1,0) – ++(0,-1) – (b7b) – ++(0,0.8); at (5.5,-1.5) [$+$]{}; (b8) at (6,-1); (b8b) \[line width=1pt,circle,draw,inner sep=1pt\] at (6,-2)[$\boldsymbol{\uparrow}$]{}; (b8) – ++(1,0) – ++(0,-1) – (b8b) – ++(0,0.8);
Discretized equations for fluctuations
--------------------------------------
After these preliminaries, let us move to the lattice equations of motion for the small fluctuations. Naturally, there is a certain freedom in writing down the discretized equations; here, we choose to construct the discretized equations so that they satisfy the following requirements:
1. Reduction to the continuum equations of motion [(\[eq:contaeom\])]{}, [(\[eq:conteeom\])]{} in the limit $a \to 0, \ {\mathrm{d}}t \to 0$.
2. \[it:gt\] Gauge covariance under the transformations [(\[eq:gaugetrlink\])]{}, [(\[eq:gaugetre\])]{}.
3. Linearity in $a_i$ and $e^i$.
4. An exact conservation of a lattice version of a Gauss’s law that reduces to [(\[eq:contgauss\])]{} in the limit $a \to 0, \ {\mathrm{d}}t\to 0$ at every time step.
5. Time reversal invariance (under ${\mathrm{d}}t \to -{\mathrm{d}}t$).
We choose here to start from condition \[it:gt\] by defining the required gauge transformation properties as those of an adjoint representation scalar field: $$\begin{aligned}
a_i({{\mathbf{x}}}) &\to& V({{\mathbf{x}}}) a_i({{\mathbf{x}}}) V^\dag({{\mathbf{x}}})
\\
e^i({{\mathbf{x}}}) &\to& V({{\mathbf{x}}}) e^i({{\mathbf{x}}}) V^\dag({{\mathbf{x}}}).\end{aligned}$$ From these it follows that $a_i$ must correspond to a variation of the link matrix $U_i({{\mathbf{x}}})$ on the left: $$U_i({{\mathbf{x}}})_\textup{bkg + fluct} = e^{i a_i({{\mathbf{x}}})}U_i({{\mathbf{x}}}) \approx
U_i({{\mathbf{x}}}) + i a_i({{\mathbf{x}}})U_i({{\mathbf{x}}}).$$ In the continuum limit, the fluctuation field on the lattice is related to the continuum equivalent though $a_i^\textnormal{lat} = a g a_i^\textnormal{cont}$.
We then *choose* to discretize the perturbation of the electric field by linearizing the r.h.s. of [(\[eq:clestep\])]{}, so that
$$\begin{aligned}
\label{eq:estep}
a^2 e^i(t+{\mathrm{d}}t) = &a^2 e^i(t) - {\mathrm{d}}t \sum_{j\neq i}\Bigg[ i \Big(
a_i({{\mathbf{x}}})\Box_{i,j}({{\mathbf{x}}})
+ a_j({{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}\to {{\mathbf{x}}})\Box_{i,j}({{\mathbf{x}}})
- \Box_{i,j}({{\mathbf{x}}})a_i({{\mathbf{x}}}+{{\boldsymbol{\hat{\j}}}}\to {{\mathbf{x}}})
- \Box_{i,j}({{\mathbf{x}}})a_j({{\mathbf{x}}})
\\
& + a_i({{\mathbf{x}}})\Box_{i,-j}({{\mathbf{x}}})
- a_j({{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}-{{\boldsymbol{\hat{\j}}}}\to {{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}\to {{\mathbf{x}}})\Box_{i,-j}({{\mathbf{x}}})
- \Box_{i,-j}({{\mathbf{x}}}) a_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\j}}}}\to {{\mathbf{x}}})
+\Box_{i,-j}({{\mathbf{x}}})a_j({{\mathbf{x}}}-{{\boldsymbol{\hat{\j}}}}\to {{\mathbf{x}}})
\Big) \Bigg]_{\mathrm{ah}}, \nonumber\end{aligned}$$
which is easily seen to be gauge covariant. Figure \[fig:estep\] illustrates the ordering of the plaquettes and the field fluctuations in [Eq. ]{}[(\[eq:estep\])]{}. Here we denote by the fluctuation parallel transported from site ${{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}$ to site ${{\mathbf{x}}}$ by $$a_j({{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}\to {{\mathbf{x}}})
\equiv
U_i({{\mathbf{x}}})a_j({{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}})U^\dag_i({{\mathbf{x}}}),$$ and similarly for the fields parallel transported over two links[^3] $$\begin{aligned}
a_j({{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}-{{\boldsymbol{\hat{\j}}}}\to {{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}& \to {{\mathbf{x}}}) \\
\equiv U_i({{\mathbf{x}}})&a_j({{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}-{{\boldsymbol{\hat{\j}}}}\to {{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}})U^\dag_i({{\mathbf{x}}})\nonumber,\end{aligned}$$ and so on. The links and gauge field fluctuations $a_i$ in [(\[eq:estep\])]{} are evaluated according to the leapfrog scheme at time $t+{\mathrm{d}}t/2$. We emphasize that the choice [(\[eq:estep\])]{} is not unique, but one could add terms proportional to $({\mathrm{d}}t)^2$ or higher powers.
Similarly to the timestep of $E^i$, Gauss’s law [(\[eq:clgauss\])]{} is linear in the chromoelectric field. The natural choice is then to derive Gauss’s law for the fluctuations by replacing $E^i$ with $E^i + e^i$, $U_i({{\mathbf{x}}})$ with $U_i({{\mathbf{x}}}) + i a_i({{\mathbf{x}}})U_i({{\mathbf{x}}})$, and taking the linear terms in the fluctuation fields. This yields $$\begin{aligned}
\label{eq:gauss}
c({{\mathbf{x}}}&,t)
= \sum_i \frac{1}{a^2} \Big\{ e^i({{\mathbf{x}}})- U_i^\dagger({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}}) e^i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}}) U_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})\nonumber
\\
&+ i U_i^\dagger({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})[a_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}}),E^i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})]U_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}}) \Big\} .\end{aligned}$$ We now have equations for the timestep of the electric field fluctuation $e^i$ and Gauss’s law for the fluctuations. To complete the set of equations, we need also to specify the timestep for $a_i({{\mathbf{x}}})$. The first guess would be a straightforward discretization of the continuum $\dot{a}_i = e^i$. However, this naive discretization is inadmissible, since it does not conserve the linearized Gauss’s law [(\[eq:gauss\])]{}. Physically this would mean an unphysical creation of “charges” in the lattice. This can be traced to the fact that Gauss’s law involves a covariant derivative using links $U_i$ that advance in time simultaneously as their fluctuations $a_i$, and the timestep must reflect this change. Another hint of the subtlety of the step for $a_i$ is to see that a linearization of the timestep for the link $U_i({{\mathbf{x}}})$ in [Eq. ]{}[(\[eq:cllinkstep\])]{} would involve developing the exponential $e^{i(E^i+e^i){\mathrm{d}}t}$ to linear order in $e^i$, which is a rather complicated expression when $E^i$ and $e^i$ do not commute.
We may, however, construct a valid update for the gauge fields by demanding the Gauss’s law constraint to be conserved, $$\begin{aligned}
c({{\mathbf{x}}},t) = c({{\mathbf{x}}},t+{\mathrm{d}}t).\end{aligned}$$ It is straightforward to see that this condition holds if the update satisfies[^4] $$\label{eq:genstep}
\left[ E^i,a_i(t+{\mathrm{d}}t) \right] = -i \left(\Box_{0i}e^i \Box^\dagger_{0i} - e^i\right)
+
\left[E^i,\Box_{0i} a_i(t)\Box^\dagger_{0i}\right],$$where we use a shorthand for the “timelike plaquette” $\Box_{0i}=e^{i E^i{\mathrm{d}}t}$. Imposing this condition on the gauge field update leads by construction to a time step that conserves Gauss’s law.
It is convenient here to separate the parts of $a_i,e^i$ that are parallel and perpendicular to $E^i$ in color space. Denoting $$\begin{aligned}
f^\parallel &=& \frac{{\, \mathrm{Tr} \, }\left[f E^i\right]}{{\, \mathrm{Tr} \, }\left[ E^i E^i\right]}E^i,
\\
f^\perp &=& f - \frac{{\, \mathrm{Tr} \, }\left[f E^i\right]}{{\, \mathrm{Tr} \, }\left[ E^i E^i\right]}E^i,\end{aligned}$$ [Eq. ]{}[(\[eq:genstep\])]{} can be solved for $a^\perp_i(t+{\mathrm{d}}t)$ in terms of $a^\perp_i(t)$ and $e^{i \perp}$ giving the equation of motion for the perpendicular component. Because of the commutator, Eq. [(\[eq:genstep\])]{} gives no condition for the parallel component, and we may complete the equations of motion with the naive discretization $$a^\parallel_i(t+{\mathrm{d}}t)=a^\parallel_i(t) + {\mathrm{d}}t e^{i \parallel }(t+{\mathrm{d}}t/2)$$ that already satisfies [Eq. ]{}[(\[eq:genstep\])]{}.
For a practical algorithm, it still remains a technical problem to solve the perpendicular components of the gauge field fluctuations from Eq. [(\[eq:genstep\])]{}. For a general gauge group we can write the solution more compactly in the adjoint representation: in terms of the unitary matrix $\left(\widetilde{\Box}_{0i}\right)^{ab} = 2 {\, \mathrm{Tr} \, }\left[t_a \Box_{0i} t_b \Box_{0i}^\dag \right]$, the hermitian matrix $\left(\widetilde{E}^i\right)^{ab} = E^i_c \left(T^c\right)^{ab}
= -if_{cab}E^i_c$ and the $N_c^2-1$ component vectors $\underline{a}_i$ and $\underline{e}^i$ with components $\left(a_i\right)^a$ and $(e^i)^a$. In this notation [Eq. ]{}[(\[eq:genstep\])]{} becomes $$\widetilde{E}^i \underline{a}_i(t+{\mathrm{d}}t) = -
i \left( \widetilde{\Box}_{0i} - \mathbb{1} \right) \underline{e}^i
+ \widetilde{E} \widetilde{\Box}_{0i}\underline{a}_i(t)$$ The parallel components are the null space of the matrix $\widetilde{E}^i$, and in this subspace the timelike plaquette acts like the identity: $\widetilde{\Box}^\dag_{0i} \underline{f}^\parallel = \underline{f}^\parallel$. Thus parallel components of $\underline{a}^i(t)$ and $\underline{e}_i$ only generate parallel components of $\underline{a}^i(t+{\mathrm{d}}t)$. In the perpendicular color directions, on the other hand, the matrix $\widetilde{E}^i$ is invertible, and we can write the gauge field timestep as $$\begin{gathered}
\label{eq:finalastep}
\underline{a}_i(t+{\mathrm{d}}t) = \left(\widetilde{E}^i\right)^{-1}_{\perp}\left[
-i \left( \widetilde{\Box}_{0i} - \mathbb{1} \right) \underline{e}^{i \perp}
+ \widetilde{E}^i \widetilde{\Box}_{0i}\underline{a}_i^\perp(t)
\right]
\\
+
\underline{e}^{i \parallel} {\mathrm{d}}t
+
\underline{a}_i^\parallel(t),\end{gathered}$$ where the notation $()^{-1}_\perp$ means a projection to the subspace where the matrix $\widetilde{E}^i$ is invertible followed by an inversion in that subspace. This equation is our general result for the timestep of the gauge field fluctuation.
In the small ${\mathrm{d}}t$ limit $\widetilde{\Box}_{0i} \approx \mathbb{1} + i \widetilde{E}^i {\mathrm{d}}t$ and we see that [Eq. ]{}[(\[eq:finalastep\])]{} reduces to $\underline{a}_i(t+{\mathrm{d}}t) = \underline{e}^i {\mathrm{d}}t + \underline{a}_i(t)$ as desired. It may seem like a disproportionate amount of trouble to formulate the equation in this way, when the result reduces to the naive discretization in the limit ${\mathrm{d}}t\to 0$ which one wants to take in the end. However, we have found that in practical computations it is essential for a good precision to conserve Gauss’s law also in discrete time and not only in the continuous time limit. At this point it is also straightforward to check that the equation is time reversal invariant, ensuring second order accuracy in ${\mathrm{d}}t$.
Note that the form [(\[eq:finalastep\])]{} of the timestep results from a choice made in writing the timestep for $e^i$ and Gauss’s law in the form [Eqs. ]{}[(\[eq:estep\])]{}, [(\[eq:gauss\])]{}. We could have resolved the ambiguity in linearizing the fluctuations of the timelike plaquette in another way by defining a different electric field fluctuation e.g. by $$\underline{e}^{i}_{\textup{mod}} =
\left({\mathrm{d}}t \widetilde{E}^i\right)^{-1}_\perp \left[
i \left( \widetilde{\Box}^\dag_{0i} - \mathbb{1} \right) \underline{e}^{i \perp}
\right]
+
\underline{e}^{i \parallel}.$$
This would make the timestep for $a_i$ simpler, but the timestep and Gauss’s law for $e^{i}_\textup{mod}$ would have a more complicated form, with the appearence of terms proportional to $\widetilde{E}^i {\mathrm{d}}t$.
The general result [(\[eq:finalastep\])]{} requires the solution of a system of ${{N_\mathrm{c}}}^2-1$ linear equations. For the special case of SU(2) we can invert the matrix $\widetilde{E}^i$ analytically using the fact that in the absence of symmetric structure constants the Fierz identity for $f^{abc}f^{ade}$ is particularly simple $\epsilon^{ijk}\epsilon^{ilm} = \delta^{jl}\delta^{km}-
\delta^{jm}\delta^{kl}$. Thus if, for the perpendicular part, $E^i_a a^\perp_{i,a}=0$, we have $\widetilde{E}^i\widetilde{E}^i \underline{a}_i^\perp = E^i_a E^i_a \underline{a}_i^\perp$, and we can write [(\[eq:finalastep\])]{} as $$\begin{gathered}
\label{eq:su2adj}
\underline{a}_i(t+{\mathrm{d}}t) = \frac{1}{E^i_a E^i_a} \widetilde{E}^i \Bigg\{ \left[
-i \left( \widetilde{\Box}_{0i} - \mathbb{1} \right) \underline{e}^{i \perp}
+ \widetilde{E}^i \widetilde{\Box}_{0i}\underline{a}_i^\perp(t)
\right]
\\
+
\underline{e}^{i \perp} {\mathrm{d}}t
+
\underline{a}_i^\perp(t) \Bigg\}, \end{gathered}$$ or in the fundamental representation as $$\begin{gathered}
\label{eq:su2fund}
a_i(t+{\mathrm{d}}t) = \frac{i}{2{\, \mathrm{Tr} \, }\left[E^i E^i\right]}
\Bigg[E^i,
\\
-i \left(\Box_{0i}e^{i \perp} \Box^\dag_{0i} - e^{i \perp}\right)
+
\left[E^i,\Box_{0i} a^\perp_i(t)\Box^\dag_{0i}\right]
\Bigg]
\\
+ {\mathrm{d}}t e^{i \parallel} + a^\parallel_i(t)\end{gathered}$$ We stress that these final versions [(\[eq:su2adj\])]{} and [(\[eq:su2fund\])]{} are valid for SU(2) only, and e.g. for SU(3) one must use [Eq. ]{}[(\[eq:finalastep\])]{}.
Numerical tests {#sec:num}
===============
![Test of the decomposition of the field in the background field and fluctuation after some finite time. All runs have been evolved to same physical time, which is ${\mathrm{d}}t N_t = 2$ here with ${\mathrm{d}}t=0.01$ and $N_t = 200.$ The upper points correspond to $\delta_{\dot E}$ and the lower $\delta_A$ (see Eqs (\[eq:de\]) and (\[eq:da\])). The straight lines are fits of the form $a x^4,$ showing that the observables decrease with the correct power law. []{data-label="fig:a_t_comparison"}](eps){width="48.00000%"}
We now have the equations of motion for the linearized fluctuation: the timestep for $e^i$ ([Eq. ]{}[(\[eq:estep\])]{}), for $a_i$ (general equation in [Eq. ]{}[(\[eq:finalastep\])]{} and SU(2)-specific ones in [Eqs. ]{}[(\[eq:su2adj\])]{} and [(\[eq:su2fund\])]{}) and Gauss’s law [(\[eq:gauss\])]{}. We present here some simple test results from an implementaion of these equations for the SU(2) gauge group.
We construct initial conditions for a background field configuration by setting the gauge fields $A_i$ to random values uniformly distributed in the interval $[0,0.9]$. The electric fields are set to zero initially in order to satisfy the Gauss’s law $C({{\mathbf{x}}},0)=0$. We then construct the link matrices by exponentiating the gauge fields $U_i = e^{iag A_i}$. We similarly construct initial fluctuation fields. We then choose a small parameter $\varepsilon$ ranging from 0.5 to 0.0001 and multiply the fluctuations by $\varepsilon$, effectively setting $\varepsilon$ as the scale of these fluctuations, i.e. $e^i, a_i \sim \varepsilon$. We can now evolve separately in time:
1. The system of the background field and linearized fluctuations $E^i, A_i, e^i, a_i$ and
2. A different pure background field configuration initialized as $\hat E^i(t=0) = E^i(t=0) + e^i(t=0)$ and $\hat A^i(t=0) = A^i(t=0) + a^i(t=0)$.
![Violation of Gauss’s law as a function of time, in single and double precision for the background field only compared to the fluctuations. We have performed a random gauge transformation on every timestep and fixed Coulomb gauge on every tenth timestep to verify the gauge invariance also numerically. Here $\varepsilon=0.1$ and $a_t=0.01.$ The expressions used to measure the violations are given by equations (\[eq:bggaussviol\]) and (\[eq:flucgaussviol\]).[]{data-label="fig:gausslaw"}](floatvsdoubleGaussGT){width="48.00000%"}
If we have now successfully linearized the classical equations of motion, the squared differences $$\label{eq:de}
\delta_E = \sum \limits_{x,i}{\, \mathrm{Tr} \, }(\hat E^i-E^i-e^i)^2$$ and $$\begin{aligned}
\label{eq:da}
\delta_A & = \dfrac{1}{2} \sum \limits_{x,i,a}\left(2 {\rm Im}\mathrm{Tr}\left( t^a \hat U_{i}U_{i}^\dagger\right) - a_i^a\right)^2\\
&\approx\sum \limits_{x,i} {\, \mathrm{Tr} \, }(\hat A^i-A^i-a^i)^2
\end{aligned}$$ should scale as $\varepsilon^4$ with the magnitude of the fluctuation. For numerical convenience it is easier for us to plot the corresponding differences for the time derivatives $E^i(t+{\mathrm{d}}t) - E^i(t)$ etc. as the expression involving time derivatives is easily obtained during the time-evolution. In [Fig. ]{} \[fig:a\_t\_comparison\] we show that indeed these differences scale in the correct way for a large range of $\varepsilon$. We can note here that for a naive $a_i$ timestep $a_i(t+{\mathrm{d}}t)= a_i(t)+ e^i$ the correct $\varepsilon$-scaling for small fluctuations is only obtained for prohibitively expensive small values of ${\mathrm{d}}t$ because the correct scaling of $\delta_E$ and $\delta_A$ is violated by terms of the order $\varepsilon^2 {\mathrm{d}}t^4$. We have verified this numerically by observing that for larger ${\mathrm{d}}t$ $\delta_E$ and $\delta_A$ begin to scale as $\varepsilon^2$. This means that the naive timestep for $a_i$ does not correctly capture the difference $\hat A^i-A^i$ even to leading order in $\varepsilon$.
We also show, in [Fig. ]{}\[fig:gausslaw\], the violation of Gauss’s law constraint as a function of time. To quantify the violation we consider for the background field
$$\label{eq:bggaussviol}
\frac{2\sum \limits_x \mathrm{Tr}\left(\sum \limits_{i}\left[E^i({{\mathbf{x}}}) - U^\dag_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})E^i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})U_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})\right]\right)^2}{2 \sum \limits_{x,i} \mathrm{Tr}\left[E^i({{\mathbf{x}}}) - U^\dag_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})E^i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})U_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})\right]^2},$$
and similarly for the fluctuations $$\label{eq:flucgaussviol}
\frac{2 \sum \limits_x \mathrm{Tr}\left(\sum \limits_i \left[
e^i({{\mathbf{x}}})- U_i^\dagger({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}}) e^i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}}) U_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})
+ i U_i^\dagger({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})[a_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}}),E^i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})]U_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})
\right]\right)^2
}{
2 \sum \limits_{x,i} \mathrm{Tr} \left[
e^i({{\mathbf{x}}})- U_i^\dagger({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}}) e^i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}}) U_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})
+ i U_i^\dagger({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})[a_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}}),E^i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})]U_i({{\mathbf{x}}}-{{\boldsymbol{\hat{\i}}}})
\right]^2}.$$
These two quantities measure how well the components in different spatial directions cancel each other. Due to numerical roundoff error, Gauss’s law is never satisfied exactly. However, our algorithm preserves it equally well for the fluctuations as for the background field. Also, the fact that Gauss’s law remains satisfied orders of magnitude more precisely in double than single precision shows that that the remaining values are purely due to the limited machine precision.
Conclusions and outlook {#sec:conc}
=======================
Unstable fluctuations seeded by quantum effects around a boost invariant classical background field play an important part in the pre-equilibrium evolution of heavy-ion collisions. Until now, the Classical Statistical Approximation has been common tool to study these phenomena. However, the very UV dominated spectrum of vacuum fluctuations in field theory makes attaining the continuum limit in CSA calculations very difficult if not impossible.
We have argued in this paper that it would be desirable to address these issues by real time lattice calculations with an explicitly linearized fluctuation around the classical field. We have here explicitly derived and tested equations of motion for these fluctuations, showing that satisfying Gauss’s law for the fluctuations requires a careful treatment in the discretization of the timestep. By giving up the attempt to resum asymptotical long time “secular divergences,” which are not a problem with a matching to kinetic theory, one stands to gain better control of the UV dynamics in the classical gauge field calculation. We expect this formalism to have several interesting applications, which we plan to return to in future work.
We thank K. Boguslavski, S. Schlichting and Y. Zhu for discussions. T. L. is supported by the Academy of Finland, projects 267321, 273464 and 303756, and J.P. by the Jenny and Antti Wihuri Foundation.
[^1]: Note that a gauge theory (unlike scalar theory studied in the cosmological context) is particularly sensitive to UV modes as the inelastic collisions of the modes can rapidly move the energy towards the IR.
[^2]: Note that in the discrete formulation from now on we abandon the summation convention for spatial indices $i,j,\dots$ (but not for color indices).
[^3]: Note that, in our notation there are two identical ways of writing the most complicated terms involving parallel transports over two links $$\begin{aligned}
& a_j({{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}-{{\boldsymbol{\hat{\j}}}}\to {{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}\to {{\mathbf{x}}})\Box_{i,-j}({{\mathbf{x}}})
\\ \nonumber & \quad
=
\Box_{i,-j}({{\mathbf{x}}})a_j({{\mathbf{x}}}+{{\boldsymbol{\hat{\i}}}}-{{\boldsymbol{\hat{\j}}}}\to {{\mathbf{x}}}-{{\boldsymbol{\hat{\j}}}}\to {{\mathbf{x}}}).\end{aligned}$$
[^4]: We drop the explicit time argument for the electric field from now on; this will always be dictated by the leapfrog scheme.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present U, V, and I-band images of the host galaxy of Hercules A (3C 348) obtained with HST/WFC3/UVIS. We find a network of dusty filaments which are more complex and extended than seen in earlier HST observations. The filaments are associated with a faint blue continuum light (possibly from young stars) and faint H$\alpha$ emission. It seems likely that the cold gas and dust has been stripped from a companion galaxy now seen as a secondary nucleus. There are dusty filaments aligned with the base of the jets on both eastern and western sides of the galaxy. The morphology of the filaments is different on the two sides - the western filaments are fairly straight, while the eastern filaments are mainly in two loop-like structures. We suggest that despite the difference in morphologies, both sets of filaments have been entrained in a slow moving boundary layer outside the relativistic flow. As suggested by @fabian08, magnetic fields in the filaments may stabilize them against disruption. We consider a speculative scenario to explain the relation between the radio source and the shock and cavities in the hot ICM seen in the Chandra data [@nulsen05]. We suggest the radio source originally ($\sim 60$ Myr ago) propagated along a position angle of $\sim 35\deg$ where it created the shock and cavities. The radio source axis changed to its current orientation ($\sim 100\deg$) possibly due to a supermassive black hole merger and began its current epoch of activity about 20 Myr ago.'
author:
- 'Christopher P. O’Dea'
- 'Stefi A. Baum'
- 'Grant R. Tremblay'
- Preeti Kharb
- 'William D. Cotton'
- 'Rick A. Perley'
title: |
[*Hubble Space Telescope*]{} Observations of Dusty Filaments in Hercules A:\
Evidence for Entrainment
---
Introduction
============
(3C 348) has the fourth highest flux density at 178 MHz. The two radio jets, extended over $\sim 400$ kpc, have remarkably different morphology [@dreher84; @gizani03; @cotton13] with the Eastern side showing a twisting jet, and the Western side showing a series of ring-like features. There is no consensus explanation for the difference in radio morphology [@mason88; @meier91; @sadun02; @saxton02; @nakamura08].
The host galaxy is the dominant cD in a poor cluster at redshift z=0.154 [@greenstein62; @owen89; @allington93; @zirbel96] and contains a secondary nucleus [e.g. @smith89; @sadun93; @baum96; @ramos11]. Although poor optically, the cluster is luminous in X-rays and contains a cooling flow [@gizani04; @nulsen05]. There are two $\sim 40$ kpc-scale cavities in the X-ray emitting gas that are nearly perpendicular to the radio axis [@nulsen05]. There is a strong shock in the hot gas indicating the release by the radio source of $\sim 10^{61}$ ergs of mechanical energy [@nulsen05].
There are dusty filaments in the host galaxy seen in HST [@baum96; @dekoff96] and ground-based [@ramos11] images. In this paper we present deeper HST/WFC3 observations, examine the nature of the filaments in more detail and discuss their relation to the radio source. We adopt a cosmology with H$_o = 71$ km sMpc, $\Omega_M = 0.27$ and $\Omega_\Lambda = 0.73$ which gives a scale of 2.64 kpc/arcsec at the distance of Hercules A (D$_L = 726$ Mpc).
HST Observations\[sec\_obs\]
============================
The HST observations were obtained on October 8, 2012 during a 3-orbit visit. We obtained U, V, and I images with the WFC3/UVIS camera. The observation parameters are given in Table 1. The observations were dithered to allow removal of cosmic rays and bad pixels and to cover the WFC3 chip gap. The HST calibration pipeline was applied to the raw data. Figure \[fig\_JVLA\] shows theV-band image and the full extent of the deep Jansky Very Large Array (JVLA) image from @cotton13. The U, V, and I images are shown in Figures \[fig\_U\], \[fig\_V\], and \[fig\_I\].
The optical emission lines are faint and low excitation [@tadhunter93; @buttiglione09]. H$\alpha$+\[NII\] and \[OII\] are the brightest emission lines in the optical spectrum [@buttiglione09]. The U-band (F336W) filter lies blue-ward of \[OII\] and the V-band (F606W) filter sits between H$\alpha$+\[NII\] and \[OII\]. Thus the U and V filters are dominated by the continuum in the host galaxy. There will be some contamination of the I-band (F814W) filter by H$\alpha$+\[NII\]. Our rough estimate suggests that contamination from H$\alpha$+\[NII\] to the total flux is less than 20%, though this will be spatially dependent.
Galactic (Milky Way) extinction was corrected using an E(B-V) = 0.25. Internal extinction was corrected using an E(B-V) = 0.34. This was calculated from the dereddened (H$\alpha$/H$\beta$) Balmer decrement ratio from the optical spectroscopy of @buttiglione09, using
$E\left(B-V\right)_{H\alpha / H\beta} = \frac{2.5 \times \log \left( 2.86 / R_{\mathrm{obs}}\right)}{k\left(\lambda_\alpha\right) - k\left(\lambda_\beta\right)}$
where R$_\mathrm{obs}$ is the Balmer decrement ($\sim 4.0$) from @buttiglione09, and K($\alpha$) = 2.54, K($\beta$)=3.61 for the R$_\mathrm{V}\ $= 3.1 extinction law of @cardelli89. The E(B-V) derived from the Balmer Decrement was then converted into extinctions at U- and I-band to correct our U- and I-band photometry for internal extinction, using Table 3 of @cardelli89. Final extinctions were A$_\mathrm{Uband-internal}$ = 1.654 and A$_\mathrm{Iband-internal}$ = 0.504.
[lrr]{} U-Band & F336W & 3656\
V-band & F606W & 1760\
I-band & F814W & 1928\
Results
=======
The Nucleus {#sec_nucleus}
-----------
In our I-band image (Figure \[fig\_I\]) we see a compact nucleus, which is not seen in our V-band image. The detection in I-band allows registration of the HST images with the JVLA image. The I-band flux of the nucleus corrected for Galactic extinction is $3.34 \pm 0.77 \times 10^{-18}$ ergs s$^{-1}$ cm$^{-2}$ $\AA^{-1}$. In contrast, the nucleus was not detected in the NICMOS H-band [@baldi10] with an upper limit on the monochromatic power of $2.2 \times 10^{28}$ ergs s Hz which corresponds to a limit on flux of $4.1\times 10^{-18}$ s$^{-1}$ cm$^{-2}$ $\AA^{-1}$. We convolved the optical spectrum from @buttiglione09 with the WFC3 U, V, and I bandpasses. The continuum in the spectrum is fairly flat between V and I; while the H$\alpha$+\[NII\] lines (redshifted to $\sim 7575$Å and with equivalent width $\sim 50$Å) make a strong contribution to the I band flux. Thus, the detection of the nucleus in I band only seems likely to be due to the H$\alpha$+\[NII\] lines. Our images also clearly show the secondary nucleus previously detected by e.g., @smith89and @sadun93. We are not aware of a redshift for the secondary nucleus.
![Hercules A. HST/WFC3/UVIS U-band (F336W) image smoothed slightly with low level radio contours of the JVLA image superposed. This image is not sensitive to the older stellar population. The compact feature 4 to the north-west is the secondary nucleus at RA = 16h 51m 07.98s Dec = +04$^o$ 59$\arcmin$ $35\farcs 6$ in the HST WCS. \[fig\_U\]](Figure2.pdf)
[lr]{} Inner radio jets & 100\
cD Major Axis & 120\
Secondary Nucleus & 135\
Overall Dust Filaments & $\sim 100$\
Inner eastern filament & 87\
Outer eastern filament & 111\
X-ray Cavities & $\sim 35$\
The Dusty Filaments
-------------------
The dusty filaments can be seen in Figures \[fig\_V\] and \[fig\_dashed\] in the regular and unsharp-masked V-band images. The deeper HST images presented here show a network of dusty filaments which are more extended and more complex than the structures seen in the previous HST data [@baum96; @dekoff96]. The largest angular size of the filament web is $\sim 10\arcsec$ ($\sim 26$ kpc). Position angles of the cD galaxy, secondary nucleus, and inner radio jets are given in Table 2. The overall position angle of the filaments is oriented along the radio jet at PA $\sim 100\deg$, though there is considerable sub structure. Of course, the morphology of the filaments differs on the two sides of the source. On the western side, the filaments tend to be relatively straight. Two filaments seem to bracket the location of the western radio jet (projected in towards the nucleus), while a third extends northwest up to the east of the secondary nucleus. There is a blob of obscuration just to the east of the secondary nucleus. On the eastern side the filaments are in two curved structures resembling parts of two loops - one extending $\sim 2\arcsec$ and the other $\sim 4\arcsec$ from the nucleus.
Figure \[fig\_RGB\] shows a RGB image based on the I, V, and U-band images. We see that there are faint patches of blue light centered on the galaxy nucleus with a total extent of $\sim 3\arcsec$ (8 kpc) with some additional small patches further out. Some of the blue patches are spatially adjacent to the dusty filaments.
![Hercules A. HST/WFC3/UVIS V-band (F606W) image with low level radio contours of the JVLA image superposed in white. U-band (F336W) contours are superposed in black. \[fig\_V\] ](Figure3.pdf)
Discussion
==========
Origin of the dusty filaments \[sec\_origin\]
---------------------------------------------
Hercules A is in a cooling flow cluster [@nulsen05] with an estimated mass accretion rate of 90 M$_\odot$ yr (§\[sec\_SFR\]). Cooling flows are thought to deposit cold gas in the central dominant galaxies though at rates of about $\sim 10\%$ of those in the original cooling flow models [e.g., @edge01; @peterson06; @odea08]. Thus, it is possible that the filaments are the result of gas condensing out of the cooling flow.
On the other hand, there is a secondary nucleus which in projection lies within the web of filaments (Figure \[fig\_V\]) and which may be experiencing a close encounter with or possibly merging with the host galaxy and providing the gas/dust for the filaments. Further, two of the western filaments extend out to near the position of the secondary nucleus. There is a blob of obscuration just to the east of the secondary nucleus. There is also a blue continuum feature which extends from the nucleus towards the secondary nucleus (Figs \[fig\_U\],\[fig\_RGB\]) which might be due to star formation in the acquired gas. The $I-V$ color map (Fig. \[fig\_I-V\]) shows that the secondary nucleus and the host galaxy have similar colors. @sadun93 suggested tentatively that the secondary nucleus is an S0 galaxy with an embedded disk and that the galaxy profile has been truncated. Thus, we favor the interpretation that the dusty filaments are due to gas acquired from an ongoing merger/encounter with the secondary nucleus.
None of the dusty filaments seem to lie in a preferred plane of the host galaxy. Combined with the “web-like” appearance of the filaments this suggests that the cold gas has not had time to settle in the galaxy, or alternatively the gas has been disturbed, e.g., by the hot ISM or the radio jets.
![Hercules A. HST/WFC3/UVIS I-band (F814W) image. We detect the host galaxy nucleus at RA = 16h 51m 08.14s Dec = +04$^o$ 59$\arcmin$ $33\farcs 6$ in the HST WCS. The nucleus seems likely to be dominated by H$\alpha$+\[NII\] (§\[sec\_obs\],\[sec\_nucleus\]). The secondary nucleus is 4 to the north-west of the nucleus of the host galaxy. Low level radio contours of the JVLA image are superposed in white. U-band (F336W) contours are superposed in black. \[fig\_I\]](Figure4.pdf)
Estimated Masses of Dust and Gas in the Filaments
-------------------------------------------------
We use the $I-V$ color map (Fig. \[fig\_I-V\]) to roughly quantify the physical properties of the gas and dust in the filaments. Following the methodology of @sadler85 as adapted by @dekoff00, the lower-limit filamentary dust mass $M_\mathrm{dust}$ can be estimated by
$$M_\mathrm{dust} = \Sigma \langle A_\lambda \rangle \Gamma^{-1}_\lambda,
\label{equation:dustmass}$$
where $\Sigma$ is the area covered by the dusty filaments, $\langle A_\lambda \rangle$ is the mean extinction due to dust within this area, and $\Gamma_\lambda$ is the mass absorption coefficient, for which we adopt the Galactic value at $V$-band from @vandokkum95;
$$\Gamma_V \approx 6 \times 10^{-6} \mathrm{mag~kpc}^2 M^{-1}_\odot .
\label{equation:massabsorption}$$
A mean “off-filament” $I-V$ galaxy color is subtracted from a mean “on-filament” dust color, yielding a color excess $E(I-V) \approx -0.4$ associated with the filaments that is (roughly) independent of Milky Way and other internal extinction effects. After converting this color excess to an extinction at $V$-band ($A_{V, \mathrm{dust}} \approx 0.775$ mag) following the Galactic $R_V=3.1$ law [@cardelli89], equations \[equation:dustmass\] and \[equation:massabsorption\] are used to estimate a rough lower limit dust mass of $M_\mathrm{dust} \approx 3.8 \times 10^{6}~M_\odot$. If the filaments follow Galactic gas-to-dust ratios, the gas mass in the filaments would be roughly 100 times higher, or $\sim 4\times10^{8} M_\odot$. Finally, the extinction due to dust ($A_{V, \mathrm{dust}}$) can be used with the relation from @predhel95 to estimate an associated neutral hydrogen column density of $\sim 1.4 \times 10^{21}$ cm$^{-2}$.
Star Formation \[sec\_SFR\]
---------------------------
@nulsen05 note that the high central electron density determined from Chandra observations gives a central cooling time consistent with cooling flow clusters. Additional details are provided by Nulsen (2012, private communication) and @tremblay13. The cooling radius (where $t_{\rm cool} = 7.7$ Gyr) is at roughly 120 kpc (the number is not very well defined because the density profile is quite flat on the inside of the shock, so $t_{\rm cool}$ varies slowly there). These numbers imply a cooling power of about $10^{44}$ erg s which gives a cooling rate of 90 M$_\odot$ yr (for kT = 4.5 keV). However, the cooling times will change after the shock moves on, so the number is going to change on a timescale that is short compared to the cooling time.
The blue continuum light seen in Figures \[fig\_U\] and \[fig\_RGB\] may be due to low levels of star formation in the cold gas. High levels of star formation, i.e., a star burst, has already been ruled out [e.g., @dicken09]. We can estimate star formation rates (SFRs) using UV observations of photons from hot young stars, H$\alpha$ observations of gas photoionized by the young stars, and by Infrared emission from dust heated by the young stars [e.g., @kennicutt98; @schmitt06]. The H$\alpha$ luminosity is L(H$\alpha$) $\simeq 1.9 \times 10^{41}$ erg s [@buttiglione09]. Using the calibration of @kennicutt98 gives a SFR(H$\alpha$) $\sim 1.5 $ M$_\odot$ yr. Because of the “contamination" of the I-band filter with H$\alpha$+\[NII\] (§\[sec\_obs\], \[sec\_nucleus\]) there may be structure in the I-band image due to the emission line gas. In Figure \[fig\_I\_unsharp\] we see faint structure in the I-band unsharp-masked image which is spatially coincident with the blue continuum seen in the U-band image. We expect the I-band continuum, extending from the old stellar component, to be smoothly distributed and mostly follow the smooth H-band surface brightness profile. Nevertheless, the I-band unsharp mask reveals clumpy and filamentary structures that we expect arise primarily from the H$\alpha$+\[NII\] contamination in the bandpass, which is not expected to be as smoothly distributed. It is therefore possible that the structure seen in Fig. \[fig\_I\_unsharp\] are H$\alpha$ filaments, which are cospatial with the U-band filament, consistent with H-alpha that has been excited by young stars. The Spitzer 24 $\mu$m flux density is $2.0\pm0.2$ mJy [@dicken08]. Using the conversion of @rieke09 gives a star formation rate of SFR(24) $\simeq 4.1$ M$_\odot$ yr. Using the U-band flux density corrected for galactic and internal extinction (using the Balmer decrement) ($1.4 \times 10^{-16}$ erg sec cm Å) and converting to luminosity gives L(U) $ \sim 3.3 \times 10^{21}$ W Hz. Using the relation for U-band derived star formation rates from massive stars ($> 5$ M$_\odot$) [@cowie97; @hopkins98] gives a star formation rate of $\sim 0.6$ M$_\odot$ yr. The values of SFR from the H$\alpha$, IR, and UV are consistent with star formation rates of a few percent of the mass accretion rate derived from the cooling power as typically found in cluster cooling flows [e.g., @mcnamara89; @peterson06; @odea08].
Radio and Optical Emission Line Properties \[sec\_line\]
--------------------------------------------------------
The emission line nebula is faint and low excitation [@tadhunter93; @buttiglione09; @buttiglione10]. @buttiglione10 suggest that Hercules A is a member of a small class of Extremely Low Excitation Galaxies (ELEG). Radio powers as large as that of Her A (Log P$_{408} \simeq 28.3$ W Hz) are generally only found in FRII radio galaxies. The H$\alpha$+\[NII\] emission line luminosity (Log $ L \simeq 34.6$ W), using the measurements of @buttiglione09 is about 1.5 orders of magnitude low for a FRII radio galaxy with that radio power. However, the emission line luminosity lies on an extrapolation to high radio power of the emission line vs radio relation for FRI radio galaxies found by @baum95. Thus, the ratio of emission line to radio power in Herc A is consistent with that of an FRI radio galaxy.
Adopting a 1400 MHz core flux density from @gizani03 and a total flux density from @kuhr81 gives the ratio of radio core to extended flux density $R \sim 8.5 \times 10^{-4}$ (log $R \simeq -3.1$). For comparison, sources with a similar high total radio power have a median log $R \simeq -2.4$ [@baum95]. The value found in Hercules A is on the low end of the distribution of R values. @capetti11 speculate that the low \[OIII\]/H$\beta$ combined with the low radio core to extended radio emission ratio suggest that the nucleus of Hercules A has turned off. This is an interesting suggestion since intermittent radio activity could help to explain the unusual radio properties of Hercules A, especially the series of shells/rings in the western lobe [@gizani03].
@gizani02 present EVN observations of the nucleus of Hercules A with 18 mas resolution which detect a compact radio source with a flux density of 14.6 mJy at 1.6 GHz. In addition, we detect the nucleus in our I-band image. The detection of the nucleus in the radio and I-band suggests that the nucleus remains active though possibly at a much lower level than previously.
Hercules A is in a cooling flow cluster [@nulsen05]. The emission line nebulae in central dominant galaxies in cooling flows are almost always low excitation, and \[OIII\]/H$\beta \sim 0.5$ is within the range of commonly observed values [e.g., @heckman89; @crawford99; @quillen08]. It seems probable that the emission line nebula in Herc A is low excitation because of the lack of a strong ionizing AGN continuum and so the excitation is dominated by e.g., star formation, shocks, or processes occurring in the cooling flow environment.
However, radio galaxies which are fed by cold gas acquired in a merger tend to have radiatively efficient accretion and high excitation emission line nebulae [e.g., @baum95; @hardcastle09; @buttiglione10; @best12]. The fact that the nebula in Hercules A is low excitation suggests that the dusty filaments are not providing significant amounts of fuel to the central supermassive black hole. This might be because (1) the gas has not yet settled deep into the galaxy nucleus, or (2) the cold gas in the nucleus is being removed, e.g., via entrainment in the radio jets.
![Hercules A. HST/WFC3/UVIS. I-band unsharp-masked image with contours of the U-band image superposed. \[fig\_I\_unsharp\]](Figure7.pdf)
Timescales {#sec_time}
----------
@nulsen05 discovered a shock front in the hot ICM with a radius of $\sim 160$ kpc. Based on fits to the jump in the X-ray surface brightness profile they estimate a shock Mach number M $\simeq 1.65$. This gives an age for the outburst that created the shock of t$_s \sim 5.9 \times 10^7$ yr. If we assume a lobe expansion velocity of 0.03c [e.g., @alexander87; @scheuer95; @odea01; @tremblay10], the 200 kpc radio source radius implies a dynamical age of t$_d \sim2 \times 10^7$ yr. For comparison, the spectral age of the electrons (due to radiative losses) at the end of the lobes is $\gae 1.3 \times 10^7$ yr [@gizani03] roughly consistent with the dynamical age estimate. The dynamical and radiative loss ages for the radio source are factors of $\sim 3$ and $\sim 4.5$ smaller than the shock age. This might be due to the uncertainties in the age estimates or it may suggest that there was a previous epoch of radio source activity $\sim 60$ Myr ago.
If the source has been recently reborn, the restarting jets would propagate faster through the evacuated lobe resulting in a shorter (model dependent) age of $1-4 \times 10^6$ yr [@gizani03]. If the dusty filaments are from the merger/interaction, they will need at least a dynamical time ($\gae 10^8$ yr) to reach the center. The lack of a bright accretion disk suggests that the dusty filaments are not yet fueling the radio source and that the fueling is being done by the cooling flow (§\[sec\_line\]). If this is the case, the stopping and restarting of the radio source would be associated with feedback and cooling timescales in the cooling flow. The buoyant rise (75% sound speed) and refilling times for both X-ray cavities are about $ 7 \times 10^7$ yr [@tremblay13] which is comparable to the age of the shock, but longer than the age of the current radio source. The difference in the ages of the X-ray cavities and the current radio source combined with the fact that the X-ray cavities are misaligned with the current radio source by $\sim 65\deg$ would be consistent with the hypothesis that the cavities were created by the previous epoch of radio source activity (assuming that the cavities are created by the radio source and not by some other phenomena). However, that would suggest that the axis of the radio source changed by $\sim 65\deg$ in between the two epochs of activity. Such a large change in orientation may require a merger between supermassive black holes [e.g., @merritt02]. The black hole required for this merger may have been acquired in a merger prior to the current encounter/merger with the secondary nucleus. In addition, if the previous radio source is 60 Myr old, there should be some evidence for low frequency radio emission along the position angle of the X-ray cavities. @nulsen05 note a buoyant bubble is formed in their model for the shock. If this mechanism would produce cavities misaligned with the radio source axis, then a swing in jet axis is not required.
A possible scenario would be the following. About 60 Myr ago, a powerful radio source was generated propagating along a position angle of $35\deg$. This radio source created the shock and the cavities in the hot ICM. A black hole merged with the central black hole changing the jet axis by $65\deg$ (in projection). Either the source was ongoing and swung in position angle, or it had turned off and restarted at the new position angle of $ 100\deg$ about 20 Myr ago. The source has maintained its position angle but the activity has been repetitive/unsteady on time scales of several Myr in order to account for the structure in the lobes [@gizani03].
Constraints on Entrainment
==========================
Our HST observations show dusty filaments aligned with the bases of the radio jets (e.g., Figures \[fig\_V\] and \[fig\_dashed\]). Jets in FRI radio galaxies are launched with relativistic velocities but decelerate to sub-relativistic velocities by the time they reach kpc scales [e.g., @bp84; @odea85; @laing93; @urry95; @giovannini01; @kharb12]. It is widely thought that the deceleration is caused by entrainment of thermal gas [e.g., @baan80; @deyoung81; @begelman82]. The gas entrained could be either (1) ambient gas entrained through a turbulent boundary layer [e.g., @bicknell84; @bicknell94; @deyoung86; @deyoung96; @komissarov90; @perucho07; @rossi08; @wang09], (2) winds from stars within the volume of the jet [e.g., @komissarov94; @bowman96], or (3) dense clouds which enter the jet [@fedorenko96; @bosch12]. Laing and collaborators have modeled the collimation, surface brightness, and polarization structure of jets in FRIs in terms of entraining and decelerating relativistic flows [@laing02a; @laing02b; @canvin04; @canvin05; @laing06]. Previous evidence for entrainment has been suggested for 3C 277.3 [@vanbreugel85] and Centaurus A (@graham81; but cf @morganti91 [@graham98]).
The boundary layer develops at the interface between the jet and ambient medium and then spreads into both the ambient medium and the jet [@wang09]. Using our HST observations we may be able to constrain how quickly the boundary layer develops, and how far it extends into the ambient medium. If the boundary layer extends far into the ambient medium, this would suggest that there is a broader “wind" which surrounds the jet and which can influence the host galaxy. Here we consider the hypothesis that the dusty filaments are entrained by the radio jet and examine the implications. Because of the difference in morphology, we consider the two sides separately. Disclaimer: There could be a different physical relationship between the jet and filaments or none at all if the apparent spatial coincidence is due to chance projection. @deyoung86 finds that entrainment can occur in the radio source bow shock. Thus, an alternative scenario could be that filaments were not entrained in a jet boundary layer, but were entrained in the bow shock of the radio source when it originally propagated through the host galaxy about 20 Myr ago (§\[sec\_time\]). However, it is not clear that the dusty filaments would survive that long.
The Western Jet
---------------
On the western side of the source, two dusty filaments align with the radio jet. The northern of the two filaments extends about 4.7 from the nucleus, while the southern extends about 2 from the nucleus. The radio jet does not become detectable until just beyond the extent of the western filaments. This “gap“ is a common feature of FRI radio galaxies [e.g., @bridle84; @laing02a; @laing02b; @canvin04; @canvin05; @laing06] and in the Laing & Bridle models is due to dimming caused by Doppler beaming of the radiation along the jet axis and thus out of our line-of-sight. We assume that there is an undetected jet which is continuous from the nucleus out to where the jet becomes detectable. The continuity of the filament from the nucleus out to $\sim 4\arcsec$ suggests that the jet is entraining gas during the ”gap" phase of the jet (i.e., the inner 10 kpc).
If the dusty gas is distributed in a cylindrical sheath, the higher column densities along our line of sight through the sides of the sheath will cause greater obscuration along the edges of the cylinder and thus the edges of the radio jets. However, the asymmetry in the two filaments suggests that the dusty gas is not distributed homogeneously in the sheath around the radio jet axis. In order to emphasize the spatial correlation between the filaments and radio jet we extrapolate the jet contours inwards towards the nucleus the radio jet contours (Figure \[fig\_dashed\]). We see that the filaments in projection lie along the outside of the extrapolated radio jet. The separation between the two filaments (1or 2.7 kpc) is slightly larger than the width of the extrapolated jet. Jets on these scales in twin-jet sources tend to be expanding with distance from the nucleus [e.g., @bridle84; @laing02a; @laing02b; @canvin04; @canvin05; @laing06]. Thus, the “true” inner jet at the location of the dusty filaments is likely to be narrower than the jet at 4 from the nucleus. The dusty filaments are seen outside the visible extent of the jets at a distance of $\sim 1.3$ kpc from the jet axis. The width of the northern filament is $\lae 0.1\arcsec$ (270 pc). The location of the observed dusty filaments relative to the jet axis depends on (1) where the gas is entrained, and (2) where it survives the process of entrainment and transport so that the filament is visible in absorption against the stellar background. The lifetime of cold dusty gas in high velocity flows is uncertain. Shocks which develop in such flows would heat/ionize gas and destroy grains. Thus, observational evidence for such cool gas is interesting. A possible relevant result is that HI absorption in jet driven outflows in compact radio sources reveals the presence of an atomic component to outflows with velocities up to $\sim 1000$ km s [e.g., @holt08]. The column density through the dusty filament determines its detectability in absorption against the stellar background. The filaments can be made less detectable by mixing the cold gas and dust into a larger volume or by destroying the dust grains. Dust grains will be destroyed by shocks with velocities greater than $\sim 100$ km s [e.g., @jones96; @welty02; @jones11]. So, if the boundary layer is turbulent, the dust may trace a low velocity ($v \lae 100$ km s) outer edge in the boundary layer. We don’t see the dusty filaments widening or the filaments changing their projected distance from the axis of the jet as a function of distance from the nucleus as expected for a fully turbulent boundary layer. (Though there is a hint of a shortening in the distance of the northern filament from the jet axis around 4 from the nucleus.) Thus, in this early stage of entrainment, perhaps the entrainment is fairly laminar, or we preferentially see the dusty gas in regions of the flow which are laminar.
The Eastern Jet
---------------
Figures \[fig\_V\] and \[fig\_dashed\] show that unlike the relatively straight filaments which bracket the jet on the western side, the eastern filaments are arc-like suggesting they are part of a bubble/shell or loop. Further the eastern filaments are slightly misaligned from the jet axis (Figure \[fig\_dashed\]). The inner filament is at PA $\sim 87\deg$ and the outer filament is at PA $\sim 111\deg$. These arc-like structures resemble the bubbles seen in galactic super winds such as e.g., NGC 3079 [e.g., @cecil01] and NGC 6764 [e.g., @hota06]. However, it is not clear what the energy source would be for winds in Hercules A. There is no bright accretion disk to drive a wind by radiation pressure (§\[sec\_line\]), and the star formation rates are too small to drive a starburst-driven wind (§\[sec\_SFR\]). The only outflow that we know exists is the radio jet. On the other hand, if the structures we see on both sides are due to entrainment, then it is not clear why the morphologies on the two sides are so different.
Although we don’t understand the difference in morphology of the filaments on the two sides, for simplicity, we hypothesize that the eastern filaments are also dragged out via entrainment in the outflowing radio jet. In this scenario, the dusty arcs are not pieces of shells blown out by a wind, but are instead loops of dusty filaments dragged out via entrainment. @hatch06 suggest that in the central cooling flow source NGC1275 (Perseus A), optical emission line filaments are dragged to distances of order 20 kpc by buoyantly rising bubbles generated by the radio source. @fabian08 have suggested that the Perseus filaments are stabilized from disruption during their transport because they are threaded by magnetic fields. If this magnetic stabilization model is correct, it may also apply to the filaments in Herc A. The presence of blue continuum and H$\alpha$ emission associated with the filaments (§\[sec\_SFR\]) suggests that the necessary free electrons will be available to freeze the magnetic field to the filaments. If the filaments were stripped from the secondary nucleus, the linking of the magnetic field to the gas must have occurred in the secondary galaxy.
Occam’s Razor favors a single explanation for the relation between the jets and dusty filaments on the two sides of the source. However, because of the difference in morphology, it is possible that a different explanation is required for the two sides. One such scenario could be that only the western filaments are entrained - producing their linear structure. In this alternate scenario, the eastern filaments are loops of dusty gas stripped from the secondary nucleus and merely seen in projection against the jet. In this scenario the entrainment process is not required to produce two different filament morphologies. Though this would seem to require the secondary nucleus to pass through the plane containing the jets in order to produce a fortuitous alignment.
As discussed in § \[sec\_SFR\] the unsharp-masked I band image (Figure \[fig\_I\_unsharp\]) suggests that there is H$\alpha$ emission associated with the dusty filaments. If the filaments are indeed entrained in the jet boundary layer we might expect to see velocity structure perhaps along and/or across the filaments. In addition, there might be evidence for the ionization of the gas by shocks due to turbulence or velocity sheer in the boundary layer.
For completeness, we note that if there are currents associated with the filaments and the radio jets [e.g., @benford78], then Lorentz force type interactions might also be possible.
Implications for the Radio Morphology
-------------------------------------
The differences in the structure of the radio jets on the two sides might be due to (1) differences in the way the outflows are generated by the supermassive black hole; or (2) differences in the way the outflows interact with their environments as they travel outward away from the galaxy. It is not clear whether the mechanisms which produce the radio morphology are related to the mechanism which produce the morphology of the dusty filaments. If we assume that they are related, then it is possible to make the following argument. If the Hubble data showed that the morphology of the cold gas and dust mimicked that of the radio jet on the same side (i.e., dust shells on the side with radio shells, and straight filaments on the side with a collimated jet), that would be consistent with the idea that the differences in jet properties were established on small scales. However, the fact that the cold gas and dust has a different morphology than the radio emission on the same side supports the idea that the differences in radio properties are established on larger scales as the jets interact with their environments [e.g. @gendre13].
SUMMARY
=======
We present U, V, and I band images of the host galaxy of Hercules A obtained with HST/WFC3/UVIS. We find a network of dusty filaments which are more complex and extended than seen in earlier HST observations. The filaments are associated with a faint blue continuum light (possibly from young stars) and faint H$\alpha$ emission. Estimated star formation rates in the filaments are a few M$_\odot$ yr which is a few percent of the mass deposition rate inferred from the Chandra-derived cooling power. The total dust mass in the filaments is about $3.8 \times 10^6$ M$_\odot$.
It seems likely that the cold gas and dust has been stripped from a companion galaxy now seen as a secondary nucleus. There are dusty filaments aligned with the base of the jets on both eastern and western sides of the galaxy. The morphology of the filaments is different on the two sides - the western filaments are fairly straight, while the eastern filaments are mainly in two loop-like structures. We suggest that despite the difference in morphologies, both sets of filaments have been entrained in a slow moving boundary layer outside the relativistic flow. In the scenario, the entrainment has occurred during the “gap" phase of jet evolution. As suggested by @fabian08 for NGC1275, magnetic fields in the filaments may stabilize them against disruption.
We suggest a speculative scenario to explain the relation of the radio source to the shock and cavities in the hot ICM. About 60 Myr ago, a powerful radio source was generated propagating along a position angle of $35\deg$. This radio source created the shock and the cavities in the hot ICM. A black hole merged with the central black hole changing the jet axis by $65\deg$ (in projection). Either the source was ongoing and swung in position angle, or it had turned off and restarted at the new position angle of $ 100\deg$ about 20 Myr ago. The source has maintained its position angle but the activity has been repetitive on time scales of several Myr in order to account for the structure in the lobes [@gizani03].
We are grateful to J. Stoke, M. Mutchler, Z. Levay, and L. Frattare for help with the HST proposal and Phase II file. We thank the referee for helpful comments. This work is based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program \#13065. The National Radio Astronomy Observatory (NRAO) is operated by Associated Universities Inc. under cooperative agreement with the National Science Foundation.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a thorough description of the physical regimes for ultracold bosons in double wells, with special attention paid to macroscopic superpositions (MSs). We use a generalization of the Lipkin-Meshkov-Glick Hamiltonian of up to eight single particle modes to study these MSs, solving the Hamiltonian with a combination of numerical exact diagonalization and high-order perturbation theory. The MS is between left and right potential wells; the extreme case with all atoms simultaneously located in both wells and in only two modes is the famous NOON state, but our approach encompasses much more general MSs. Use of more single particle modes brings dimensionality into the problem, allows us to set hard limits on the use of the original two-mode LMG model commonly treated in the literature, and also introduces a new mixed Josephson-Fock regime. Higher modes introduce angular degrees of freedom and MS states with different angular properties.'
address:
- '$^1$Department of Physics, Colorado School of Mines, Golden, CO 80401, U.S.A.'
- '$^2$Department of Physics, University College Cork, Cork, Ireland'
- '$^3$Department of Physics, University of California, Berkeley, California 94720, USA'
- '$^4$Universität Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany'
author:
- 'M. A. Garcia-March$^{1,2}$, D. R. Dounas-Frazer$^{3}$, and Lincoln D. Carr$^{1,4}$'
title: 'Macroscopic superposition states of ultracold bosons in a double-well potential'
---
Introduction {#sec:introduction}
============
The superposition of quantum states Is a postulate in quantum mechanics. When these states can be distinguished macroscopically it leads to the fundamental question of how the theory describing the physics of point particles or atoms is connected with macroscopic objects [@2002LeggettJPCM]. Ultracold bosons in double wells provide a useful realization of macroscopic superposition (MS) states, in which the distinguishable macroscopic property is the localization of atoms in one of the wells [@2007DounasFrazerPRL; @2009XiaoCueCTP; @2011MazzarellaPRA]. In this system, it is possible to restrict the atoms to occupy only two single particle modes, corresponding to their being condensed in the lowest energy state and localized spatially in one of the wells [@1999DalfovoRMP; @2001LeggetRMP]. In this simplified two mode picture, the main two processes are the tunneling of atoms between wells and their interaction in pairs in one of the wells. When the tunneling energy is larger than the interaction energy, this is an ideal system for studying Josephson effects [@1986JavanainenPRL; @1997MilburnPRA; @1997SmerziPRL; @1998ZapataPRA; @2000OstrovskayaPRA; @2000Giovanazzi; @2005MahmudPRA; @2005AlbiezPRL; @2006AnanikianPRA; @2006WangPRA; @2006LiPRA; @2007LevyNature; @2010JuliaDiazPRA]. Conversely, when interactions dominate over tunneling, MSs can be obtained [@1998SteelPRA; @2002KalosakasPRA; @2007DounasFrazerPRL]. When all $N$ atoms occupy simultaneously the two single particle states localized in each well, these MSs are known as NOON states. The tunneling dominated regime is known as the *Josephson regime*, while the interaction regime is known as the *Fock regime*. In the latter, the two mode approach can cease to be sufficient to describe the system, due to large interactions populating single particle excited states. Here, we offer a detailed study of the eigenvector spectra in one dimensional (1D) and three dimensional (3D) double wells, with special attention to the appearance of MS states which occur in the Fock regime, and we include more modes in our approach, as appropriate for this regime.
![(Color online) [*Schematic of the Double Well.*]{} (a) 1D: the $J$’s indicate single-particle tunneling processes while the $U$’s indicate two-particle tunneling and interaction processes. (b) 3D: the spherical harmonics $Y_{\ell m}$, with total and $z$-component of the angular momentum $\ell$ and $m$ respectively, are distorted by the coupling to the nearby well. \[fig0\]](fig0.pdf){width="6.cm"}
To model 1D double wells we use a four-mode generalization of the Lipkin-Meshkov-Glick Hamiltonian (LMGH) [@1965LipkinJNP; @2004VidalPRA], where the atoms are allowed to occupy two excited on-well (sometimes called on-site) localized single particle eigenstates (see Fig. \[fig0\](a)). Due to the new two excited modes, there are three new processes in the model. On the one hand, pairs of atoms in the ground state in one well can be excited to the first level, or decay from this level to the ground state, as sketched in Fig. \[fig0\](a). Also, the atoms excited to the first level of eigenenergies can interact on-well or tunnel to the other well, also sketched in Fig. \[fig0\](a). Then, in this picture, the conventional division into the Josephson and Fock regimes must be amended to include a new *mixed regime,* in which the atoms behave differently depending on their energy level, as discussed in Sec. \[sec:bounds\]. We use perturbation theory to characterize the eigenstates and eigenvalues in the Fock regime and we also characterize them analytically in the non-interacting regime. In between these two regimes, we use numerical exact diagonalization of the LMGH to show how the transition between these regimes occurs. We identify clearly the regimes in which the original NOON states as well as novel MS states can be obtained. We characterize the limits of the model, the occurrence of crossings in the spectra, and couplings between states with atoms in different levels. Our crossing criteria clarify when the *one-level approximation* (LMGH with only two modes) ceases to be sufficient, and one must use at least the *two-level approximation.*
The four-mode, two-level generalization of the LMGH is the natural way to introduce dimensionality in the problem. When the same two levels are generalized to 3D double wells, an eight-mode generalization of the LMGH is obtained as sketched in Fig. \[fig0\](b). Then, a new quantum number, the orbital degree of freedom, is introduced and it plays a key role [@2011GarciaMarchPRA]. Since new parameters arise in the 3D case, new dynamical regimes are found. Moreover, new MS states involving angular degrees of freedom can occur. In a previous work we characterized the limits of the one- and two- level approximations, the occurrence of crossings in different regimes, and the coupling between states with atoms in different levels [@2011GarciaMarchPRA]. In this Article we review, complete, and extend the 1D picture obtained in [@2007DounasFrazerPRL; @2007DounasFrazerMT]. We also compare with the 3D case, and summarize the derivation of the 3D two-level LMGH and the routes to new physics in 3D which are currently being explored.
By means of a semiclassical, mean-field approximation macroscopic quantum tunneling has been predicted in the Josephson regime, while macroscopic self-trapping in one of the wells has been shown to occur when the interactions grow [@1986JavanainenPRL; @1997MilburnPRA; @1997SmerziPRL; @1998ZapataPRA; @2000OstrovskayaPRA; @2006AnanikianPRA; @2006WangPRA; @2006LiPRA; @2005MahmudPRA; @2010JuliaDiazPRA]. The macroscopic quantum tunneling and self-trapping were observed in a recent experiment [@2005AlbiezPRL]. The former can be understood as the d.c. Josephson effect for superconductors, while the a.c. Josephson effect corresponds to small oscillations in the difference between the populations in both wells obtained for bigger interactions [@2000Giovanazzi]. The a.c. Josephson effect was recently demonstrated experimentally [@2007LevyNature]. One perspective is that the energy levels depend on the occupation of one of the wells. Then one can analyze the double well problem in terms of Landau-Zener tunneling. In this case the energy between levels changes with a constant small rate, leading to an adiabatically driven Josephson junction [@2000PRAWu; @2002PRALiu; @2006WuPRL].
In the Fock regime the semiclassical approach ceases to be appropriate, and other approaches can be useful; the most numerically accurate but also the most computationally intensive is the multiconfigurational Hartree method [@2005MasielloPRA; @2006StreltsovPRA; @2008AlonPRA; @2008ZollnerPRA; @2009SakmanPRL]. Methods based on the LMGH [@1965LipkinJNP; @2004VidalPRA], although limited to weaker interactions, have the advantage that one can treat more particles in higher dimensions, as well as make use of perturbation theory to obtain analytical results [@2007DounasFrazerPRL; @2010CarrEPL]. In this Article we use a modified LMGH for these reasons. The LMGH was first introduced in the framework of nuclear physics [@1965LipkinJNP]. It has applications in a wide range of other fields, like ultracold bosons, and it is a perfect simple Hamiltonian to study a vast range of quantum effects, from quantum phase transitions [@2004VidalPRA; @2005DusuelPRC] to Josephson oscillations. The LMGH also applies in cold atoms for two hyperfine boson species in a single well, where MS states are predicted to occur [@1998CiracPRA; @2000DalvitPRA]. In this Fock regime the two mode approach could be insufficient to characterize the problem, since crossings and coupling to other more excited single particle states may be possible. Indeed, it is necessary to consider a second level in 1D for the physical parameters found in typical BEC experimental systems [@2007DounasFrazerPRL]. To consider this possibility we use a four-mode generalization of the LMGH for 1D double wells and an eight mode characterization in the 3D case.
Besides being a good system for fundamental studies of quantum many body effects, ultracold bosons in double wells also have many technological applications, for example, in quantum high precision measurements of inertial and gravitational fields [@2005SchumNatPhys; @2007HallPRL], as a primary thermometer [@2006GatiNJP], or in quantum computing [@2004CalarcoPRA; @2006SebbyStrableyPRA; @2008StrauchPRA].
The Article is organized as follows. In Sec. \[sec:secondQuantizedHamiltonian\] we introduce the initial second quantized Hamiltonian. In Sec. \[sec:1DLMG\] we obtain the 1D four-mode LMGH. Then, in Sec. \[sec:characterization\] we characterize the eigenvectors and eigenvalues in 1D for the Josephson and Fock regimes. In Sec. \[sec:bounds\] we obtain the limits of the model and the occurrence of crossings, we discuss all possible regimes, and we offer numerical examples to illustrate the results. Finally, we derive the 3D Hamiltonian and compare with the 1D case in Sec. \[sec:3D\]. We conclude, summarize, and discuss lines of future research in Sec. \[sec:conclusion\].
Second Quantized Hamiltonian {#sec:secondQuantizedHamiltonian}
============================
The second quantized Hamiltonian for $N$ interacting bosons of mass $\mathcal{M}$ confined by an external potential $V(\mathbf{x})$ with $\mathbf{x}\in{\mathbb R}^{3}$ in terms of the bosonic creation and annihilation field operators $\hat{\Psi}(\mathbf{x})$, $\hat{\Psi}^{\dagger}(\mathbf{x})$, obeying the usual commutation relations, is $$\begin{aligned}
&\hat{H} = \int\!\!
d\mathbf{x}\,\hat{\Psi}^{\dagger}(\mathbf{x})\left[-\frac{\hbar^{2}}{2\mathcal{M}}\nabla^{2}+V(\mathbf{x})\right]\hat{\Psi}(\mathbf{x})\label{eq:second-quantized1}\\
& + \frac{1}{2}\int\!\! d\mathbf{x}\,\hat{\Psi}^{\dagger}(\mathbf{x})\left[\int
d\mathbf{x'}\hat{\Psi}^{\dagger}(\mathbf{x}')V_{\mathrm{int}}(\mathbf{x}-\mathbf{x}')\hat{\Psi}(\mathbf{x}')\right]\hat{\Psi}(\mathbf{x})\,,\nonumber\end{aligned}$$ where $V_{\mathrm{int}}(\mathbf{x}-\mathbf{x}')$ stands for the two-body interaction and $V(\mathbf{x})$ is a three dimensional double-well potential with minima at $\mathbf{x}=\pm\mathbf{a}\in\mathbb{R}^{3}$ and a local maximum at $\mathbf{x}=\mathrm{{\bf 0}}$. Without loss of generality, we assume $V(\mathbf{x})=V(x)+V(y)+V(z)$. We take the 1D potentials in the $x$ and $y$ directions as isotropic harmonic oscillators, that is, $V_{x}=\frac{1}{2} \omega_{x}^2 x^2 $ and $V_{y}=\frac{1}{2} \omega_{y}^2 y^2 $, where both frequencies are of the same order. For simplicity, we consider $\omega_x=\omega_y=\omega_{\perp}$ in the following. A conventional 1D double well potential $ V(z)$ of barrier height $V_0$ can be approximated near its minima at $z=\pm a$ by $$V(z\pm a)\approx\frac{1}{2}\mathcal{M}\omega^{2}z^{2}\,,\label{eq:V(z)}$$ where $$\omega\equiv\left(\frac{1}{\mathcal{M}}\frac{\partial^2 V}{\partial z^2}\right)^{1/2}_{z=a}$$ is the local trapping frequency in each well. The recoil energy, defined as $E_r=\hbar^2/\mathcal{M} a^2$, is used through the dimensionless parameter $V_0/E_r$ to determine the barrier height in most of the experiments, with $\lambda=2a$ an effective wavelength.
We assume first that $\omega$ is of the same order as $\omega_{\perp}$ and we consider low densities, for which only binary collisions are relevant; also, we consider low energies, when these collisions are characterized by the *s*-wave scattering length of the atoms, $a_s$. The diluteness condition for a weakly interacting Bose gas is $$\sqrt{|\bar{n}\, a_{s}^{3}|}\ll 1 \,,$$ where $\bar{n}$ is the average density of the gas. In the context of the double-well potential, the maximum density of the gas is approximately $\bar{n}=N/(\sqrt{2\pi}\, a_{\mathrm{ho}})^{3}$, where $a_{\mathrm{ho}}$ is the harmonic oscillator length given by $a_{\mathrm{ho}}=\sqrt{\hbar/\mathcal{M}\omega}$ (a similar condition must hold in the other two directions). Correspondingly, we restrict our discussion to the regime $$N^{1/3}\ll\left|\sqrt{2\pi}\, a_{\mathrm{ho}}/a_{s}\right|\,.\label{eq:diluteness}$$ Although the system is said to be weakly interacting when condition (\[eq:diluteness\]) is met, the interaction energy can be on the order of the kinetic energy and dilute gases can therefore exhibit non-ideal behavior [@1999DalfovoRMP; @2001LeggetRMP]. Under these conditions the second quantized Hamiltonian can be approximated as $$\begin{aligned}
\hat{H} =& \int\! d^3{\mathrm{\bf x}}\, \hat{\Psi}^{\dagger}({\mathrm{\bf x}})\left[ -\frac{\hbar^2}{2\mathcal{M}}\nabla^2 + V({\mathrm{\bf x}})\right]\hat{\Psi}({\mathrm{\bf x}})\,\nonumber\\
& + \frac{g}{2}\int \! d^3{\mathrm{\bf x}}\, \hat{\Psi}^{\dagger}({\mathrm{\bf x}})\hat{\Psi}^{\dagger}({\mathrm{\bf x}})\hat{\Psi}({\mathrm{\bf x}})\hat{\Psi}({\mathrm{\bf x}})\,.
\label{eq:second-quantized}\end{aligned}$$ The coupling constant $g$ is proportional to the $s$-wave scattering length, $ g = 4\pi\hbar^2 a_s/\mathcal{M}$; in Eq. (\[eq:second-quantized\]) we took the two-body potential from Eq. (\[eq:second-quantized1\]) to be approximated by an effective local interaction $V_{\mathrm{int}}(\mathbf{x}-\mathbf{x}')=g\,\delta^{(3)}( \mathbf{x}-\mathbf{x}')$.
One-dimensional Lipkin-Meshkov-Glick Hamiltonian {#sec:1DLMG}
================================================
Let us consider first the reduced 1D case, taking $\omega\ll\omega_{\perp}$. The double-well potential can be reduced to one spatial dimension in extremely anisotropic traps, where the transverse trapping frequencies must be sufficiently high to reduce the dimensionality of the single-particle wavefunctions. However, one must avoid potential resonances by not squeezing the trap too tightly [@1998OlshaniiPRL]: it is sufficient that the transverse harmonic oscillator length $a_{\mathrm{ho},\perp} \equiv \sqrt{\hbar/\mathcal{M}\omega_{\perp}}\gg a_s$. Under these assumptions, we can reduce the dimensionality of the second quantized Hamiltonian, by considering that the particles interact and tunnel only in one-dimension. Then, we can use appropriate superpositions of the eigenfunctions of the one-dimensional single particle Hamiltonian $$H_{\mathrm{sp}}=-\frac{\hbar^{2}}{2\mathcal{M}}\nabla^{2}+V(z)\,,\label{eq:SPH1}$$ to obtain a set of on-well localized wavefunctions, in an analogous way that on-site Wannier states are obtained from Bloch functions on a lattice [@1976Ashcroft]. We use this set of functions to expand the field operator: $$\hat{\Psi}(z)=\sum_{j,l}\hat{b}_{j\ell}\psi_{\ell}(z-z_{j})\,,\label{eq:hatPsi}$$ where $\psi_{\ell}(z-z_{j})$ are the on-well localized functions. Here, $z_{L}\equiv-a$ and $z_{R}\equiv a$ are the minima of the left and right wells, respectively, $j\in\{L,R\}$ is the *well index*, and the label $\ell$ is the *level index*. The level index increases with single particle energies in each well. For the two level approach considered here, $\ell \in \{0,1\}$. The operators $\hat{b}_{j\ell}$ and $\hat{b}_{j\ell}^{\dagger}$ satisfy the usual bosonic annihilation and creation commutation relations, $$\begin{aligned}
[\hat{b}_{j\ell}^{\dagger} ,\hat{b}_{j'\ell'}] & =\delta_{jj'}\delta_{\ell\ell'}\,,\nonumber \\
{}[\hat{b}_{j\ell},\hat{b}_{j'\ell'}] & =[\hat{b}_{j\ell}^{\dagger},\hat{b}_{j'\ell'}^{\dagger}]=0\,.\end{aligned}$$ Let $\varphi^{n}(z)$ be the $n$th eigenfunction of Hamiltonian (\[eq:SPH1\]), with eigenvalue $\epsilon_{n}$. Then, the localized functions at well $j$ are $$\begin{aligned}
& \psi_{j0}(z)=\frac{1}{\sqrt{2}}\left(\varphi^{1}(z)\pm\varphi^{2}(z)\right)\,,\label{eq:sup1}\\ &
\psi_{j1}(z)=\frac{1}{\sqrt{2}}\left(\varphi^{3}(z)\pm\varphi^{4}(z)\right)\,,\label{eq:sup2}\end{aligned}$$ with $j=L$ for the plus sign and $j=R$ for the minus sign. The corresponding eigenvalues are $E_{0}=(\epsilon^{1}+\epsilon^{2})/2$ and $E_{1}=(\epsilon^{3}+\epsilon^{4})/2$. These on-well localized eigenfunctions are represented schematically in Fig. \[fig0\](a).
Substituting Eq. (\[eq:hatPsi\]) into the second quantized Hamiltonian (\[eq:second-quantized\]) yields the two-level Hamiltonian: $$\hat{H}=\hat{H}_{0}+\hat{H}_{1}+\hat{H}_{01}\label{eq:two-level}$$ with $$\label{Eq:onelevelH}
\hat{H}_{\ell}=-J_{\ell}\sum_{j\ne
j'}\hat{b}_{j\ell}^{\dagger}\hat{b}_{j'\ell}+U_{\ell\ell}\sum_{j}\hat{n}_{j\ell}\left(\hat{n}_{j\ell}-1\right) +E_{\ell}\sum_{j}\hat{n}_{j\ell}\,,$$ $$\hat{H}_{01}=U_{01}\sum_{j,\ell\ne
\ell'}\left(2\hat{n}_{j\ell}\hat{n}_{j\ell'}+\hat{b}_{j\ell}^{\dagger}\hat{b}_{j\ell}^{\dagger}\hat{b}_{j\ell'}\hat{b}_{j\ell'}\right)\,,$$ where $\hat{n}_{j\ell} \equiv \hat{b}_{j\ell}^{\dagger}\hat{b}_{j\ell}$ is the number operator. The hopping and interaction terms are $$J_{\ell}=-\int dz \psi_{\ell}^{\ast}(z-a)\left[-\frac{\hbar^{2}}{2\mathcal{M}}\!\nabla^{2}+V(z)\right]\psi_{\ell}(z+\! a)\,
,\label{eq:J_l}$$ and $$U_{\ell\ell'}=\frac{g_{1}}{2}\int dz|\psi_{\ell}(z)|^{2}|\psi_{\ell'}(z)|^{2}\, ,\label{eq:U_l_l'}$$ respectively. The coupling constant obtained after reducing the dimensionality of the problem is $g_{1}=2\hbar\omega_{\perp}a_{s}$. In the following, we denote $U_{\ell\ell}$ simply as $U_{\ell}$.
The functions $\psi_{\ell}(x)$ resemble roughly the eigenfunctions of the harmonic oscillator potential: $$\begin{aligned}
\psi_{0}(z)&=a_{\mathrm{ho}}^{1/2}\pi^{-1/4}e^{-z^{2}/2a_{\mathrm{ho}}^2}\,,\label{eq:EF0}\\
\psi_{1}(z)&=a_{\mathrm{ho}}^{1/2}\sqrt{2}\,\pi^{-1/4}(z/a_{\mathrm{ho}})e^{-z^{2}/2a_{\mathrm{ho}}^2}\,.\label{eq:EF1}\end{aligned}$$ The energy of an atom associated to the wavefunction $\psi_{\ell}(z-z_{j})$ is $E_{l}\approx\hbar\omega(\ell+1/2)$. This approximation fails when the overlap between different on-well localized eigenfunctions is small. Then, in general, it gives a poor approximation of the $J_{\ell}$ coefficients given in (\[eq:J\_l\]). Nevertheless, they can be used to obtain a good approximation for the overlap integrals purely on-well, like those defining the $U_{\ell\ell'}$ coefficients, Eq. (\[eq:U\_l\_l’\]). This is valid for both regimes, though it gives only a rough idea of these coefficients when the actual single particle eigenfunctions are highly distorted, as occurs in the Josephson regime. In this case, it can only be used to obtain scaling relations in the problem. By solving these integrals analytically using Eq. (\[eq:EF1\]) to approximate the single particle eigenfunctions, we obtain $$\label{eq:U0}
U_{0}=\hbar\omega\frac{g_1}{\sqrt{2\pi}}\left(\frac{a_{\mathrm{ho}}}{a}\right)\,.$$ Moreover, we can relate the other interaction coefficients to $U_0$ by solving the corresponding integrals, which gives $U_{1}=(1/2)U_{0}$ and $U_{01}=(3/4)U_{0}$. Here we use these expressions in the illustrative examples presented in Sec. \[sec:bounds\], though for actual potentials to obtain accurate values for these coefficients, it is necessary to calculate the single particle eigenfunctions numerically to solve the integrals (\[eq:U\_l\_l’\]).
Finally, to deduce the Hamiltonian (\[eq:two-level\]) we neglected off-site interactions. These correspond to interaction terms like: $$U_{\ell \ell'}^{j
j'}=\frac{g_{1}}{2}\int dz |\psi_{\ell}(z-z_j)|^{2}|\psi_{\ell'}(z-z_{j'})|^{2}\,,\label{eq:U_llp}$$ where $j\ne j'$. If the barrier is infinitely high these integrals vanish. If the single particle eigenfunctions are approximated by Eq. (\[eq:EF1\]) this integral gives $U_{0}\exp[-(a/a_{\mathrm{ho}})^2]$. Then, one could assume that these terms can be neglected for sufficiently high or separated wells, that is, if $ a_{\mathrm{ho}}\ll a$. Nevertheless, these are not on-well integrals and then the expressions of the harmonic oscillator eigenfunctions, Eq. (\[eq:EF1\]) do not give accurate results. Then these integrals have to be evaluated numerically for the actual potential and for the correct numerically evaluated single particle eigenfunctions. Moreover, these coefficients have to be compared with $J_{\ell}$ to justify their being neglected, since both quantities could be comparable in the Fock regime. In the Josephson regime we can safely neglect these coefficients since they are much smaller than any other interaction coefficient, which in turn are smaller than the tunneling ones. For the Fock regime, we assume that the tunneling coefficients are always bigger that these cross-terms, and have verified that our assumption holds for a number of specific cases. We refer the reader to Ref. [@1999SpekkensPRA] for a thorough discussion of the effect of these terms.
Characterization of the Eigenstates {#sec:characterization}
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General Considerations
----------------------
An arbitrary state vector $|\Psi\rangle$, in the solution space of the two-level Hamiltonian Eq. (\[eq:two-level\]), can be expressed in Fock space as $$|\Psi\rangle=\sum_{i=0}^{\Omega-1}c_{i}|i\rangle,\quad|i\rangle=\bigotimes_{j,\ell}|n_{j\ell}^{(i)}\rangle\,,\label{eq:PsiFock}$$ where $$|n_{j\ell}^{(i)}\rangle=\frac{1}{\sqrt{n_{j\ell}!}}\left(\hat{b}_{j\ell}^{\dagger}\right)^{n_{j\ell}^{(i)}}|0\rangle\,.$$ Here $\Omega$ is the dimension of the Hilbert space $\{|i\rangle\}$, $i$ is the *Fock index*, and $|c_{i}|^{2}$ is the probability of finding the atoms distributed between different energy levels and wells in the Fock vector $|i\rangle$, when the system is described by state $|\Psi\rangle$, that is $c_i=\langle i|\Psi\rangle$; henceforth we refer to $c_i$ as the *Fock-space amplitude*. The Fock vector $|i\rangle$ can also be written in longer form as $$|i\rangle = |n_{L0}^{(i)},n_{R0}^{(i)}\rangle|n_{L1}^{(i)},n_{R1}^{(i)}\rangle\,,$$ indicating the occupation of each well (L,R) and each level (0,1) associated with Fock index $i$. We work in the canonical ensemble, i.e., we require the total number of particles $$N=\sum_{j\ell}n_{j\ell}\,,\label{eq:Ntotal}$$ to be constant.
The Fock index $i$ is chosen to increase starting with the number of atoms in well $j=L$, and then subsequently increasing with the number of atoms in the same, left well moving up into the excited level $\ell=1$. Therefore, for the first $N+1$ Fock vectors $i=1+n_{L0}$ and they correspond with vectors with no occupation of the excited level. Then, they satisfy $$|i\rangle=\frac{1}{\sqrt{n_{L0}!n_{R0}!}}\!\left(\hat{b}_{L0}^{\dagger}\right)^{n_{L0}}
\!\left(\hat{b}_{R0}^{\dagger}\right)^{n_{R0}}|0\rangle\,,$$ for $i=0,\;1,\;\cdots,\; N$. For example, for $N=2$, the Fock vectors with index $i=1,2,3$ are $|0,2\rangle|0,0\rangle$, $|1,1\rangle|0,0\rangle$, and $|2,0\rangle|0,0\rangle$, respectively. The one-level approximation can easily be recovered from Eq. (\[eq:two-level\]) by requiring $i\leq N+1$. In this truncated space, the dimension of the Hilbert space reduces to that of the one-level approximation, namely, $N+1$, and the two-level Hamiltonian $\hat{H}$ reduces to the one-level Hamiltonian $\hat{H}_{0}$.
A general expression for the Fock index is: $$\begin{aligned}
i & = 1+n_{L0}+\sum_{p=-1}^{N_{1}-1}(N+1-p)(1+p)\nonumber \\ & +(N+1-N_{1})n_{L1}\,,
\label{Eq:Fockindexordering}\end{aligned}$$ where $N_{1}$ is the number of atoms in the excited level. Then, the following Fock vectors for $N=2$ are $|0,1\rangle|0,1\rangle$, $|1,0\rangle|0,1\rangle$, $|0,1\rangle|1,0\rangle$, and $|1,0\rangle|1,0\rangle$ with indices $i=4,5,6,7$. The last three show occupation only of the excited level and they are $|0,0\rangle|0,2\rangle$, $|0,0\rangle|1,1\rangle$, and $|0,0\rangle|2,0\rangle$, for $i = 8, 9, 10$. Thus for just two atoms there are already 10 states in our two-level problem, and with increasing $N$ it is necessary to use numerical matrix methods to keep track.
The eigenstates $|\phi^{(k)}\rangle$ of the two-level Hamiltonian (\[eq:two-level\]) satisfy: $$\hat{H}|\phi^{(k)}\rangle=\varepsilon^{(k)}|\phi^{(k)}\rangle\,,$$ where $\varepsilon^{(k)}$ is the energy eigenvalue corresponding to the state $|\phi^{(k)}\rangle$. The eigenstate label $k$ is chosen to increase with $\varepsilon^{(k)}$. In order to describe these states, we will use the previously introduced Fock-space amplitudes $$c_{i}^{(k)}=\left\langle i\right.|\phi^{(k)}\rangle\,,$$ now containing two separate indices for clarity: $i$ is the Fock index, describing the ordering of the Fock basis; $k$ is the energy index, describing the ordering of the energy eigenvalues.
When interlevel effects are not relevant, the eigenstates fall into one of two categories: *harmonic-oscillator-like states*, and *macroscopic superposition states* (MS states). When the barrier between wells is low, $J_{0}\gg NU_{0}$, all states are harmonic oscillator-like (HO) states. This is the Josephson regime and it is characterized by $$\xi_{J_{\ell}}=N/\zeta_{\ell}\ll 1\,,\label{eq:CA}$$ where $
\zeta_{\ell}=J_{\ell}/U_{\ell}$. This criterion must be evaluated separately for both levels, since $J_1>J_0$, while the interaction terms are of the same order.
On the other hand, MS states dominate the spectrum in the high barrier limit. This is the Fock regime, in which $$\xi_{U_{\ell}}=\zeta_{\ell}\ll 1\,.\label{eq:CB}$$ Again, this criterion must be evaluated separately for the two levels.
When the level spacing is comparable to $NU_{0}$, interlevel effects can no longer be neglected, and a third category of eigenstates emerges. These states show weak coupling between states with particles only in the lowest energy level and others with particles in the excited one. We name these coupled excited states *shadows of the MS states*, because in a surface plot of the Fock-space amplitudes as a function of both energy and Fock index, they appear as faint copies of the MS states with occupation only of the lower level at higher Fock index [@2011GarciaMarchPRA]. In the following two subsections we characterize the eigenstates in the Josephson and Fock regimes.
Non-interacting Regime {#subsec:HO}
----------------------
Let us consider the extreme Josephson regime, or noninteracting regime, for which $U_{\ell}=0$. The Hamiltonian reduces to $\hat{H}=\hat{H}_{0}+\hat{H}_{1}$, with $$\hat{H}_{\ell}=-J_{\ell}\sum_{j\ne j'}\hat{b}_{j}^{\ell\dagger}\hat{b}_{j'}^{\ell}+E_{\ell}\sum_j\hat{n}_{j\ell}\,.
\label{eq:HOHam}$$ Since the Hamiltonian is clearly separable, its eigenstates $|\phi^{(k)}\rangle$ are a direct product of the one-level ones, $|\phi^{(K_{\ell}^{(k)})}\rangle$: $$\label{eq:HO_states}
|\phi^{(k)}\rangle = \bigotimes_{\ell}|\phi^{(K_{\ell}^{(k)})}\rangle\,.$$ where $K_{\ell}^{(k)}=0,1,\dots,N_{\ell}^{(k)}$ is the one-level eigenstate label and for the $k$th eigenstate there are $N_{\ell}^{(k)}$ atoms at level $\ell$. Since the total number of atoms is conserved, $\sum_{\ell=0,1}N_{\ell}^{(k)}=N$. The one-level eigenstates can be expressed as $|\phi^{(K_{\ell}^{(k)})}\rangle=\sum
c_{i\ell}^{(k)} |n_{L\ell}^{(k)},n_{R\ell}^{(k)}\rangle $ with $$\label{eq:cilmlcoeff}
c_{i\ell}^{(k)} \!\!= \! a\!_{K_{\ell}^{(k)}}\!\!\left(\!N_{\ell}^{(k)}\!\right)\!
h\!_{K_{\ell}^{(k)}}\!\!\left(\!n_{L\ell}^{(k)}\!\left|N_{\ell}^{(k)}\right.\!\right)\!
p\!\left(\!n_{L\ell}^{(k)}\!\left|N_{\ell}^{(k)}\right.\!\right)\!,$$ where $n_{L\ell}^{(k)}$ is the occupation of the left well for the corresponding Fock vector, $ h_{K_{\ell}^{(k)}}(n_{L\ell}^{(k)}|N_{\ell}^{(k)})$ is a $K_{\ell}^{(k)}$th order discrete Hermite polynomial, $p(n_{L\ell}^{(k)}|N_{\ell}^{(k)})$ is the square root of the binomial distribution, $$p\left(\!n_{L\ell}^{(k)}\!\left|N_{\ell}^{(k)}\right.\!\right)=
\frac{1}{2^{N_{\ell}^{(k)}/2}}\sqrt{\frac{N_{\ell}^{(k)}!}{n_{L\ell}!(N_{\ell}^{(k)}-n_{L\ell}^{(k)})!}}$$ and $a_{K_{\ell}^{(k)}}(N_{\ell}^{(k)})$ is the normalization factor $$a_{K_{\ell}^{(k)}}(N_{\ell}^{(k)}) = \sqrt{\frac{(N_{\ell}^{(k)}-K_{\ell}^{(k)})!}{N_{\ell}^{(k)}!\,K_{\ell}^{(k)}!}}\,.$$ The eigenenergies $\varepsilon^{(k)}$ can be expressed in terms of the one-level eigenenergies as $$\label{eq:eigenval}
\varepsilon^{(k)} = \sum_{\ell}\varepsilon_{(k) \ell}\,,$$ where $$\label{eq:eigenv_OSC}
\varepsilon_{(k), \ell } = -J_{\ell }\left(N_{\ell}^{(k)}-2K_{\ell}^{(k)}\right) + E_{\ell} N_{\ell}^{(k)}\,.$$ The one-level Hamiltonians, expressed in the Fock basis is, in the non-interacting limit, a harmonic oscillator potential truncated at hard walls. The analytical expressions of the Fock-space amplitudes $ c_{i\ell}^{(k)} $ resemble the probability amplitudes of observing the particle in position $x$ for the 1D harmonic oscillator problem, obtained from the expression of its eigenfunctions in the position representation. Moreover, in both cases the eigenvalues $\varepsilon_{k\ell }$ are linear in $K_{\ell}^{(k)}$. The analogy can be established between occupation of the wells in the double well and the positions in the Harmonic oscillator problem. Since one variable is discrete while the other is continuous the analogy is valid for the limit of infinite particles. Let us show this for the one level approach, which we obtain considering $\hat{H}=\hat{H}_0$, where $\hat{H}_0$ is given by Eq. (\[eq:HOHam\]). In the Heisenberg picture the evolution of the operators $\hat{b}_{j 0}$ for $j=L,R$ is given by $$i\frac{d \hat{b}_{j 0}}{d t}=[\hat{b}_{j 0},\hat{H}]=-J \hat{b}_{j' 0}\,,
\label{eq:HeisPict}$$ with $j'\neq j$, and where we considered $E_0=0$ for simplicity. In the following we omit the level index $\ell$ because we are considering only the one level approximation. For $N\rightarrow \infty$, we can approach the operators by $c$-numbers $\alpha_j=\sqrt{N_j}e^{i\theta_j}$. Substituting we obtain the two corresponding equations of motion which in turn can be obtained from the pendulum Hamiltonian $$\mathcal{H}=-2J\sqrt{N_L\,N_{R}}\cos(\theta_L-\theta_{R})=-J\,N\sqrt{1-z^2}\cos(\theta)\,,$$ where $N=N_L+N_R$, $ z=(N_L-N_R)/N$, $\theta=\theta_L-\theta_{R}$. The previous approach is widely known to be extended to the small interacting regime, where it gives a Hamiltonian analogous to that of a non-rigid pendulum. The study of the dynamics of such a Hamiltonian permits one to predict macroscopic quantum tunneling and self-trapping within the semiclassical approach [@1986JavanainenPRL; @1997MilburnPRA; @1997SmerziPRL]. In our case, this can be extended to a harmonic oscillator Hamiltonian, where the angular variables can take any value. The analogy here is with the harmonic oscillator truncated at hard walls, because the number of particles is finite. Due to this analogy, the eigenstates (\[eq:HO\_states\]) are said to be harmonic-oscillator-like. In Sec. \[sec:bounds\] we obtain numerically the eigenvectors and eigenvalues for small interactions, showing that for non-zero interactions the eigenvectors and eigenvalues closely resemble the ones characterized for the non-interacting regime.
Fock Regime {#sec:MS}
-----------
In the extreme Fock regime, or the infinite-barrier limit, $J_0=J_1=0$. In this regime, the coefficient $ U_{01 } $ cannot be neglected. Then, the eigenvectors of Hamiltonian (\[eq:two-level\]) are Fock vectors, if we further assume that $ 2\triangle E\gg N^2 U_0$. The latter assumption is needed because the term $\hat{b}_{jl}^{\dagger}\hat{b}_{jl}^{\dagger}\hat{b}_{jl'}\hat{b}_{jl'}$ couples Fock vectors with atoms in different levels. As demonstrated in Sec. \[sec:bounds\], this coupling is small under that assumption. In this limit, the eigenvalues are $$\begin{aligned}
\label{eq:eigenHB}
\varepsilon^{(k)}&= \sum_{\ell}\Bigg\{E_{\ell} N_{\ell }^{(k)}+U_{\ell }\left[2\left(n_{L\ell }^{(k)}-\frac{N_{\ell
}^{(k)}}{2}\right)^{2}\right.\nonumber\\ &+\left.N_{\ell }^{(k)}\left(\frac{N_{\ell }^{(k)}}{2}-1\right)\right]\Bigg\}+U_{01}\sum_{j\ell}2\,n_{j\ell
}^{(k)}n_{j\ell' }^{(k)}\,. \end{aligned}$$
Here, the spectra reflects the main symmetry in the problem: that of discrete rotations of an angle $\pi$ around an axis perpendicular to the $x-y$ plane that intersects this plane in the origin. This in turn means that the eigenstates, which are the Fock vectors $ |n_{L0},n_{R0}\rangle|n_{L1},n_{R1}\rangle$, are degenerate in pairs with those obtained for $L \to R$. Small but non-negligible hopping terms, $J_0 \ll U_0$ and $J_1 \ll U_0$, break this degeneracy, as can be shown by non-degenerate high-order perturbation theory. Then, in the small tunneling limit, that is, in the Fock regime, the eigenvectors are quasi-degenerate symmetric and antisymmetric superpositions of those pairs of Fock vectors that are degenerate in the infinite barrier case. For the particular cases in which all atoms occupy the same level the eigenstates are direct products of one-level vectors $|\phi_{\ell}^{\pm}\rangle|0,0\rangle_{\ell'} $ with $$\begin{aligned}
& |\phi_{\ell}^{\pm}\rangle
\equiv\frac{e^{i\theta_0}}{\sqrt{2}}\left[\frac{1}{\sqrt{\nu!(N-\nu)!}}
\left({\hat{b}_{L \ell }^{\dagger}}\right)^{\nu}
\left({\hat{b}_{R \ell }^{\dagger}}\right)^{N-\nu}\right.\nonumber\\
& \pm
\left.\frac{1}{\sqrt{\nu!(N-\nu)!}}
\left({\hat{b}_{L \ell }^{\dagger}}\right)^{N-\nu}
\left({\hat{b}_{R \ell }^{\dagger}}\right)^{\nu}
\right]|0\rangle\,,\end{aligned}$$ for $0\leq \nu < N/2$. These are MS states in which $\nu$ and $N-\nu$ atoms simultaneously occupy the $\ell$th energy level of both wells; although the expression might appear complicated, in fact it is just a two-state approximation. We have neglected terms on the order of $(J_{\ell}/U_{\ell})^{N-2\nu}$ and smaller. Here $ \theta_0$ is the usual arbitrary phase associated with vectors in a Hilbert space. We will set $ \theta_0=0$ for the rest of this Article.
The special case $\nu=0$ represents an extreme MS state, or *NOON state*, in which all $N$ atoms simultaneously occupy the left and right wells, which can be written also as $ (1/\sqrt{2})\left(|N,0\rangle|0,0\rangle\pm |0,N\rangle|0,0\rangle\right)$. These symmetric $(+)$ or antisymmetric $(-)$ MS states are nearly degenerate, with a splitting $\Delta\varepsilon_{\ell}(\nu)$ between the states $|\phi_{\ell}^{+}\rangle$ and $|\phi_{\ell}^{-}\rangle$ given by
$$\Delta\varepsilon_{\ell}(\nu) =
\frac{4U_{\ell }[J_{\ell }/(2U_{\ell})]^{N-2\nu}(N-\nu)!}
{\nu![(N-2\nu-1)!]^2}\,,$$
up to $(N-2\nu)$th order in $J_{\ell}/U_{\ell}$. General MS states are symmetric/antisymmetric superpositions of Fock vectors with atoms in both levels, that is, $|\nu,N_{0}-\nu\rangle|\nu',N_{1}-\nu'\rangle $ and $|N_{0}-\nu,\nu\rangle|N_{1}-\nu',\nu'\rangle $, with splittings that are proportional to $(J_0/U_0)^{N_0-2\nu} $ $(J_1/U_1)^{N_1-2\nu'}$.
Since $J_1>J_0$, it is also possible that $J_0 \ll U_0$ but $J_1 > U_0$; this is the mixed regime. The atoms in the lower level behave as in the Josephson regime while the ones in the excited level behave as in the Fock regime. We will show numerical examples of the mixed regime in the following section.
Characterization of the Bounds of the Model, Regimes, and Crossings {#sec:bounds}
===================================================================
Limits of the Two-level Model {#sec:limits}
-----------------------------
Let us find criteria in terms of the characteristic parameters of the problem to describe the limits of all regimes, the occurrence of shadows of the MS states, crossings, and the limits of applicability of the Hamiltonian, Eq. (\[eq:two-level\]). We begin by seeking criteria of validity for the model. On the one hand, the consideration of quantum tunneling associated with a second level of energies is meaningful if $V_0>E_1$, as evident from Fig. \[fig0\](a). If the eigenfunctions are approximated by the eigenfunctions of the harmonic oscillator, then this criterion can be approximated by $V_0> (3/2)\hbar\omega$. Secondly, the diluteness condition given by Eq. (\[eq:diluteness\]) can be written in three dimensions as $N^{1/3}U_0^{3D}/\Delta E\ll 1$ [@2011GarciaMarchPRA]. In 1D, this becomes $$\frac{\omega N^{1/3}U_0^{3D}}{\omega_{\perp}\Delta E} \ll 1\,,$$ which, contrary to the 3D case, accounts also for the geometrical compression in one of the dimensions through the term $\omega/\omega_{\perp}$. Here, it is also important to consider the restrictions on $\omega_{\perp}$ discussed at the beginning of Sec. \[sec:1DLMG\]. Finally, a last limit of the model is obtained when the coupling between levels is big enough to require more levels to characterize the eigenvectors. We obtain below the criterion delimiting when this coupling is too big. If this criterion is not fulfilled, more levels are required in the approximation or one should use the MCTDH method [@2005MasielloPRA; @2006StreltsovPRA; @2008AlonPRA], depending on numerical efficiency requirements and spatial dimensionality. Other methods such as matrix product state algorithms, for example, time-evolving block decimation, can also of utility in this regime [@2011schollwoeckAP].
Discussion of the Different Regimes {#sec:shadows}
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Fock index $i$ $N_0$ $N_1$
------------------ ------- -------
1 to $N+1$ $N$ 0
$N+2$ to $3 N+2$ $N-1$ 1
$3 N+3$ to
$3 N+3+3(N-1)$ $N-2$ 2
: Fock indices for the the Fock vectors up to the $(3 N+3+3(N-1))$-th resulting from the ordering given in Eq. (\[Eq:Fockindexordering\]). These include all combinations with up to two particles excited.[]{data-label="tab"}
The Josephson and Fock regimes are characterized by the criteria discussed above, Eqs. (\[eq:CA\]) and (\[eq:CB\]). Next we will show some numerical examples of the direct diagonalization of the Hamiltonian (\[eq:two-level\]) for different sets of parameters, corresponding to different regimes. In Fig. \[fig1\] we show an example of the Josephson regime, $U_1=U_0/2$, $U_{01}=(3/4) U_0$, $\zeta_0=10^2$, $J_1=5J_0$, $\Delta E =
\sigma N U_0$, with $N=10$ atoms and $\sigma=2\cdot10^3$. In Fig. \[fig1\](a) we show all the Fock-space amplitudes $|c_i|^2$ for all the eigenvectors, while Fig. \[fig1\](b) shows a zoom of these for the lowest excited $3 N+3(N-1)+1$ states. The first $N+1$ are eigenstates with no coupling to Fock vectors with atoms in the excited level. Then, the following $2N$ are eigenvectors where the Fock vectors show $N-1$ atoms in the lower level and one atom in the excited level. Finally, the following $3(N-1)$ show two atoms excited and $N-2$ in the lower level of eigenenergies (see table \[tab\]). In Fig. \[fig1\](c) we show the Fock-space amplitudes $|c_i|^2$ for the ground state, $(N+1)$th, and $(N+2)$th excited eigenstates. These two figures show the HO behavior discussed in Sec. \[subsec:HO\]. In Fig. \[fig1\](d) we represent all the eigenvalues while in the inset in this figure we represent a zoom for the lowest excited $N+1$ eigenvectors, where the linear behavior described by Eq. (\[eq:eigenv\_OSC\]) is shown. Every step in the values of the eigenvalues in Fig. \[fig1\](d) occurs when the corresponding eigenstate is the superposition of Fock vectors with one more atom occupying the second level of energies. The increase in the slope in every step is due to the fact that $J_1>J_0$. Indeed, for the last $N+1$ eigenvectors, for which all atoms are excited to the higher level, the slope is $J_1$.
If we reduce $\zeta_0$ to one, we cannot assume that the parameters correspond clearly to the Fock or to the Josephson regime, as represented in Fig. \[fig2\](a). The Fock-space amplitudes $|c_i|^2$ of the $N+1$ lowest excited eigenvectors show different behavior for the ground state and for the $(N+1)$th excited state, as represented in Fig. \[fig2\](b) and (c). The ground state is still a HO eigenstate, while the $(N+1)$th excitation is a NOON-like state, as corresponding to the Fock regime. Fig. \[fig2\](d) shows a different behavior than in the previous case. Particularly, as shown in the inset in Fig. \[fig2\](d), the eigenvalues behave linearly for the five lowest excited eigenvectors. Then, the next six appear in quasidegenerate pairs with small splittings, as corresponds for the Fock regime. Notice that, since $J_1> J_0$, the less excited $N+1$ eigenvectors behave as HO eigenstates. Then in this mixed regime, states with atoms occupying only the lower level can behave as in the Fock regime, while states with atoms occupying the excited level behave as in the Josephson one.
If we decrease $\zeta$ further, to $\zeta=10^{-2}$, we enter clearly in the Fock regime, as represented in Fig. \[fig3\]. Now, the eigenstates are MS states, as shown in Fig. \[fig3\](b) and (c), while the ground state is sharply peaked in $
|N/2,N/2\rangle |0 ,0\rangle $. Also, Fig. \[fig3\](d) shows that the MS states appear in quasidegenerate symmetric/antisymmetric pairs, with small splittings in the eigenvalues (see also inset in this figure). Notice that, when atoms are excited, the presence of the term $2 U_{01}\sum_{j, \ell\neq\ell'} \hat{n}_{j \ell}\hat{n}_{j
\ell'}$ breaks the degeneracy in the $J_{\ell}=0$ case between the vectors $ |N_0-\nu,\nu\rangle |N_1-\nu' ,\nu'\rangle $ and $ |N_0-\nu,\nu\rangle
|\nu',N_1-\nu' \rangle $. For small $J_{\ell}$, this in turn makes the MS states with $N_1\ne0$ appear also in pairs, and not as the superposition of four Fock vectors. These pairs can be observed in Fig. \[fig3\](b).
Finally, the last regime is characterized by the presence of shadows of the MS states. Using perturbation theory we can show that the MS states $
|n_{L,0},n_{R,0}\rangle|0,0\rangle\pm|n_{R,0},n_{L,0}\rangle|0,0\rangle $ couple to MS states showing two excited atoms $|n_{L,0}-2,n_{R,0}\rangle|2,0\rangle\pm|n_{R,0},n_{L,0}-2\rangle|0,2\rangle $ and $|n_{L,0},n_{R,0}-2\rangle|0,2\rangle\pm|n_{R,0}-2,n_{L,0}\rangle|2,0\rangle $ with coefficients $$\begin{aligned}
c_L & = U_{01}\frac{\sqrt{2 n_{L,0}(n_{L,0}-1)}}{U_0(6-4 n_{L,0})+2U_1+2\Delta E}\,,\nonumber\\ c_R & = U_{01}\frac{\sqrt{2
n_{R,0}(n_{R,0}-1)}}{U_0(6-4 n_{R,0})+2U_1+2\Delta E}\,. \end{aligned}$$ For the most excited MS state, $n_{L,0}=N$, and $$\label{eq:shad}
c_{N} = U_{01}\frac{\sqrt{2 N(N-1)}}{U_0(6-4 N)+2U_1+2\Delta E}\,.$$ If this coupling is not negligible, shadows of the MS states are coupled, and the upper level cannot be neglected, as discussed below.
Crossings of the Eigenvalues
----------------------------
Let us show, for both regimes, bounds on the one- and two-level approximations other than criterion (\[eq:shad\]). With our choice of indexing states, the one-level approximation corresponds to truncating the size of the Hilbert space to $N+1$. With this definition of the one-level approximation, the bounds we present below will describe the regime in which this truncation is valid.
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Let us consider first the crossings in the Josephson regime. The energy levels are coupled by the interaction energy $U_{01}$. Energy levels only become completely decoupled when $U_{01}=0$. In this regime $U_{01}$ is small, and then the coupling between levels is weak. Criterion (\[eq:shad\]) gives a good estimate of this coupling. Let us show that, although the energy levels are weakly coupled, eigenvalue crossings are induced by the presence of the excited level.
We consider first $U_{0}=0$, $U_{1}=0$ and $U_{01}=0$. We find that the maximum of the eigenvalues for the first $N+1$ eigenstates with no occupation of the excited level coincides with the minimum of the eigenvalues of the states with one atom in the excited level if $$\chi_{\mathrm{Jos}}=\frac{\triangle E}{J_{0}\,(2N-1)+J_{1}}=1\,.\label{eq:CC}$$ Equation (\[eq:CC\]), determines the first eigenvalue crossing in this regime. For $\chi_{\mathrm{Jos}}>1$ no crossing occurs. In Fig. \[fig4\] we show an example where all the parameters are the same as in Fig. \[fig1\], that is $\zeta_0=10^2$, but with a smaller gap between levels, since $\sigma=235$. As shown in Fig. \[fig4\](b) and (c) the $(N+1)$th eigenvector shows occupation of the excited level. Notice that for small interactions, the presence of the excited level can induce this crossing. Fig. \[fig4\](d) shows that the steps in the eigenvalues due to the excited atoms observed in Fig. \[fig1\](d) are now absent, since the excited atoms have now a much smaller energy. On the other hand, the inset in Fig. \[fig1\](d) shows that the behavior of the eigenvalues is still linear for all the eigenstates which are formed by superpositions of Fock vectors with the same number of excited atoms. For $$\chi_{\mathrm{Jos,gs}}=\frac{\triangle E}{J_{1}-J_{0}}=1\,,\label{eq:CD}$$ the first crossing involving the ground state occurs, i.e., the ground state is a state with non-zero occupation of the excited level if $\chi_{\mathrm{Jos,gs}}<1$. This is, indeed, a limiting criterion for the validity of the model.
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![(Color online) [*Crossings as $\zeta$ is decreased.*]{} Shown are the eigenvalues of the $N+1$ eigenvectors with no occupation of the excited level (solid line) and the $2(N-1)$ eigenvalues of the eigenvectors showing only one excited atom. The ground state is non-degenerate, while the rest appear in quasidegenerate pairs. The dash-dotted line corresponds to the critical value of $\zeta$ at which the first crossing occurs, as calculated with (\[eq:CE\]). As $\zeta_0$ is decreased, the eigenstates change from the Josephson regime, where they have are HO states, to the Fock regime, where they are MS states. If $\zeta$ is decreased further, crossings appear, and eventually, shadows of the MS states. \[fig7\]](fig7.pdf){width="\columnwidth"}
Analogously, in the Fock regime, the first eigenvalue crossings occurs when the condition $$\chi_{\mathrm{Fock}}=\frac{\Delta E}{U_0(N^{2}/2+N-2)-2U_{01}(N/2-1)}\label{eq:CE}$$ is met. This condition is obtained equating the maximum eigenvalue for states with no occupation of the excited level, to the minimum eigenvalue for states with one particle in the excited level. For $\chi_{\mathrm{Fock}}>1$ no crossing takes place. In Fig. \[fig5\] we show an example for which all the parameters are the same as in Fig. \[fig3\], corresponding to the Fock regime, but now $\sigma=5.1$. Here, the crossing appeared in the Fock regime, which can be due to the reduction of the gap between levels or to the increasing of the interactions. Figure \[fig4\](d) and its inset show that the eigenvectors still appear as quasidegenerate symmetric/antisymmetric MS states, as discussed in Sec. \[sec:MS\].
If we increase further the interactions, or reduce the energy gap, shadows of the MS states appear, as discussed at the end of Sec. \[sec:shadows\]. For example, if we reduce the $\Delta E$ by taking $\sigma=2.5$, the coupling between the MS states with no excited atoms and MS states with excited atoms is not negligible, as shown in Fig. \[fig6\](b) and (c).
Finally, the first crossing involving the ground state occurs when the condition $$\chi_{\mathrm{Fock,gs}}=\frac{\Delta E}{U_0(N-2)-2U_{01}(N/2-1)}\label{eq:CF}$$ is satisfied. For $\chi_{\mathrm{Fock,gs}}>1$ the ground state shows non-zero occupation of the excited level. This criterion should be also considered as a limit for the validity of the model. As discussed in Sec. \[sec:limits\], when this limit is reached one must consider more energy levels or an MCTDH [@2005MasielloPRA; @2006StreltsovPRA; @2008AlonPRA] or other many-body method [@2011schollwoeckAP].
In Fig. \[fig7\], the first $N+1+2(N-1)$ eigenvalues are represented when $\zeta$ is reduced, for $\sigma=100$. The eigenvectors behave as HO-like states for certain region, then as MS states, and finally, the most excited eigenstate with occupation only of the lower level crosses the least excited state with one atom excited when criterion (\[eq:CE\]) is reached. If $\zeta$ is increased further, shadows of the MS states appear, and eventually, the limit stated by criterion (\[eq:CF\]) is reached.
Therefore, the bounds to the one level approximation are given by criteria (\[eq:CC\]) and (\[eq:CE\]). We are interested in describing cat-like states which are typically excited eigenstates. Therefore, a characterization of eigenvalue crossings of energies other than the ground state are also relevant. These crossings appear when criteria (\[eq:CD\]) and (\[eq:CF\]) are satisfied.
Three-dimensional Double Well {#sec:3D}
=============================
The three dimensional (3D) double well was extensively studied in [@2011GarciaMarchPRA]. In this case $\omega \sim \omega_{\perp}$. The field operators in Hamiltonian (\[eq:second-quantized\]) can be expanded in a fixed well localized single-particle basis obtained from the delocalized eigenfunctions of the single particle Hamiltonian $H_{\mathrm{sp}}=-\frac{\hbar^2}{2\mathcal{M}}\nabla^2 + V({\mathrm{\bf x}})$. These eigenfunctions are distortions of the eigenfunctions of the harmonic oscillator potential $V({\mathrm{\bf x}})=\frac{1}{2}\omega^2{\mathrm{\bf x}}^2$, given by $$\label{eq:psi_nlm}
\psi_{n\ell m}({\mathrm{\bf x}}) \approx R_{n\ell}(r)Y_{\ell m}(\theta,\phi)\,,$$ with $n \in \{0,1,2,\hdots\}$, $\ell \in \{n,n-2,n-4,\hdots,\ell_{\min}\}$, and $m \in \{-\ell,\;-\ell+1,\;\hdots,\;\ell-1,\;\ell\}$. Here $R_{n\ell}(r)$ is the radial part of the wavefunction, $Y_{\ell m}(\theta,\phi)$ are the familiar spherical harmonics, and $\ell_{\min}$ is 0 when $n$ is even and 1 when $n$ is odd. Then, $n$ is the single-particle energy level, $\ell$ is the orbital angular momentum in 3D, and $m$ is its $z$-projection. For the two level approximation $n=\ell$, and therefore the index $n$ is superfluous and hereafter suppressed. These functions can be explicitly written in spherical coordinates for $\ell=0,1$ as $$\begin{aligned}
\psi_{00}(r,\theta,\varphi)&=a_{\mathrm{ho}}^{-3/2}\pi^{-3/4}e^{-r^{2}/2a_{\mathrm{ho}}^2}\,,\\
\psi_{10}(r,\theta,\varphi)&=a_{\mathrm{ho}}^{-3/2}\sqrt{2}\pi^{-3/4}(r/a_{\mathrm{ho}})e^{-r^{2}/2a_{\mathrm{ho}}^2}\cos(\theta)\,,\\
\psi_{1\pm1}(r,\theta,\varphi)&=a_{\mathrm{ho}}^{-3/2}\pi^{-3/4}(r/a_{\mathrm{ho}})e^{-r^{2}/2a_{\mathrm{ho}}^2}\sin(\theta)e^{\pm i\varphi}\,. \end{aligned}$$ The energy of an atom associated with the wavefunction $\psi_{\ell m}({\mathrm{\bf x}}-{\mathrm{\bf x}}_j)$ is $$\label{eq:E_nlm3}
E_{n} \approx \hbar\omega(n+3/2)\,.$$ In general, the spherical harmonics cannot be accurately used as a basis. In particular, they give poor approximations of the overlap between functions localized in different wells. Then, the localized functions at well $j$ can be obtained numerically as $$\begin{aligned}
\psi_{j00}({\mathrm{\bf x}})&=\phi(x)^0\phi(y)^0\psi_{j0}(z)\,,\\
\psi_{j10}({\mathrm{\bf x}})&=\phi(x)^0\phi(y)^0\psi_{j1}(z)\,,\\
\psi_{j,1\pm1}(r,\theta,\varphi)({\mathrm{\bf x}})&= \frac{1}{\sqrt{2}}\left[\phi^{1}(x)\phi^{0}(y)\psi_{j0}(z)\right.\\
&\pm \left. i\phi^{0}(x)\phi^{1}(y)\psi_{j0}(z)\right]\,, \end{aligned}$$ where $\phi^{\ell}(x)$ and $\phi^{\ell}(y)$ are two lowest excited eigenfunctions of the harmonic oscillator in the $x$ and $y$ directions, and $\psi_{j\ell}(z)$ are the on-well localized eigenfunctions of the double well given in (\[eq:sup1\]) and (\[eq:sup2\]), respectively. The four modes localized in one of the wells are represented schematically in Fig. \[fig0\](b).
Then, we can expand the field operator in (\[eq:second-quantized1\]) in terms of this eight mode basis as $$\label{eq:hatPsi3}
\hat{\Psi}({\mathrm{\bf x}})
= \sum_{j,\ell,m}{\hat{b}_{j \ell m }^{}}\psi_{\ell m}({\mathrm{\bf x}}-{\mathrm{\bf x}}_j)\,,$$ where ${\mathrm{\bf x}}_{1} \equiv -{\mathrm{\bf x}}_{\mathrm{min}}$ and ${\mathrm{\bf x}}_2 \equiv {\mathrm{\bf x}}_{\mathrm{min}}$ are the minima of the left and right wells. The operators ${\hat{b}_{j \ell m }^{\dagger}}$ and ${\hat{b}_{j \ell m }^{}}$ satisfy the usual bosonic annihilation and creation commutation relations, $$\begin{aligned}
[{\hat{b}_{j \ell m }^{}},\;{\hat{b}_{j' \ell' m' }^{\dagger}}]&=\delta_{jj'}\delta_{\ell\ell'}\delta_{mm'}\,,\nonumber\\
[{\hat{b}_{j \ell m }^{\dagger}},\;{\hat{b}_{j' \ell' m' }^{\dagger}}]&=[{\hat{b}_{j \ell m }^{}},\;{\hat{b}_{j' \ell' m' }^{}}]=0\,.\end{aligned}$$
By using this eight-mode expansion of the field operator, the 3D Hamiltonian was obtained: $$\hat{H} = \sum_{\ell,m} \hat{H}_{\ell m} + \hat{H}_{\mathrm{int}}\, .$$ The first term is a sum over LMGHs for each level, similar to the one discussed in Sec. \[sec:1DLMG\] \[See Eq. (\[Eq:onelevelH\])\]: $$\begin{aligned}
\label{eq:LMGHpart}
H_{\ell m}&= U_{\ell m}\sum_j {\hat{n}_{j \ell m}}({\hat{n}_{j \ell m}}-1)-J_{\ell m}\sum_{j'\neq j} {\hat{b}_{j \ell m }^{\dagger}}{\hat{b}_{j' \ell m }^{}}\nonumber\\
&+E_{\ell}\sum_j{\hat{n}_{j \ell m}}\,.
\end{aligned}$$ In Eq. (\[eq:LMGHpart\]), the term $E_{\ell}\sum_j{\hat{n}_{j \ell m}}$ accounts for the energy of the atoms at level $\ell$ and $z$ component of angular momentum $m$, with ${\hat{n}_{j \ell m}}={\hat{b}_{j \ell m }^{\dagger}}{\hat{b}_{j \ell m }^{}}$ the number operator. The term $$\label{eq:Hinter}
\hat{H}_{\mathrm{int}} =\sum_m \hat{H}_{\mathrm{inter}}^m+\hat{H}_{\mathrm{intra}}\,,$$ accounts for processes among atoms in different levels and with different $z$ component of the angular momentum.
There are three relevant processes in Eq. (\[eq:Hinter\]). The first of these is given by $$\begin{aligned}
\label{eq:Hinter1}
\hat{H}_{\mathrm{inter}}^0 & = \!\sum_{j} \Big{\{}U_{0 1}^{0 0} \left[\!\left(\!{\hat{b}_{j 0 0 }^{\dagger}}\!\right)^2\!
\left({\hat{b}_{j 1 0 }^{}}\right)^2+{\mathrm{h.c.}}\right]\nonumber\\
&+4\,U_{0 1}^{0 0}\,{\hat{n}_{j 0 0}}\,{\hat{n}_{j 1 0}}\Big{\}}\,.\end{aligned}$$ This process describes the excitation of two atoms from the ground state to an orbital with $m=0$, or conversely, their decay from an excited state with $m=0$ to the ground state. We name this process [*zero-vorticity interlevel hopping*]{}. The second is $$\begin{aligned}
\label{eq:Hinter2}
\hat{H}_{\mathrm{inter}}^1 & = \!\sum_{j} \Big{\{} U_{0 1}^{0 1} \left[\!\left({\hat{b}_{j 0 0 }^{\dagger}}\!\right)^2
{\hat{b}_{j 1 1 }^{}}{\hat{b}_{j 1, -1 }^{}}+{\mathrm{h.c.}}\right]\nonumber\\
& +4\,U_{0 1}^{0 1}\,({\hat{n}_{j 0 0}}\,({\hat{n}_{j 1 1}}+{\hat{n}_{j 1, -1}}))\Big{\}}\,.\end{aligned}$$ This process is similar to the first one and we call it [*vortex-antivortex interlevel hopping*]{}. It permits an atom to change its level and also its $z$ component of its angular momentum, $m$. Through this process two atoms in the ground state can be excited, one with $m=1$ (a vortex) and the other with $m=-1$ (an anti-vortex). Conversely, one atom with $m=+1$ and another with $m=-1$ can decay to the ground state. Finally, the third process is $$\begin{aligned}
\label{eq:Hinter3}
\hat{H}_{\mathrm{intra}} & = \!\sum_{j} \Big{\{}U_{1 1}^{0 1}\
\left[\!\left(\!{\hat{b}_{j 1 0 }^{\dagger}}\!\right)^2\!
{\hat{b}_{j 1 1 }^{}}{\hat{b}_{j 1 ,-1 }^{}}
+{\mathrm{h.c.}}\right]\\
& + 2\,U_{1 1}^{0 1}\,({\hat{n}_{j 1 0}}\,({\hat{n}_{j 1 1}}+{\hat{n}_{j 1, -1}})
+2\,U_{11}\,({\hat{n}_{j 1 1}}{\hat{n}_{j 1, -1}})\Big{\}}\,.\nonumber\end{aligned}$$ Through the third process, [*vortex-antivortex intralevel hopping*]{}, the atoms can only change their angular properties, but not their energy level. Then, two excited atoms with $m=0$ can generate a pair of atoms, one with $m=1$ and the other with $m=-1$ or vice versa. These three processes are represented schematically in Fig. \[fig0\](b). Again, in obtaining this Hamiltonian, the off-site interaction coefficients were neglected, as discussed in Sec. \[sec:1DLMG\] for the 1D case.
Notice that the 3D interaction and tunneling coefficients that appear in the LMGH part of this Hamiltonian, Eq. (\[eq:LMGHpart\]), have to be evaluated independently for atoms in the same level and with the same $m$. The tunneling coefficients in Eq. (\[eq:LMGHpart\]) are defined as $$\begin{aligned}
J_{\ell m}&=-\int d^3{\mathrm{\bf x}}\psi^{\ast}_{\ell m}({\mathrm{\bf x}}-{\mathrm{\bf x}}_{\mathrm{min}}) \left[-\frac{\hbar^{2}}{2\mathcal{M}}\nabla^{2}+V({\mathrm{\bf x}})\right] \nonumber\\
&\times \psi_{\ell m}({\mathrm{\bf x}}+{\mathrm{\bf x}}_{\mathrm{min}})\,.\label{eq:J_lm}\end{aligned}$$ The interaction coefficients are $$U_{\ell m}^{\ell' m'}=\frac{g}{2}\int d^3{\mathrm{\bf x}}|\psi_{\ell m}({\mathrm{\bf x}})|^{2}|\psi_{\ell' m'}({\mathrm{\bf x}})|^{2}\,.
\label{eq:U_lm_l'm'}$$ Notice that the coefficients in the coupling part of the Hamiltonian, Eqs. (\[eq:Hinter1\])-(\[eq:Hinter3\]), are related to the interaction coefficients, and not to tunneling coefficients.
The presence of many new coefficients makes necessary a wider characterization of regimes, further away from the 1D Josephson, intermediate, and Fock regimes (where the eigenvectors and eigenvalues behave as in Figs. \[fig1\], \[fig2\], and \[fig3\]). This was accomplished in [@2011GarciaMarchPRA], where criteria for the crossings and limits of the model were also given, and represented numerically for the particular case of the Duffing double well potential $V(z)=V_0(-8 z^2/a^2+16 z^4/a^4+1)$. A Josephson and Fock regime can be described, for which the eigenvectors can be characterized as HO-like or MS states, respectively. These MS states can show occupation of the excited level, resulting in an MS of angular degrees of freedom. The presence of these orbitals are the fundamental difference from the 1D case. Moreover, the three coupling processes described above give rise to interesting dynamical phenomena, like vortex tunneling and vortex/antivortex creation/annihilation along with the tunneling of atoms with non-zero $m$. These new dynamical phenomena are the subject of our upcoming work [@2011GarciaMarchpre].
Conclusions {#sec:conclusion}
===========
Ultracold bosons in double well potentials are a simple system to study a great variety of physical phenomena. There are two main processes: the atoms can interact in pairs on-well with energy $U_0$ or tunnel to the other well with energy $J_0$. When the interactions dominate over the tunneling energies in the system, the spectra of the eigenvectors are characterized by the presence of MS states. But also in this case it is necessary to consider the possibility that the atoms populate an excited level. Then, other energies are relevant, including the interaction energy of the atoms in the excited level $U_1$, the tunneling energy in the excited level $J_1$, and the single-particle energy gap between levels, $\Delta E$.
In this Article, we used a two level approach to describe the possible physical scenarios in one- and three-dimensional double wells. For the 1D case, we clearly identified the Josephson regime, for which quantum tunneling was experimentally demonstrated. We characterized the eigenvectors and eigenvalues in the noninteracting regime, and we showed through direct diagonalization of the two level LMGH that this behavior can be extended to the Josephson regime. Our interest was mainly in the occurrence of MS states of atoms localized in either one or the other well, which appear for bigger interactions, in the Fock regime. We characterized the appearance of crossings in the spectra, which serves as a limiting criterion for the validity of the one-level or LMGH approximation. We obtained also the limits of the two-level approximation. In particular, we found when the interactions are sufficiently large that coupling to states with non-zero occupation of the excited level is relevant. In this case, the interaction coefficient can overcome the energy gap between levels, and then the states show non-zero occupation of the excited level. We described numerically the transition from the Josephson regime to the Fock regime. We introduced a new Josephson-Fock *mixed regime*, in which the most excited eigenvectors with no occupation of the excited level behave as MS states while the less excited ones are HO-like states. Moreover, since $J_1\gg J_0$, when the eigenvectors involve only atoms in the excited level, they behave as HO-like states.
MS states are, unfortunately, highly excited states and fragile against decoherence processes [@2000DalvitPRA; @2010PichlerPRA]. Thus they are difficult to observe experimentally. Quantum superpositions of matter waves have been observed for few particles, like electrons, but remains a challenging problem for larger objects; experiments with $C_{60}$ molecules are in the lead at present in such efforts (see [@1999ArndtNature; @2002BrezgerPRL] and references therein), but ultracold bosons have the potential to go to hundreds or thousands of particles in an MS state. Therefore, there are many theoretical proposals for realizing them in a BEC experiment [@2001MenottiPRA; @2004HigbiePRA; @2005MahmudPRA; @2006HuangPRA; @2006DunninhamNJP; @2008PiazzaPRA; @2008FerriniPRA; @2008MazetsEPL; @2010watanabePRA; @2010XuPRA; @2010HallwoodPRA; @2010CarrEPL; @2011KanamotoJO], and this remains an appealing research topic with deep physical implications. The two-level scenario introduces the possibility to study other initial states with non-zero occupation of the excited level, which could be the key to realize MS states experimentally, as discussed in [@2010CarrEPL]. Also, the two-level approach in three dimensions permits one to consider angular degrees of freedom in the problem [@2011GarciaMarchPRA]. The angular degrees of freedom are a very interesting topic in systems of ultracold atoms in optical lattices [@2005IsacssonPRA; @2005ScarolaPRL; @2006LiuPRA; @2007XuPRB; @2010CollinPRA; @2011LewensteinNatPhys], where the excitation of part of the population to a $p$ level was demonstrated experimentally [@2007MullerPRL; @2011WirthNatPhys]. Here, we summarized the derivation of the Hamiltonian and the MS states. Then, the study of the dynamics of ultracold atoms in three dimensional double wells where orbital degrees of freedom play a relevant role is an interesting topic for future research.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The analytic properties in the energy variable $k_{0}$ of finite-temperature self-energies are investigated. A typical branch cut results from $n$ particles being emitted into the heat bath and $n''$ being absorbed from the heat bath. There are three main results: First, in addition to the branch points at which the cuts terminate, there are also branch points attached to the cuts along their length. Second, branch points at $k_{0}=\pm k$ are ubiquitous and for massive particles they are essential singularities. Third, in a perturbative expansion using free particle propagators or in a resummed expansion in which the propagator pole occurs at a real energy, the self-energy will have a branch point at the pole location.'
address: 'Department of Physics, West Virginia University, Morgantown, West Virginia 26506-6315'
author:
- 'H. Arthur Weldon'
date: 'December 24, 2001'
title: 'Analytic Properties of Finite-Temperature Self-Energies'
---
Introduction
============
At non-zero temperature most examinations of the self-energy have emphasized one-loop results. For massless gauge theories Braaten and Pisarski showed that all diagrams with one loop (but any number of external lines) will produce effects as large as the tree diagrams and must therefore be resummed [@Pisarski; @BP]. In a resummed expansion the discontinuities of various one-loop and two-loop diagrams have been computed in QCD in order to predict processes relevant for quark gluon plasmas such as dilepton production, real photon production, and vector meson production.
There have been few investigations of the analytic properties of finite-temperature self-energies [@Evans; @BB] and much of the emphasis has been on the behavior at zero four-momentum [@zero].
The development of the questions investigated in this paper and of the approach is best illustrated by considering how a familiar zero-temperature calculation changes at non-zero temperature.
### T=0 example {#t0-example .unnumbered}
A typical example arises for a massive, self-interacting scalar field with ${\cal H}_{I}=g^{2}\phi^{4}/4!$. One of the two-loop contributions to the zero-temperature self-energy is $$\begin{aligned}
\Pi_{F}(K)=-ig^{4}\int{d^{4}P_{1}\over (2\pi)^{4}}
{d^{4}P_{2}\over (2\pi)^{4}}&&\;{1\over
\big(P_{1}^{2}\!-\!m^{2}+i\epsilon\big)}\nonumber\\
\times
{1\over\big(P_{2}^{2}\!-\!m^{2}+i\epsilon\big)}&&
{1\over \big(P_{3}^{2}\!-\!m^{2}+i\epsilon\big)},\nonumber\end{aligned}$$ where $P_{3}=K\!-\!P_{1}\!-\!P_{2}$. Direct integration of the energy variables $P_{01}$ and $P_{02}$ gives $$\begin{aligned}
\Pi_{F}(K)=-g^{2}\int\!{d^{3}p_{1}\over (2\pi)^{2}}
{d^{3}p_{2}\over (2\pi)^{3}} &&\;{1\over
2E_{1}2E_{2}2E_{3}}\label{piT=0}\\
\times\bigg[{1\over
k_{0}\!-\!E_{1}\!-\!E_{2}\!-\!E_{3}\!+\!i\eta}
-&&{1\over
k_{0}\!+\!E_{1}\!+\!E_{2}\!+\!E_{3}\!-\!i\eta}\bigg],
\label{eq1}\nonumber\end{aligned}$$ with $E_{j}=(p_{j}^{2}+m^{2})^{1/2}$. The first denominator produces a branch point in the self-energy at $k_{0}=3E(k/3)=(k^{2}+9m^{2})^{1/2}$ and a branch cut along the positive $k_{0}$ axis for $3E(k/3)\le k_{0}\le\infty$. The second denominator produces a branch cut for $-\infty\le k_{0}\le
-3E(k/3)$. From the operator point of view these two contributions arise from inserting a three-particle intermediate state in the two time-orderings contained in $\langle 0|T\big(\phi^{3}(x)\phi^{3}(y)\big)|0\rangle$. At higher orders in perturbation theory the self-energy continues to be real and analytic in the open interval $-3E(k/3)< k_{0} < 3E(k/3)$.
### The same example at T$\neq$0 {#the-same-example-at-tneq0 .unnumbered}
At non-zero temperature there are four real-time propagators organized into a $2\times 2$ matrix $D_{ij}$ with $i,j=1,2$ [@LeBellac; @Das]. The proper self-energy becomes a matrix $\Pi_{ij}$. In the same $g^{2}\phi^{4}/4!$ theory the two-loop contribution to the time-ordered self-energy $\Pi_{11}$ is $$\Pi_{11}(K)=-ig^{4}\!\int{d^{4}P_{1}\over (2\pi)^{4}}
{d^{4}P_{2}\over (2\pi)^{4}}\,
D_{11}(P_{1})D_{11}(P_{2})D_{11}(P_{3}).$$ The finite-temperature propagators are $$D_{11}(P)={1+f_{BE}\over P^{2}-m^{2}+i\epsilon}
-{f_{BE}\over P^{2}-m^{2}-i\epsilon},$$ where $f_{BE}=\big[\exp(\beta|p_{0}|)-1\big]^{-1}$ is the Bose-Einstein function. Performing the integrations over $P_{01}$ and $P_{02}$ leads to an integrand much more complicated than in Eq. (\[piT=0\]). The new integrand can be expressed as a linear combination of eight terms each of which has a different $k_{0}$ dependence in the denominator. The denominators are of the form $k_{0}\pm E_{1}\pm E_{2}\pm E_{3}+i\eta(\vec{p}_{j})$, in which all possible sign combinations occur. In the various denominators the sign of the infinitesimal imaginary parts are momentum-dependent, which makes the calculation tedious and the analytic properties obscure. The complications of the $i\eta$’s is a result of the absolute value bars in $f_{BE}$.
A much simpler analytic structure is enjoyed by the retarded self-energy $\Pi_{R}$. All four $\Pi_{ij}$ can be expressed in terms of $\Pi_{R}$ and $\Pi_{A}$, where $\Pi_{A}(K)=\Pi_{R}(-K)$. All four propagators $D_{ij}$ can be expressed in terms of the retarded and advanced propagators $D_{R}$ and $D_{A}$ [@Aurenche]. For example, $\Pi_{11}(K)$ is can be expressed as $$\Pi_{11}(K)=[1+f]\,\Pi_{R}(K)-f\,\Pi_{A}(K),$$ where $f=\big[\exp(\beta k_{0})-1\big]^{-1}$ has no absolute value bars and is an analytic function of $k_{0}$. The retarded and advanced self-energies have simple analytic structure: $\Pi_{R}(K)$ is holomorphic for ${\rm Im}(k_{0})>0$ and $\Pi_{A}(K)$ is holomorphic for ${\rm Im}(k_{0})<0$.
To compute the retarded self-energy directly without using the $\Pi_{ij}$, one can either use the real-time Feynman rules expressed in terms of $D_{R}$ and $ D_{A}$[@Aurenche; @EKW] or use the imaginary-time Feynman rules [@Kapusta] and then analytically continue in energy. For the above two-loop example in in $g^{2}\phi^{4}/4!$ theory the result for the retarded self-energy can be expressed as as sum of eight contributions: $$\begin{aligned}
\Pi_{R}(K)=\int {d^{3}p_{1}\over (2\pi)^{3}}
{d^{3}p_{2}\over (2\pi)^{3}}
\sum_{A=1}^{8}{f_{A}(\vec{p}_{1},\vec{p}_{2},\vec{p}_{3})
\over k_{0}+i\eta- \psi_{A}},\label{twolooppi}\end{aligned}$$ where now all the denominators depend on $k_{0}+i\eta$ so that $\Pi_{R}(K)$ is manifestly holomorphic in the upper-half of the complex $k_{0}$ plane. Each $\psi_{A}$ is a sum or difference of the three particle energies: $$\psi_{A}=\pm E_{1}\pm E_{2}\pm E_{3},\label{psi}$$ where the eight combinations of the $\pm$ signs account for the eight different $\psi$’s. The physical interpretation of all these possibilities is standard [@AW1]: The energies that appear in Eq. (\[psi\]) with a positive sign correspond to particles emitted into the heat bath; the energies that appear in with a negative sign correspond to particles absorbed from the heat bath.
### Approximate dispersion relations that are real {#approximate-dispersion-relations-that-are-real .unnumbered}
Using a conventional propagator which has poles at $p_{0}=\pm(p^{2}+m^{2})^{1/2}$ may not be a good procedure because thermal corrections will shift the pole to a different location. The shift in the pole location is most dramatic in massless theories. For example, in massless $g^{2}\phi^{4}/4!$ the one-loop correction will shift the pole to $p_{0}=\pm(p^{2}+g^{2} T^{2}/24)^{1/2}$. In massless QED and QCD the one-loop corrections to the fermion and gauge-boson self-energies shift the locations of the poles to $p_{0}=\pm E(p)$, where $E(p)$ is a complicated, real transcendental function of momentum. The retarded propagator for any real dispersion relation is $$D_{R}(P)={1\over (p_{0}\!+\!i\epsilon)^{2} \!-\!E^{2}(p)}.
\label{retprop}$$ The analysis in this paper will apply to propagators of the form $D_{R}$. This propagator is not as complicated as the hard-thermal-loop propagators $^{*}S(P)$ and $^{*}D_{\mu\nu}(P)$ for fermion and gauge bosons [@Pisarski; @LeBellac; @Blaizot; @Smilga1]. The hard-thermal-loop resummed propagators have the structure $$^{*}D_{R}(P)={1\over
(p_{0}\!+\!i\epsilon)^{2}\!-\!p^{2}\!-\, ^{*}\Pi_{R}(p_{0},p)}.$$ Although $^{*}D_{R}(P)$ has the same poles on the real axis at $p_{0}=\pm E(p)$ as Eq. (\[retprop\]), it also has a branch cut for space-like momentum, $-p\le p_{0}\le p$. It can be written as $$^{*}D_{R}(P)={N(p_{0},p)\over (p_{0}\!+\!i\epsilon)^{2}
\!-\!E^{2}(p)},$$ where the numerator function $N$ contains the branch cut but has no poles. It seems quite likely that the self-energies computed with $^{*}D_{R}$ would contain all the branch points that will be found using $D_{R}$ and would contain additional branch cuts directly related to the branch cut in $N$. However, this diversion will not be pursued.
### Generalization to complex dispersion relations {#generalization-to-complex-dispersion-relations .unnumbered}
From physical considerations one knows that particle propagation is damped at finite temperature. One-loop calculations are misleading in that the the solution to $E^{2}=p^{2}+m^{2}+\Pi_{R}^{\rm
1\;loop}(E,p)$ is always a real energy $E(p)$. If one calculates the self-energy to two-loop accuracy then the pole in the propagator to two-loop accuracy will be complex. The location of this complex pole in the retarded propagator will be denoted by ${\cal E}(p)$: $${\cal E}(p)=\omega(p)-{i\over 2}\gamma(p),$$ where $\omega$ and $\gamma$ both real and both positive. The corresponding propagator in the pole approximation is $$D_{R}(P)={1\over \big(p_{0}-{\cal E}(p)\big)
\big(p_{0}+{\cal E}^{*}(p)\big)}.\label{retprop2}$$ It has no singularities for ${\rm Im} (p_{0})>0$. When $p_{0}$ is complex but $p$ is real, it satisfies the condition $D_{R}(p_{0},p)=\big[D_{R}(-p_{0}^{*},p)\big]^{*}$.
It is important to emphasize that the damping function $\gamma(p)$ cannot just be invented for phenomenological purposes, because when the three-momentum is allowed to become complex, $p_{c}=p+ip'$, then ${\cal E}(p_{c})$ must be an analytic function of the variable $p_{c}$: $${\cal E}(p\!+\!ip')=\omega(p,p')-{i\over
2}\gamma(p,p').$$ In particular $\omega$ and $\gamma$ must satisfy the Cauchy-Riemann conditions: $$\begin{aligned}
{\partial \omega(p,p')\over\partial p}=&&-{1\over 2}\, {\partial
\gamma(p,p')\over\partial p'}\nonumber\\
{\partial \omega(p,p')\over\partial p'}=&& {1\over 2}\,{\partial
\gamma(p,p')\over\partial p}.\nonumber\end{aligned}$$ This guarantees that, in computing a self-energy correction, the integration contours for the momentum variables can be distorted into the complex plane. Without this property the locus of non-analyticity for self-energy corrections would be continuous lines rather than isolated branch points.
### Form of the self-energy {#form-of-the-self-energy .unnumbered}
The following analysis will employ propagators of the general form in Eq. (\[retprop2\]). Obviously Eq. (\[retprop\]) can be considered as a special case. A particular multi-loop self-energy diagram will have many branch cuts and can be written as the sum of various integrals, each of which displays a unique branch cut [@AW2]. The branch cut of the most general integral results from an intermediate state in which $n$ particles are emitted into the heat bath and $n'$ particles are absorbed from the heat bath and is of the form $$\Pi_{R}(K)=\int\prod_{j=1}^{n-1}
d^{3}p_{j}
\prod_{\ell=1}^{n'}d^{3}q_{\ell}\;\;{f\big(\vec{p},\vec{q}\big)
\over k_{0}-\psi}.
\label{defPi}$$ This definition applies in the region ${\rm Im} (k_{0})>0$, where the retarded self-energy is holomorphic. Branch point will be sought by analytically continuing away from this region. A particular Feynman diagram will be the sum of several integrals of this form, involving different values of $n$ and $n'$ and often different values of the sum $n+n'$.
Although there are $n+n'$ momenta, one of them is determined by momentum conservation: $$\vec{k}
=\sum_{j=1}^{n}\vec{p}_{j}+\sum_{\ell=1}^{n'}\vec{q}_{\ell}.
\label{defmom}$$ The denominator function $\psi$ sums over the energies of the emitted particles positively and over the energies of the absorbed particles negatively: $$\psi=\sum_{j=1}^{n}{\cal E}(\vec{p}_{j})-\sum_{\ell=1}^{n'}
{\cal E}^{*}(\vec{q}_{\ell}).\label{defpsi1}$$ The complex energy ${\cal E}$ has a negative imaginary part and so $\psi$ automatically has a negative imaginary part. This guarantees that $\Pi_{R}(K)$ in Eq. (\[defPi\]) is holomorphic for ${\rm Im} (k_{0})>0$.
The range of the function $\psi$ depends on the values of $n$ and $n'$. For $n\ge 2$ and $n'\ge 2$ then $-\infty\le\psi\le\infty$ and so the self-energy will have a branch cut along the entire length of the real $k_{0}$ axis. For $n\ge 2$ and $n'=1$, momentum conservation forces $\psi$ to be bounded from below but not from above. For $n=1$ and $n'\ge 2$ momentum conservation forces $\psi$ to be bounded from above but not from below. For $n=1$ and $n'=1$ the range of $\psi$ will be finite.
Most points at which $k_{0}=\psi$ will not produce singularities in $\Pi_{R}$ because the integration contours can generally be distorted so that the integration does not pass over the singularity. (Provided that ${\cal
E}(p)$ is analytic in $p$.) There are two situations which do produce singularities [@ELOP; @IZ]. The first, called pinch singularities, occur at values of $k_{0}$ at which two or more singularities of the integrand pinch the integration contour from opposite sides. The necessary condition for a pinch is the simultaneous vanishing of $k_{0}-\psi$ and of the derivatives of $\psi$ with respect to the $\vec{p}$ ’s and $\vec{q}$ ’s. The sufficient conditions require more detailed study of the integrand. The second, called end-point singularities, occurs at values of $k_{0}$ at which the singularities of the integrand occur at end-points of the integration region, in this case from $\vec{p}$ and $ \vec{q}$ taking on values $\pm\infty$.
It is perhaps worth emphasizing that it is the location of branch points that is under investigation and not the value of the discontinuity across the branch cut. For the present purposes it does not matter if the discontinuity can be grouped as a product of factors from one side of the cut or the other [@cuts].
### Applicability to QCD {#applicability-to-qcd .unnumbered}
In QCD the quark dispersion relations are different from the gluon dispersion relations at the one-loop level and certainly at higher loops. The analysis presented here applies to any self-energy contribution in which all of the cut propagators have the same dispersion relation. Thus, for quark self-energies it applies to cuts across intermediate states with all quarks but no gluons. Similarly, for gluon self-energies it applies to cuts across intermediate states that are composed entirely of gluons or entirely of quarks. The inability to treat intermediate states with mixtures of particle species is obviously a limitation and it will require more work to overcome. Even at $T=0$, unequal masses are difficult to treat.
### Sample result {#sample-result .unnumbered}
A simple but interesting example of the results that will be derived occurs for self-energy diagrams with a three-particle intermediate state. Three-particle intermediate states in which all three particles are the same species occur at two-loop order in the following cases: (a) scalar field self-energy with $\phi^{4}$ interaction as already discussed; (b) gluon self-energy in QCD with three-gluon intermediate state; (c) quark self-energy in QCD with a three-quark intermediate state. In all these examples $n+n'=3$. One contribution allows two particles to be emitted into the heat bath and one particle to be absorbed from the heat bath ($n=2, n'=1$). Subsequent analysis will show that there will be three branch points: viz. at $k_{0}=\pm k$ and at $k_{0}=\infty$ with branch cuts connecting them. The branch points at $k_{0}=\pm k$ will be essential singularities with behavior $\exp(3m^{2}k/(K^{2} T))$ as $K^{2}$ approaches zero from the negative region. Here $m$ is the effective thermal mass from the large-momentum expansion of the dispersion relation.
If the single-particle energy ${\cal E}(p)$ used to define the loop expansion is complex, then only the above three branch points occur. If, however, a real energy $E(p)$ is used then the self-energy will have a fourth branch point at $k_{0}=E(k)$. In this situation the propagator which was assumed to have a simple pole at $k_{0}=E(k)$ turns out to generate a branch point also at $k_{0}=E(k)$. This ugliness infects any perturbative expansion built on a real dispersion relation.
### Organization {#organization .unnumbered}
It is assumed throughout that the branch cuts of thermal self-energies come entirely from the denominators of the propagator functions and are not affected by the spin of the particles.
Sec. II presents three toy examples of functions with branch cuts that extend from $-\infty$ to $+\infty$, as this does not occur for zero-temperature self-energies. Two of these examples have additional branch points on the real axis that illustrate features that will be found in the actual self-energy.
Sec. III analyzes the branch points that occur for the general intermediate state consisting of $n$ particles emitted into the heat bath and $n'$ absorbed from the heat bath. Sec. III.D summarizes the results and may be read independently of the development sections.
Sec. IV discusses some implications of the results.
There are three appendixes. Appendix A and B contain detailed proofs that complete the arguments given in Sec. III. Appendix C is an explicit one-loop example that displays the essential singularity at $k_{0}=\pm k$. Appendix D is an explicit two-loop example from Wang and Heinz [@Heinz] that shows both the essential singularity at the light cone and the branch point at the mass shell.
Simple functions with branch points at $\pm\infty$
==================================================
One of the main results of this paper will be that self-energies at $T\neq 0$ not only have branch points at the ends of their branch cuts but also have extra branch points not at the ends but attached to the cuts. Although this is unfamiliar from $T=0$ physics, it is not very exotic mathematics. This section contains three toy examples involving one-dimensional integrals that can be computed exactly.
Each example concerns a function defined by an integral of the form $$F(\omega)=\int_{-\infty}^{\infty}\!dz\;
{1\over\omega-\psi(z)},$$ where $\psi(z)$ is a real function when $z$ is real. When ${\rm
Im}\,\omega>0$ the function is holomorphic and defines the retarded form of $F(\omega)$. When ${\rm Im}\,\omega<0$ the function is holomorphic and defines the advanced form of $F(\omega)$.
In the following examples the function $\psi(z)$ will be real and chosen so that $F(\omega)$ have a branch cut running from $\omega=-\infty$ to $\omega=\infty$, which separates the two regions of holomorphicity. Such a cut requires that $\psi(z)$ takes on all real values.
The branch points in the three examples can be found by examining the integrands and are confirmed by explicit integration. The discontinuity across the branch cut is pure imaginary $F(\omega_{r}\!+\!i\epsilon)-F(\omega_{r}\!-\!i\epsilon)=2i{\rm
Im} F(\omega_{r})$, where $${\rm Im} F(\omega_{r})=
-\pi \int_{-\infty}^{\infty}\!\!dz\;\delta\,\Big[\omega_{r}-
\psi(z)\Big].\label{disc}$$ The discontinuity formula is not the best way to answer the question of whether $F(\omega)$ has any branch points at finite real values of $\omega$ that are attached to the branch cut.
#### Example 1: {#example-1 .unnumbered}
The first example is $$f(\omega)=\int_{-\infty}^{\infty}\!
dz\,{1\over \omega-\sinh z}.$$ For any real value of $\omega$, positive or negative, there is a real value of $z$ at which the denominator of the integrand vanishes and this leads to a branch cut along the entire real axis. The end-points $z=\pm\infty$ of the integration produce the branch points at $\omega=\pm\infty$. Explicit integration confirms this: $$f(\omega)={1\over\sqrt{\omega^{2}\!+\!1}}
\ln\bigg[{\omega\!+\!\sqrt{\omega^{2}\!+\!1} \over
\omega\!-\!\sqrt{\omega^{2}\!+\!1} }\bigg].\label{f2}$$ There are branch points at $\omega=\pm\infty$, where the argument of the logarithm vanishes. Inspection shows that Eq. (\[f2\]) is discontinuous across the real axis. If $\omega$ approaches the real axis from above, then the argument of the logarithm approaches $e^{-i\pi}|R|$; if $\omega$ approaches the real axis from below, then the argument of the logarithm approaches $e^{i\pi}|R|$. In this example there are no branch points at finite values of $\omega$. This is the type of behavior that is usually thought to be typical of finite temperature field theory.
#### Example 2: {#example-2 .unnumbered}
Next consider $$g(\omega)=\int_{-\infty}^{\infty}\!\!dz
\;{1\over \omega-z^{3}}.$$ This integral has end-point singularities at $\omega=\pm\infty$ and also a pinch singularity at $\omega=0$. The pinch occurs because at $z=0$ both $\psi(z)$ and $d\psi/dz$ vanish [@ELOP; @IZ].
The integral may easily be evaluated by Cauchy’s theorem. For any $\omega$ the integrand contains three simple poles as a function of $z$. When $\omega$ is in the upper half-plane, there are two poles in $z$ above the real axis and one pole below. Integration gives $${\rm Im}
\omega>0:\hskip1cm
g(\omega)={2\pi i\over 3}\;{e^{-i2\pi/3}\over
\omega^{2/3}}.\label{ex2}$$ When $\omega$ is in the lower half-plane, then $g(\omega)$ is the complex conjugate of the above: $${\rm Im}\omega<0:\hskip 1cm
g(\omega)={-2\pi i\over 3}\;{e^{i2\pi/3}\over
\omega^{2/3}}.$$ As expected, $g(\omega)$ has a branch cut along the full length of the real axis with branch points at $\omega=\pm\infty$. The new feature is the third branch point at $\omega=0$.
It is useful to investigate the analytic structure a bit more. Let $\omega_{0}$ lie in the upper half-plane, where $g(\omega)$ given by Eq. (\[ex2\]) is analytic. To explore $g(\omega)$ in the neighborhood of $\omega_{0}$, put $\omega=\omega_{0}+re^{i\phi}$ with $r$ real. As $\phi$ increases from $0$ to $2\pi$, $\omega$ moves in a circle of radius $r$ centered on $\omega_{0}$. This circle can pass into the lower half-plane since Eq. (\[ex2\]) can be analytically continued into the lower half-plane. If $r<|\omega_{0}|$ the the circle will not pass around the origin and the function $(\omega_{0}+re^{i\phi})^{2/3}$ will have the same value at $\phi=0$ and at $\phi=2\pi$. However, if $r>|\omega_{0}|$ then $\omega$ will encircle the origin and $g(\omega)$ will not return to its original value. To clarify this, choose $\omega_{0}=0$ so that $\omega=re^{i\phi}$. Then when $\phi$ increases from $0$ to $2\pi$, $\omega^{2/3}$ will return to the value $e^{4\pi i/3}\omega^{2/3}$ and $g(\omega)$ will return to the value $$g_{II}(\omega)={2\pi i\over 3}\;{1\over\omega^{2/3}}.$$ This shows that the function $g(\omega)$ has a branch point at $\omega=0$ in addition to those at $\omega=\pm\infty$. (If $\omega$ encircles the origin two more times in a counter clock-wise direction then $g(\omega)$ will return to the original value in Eq. (\[ex2\]).)
#### Example 3: {#example-3 .unnumbered}
The third example is $$h(\omega)=
\int_{-\infty}^{\infty}\!\!dz
\;{1\over 2\omega-z^{3}+3z}.$$ In addition to end-point singularities at $\omega=\pm\infty$, this integral has pinch singularities at $\omega=\pm 1$. The pinch singularities arise because $\psi(z)=z^{3}\!-\!3z$ has a local maximum at $z=1$ and a local minimum at $z=-1$. Consequently the denominator has a double zero at $\omega=\pm 1$ [@ELOP; @IZ].
Since the integrand has three simple poles, the integral can be performed using Cauchy’s theorem. For ${\rm
Im}\,\omega>0$ the result is $${\rm Im}\omega>0:\hskip0.2cm h(\omega)={2\pi i\over 3}\;{1\over
e^{2\pi i/3}A^{2/3}
+e^{-2\pi i/3}B^{2/3}+1},\label{ex3}$$ where $$A=\omega+\sqrt{\omega^{2}\!-\!1}
\hskip1cm B=\omega-\sqrt{\omega^{2}\!-\!1}.$$ As expected, $h(\omega)$ has four branch points on the real axis: at $\omega=-\infty$ where A vanishes; at $\omega=\infty$, where B vanishes; and at $\omega=\pm1$, where A and B have branch points.
The discontinuity of $h(\omega)$ across the real axis can be computed either directly from Eq. (\[ex3\]) or by using Eq. (\[disc\]). By either method the result is $$\begin{aligned}
\omega^{2}>1:\hskip0.3cm {\rm Im}\, h(\omega)=&&
-{\pi \over 3}\,{1\over A^{2}+B^{2}+1}\\
\omega^{2}<1:\hskip0.3cm {\rm Im}\, h(\omega)=&&
{2\pi \over 3}\,{1\over 2\cos\big[(2\theta+2\pi)/3\big]+1},\end{aligned}$$ where for $-1<\omega<1$ the angle $\theta$ is defined by $\omega=\cos\theta$. The imaginary part has a different value as $\omega$ approaches 1 from above or from below. For infinitesimal $\epsilon$, $$\begin{aligned}
\omega=1+\epsilon:\hskip0.7cm {\rm Im} h(1+\epsilon)=&&-{\pi\over
9}\\
\omega=1-\epsilon:\hskip0.7cm
{\rm Im} h(1-\epsilon)=&&-\infty.\end{aligned}$$ Thus the imaginary part of $h(\omega)$ is discontinuous at at the branch point. This method will be used in Appendix D.
Branch points of self-energies
==============================
This section will examine the general problem of a perturbative expansion based on propagators of the form Eq. (\[retprop2\]) in which ${\cal E}(p)$ is any single-particle energy, real or complex. A summary of this section is given in III.D.
Branch points for $n$ emissions with no absorptions
---------------------------------------------------
The simplest type of branch cuts are those that come from the production of $n$ particles. After integrations over the time-like components of the loop momenta, the retarded thermal, self-energy can be written as an integral over $n\!-\!1$ independent three momenta: $$\Pi_{R}(k_{0},k)=\int\!{ d^{3}p_{1}d^{3}p_{2}\dots
d^{3}p_{n-1}\;\;f(\vec{p}_{j})\over
k_{0}-\psi}.
\label{NpirealE}$$ The numerator $f(\vec{p}_{j})$ will depend on temperature and and on the spins of the particles. The denominator function $\psi$ is $$\psi=\sum_{j=1}^{n}{\cal E}(\vec{p}_{j}).$$ The momentum of the last particle, viz. $\vec{p}_{n}$, is determined by momentum conservation $$\vec{k}=\sum_{j=1}^{n}\vec{p}_{j}.$$
A value of $k_{0}$ that makes the denominator of the integrand in Eq. (\[NpirealE\]) vanish will rarely produce a singularity in $\Pi_{R}$ because the integration contour can be distorted into the complex plane so as to avoid the point at which the denominator vanishes. Another way to describe this situation is to focus on the values of $\vec{p}_{j}$ that make the denominator vanish for a particular $k_{0}$. As $k_{0}$ varies, the location of the critical $\vec{p}_{j}$ varies. At a particular $k_{0}$ the singularity may move onto the real $\vec{p}_{j}$ axis. This will generally not produce a singularity of the function $\Pi_{R}$ because the contour can be distorted so as to avoid the singularity [@ELOP; @IZ]. However if two singularities move so as to pinch the contour between them at a particular $k_{0}$ then the function $\Pi_{R}$ will have a singularity at that $k_{0}$. The necessary condition for the denominator of Eq. (\[NpirealE\]) to have a double pole at some particular $k_{0}$ requires that both the denominator and its first derivative vanish [@ELOP; @IZ].
It is convenient to implement momentum conservation by employing a Lagrange multiplier $\vec{v}$ and defining a new function $\Psi$ as $$\Psi=
\sum_{j=1}^{n}{\cal E}(\vec{p}_{j})
+\vec{v}\cdot\Big(\vec{k}-\sum_{j=1}^{n}\vec{p}_{j}
\Big).$$ Any point at which the derivatives of $\Psi$ with respect to $\vec{p}_{1},\dots,\vec{p}_{n}$ and $\vec{v}$ all vanish, will be a point at which the derivatives of $\psi$ with respect to $\vec{p}_{1},\dots,\vec{p}_{n-1}$ vanish while keeping momentum conserved.
To proceed further it is helpful to introduce the group velocity $$V(p)={d{\cal E}(p)\over dp},$$ which may be complex when ${\cal E}$ is complex. The pinch conditions $$0={\partial\Psi\over\partial\vec{p}_{j}}=\hat{p}_{j}V(p_{j})-\vec{v},$$ imply that all the $\vec{p}_{j}$ are equal. The common value of $\vec{p}_{j}$ is determined by extremizing with respect to the Lagrange multiplier: $$0={\partial\Psi\over\partial\vec{v}}=\vec{k}-\sum_{j=1}^{n}\vec{p}_{j},$$ and this fixes $\vec{p}_{j}=\vec{k}/N$. The extreme value of $\psi$ is $$\psi_{\rm ext}=n\,{\cal E}\big(\vec{k}/n\big).\label{psiextreme}$$ Thus there will be branch point at $k_{0}=n{\cal E}(\vec{k}/n)$. For the free-particle dispersion relation the branch point is at $k_{0}=\sqrt{k^{2}+(nm)^{2}}$. When ${\cal E}$ is an effective thermal energy, the branch point at $k_{0}=n\,E(k/n)$ will be temperature-dependent. In either case, the branch cut runs parallel to the positive $k_{0}$ axis and terminates with a branch point at $k_{0}=\infty$.
For the related situation of $n$ particles absorbed from the thermal bath, then $\psi=-\sum_{j=1}^{n}{\cal
E}^{*}(\vec{p}_{j})$. This produces a branch point at $k_{0}=-n{\cal
E}^{*}(\vec{k}/n)$ and a branch cut which runs parallel to the negative $k_{0}$ axis and terminates at $-\infty$.
Branch points that only occur for real group velocities
-------------------------------------------------------
The part of the self-energy which has an intermediate state consisting of $n$ emitted particles and $n^{\prime}$ absorbed particles has the form $$\Pi_{R}(K)=\int\!\prod_{j=i}^{n-1} d^{3}p_{j}
\prod_{\ell=1}^{n'}d^{3}q_{\ell}\;\;{f\big(\vec{p},\vec{q}\big)
\over k_{0}-\psi}$$ where the momentum $\vec{p}_{n}$ is determined by momentum conservation, Eq. (\[defmom\]), and $\psi$ is given by Eq. (\[defpsi1\]).
To examine for pinch singularities in momentum space subject to the constraint of momentum conservation, it is again convenient to introduce a Lagrange multiplier $\vec{v}$ and define a new function $$\Psi=
\sum_{j=1}^{n}{\cal E}(\vec{p}_{j})
\!-\!\sum_{\ell=1}^{n'}{\cal E}^{*}(\vec{q}_{\ell})+\vec{v}\cdot\Big(
\vec{k}-\sum_{j=1}^{n}\vec{p}_{j}-\sum_{\ell=1}^{n'}
\vec{q}_{\ell}\Big).\label{Bpsi}$$ The pinch condition $$0={\partial\Psi\over\partial \vec{p}_{j}}=
\hat{p}_{j}V(p_{j})-\vec{v}$$ implies that all $\vec{p}_{j}$ are equal. The condition $$0= {\partial\Psi\over\partial \vec{q}_{\ell}}=
-\hat{q}_{\ell}V^{*}(q_{\ell})-\vec{v}$$ implies that all the $\vec{q}_{\ell}$ are equal. Eliminating the Lagrange multiplier $\vec{v}$ in these last two conditions gives $$\hat{p}_{j}V(p_{j})=-\hat{q}_{\ell}V^{*}(q_{\ell}).\label{cond12}$$ The third condition is $$0={\partial\Psi\over\partial \vec{v}}=\vec{k}
-\sum_{j=1}^{n}\vec{p}_{j}-\sum_{\ell=1}^{n'}\vec{q}_{\ell}.\label{cond3}$$
#### Case 1. ${\cal E}$ real: {#case-1.-cal-e-real .unnumbered}
When the single particle energy ${\cal E}$ is real, it is denoted by $E$. The group velocity $V$ is real. Eq. (\[cond12\]) implies first that $\hat{p}_{j}=-\hat{q}_{\ell}$ and second that $V(p_{j})=V(q_{\ell})$. This is solved by $\vec{p}_{j}=-\vec{q}_{\ell}$. Eq. (\[cond3\]) can then be solved for $n\neq n'$: $$\vec{p}_{j}={\vec{k}\over n\!-\!n'}\,;
\hskip1cm \vec{q}_{\ell}={-\vec{k}\over
n\!-\!n'}.$$ (For $n\neq n'$ there is no solution.) The value of Eq. (\[Bpsi\]) at the extremum is $$\psi_{\rm ext}= [n\!-\!n']\,E\Big(k\big/[n\!-\!n']\Big).$$ The necessary conditions for a branch point at $k_{0}=\psi_{\rm ext}$ are thus satisfied. Because this extrememum is a saddle point and not a local maximum or minimum, the conventional experience does not apply. To demonstrate that there actually is a pinch of the integration contour requires more analysis. This analysis is done in Appendix A and confirms that there is a branch point at $k_{0}=\psi_{\rm ext}$ and also shows that the branch point has infinitely many sheets.
For the free particle dispersion relation the branch point is at $k_{0}=\sqrt{k^{2}+((n-n')m)^{2}}$. If $E$ is a temperature-dependent effective energy, then the location of the branch cut will be temperature-dependent.
The most surprising consequence of this is that when $n\!-\!n'=1$, there is a branch point at $k_{0}=E(k)$, which is precisely at the location of the pole in the propagator that was used to define the perturbative series. An example of this phenomena occurs in the self-energy in $\phi^{4}$ theory. Wang and Heinz [@Heinz] have calculated the imaginary part of the two-loop self-energy. Appendix D shows explicitly that the two-loop self-energy has a branch point at the mass-shell.
When $n\!-\!n'=2$ there is a branch point at $k_{0}=2E(\vec{k}/2)$ that occurs by cutting $n+n'$ propagators. For $n'=0$ this is the two-particle normal threshold already displayed in Eq. (\[psiextreme\]). But for $n'\neq 0$ the branch point occurs in more complicated diagrams than in Eq. (\[psiextreme\]). Similarly, for $n\!-\!n'=7$ the branch point at $k_{0}=7E(\vec{k}/7)$ occurs in diagrams with $n+n'\ge 7$.
#### Case 2. ${\cal E}$ complex but $V$ real: {#case-2.-cal-e-complex-but-v-real .unnumbered}
It is possible to have ${\cal E}(p)=E(p)-i\gamma/2$ but $\gamma$ is a non-zero constant. The true damping cannot be constant, but the constant $\gamma$ approximation is sometimes useful [@Henning]. The group velocity $V=dE/dp$ is real so that the pinch condition is satisfied at the same momenta $\vec{p}_{j}$ and $\vec{q}_{\ell}$ as in case 1. The only difference is that the extremum of Eq. (\[Bpsi\]) is $$\psi_{\rm
ext}=[n\!-\!n']\,E\Big(k\big/[n\!-\!n']\Big)-i(n\!+\!n'){\gamma\over
2}.$$ When $n\!-\!n'=1$ the various branch points at $k_{0}=E(k)-i(n\!+\!n')\gamma/2$ do not coincide with the single-particle pole at $k_{0}=E(k)-i\gamma/2$. The proof in Appendix A includes this case.
#### Case 3. ${\cal E}$ and $V$ complex: {#case-3.-cal-e-and-v-complex .unnumbered}
When the single-particle energy ${\cal E}$ is complex, it is difficult to solve Eqs. (\[cond12\]) and (\[cond3\]). The first equation implies that $\hat{p}_{j}=\pm\hat{q}_{\ell}$. Let us examine the case $\hat{p}_{j}=-\hat{q}_{\ell}$. Then $$V(p_{j})=V^{*}(q_{\ell}).$$ It is possible to invent an analytic function $V(p)$ that satisfies this condition at special momentum. However, the branch points would then be artifacts of the approximation scheme. The exact value of the single-particle pole energy, ${\cal E}_{\rm pole}$, the imaginary part is negative and vanishes at zero momentum and at infinite momentum. Therefore its first derivative must be negative at small momentum and positive at large momentum. If $p_{j}$ is small and $q_{\ell}$ is large, it may be possible for ${\rm Im} V(p_{j})=-{\rm Im} V(q_{\ell})$. Whether the real parts would satisfy ${\rm Re} V(p_{j})={\rm Re} V(q_{\ell})$ seems unlikely.
Essential singularities at $k_{0}=\pm k$
----------------------------------------
The branch points discussed above occur when $\vec{p}_{j}$ and $\vec{q}_{\ell}$ all have a finite magnitude. Additional branch points can result from pinches at infinite values of $\vec{p}_{j}$ and $\vec{q}_{\ell}$. To investigate these it is necessary to make some assumption about the behavior of the dispersion relation ${\cal E}(p)$ at large momenta. It will be assumed that $$p\to\infty:\hskip0.7cm {\cal E}(p)\to p+{m^{2}\over 2p}+\dots
\label{asympt}$$ and that the imaginary part of ${\cal E}(p)$ falls faster than $1/p$. This is obviously the correct asymptotic behavior for any theory that is massive at zero temperature. Theories that are massless at zero temperature require a resummation to obtain a sensible dispersion relation ${\cal E}(p)$. In this case the parameter $m$ plays acts as an effective thermal mass at large momentum. The asymptotic behavior of one-loop dispersion relations in massless gauge theories is well-known [@LeBellac; @Blaizot; @Smilga1]. The asymptotic behavior Eq. (\[asympt\]) applies to spinless fields, to the spinor field components which have the same helicity as chirality, and to the vector field components that are transversely polarized. It does not apply to the spinor field components that have the helicity opposite to the chirality nor to the longitudinally polarized vector bosons. (Both these cases have asymptotic behavior ${\cal E}(p)\to p+Ap\exp(-p^{2}/m^{2})$. However, in these two cases the residue of the pole vanishes at large momentum like $\exp(-p^{2}/m^{2})$.)
To investigate the branch points that can occur at large momenta, it is useful to introduce three Lagrange multiplier vectors: $\vec{v}_{1},\vec{v}_{2},\vec{P}$ and define $$\begin{aligned}
\Psi=
\sum_{j=1}^{n}{\cal E}(\vec{p}_{j})
-&&\sum_{\ell=1}^{n'}{\cal E}^{*}(\vec{q}_{\ell})
+\vec{v}_{1}\cdot\Big({1\over
2}\vec{k}+\vec{P}-\sum_{j=1}^{n}\vec{p}_{j}\Big)\nonumber\\
+&&\vec{v}_{2}\cdot\Big({1\over 2}\vec{k}-\vec{P}-\sum_{\ell=1}^{n
'}\vec{q}_{\ell}\Big).\label{Cpsi}\end{aligned}$$ This is equivalent to Eq. (\[Bpsi\]) because extremizing with respect to $\vec{P}$ sets $\vec{v}_{1}=\vec{v}_{2}$. However we will compute the extrema of Eq. (\[Cpsi\]) by computing the derivatives in a different order. The pair of conditions $$\begin{aligned}
0=&&{\partial\Psi\over\partial\vec{p}_{j}}=
V(p_{j})\,\hat{p}_{j}-\vec{v}_{1}\nonumber\\
0=&&{\partial
\Psi\over\partial\vec{v}_{1}}={1\over
2}\vec{k}+\vec{P}-\sum_{j=1}^{n}\vec{p}_{j},\nonumber\end{aligned}$$ imply that all $\vec{p}_{j}$ are equal and that they have the common value $$\vec{p}_{j}=({1\over 2}\vec{k}+\vec{P})/n.
\label{commonp}$$ The pair of conditions $$\begin{aligned}
0= &&{\partial\Psi\over\partial \vec{q}_{\ell}}=
-V^{*}(q_{\ell})\,\hat{q}_{\ell}-\vec{v}_{2}\nonumber\\
0=&&{\partial\Psi\over\partial\vec{v}_{2}}={1\over 2}\vec{k}-\vec{P}-
\sum_{\ell=1}^{n'}\vec{q}_{\ell}\nonumber\end{aligned}$$ imply that all the $\vec{q}_{\ell}$ are equal and have the common value $$\vec{q}_{\ell}=({1\over 2}\vec{k}-\vec{P})/n'.\label{commonq}$$ As a result, $$\Psi=n\,{\cal E}\bigg({1\over
n}\big|\vec{P}+\vec{k}/2\big|\bigg)-n'\,
{\cal E}^{*}\bigg({1\over n'}\big|\vec{P}-\vec{k}/2\big|\bigg).$$
The condition $0=\partial\Psi/\partial
\vec{P}$ requires $${\vec{P}+\vec{k}/2\over
|\vec{P}+\vec{k}/2|}\;V\bigg({1\over n}\Big|\vec{P}+{\vec{k}\over
2}\Big|\bigg)=
{\vec{P}-\vec{k}/2\over
|\vec{P}-\vec{k}/2|}\;V^{*}\bigg({1\over n'}\Big|\vec{P}-{\vec{k}\over
2}\Big|\bigg).$$ Regardless of the value of the group velocity in this equation, the two vectors $\vec{P}+\vec{k}/2$ and $\vec{P}-\vec{k}/2$ can only be proportional to each when $\vec{P}_{\bot}=0$, where $\vec{P}=\hat{k}P_{\|}+\vec{P}_{\bot}$. When $\vec{P}_{\bot}=0$ the vectors multiplying $V$ and $V^{*}$, respectively, both are equal to the unit vector $\hat{k}$. Thus the condition reduces to $$V\bigg({1\over n}\Big|P_{\|}+{k\over
2}\Big|\bigg)=
V^{*}\bigg({1\over n'}\Big|P_{\|}-{k\over
2}\Big|\bigg).$$ Because of the presumed asymptotic behavior in Eq. (\[asympt\]), this is satisfied in the limit $P_{\|}\to\pm\infty$. If $V$ is complex, this is the only possible solution. If $V$ is real, then in addition to the solution for infinite $P_{\|}$ there is also a finite solution when $n\neq n'$, namely $\vec{P}_{\|}+k/2=nk /(n\!-\!n')$. The finite solutions was already treated in Sec. III.B and requires no further discussion.
Thus, regardless of the values of $n$ and $n'$, at $P_{\|}\to\infty$ the necessary conditions are satisfied for a branch point. From Eqs. (\[commonp\]) and (\[commonq\]) and the fact that $\vec{P}_{\bot}=0$, the important region of integration is $$\vec{p}_{j}={\hat{k}\over n}({k\over 2}+P_{\|});
\hskip0.8cm
\vec{q}_{\ell}={\hat{k}\over n}({k\over 2}-P_{\|}).\label{pandq}$$ The denominator function is $$\psi=n\,{\cal E}\Big({1\over n}\Big|P_{\|}+{k\over 2}\Big|\Big)
-n'\,{\cal E}^{*}\Big({1\over n'}\Big|P_{\|}-{k\over 2}\Big|\Big).$$ The asymptotic behavior assumed in Eq. (\[asympt\]) implies $$P_{\|}\to +\infty:\hskip0.3cm \psi\to k+{(nm)^{2}\over 2P_{\|}+k}
-{(n'm)^{2}\over 2P_{\|}-k}+\dots$$ Therefore the branch point in $\Pi_{R}(K)$ produced by the denominator $k_{0}-\psi$ will occur at $k_{0}=k$. The region $P_{\|}\to -\infty$ produces a branch point at $k_{0}=-k$. As in Sec. III.B the arguments thus far presented are only necessary conditions for a branch point. To show sufficiency requires a more detailed analysis and this is provided in Appendix B.
### Why an essential singularity {#why-an-essential-singularity .unnumbered}
In Sec. A and B the branch points were produced by particle momenta that were finite. Here the branch points at $k_{0}=\pm k$ are produced by momenta that are infinite and this makes it possible to show that the branch points are essential singularities.
The effect comes from the statistical factor, $S$, in the integrand of the self-energy contribution that contains $n$ emitted particles and $n'$ absorbed particles: $$S=S_{\rm direct}-\sigma S_{\rm inverse},\label{statistical1}$$ where $\sigma=1$ for a boson self-energy and $\sigma=-1$ for a fermion self-energy. The statistical factors are $$\begin{aligned}
S_{\rm
direct}=&&\prod_{j=1}^{n}[1+\sigma_{j}N_{j}]\,\prod_{\ell=1}^{n'}
N^{*}_{\ell}
\\
S_{\rm
inverse}=&&\prod_{j=1}^{n}N_{j}\,\prod_{\ell=1}^{n'}
[1+\sigma_{\ell}N^{*}_{\ell}]\end{aligned}$$ where $\sigma=\pm 1$ for bosons and fermions and $$\begin{aligned}
N_{j}=&&1/[\exp(\beta {\cal E}(\vec{p}_{j}))-\sigma_{j}]\nonumber\\
N^{*}_{\ell}=&&1/[\exp(\beta
{\cal E}^{*}(\vec{q}_{\ell}))-\sigma_{\ell}]\nonumber.\end{aligned}$$ Because of Eq. (\[asympt\]) and (\[pandq\]), in the region $P_{\|}\to\infty$ the statistical factor becomes $$S\to \big( e^{\beta k/2}-\sigma e^{-\beta k/2}\big)\,
\exp\big(-\beta P_{\|}\big).\label{statistical2}$$
By finding the way in which $P_{\|}$ approaches infinity as $k_{0}\to k$, one can be more specific about the nature of the branch point. Near the branch point, $k_{0}-k$ is very small but non-zero, and the self-energy denominator $k_{0}-\psi$ will vanish when $P_{\|}$ is very large but not actually infinite. The condition $k_{0}-\psi=0$ gives a quadratic equation for $P_{\|}$. The two roots are $$\begin{aligned}
P_{\|\pm}=&&{1\over 4(k_{0}-k)}
\bigg[(n^{2}\!-\!n'^{2})m^{2}\nonumber\\
\pm&&\Big(\big[2k(k_{0}\!-\!k)
-(n^{2}\!+\!n'^{2})m^{2}]^{2}
-[2nn'm^{2}]^{2}\Big)^{1/2}\bigg].\nonumber\end{aligned}$$ There are two cases to be distinguished.
#### Case 1. $n=n'$: {#case-1.-nn .unnumbered}
If $k_{0}$ is real and approaches $k$ from below, the root that approaches $+\infty$ is $$P_{\|-}={k\over 2}\sqrt{1-{2(nm)^{2}\over k(k_{0}-k)}}.
\label{root}$$ The statistical factor Eq. (\[statistical2\]) becomes $$S\to \big( e^{\beta k/2}-\sigma e^{-\beta k/2}\big)
\exp\Bigg(\!-{\beta k\over 2}\sqrt{1-{2(nm)^{2}\over
k(k_{0}-k)}}\;\Bigg).\label{statistical3}$$ This is an essential singularity at $k_{0}=k$. If $k_{0}$ is real and approaches $k$ from below, then $S\to 0$. More generally the behavior depends on how $k_{0}-k$ approaches zero in the complex plane. Appendix C provides a one-loop calculation with $n=n'=1$ that displays this behavior in Eq. (\[essential1\]).
#### Case 2. $n\neq n'$: {#case-2.-nneq-n .unnumbered}
For definiteness take $n>n'$. Then if $k_{0}$ is real and approaches $k$ from above, the root that approaches $+\infty$ is $$P_{\|+}\to{(n^{2}-n'^{2})m^{2}\over 2(k_{0}-k)}.$$ The statistical factor Eq. (\[statistical2\]) becomes $$S\to \big( e^{\beta k/2}\!-\!\sigma e^{-\beta k/2}\big)
\exp\Bigg(\!-{\beta(n^{2}-n'^{2})m^{2}\over 2(k_{0}-k)}\Bigg).
\label{statistical4}$$ This, again, is an essential singularity at $k_{0}=k$, whose behavior naturally depends upon the direction from which $k_{0}$ approaches $k$. Appendix D provides a two-loop calculation with $n=2$, $n'=1$, and $k=0$. The exponent is predicted to be $-\beta 3m^{2}/2k_{0}$, and this is just what is found in Eq. (\[essential2\]).
#### Comment on Hard Thermal Loops: {#comment-on-hard-thermal-loops .unnumbered}
The one-loop gluon self-energy has branch points at $k_{0}=\pm k$ [@LeBellac; @Blaizot; @Smilga1]. These come from intermediate states with $n=1, n'=1$ (either two gluons or two fermions). The one-loop calculations are done using a massless dispersion relation for the intermediate particles. Therefore $m=0$ in Eq. (\[asympt\]) so that $P_{\|+}=k/2$ in Eq. (\[root\]). Since this momenta is finite, the statistical factor $S$ is unremarkable and cannot produce an essential singularity. Explicit calculations show that the branch points at $k_{0}=\pm k$ are logarithmic for the hard thermal loops.
Summary
-------
In the following summary the single-particle energies ${\cal
E}(p)$ can be complex or real. For results that only apply when the energies are real, the block $E$ will be used instead of the script ${\cal E}$. Real $E$ produces the exceptional branch points discussed in Sec. III.B and they will be described in parentheses in the summary below. The ubiquitous branch points at $k_{0}=\pm k$ are always essential singularities and this will not be repeated each time. As noted following Eq. (\[defpsi1\]), when $n$ and/or $n'$ have the value $1$, the range of $\psi$ is constrained by momentum conservation and this often determines the end points of the branch cuts.
### Organized by $n'$, the number of absorptions
The most concise way to summarize the previous results is by $n'$, the number of particles absorbed from the heat bath.
#### (a) No absorptions: $n'=0$. {#a-no-absorptions-n0. .unnumbered}
This is the simplest case and directly analogous to zero temperature. The branch cut is semi-infinite: $ n\,{\cal E}(k/n)\le
k_{0}\le\infty$.
#### (b) One absorption: $n'=1$. {#b-one-absorption-n1. .unnumbered}
There are two subcases. If $n=1$ then the branch cut is only for space-like four momenta: $-k\le
k_{0}\le k$. If $n\ge 2$ then the branch cut is semi-infinite: $-k\le k_{0}
\le\infty$; and there is a branch point at $k_{0}=k$. (If the single-particle energies are real, there is an exceptional branch point at $k_{0}=[n-1]E(k/[n-1])$. For $n=2$ the last branch point coincides with the free particle pole at $k_{0}=E(k)$.)
#### (c) Two or more absorptions: $n'\ge 2$. {#c-two-or-more-absorptions-nge-2. .unnumbered}
There are three subcases. If $n=0$ the branch cut is semi-infinite: $-\infty\le k_{0}\le -n'{\cal E}^{*}(k/n')$. If $n=1$ the branch cut is also semi-infinte: $-\infty\le k_{0}\le
k$, and there is an additional branch point at $k_{0}=-k$. If $n\ge 2$ the branch cut runs the full length of the real axis: $-\infty\le k_{0}\le\infty$; and there are two additional branch points at $k_{0}=\pm k$. (If the single-particle energies are real, there are exceptional branch points for $n\neq n'$ at $k_{0}=[n-n']E(k/[n-n'])$. Whenever $n-n'=\pm 1$ this last branch point coincides with the free particle poles at $k_{0}=\pm E(k)$.)
### Organized by $n+n'$, the number of particles in the intermediate state
A particular diagram can generally be cut in several possible ways. Each cut is through a particular number of propagators or equivalently through an intermediate state with a particular number of particles. As a practical matter, this is perhaps the most useful way to summarize the cut structure.
#### (a) Two-particle states: {#a-two-particle-states .unnumbered}
Two-particle intermediate states, $n+n'=2$, are possible with a cubic coupling but not with a quartic coupling. There are three types of branch cuts. For $n=2, n'=0$ there will be a semi-infinite cut $2{\cal E}(k/2)\le k_{0}\le\infty$. For $n=n'=1$ there will be a finite length branch cut $-k\le k_{0}\le k$. For $n=0,n'=2$ there will be a semi-infinite branch cut $-\infty\le k_{0}\le -2{\cal E}^{*}(k/2)$.
#### (b) Three-particle states. {#b-three-particle-states. .unnumbered}
The possibility $n+n'=3$ occurs with both cubic and quartic coupling. There are four types of branch cuts. For $n=3, n'=0$ there will be a semi-infinite branch cut $3{\cal E}(k/3)\le k_{0}\le\infty$. For $n=2, n'=1$ there will be a semi-infinite branch cut for $-k\le k_{0}\le\infty$ and in addition there will be a branch point attached to the cut at $k_{0}=k$. (If the single-particle energies are real, there will be an exceptional branch point at $k_{0}=E(k)$.) For $n=1$, $n'=2$ the range of the branch cut is $-\infty\le k_{0}\le k$ with an attached branch point at $k_{0}=-k$. (If the single-particle energies are real, there will be an exceptional branch point at $k_{0}=-E(k)$.) For $n=0$, $n'=3$ the extent of the branch cut will be $-\infty\le k_{0}\le
-3{\cal E}^{*}(k/3)$.
#### (c) Four-particle states: {#c-four-particle-states .unnumbered}
For a cut through $n+n'=4$ propagators, there are five different types of branch cuts. For $n=4$, $n'=0$ there is only a four-particle production cut for $4{\cal E}(k/4)\le k_{0}\le \infty$. For $n=3$, $n'=1$ the range of the branch cut is $-k\le k_{0}\le\infty$ with an additional branch point at $k_{0}=k$. (If the single-particle energies are real, there will be an exceptional branch point at $k_{0}=2E(k/2)$.) For $n=n'=2$ the branch cut runs the full length of the real axis, $-\infty\le k_{0}\le \infty$ with two additional branch points at $k_{0}=-k$ and $k_{0}=k$. For $n=1$, $n'=3$ the range of the branch cut is $-\infty\le k_{0}\le k$ with an additional branch point at $k_{0}=-k$. (If the single-particle energies are real, there will be an exceptional branch point at $k_{0}=-2E(k/2)$.) For $n=0$, $n'=4$ there is only a four-particle absorption cut for $-\infty\le k_{0}\le -4{\cal E}^{*}(k/4)$.
Comments
========
Expectations for the exact self-energy
--------------------------------------
Section III.A showed that there will a branch point in the retarded self-energy $\Pi_{R}(k_{0},\vec{k})$ at $k_{0}=n{\cal E}(\vec{k}/n)$ that results from the emission of $n$ particles and, likewise, a branch point at $k_{0}=-n{\cal
E}^{*}(\vec{k}/n)$ that results from the absorption of $n$ particles from the heat bath. The effective single-particle energies will generally be temperature-dependent and complex and so the location of the branch points will generally be temperature-dependent and complex. Furthermore, the location of the branch points is model-dependent in the sense that one can change the single-particle energies ${\cal
E}$ and thus change the location of the branch points in the perturbative expansion.
However the location of the branch points in the exact self-energy cannot be model-dependent. If one summed the perturbative self-energy contributions to all orders, the model-dependence of the branch points would disappear just as the model-dependence of the propagator pole would disappear. If the exact propagator has a pole at ${\cal E}_{\rm exact}(p)$, the exact self-energy should have normal-threshold branch points at $k_{0}=n\,{\cal E}_{\rm exact}(\vec{k}/n)$ and at $k_{0}=-n\,{\cal
E}_{\rm exact}^{*}(\vec{k}/n)$. In addition, there will be essential singularities at $k_{0}=\pm k$.
Bad features of real dispersion relations
-----------------------------------------
Performing perturbative calculations using the free thermal propagator or indeed any thermal propagator containing a real dispersion relation $E(p)$ leads to a self-energy that has branch points at the perturbative mass-shell, $k_{0}=E(k)$. If $\Pi_{R}(k_{0})$ is the retarded self-energy computed beyond one-loop order using a real energy $E(p)$ to define the perturbation series, then to improve on the value of $E(p)$ one needs to solve perturbatively for ${\cal E}_{\rm pole}$ $${\cal E}_{\rm pole}=\sqrt{E^{2}+\Pi_{R}({\cal E}_{\rm pole})}.
\label{pole1}$$ A perturbative expansion means that $\Pi_{R}(E)$ and its derivatives at $E$ are small compared to $E$. Thus the lowest-order contribution to the damping rate should be $${\rm Im}\,{\cal E}_{\rm pole}\approx {\rm Im} \big(\Pi_{R}(E)\big)/2E.$$ However this will fail at two-loop order because $\Pi_{R}(k_{0})$ has a branch point precisely at $k_{0}=E$. This was first encountered in calculations of the fermion damping rate in QCD, where it was found that even with a magnetic mass to eliminate the infrared divergence there is a branch point at $k_{0}=E$ that comes from the intermediate state with $n=2, n'=1$ [@damping]. Appendix D contains an explicit calculation in $\phi^{4}$ theory that shows that the self-energy is not differentiable at the mass-shell.
If one pretends that the self-energy is differentiable at $E$ then the solution to Eq. (\[pole1\]) would be $${\cal E}_{\rm pole}=E
+\sum_{s=1}^{\infty}{(-1)^{s}\over s!}\big[\Pi_{R}(E)\big]^{s}
\Bigg[{d^{s-1}\over dk_{0}^{s-1}}{1\over \big[f(k_{0})\big]^{s}}
\Bigg]_{k_{0}=0},$$ where $f(k_{0})$ is the function $$f(k_{0})=-k_{0}-2E
+\sum_{\ell=1}^{\infty}{(k_{0})^{\ell-1}\over \ell !}
{d^{\ell}\Pi_{R}(E)\over dE^{\ell}}.$$ However $d^{\ell}\Pi_{R}(E)/dE^{\ell}$ does not exist and thus the perturbative calculation of ${\cal E}_{\rm pole}$ fails.
There is another consequence of real dispersion relations that is curious, though perhaps not as dire. Diagrams in which two particles are emitted and none absorbed ($n=2, n'=0$) will have the usual normal threshold branch point at $k_{0}=2E(\vec{k}/2)$. All contributions in which $n'\!+\!2$ particles are emitted and $n'$ are absorbed will also have a branch point at $k_{0}=2E(\vec{k}/2)$. Similarly, All contributions in which $n'\!+\!3$ particles are emitted and $n'$ are absorbed will also have a branch point at $k_{0}=3E(\vec{k}/3)$. These coincidences will be absent if a complex dispersion relation is employed.
Good feature of complex dispersion relations
--------------------------------------------
Any complex dispersion relation (even if the group velocity is real) will generate a perturbative expansion that does not have a branch point in the higher order self-energy at $k_{0}={\cal E}(k)$ as show in Sec. III.B. Let $\Pi^{\rm eff}_{R}(k_{0})$ be the self-energy computed beyond one-loop order using the free retarded propagator in Eq. (\[retprop2\]). The radiatively-corrected retarded propagator is $$D^{\rm eff}_{R}(k_{0})={1\over \big(k_{0}-{\cal E}\big)
\big(k_{0}+{\cal E}^{*}\big)-\Pi^{\rm eff}_{R}(k_{0})}.$$ The pole in this propagator satisfies $$\big({\cal E}_{\rm pole}-{\cal E}\big)
\big({\cal E}_{\rm pole}+{\cal E}^{*}\big)
=\Pi^{\rm eff}_{R}({\cal E}_{\rm pole}).\label{pole4}$$ Since the self-energy $\Pi^{\rm eff}_{R}(k_{0})$ does not have a branch point at $k_{0}={\cal E}$, it is infinitely differentiable there. The perturbative solution to Eq. (\[pole4\]) is $${\cal E}_{\rm pole}= {\cal E}
+\sum_{s=1}^{\infty}{(-1)^{s}\over
s!}\big[\Pi^{\rm eff}_{R}({\cal E})\big]^{s}
\Bigg[{d^{s-1}\over dk_{0}^{s-1}}{1\over \big[g(k_{0})\big]^{s}}
\Bigg]_{k_{0}=0},$$ where $g(k_{0})$ is the function $$g(k_{0})=-k_{0}\!-\!2\omega
+\sum_{\ell=1}^{\infty}{(k_{0})^{\ell-1}\over \ell !}
{d^{\ell}\Pi^{\rm eff}_{R}({\cal E})\over d{\cal E}^{\ell}}.$$ In this case, $g(k_{0})$ does exist and the perturbative expansion for ${\cal E}_{\rm pole}$ is valid.
This work was supported in part by National Science Foundation grant PHY-0099380.
Detailed proof of branch point at E( for
=========================================
In Sec. III.B it was shown that if the single-particle energy $E$ is real, the necessary conditions for a singularity are satisfied at $$k_{0}=(n\!-\!n')\, E\big(\vec{k}/[n\!-\!n']\big).$$ This Appendix proves sufficiency, viz. that there really is a branch point. The momenta which trap the integration contour are near a saddle point and this makes the analysis different than at zero temperature.
Taylor series expansion of $\psi$ near the saddle point
-------------------------------------------------------
It is convenient to label all the loop momenta as $\vec{p}_{j}$ so that the denominator function is $$\psi=\sum_{j=1}^{n}E(\vec{p}_{j})-
\sum_{j=n+1}^{n+n'}E(\vec{p}_{j})\label{Apsi1}.$$ Define $\vec{s}=\vec{k}/(n-n')$ and put $$\vec{p}_{j}=\cases{+\vec{s}+\vec{\alpha}_{j},
& $1\le j\le n$\cr
-\vec{s}+\vec{\alpha}_{j}, & $n+1\le j\le n+n'$}.$$ The stationary point of $\psi$ that was found in Sec. III.B occurs when all $\alpha_{j}=0$. Momentum conservation requires that $$0=\sum_{j=1}^{n+n'}\vec{\alpha}_{j}.\label{Amomentum}$$
A typical energy can be expanded in a Taylor series to second order in the small quantities $\vec{\alpha}_{j}$. In doing this it is convenient to decompose the vectors into components parallel and perpendicular to $\vec{k}$ or equivalently to $\vec{s}$. Thus $\vec{\alpha}_{j}=\hat{k}\alpha_{j\|}+\vec{\alpha}_{j\bot}$. The Taylor series can then be written to second order as $$E(\vec{s}+\vec{\alpha}_{j})=E(\vec{s})+{dE\over ds}
\alpha_{j\|}
+{1\over
2}A_{\|}\alpha_{j\|}^{2}+{1\over
2}A_{\bot}\vec{\alpha}_{j\bot
}^{2}+\dots$$ where
$$\begin{aligned}
A_{\|}= &&{d^{2}E\over ds^{2}} \\
A_{\bot}=&& {1\over s}\,{dE\over ds}.\end{aligned}$$
\[defA\]
For a free particle, $E(s)=(s^{2}+m^{2})^{1/2}$, and $A_{\|}=m^{2}/E^{3}$ and $A_{\bot}=1/E$. For a quasiparticle dispersion relation, one or both of $A_{\|}$ and $A_{\bot}$ could be negative. This will not alter the following argument.
When this expansion is inserted into Eq. (\[Apsi1\]), the terms linear in $\vec{\alpha}_{j}$ cancel because of momentum conservation and leave $$\begin{aligned}
\psi=(n\!-\!n')E(s)+&&{1\over
2}A_{\|}\bigg(\sum_{j=1}^{n}\alpha_{j\|}^{2}
-\sum_{j=n+1}^{n+n'}\alpha_{j\|}^{2}\bigg)\nonumber\\
+&&{1\over
2}A_{\bot}\bigg(\sum_{j=1}^{n}\vec{\alpha}_{j\bot}^{2}-
\sum_{j=n+1}^{n+n'}\vec{\alpha}_{j\bot}^{2}\bigg)
\label{Apsi2}.\end{aligned}$$ The constraint in Eq. (\[Amomentum\]) means that there are only $n+n'-1$ linearly independent momentum vectors. One can eliminate the last momentum, $\vec{\alpha}_{n+n'}$, by expressing it as the sum of all the other $\vec{\alpha}$’s. When this is done, Eq. (\[Apsi2\]) is no longer diagonal, but it will be real and symmetric and can therefore be diagonalized by a real rotation. The rotation that diagonalizes the terms proportional to $A_{\|}$ will also diagonalize the terms proportional to $A_{\bot}$.
Diagonalization of $\psi$
-------------------------
The simplest case is $n=2, n'=1$, i.e. a three particle intermediate state in which two particles are emitted and one is absorbed. From the constraint $\vec{\alpha}_{1}+\vec{\alpha}_{2}+\vec{\alpha}_{3}=0$, express $\vec{\alpha}_{3}$ in terms of the other two. Then $$\begin{aligned}
\psi=E(k)+ &&{1\over 2}A_{\|}\big(\alpha_{1\|}^{2}+
\alpha_{2\|}^{2}
-(\alpha_{1\|}+\alpha_{2\|})^{2}\big)
\nonumber\\
+&&{1\over 2}A_{\bot}\big(\vec{\alpha}_{1\bot}^{2}
+\vec{\alpha}_{2\bot}^{2}-(\vec{\alpha}_{1\bot}
+\vec{\alpha}_{2\bot})^{2}\big)\nonumber\\
= E(k)- && A_{\|}\alpha_{1\|}\alpha_{2\|}
-A_{\bot}\vec{\alpha}_{1\bot}\cdot\vec{\alpha}_{2\bot}.
\nonumber\end{aligned}$$ This can be easily diagonalized by defining new momenta $\vec{u}=(\vec{\alpha}_{1}+\vec{\alpha}_{2})/2$ and $\vec{v}=(\vec{\alpha}_{1}-\vec{\alpha}_{2})/2$ so that $$\psi=E(k)+A_{\|}\big(-u_{\|}^{2}+v_{\|}^{2}\big)
+A_{\bot}\big(-u_{\bot}^{2}+v_{\bot}^{2}\big).$$ The diagonal form is a function of the six Cartesian components of $\vec{u}$ and $\vec{v}$. The diagonal form is traceless with three positive terms and three negative terms regardless of the signs of $A_{\|}$ and $A_{\bot}$.
For general $n$ and $n'$ express $\vec{\alpha}_{n+n'}$ in terms of the other $\vec{\alpha}'s$ using Eq. (\[Amomentum\]). Then $\psi$ has the form $$\begin{aligned}
\psi=(n\!-\!n')E(s)- &&A_{\|}\sum_{i,j=1}^{n+n'-1}
M_{ij}\;\alpha_{i\|}\alpha_{j\|}\nonumber\\
-&&A_{\bot}\sum_{i,j=1}^{n+n'-1}M_{ij}\;
\vec{\alpha}_{i\bot}\cdot\vec{\alpha}_{j\bot},\end{aligned}$$ where $M_{ij}$ is a simple numerical matrix. The eigenvalues of this matrix will be called $\lambda_{j}$. The matrix can be diagonalized by by a real rotation to a new basis set $\vec{\beta}_{j}$. In the new basis $\psi$ will be diagonal $$\begin{aligned}
\psi=(n\!-\!n')E(s)- &&A_{\|}\sum_{j=1}^{n+n'-1}
\lambda_{j}\;\beta_{j\|}^{2}\nonumber\\
- &&A_{\bot}\sum_{j=1}^{n+n'-1}\lambda_{j}\;
\vec{\beta}_{j\bot}^{2}\end{aligned}$$ None of the eigenvalues $\lambda_{j}$ vanish as the following analysis will show.
#### Case 1. $n'=1$: {#case-1.-n1 .unnumbered}
When there is only one absorption $n'=1$ and any number of emissions $n\ge 2$, the matrix elements $M_{ij}$ have the following values $$M_{ij}=\cases{ 1, & $i\neq j$\cr
0, & $i=j$.}$$ There are $n$ eigenvalues of this matrix: $$\lambda=\cases{n-1, & degeneracy $=1$ \cr
-1, & degeneracy $= n-1$.}$$ Thus the diagonal form of $\psi$ is $$\begin{aligned}
\psi=(n\!-\!n')E(s)- &&A_{\|}
\bigg((n\!-\!1)\beta_{\|}^{2}-
\sum_{j=1}^{n-1}\beta_{j\|}^{2}\bigg)\nonumber\\
-&&
A_{\bot}\bigg((n\!-\!1)\vec{\beta}_{\bot}^{2}
-\sum_{j=1}^{n-1}\vec{\beta}_{j\bot}^{2}\bigg).\end{aligned}$$
#### Case 2. $n'\ge 2$: {#case-2.-nge-2 .unnumbered}
When there are $n'\ge 2$ absorptions and a larger number $n>n'$ of emissions, the matrix elements are $$M_{ij}=\cases{ 1, & $i\neq j$\cr
0, & $i=j\leq n$\cr
2, & $i=j\ge n+1$.}$$ The matrix has two non-trivial eigenvalues: $$\lambda_{\pm}={1\over
2}\bigg[n+n'-1\pm\sqrt{(n+n'-1)^{2}-4(n-n')}\bigg].$$ Both $\lambda_{+}$ and $\lambda_{-}$ are positive. The complete set of eigenvalues are $$\lambda=\cases{\lambda_{+}, & degeneracy $=1$\cr
\lambda_{-}, & degeneracy $=1$\cr
1, & degeneracy $=n'-2$\cr
-1, & degeneracy $=n-1$.}$$ Thus $n'$ eigenvalues are positive and $n-1$ are negative.
Thus there are always $n'$ positive eigenvalues, which will be labeled $|\lambda_{1}|,\dots, |\lambda_{n'}|$, and $n-1$ negative eigenvalues, which will be labeled $-|\lambda_{n'+1}|,\dots ,-|\lambda_{n'+n-1}|$. Then $\psi$ has the form $$\begin{aligned}
\psi=(n\!-\!n')E(s) &&-\sum_{j=1}^{n'}
|\lambda_{j}|\;\Big(A_{\|}\beta_{j\|}^{2}
+A_{\bot}\vec{\beta}_{j\bot}^{2}\Big)\nonumber\\
+ &&\sum_{j=n'+1}^{n+n'-1}|\lambda_{j}|\;
\Big(A_{\|}\beta_{j\|}^{2}
+A_{\bot}\vec{\beta}_{j\bot}^{2}\Big).\label{Apsi}\end{aligned}$$ The contribution to the retarded self-energy of this small region of momentum space is $$\Pi_{R}(k_{0})=\int\Big(\prod_{j=1}^{n+n'-1}\!d^{3}\beta_{j}
\Big)\;{f(\vec{\beta}_{j})\over k_{0}-\psi}$$ In the quadratic approximation, $\psi$ does not depend separately on all the vectors $\vec{\beta}_{j}$ but only on two real variables $u$ and $v$ such that $$\psi=(n\!-\!n')E(s)-u^{2}+v^{2}.$$ As noted earlier, $A_{\|}$ and $A_{\bot}$ are expected to be positive. If that is the case, define $u$ and $v$ by $$\begin{aligned}
u=&&\bigg[\sum_{j=1}^{n'}
|\lambda_{j}|\;\big(A_{\|}\beta_{j\|}^{2}
+A_{\bot}\vec{\beta}_{j\bot}^{2}\big)\bigg]^{1/2}\\
v=&&\bigg[\sum_{j=n'+1}^{n'+n-1}
|\lambda_{j}|\;\big(A_{\|}\beta_{j\|}^{2}
+A_{\bot}\vec{\beta}_{j\bot}^{2}\big)\bigg]^{1/2}.\end{aligned}$$ However, if one or both of $A_{\|}$ and $A_{\bot}$ are negative, then define $-u^{2}$ as the sum of the terms that enter negatively in Eq. (\[Apsi\]) and $v^{2}$ as the sum of the terms that enter positively in Eq. (\[Apsi\]). Obviously $u$ and $v$ are real and positive. The relevant integration is only over these two variables and is of the form $$\int_{0}^{u_{\rm max}}\!du\int_{0}^{v_{\rm max}}\!dv
\;{h(u,v)\over \omega-u^{2}+v^{2}},\label{numeratorh}$$ where $$\omega=k_{0}-(n-n')E(k/[n-n']).\label{defomega}$$ A branch point in the self-energy is now reduced to the question of showing that Eq. (\[numeratorh\]) has a branch point at $\omega=0$. The limits $u_{\rm max}$ and $v_{\rm max}$ represent the region of validity of the second order Taylor series expansion. The integral as written has a branch cut on the real $\omega$ axis for $-v_{\rm
max}^{2}<\omega <u_{\rm max}^{2}$.
Existence of the branch point
-----------------------------
The question at hand is whether the integral Eq. (\[numeratorh\]) has, in addition to the branch cut along the real axis, a branch point at $\omega=0$. The putative existence of such a branch point clearly comes from the region $u\approx v$ and has nothing to do with the upper limits of integration and nothing to do with the numerator function $h(u,v)$. To complete the analysis it is therefore sufficient to examine the function $$f(\omega)=\int_{0}^{M}\!du\int_{0}^{M}\!dv\;{1\over\omega-u^{2}+v^{2}}
.$$ Although this integral cannot be performed explicitly, it is possible to prove the existence of a branch point at $\omega=0$. To analyze this integral it is useful to split the integration over $v$ into two parts: $$\begin{aligned}
f(\omega)=&&\int_{0}^{M}\!du\int_{0}^{u}\!dv\;{1\over
\omega-u^{2}+ v^{2}}\nonumber\\
+&&\int_{0}^{M}\!du\int_{u}^{M}\!dv\;{1\over
\omega-u^{2}+ v^{2}}.\nonumber\end{aligned}$$ In the first integral, replace $v$ by $x=\sqrt{u^{2}-v^{2}}$. In the second, replace $v$ by $x=\sqrt{v^{2}-u^{2}}$: $$\begin{aligned}
f(\omega)=&&\int_{0}^{M}\!du\int_{0}^{u}\! dx
\:{x\over\sqrt{u^{2}-x^{2}}}\;{1\over \omega-x^{2}}\nonumber\\
+&&\int_{0}^{M}\!du\int_{0}^{\sqrt{M^{2}-u^{2}}}\!\!dx
\;{x\over\sqrt{u^{2}+x^{2}}}\;{1\over \omega+x^{2}}.\nonumber\end{aligned}$$ Now interchange the order of integration in both and perform the integrations over $u$ to obtain $$f(\omega)\!=\!\int_{0}^{M}\! dx\bigg({x\over\omega-x^{2}}
+{x\over\omega+x^{2}}\bigg)
\ln\!\bigg[{M+\sqrt{M^{2}-x^{2}}\over x}\bigg].$$ Note that $f(0)$ is finite, but $\big[df(\omega)/d\omega\big]_{\omega=0}$ is divergent as are all the odd derivatives. This already shows that $f(\omega)$ is not analytic at $\omega=0$. (Note that the original function in Eq. (\[numeratorh\]) contains a numerator $h(u,v)$ which could vanish at $x=0$. Consequently the divergence of the first derivative of $f(\omega)$ may not hold when the numerator is included. However, higher derivatives will diverge.)
To confirm the branch point at $\omega=0$ the best procedure is to analytically continue $\omega$ in a small circle enclosing the origin. The integrand of $f(\omega)$ has simple poles at $x_{1}=\sqrt{\omega}$, $x_{2}=-\sqrt{\omega}$, $x_{3}=i\sqrt{\omega}$, and $x_{4}=-i\sqrt{\omega}$. When $\omega$ has a small, positive imaginary part, these singularities are off the real axis. To expose the branch point at $\omega=0$, set $\omega=re^{i\phi}$. Then as $\phi$ increases from $0^{+}$ to $2\pi^{+}$, all four $x_{j}$ move in small counter-clockwise circles and return to different values: $x_{1}$ moves to the negative real axis without coming near the integration contour; $x_{2}$ moves counter-clockwise through the integration contour into the upper half-plane at the position originally occupied by $x_{1}$. The change in the value of the integral can be computed by integrating in a small circular contour $C_{1}$ around the position originally occupied by $x_{1}$: $$\begin{aligned}
\oint_{C_{1}}\!dx\,{x\over
\omega-x^{2}}\ln\!\bigg[{M+\sqrt{M^{2}-x^{2}}\over
x}\bigg]&&\nonumber\\
=i\pi \ln\!\bigg[{M+\sqrt{M^{2}-\omega}\over
\sqrt{\omega}}\bigg].&&\label{c1}\end{aligned}$$ Likewise, as $\phi$ increases from $0^{+}$ to $2\pi^{+}$, $x_{3}$ moves from the positive imaginary axis clockwise to the negative imaginary axis without touching to the real axis. However, $x_{4}$ moves counter clockwise from the negative real axis to the positive real axis and drags the $x$ contour with it. The change in the value of the integral from this distortion can be computed by integrating in a small circular contour $C_{3}$ around the position originally occupied by $x_{3}$: $$\begin{aligned}
\oint_{C_{3}}\!dx\,{x\over
\omega+x^{2}}\ln\!\bigg[{M+\sqrt{M^{2}-x^{2}}\over
x}\bigg]&&\nonumber\\
=-i\pi \ln\!\bigg[{M+\sqrt{M^{2}+\omega}\over
i\sqrt{\omega}}\bigg].&& \label{c2}\end{aligned}$$ The change in $f(\omega)$ resulting from encircling the origin is the sum of (\[c1\]) and (\[c2\]): $$f\big(e^{2\pi i}\omega\big)\!-\!f(\omega)=
i\pi \ln\!\bigg[{i\big(M+\sqrt{M^{2}-\omega}\big)\over
M+\sqrt{M^{2}+\omega}}\bigg].$$ The non-vanishing of the right hand side confirms that there is a branch point at $\omega=0$ and completes the proof.
The branch point has infinitely many sheets because if one rotates the phase of $\omega$ by $2\pi N$ the result is $$f\big(e^{2\pi N i}\omega\big)\!-\!f(\omega)=
iN\pi \ln\!\bigg[{i\big(M+\sqrt{M^{2}-\omega}\big)\over
M+\sqrt{M^{2}+\omega}}\bigg].$$
Detailed proof of branch point at
==================================
In Sec. III.C it was shown that for a real or complex single-particle energy with the asymptotic behavior in Eq. (\[asympt\]) the necessary conditions for an singularity are satisfied at $k_{0}=\pm k$. This Appendix proves that there is a branch point.
As before, it is necessary to examine the integration in the region at which the contour is trapped. The denominator function is $$\psi=\sum_{j=1}^{n}{\cal E}(\vec{p}_{j})-
\sum_{j=n+1}^{n+n'}{\cal E}(\vec{p}_{j})\label{Bpsi1}.$$ Define the two momenta $$\begin{aligned}
\vec{s}={\hat{k}\over n}\big({k\over
2}+P_{\|}\big);\hskip1cm
\vec{s}^{\,\prime}={\hat{k}\over n'}\big({k\over
2}-P_{\|}\big),\end{aligned}$$ and put $$\vec{p}_{j}=\cases{\vec{s}+\vec{\alpha}_{j}\sqrt{P_{\|}},
& $1\le j\le n$\cr
\vec{s}^{\,\prime}+\vec{\alpha}_{j}\sqrt{P_{\|}}, & $n+1\le j\le
n+n'$.}$$ Momentum conservation requires that $$0=\sum_{j=1}^{n+n'}\vec{\alpha}_{j}.\label{Bmomentum}$$ When all the $\vec{\alpha}_{j}=0$ $$\psi\big|_{\vec{\alpha}_{j}=0}=n{\cal E}(\vec{s})-n'{\cal
E}(\vec{s}').$$ To demonstrate that there is a branch point it is necessary to expand $\psi$ in a Taylor series for $|\vec{\alpha}_{j}|\sqrt{P_{\|}}$ small compared to $|\vec{s}|$ and $|\vec{s}'|$: $$\begin{aligned}
\psi=&&nE(s)+{P_{\|}\over 2}\sum_{j=1}^{n}\bigg(
A_{\|}\alpha_{j\|}^{2}+
A_{\bot}\vec{\alpha}_{j\bot}^{2}\bigg)\nonumber\\
-&&n'E(s')-{P_{\|}\over 2}\sum_{j=n+1}^{n+n'}\bigg(
A_{\|}^{\prime}\alpha_{j\|}^{2}+
A_{\bot}^{\prime}\vec{\alpha}_{j\bot}^{2}\bigg).\nonumber\end{aligned}$$ The terms linear in $\vec{\alpha}_{j}$ canceled by momentum conservation as in Appendix A. The coefficients $A_{\|}$ and $A_{\bot}$ are as defined in Eq. (\[defA\]).
The branch point we are seeking occurs when $P_{\|}\to\infty$. In this limit both $s$ and $s'$ approach infinity so that $$\begin{aligned}
{\cal E}(s)\to && s+{m^{2}\over 2s}+\dots\nonumber\\
{d{\cal E}\over ds}\to && 1-{m^{2}\over 2s^{2}}+\dots\nonumber\\
A_{\|}\to && {m^{2}\over 3s^{2}}+\dots\nonumber\\
A_{\bot}\to && {1\over s}-{m^{2}\over 2s^{3}}+\dots\nonumber\end{aligned}$$ At large $P_{\|}$ the denominator function $\psi$ behaves as $$\begin{aligned}
\psi\to && k+{(mn)^{2}\over 2P_{\|}+k}-{(mn')^{2}\over 2P_{\|}-k}
\nonumber\\
+&&{nP_{\|}\over 2P_{\|}+k}\sum_{j=1}^{n}\vec{\alpha}_{j\bot}^{2}
-{n'p_{\|}\over 2P_{\|}-k}\sum_{j=n+1}^{n+n'}\vec{\alpha}_{j\bot}^{2}.
\nonumber\end{aligned}$$ Now take $P_{\|}\to\infty$ so that $$P_{\|}\to\infty:
\hskip0.5cm \psi=k+{n\over 2}\sum_{j=1}^{n}\vec{\alpha}_{j\bot}^{2}
-{n'\over 2}\sum_{j=n+1}^{n+n'}\vec{\alpha}_{j\bot}^{2}.$$ This is of the same form as Eq. (\[Apsi2\]) except that $\vec{\alpha}_{j\|}$ do not enter. Therefore, the proof from Appendix A applies. The contribution to the retarded self-energy is $$\Pi_{R}(k_{0})=\int\Big(\prod_{j=1}^{n+n'}\!d^{3}\alpha_{j}
\Big)\;{f(\vec{\alpha}_{j})\over k_{0}-\psi}$$ The proof in Appendix A shows that if $k_{0}=k+re^{i\phi}$ then the value of $\Pi_{R}(k_{0})$ does not return to the same value when $\phi$ increases from 0 to $2\pi$.
$\phi^{3}$ Example to one loop
==============================
The simplest example of an essential singularity at $k_{0}=\pm k$ occurs in a theory with ${\cal H}_{I}=g\phi^{3}/3!$. With the free-particle dispersion relation $E(p)=(p^{2}+m^{2})^{1/2}$, the one-loop self-energy has a branch cut for $-k\le k_{0}\le k$ that results from an intermediate state with $n=n'=1$. The following calculation will expose the essential singularity at $k_{0}=\pm k$ in the self-energy $$\Pi_{R}(k_{0})=g^{2}\int\!{d^{3}p\over (2\pi)^{3}}
{n(p)-n(\vec{p}+\vec{k})\over
\big(k_{0}+i\epsilon-\psi\big)2E(p)2E(\vec{p}+\vec{k})}.$$ The denominator function is $$\psi=E(\vec{p}+\vec{k})-E(p).$$ Rather that calculate $\Pi_{R}$ itself, it is easier to calculate the imaginary part: $${\rm Im}\Pi_{R}(K)=-{g^{2}\pi\over 4}\int\! {d^{3}p\over
(2\pi)^{3}}\,\delta[\,k_{0}-\psi\,]{n(p)-n(\vec{p}\!+\!\vec{k})\over
E(p)E(\vec{p}\!+\!\vec{k})}.$$ With the decomposition $\vec{p}=\hat{k}p_{\|}+\vec{p}_{\bot}$, the integral over $\vec{p}_{\bot}$ can be performed using the Dirac delta function: $$\int\! d^{2}p_{\bot}\,\delta[k_{0}-\psi]={2\pi p_{\bot}\over
d\psi/dp_{\bot}}\bigg|_{\psi=k_{0}}
=2\pi{E(\vec{p})E(\vec{p}\!+\!\vec{k})\over k_{0}}.$$ Consequently $${\rm Im}\Pi_{R}(K)=-{g^{2}\over 16\pi k_{0}}\int_{p_{\|}^{\rm
min}} ^{\infty}\!dp_{\|}
\Big[n(p)-n(p+k)\Big].\label{impi}$$ The condition $k_{0}=\psi$ can only be satisfied for $K^{2}<0$ and it makes $E(\vec{p})$ a linear function of $p_{\|}$: $$E(\vec{p})=
{k\over k_{0}}p_{\|}-{K^{2}\over 2k_{0}},$$ where $$p_{\|}=
-{k\over 2}+{k_{0}\over 2}\sqrt{1-{4(m^{2}+p_{\bot}^{2})\over K^{2}}}.$$ Since $0\le p_{\bot}\le \infty$, the minimum value of the parallel momentum is $$p_{\|}^{\rm min}=-{k\over 2}+{k_{0}\over
2}\sqrt{1-{4m^{2}\over K^{2}}}.$$ The remaining integration in (\[impi\]) is elementary: $${\rm Im}\Pi_{R}(K)=-{g^{2}T\over 16\pi k}\ln\Bigg[{1-e^{-\beta
E(k+p_{\|}^{\rm min})}\over 1-e^{-\beta
E(p_{\|}^{\rm min})}}\Bigg],\label{impi2}$$ where the energies that enter are $$\begin{aligned}
E(p_{\|}^{\rm min})=&&
-{k_{0}\over 2}+{k\over 2}\sqrt{1-{4m^{2}\over K^{2}}}\nonumber\\
E(k\!+\!p^{\rm min}_{\|})=&&
{k_{0}\over 2}+{k\over 2}\sqrt{1-{4m^{2}\over K^{2}}}.
\nonumber\end{aligned}$$ Both energies are positive since $K^{2}<0$. Thus the imaginary part is $${\rm Im}\Pi_{R}(K)=-{g^{2}T\over 16\pi k}\ln\Bigg[{1-e^{-\beta
\big(k_{0}+k\sqrt{1-4m^{2}/K^{2}}\big)/2}\over 1-e^{-\beta
\big(-k_{0}+k\sqrt{1-4m^{2}/K^{2}}\big)/2}}\Bigg].$$ The imaginary part is an odd function of $k_{0}$. As expected, there is an essential singularity at $k_{0}=\pm k$. The leading behavior as $k_{0}\to k$ is $$\begin{aligned}
{\rm Im}\Pi_{R}(K)\to - &&{g^{2}T\over 16\pi k}\big( e^{\beta
k/2}-e^{-\beta k/2}\big)\nonumber\\
\times&&\exp\Bigg(\!-{\beta k\over 2}\sqrt{1-{2(nm)^{2}\over
k(k_{0}-k)}}\;\Bigg).\label{essential1}\end{aligned}$$ This agrees perfectly with Eq. (\[statistical3\]) for $n=n'=1$.
$\phi^{4}$ example at two loops
===============================
This appendix will explicitly show the branch points on the light cone and on the mass shell in $\phi^{4}$ theory at two-loop order. The analysis of this section will be based on the work of Wang and Heinz [@Heinz], who calculated the imaginary part of the self-energy to two-loop order as a function of energy. Previous works had computed the imaginary part just on the mass-shell [@Parwani; @Jeon].
The notation in this appendix will be that of Wang and Heinz [@Heinz]. The interaction Hamiltonian is $g^{2}\phi^{4}/4!$. The zero-temperature particles are taken as massless, but thermal resummation leads to propagators with poles at $p_{0}=\pm(p^{2}+m_{P}^{2})^{1/2}$, where $m_{P}$ is a resummed plasmon mass $$m_{P}^{2}={g^{2}T^{2}\over 24}\bigg(1-{g\over 2\pi}\sqrt{{3\over
2}} \bigg).$$ The imaginary part of the two-loop self-energy is grouped as $${\rm Im}\Sigma(\omega,\vec{p})={\rm Im} g_{1}(\omega,\vec{p})
+{\rm Im} g_{2}(\omega,\vec{p}).$$ Here $g_{1}$ contains the usual three-particle cuts (i.e. $n=3$, $n'=0$ and $n=0$, $n'=3$) and will not be discussed here; $g_{2}$ contains the cut for two emissions and one absorption ($n=2$, $n'=1$) and the cut for one emission and two absorption ($n=1$, $n'=2$). As demonstrated in Section III, the function $g_{2}(\omega,\vec{p})$ should have branch points at $\omega=\pm(p^{2}+m_{P}^{2})^{1/2}$ and essential singularities at $\omega=\pm p$. The results of Wang and Heinz for ${\rm Im}\,
g_{2}(\omega,\vec{p})$ are extremely complicated double integrals when $\vec{p}\neq 0$. Consequently, this Appendix will only examine $\vec{p}=0$ and will demonstrate a branch point at $\omega=m_{P}$ and an essential singularity at $\omega=0$.
At $\vec{p}=0$, Wang and Heinz express the imaginary part of the self-energy as $$\begin{aligned}
\omega<m_{P}:\hskip0.6cm {\rm Im} g_{2}(\omega,0)= &&
\int_{\varepsilon}^{\infty}\!\! dv\, F(w, v)\label{less}\\
\omega>m_{P}:\hskip0.6cm {\rm Im}
g_{2}(\omega,0)= &&\int_{a}^{\infty}\!\! dv\, F(w, v)
.\label{more}\end{aligned}$$ In the integrals, $v$ and $w$ are Latin letters: $v$ is a dimensionless variable (an energy divided by T) and $w$ is the dimensionless ratio $$w={\omega\over T}.$$ The lower limit of the first integral is $$\varepsilon=\bigg[a^{2}+{(a^{2}-w^{2})
(9a^{2}-w^{2})\over 4w^{2}}\bigg]^{1/2},\label{varepsilon}$$ and the lower limit of the second integral is $$a={m_{P}\over T}.$$ Note that $\varepsilon\to a$ when $w\to a$.
Branch point at $\omega=m_{P}$
------------------------------
At the mass-shell $\omega=m_{P}$ (equivalently $w=a$) the lower limits are equal ($\varepsilon=a$) and thus ${\rm
Im}g_{2}(\omega,0)$ is continuous at $\omega=m_{P}.$
The first derivative of Eq. (\[less\]) is $$\begin{aligned}
\omega<m_{P}:\hskip0.3cm T\,{d\,{\rm Im} g_{2}(\omega,0)
\over d\omega}= &&-{d\varepsilon\over
dw}\,F(w,\varepsilon)\nonumber\\ +&&\int_{\varepsilon}^{\infty}\!\!
dv\,{\partial F(w, v)
\over \partial w}.\end{aligned}$$ An essential property of the integrand is [@Heinz] $$F(w,v)\big|_{v=a}=0.\label{essential}$$ As $\omega\to m_{P}$, the lower limit $\varepsilon\to a$. Since $F(a,a)=0$ by Eq. (\[essential\]), the first derivative of Eq. (\[less\]) at $\omega=m_{P}$ is the same as the first derivative of Eq. (\[more\]) at $\omega=m_{P}$.
The second derivative of Eq. (\[less\]) is $$\begin{aligned}
\omega<m_{P}:\hskip0.3cm T^{2}\,{d^{2}{\rm Im} g_{2}(\omega,0)
\over d\omega^{2}}= &&-{d\over dw}\bigg[{d\varepsilon\over
dw}\,F(w,\varepsilon)\bigg]
\nonumber\\ -{d\varepsilon\over dw}
\,{\partial F(w, \varepsilon)
\over \partial w}
+&&\int_{\varepsilon}^{\infty}\!\!
dv\,{\partial^{2} F(w, v)
\over \partial w^{2}}.\end{aligned}$$ The second term on the right hand side vanishes at $w=a$ because of Eq. (\[essential\]). The first term on the right hand side simplifies to $$-{d^{2}\varepsilon\over dw^{2}}\,F(w,\varepsilon)-{d\varepsilon
\over dw}\,{\partial F(w,\varepsilon)\over \partial w}
-\bigg[{d\varepsilon\over dw}\bigg]^{2}{\partial
F(w,\varepsilon)\over d\varepsilon}.$$ The first two terms of this vanish at $w=a$ because of Eq. (\[essential\]), but the third does not. Thus as $\omega$ approaches $m_{P}$ from below, the second derivative is $$\begin{aligned}
T^{2}\,{d^{2}{\rm Im} g_{2}(\omega,0)
\over d\omega^{2}}\bigg|_{\omega=m_{P}^{-}}= &&-
\bigg[{d\varepsilon\over dw}\bigg]_{\omega =a}^{2}
{\partial F(a,v)\over\partial v}\Bigg|_{v=a}
\nonumber\\ \nonumber\\
&&+\int_{a}^{\infty}\!\!
dv\,{\partial^{2} F(w, v)
\over \partial w^{2}}.\end{aligned}$$ At $\omega=a$, Eq. (\[varepsilon\]) gives $d\varepsilon/dw=-2$. From Wang and Heinz [@Heinz] the function $F(w,v)$ simplifies at $w=a$ to $$\begin{aligned}
F(a,v)={g^{4}T^{2}\over 128\pi^{3}}{e^{v}\over e^{v}-1}
{e^{a}-1\over e^{a+v}-1}
2\ln\bigg[{\sinh(v/2)\over\sinh(a/2)}\bigg].\end{aligned}$$ As expected, this vanishes at the lower limit $v=a$. However $\partial F(a,v)/\partial v$ does not vanish at $v=a$. Thus the second derivative is discontinuous at $w=m_{P}$ with a discontinuity given by $$\begin{aligned}
\Bigg[{d^{2}{\rm Im} g_{2}(\omega,0)
\over d\omega^{2}}\Bigg]_{\omega=m_{P}^{-}}^{\omega=m_{P}^{+}}
={g^{4}\over 32\pi^{3}}{e^{a}\over
(e^{a}-1)^{2}}.\label{2derivative}\end{aligned}$$ This confirms the existence of a branch point at the mass-shell $\omega=m_{P}$.
There is a further check of Eq. (\[2derivative\]). In the specific calculation of Wang and Heinz, the mass was entirely thermal so that $a=m_{P}/T$ is independent of temperature. However, the calculation of the two-loop discontinuity would also apply in a theory with a non-thermal mass $m$. Then the right hand side of Eq. (\[2derivative\]) would be temperature-dependent with $a=m/T$. In the zero-temperature limit, $a\to\infty$ and discontinuity in the second derivative vanishes as expected.
Essential singularity at $\omega=0$
-----------------------------------
For $\vec{p}\neq 0$, the function $g_{2}(\omega,\vec{p})$ will have essential singularities at $\omega=\pm p$. In the case considered here, viz. $\vec{p}=0$, these collapse to an essential singularity at $\omega=0$. In the vicinity of $\omega\approx 0$, the imaginary part is given by Eq. (\[less\]). The lower limit of the integral grows as $\omega\to 0$: $$\omega\to 0:\hskip0.75cm \varepsilon\to {3a^{2}\over
2w}+\dots$$ It is convenient to change variables from $v$ to $\overline{v}$, $$v={3a^{2}\over 2w}\big[1+\overline{v}\big],$$ where $0\le\overline{v}\le\infty$. Then Eq. (\[less\]) becomes $$\omega< m_{P}:\hskip0.5cm {\rm Im} g_{2}(\omega,0)={3a^{2}\over
2w}\int_{0}^{\infty}
\!d\overline{v}\;F(w,v).$$ From [@Heinz] the integral becomes in the limit $\omega\to
0$: $$\begin{aligned}
&&{\rm Im} g_{2}(\omega,0)\to {9g^{4}T^{2}\over 1024 \pi^{3}}
\,{a^{4}\over w}\,e^{-3a^{2}/2w}\;\; I(\omega)\\
\nonumber\\
&&I(\omega)=
\int_{0}^{\infty}\!d\overline{v}\,e^{-3a^{2}\overline{v}
/2\omega}\big(1+\overline{v}\big)\Big(1+\sqrt{{3\overline{v}\over
4+3\overline{v}}}
\Big).\nonumber\end{aligned}$$ As $w\to 0$, the integrand of $I(\omega)$ is exponentially small for any $\overline{v}$ that is not infinitesimal. The dominant contribution comes from the region $0\le\overline{v}\ll
2w/3a^{2}$ and gives $I(\omega)\to 2\omega/3a^{2}$. Thus $$\omega\to 0:\hskip0.7cm{\rm Im} g_{2}(\omega,0)\to
{3g^{4}m_{P}^{2}\over 512
\pi^{3}}
\;e^{-3a^{2}/2w}.\label{essential2}$$ The exponent here, $-3a^{2}/2w$, agrees precisely with that anticipated in Eq. (\[statistical4\]) for $n=2, n'=1$.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Given a contraction $A$ on a Hilbert space $\mathcal{H}$, an operator $T$ on $\mathcal{H}$ is said to be $A$-invariant if $\langle Tx,x\rangle=\langle TAx,Ax\rangle$ for every $x\in\mathcal{H}$ such that $\|Ax\|=\|x\|$. In the special case in which both defect indices of $A$ are equal to $1$, we show that every $A$-invariant operator is the compression to $\mathcal{H}$ of an unbounded linear transformation that commutes with the minimal unitary dilation of $A$. This result was proved by Sarason under the additional hypothesis that $A$ is of class $C_{00}$, leading to an intrinsic characterization of the truncated Toeplitz operators. We also adapt to our more general context other results about truncated Toeplitz operators.'
address:
- |
Mathematics Department,\
Indiana University,\
Bloomington, IN 47405, USA
- |
Simion Stoilow Institute of Mathematics\
Romanian Academy,\
Calea Griviţei 21,\
Bucharest, Romania
author:
- 'H. Bercovici and D. Timotin'
title: Operators invariant relative to a completely nonunitary contraction
---
[^1]
Introduction
============
Suppose that $A$ is a completely nonunitary contraction acting on a Hilbert space $\mathcal{H}$ and $U$ is the minimal unitary dilation of $A$ acting on $\mathcal{K}\supset\mathcal{H}$. Thus, $A^{n}=P_{\mathcal{H}}U^{n}|\mathcal{H}$ is the compression of $U^{n}$ to $\mathcal{H}$ for every positive integer $n$. It is of interest to consider operators of the form $P_{\mathcal{H}}X|\mathcal{H}$, where $X$ is in the commutant $\{U\}'$ of $U$. The commutant lifting theorem [@sar-H-infty; @harmonic] shows that every element of $\{A\}'$ is of this form. When $A$ is the unilateral shift on the Hardy space $H^{2}$, the collection $\{P_{\mathcal{H}}X|\mathcal{H}:X\in\{U\}'\}$ consists precisely of the Toeplitz operators on $H^{2}$. When $A$ is an operator of class $C_{00}$ with defect indices equal to $1$, the collection $\{P_{\mathcal{H}}X|\mathcal{H}:X\in\{U\}'\}$ is hard to characterize intrinsically. However, a larger collection, obtained by considering closed unbounded linear transformations $X$ that commute with $U$, has been identified in [@sar-TTO] with the class of those bounded operators $Y$ on $\mathcal{H}$ that are $A$-invariant in the sense that they satisfy the identity $$\langle YAx,Ax\rangle=\langle Yx,x\rangle$$ for every vector $x\in\mathcal{H}$ such that $\|Ax\|=\|x\|$. Of course, operators $A$ of the type just described can be identified up to unitary equivalence with compressions of the unilateral shift to co-invariant subspaces, and the class of operators $Y$ described above is in that case the class of truncated Toeplitz operators [@sar-TTO].
Our purpose in this paper is to consider arbitrary operators $A$ with defect indices equal to $1$ and the class of bounded operators on $\mathcal{H}$ that can be obtained as compressions of (possibly) unbounded linear transformations that commute with $U$. We call these operators *truncated multiplication operators* and we show, in particular, that operators in this class are characterized by the fact that they are $A$-invariant. Operators $A$ with defect indices equal to $1$ are always complex symmetric and, in the $C_{00}$ case, it is known [@sar-TTO] that the corresponding $A$-invariant operators satisfy the same complex symmetry. This result no longer persists if $A$ is not of class $C_{00}$. In this case, the complex symmetric truncated $A$-invariant operators belong, roughly speaking, to the linear space generated by $\{A\}'$ and $\{A^{*}\}'$.
The remainder of the paper is organized as follows. Section \[sec:Preliminaries\] contains a description of the functional models of contractions with defect indices equal to one, as well as the definition of truncated multiplication operators and their symbols in this context. In Section \[sec:TTO=00003Dinv\], we characterize the class of truncated multiplcation operators by $A$-invariance. The main result of Section \[sec:Nonuniqueness-of-the symbol\] establishes the extent to which the symbol of an $A$-invariant operator is uniquely determined. In Section \[sec:An-analog-of Crofoot\] we describe some useful and explicit unitary equivalences between model spaces. Finally, in Section \[sec:Complex-symmetries\] we discuss complex symmetries, in particular the decomposition of $A$-symmetric operators into complex symmetric and complex skew-symmetric summands.
Preliminaries\[sec:Preliminaries\]
==================================
We denote by $\mathbb{C}$ the complex plane, by $\mathbb{D}=\{\lambda\in\mathbb{C}:|\lambda|<1\}$ the open unit disk, by $\mathbb{T}=\partial\mathbb{D}$ the unit circle, and by $\chi$ the identity function $\chi(\lambda)=\lambda$. Normalized arclength defines a Borel probability measure $m$ on $\mathbb{T}$, $L^{p}$ stands for the corresponding space $L^{p}(\mathbb{T},m)$, and $H^{p}\subset L^{p}$ is the Hardy space for $p\in[1,+\infty]$. We recall that an element $h\in H^{p}$ can also be considered to be an analytic function on $\mathbb{D}$, and the values of $u$ on $\mathbb{T}$ can be recovered as radial limits almost everywhere with respect to $m$.
As noted in the introduction, we focus on contractions $A$ acting on a Hilbert space $\mathcal{H}$ with the property that the operators $I_{\mathcal{H}}-A^{*}A$ and $I_{\mathcal{H}}-AA^{*}$ have rank equal to one. In a different terminology, $T$ has *defect indices* $1$ and $1$, where the defect indices are a measure of how far $A$ and $A^{*}$ are from being isometric. In addition, we impose the condition that $A$ has no nonzero reducing subspace $\mathcal{K}$ with the property that the restriction $A|\mathcal{K}$ is a unitary operator. In other words, $A$ is supposed to be *completely nonunitary*.
Sz.-Nagy and Foias have developed a functional model for completely nonunitary contractions, showing for instance that such a contraction $A$ is uniquely determined, up to unitary equivalence, by a purely contractive analytic function $\Theta_{A}$ whose values are operators between two Hilbert spaces with dimensions equal to the defect indices of $A$. The function $\Theta_{A}$ is called the *characteristic function* of $A$, and it plays an analogous role to that of the characteristic matrix of a linear operator on a finite dimensional space. In our case, the defect indices are both equal to $1$, so the characteristic function of $A$ can be thought of simply as a function $u\in H^{\infty}$ such that $\|u\|_{\infty}\le1$. Such a function is purely contractive precisely when $|u(0)|<1$, that is, when $u$ is not identically equal to a constant of modulus one. Thus, throughout this paper, we work with a purely contractive function $u\in H^{\infty}$. When the characteristic function of $A$ is an inner function in $H^{\infty}$, the minimal unitary dilation of $A$ is a bilateral shift, and this allows for the construction of a particularly simple functional model for $A$. In our more general setting, this dilation is a unitary operator with spectral multiplicity at most $2$.
We now describe the functional model associated to a given purely contractive function in $H^{\infty}$. Fix $u\in H^{\infty}$ such that $\|u\|_{\infty}\le1$ and $|u(0)|<1$, and define the function $\Delta\in L^{\infty}$ by $$\Delta(\zeta)=(1-|u(\zeta)|^{2})^{1/2},\quad\zeta\in\mathbb{T}.$$ Using this function, we construct spaces $$\mathbf{K}=L^{2}\oplus(\Delta L^{2})^{-},\quad\mathbf{K}_{+}=H^{2}\oplus(\Delta L^{2})^{-},\quad\mathbf{G}=\{uf\oplus\Delta f:f\in H^{2}\},$$ and finally, $$\mathbf{H}_{u}=\mathbf{K}_{+}\ominus\mathbf{G}.$$ Note for further use that a function $f\oplus g\in\mathbf{K}_{+}$ belongs to $\mathbf{H}_u$ if and only if $$\overline{u}f+\Delta g\in L^{2}\ominus H^{2}.$$ We define now operators $U\in\mathcal{B}(\mathbf{K}),$ $U_{+}\in\mathcal{B}(\mathbf{K}_{+})$, and $S_{u}\in\mathcal{B}(\mathbf{H}_{u})$ by $$U(f\oplus g)=\chi f\oplus\chi g,\quad f\oplus g\in\mathbf{K},\quad U_{+}=U|\mathbf{K}_{+},$$ and $$S_{u}=P_{\mathbf{H}_{u}}U|\mathbf{H}_{u}=(U_{+}^{*}|\mathbf{H}_{u})^{*}.\label{eq:definition of S_u}$$ Then the operator $S_{u}$ is completely nonunitary, it has defect indices equal to $1$, and its characteristic function coincides with $u$. Moreover $U_{+}$ is the minimal isometric dilation of $S_{u}$, and $U$ is the minimal unitary dilation of $S_{u}$. We refer to [@harmonic] or [@nik] for an exposition of these facts.
Observe that the operator $S_{u}$ is of class $C_{00}$, that is, $$\lim_{n\to\infty}\|S_{u}^{n}h\|=\lim_{n\to\infty}\|S_{u}^{*n}h\|=0,\quad h\in\mathcal{H},$$ if and only if $u$ is an inner function, that is, $\Delta=0$. In this case $\mathbf{H}_{u}=H^{2}\ominus uH^{2}$. In this paper we concern ourselves primarily with the case in which $u$ is not inner. All of the arguments in the paper, with the exception of the proof of Proposition \[prop:symbols for the zero operator\], work equally well if $u$ is an inner function. However, these results were already known in the inner case. We refer to [@sar-TTO] for a detailed discussion.
We record for further use the formula $$U_{+}^{*}(f\oplus g)=\overline{\chi}(f-f(0))\oplus\overline{\chi}g,\quad f\oplus g\in\mathbf{K}_{+}.$$
We use the linear manifolds $$\mathbf{K}^{\infty}=\{f\oplus g:f\in L^{\infty},g\in L^{\infty}\cap(\Delta L^{2})^{-}\},\quad\mathbf{K}_{+}^{\infty}=\mathbf{K_{+}\cap\mathbf{K}}^{\infty},$$ and $$\mathbf{H}_{u}^{\infty}=\mathbf{H}_{u}\cap\mathbf{K}^{\infty}.$$ It is clear that $\mathbf{K}^{\infty}$ is dense in $\mathbf{K}$ and $\mathbf{K}_{+}^{\infty}$ is dense in $\mathbf{K}_{+}$. To show that $\mathbf{H}_{u}^{\infty}$ is also dense in $\mathbf{H}_{u}$, we consider the vectors $\chi^{n}\oplus0$ and $\chi^{-n}u\oplus\chi^{-n}\Delta$, $n\in\mathbb{Z}.$ These elements of $\mathbf{K}^{\infty}$ span a dense linear manifold in $\mathbf{K},$ and therefore their orthogonal projections onto $\mathbf{H}_{u}$ span a dense linear manifold in $\mathbf{H}_{u}$. These orthogonal projections are again bounded functions. In fact, $P_{\mathbf{H}_{u}}(\chi^{n}\oplus0)=0$ for $n<0$, and $$P_{\mathbf{H}_{u}}(\chi^{n}\oplus0)=\chi^{n}\oplus0-P_{\mathbf{G}}(\chi^{n}\oplus0),\quad n\ge0.$$ The second projection is easily calculated as $$P_{\mathbf{G}}(\chi^{n}\oplus0)=\sum_{j=0}^{n}\overline{\alpha_{n-j}}(\chi^{j}u\oplus\chi^{j}\Delta),\quad\alpha_{n-j}=\langle u,\chi^{n-j}\rangle,\quad j=0,\dots,n.$$ Similarly, $P_{\mathbf{H}_{u}}(\chi^{-n}u\oplus\chi^{-n}\Delta)=0$ for $n\le0$, and $$P_{\mathbf{H}_{u}}(\chi^{-n}u\oplus\chi^{-n}\Delta)=P_{H^{2}}(\chi^{-n}u)\oplus\chi^{-n}\Delta,\quad n>0,$$ where $P_{H^{2}}:L^{2}\to H^{2}$ denotes the orthogonal projection, so $$P_{H^{2}}(\chi^{-n}u)=\chi^{-n}u-\sum_{j=0}^{n-1}\alpha_{j}\chi^{j-n},\quad\alpha_{j}=\langle u,\chi^{j}\rangle,\quad j=0,\dots,n-1.$$
Two particularly important vectors in $\mathbf{H}_{u}^{\infty}$ are defined by $$k_{0}=k_{0}^{u}=P_{\mathbf{H}_{u}}(1\oplus0)=(1-\overline{u(0)}u)\oplus(-\overline{u(0)}\Delta)$$ and $$\widetilde{k}_{0}=\widetilde{k}_{0}^{u}=P_{\mathbf{H}_{u}}(\overline{\chi}u\oplus\overline{\chi}\Delta)=(\overline{\chi}(u-u(0))\oplus(\overline{\chi}\Delta).$$ The operator $S_{u}$ maps $\mathbf{H}_{u}\ominus\mathbb{C}\widetilde{k}_{0}$ isometrically onto $\mathbf{H}_{u}\ominus\mathbb{C}k_{0}$, $S_{u}h=Uh$ for $h\in\mathbf{H}_{u}\ominus\mathbb{C}\widetilde{k}_{0}$, and $$S_{u}\widetilde{k}_{0}=-u(0)k_{0},\quad S_{u}^{*}k_{0}=-\overline{u(0)}k_{0}.$$ Using these facts and the equalities $\|k_{0}\|^{2}=\|\widetilde{k}_{0}\|^{2}=1-|u(0)|^{2}$, it is easy to verify the identities $$I_{\mathbf{H}_{u}}-S_{u}S_{u}^{*}=k_{0}\otimes k_{0},\quad I_{\mathbf{H}_{u}}-S_{u}^{*}S_{u}=\widetilde{k}_{0}\otimes\widetilde{k}_{0},\label{eq:defect of S_u}$$ where we use the notation $v\otimes w$ for the rank one operator $h\mapsto\langle h,w\rangle v$.
The linear manifolds $\mathbf{K}^{\infty}$ and $\mathbf{K}_{+}^{\infty}$ are clearly invariant under $U$. The linear manifold $\mathbf{H}_{+}^{\infty}$ is also invariant under $S_{u}$, as seen from the formula $$S_{u}(f\oplus g)=\chi f\oplus\chi g-\langle f\oplus g,\widetilde{k}_{0}\rangle(u\oplus\Delta),\quad f\oplus g\in\mathbf{H}_{u}.$$ Similarly, $\mathbf{H}_{u}^{\infty}$ is invariant under $S_{u}^{*}$ because $$S_{u}^{*}(f\oplus g)=\overline{\chi}f\oplus\overline{\chi}g-\langle f\oplus g,k_{0}\rangle(1\oplus0),\quad f\oplus g\in\mathbf{H}_{u}.$$
It is well known that the commutant $\{U\}'$ consists of multiplication operators by matrix functions $$\left[\begin{array}{cc}
a & b\\
c & d
\end{array}\right],$$ where $a,b,c,d\in L^{\infty}.$ We require a larger class of unbounded linear transformations that commute with $U$. Suppose that we are given functions $a,b,c,d\in L^{2}$. We consider the matricial function $$F=\left[\begin{array}{cc}
a & b\\
c & d
\end{array}\right]$$ and the linear transformation $M_{F}:\mathbf{K}^{\infty}\to\mathbf{K}$ given by $$M_{F}(f\oplus g)=(af+bg)\oplus(cf+dg),\quad f\oplus g\in\mathbf{K}^{\infty}.$$ In other words, $M_{F}$ is the operator of multiplication by $F$. (Observe that modifying the values of $b,c,$ and $d$ on $\{\zeta\in\mathbb{T}:\Delta(\zeta)=0\}$ does not alter the operator $M_{F}$. It is useful however to allow for arbitrary $b,c,d\in L^{2}$.) Generally, $M_{F}$ is not continuous but it is closable, as can be seen from the inclusion $M_{F^{*}}\subset(M_{F})^{*}$, where $$F^{*}=\left[\begin{array}{cc}
\overline{a} & \overline{c}\\
\overline{b} & \overline{d}
\end{array}\right].$$ The equality $M_{F}Uv=UM_{F}v$ holds for every $v\in\mathbf{K}_{u}^{\infty}$. The operator $M_{F}$ is bounded if and only if $a\in L^{\infty}$ and the functions $b,c,d$ are essentially bounded on $\{\zeta\in\mathbb{T}:\Delta(\zeta)\ne0\}$. We define a linear transformation $A_{F}:\mathbf{H}_{u}^{\infty}\to\mathbf{H}_{u}$ by $$A_{F}v=P_{\mathbf{H}_{u}}M_{F}v,\quad v\in\mathbf{H}_{u}^{\infty}.$$ We also have $A_{F^{*}}\subset(A_{F})^{*},$ so $A_{F}$ is always closable. In particular, $A_{F}$ is bounded if and only if $A_{F}^{*}$ is bounded. If $M_{F}$ is bounded then, of course, $A_{F}$ is bounded as well, but not conversely [@Baranov-c-f-m-t; @baranov-b-k].
\[def:TTO\]A bounded linear operator $T\in\mathcal{B}(\mathbf{H}_{u})$ is called a *truncated multiplication operator* if there exists a function $F$ as above such that $Tv=A_{F}v$ for every $v\in\mathbf{H}_{u}^{\infty}$. The collection of all truncated multiplication operators is denoted $\mathcal{T}_{u}$.
The collection $\mathcal{T}_{u}$ is a linear space, closed under taking adjoints. In other words, $\mathcal{T}_{u}$ is an *operator system*.
\[rem:symbol not unique\]In the preceding definition, it seems natural to view $F$ as the *symbol* of the truncated Toeplitz operator $T$. Note however that there are nonzero functions $F$ such that $A_{F}=0$, and thus a given operator in $\mathcal{T}_{u}$ may have more than one symbol. The symbols $F$ with the property that $A_{F}=0$ are described in Proposition \[prop:symbols for the zero operator\].
\[exa:S\_u is TMO\]The operator $S_{u}$ itself belongs to $\mathcal{T}_{u}$. One symbol for $S_{u}$ is the function $$\left[\begin{array}{cc}
\chi & 0\\
0 & \chi
\end{array}\right],$$ as can be seen directly from (\[eq:definition of S\_u\]).
\[exa:a TTO of rank one\]The function $$F=\left[\begin{array}{cc}
\chi\overline{u} & \chi\Delta\\
0 & 0
\end{array}\right]$$ is the symbol of the rank one operator $T=(1-|u(0)|^2)k_0\otimes
\widetilde{k}_0
\in\mathcal{T}_{u}$. To see this, consider an arbitrary vector $x=h\oplus g\in\mathbf{H}_{u}$, so $$Tx=P_{\mathbf{H}_{u}}(\chi(\overline{u}h+\Delta g)\oplus0)=P_{\mathbf{H}_{u}}(P_{H^{2}}(\chi(\overline{u}h+\Delta g))\oplus0).$$ It was noted earlier that $\overline{u}h+\Delta g\in L^{2}\ominus H^{2}$, and therefore $P_{H^{2}}(\chi(\overline{u}h+\Delta g))=P_{H^{2}}(\rho\oplus0)$ for some $\rho\in\mathbb{C}$. The constant $\rho$ is equal to zero if $x\perp\mathbb{C}\widetilde{k}_{0}$. For $x=k_{0}$, we have $$\chi(\overline{u}h+\Delta g)=\overline{u}(u-u(0))+\Delta^{2}=1-u(0)\overline{u},$$ and thus $P_{H^{2}}(\chi(\overline{u}h+\Delta g))=1-|u(0)|^{2}$.
There is a special class of matrix functions $F$ with the property that $A_{F}$ commutes with $S_{u}$ on the space $\mathbf{H}_{u}^{\infty}$. These functions are of the form $$F=\left[\begin{array}{cc}
a & 0\\
\Delta c & a-uc
\end{array}\right],\label{eq:commutant F}$$ where $a\in H^{2}$ and $c\in L^{2}.$ It is easily seen that functions of this form satisfy $M_{F}(\mathbf{K}_{+}^{\infty})\subset\mathbf{K}_{+}^{\infty}$ and $M_{F}(\mathbf{K}_{+}^{\infty}\cap\mathbf{G})\subset\mathbf{G}$. Thus, if $x\in\mathbf{H}_{u}^{\infty}$, we have $P_{\mathbf{H}_{u}}(M_{F}P_{\mathbf{G}}Ux)=0$ and $P_{\mathbf{H}_{u}}(UP_{\mathbf{G}}M_{F}x)=0$ and therefore $$\begin{aligned}
A_{F}S_{u}x & = & P_{\mathbf{H}_{u}}M_{F}S_{u}x=P_{\mathbf{H}_{u}}(M_{F}Ux-M_{F}P_{\mathbf{G}}Ux)=P_{\mathbf{H}_{u}}M_{F}Ux\\
& = & P_{\mathbf{H}_{u}}UM_{F}x=P_{\mathbf{H}_{u}}(UA_{F}x)+P_{\mathbf{H}_{u}}(UP_{\mathbf{G}}M_{F}x)=S_{u}A_{F}x.\end{aligned}$$ In the case in which $u\not\equiv0$, the commutant lifting theorem implies that every bounded operator $T\in\{S_{u}\}'$ is of the form $A_{F}$, where $F$ is a function of the form \[eq:commutant F\] with $a\in H^{\infty}$ and $c\in L^{\infty}$ [@intertwining Lemma 2.1].
\[lem:commutant commutative\] Suppose that the function $u$ is not identically zero. Then the commutant $\{S_{u}\}'$ is commutative.
Suppose that the operators $T,T'\in\{S_{u}\}'$ are determined by the functions $$F=\left[\begin{array}{cc}
a & 0\\
\Delta c & a-uc
\end{array}\right],\quad F'=\left[\begin{array}{cc}
a' & 0\\
\Delta c' & a'-uc'
\end{array}\right],$$ respectively, for some $a,a'\in H^{\infty}$ and $c\in L^{\infty}$. A calculation shows that $$FF'=F'F=\left[\begin{array}{cc}
a'' & 0\\
\Delta c'' & a''-uc''
\end{array}\right],$$ where $a''=aa'$ and $c''=ac'+a'c-ucc'$. Suppose that $x$ is an arbitrary vector in $\mathbf{H}_{u}$. Then $$TT'x=P_{\mathbf{H}_{u}}(FP_{\mathbf{H}_{u}}(F'x))=P_{\mathbf{H}_{u}}(FF'x)-P_{\mathbf{H}_{u}}(FP_{\mathbf{G}}(F'x))=P_{\mathbf{H}_{u}}(FF'x),$$ since $F\mathbf{G}\subset\mathbf{G}\subset\mathbf{H}_{u}^{\perp}$. It follows that $A_{F''}$ is a symbol for $TT'$. Similarly, $A_{F''}$ is a symbol for $T'T$, and thus $TT'=T'T$.
The commutant $\{S_{u}\}'$ is not commutative if $u\equiv0$; see Example \[exa:u=00003D0\].
\[sec:TTO=00003Dinv\]Characterization of truncated multiplication operators by invariance
=========================================================================================
In this section, we show that truncated multiplication operators are characterized intrinsically by their properties as operators, without reference to a symbol.
\[def:A-invariant\] Suppose that $A$ is a contraction on a Hilbert space $\mathcal{H}.$ A bounded linear operator $T\in\mathcal{B}(\mathcal{H})$ is said to be *$A$-invariant* if the equality $$\langle Tx,y\rangle=\langle TAx,Ay\rangle$$ holds for every pair of vectors $x,y\in\ker(I_{\mathcal{H}}-A^{*}A)$.
\[lem:about invariance\]Suppose that $A\in\mathcal{B}(\mathcal{H})$ is a contraction and $T\in\mathcal{B}(\mathcal{H})$ is an arbitrary operator. Denote by $\mathcal{D}_{A}=[(I_{\mathcal H}-A^{*}A)\mathcal{H}]^{-}$ and $\mathcal{D}_{A^{*}}=[(I_{\mathcal H}-AA^{*})\mathcal{H}]^{-}$ the defect spaces of $A$, and by $P_{\mathcal{D}_{A}}$ and $P_{\mathcal{D}_{A^{*}}}$ the corresponding orthogonal projections. Then the following conditions are equivalent*:*
1. $T$ is $A$-invariant.
2. $T$ is $A^{*}$-invariant.
3. There exist $X,Y\in\mathcal{B}(\mathcal{H})$ such that $T-ATA^{*}=XP_{\mathcal{D}_{A^{*}}}+P_{\mathcal{D}_{A^{*}}}Y$.
4. There exist $X,Y\in\mathcal{B}(\mathcal{H})$ such that $T-A^{*}TA=XP_{\mathcal{D}_{A}}+P_{\mathcal{D}_{A}}Y$.
The operator $A$ maps the space $\ker(I-A^{*}A)=\mathcal{D}_{A}^{\perp}$ isometrically onto $\ker(I-AA^{*})=\mathcal{D}_{A^{*}}^{\perp}$. Thus, given arbitrary vectors $u,v\in\ker(I-AA^{*})$, there exist unique $x,y\in\ker(I-A^{*}A)$ such that $Ax=u$, $Ay=v$, $A^{*}u=x$, and $A^{*}v=y$. If $T$ is $A$-invariant, we see that $$\langle Tu,v\rangle=\langle TAx,Ay\rangle=\langle Tx,y\rangle=\langle TA^{*}u,A^{*}v\rangle,$$ and this shows that $T$ is $A^{*}$-invariant. This establishes that (1) implies (2) and the equivalence of (1) and (2) follows by symmetry.
Suppose now that $T$ is $A$-invariant and observe that $$\langle(T-A^{*}TA)x,y\rangle=\langle Tx,y\rangle-\langle TAx,Ay\rangle=0$$ for every $ x,y\in\ker(I_{\mathcal H }-A^{*}A)=\mathcal{D}_{A}^{\perp}$. It follows that $(I_{\mathcal H }-P_{\mathcal{D}_{A}}) (T-A^*TA)(I_{\mathcal H }-P_{\mathcal{D}_{A}})=0$, and thus (4) is satisfied with $$\begin{aligned}
X & = & T-A^*TA,\\
Y & = & (T-A^*TA)(I_{\mathcal H }-P_{\mathcal{D}_{A}}).\end{aligned}$$ Conversely, if (4) is satisfied, the identity $(I_{\mathcal H }-P_{\mathcal{D}_{A}}) (T-A^*TA)(I_{\mathcal H }-P_{\mathcal{D}_{A}})=0$ follows immediately, thus showing that $T$ is $A$-invariant. We conclude that (1) is equivalent to (4). The equivalence of (2) and (3) is proved the same way, replacing $A$ by $A^{*}$.
Suppose now that $u\in H^\infty$ is purely contractive. In the special case of the operator $A=S_{u}$, (\[eq:defect of S\_u\]) shows that $\ker(I_{\mathbf{H}_{u}}-S_{u}^{*}S_{u})=\mathbf{H}_{u}\ominus\mathbb{C}\widetilde{k}_{0}$. Moreover, given $x\in\mathbf{H}_{u}$, we have $Ux=S_{u}x\in\mathbf{H}_{u}$ precisely when $x\in\mathbf{H}_{u}\ominus\mathbb{C}\widetilde{k}_{0}$. Thus an operator $T\in\mathcal{B}(\mathbf{H}_{u})$ is $S_{u}$-invariant if and only if $$\langle Tx,y\rangle=\langle TUx,Uy\rangle,\quad x,y\in\mathbf{H}_{u}\ominus\mathbb{C}\widetilde{k}_{0}.$$ The polarization identity shows that an operator $T\in\mathcal{B}(\mathbf{H}_{u})$ is $S_{u}$-invariant if and only $$\langle Tx,x\rangle=\langle TUx,Ux\rangle,\quad x\in\mathbf{H}_{u}\ominus\mathbb{C}\widetilde{k}_{0}.\label{eq:invariance-one}$$ The invariance condition can be written equivalently as $$\langle Tx,x\rangle=\langle TU^{*}x,U^{*}x\rangle,\quad x\in\mathbf{H}_{u}\ominus\mathbb{C}k_{0}.\label{eq:invariance-2}$$
We now state the main result in this section.
\[teorema principala\]The following four conditions on an operator $T\in\mathcal{B}(\mathbf{H}_{u})$ are equivalent*:*
1. $T\in\mathcal{T}_{u}$.
2. $T$ is $S_{u}$-invariant.
3. There exist vectors $v,w\in\mathbf{H}_{u}$ such that $T-S_{u}TS_{u}^{*}=v\otimes k_{0}+k_{0}\otimes w$.
4. There exist vectors $\widetilde{v,}\widetilde{w}\in\mathbf{H}_{u}$ such that $T-S_{u}^{*}TS_{u}=\widetilde{v}\otimes\widetilde{k}_{0}+\widetilde{k}_{0}\otimes\widetilde{w}$.
The equations (\[eq:defect of S\_u\]) show that $P_{\mathcal{D}_{S_{u}^{*}}}$ and $P_{\mathcal{D}_{S_{u}}}$ are constant multiples of $k_{0}\otimes k_{0}$ and $\widetilde{k}_{0}\otimes\widetilde{k}_{0}$, respectively. Since $X(k_{0}\otimes k_{0})=(Xk_{0})\otimes k_{0}$ and $(k_{0}\otimes k_{0})Y=k_{0}\otimes(Y^{*}k_{0})$ for every $X,Y\in\mathcal{B}(\mathbf{H}_{u})$, the equivalence of (2), (3), and (4) follows immediately from Lemma \[lem:about invariance\].
Suppose now that (1) holds, and thus $Tv=A_{F}v,$ $v\in\mathbf{H}_{u}^{\infty}$, for some matrix $F$. Since $\widetilde{k}_{0}\in\mathbf{H}_{u}^{\infty}$, it follows that $\mathbf{H}_{u}^{\infty}\cap$($\mathbf{H}_{u}\ominus\mathbb{C}\widetilde{k}_{0}$) is dense in $\mathbf{H}_{u}\ominus\mathbb{C}\widetilde{k}_{0}$. Thus, it suffices to verify (\[eq:invariance-one\]) for $v\in\mathbf{H}_{u}^{\infty}\cap(\mathbf{H}_{u}\ominus\mathbb{C}\widetilde{k}_{0})$. For such a vector $v$ we have $$\begin{aligned}
\langle TUv,Uv\rangle & = & \langle P_{\mathbf{H}_{u}}M_{F}Uv,Uv\rangle=\langle M_{F}Uv,Uv\rangle=\langle UM_{F}v,Uv\rangle\\
& = & \langle M_{F}v,v\rangle=\langle P_{\mathbf{H}_{u}}M_{F}v,v\rangle=\langle Tv,v\rangle,\end{aligned}$$ where we used the facts that $U$ is unitary and $M_{F}$ commutes with $U$. We conclude that (1) implies (2).
We come now to the heart of the proof by showing that (3) implies (1). Suppose that (3) holds for some vectors $v=a_{1}\oplus c$ and $w=a_{2}\oplus b$ in $\mathbf{H}_{u}$. We define a matrix function $F$ by $$F=\left[\begin{array}{cc}
a_{1}+\overline{a_{2}} & \overline{b}\\
c & 0
\end{array}\right].$$ We show first that:
1. the operator $A_{F}$ is bounded,
2. the sequence $\{S_{u}^{n}TS_{u}^{*n}\}_{n\in\mathbb{N}}$ converges in the weak operator topology to an operator $T'$ such that $T'=S_{u}T'S_{u}^{*}$, and
3. $T=A_{F}+T'$ on $\mathbf{H}_{u}^{\infty}$.
To do this, fix a vector $x=g\oplus h\in\mathbf{H}_{u}^{\infty}$ and iterate the relation $T-S_{u}TS_{u}^{*}=v\otimes k_{0}+k_{0}\otimes w$ to obtain $$Tx=S_{u}^{n}TS_{u}^{*}x+\sum_{j=0}^{n-1}[S_{u}^{j}v\otimes S_{u}^{j}k_{0}+S_{u}^{j}k_{0}\otimes S_{u}^{j}w]x,\quad n\in\mathbb{N}.\label{eq:alta ecuatie}$$ We show that the sum above converges weakly to $A_{F}x$. We calculate first $$\begin{aligned}
\sum_{j=0}^{n-1}[S_{u}^{j}k_{0}\otimes S_{u}^{j}w]x & = & \sum_{j=0}^{n-1}\langle x,S_{u}^{j}w\rangle S_{u}^{j}k_{0}=P_{\mathbf{H}_{u}}\sum_{j=0}^{n-1}\langle x,\chi^{j}w\rangle(\chi^{j}\oplus0),\end{aligned}$$ where $$\langle x,\chi^{j}w\rangle=\langle\overline{a_{2}}g+\overline{b}h,\chi^{n}\rangle.$$ Therefore, the sum $\sum_{j=0}^{n-1}\langle x,\chi^{j}w\rangle\chi^{j}$ converges in $L^{2}$ to $P_{H^{2}}(\overline{a_{2}}g+\overline{b}h)$ and thus $\sum_{j=0}^{n-1}[S_{u}^{j}k_{0}\otimes S_{u}^{j}w]x$ converges in norm to $$P_{\mathbf{H}_{u}}(P_{H^{2}}(\overline{a_{2}}g+\overline{b}h)\oplus0)=P_{\mathbf{H}_{u}}((\overline{a_{2}}g+\overline{b}h)\oplus0).$$ Similarly, $$\begin{aligned}
\sum_{j=0}^{n-1}[S_{u}^{j}v\otimes S_{u}^{j}k_{0}]x & = & \sum_{j=0}^{n-1}\langle x,S_{u}^{j}k_{0}\rangle S_{u}^{j}v=\sum_{j=0}^{n-1}\langle x,\chi^{j}\oplus0\rangle S_{u}^{j}v\\
& = & P_{\mathbf{H}_{u}}\left[\sum_{j=0}^{n-1}\langle g,\chi^{j}\rangle\chi^{j}\right]v.\end{aligned}$$ Moreover, since $$\sum_{j=0}^{n-1}[S_{u}^{j}v\otimes S_{u}^{j}k_{0}]x=Tx-S_{u}^{n}TS_{u}^{*n}x-\sum_{j=0}^{n-1}[S_{u}^{j}k_{0}\otimes S_{u}^{j}w]x,$$ it follows that the vectors on the left hand side of this equation are bounded in $\mathbf{H}_{u}$. To show that they have a weak limit in $\mathbf{H}_{u}$, it suffices to consider their scalar product with another element $x'=g'\oplus h'\in\mathbf{H}_{u}^{\infty}$. We have $$\left\langle \sum_{j=0}^{n-1}[S_{u}^{j}k_{0}\otimes S_{u}^{j}w]x,x'\right\rangle =\left\langle \left[\sum_{j=0}^{n-1}\langle g,\chi^{j}\rangle\chi^{j}\right]v,x'\right\rangle ,$$ and the functions $\sum_{j=0}^{n-1}\langle g,\chi^{j}\rangle\chi^{j}$ converge to $g$ in $H^{2}$ as $n\to\infty$. Since $v\in\mathbf{H}_{u}$ and $x'$ is bounded, the scalar products above tend to $\langle gv,x'\rangle=\langle P_{\mathbf{H}_{u}}(a_{1}g\oplus cg),x'\rangle$ as $n\to\infty$. We conclude that the sum $$\sum_{j=0}^{n-1}[S_{u}^{j}v\otimes S_{u}^{j}k_{0}+S_{u}^{j}k_{0}\otimes S_{u}^{j}w]x$$ converges weakly to $P_{\mathbf{H}_{u}}(((a_{1}+\overline{a_{2}})g+\overline{b}h)\oplus cg)=A_{F}x$. The identity (\[eq:alta ecuatie\]) shows that $\|A_{F}x\|\le2\|T\|$, thus proving (i). Rewriting (\[eq:alta ecuatie\]) as $$S_{u}^{n}TS_{u}^{*n}=T-\sum_{j=0}^{n-1}[S_{u}^{j}v\otimes S_{u}^{j}k_{0}+S_{u}^{j}k_{0}\otimes S_{u}^{j}w],$$ we see that $S_{u}^{n}TS_{u}^{*n}x$ converges weakly to $Tx-A_{F}x$ for $x\in\mathbf{H}_{u}^{\infty}$, so the weak convergence of $\{S_{u}^{n}TS_{u}^{*n}\}_{n\in\mathbb{N}}$ follows from the fact that the sequence $\{\|S_{u}^{n}TS_{u}^{*n}\|\}_{n\in\mathbb{N}}$ is bounded. Also, $S_{u}T'S_{u}^{*}$ is the weak limit of the sequence $\{S_{u}^{n+1}TS_{u}^{*n+1}\}_{n\in\mathbb{N}}$, so it is equal to $T'$. This proves (ii) and (iii).
To conclude the proof of (1), it suffices to show that $T'\in\mathcal{T}_{u}$. To do this, we observe that for every $n\in\mathbb{N}$, we have $U_{+}^{*n}\mathbf{H}_{\chi^{n}u}=\mathbf{H}_{u}$ and $U_{+}^{n}\mathbf{H}_{u}\subset\mathbf{H}_{\chi^{n}u}$. We define an operator $T_{n}\in\mathcal{B}(\mathbf{H}_{\chi^{n}u})$ by $$T_{n}x=U_{+}^{n}T'U_{+}^{*n}x,\quad x\in\mathbf{H}_{\chi^{n}u},n\in\mathbb{N}.$$ Given $n\in\mathbb{N}$ and $x\in\mathbf{H}_{\chi^{n}u}\subset\mathbf{H}_{\chi^{n+1}u}$, we have $$\begin{aligned}
T_{n+1}x & = & U_{+}^{n}U_{+}T'S_{u}^{*}U_{+}^{*n}x\\
& = & T_{n}x+U_{+}^{n}(U_{+}-S_{u})T'S_{u}^{*}U_{+}^{*n}x.\end{aligned}$$ The vector $U_{+}^{n}(U_{+}-S_{u})T'S_{u}^{*}U_{+}^{*n}x$ belongs to $\mathbf{H}_{\chi^{n+1}u}^{\perp}$, and thus $$T_{n}x=P_{\mathbf{H}_{\chi^{n}u}}T_{n+1}x,\quad x\in\mathbf{H}_{\chi^{n}u}.$$ In particular, $T'=P_{\mathbf{H}_{u}}T_{n}|\mathbf{H}_{u}$ for every $n\in\mathbb{N}.$ Since $\|T_{n}\|\le\|T\|$, $n\in\mathbb{N}$, it follows that there exists an operator $X\in\mathcal{B}(\mathbf{K}_{+})$ with the property that $T_{n}=P_{\mathbf{H}_{\chi^{n}u}}X|\mathbf{H}_{\chi^{n}u}$ for every $n\in\mathbb{N}$. In fact, $\bigcup_{m\in\mathbb{N}}\mathbf{H}_{\chi^{m}u}$ is dense in $\mathbf{K}_{+}$, and $$Xx=\lim_{n\to\infty}T_{n}x$$ if $x\in\mathbf{H}_{\chi^{m}u}$ for some $m\in\mathbb{N}$. The operator $X$ satisfies the identity $X=U_{+}XU_{+}^{*}.$ This implies that $$X(g\oplus0)=\lim_{n\to\infty}U_{+}^{n}XU_{+}^{*n}(g\oplus0)=0,\quad g\in H^{2}.$$ Analogously, the equality $X^{*}=U_{+}X^{*}U_{+}^{*}$ yields $X^{*}(g\oplus0)=0$ for $g\in H^{2}$. Thus, $X$ is of the form $X=0_{H^{2}}\oplus Y$, where $Y\in\mathcal{B}((\Delta L^{2})^{-})$. Since $X$ commutes with $U_{+}$, it follows that $Y$ commutes with multiplication by $\chi$. Thus $Y$ must be the operator of multiplication by some bounded measurable function $d$, and therefore $$X(g\oplus h)=0\oplus dh,\quad g\oplus h\in\mathbf{K}_{+}.$$ The equality $T'=P_{\mathbf{H}_{u}}X|\mathbf{H}_{u}$ shows that $T'$ is a truncated multiplication operator with symbol $$\left[\begin{array}{cc}
0 & 0\\
0 & d
\end{array}\right].$$ Putting these facts together, we have shown that $$T=A_{F}+T'=A_{F'},$$ where $$F'=\left[\begin{array}{cc}
a_{1}+\overline{a_{2}} & \overline{b}\\
c & 0
\end{array}\right]+\left[\begin{array}{cc}
0 & 0\\
0 & d
\end{array}\right].$$ We have established that (3) implies (1), thus concluding the proof.
\[cor:WOT closed system\]For every purely contractive function $u\in H^{\infty}$, the operator system $\mathcal{T}_{u}$ is closed in the weak operator topology.
By Theorem \[teorema principala\], membership of an operator $T$ in $\mathcal{T}_{u}$ is characterised by the system of equations $$\langle TS_{u}x,S_{u}x\rangle=\langle Tx,x\rangle,\quad x\in\mathbf{H}_{u}\ominus\mathbb{C}\widetilde{k}_{0}.$$ Each of these equations is given by a continuous linear functional in the weak operator topology.
\[exa:the rank one perturbations\]Given an arbitrary scalar $\mu\in\mathbb{C}$, we define a bounded linear operator $X_{\mu}\in\mathcal{B}(\mathbf{H}_{u})$ by $$X_{\mu}x=\begin{cases}
Ux=S_{u}x, & x\in\mathbf{H}_{u}\ominus\mathbb{C}\widetilde{k}_{0},\\
\mu k_{0}, & x=\widetilde{k}_{0}.
\end{cases}$$ It is easily verified using Theorem \[teorema principala\](2) that $X_{\mu}$ is a truncated multiplication operator. We show in Corollary \[cor:commutants a la Sedlock\] that the commutant of $X_{\mu}$ consists entirely of truncated multiplication operators.These rank one perturbations of $S_{u}$ have been considered earlier in [@clark] (when $u$ is inner) and [@Ball-Lubin] (see also [@livsic; @brang]). The following result follows from [@Ball-Lubin].
\[prop:about the various rank one perturbations\]Fix $\mu\in\mathbb{C}$, a purely contractive function $u\in H^{\infty}$, and let $X_{\mu}$ be defined as in Example *\[exa:the rank one perturbations\]*. Then*:*
1. For $|\mu|<1$, the operator $X_{\mu}$ is a completely nounitary contraction with defect indices equal to $1$.
2. For $|\mu|>1$, the operator $X_{\mu}$ is invertible and $X_{\mu}^{-1}$ is a completely nounitary contraction with defect indices equal to $1$.
3. For $|\mu|=1$, the operator $X_{\mu}$ is unitary with spectral multiplicity equal to $1$.
\[cor:commutant of perturbation\] With the notation of Proposition *\[prop:about the various rank one perturbations\]*, the commutant of the operator $X_{\mu}$ is commutative for all $\mu\in\mathbb{C}\setminus\{0\}$. The commutant of $X_{0}$ is also commutative if $u$ is not a constant function.
If $|\mu|\ne1$, the corollary follows from parts (1) and (2) of Proposition \[prop:about the various rank one perturbations\] and from Lemma \[lem:commutant commutative\]. The case $|\mu|=1$ is a consequence of the general description of commutants of normal operators. The case $\mu=0$ follows from the fact that the characteristic function of $X_{0}$ is zero precisely when $u$ is a constant function.
\[cor:commutants a la Sedlock\] Let $\mu\in\mathbb{C}$, and let $X_{\mu}\in\mathcal{B}(\mathbf{H}_{u})$ be the operator defined in Example *\[exa:the rank one perturbations\]*. Then every operator $T\in\mathcal{B}(\mathbf{H}_{u})$ that commutes with either $X_{\mu}$ or with $X_{\mu}^{*}$ is a truncated multiplication operator.
We observe first that $\langle X_{\mu}h,Uk\rangle=\langle h,k\rangle$ if $h,k,Uk\in\mathbf{H}_{u}$. This is immediate if $Uh\in\mathbf{H}_{u}$ as well. On the other hand, if $h=\widetilde{k}_{0}$, then $\langle X_{\mu}h,Uk\rangle=\langle h,k\rangle=0$. Suppose now that $TX_{\mu}=X_{\mu}T$ and $k,Uk\in\mathbf{H}_{u}$. Then $$\langle TUk,Uk\rangle=\langle TX_{\mu}k,Uk\rangle=\langle X_{\mu}Tk,Uk\rangle=\langle Tk,k\rangle,$$ by the preceding observation applied to $h=Tk$. Thus $T$ is $S_{u}$-invariant and $T\in\mathcal{T}_{u}$ by Theorem \[teorema principala\]. If $TX_{\mu}^{*}=X^{*}T$ then the above argument shows that $T_{\mu}^{*}\in\mathcal{T}_{u}$ and thus $T\in\mathcal{T}_{u}$ because $\mathcal{T}_{u}$ is a selfadjoint space.
\[rem:Sedlock algebras\] In the case in which $u$ is an inner function, it was shown in [@sedlo] that every algebra contained $\mathcal{T}_{u}$ is commutative and is contained either in $\{X_{\mu}\}'$ or in $\{X_{\mu}^{*}\}'$ for some $\mu\in\mathbb{C}.$ It would be interesting to see whether this result remains true if $u$ is not inner. Note, incidentally, that $\mathcal{T}_{u}$ does contain a noncommutative algebra if $u$ is a constant function, namely the commutant of $X_{0}$ (see Example \[exa:u=00003D0\]).
\[rem:deB-Rov\]In case $u$ is an extreme point of the unit ball of $H^{\infty}$, it is known (see, for instance, [@sub-hardy Chapter IV]) that the projection onto the first component yields a unitary operator $J:\mathbf{H}_{u}\to\mathcal{H}(u)$, where $\mathcal{H}(u)$ is the de Branges-Rovnyak space associated to $u$. The operator $X=JS_{u}^{*}J^{*}$ is precisely the restriction to $\mathcal{H}(u)$ of the backward shift $f\mapsto\overline{\chi}(f-f(0))$. Therefore, Theorem \[teorema principala\] yields a characterization of those operators in $\mathcal{B}(\mathcal{H}(u))$ that are $X$-invariant.
\[sec:Nonuniqueness-of-the symbol\]Nonuniqueness of the symbol of a truncated multiplication operator
=====================================================================================================
As noted earlier, the symbol of an operator in $\mathcal{T}_{u}$ is not unique. The proof of Theorem \[teorema principala\] shows that a certain sequence related with an operator $T\in\mathcal{T}_{u}$ converges in the weak operator topology. The following result identifies that limit in terms of an arbitrary symbol for $T$.
\[prop:limit for uniqueness\]Suppose that $T\in\mathcal{B}(\mathbf{H}_{u})$ is a truncated multiplication operator with symbol $$\left[\begin{array}{cc}
a & b\\
c & d
\end{array}\right].$$ Then $d$ is essentially bounded on $\{\zeta\in\mathbb{T}:\Delta(\zeta)\ne0\}$, and the sequence $\{S_{u}^{n}TS_{u}^{*n}\}_{n\in\mathbb{N}}$ converges in the weak operator topology to the truncated multiplication operator with symbol $$\left[\begin{array}{cc}
0 & 0\\
0 & d
\end{array}\right].$$ In particular, the function $d$ is uniquely determined almost everywhere on $\{\zeta\in\mathbb{T}:\Delta(\zeta)\ne0\}$.
Let $x=g\oplus h$ and $x'=g'\oplus h'$ be two vectors in $\mathbf{H}_{u}$. We have $$\langle S_{u}^{n}TS_{u}^{*n}x,x'\rangle=\langle TS_{u}^{*n}x,S_{u}^{*n}x'\rangle,\quad n\in\mathbb{N},$$ and $S_{u}^{*n}x=P_{+}(\overline{\chi}^{n}g)\oplus\overline{\chi}^{n}h$, where $P_{+}:L^{2}\to H^{2}$ denotes the orthogonal projection. By the M. Riesz theorem, $P_{+}$ also defines a bounded operator on $L^{p}$ for $p\in(2,+\infty)$. We have $g\in L^{\infty}\subset L^{6},$ $\lim_{n\to\infty}\|P_{+}(\overline{\chi}^{n}g)\|_{2}=0$, and an application of the Hölder inequality shows that $$\|P_{+}(\overline{\chi}^{n}g)\|_{4}\le\|P_{+}(\overline{\chi}^{n}g)\|_{2}^{1/4}\|P_{+}(\overline{\chi}^{n}g)\|_{6}^{3/4},\quad n\in\mathbb{N}.$$ We deduce that $\lim_{n\to\infty}\|P_{+}(\overline{\chi}^{n}g)\|_{4}=0$. Similarly, $\lim_{n\to\infty}\|P_{+}(\overline{\chi}^{n}g')\|_{4}=0$. Expand now $$\begin{aligned}
\langle TS_{u}^{*n}x,S_{u}^{*n}x'\rangle & = & \langle aP_{+}(\overline{\chi}^{n}g),P_{+}(\overline{\chi}^{n}g')\rangle+\langle b\overline{\chi}^{n}h,P_{+}(\overline{\chi}^{n}g')\rangle\\
& & +\langle cP_{+}(\overline{\chi}^{n}g),\overline{\chi}^{n}h'\rangle+\langle d\overline{\chi}^{n}h,\overline{\chi}^{n}h'\rangle.\end{aligned}$$ The fourth term on the right hand side is equal to $\langle dh,h'\rangle=\langle P_{\mathbf{H}_{u}}(0\oplus dh),x'\rangle$ for every $n\in\mathbb{N}$, and we show that the remaining three terms converge to zero as $n\to\infty$. The Hölder inequality yields $$\begin{aligned}
|\langle aP_{+}(\overline{\chi}^{n}g),P_{+}(\overline{\chi}^{n}g')\rangle| & \le & \|a\|_{2}\|P_{+}(\overline{\chi}^{n}g)\|_{4}\|P_{+}(\overline{\chi}^{n}g')\|_{4},\\
|\langle b\overline{\chi}^{n}h,P_{+}(\overline{\chi}^{n}g')\rangle| & \le & \|b\|_{2}\|h\|_{4}\|cP_{+}(\overline{\chi}^{n}g')\|_{4},\\
|\langle cP_{+}(\overline{\chi}^{n}g),\overline{\chi}^{n}h'\rangle| & \le & \|c\|_{2}\|cP_{+}(\overline{\chi}^{n}g)\|_{4}\|h'\|_{4},\end{aligned}$$ and the sequences in the right hand side tend to zero, as shown above. We also see that $|\langle dh,h'\rangle|\le\|T\|\|h\|_{2}\|h'\|_{2}$.
To conclude the proof, we deduce from this inequality that $d$ is essentially bounded on $\{\zeta\in\mathbb{T}:\Delta(\zeta)\ne0\}$. We observe that $$U_{+}^{*n}(u\oplus\Delta)=P_{+}(\overline{\chi}^{n}u)\oplus\overline{\chi}^{n}\in\mathbf{H}_{u}^{\infty},$$ and thus $|\langle dh,h'\rangle|\le\|T\|\|h\|_{2}\|h'\|_{2}$ if $h$ and $h'$ are of the form $$h=\sum_{j=1}^{n}\alpha_{j}\overline{\chi}^{j},\quad h'=\sum_{j=1}^{n}\alpha'_{j}\overline{\chi}^{j}$$ for some $n\in\mathbb{N}$ and $\alpha_{1},\dots,\alpha_{n},\alpha'_{1},\dots,\alpha'_{n}\in\mathbb{C}$. Moreover, since $\langle dh,h'\rangle=\langle d\chi^{m}h,\chi^{m}h'\rangle$ for every $m\in\mathbb{N},$ we must have $|\langle dh,h'\rangle|\le\|T\|\|h\|_{2}\|h'\|_{2}$ whenever $h=p\Delta$ and $h'=q\Delta$ for some trigonometric polynomials $p$ and $q$. Since the trigonometric polynomials form a dense linear manifold in $L^{2}$, we see that $$|\langle df\Delta,g\Delta\rangle|\le\|T\|\|f\Delta\|_{2}\|g\Delta\|_{2}\label{eq:o inegalitate}$$ for every pair $f,g$ of functions in $L^{2}$. This finally implies that $|d|\le\|T\|$ almost everywhere on $\{\zeta\in\mathbb{T}:\Delta(\zeta)\ne0\}$. Indeed, in the contrary case, there exist $\varepsilon,M>0$ such that $\|T\|+\varepsilon\le|d|\le M$ on a set $\sigma\subset\{\zeta\in\mathbb{T}:\Delta(\zeta)\ne0\}$ of positive arclength. Then the choice $f=\overline{d}\chi_{\sigma},g=1$ contradicts (\[eq:o inegalitate\]).
We can now describe all the symbols associated to the zero operator.
\[prop:symbols for the zero operator\]Suppose that $T\in\mathcal{B}(\mathbf{H}_{u})$ is a truncated multiplication operator with symbol $$\left[\begin{array}{cc}
a & b\\
c & d
\end{array}\right],$$ where $a,b,c\in L^{2}$ and $d\in L^{\infty}$. Then $T=0$ if and only the following two conditions are satisfied*:*
1. $d=0$ almost everywhere on the set $\{\zeta\in\mathbb{T}:\Delta(\zeta)\ne0\}$.
2. There exist functions $f_{1},f_{2}\in H^{2}$ such that*:*
1. $a=uf_{1}+\overline{uf_{2}},$
2. $c=\Delta f_{1}$ and $b=\Delta\overline{f}_{2}$ almost everywhere on the set $\{\zeta\in\mathbb{T}:\Delta(\zeta)\ne0\}$.
The case in which $u$ is an inner function is proved in [@sar-TTO Theorem 3.1]. Therefore we may, and do, assume that $u$ is not inner, and thus the set $\{\zeta\in\mathbb{T}:\Delta(\zeta)\ne0\}$ has positive arclength.
Suppose first that conditions (1) and (2) are satisfied and set $$F_{1}=\left[\begin{array}{cc}
uf_{1} & 0\\
\Delta f_{1} & 0
\end{array}\right],\quad F_{2}=\left[\begin{array}{cc}
uf_{2} & 0\\
\Delta f_{2} & 0
\end{array}\right],\label{eq:F_j}$$ so $F=F_{1}+F_{2}^{*}$. If $x=h\oplus g$ is an arbitrary element of $\mathbf{H}_{u}^{\infty},$ we have $$A_{F_{1}}x=uf_{1}h\oplus\Delta f_{1}h\in\mathbf{G},$$ and thus $A_{F_{1}}=0$. Similarly, $A_{F_{2}}^{*}=0$, and therefore $T|\mathbf{H}_{u}^{\infty}=A_{F}=0$.
Conversely, suppose that $T=0$. Condition (1) follows from Proposition \[prop:limit for uniqueness\]. In addition, we have $Tk_{0}=T^{*}k_{0}=0$. Since $k_{0}=(1-\overline{u(0)}u)\oplus(-\overline{u(0)}\Delta)$, the vectors $$\begin{aligned}
Fk_{0} & = & [a(1-\overline{u(0)}u)-b\overline{u(0)}\Delta]\oplus[c(1-\overline{u(0)}u)],\\
F^{*}k_{0} & = & [\overline{a}(1-\overline{u(0)}u)-\overline{cu(0)}\Delta]\oplus[\overline{b}(1-\overline{u(0)}u)],\end{aligned}$$ must belong to $\mathbf{H}_{u}^{\perp}=[H^{2\perp}\oplus\{0\}]+\mathbf{G},$ that is, $$\begin{aligned}
Fk_{0} & = & (g_{1}+uh_{1})\oplus(\Delta h_{1}),\\
F^{*}k_{0} & = & (g_{2}+uh_{2})\oplus(\Delta h_{2}),\end{aligned}$$ for some $g_{1},g_{2}\in H^{2\perp}$ and $h_{1},h_{2}\in H^{2}$. Equating the second components, we see that $$c=\frac{\Delta h_{1}}{1-\overline{u(0)}u}=\Delta f_{1},\quad\overline{b}=\frac{\Delta h_{2}}{1-\overline{u(0)}u}=\Delta f_{2},$$ where $f_{j}=h_{j}/(1-\overline{u(0)}u)\in H^{2}$ for $j=1,2$. Define now $F_{1}$ and $F_{2}$ by (\[eq:F\_j\]) and set $$F_{0}=F-F_{1}-F_{2}^{*}=\left[\begin{array}{cc}
a_{0} & 0\\
0 & 0
\end{array}\right],$$ where $a_{0}=a-uf_{1}-\overline{uf_{2}}$. The hypothesis and the first part of the proof show that $F_{0}$ is also a symbol of the zero operator, and thus the vectors $$F_{0}k_{0}=a_{0}(1-\overline{u(0)}u)\oplus0,\quad F_{0}^{*}k_{0}=\overline{a_{0}}(1-\overline{u(0)}u),$$ must belong to $\mathbf{H}_{u}^{\perp}$. Observe that, given $f\in H^{2}$, the equality $\Delta f=0$ implies that $f$ vanishes almost everywhere on the set $\{\zeta\in\mathbb{T}:\Delta(\zeta)\ne0\}$, and thus $f=0$ by the F. and M. Riesz theorem. Therefore there exist functions $g_{1},g_{2}\in H^{2\perp}$ such that $$a_{0}(1-\overline{u(0)}u)=g_{1},\quad\overline{a_{0}}(1-\overline{u(0)}u)=g_{2}.$$ We have then $$a_{0}=\frac{g_{1}}{1-\overline{u(0)}u}=\frac{\overline{g_{2}}}{1-u(0)\overline{u}},$$ so $$g_{1}(1-u(0)\overline{u})=\overline{g_{2}}(1-\overline{u(0)}u).$$ The left hand side of this equality has a Fourier series with no analytic terms, while the right hand side has only analytic terms. We conclude that $g_{1}=g_{2}=0$, thus establishing that $F_{0}=0$. The proposition follows.
\[exa:u=00003D0\]We examine the special case $u=0$. In this case, $\mathbf{K}_{+}=H^{2}\oplus L^{2}$, $\mathbf{G}=\{0\}\oplus H^{2}$, $\mathbf{H}_{u}=H^{2}\oplus(L^{2}\ominus H^{2})$, and thus the operator $S_{u}$ is of the form $A\oplus B$, where $A$ is the forward shift on $H^{2}$ and $B$ is the forward (co-isometric) shift on $L^{2}\ominus H^{2}$. (In other words, $S_{u}$ is unitarily equivalent to $A\oplus A^{*}$.) A symbol $$F=\left[\begin{array}{cc}
a & b\\
c & d
\end{array}\right]$$ represents the zero operator in $\mathcal{T}_{u}$ precisely when $a=d=0$ and $\overline{b},c\in H^{2}$. In particular, the (1,1) and (2,2) entries of the symbol of an operator $T\in\mathcal{T}_{u}$ are uniquely determined by $T$. The operators that commute with $S_{u}$ are described, using the commutant lifting theorem, as the truncated multiplication operators with a symbol of the form $$F=\left[\begin{array}{cc}
a & 0\\
c & d
\end{array}\right],$$ where $a,d\in H^{\infty}$ and $b\in L^{\infty}$. The commutant of $S_{u}$ is not commutative, as illustrated by the operators $T_{1}$ and $T_{2}$ with symbols $$F_{1}=\left[\begin{array}{cc}
1 & 0\\
0 & 0
\end{array}\right]\text{ and }F_{2}=\left[\begin{array}{cc}
0 & 0\\
\overline{\chi} & 0
\end{array}\right],$$ for which $T_{1}T_{2}=0$ and $T_{2}T_{1}=T_{2}\ne0$.
An analog of the Crofoot operator\[sec:An-analog-of Crofoot\]
=============================================================
Suppose that $u\in H^{\infty}$ satisfies $\|u\|_{\infty}\le1$ and $|u(0)|<1$. The operator $X_{\mu}$ introduced in Example \[exa:the rank one perturbations\] is a completely nonunitary contractions with defect indices equal to $1$ provided that $|\mu|<1$. The caracteristic function of $X_{\mu}$ is equal to $$u_{\alpha}=\frac{u-\alpha}{1-\overline{\alpha}u}\in H^{\infty},$$ where $\alpha\in\mathbb{D}$ is chosen such that $u_{\alpha}(0)=-\mu$. Thus, there exists a unitary operator in $\mathcal{B}(\mathbf{H}_{u},\mathbf{H}_{u_{\alpha}})$, uniquely determined up to a constant factor of modulus $1$, that intertwines $X_{\mu}$ and $S_{u_{\alpha}}$. This unitary operator was first written explicitly by Crofoot [@crof] for the case in which $u$ is inner, and thus $u_{\alpha}$ is inner as well. He showed that it is the restriction to $\mathbf{H}_{u}$ of the multiplication operator by a function in $H^{\infty}$. We prove an analogous result for arbitrary purely contractive functions $u\in H^{\infty}$. To begin with, a simple calculation shows that the function $\Delta_{\alpha}=(1-|u_{\alpha}|^{2})^{1/2}$ satisfies $$\Delta_{\alpha}=\frac{(1-|\alpha|^{2})^{1/2}}{|1-\overline{\alpha}u|}\Delta,$$ so $(\Delta_{\alpha}L^{2})^{-}=(\Delta L^{2})^{-}$, and therefore $\mathbf{H}_{u}$ and $\mathbf{H}_{u_{\alpha}}$ are both subspaces of $\mathbf{K}$. More precisely, $$\mathbf{H}_{u_{\alpha}}=\mathbf{K}_{+}\ominus\mathbf{G}_{\alpha},$$ where $$\mathbf{G}_{\alpha}=\{u_{\alpha}f\oplus\Delta_{\alpha}f:f\in H^{2}\}.$$ We consider the bounded measurable function $F_{\alpha}$ defined by $$\begin{aligned}
F_{\alpha} & = & \left[\begin{array}{cc}
(1-|\alpha|^{2})^{1/2}(1-\overline{\alpha}u)^{-1} & 0\\
\overline{\alpha}\Delta|1-\overline{\alpha}u|^{-1} & (1-\overline{\alpha}u)|1-\overline{\alpha}u|^{-1}
\end{array}\right]\\
& = & \left[\begin{array}{cc}
(1-|\alpha|^{2})^{1/2}(1-\overline{\alpha}u)^{-1} & 0\\
\overline{\alpha}(1-|\alpha|^{2})^{-1/2}\Delta_{\alpha} & (1-\overline{\alpha}u)|1-\overline{\alpha}u|^{-1}
\end{array}\right].\end{aligned}$$ Since the $(1,1)$ entry of $F_{\alpha}$ belongs to $H^{\infty}$, it follows that $M_{F_{\alpha}}$ leaves $\mathbf{K}_{+}$ invariant.
\[prop:crofoot\]The operator $M_{F_{\alpha}}$ maps $\mathbf{H}_{u}$ isometrically onto $\mathbf{H}_{u_{\alpha}}$.
Suppose that $f\oplus g\in\mathbf{H}_{u}$ and thus $\overline{u}f+\Delta g\in L^{2}\ominus H^{2}$. As noted above, the vector $f_{\alpha}\oplus g_{\alpha}=M_{F_{\alpha}}(f\oplus g)$ belongs to $\mathbf{K}_{+}$. A direct calculation shows that $$\overline{u_{\alpha}}f_{\alpha}+\Delta_{\alpha}g_{\alpha}=(1-|\alpha|^{2})^{1/2}(\overline{u}f+\Delta g)(1-\alpha\overline{u})^{-1},$$ and this function belongs to $L^{2}\ominus H^{2}$ because $\overline{u}f+\Delta g\in L^2\ominus H^2$ and $(1-\alpha\overline{u})^{-1}$ is a bounded, conjugate analytic function. We conclude that $f_{\alpha}\oplus g_{\alpha}\in\mathbf{H}_{u_{\alpha}}$. In order to calculate the norm of $f_{\alpha}\oplus g_{\alpha}$ we observe that $$w=(\overline{\alpha}(1-|\alpha|^{2})^{-1/2}u_{\alpha}f)\oplus(\overline{\alpha}(1-|\alpha|^{2})^{-1/2}\Delta_{\alpha}f)\in\mathbf{G}_{\alpha}$$ and thus $$\|f_{\alpha}\oplus g_{\alpha}\|^{2}=\|(f_{\alpha}\oplus g_{\alpha})-w\|^{2}+\|w\|^{2}=\|(f_{\alpha}\oplus g_{\alpha})-w\|^{2}+|\alpha|^{2}(1-|\alpha|^{2})^{-1}\|f\|^{2}.$$ Since $$(f_{\alpha}\oplus g_{\alpha})-w=((1-|\alpha|^{2})^{-1/2}f)\oplus((1-\overline{\alpha}u)|1-\overline{\alpha}u|^{-1}g),$$ it follows that $\|(f_{\alpha}\oplus g_{\alpha})-w\|^{2}=(1-|\alpha|^{2})^{-1}\|f\|^{2}+\|g\|^{2}$ and hence that $\|f_{\alpha}\oplus g_{\alpha}\|^{2}=\|f\oplus g\|^{2}.$ The fact that $M_{F_{\alpha}}$ maps $\mathbf{H}_{u}$ onto $\mathbf{H}_{u_{\alpha}}$ follows from the above considerations applied to the operator $M_{F_{\alpha}}^{-1}$ because $$F_{\alpha}^{-1}=\left[\begin{array}{cc}
(1-|\alpha|^{2})^{1/2}(1+\overline{\alpha}u_{\alpha}){}^{-1} & 0\\
\overline{\alpha}\Delta_{\alpha}|1-\overline{\alpha}u_{\alpha}|^{-1} & (1-\overline{\alpha}u_{\alpha})|1-\overline{\alpha}u_{\alpha}|^{-1}
\end{array}\right],$$ and $u=(u_{\alpha}+\alpha)(1+\overline{\alpha}u_{\alpha})^{-1}$.
We denote by $V_{\alpha}\in\mathcal{B}(\mathbf{H}_{u},\mathbf{H}_{u_{\alpha}})$ the unitary operator defined by $V_{\alpha}x=M_{F_{\alpha}}x$, $x\in\mathbf{H}_{u}$. In the case in which $u$ is inner, $V_{\alpha}$ is precisely the operator constructed in [@crof].
\[prop:TTO preserved by Crofoot\]An operator $T\in\mathcal{B}(\mathbf{H}_{u})$ is a truncated multiplication operator if and only if $V_{\alpha}TV_{\alpha}^{*}\in\mathcal{B}(\mathbf{H}_{u_{\alpha}})$ is a truncated multiplication operator. Thus, $\mathcal{T}_{u_{\alpha}}=\{V_{\alpha}TV_{\alpha}^{*}:T\in\mathcal{T}_{u}\}$.
Fix $T\in\mathcal{B}(\mathbf{H}_{u})$ and define $T_{\alpha}=V_{\alpha}TV_{\alpha}^{*}$, so $$\langle T_{\alpha}V_{\alpha}x,V_{\alpha}x\rangle=\langle Tx,x\rangle,\quad x\in\mathbf{H}_{u}.$$ Since $M_{F_{\alpha}}U=UM_{F_{\alpha}}$, it follows that $Ux\in\mathbf{H}_{u}$ if and only if $UV_{\alpha}x\in\mathbf{H}_{u_{\alpha}}$. We conclude from the preceding identity that $T$ is $U$-invariant if and only if $T_{\alpha}$ is $U$-invariant. The proposition follows from Theorem \[teorema principala\].
\[sec:Complex-symmetries\]Complex symmetries
============================================
Suppose that $\mathcal{H}$ is a (complex) Hilbert space. A map $C:\mathcal{H}\to\mathcal{H}$ is called a *conjugation* if it is conjugate linear, isometric, and $C^{2}=I_{\mathcal{H}}$. A bounded operator $T\in\mathcal{B}(\mathcal{H})$ is said to be $C$-*symmetric* (respectively, $C$-*skew-symmetric*) if $CTC=T^{*}$ (respectively, $CTC=-T^{*}$). The operator $T$ is said to be *complex symmetric* if it is *$C$-symmetric* for some conjugation $C$. ** An operator $T$ can be complex symmetric relative to several conjugations. For instance, suppose that $U\in\mathcal{B}(L^{2})$ is the bilateral shift, that is, $Uf=\chi f$, $f\in L^{2}$. Given an arbitrary function $v\in L^{\infty}$ such that $|v|=1$ almost everywhere, the formula $$C_{v}f=v\overline{f},\quad f\in L^{2},$$ defines a conjugation on $L^{2}$ such that $U$ is $C_{v}$-symmetric. (It easy to see that these are all the conjugations relative to which $U$ is symmetric.)
\[prop:symmetries, not unique\] Suppose that $T\in\mathcal{B}(\mathcal{H})$, and $C$ and $D$ are two symmetries such that $T$ is both $C$-symmetric and $D$-symmetric. Then at least one of the following is true*:*
1. There exists a constant $\gamma\in\mathbb{T}$ such that $D=\gamma C$.
2. There exists a proper reducing subspace $\mathcal{K}$ for $T$ such that both $T|\mathcal{K}$ and $T|\mathcal{K}^{\perp}$ are complex symmetric.
Suppose that (1) is not true, and therefore the operator $V=DC$ is not a scalar multiple of $I_{\mathcal{H}}$. The operator $V$ is unitary and $$VT=DCT=DT^{*}C=TDC=TV.$$ Moreover, we have $$CVC=CD=(DC)^{-1}=V^{-1}=V^{*},$$ so $V$ is $C$-symmetric. If $E_{V}$ denotes the spectral measure of $V$, it follows that $E_{V}(\omega)$ is also $C$-symmetric for every Borel set $\omega\subset\mathbb{T}$, and therefore $E(\omega)TE(\omega)$ is also $C$-symmetric. To show that (2) is true, simply choose $\omega$ such that $0\ne E(\omega)\ne I_{\mathcal{H}}$ and set $\mathcal{K}=E(\omega)\mathcal{H}$. Then $T|\mathcal{K}$ is $C|\mathcal{K}$-symmetric and $T|\mathcal{K}^{\perp}$ is $C|\mathcal{K}^{\perp}$-symmetric.
Given a function $u\in H^{\infty}$ such that $\|u\|_{\infty}\le1$ and $|u(0)|<1$, the operator $S_{u}$ does not have any nontrivial reducing subspaces unless $u=0$. For $u=0$, $S_{u}$ has exactly one pair of complementary nontrivial reducing subspaces, and the restrictions of $S_{u}$ to these spaces are a unilateral shift and the adjoint of a unilateral shift, neither of which is complex symmetric relative to any conjugation. It follows that, up to a constant multiple of modulus one, there is at most one conjugation $C$ such that $S_{u}$ is $C$-symmetric. If $u$ is inner or, more generally, if $u$ is an extreme point of the unit ball of $H^{\infty}$, it follows from [@lot-sar] that $S_{u}$ is complex symmetric (see also [@garcia-p]). More general results about functional models [@che-fri-tim] show that $S_{u}$ is always complex symmetric. We describe below the essentially unique conjugation $C_{u}$ such that $S_{u}$ is $C_{u}$-symmetric.
The spaces $\mathbf{K},\mathbf{G}$, and $\mathbf{H}_{u}$ in the following statement were defined in Section \[sec:Preliminaries\].
Let $u\in H^{\infty}$ be such that $\|u\|_{\infty}\le1$ and $|u(0)|<1$. Then the operator $C:\mathbf{K}\to\mathbf{K}$ defined by $$C(f\oplus g)=(\overline{\chi}u\overline{f}+\overline{\chi}\Delta\overline{g})\oplus(\overline{\chi}\Delta\overline{f}-\overline{\chi}\overline{u}\overline{g}),\quad f\oplus g\in\mathbf{K},$$ is a conjugation such that $U$ is $C$-symmetric. Moreover, we have $C\mathbf{H}_{u}=\mathbf{H}_{u}$ and the operator $C_{u}=C|\mathbf{H}_{u}$ is a conjugation such that $S_{u}$ is $C_{u}$-symmetric.
The operator $C$ is simply complex conjugation followed by multiplication by the matrix function $$\overline{\chi}\left[\begin{array}{cc}
u & \Delta\\
\Delta & -\overline{u}
\end{array}\right].$$ It is easily seen that the matrix $$\left[\begin{array}{cc}
u(\zeta) & \Delta(\zeta)\\
\Delta(\zeta) & -\overline{u(\zeta)}
\end{array}\right]$$ is unitary for $\zeta\in\mathbb{T}$, and thus $C$ is an isometry. The operator $C^{2}$ is the multiplication operator by the matrix function $$\left[\begin{array}{cc}
\overline{u} & \Delta\\
\Delta & -u
\end{array}\right]\left[\begin{array}{cc}
u & \Delta\\
\Delta & -\overline{u}
\end{array}\right]=\left[\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right],$$ and thus $C^{2}=I_{\mathbf{K}}.$ The identity $U^{*}C=CU$ is also immediate. Observe next that $$C(uf\oplus\Delta f)=\overline{\chi}\overline{f}\oplus0,\quad f\in H^{2},$$ which shows that $C(\mathbf{G})=H^{2}\oplus\{0\}$, and thus $C(H^{2}\oplus\{0\})=\mathbf{G}$ as well. We conclude that $C(\mathbf{H}_{u}^{\perp})=\mathbf{H}_{u}^{\perp}$, $C(\mathbf{H}_{u})=\mathbf{H}_{u}$, and $C_{u}$ is indeed a conjugation on $\mathbf{H}_{u}$. Finally, $$S_{u}C_{u}=P_{\mathbf{H}_{u}}UC|\mathbf{H}_{u}=P_{\mathbf{H_{u}}}CU^{*}|\mathbf{H}_{u}=CP_{\mathbf{H}_{u}}U^{*}|\mathbf{H}_{u}=C_{u}S_{u}^{*},$$ showing that $S_{u}$ is $C_{u}$-symmetric.
We note for further use the equality $$C_{u}k_{0}=\widetilde{k}_{0}.\label{eq:C of k_0}$$
The linear manifold $\mathbf{K}^{\infty}$ is invariant under the conjugation $C$. It is not the case that every multiplication operator $M_{F}$ satisfies the equation $M_{F}v=CM_{F^{*}}Cv$ for every $v\in\mathbf{K}^{\infty}$.
\[prop:MF symmetric\]The multiplication operator $M_{F}$ by the matrix function $$F=\left[\begin{array}{cc}
a & b\\
c & d
\end{array}\right]$$ satisfies the equation $M_{F}=CM_{F^{*}}C|\mathbf{K}^{\infty}$ if and only if the equality $$\Delta(d-a)=-uc-\overline{u}b\label{eq:C-symmetric matrix}$$ holds almost everywhere on $\{\zeta\in\mathbb{T}:\Delta(\zeta)\ne0\}$.
The operator $CM_{F^{*}}C|\mathbf{K}^{\infty}$ is the operator of multiplication by the matrix $$\left[\begin{array}{cc}
u & \Delta\\
\Delta & -\overline{u}
\end{array}\right]\left[\begin{array}{cc}
a & c\\
b & d
\end{array}\right]\left[\begin{array}{cc}
\overline{u} & \Delta\\
\Delta & -u
\end{array}\right],$$ and a calculation shows that $$\left[\begin{array}{cc}
u & \Delta\\
\Delta & -\overline{u}
\end{array}\right]\left[\begin{array}{cc}
a & c\\
b & d
\end{array}\right]\left[\begin{array}{cc}
\overline{u} & \Delta\\
\Delta & -u
\end{array}\right]-F=\left[\begin{array}{cc}
\Delta h & -uh\\
-\overline{u}h & -\Delta h
\end{array}\right],$$ where $$h=\Delta(d-a)+uc+\overline{u}b.$$ The desired conclusion follows from Proposition \[prop:symbols for the zero operator\](1).
\[cor:AF symmetric\]If the matrix $$F=\left[\begin{array}{cc}
a & b\\
c & d
\end{array}\right]$$ satisfies the equality $\Delta(a-d)=uc+\overline{u}b$ almost everywhere on $\{\zeta\in\mathbb{T}:\Delta(\zeta)\ne0\}$, then $A_{F}=C_{u}A_{F^{*}}C_{u}|\mathbf{H}_{u}^{\infty}$.
In the particular case in which $u$ is an inner function, the function $\Delta$ is equal to zero almost everywhere. Thus, the preceding corollary shows that every operator in $\mathcal{T}_{u}$ is $C_{u}$-symmetric. This result [@sar-TTO Section 2.3] plays an important role in the study of truncated Toeplitz operators. If $u$ is not inner, there are operators in $\mathcal{T}_{u}$ that are not $C_{u}$-symmetric. For instance, the operator with symbol $$\left[\begin{array}{cc}
0 & 0\\
0 & 1
\end{array}\right]$$ is not $C_{u}$-symmetric.
Suppose that $T\in\mathcal{B}(\mathbf{H}_{u})$ is a truncated multiplication operator. Then the operator $C_{u}T^{*}C_{u}$ is easily seen to be a truncated multiplication operator as well. It follows that $T$ can be written in a unique way as a sum $T=T_{1}+T_{2}$, where $T_{1}=(1/2)(T+C_{u}T^{*}C_{u})$ is a $C_{u}$-symmetric truncated multiplication operator and $T_{2}=(1/2)(T-C_{u}T^{*}C_{u})$ is a $C_{u}$-skew-symmetric operator. The above calculations allow us to show that the operators $T_{1}$ and $T_{2}$ have symbols of a special form.
\[prop:symbols of symmetric and skew-symmetric TTO\]Suppose that $T\in\mathcal{T}_{u}$. Then*:*
1. If $T$ is $C_{u}$-symmetric then it has a symbol of the form $$\left[\begin{array}{cc}
a & b\\
c & a-(\overline{u}b+uc)/\Delta
\end{array}\right]$$ for some $a,b,c\in L^{2}$.
2. If $T$ is $C_{u}$-skew-symmetric then it has a symbol of the form $$\left[\begin{array}{cc}
-\Delta f & uf\\
\overline{u}f & \Delta f
\end{array}\right]$$ for some $f\in L^{2}$.
By Theorem \[teorema principala\] and Proposition \[prop:limit for uniqueness\], $T$ has a symbol of the form $$G=\left[\begin{array}{cc}
\alpha & \beta\\
\gamma & \delta
\end{array}\right]$$ with $\alpha,\beta,\gamma\in L^{2}$ and $\delta\in L^{\infty}.$ If $T$ is $C_{u}$ symmetric, the function $F$ such that $M_{F}=(1/2)(M_{G}+CM_{G}^{*}C)$ is again a symbol for $T$ and it has the form specified in (1) by Proposition of \[prop:MF symmetric\]. If $T$ is $C_{u}$-skew-symmetric, we use instead the operator $M_{H}=(1/2)(M_{G}-CM_{G}^{*}C)$. The proof of Proposition \[prop:MF symmetric\] shows that $H$ has the form specified in (2).
[10]{} J. A. Ball and A. Lubin, On a class of contractive perturbations of restricted shifts, *Pacific J. Math.* **63** (1976), no. 2, 309–323.
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A. Baranov, R. Bessonov, and V. Kapustin, Symbols of truncated Toeplitz operators, *J. Funct. Anal.* **261** (2011), no. 12, 3437–3456.
N. Chevrot, E. Fricain, and D. Timotin, The characteristic function of a complex symmetric contraction, *Proc. Amer. Math. Soc.* **135** (2007), no. 9, 2877–2886.
D. N. Clark, One dimensional perturbations of restricted shifts, *J. Analyse Math.* **25** (1972), 169–191.
R. B. Crofoot, Multipliers between invariant subspaces of the backward shift, *Pacific J. Math*. **166** (1994), no. 2, 225–246.
L. de Branges, Some Hilbert spaces of entire functions. II, *Trans. Amer. Math. Soc*. **99** (1961), 118–152.
S. R. Garcia and M. Putinar, Complex symmetric operators and applications, *Trans. Amer. Math. Soc.* **358** (2006), no. 3, 1285–1315.
M. S. Liv¨ic, On the theory of isometric operators with equal deficiency indices, *Doklady Akad. Nauk SSSR (N.S.)* **58**, (1947). 13–15.
B. A. Lotto and D. Sarason, Multiplicative structure of de Branges’s spaces, *Rev. Mat. Iberoamericana* **7** (1991), no. 2, 183–220.
N. K. Nikolski, *Operators, functions, and systems: an easy reading. Vol. 2. Model operators and systems*, Translated from the French by Andreas Hartmann and revised by the author, Mathematical Surveys and Monographs, 93, American Mathematical Society, Providence, RI, 2002.
D. Sarason, Generalized interpolation in $H^{\infty}$, *Trans. Amer. Math. Soc*. **127** (1967) 179–203.
———, *Sub-Hardy Hilbert spaces in the unit disk*, John Wiley and Sons, New York, 1994.
———, Algebraic properties of truncated Toeplitz operators, *Oper. Matrices* **1** (2007), no. 4, 491–526.
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[^1]: HB was supported in part by a grant of the National Science Foundation.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Alexander Grabner$^1$\'
- 'Peter M. Roth$^1$\'
- 'Vincent Lepetit$^{2,1}$\'
- '$^1$Institute of Computer Graphics and Vision, Graz University of Technology, Austria\'
- '$^2$Laboratoire Bordelais de Recherche en Informatique, University of Bordeaux, France\'
- '[{alexander.grabner,pmroth,lepetit}@icg.tugraz.at]{}'
bibliography:
- 'string.bib'
- 'references.bib'
title: |
Location Field Descriptors:\
Single Image 3D Model Retrieval in the Wild
---
#### Acknowledgement
This work was supported by the Christian Doppler Laboratory for Semantic 3D Computer Vision, funded in part by Qualcomm Inc. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research.
In the following, we provide additional details and qualitative results of our novel 3D model retrieval approach called Location Field Descriptors.
Datasets and Evaluation Setup
=============================
We evaluate our proposed approach for 3D model retrieval in the wild on three challenging real-world datasets with different object categories: Pix3D [@Sun2018pix3d] (*bed*, *chair*, *sofa*, *table*), Comp [@Wang2018fine] (*car*), and Stanford [@Wang2018fine] (*car*). These datasets have only been released recently and, to the best of our knowledge, we are the first to report results for 3D model retrieval on all of them. In addition to retrieval from 3D models provided by these datasets, we also retrieve 3D models from ShapeNet [@Shapenet2015]. Table \[table:datasets\] presents an overview of the object categories, the number of RGB images, and the number of 3D models in the evaluated datasets.
Dataset Category Train Images Test Images 3D Models from Dataset 3D Models from ShapeNet
---------- ---------- -------------- ------------- ------------------------ -------------------------
bed 203 191 19 254
chair 1506 1388 221 6778
sofa 552 540 20 3173
table 387 351 62 8443
Comp car 3798 1898 98 7497
Stanford car 8144 8041 134 7497
The Pix3D dataset provides multiple categories, however, we only train and evaluate on categories which have more than 300 non-occluded and non-truncated samples (*bed*, *chair*, *sofa*, *table*). Further, we restrict the training and evaluation to samples marked as non-occluded and non-truncated, because we do not know which objects parts are occluded nor the extent of the occlusion, and many objects are heavily truncated. For each 3D model, we randomly choose 50% of the corresponding images for training and the other 50% for testing.
The Comp and Stanford datasets only provide one category (*car*). Most images show one prominent car which is non-occluded and non-truncated. The two datasets already provide a train-test split. Thus, we use all available samples from Comp and Stanford for training and evaluation.
In contrast to these datasets which provide a large number of RGB images and a moderate number of 3D models with corresponding annotations, ShapeNet does not provide RGB images but a large number of 3D models. Due to its enormous size, ShapeNet does not only cover many different object categories but also presents a large variety in 3D model geometry. If 3D models are present in the respective dataset and in ShapeNet, we exclude them for retrieval from ShapeNet to evaluated retrieval from an entirely unseen database.
We consistently orient, scale and translate all 3D models. In particular, we rotate all 3D models to have common front facing, up, and starboard directions. Additionally, we scale and translate all 3D models to fit inside a unit cube centered at the coordinate origin $(0,0,0)$ while preserving the aspect-ratio of the 3D dimensions. This 3D model alignment is not only important for training our approach but also for the evaluated metrics. For example, computing the modified Hausdorff distance and the 3D IOU between two 3D models is only meaningful if they are consistently oriented, scaled and centered.
Implementation and Training Details
===================================
For our Location Field CNN, we use a Feature Pyramid Network [@Lin2017feature] on top of a ResNet-101 backbone [@He2016deep; @He2016identity]. For our Descriptor CNN, we use a DenseNet-50 architecture [@Huang2017densely] with 3 dense blocks and a growth rate of 24. For our implementation, we resize and pad RGB images to a spatial resolution of $512\times512\times3$ maintaining the aspect ratio. For the location fields, we employ a resolution of $58\times58\times3$. In this configuration, the Descriptor CNN maps low-resolution location fields to a 270-dimensional embedding space.
We initialize the convolutional backbone and the detection branches of the Location Field CNN with weights trained for instance segmentation [@He2017mask] on COCO [@Lin2014microsoft]. The location field branch and the Descriptor CNN are trained from scratch. We train our networks for 300 epochs using a batch size of 32. The initial learning rate of $1e^{-3}$ is decreased by a factor of 5 after 150 and 250 epochs.
We employ different forms of data augmentation. For RGB images, we use mirroring, jittering of location, scale, and rotation, and independent pixel augmentations like additive noise. For rendered location fields, we additionally use different forms of blurring to simulate predicted location fields. During training of the Descriptor CNN, we further leverage synthetic data and train on predicted and rendered location fields using a ratio of $1:3$.
To balance the individual terms in the system loss $$L = L_{\text{D}} + L_{\text{softmax}} + \alpha L_{\text{C}} + \beta L_{\text{TC}} + \gamma L_{\text{LF}} + \delta L_{\text{FM}},
\label{eq:descriptor_loss}$$ we assign unit weights to classification losses and non-unit weights to regression losses. Thus, we combine the unmodified Detection losses ($L_{\text{D}}$) of the generalized Faster/Mask R-CNN framework and the Descriptor CNN softmax loss ($L_{\text{softmax}}$) with the weighted Center ($L_{\text{C}}$), Triplet-Center ($L_{\text{TC}}$), Location Field ($L_{\text{LF}}$) and Feature Mapping ($L_{\text{FM}}$) losses. We experimentally set $\alpha=0.01$, $\beta=0.1$, $\gamma=10$, $\delta=0.01$, and use a margin of $m=1$. For the Huber distance [@Huber1964robust], we set the threshold to $1$.
Failure Cases
=============
Fig. \[fig:fails\] shows failure cases of our approach. Most failure cases relate to incorrect location field predictions. For example, if the 3D pose of the object in the image is far from the 3D poses seen during training, or if multiple objects are detected as a single object in a complex occlusion scenario, we cannot predict an accurate location field. In other failure cases, we predict an accurate location field, but retrieve a 3D model from a different category due to ambiguous 3D geometries, , *table* instead of *chair*. While the detection branch of the generalized Faster/Mask R-CNN predicts a category for each detected object in an RGB image, we do not use this information during retrieval, because there is a category ambiguity for many objects. For example, it is unclear if a couch with sleeping functionality is a *sofa* or a *bed* (see Fig. \[fig:ret\_pose1\], left column, last example).
Additional Qualitative Results
==============================
Finally, we present additional qualitative results which complement those presented in the main paper.
Fig. \[fig:pred\_lfs2\] presents further qualitative examples of our predicted location fields. We upscale and pad the predicted location fields to match the input image resolution. The overall 3D shape is recovered well in the location fields, but fine-grained details like the side mirrors of cars or thin structures like the frame ornaments of tables and beds are missed.
Fig. \[fig:top10C\] shows additional qualitative results for 3D model retrieval from ShapeNet. Considering the top ten ranked 3D models, we observe that the retrieved models have a consistent and accurate overall 3D shape and geometry.
Figs. \[fig:ret\_pose1\] and \[fig:ret\_pose2\] present more qualitative results for 3D model retrieval from both seen and unseen databases. In addition, we show that our predicted location fields provide all relevant information to also compute the 3D pose of objects. For this purpose, we sample 2D-3D correspondences from the location field and solve a [P*n*P]{} problem during inference. The projections onto the image show that both our retrieved 3D models and our computed 3D poses are highly accurate. Our approach naturally handles multiple objects in a single image, however, all evaluated datasets only provide annotations for a single instance per image, as shown in Fig. \[fig:ret\_pose2\].
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{
"pile_set_name": "ArXiv"
}
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---
abstract: |
The variance of a bounded linear operator $a$ on a Hilbert space $\h$ at a unit vector $\xi$ is defined by ${D_{\xi}(a)}=\|a\xi\|^2-|{\langle a\xi,\xi\rangle}|^2$. We show that two operators $a$ and $b$ have the same variance at all vectors $\xi\in\h$ if and only if there exist scalars $\sigma,\lambda\in\bc$ with $|\sigma|=1$ such that $b=\sigma a+\lambda1$ or $a$ is normal and $b=\sigma a^*+\lambda1$. Further, if $a$ is normal, then the inequality ${D_{\xi}(b)}\leq\kappa {D_{\xi}(a)}$ holds for some constant $\kappa$ and all unit vectors $\xi$ if and only if $b=f(a)$ for a Lipschitz function $f$ on the spectrum of $a$. Variants of these results for C$^*$-algebras are also proved, where vectors are replaced by pure states.
We also study the related, but more restrictive inequalities $\|bx-xb\|\leq
\|ax-xa\|$ supposed to hold for all $x\in{{\rm B}(\mathcal{H})}$ or for all $x\in{\rm B}(\h^n)$ and all $n\in\bn$. We consider the connection between such inequalities and the range inclusion $d_b({{\rm B}(\mathcal{H})})\subseteq d_a({{\rm B}(\mathcal{H})})$, where $d_a$ and $d_b$ are the derivations on ${{\rm B}(\mathcal{H})}$ induced by $a$ and $b$. If $a$ is subnormal, we study these conditions in particular in the case when $b$ is of the form $b=f(a)$ for a function $f$.
address: |
Department of Mathematics\
University of Ljubljana\
Jadranska 21\
Ljubljana 1000\
Slovenia
author:
- Bojan Magajna
title: Variance of operators and derivations
---
[^1]
[^2]
Introduction and notation
=========================
The expected value of a quantum mechanical quantity represented by a selfadjoint operator $a$ on a complex Hilbert space $\h$ in a state $\omega$ is $\omega(a)$, while the [*variance*]{} of $a$ is defined by ${D_{\omega}(a)}=\omega(a^*a)-|\omega(a)|^2$. If $a$ is the multiplication by a bounded measurable function on $L^2(\mu)$ for a probability measure $\mu$ and $\omega$ is the state $x\mapsto{\langle x1,1\rangle}$, where $1\in L^2(\mu)$ is the constant function, these notions reduce to the classical notions of probability calculus. We may define the variance by the same formula for all (not necessarily selfadjoint) operators $a\in{{\rm B}(\mathcal{H})}$. For a general vector state $\omega(a):={\langle a\xi,\xi\rangle}$, coming from a unit vector $\xi\in\h$, the variance ${D_{\omega}(a)}=\|a\xi\|^2-|{\langle a\xi,\xi\rangle}|^2$ means just the square of the distance of $a\xi$ to the set of all scalar multiples of $\xi$. (Thus ${D_{\omega}(a)}=\eta_a(\xi)^2$, where $\eta$ is the function considered by Brown and Pearcy in [@BP].) We will prove that an operator $a$ is almost determined by its variances: if $a,b\in{{\rm B}(\mathcal{H})}$ are such that ${D_{\omega}(a)}={D_{\omega}(b)}$ for all vector states $\omega$ then $b=\alpha a+\beta1$ or $a$ is normal and $b=\alpha a^*+\beta1$ for some $\alpha,\beta\in\bc$ with $|\alpha|=1$ (Theorem \[th1\]). We will also deduce a variant of this statement for C$^*$-algebras, where vector states are replaced by pure states.
Then we will study the inequality $$\label{-1}{D_{\omega}(b)}\leq\kappa{D_{\omega}(a)},$$ where $\kappa$ is a positive constant (which may be taken to be $1$ if we replace $b$ by $\kappa^{-1/2}b$). If (\[-1\]) holds for all vector states $\omega$, then we will show that there exists a Lipschitz function $f:{\sigma_{\rm ap}(a)}\to{\sigma_{\rm ap}(b)}$, where ${\sigma_{\rm ap}(\cdot)}$ denotes the approximate point spectrum, such that if $a$ is normal then $b=f(a)$ (Theorem \[th3\]). For a general $a$, however, $f$ is perhaps not nice enough to allow the definition of $f(a)$. Therefore we will also consider stronger variants of (\[-1\]).
For $2\times 2$ matrices (\[-1\]) implies that $b=\alpha a+\beta1$ for some scalars $\alpha,\beta\in\bc$ (Lemma \[le2\]). But for general operators the condition (\[-1\]) is not very restrictive for it does not even imply that $b$ commutes with $a$. For example, if $a$ is hyponormal (\[-1\]) holds with $b=a^*$. A simple computation (Lemma \[le41\]) shows, however, that for a vector state $\omega=\omega_{\xi}$ the quantity ${D_{\omega}(a)}$ is just the square of the norm of the operator $d_a(\xi\otimes\xi^*)$, where $d_a$ is the derivation on ${{\rm B}(\mathcal{H})}$, defined by $d_a(x)=ax-xa$, and $\xi\otimes\xi^*$ is the rank one operator on $\h$, defined by $(\xi\otimes\xi^*)
\eta={\langle \eta,\xi\rangle}\xi$. Thus we will also study the condition $$\label{01}\|d_b(x)\|\leq\kappa\|d_a(x)\| \ \ (\forall x\in{{\rm B}(\mathcal{H})}),$$ where $a,b\in{{\rm B}(\mathcal{H})}$ and $\kappa>0$ are fixed. We will show (Theorem \[th42\]) that if equality holds in (\[01\]) and $\kappa=1$ then either $b=\sigma a+\lambda1$ for some scalars $\sigma,\lambda\in\bc$ with $|\sigma|=1$ or there exist a unitary $u$ and scalars $\alpha,\beta,\lambda,\mu$ in $\bc$ with $|\beta|=|\alpha|$ such that $a=\alpha u^*+\lambda1$ and $b=\beta u+\mu1$. This will also be generalized to C$^*$-algebras.
For a normal $a$ Johnson and Williams [@JW] proved that the condition (\[01\]) is equivalent to the range inclusion $d_b({{\rm B}(\mathcal{H})})\subseteq d_a({{\rm B}(\mathcal{H})})$. Their work was continued by several researchers, including Williams [@W], Fong [@F], Kissin and Shulman [@KS], Brešar [@Br] and in [@BMS] in different contexts, but still restricted to special classes of operators $a$ (such as normal, isometric or algebraic). It is known that the range inclusion $d_b({{\rm B}(\mathcal{H})})\subseteq d_a({{\rm B}(\mathcal{H})})$ does not imply (\[01\]) in general since it does not even imply that $b$ is in the bicommutant $(a)^{\prime\prime}$ of $a$ [@Ho]. However in Theorem \[th54\] we will prove that conversely (\[01\]) implies the range inclusion if $a$ is a direct sum $a_1\oplus a_2$, where the commutant ${(a_1)^{\prime}}$ of $a_1$ contains a bounded net of trace class operators converging strongly to the identity, while ${(a_2)^{\prime}}$ does not contain any nonzero trace-class operator. Examples include all normal operators, isometries and cyclic subnormal operators. The author does not know of any operators $a,b$ satisfying (\[01\]) for which the range inclusion does not hold. The corresponding purely algebraic problem for operators on an (infinite dimensional) vector space $\mathcal{V}$, where ${{\rm B}(\mathcal{H})}$ is replaced by the algebra ${\rm L}(\mathcal{V})$ of all linear operators on $\mathcal{V}$ and the condition (\[01\]) is replaced by the inclusion of the kernels $\ker d_a\subseteq\ker d_b$, has a positive answer [@M1].
By the Hahn-Banach theorem the inclusion $d_b({{\rm B}(\mathcal{H})})\subseteq{\overline{\overline{d_a({{\rm B}(\mathcal{H})})}}}$ (the norm closure) is equivalent to the requirement that for each $\rho\in{{{\rm B}(\mathcal{H})}^{\sharp}}$ (the dual of ${{\rm B}(\mathcal{H})}$) the condition $a\rho-\rho a=0$ implies $b\rho-\rho b=0$, where $a\rho$ and $b\rho$ are functionals on ${{\rm B}(\mathcal{H})}$ defined by $(a\rho)(x)=\rho(xa)$ and $(\rho a)(x)=\rho(ax)$. The operator spaces ${{\rm B}(\mathcal{H})}$ and ${{{\rm B}(\mathcal{H})}^{\sharp}}$ are quite different (if $\h$ is infinite dimensional), so in general we can not expect a strong connection between (\[01\]) and a formally similar condition $$\label{03}\|b\rho-\rho b\|\leq\kappa\|a\rho-\rho a\|\ \ (\forall x\in{{{\rm B}(\mathcal{H})}^{\sharp}}).$$
Does (\[01\]) imply at least that the centralizer $C_a$ of $a$ in ${{{\rm B}(\mathcal{H})}^{\sharp}}$ (that is, the set of all $\rho\in{{{\rm B}(\mathcal{H})}^{\sharp}}$ satisfying $a\rho=\rho a$) is contained in $C_b$?
Using C$^*$-algebraic tools, the above question can be reduced to the corresponding question in the Calkin algebra, and answered affirmatively if $a$ is essentially normal (Corollary \[co07\]). We recall that an operator $a$ is essentially normal if $a^*a-aa^*$ is compact.
A stronger condition than (\[01\]), namely that (\[01\]) holds for all $x\in{{\rm M}_n({{\rm B}(\mathcal{H})})}$ and all $n\in\bn$ (where $a$ and $b$ are replaced by the multiples $a^{(n)}$ and $b^{(n)}$ acting on $\h^n$), implies that (\[01\]) holds in any representation of the C$^*$-algebra generated by $a,b$ and $1$ (Lema \[le61\]) and that $b$ is contained in the C$^*$-algebra generated by $a$ and $1$ (Corollary \[co610\]). In a special situation (when $\h$ is a cogenerator for Hilbert modules over the operator algebra $A_0$ generated by $a$ and $1$) it follows that $b$ must be in $A_0$ (Proposition \[th62\]). If $a$ is, say, subnormal (a restriction of a normal operator to an invariant subspace), this means that $b=f(a)$ for a function $f$ in the uniform closure of polynomials on $\sigma(a)$. Perhaps for a general subnormal operator $a$ (\[01\]) does not imply that $b=f(a)$ for a function $f$, but when it does, it forces on $f$ certain degree of regularity. For example, if $a$ is the operator of multiplication on the Hardy space $H^2(G)$ by the identity function on $G$, where $G$ is a domain in $\bc$ bounded by finitely many nonintersecting analytic Jordan curves, (\[01\]) implies that $b$ is an analytic Toeplitz operator with a symbol $f$ which is continuous also on the boundary of $G$ (Proposition \[pr6101\]).
Let us call a complex function $f$ on a compact set $K\subseteq\bc$ a [*Schur function*]{} if the supremum over all (finite) sequences $\lambda=(\lambda_1,\lambda_2,\ldots)\subseteq K$ of norms of matrices $$\Lambda(f;\lambda)=\left[\frac{f(\lambda_i)-f(\lambda_j)}{\lambda_i-\lambda_j}\right],$$ regarded as Schur multipliers, is finite. (Here the quotient is interpreted as $0$ if $\lambda_i=\lambda_j$.) If $a$ is normal the work of Johnson and Williams [@JW] tells us that $b=f(a)$ satisfies (\[01\]) if and only if $f$ is a Schur function on $\sigma(a)$. In the ‘only if’ direction we extend this to general subnormal operators (Lemma \[le521\]), in the other direction only to subnormal operators with nice spectra (Theorem \[th621\]).
In the last section we will investigate the condition (\[01\]) in the case when $a$ is subnormal and $b=f(a)$ for a function $f$. If $a$ is normal, a known effective method of studying such commutator estimates is based on double operator integrals (see [@APPS] and the references there), which are defined via spectral projection valued measures. But, since invariant subspaces of a normal operator $c$ are not necessarily invariant under the spectral projections of $c$, we will use a different method. In Section 7 we will ‘construct’ for a given subnormal operator $a$ and suitable function $f$ on $\sigma(a)$ a completely bounded map $T_{a,f}$ on ${{\rm B}(\mathcal{H})}$ such that $aT_{a,f}(x)-T_{a,f}(x)a=f(a)x-xf(a)$ for all $x\in{{\rm B}(\mathcal{H})}$. For $b=f(a)$ this implies (\[01\]) and also the range inclusion $d_b({{\rm B}(\mathcal{H})})\subseteq d_a({{\rm B}(\mathcal{H})})$. Thus, by the above mentioned result from [@JW] even if $a$ is normal the functions $f$ considered here must be Schur. By [@JW] every Schur function on $\sigma(a)$ is complex differentiable relative to $\sigma(a)$ at each nonisolated point of $\sigma(a)$ (thus holomorphic on the interior of $\sigma(a)$) and $f^{\prime}$ is bounded. The construction of $T_{a,f}$ applies to a subclass that includes all functions for which $f^{\prime}$ is Lipschitz of order $\alpha>0$; only if $\sigma(a)$ is nice enough are we able to find $T_{a,f}$ for all Schur functions.
We will denote by ${\overline{\overline{S}}}$ the norm closure and by ${\overline{S}}$ the weak\* closure of a subset $S$ in ${{\rm B}(\mathcal{H})}$.
Variance of operators
=====================
For a bounded operator $a$ on a Hilbert space $\h$ and a vector $\xi\in\h$ let $${D_{\xi}(a)}=(\|a\xi\|^2\|\xi\|^2-|{\langle a\xi,\xi\rangle}|^2)\|\xi\|^{-4}.$$ Thus, if $\xi$ is a unit vector and $\omega:x\mapsto{\langle x\xi,\xi\rangle}$ is the corresponding vector state on ${{\rm B}(\mathcal{H})}$, then $${D_{\xi}(a)}=\omega(a^*a)-|\omega(a)|^2,$$ and this formula can be used to define the [*variance*]{} $D_{\omega}(a)$ of $a$ in any (not just vector) state $\omega$.
\[re0\] (i) It is clear from the definition that ${D_{\xi}(a)}$ is just the square of the distance of $a\xi$ to the set $\bc\xi$ of scalar multiples of $\xi$. Hence, if ${D_{\xi}(b)}\leq{D_{\xi}(a)}$ for all $\xi\in\h$, then in particular each eigenvector of $a$ is also an eigenvector for $b$.
\(ii) ${D_{\xi}(\alpha a+\beta1)}=|\alpha|^2{D_{\xi}(a)}$ for all $a,b\in{{\rm B}(\mathcal{H})}$ and $\alpha,\beta\in\bc$.
\(iii) ${D_{\xi}(a^*)}={D_{\xi}(a)}$ for all $\xi\in\h$ if and only if $a$ is normal.
\[th1\]If operators $a,b\in{{\rm B}(\mathcal{H})}$ satisfy ${D_{\xi}(b)}={D_{\xi}(a)}$ for all $\xi\in\h$, then there exist $\alpha,\beta\in\bc$ with $|\alpha|=1$ such that $b=\alpha a+\beta$ or $a$ is normal and $b=\alpha a^*+\beta$.
For any two vectors $\xi,\eta\in\h$ we expand the function $$f(z):={D_{\xi+z\eta}(a)}=\|a(\xi+z\eta)\|^2\|\xi+z\eta\|^2-|{\langle a(\xi+z\eta),\xi+z\eta\rangle}|^2$$ of the complex variable $z$ into powers of $z$ and $\overline{z}$, $$f(z)={D_{\xi}(a)}+2{{\rm Re}\,{(}}D_1z)+2{{\rm Re}\,{(}}D_2z^2)+D_3|z|^2+2{{\rm Re}\,{(}}D_4|z|^2z)+{D_{\eta}(a)}|z|^4.$$ Among the coefficients $D_j$ we will need to know only $D_2$, which is $$D_2={\langle a\eta,a\xi\rangle}{\langle \eta,\xi\rangle}-{\langle a\eta,\xi\rangle}{\langle \eta,a\xi\rangle}.$$ Thus, from the equality ${D_{\xi+z\eta}(a)}={D_{\xi+z\eta}(b)}$, by considering the coefficients of $z^2$ we obtain $${\langle b\eta,b\xi\rangle}{\langle \eta,\xi\rangle}-{\langle b\eta,\xi\rangle}{\langle \eta,b\xi\rangle}={\langle a\eta,a\xi\rangle}{\langle \eta,\xi\rangle}-{\langle a\eta,\xi\rangle}{\langle \eta,a\xi\rangle}.$$ From this we see that if $\eta$ is orthogonal to $\xi$ and $a\xi$ then $\eta$ must be orthogonal to $b\xi$ or to $b^*\xi$. In other words, if for a fixed $\xi$ we denote $$\h_0(\xi)=\{\xi,a\xi\}^{\perp},\ \ \ \h_1(\xi)=\{\xi,a\xi,b\xi\}^{\perp},\ \ \ \h_2=\{\xi,a\xi,b^*\xi\}^{\perp},$$ then $\h_0(\xi)=\h_1(\xi)\cup\h_2(\xi)$. Since $\h_j(\xi)$ are vector spaces, this implies that $\h_1(\xi)=\h_0(\xi)$ or else $\h_2(\xi)=\h_0(\xi)$. In the first case we have $b\xi\in\bc\xi+\bc a\xi$, while in the second case $b^*\xi\in\bc\xi+\bc a\xi$. Since this holds for all $\xi\in\h$, it follows that $\h$ is the union of the two sets $$F_1=\{\xi\in\h:\, b\xi\in\bc\xi+\bc a\xi\}\ \ \ \mbox{and}\ \ \ F_2=\{\xi\in\h:\, b^*\xi\in\bc\xi+\bc a\xi\}.$$ Since $F_1$ and $F_2$ are closed, by Baire’s theorem at least one of them has nonempty interior $\stackrel{\circ}{F_i}$. We will consider only the case when $\stackrel{\circ}{F_1}\ne\emptyset$ for the other case is similar (except that in the end the observation that ${D_{\xi}(a^*)}={D_{\xi}(a)}$ ($\forall \xi\in\h$) implies the normality of $a$ is used). We may assume that $a$ is not a scalar multiple of the identity, otherwise the proof is trivial. Then there exists a vector $\xi\in \stackrel{\circ}{F_1}$ such that $\xi$ and $a\xi$ are linearly independent. (Namely, if $a\xi=\alpha_{\xi}\xi$ for all $\xi\in\stackrel{\circ}{F_1}$, where $\alpha_{\xi}\in\bc$, then considering this equality for the vectors $\xi$, $\zeta$ and $(1/2)(\xi+\zeta)$ in $\stackrel{\circ}{F_1}$, where $\xi$ and $\zeta$ are linearly independent, it follows easily that $\alpha_{\xi}$ must be independent of $\xi$ for $\xi$ in an open subset of $\h$, hence $a$ must be a scalar multiple of $1$.) Let $$U=\{\xi\in\stackrel{\circ}{F_1}:\, \xi\ \mbox{and}\ a\xi\ \mbox{are linearly independent}\}.$$ Then for any $\xi,\eta\in U$ such that the ‘segment’ $\xi(z)=(1-z)\xi+z\eta$ ($|z|\leq1$) is contained in $U$, we have $$\label{1}b\xi(z)=\alpha(z)a\xi(z)+\beta(z)\xi(z)$$ for some scalars $\alpha(z),\beta(z)\in\bc$. To see that the coefficients $\alpha$ and $\beta$ are holomorphic functions of $z$, for any fixed $z_0$ with $|z_0|\leq1$ we take the inner product of both sides of (\[1\]) with the vectors $\xi(z_0)$ and $a\xi(z_0)$ to obtain two equations from which we compute $\alpha(z)$ and $\beta(z)$ by Cramer’s rule (if $z$ is near $z_0$). But from the condition ${D_{\xi(z)}(b)}={D_{\xi(z)}(a)}$ and (\[1\]) we also conclude that $|\alpha(z)|=1$, hence $\alpha$ must be constant (for a fixed $\xi$ and $\eta$). Setting in (\[1\]) first $z=0$ and then $z=1$ we get $$\label{2}b\xi=\alpha a\xi+\beta(0)\xi\ \ \ \mbox{and}\ \ \ b\eta=\alpha a\eta+\beta(1)\eta,$$ where the constant $\alpha$ (with $|\alpha|=1$) is the same for all vectors $\xi,\eta$ in an open subset of $\h$. Namely, the coefficients in (\[2\]) are unique since $\eta$ and $a\eta$ are linearly independent for all $\eta\in U$, hence by fixing $\eta$ and varying $\xi$ in (\[2\]) we see that $\alpha$ must be independent of $\xi$. From (\[2\]) we have now that $(b-\alpha a)\xi\in\bc\xi$ for all vectors $\xi$ in an open subset of $\h$ (with a constant $\alpha$), which easily implies that $b-\alpha a=\beta1$ for some constant $\beta\in\bc$.
\[co11\]If elements $a,b$ in a C$^*$-algebra $A\subseteq{{\rm B}(\mathcal{H})}$ satisfy ${D_{\omega}(b)}={D_{\omega}(a)}$ for all pure states $\omega$ on $A$, then there is a projection $p$ in the center $Z$ of the weak\* closure $R$ of $A$ and central elements $u_1,z_1\in Rp$, $u_2,z_2\in R{p^{\perp}}$, with $u_1,u_2$ unitary, such that $bp=u_1a+z_1$ and $b{p^{\perp}}=u_2a^*+z_2$ and $a{p^{\perp}}$ is normal.
Since the condition ${D_{\omega}(b)}={D_{\omega}(a)}$ persists for all weak\* limits of pure states on $A$ and such states are precisely the restrictions of weak\* limits of pure states on $R$ by [@G Theorem 5], the proof immediately reduces to the case $A=R$. Let $Z$ be the center of $R$, $\Delta$ the maximal ideal space of $Z$, for each $t\in \Delta$ let $Rt$ be the closed ideal of $R$ generated by $t$ and set $R(t):=R/(Rt)$. For any $a\in R$ let $a(t)$ denotes the coset of $a$ in $R(t)$. Since each pure state on $R(t)$ can be lifted to a pure state on $R$, we have $D_{\omega}(b(t))=D_{\omega}(a(t))$ for each pure state $\omega$ on $R(t)$ and each $t\in\Delta$. Since $R(t)$ is a primitive C$^*$-algebra by [@Halp], it follows from Theorem \[th1\] that there exist scalars $\alpha(t),\beta(t)$, with $|\alpha(t)|=1$, such that
$$\label{11} b(t)=\alpha(t)a(t)+\beta(t)1$$
or $$\label{12} b(t)=\alpha(t)a(t)^*+\beta(t)1\ \mbox{and}\ a\ \mbox{is normal}.$$ Let $F_1$ be the set of all $t\in\Delta$ for which (\[11\]) holds, $F_2$ the set of all those $t$ for which (\[12\]) holds and $U$ the set of all $t$ such that $a(t)$ is not a scalar. Since for each $x\in R$ the function $t\mapsto\|x(t)\|$ is continuous on $\Delta$ by [@G], it is easy to see that $U$ is open and $F_1$, $F_2$ are closed.
To show that the coefficients $\alpha$ and $\beta$ in (\[11\]) and (\[12\]) are continuous functions of $t$ on $U$, let $t\in U$ be fixed, note that the center of $R(t)$ is $\bc1$ and that $R(t)$ is generated by projections, so there is a projection $p_t\in R(t)$ such that $(1-p_t)a(t)p_t\ne0$. We may lift $1-p_t$ and $p_t$ to positive elements $x,y$ in $R$ with $xy=0$ [@KR 4.6.20]. Then from (\[11\]) $$\label{111}x(s)b(s)y(s)=\alpha(s)x(s)a(s)y(s)\ (\forall s\in\Delta),$$ and $\|(xay)(s)\|\ne0$ for $s$ in a neighborhood of $t$ by continuity. Now let $c\in Z$ be the element whose Gelfand transform is the function $s\mapsto\|x(s)a(s)y(s)\|$ and let $\phi:R\to Z$ be a bounded $Z$-module map such that $\phi(xay)=c$. (Such a map may be obtained simply as the completely bounded $Z$-module extension to $R$ of the map $Z(xay)\to Z$, $z\mapsto z(xay)$, since $Z$ is injective [@BLM].) Since $\phi$ is a $Z$ module map, $\phi$ is just a collection of maps $\phi_s:R(s)\to Z(s)=\bc$, hence from (\[111\]) we obtain $\alpha(s)c(s)=(\phi(xby))(s)$. Since $c(t)=\|x(t)a(t)y(t)\|\ne0$, it follows that $\alpha$ is continuous in a neighborhood of $t$, hence continuous on $U$. Then, denoting by $q_0$ the projection corresponding to a clopen neighborhood $U_0\subseteq U$ of $t$, we have from (\[11\]) that $\beta(t)q_0(t)=e(t)$, where $e=(bq_0-\alpha aq_0)\in Rq_0$, hence $\beta|U_0$ represents a central element of $Rq_0$ and is therefore continuous.
Since $\Delta$ (hence also ${\overline{U}}$) is a Stonean space and $\alpha$ (hence also $\beta$) are bounded continuous functions, they have continuous extensions to ${\overline{U}}$ (see [@KR p. 324]). If $q\in Z$ is the projection that corresponds to ${\overline{U}}$, then $a{q^{\perp}}$ is a scalar in $R{q^{\perp}}$, and it follows easily that $b{q^{\perp}}$ must also be a scalar. So we have only to consider the situation in $Rq$, which means that we may assume that ${\overline{U}}=\Delta$, hence that $\alpha$ and $\beta$ are defined and continuous throughout $\Delta$. The interior $F:=\stackrel{\circ}{F}_1$ of $F_1$ is a clopen subset such that (\[11\]) holds for $t\in F$. Since the complement $F^c\ (={\overline{F_1^c}}$) is contained in $F_2$, (\[12\]) holds if $t\in F^c$. Finally, let $p\in Z$ be the projection that corresponds to $F$, and let $u_1, z_1\in Zp$, $u_2,z_2\in Z{p^{\perp}}$ be elements that corresponds to functions $\alpha|F$, $\beta|F$, $\alpha|F^c$ and $\beta|F^c$ (respectively).
We note that the converse of Corollary \[co11\] also holds, the proof follows easily from the well-known fact [@KR p. 268] that if $\omega$ is a pure state on a C$^*$-algebra $R$ then $\omega(xz)=\omega(x)\omega(a)$ for all $x\in R$ and all $z$ in the center of $R$.
The inequality ${D_{\xi}(b)}\leq{D_{\xi}(a)}$
=============================================
\[le1\]For any two operators $a,b\in{{\rm B}(\mathcal{H})}$ and any state $\omega$ on ${{\rm B}(\mathcal{H})}$ the following estimate holds: $$|D_{\omega}(b)-D_{\omega}(a)|\leq 2\|b-a\|(\|a\|+\|b\|).$$
Since $|\omega(b^*b-a^*a)|\leq\|b^*b-a^*a\|=\|(b^*-a^*)b+a^*(b-a)\|\leq\|b-a\|(\|b\|+\|a\|)$ and $\left||\omega(b)|^2-|\omega(a)|^2\right|=
(|\omega(b)|+|\omega(a)|)\left| |\omega(b)|-|\omega(a)|\right|\leq(\|a\|+\|b\|)\|b-a\|$, we have $$\begin{aligned}
|D_{\omega}(b)-D_{\omega}(a)|=|\omega(b^*b-a^*a)-(|\omega(b)|^2-|\omega(a)|^2)|\\
\leq|\omega(b^*b-a^*a)+\left|(|\omega(b)|^2-|\omega(a)|^2)|\right|\leq2\|b-a\|(\|a\|+\|b\|).\end{aligned}$$
\[le2\]Let $a,b\in{\mathbb M}_2(\bc)$ ($2\times 2$ complex matrices), $0<\varepsilon<1/2$, and let $\alpha_i$ and $\beta_i$ ($i=1,2$) be the eigenvalues of $a$ and $b$ (respectively).
\(i) If ${D_{\xi}(b)}\leq{D_{\xi}(a)}$ for all unit vectors $\xi\in\bc^2$, then $b=\theta a+\tau$ for some scalars $\theta,\tau\in\bc$ with $|\theta|\leq1$.
\(ii) If ${D_{\xi}(b)}\leq{D_{\xi}(a)}+\varepsilon^8$ for all unit vectors $\xi\in\bc^2$, then $$|\beta_2-\beta_1|\leq|\alpha_2-\alpha_1|+2\varepsilon(\|a\|+2\|b\|+1).$$
\(i) Since ${D_{\xi}(a)}={D_{\xi}(a-\lambda1)}$ for all $\lambda\in\bc$, we may assume that one of the eigenvalues of $a$ is $0$, say $a\xi_2=0$ for a unit vector $\xi_2\in\bc^2$. Then from $0\leq{D_{\xi_2}(b)}\leq{D_{\xi_2}(a)}=0$ we see that $\xi_2$ is also an eigenvector for $b$, hence (replacing $b$ by $b-\lambda1$ for a $\lambda\in\bc$) we may assume that $b\xi_2=0$. So, choosing a suitable orthonormal basis $\{\xi_1,\xi_2\}$ of $\bc^2$, we may assume that $a$ and $b$ are of the form $$a=\left[\begin{array}{ll}
\alpha_1&\gamma\\
0&0
\end{array}\right],\ \ \ \ b=\left[\begin{array}{ll}
\beta_1&\delta\\
0&0\end{array}\right].$$ Now we compute for any unit vector $\xi=(\lambda,\mu)\in\bc^2$ (using $|\lambda|^2+|\mu|^2=1$) that $$\begin{aligned}
{D_{\xi}(a)}=\|a\xi\|^2-|{\langle a\xi,\xi\rangle}|^2=|\alpha_1\lambda+\gamma\mu|^2-|(\alpha_1\lambda+\gamma\mu)\overline{\lambda}|^2\\
=|\mu|^2[|\alpha_1|^2|\lambda|^2+|\gamma|^2|\mu|^2+2{{\rm Re}\,{(}}\alpha_1\overline{\gamma}\lambda\overline{\mu})].\end{aligned}$$ Using this and a similar expression for ${D_{\xi}(b)}$, the condition ${D_{\xi}(b)}\leq{D_{\xi}(a)}$ can be written as $$\label{20}(|\alpha_1|^2-|\beta_1|^2)|\lambda|^2+(|\gamma|^2-|\delta|^2)|\mu|^2+
2{{\rm Re}\,{(}}(\alpha_1\overline{\gamma}-\beta_1\overline{\delta})\lambda\overline{\mu})\geq0,$$ which means that the matrix $$M=\left[\begin{array}{ll}
|\alpha_1|^2-|\beta_1|^2&\alpha_1\overline{\gamma}-\beta_1\overline{\delta}\\
\overline{\alpha}_1\gamma-\overline{\beta}_1\delta&|\gamma|^2-|\delta|^2\end{array}\right]$$ is nonnegative. This is equivalent to the conditions $$|\beta_1|\leq|\alpha_1|,\ \ |\delta|\leq|\gamma|\ \ \mbox{and}\ \ \det M\geq0.$$ Since $\det M=-|\alpha_1\delta-\beta_1\gamma|^2$, the condition $\det M\geq0$ means that $\alpha_1\delta=\beta_1\gamma$. If $\alpha_1\ne0$, it follows that $b$ is of the form $$b=\left[\begin{array}{ll}
\beta_1&\frac{\beta_1}{\alpha_1}\gamma\\
0&0\end{array}\right]=\frac{\beta_1}{\alpha_1}a=\theta a, \ \mbox{where}\ \theta:=\frac{\beta_1}{\alpha_1},\ \mbox{hence}\ |\theta|\leq1.$$ If $\alpha_1=0$, then $\beta_1=0$ (since $|\beta_1|\leq|\alpha_1|$), hence again $b=\theta a$, where $\theta=\delta/\gamma$ if $\gamma\ne0 $.
\(ii) As above, replacing $a$ and $b$ by $a-\lambda1$ and $b-\mu1$, where $\lambda$ and $\mu$ are eigenvalues of $a$ and $b$, we may assume that $a$ and $b$ are of the form $$a=\left[\begin{array}{ll}
\alpha_1&\gamma\\
0&0\end{array}\right],\ \ \ \ b=\left[\begin{array}{ll}
\beta_1&\delta_1\\
\delta_2&0\end{array}\right].$$ The norms of the new $a$ and $b$ are at most two times greater than the norm of original ones, which will be taken into account in the final estimate. If $\xi=(1,0)$, then ${D_{\xi}(b)}=|\delta_2|^2$ and ${D_{\xi}(a)}=0$, hence the condition ${D_{\xi}(b)}\leq{D_{\xi}(a)}+\varepsilon^8$ shows that $|\delta_2|\leq\varepsilon^4$. Thus, denoting by $b_0$ the matrix $$b_0=\left[\begin{array}{ll}
\beta_1&\delta_1\\
0&0\end{array}\right],$$ we have that $\|b-b_0\|\leq\varepsilon^4$, hence by Lemma \[le1\] $$|{D_{\xi}(b)}-{D_{\xi}(b_0)}|\leq2\varepsilon^4(\|b_0\|+\|b\|)\leq4\|b\|\varepsilon^4\ \mbox{for all unit vectors}\ \xi\in\bc^2.$$ It follows that $${D_{\xi}(b_0)}\leq{D_{\xi}(a)}+4\|b\|\varepsilon^4+\varepsilon^8\leq{D_{\xi}(a)}+\varepsilon^4(4\|b\|+1).$$ The same calculation that led to (\[20\]) shows now that $$|\mu|^2[(|\alpha_1|^2-|\beta_1|^2)|\lambda|^2+(|\gamma|^2-|\delta_1|^2)|\mu|^2+
2{{\rm Re}\,{(}}(\alpha_1\overline{\gamma}-\beta_1\overline{\delta}_1)\lambda\overline{\mu})]\geq-\varepsilon^4(4\|b\|+1)$$ for all $\lambda,\mu\in\bc$ with $|\lambda|^2+|\mu|^2=1$. We may choose the arguments of $\lambda$ and $\mu$ so that $(\alpha_1\overline{\gamma}-\beta_1\overline{\delta}_1)\lambda\overline{\mu}$ is negative, hence the above inequality implies that $$t[(|\alpha_1|^2-|\beta_1|^2)(1-t)+(|\gamma|^2+|\delta_1|^2)t-2|\alpha_1\overline{\gamma}-\beta_1\overline{\delta}_1|\sqrt{t(1-t)}]
\geq-\varepsilon^4(4\|b\|+1)$$ for all $t\in[0,1]$. Setting $t=\varepsilon^2$, it follows that $$(|\alpha_1|^2-|\beta_1|^2)(1-\varepsilon^2)+(\|a\|^2+\|b\|^2)\varepsilon^2\geq-\varepsilon^2(4\|b\|+1),$$ hence $|\alpha_1|^2-|\beta_1|^2\geq-\varepsilon^2(\|a\|^2+\|b\|^2+4\|b\|+1)$, so $$|\beta_1|\leq|\alpha_1|+\varepsilon(\|a\|+2\|b\|+1).$$ Taking into account that $\alpha_2$ and $\beta_2$ were initially reduced to $0$ (by which the norms of $a$ and $b$ may have increased at most by a factor $2$, this proves (ii).
The approximate point spectrum of an operator $a$ will be denoted by ${\sigma_{\rm ap}(a)}$.
\[de1\] If $a,b\in{{\rm B}(\mathcal{H})}$ are such that ${D_{\xi}(b)}\leq{D_{\xi}(a)}$ for all $\xi\in\h$, then we can define a function $f:{\sigma_{\rm ap}(a)}\to{\sigma_{\rm ap}(b)}$ as follows. Given $\alpha\in{\sigma_{\rm ap}(a)}$, let $(\xi_n)$ be a sequence of unit vectors in $\h$ such that $\lim\|(a-\alpha1)\xi_n\|=0$. Then from the condition ${D_{\xi_n}(b)}\leq{D_{\xi_n}(a)}$ we conclude that $\lim\|(b-\lambda_n1)\xi_n\|=0$, where $\lambda_n={\langle b\xi_n,\xi_n\rangle}$. We will show that the sequence $(\lambda_n)$ converges, so we define $$f(\alpha)=\lim\lambda_n.$$
\[pr2\]The function $f$ is well-defined and Lipschitz: $|f(\beta)-f(\alpha)|\leq|\beta-\alpha|$ for all $\alpha,\beta\in{\sigma_{\rm ap}(a)}$.
Given $\varepsilon>0$, choose unit vectors $\xi,\eta\in\h$ such that $$\|(a-\alpha1)\xi\|<\varepsilon\ \ \ \mbox{and}\ \ \ \|(a-\beta1)\eta\|<\varepsilon.$$ Let $p$ be the projection onto the span of $\{\xi,\eta\}$ and let $c$ be the operator on $p\h$ defined by $c\xi=\alpha\xi$ and $c\eta=\beta\eta$. Then $$\label{31}\|a|_{p\h}-c\|^2\leq\|(a-c)\xi\|^2+\|(a-c)\eta\|^2<2\varepsilon^2.$$ Let $\lambda={\langle b\xi,\xi\rangle}$, $\mu={\langle b\eta,\eta\rangle}$ and $d$ the operator on $p \h$ defined by $d\xi=\lambda\xi$ and $d\eta=\mu\eta$. Then, using the conditions ${D_{\xi}(b)}\leq{D_{\xi}(a)}$ and $D_{\eta}(b)\leq D_{\eta}(a)$, we have $$\begin{aligned}
\label{32}\|b|_{p\h}-d\|^2\leq\|(b-\lambda1)\xi\|^2+\|(b-\mu1)\eta\|^2
\leq\|(a-\alpha1)\xi\|^2+\|(a-\beta1)\eta\|^2<2\varepsilon^2.\end{aligned}$$ Now by Lemma \[le1\] and Remark \[re0\](i) and since $\|d\|\leq\|b\|$, $\|c\|\leq\|a\|$ we infer from (\[31\]) and (\[32\]) that $${D_{\xi}(d)}\leq{D_{\xi}(b)}+4\|d-b|p\h\|\|b\|<{D_{\xi}(b)}+4\varepsilon\sqrt{2}\ {\rm and}\
{D_{\xi}(c)}>{D_{\xi}(a)}-4\varepsilon\sqrt{2},$$ hence (since ${D_{\xi}(b)}\leq{D_{\xi}(a)}$) $${D_{\xi}(d)}\leq{D_{\xi}(c)}+8\varepsilon\sqrt{2}\ \ \mbox{for all}\ \xi\in\h\ \mbox{with}\ \|\xi\|=1.$$ By Lemma \[le2\] (ii) we now conclude that $$\label{33}|\mu-\lambda|\leq|\beta-\alpha|+\kappa\varepsilon,$$ where $\kappa$ is a constant.
If $(\xi_n)$ and $(\eta_n)$ are two sequences of unit vectors in $\h$ such that $\lim\|(a-\alpha1)\xi_n\|=0$ and $\lim\|(a-\beta1)\eta_n\|=0$, we infer from (\[33\]) (since $\varepsilon$ can be taken to tend to $0$ as $n\to\infty$) that $$\label{34}\limsup|\mu_n-\lambda_n|\leq|\beta-\alpha|.$$ Further, if $\beta=\alpha$ and we put in (\[33\]) $\lambda_n={\langle b\xi_n,\xi_n\rangle}$ instead of $\lambda$ and $\lambda_m={\langle b\xi_m,\xi_m\rangle}$ instead of $\mu$, we conclude that $(\lambda_n)$ is a Cauchy sequence, hence it converges to a point $\lambda\in\bc$. From $\lim\|(b-\lambda_n1)\xi_n\|=0$ it follows now that $\lim\|(b-\lambda1)\xi_n\|=0$, hence $\lambda\in{\sigma_{\rm ap}(b)}$. Similarly the sequence $(\mu_n)=({\langle b\eta_n,\eta_n\rangle})$ converges to some $\mu$ and (\[34\]) implies that $$|\mu-\lambda|\leq|\beta-\alpha|.$$ This shows that $f$ is a well-defined Lipschitz function.
\[th3\]Let $a,b\in{{\rm B}(\mathcal{H})}$. If $a$ is normal, then there exists a constant $\kappa$ such that ${D_{\xi}(b)}\leq\kappa{D_{\xi}(a)}$ for all $\xi\in\h$ if and only if $b=f(a)$ for a Lipschitz function $f$ on $\sigma(a)$. In this case ${D_{\omega}(b)}\leq\kappa{D_{\omega}(a)}$ for all states $\omega$.
Assume that ${D_{\xi}(b)}\leq{D_{\xi}(a)}$ for all $\xi\in\h$. We may assume that $a$ is not a scalar (otherwise the proof is trivial). First consider the case when $a$ can be represented by a diagonal matrix ${\rm diag}\,
(\alpha_j)$ in some orthonormal basis $(\xi_j)$ of $\h$. If $f:\sigma(a)\to{\sigma_{\rm ap}(b)}$ is defined as in Definition \[de1\], then $b\xi_j=f(\alpha_j)\xi_j$ for all $j$, hence $b=f(a)$.
For a general normal $a$, first suppose that $\h$ is separable. Then by Voiculescu’s version of the Weyl-von Neumann-Bergh theorem [@Vo1], given $\varepsilon>0$, there exists a diagonal normal operator $c={\rm diag}\,(\gamma_j)$ such that $\|a-c\|_2<\varepsilon$, where $\|\cdot\|_2$ denotes the Hilbert-Schmidt norm. Let $(\xi_j)$ be an orthonormal basis of $\h$ consisting of eigenvectors of $c$, so that $c\xi_j=\gamma_j\xi_j$. Since ${D_{\xi_j}(b)}\leq
{D_{\xi_j}(a)}$, by Remark \[re0\](i) there exist scalars $\beta_j\in\bc$ such that $\|(b-\beta_j1)\xi_j\|
\leq\|(a-\gamma_j1)\xi_j\|=\|(a-c)\xi_j\|$, hence $$\sum_j\|(b-\beta_j1)\xi_j\|^2\leq\sum_j\|(a-c)\xi_j\|^2<\varepsilon^2.$$ In particular $\|b-d\|<\varepsilon$, where $d$ is the diagonal operator defined by $d\xi_j=\beta_j\xi_j$. Since $d$ and $c$ commute, it follows that $$\|bc-cb\|=\|(b-d)c-c(b-d)\|<2\varepsilon\|c\|\leq2\varepsilon(\|a\|+\varepsilon)\leq4\varepsilon\|a\|\ (\mbox{if}\
\varepsilon\leq\|a\|),$$ hence also $$\|ba-ab\|=\|(bc-cb)+b(a-c)-(a-c)b\|\leq4\varepsilon(\|a\|+\|b\|).$$ Since this holds for all $\varepsilon>0$, it follows that $a$ and $b$ commute. If $a$ has a cyclic vector this already implies that $b$ is in $(a)^{\prime\prime}$ hence a measurable function of $a$, but in general we need an additional argument to prove this. Let $f:\sigma(a)\to{\sigma_{\rm ap}(b)}$ be defined as in Definition \[de1\]. (Note that $\sigma(a)={\sigma_{\rm ap}(a)}$ since $a$ is normal.) Let $e(\cdot)$ be the projection valued spectral measure of $a$, $\xi\in\h$ any separating vector for the von Neumann algebra $(a)^{\prime\prime}$ generated by $a$ and $\varepsilon>0$. If $\alpha$ is any point in $\sigma(a)$, $U$ is any Borel subset of $\sigma(a)$ containing $\alpha$ and $\xi_U:=\|e(U)\xi\|^{-1}e(U)\xi$, then $\|(a-\alpha1)\xi_U\|$ converges to $0$ as the diameter of $U$ shrinks to $0$. For each $U$ let $\beta_U={\langle b\xi_U,\xi_U\rangle}$ so that $\|(b-\beta_U1)\xi_U\|\leq
\|(a-\alpha1)\xi_U\|$; then $f(\alpha)=\lim_{U\to\{\alpha\}}\beta_U$ by the definition of $f$. Thus, since by Proposition \[pr2\] $f$ is a Lipschitz function, for each $\alpha\in\sigma(a)$ there is an open neighborhood $U_{\alpha}$ with the diameter at most $\varepsilon$ such that $|f(\alpha)-\beta_U|<\varepsilon$ for all Borel subsets $U\subseteq U_{\alpha}$ and $|f(\alpha_2)-f(\alpha_1)|<\varepsilon$ if $\alpha_1,\alpha_2\in U_{\alpha}$. By compactness we can cover $\sigma(a)$ with finitely many such neighborhoods $U_{\alpha_i}$ and this covering then determines a partition of $\sigma(a)$ into finitely many disjoint Borel sets $\Delta_j$ (say $j=1,\ldots,n$) such that each $\Delta_j$ is contained in some $U_{\alpha_{i(j)}}$. Let $e_j=e(\Delta_j)$. Now we can estimate, denoting $\beta_j=\beta_{\Delta_j}$, $$\begin{aligned}
\|(b-f(a))e_j\xi\|\leq&\|(b-\beta_{j}1)e_j\xi\|+|(\beta_{j}-f(\alpha_{i(j)})|
\|e_j\xi\|+\|(f(\alpha_{i(j)})1-f(a))e_j\xi\|\\
\leq&\|(a-\alpha_j1)e_j\xi\|+|(\beta_{j}-f(\alpha_{i(j)})|
\|e_j\xi\|+\|(f(\alpha_{i(j)})1-f(a))e_j\xi\|\\
\leq&3\varepsilon\|e_j\xi\|.\end{aligned}$$ (Here we have used the spectral theorem to estimate the term $\|(f(\alpha_{i(j)})1-f(a))e_j\xi\|$ from above by $\sup_{\alpha\in\Delta_j}|f(\alpha_{i(j)}-f(\alpha)|\|e_j\xi\|\leq\varepsilon\|e_j\xi\|$.) Since $b$ commutes with $a$, hence also with all spectral projections of $a$, it follows that $$\begin{aligned}
\|(b-f(a))\xi\|^2=&\|{\sum_{j=1}^{n}}e_j(b-f(a))e_j\xi\|^2
={\sum_{j=1}^{n}}\|e_j(b-f(a))e_j\xi\|^2\\
\leq&
9\varepsilon^2{\sum_{j=1}^{n}}\|e_j\xi\|^2=9\varepsilon^2\|\xi\|^2.\end{aligned}$$ Thus $\|(b-f(a)1)\xi\|\leq3\varepsilon\|\xi\|$ and, since this holds for all $\varepsilon>0$ and separating vectors are dense in $\h$, we conclude that $b=f(a)$.
If $\h$ is not necessarily separable, $\h$ can be decomposed into an orthogonal sum of separable subspaces $\h_k$ that reduce both $a$ and $b$ and are such that $\sigma(a|\h_k)=\sigma(a)$. For each $k$ there exists a Lipschitz function $f_k$ such that $b|\h_k=f(a|\h_k)$. Since for any two $k,j$ the space $\h_k\oplus\h_j$ is also separable, there also exists a function $f$ such that $b|(\h_j\oplus\h_k)=f(a|(\h_j\oplus\h_k))$ and it follows easily that $f_k=f=f_j$. Thus $b=f(a)$.
Conversely, if $b=f(a)$ for a function $f$ such that $$|f(\alpha_2)-f(\alpha_1)|\leq \kappa|\alpha_2-\alpha_1|$$ for all $\alpha_1,\alpha_2\in\sigma(a)$ and some constant $\kappa$, then for a fixed unit vector $\xi\in\h$ denote by $\mu$ the probability measure on Borel subsets of $\sigma(a)$ defined by $\mu(\cdot)={\langle e(\cdot)\xi,\xi\rangle}$. Since ${D_{\xi}(a)}$ is just the square of the distance of $a\xi$ to $\bc\xi$ and similarly for ${D_{\xi}(b)}$, the estimate $$\begin{aligned}
\|(f(a)-f(\alpha))1\xi\|^2=&\int_{\sigma(a)}|f(\lambda)-f(\alpha)|^2\, d\mu(\lambda)\\
\leq& \int_{\sigma(a)}\kappa|\lambda-\alpha|^2\, d\mu(\lambda)=\kappa\|(a-\alpha1)\xi\|^2\end{aligned}$$ implies that ${D_{\xi}(b)}\leq\kappa{D_{\xi}(a)}$.
Finally, since any state $\omega$ is in the weak\*-closure of the set of all convex combinations of vector states and each such combination can be represented as a vector state on ${\rm B}(\h^n)$ for some $n\in\bn$, the argument of the previous paragraph (applied to $a^{(n)}$ and $b^{(n)}=f(a^{(n)})$ implies that ${D_{\omega}(b)}\leq{D_{\omega}(a)}$.
A variant of the above Theorem \[th3\] was proved in [@JW] and generalized to C$^*$-algebras in [@BMS], but both under the much stronger hypothesis that $\|[b,x]\|\leq\kappa\|[a,x]\|$ for all elements $x$, where $[a,x]$ denotes the commutator $ax-xa$. (See Lemma \[le41\] below for the explanation of the connection between the two conditions.)
The following Corollary was proved in [@BMS 5.2] for prime C$^*$-algebras, but under a much stronger assumption about the connection between $a$ and $b$ instead of the inequality ${D_{\omega}(b)}\leq{D_{\omega}(a)}$ for pure states $\omega$.
\[co31\]Let $A$ be a unital C$^*$-algebra, $a,b\in A$, $a$ normal. If ${D_{\omega}(b)}\leq{D_{\omega}(a)}$ for all states $\omega$ on $A$, then $b=f(a)$ for a function $f$ on $\sigma(a)$ such that $|f(\mu)-f(\lambda)|\leq|\mu-\lambda|$ for all $\lambda,\mu\in\sigma(a)$. If $A$ is prime, it suffices to assume the condition for pure states only.
The first statement follows immediately from Theorem \[th3\] since we may assume that $A\subseteq{{\rm B}(\mathcal{H})}$ for a Hilbert space $\h$ and each vector state on ${{\rm B}(\mathcal{H})}$ restricts to a state on $A$. For the second statement, we note that the C$^*$-algebra generated by $a$ and $b$ is contained in a separable prime C$^*$-subalgebra $A_0$ of $A$ by [@EZ 3.1] (an elementary proof of this is in [@M 3.2]), and $A_0$ is primitive by [@Ped p. 102], hence we may assume that $A_0$ is an irreducible C$^*$-subalgebra of ${{\rm B}(\mathcal{H})}$. But then each vector state on ${{\rm B}(\mathcal{H})}$ restricts to a pure state on $A_0$, and each pure state on $A_0$ extends to a pure state on $A$.
\[co32\]Let $a,b\in{{\rm B}(\mathcal{H})}$ satisfy ${D_{\xi}(b)}\leq{D_{\xi}(a)}$ for all $\xi\in\h$. If $a$ is essentially normal, then this implies that $\dot{b}=f(\dot{a})$ for a Lipschitz function $f$ on the essential spectrum of $a$, where $\dot{a}$ denotes the coset of $a$ in the Calkin algebra.
Any state $\omega$ on the Calkin algebra can be regarded as a vector state on ${{\rm B}(\mathcal{H})}$ annihilating the compact operators. By Glimm’s theorem (see [@KR 10.5.55] or [@G]) such a state $\omega$ is a weak\* limit of vector states, hence ${D_{\omega}(b)}\leq{D_{\omega}(a)}$. The conclusion follows now from Corollary \[co31\].
Let $A\subseteq{{\rm B}(\mathcal{H})}$ be a C$^*$-algebra $a,b\in R$ and $a$ normal. Denote by $R$ the weak\* closure of $A$ and by $Z$ the center of $R$. Then the inequality ${D_{\omega}(b)}\leq{D_{\omega}(a)}$ holds for all pure states $\omega$ on $A$ if and only if $b$ is in the norm closure of the set $S$ of all elements of the form $\sum_jp_jf_j(a)$ (finite sum), where $p_j$ are orthogonal projections in $Z$ with the sum $\sum_jp_j=1$ and $f_j$ are functions on $\sigma(a)$ such that $|f_j(\mu)-f_j(\lambda)|\leq
|\mu-\lambda|$ for all $\lambda,\mu\in\sigma(a)$.
Note that $g(a(t))=g(a)(t)$ for each continuous function $g$ on $\sigma(a)$. We will use the notation from the proof of Corollary \[co11\]. Similarly as in that proof, the condition that ${D_{\omega}(b)}\leq{D_{\omega}(a)}$ for all pure states $\omega$ on $A$ implies the same condition for all pure states on $R(t)$ for all $t\in\Delta$ and it follows then from Corollary \[co31\] that for each $t$ there exists a Lipschitz function $f_t$ on $\sigma(a(t))$ with the Lipschitz constant $1$ such that $b(t)=f_t(a(t))$. By Kirzbraun’s theorem each $f_t$ can be extended to a Lipschitz function on $\sigma(a)$, denoted again by $f_t$, with the same Lipschitz constant $1$. Given $\varepsilon>0$, since $\Delta$ is extremely disconnected and for each $x\in R$ the function $t\mapsto\|x(t)\|$ is continuous on $\Delta$ by [@G], each $t\in\Delta$ has a clopen neighborhood $U_t$ such that $\|f_t(a)(s)-b(s)\|\leq\varepsilon$ for all $s\in U_t$. Let $(U_j)$ be a finite covering of $\Delta$ by such neighborhoods $U_j:=U_{t_j}$ and for each $j$ let $p_j$ be the central projection in $R$ that corresponds to the clopen set $U_j$, and set $f_j:=f_{t_j}$. Then $$\|b-\sum_jf_j(a)\|\leq\varepsilon.$$ Since this can be done for all $\varepsilon>0$, $b$ is in the closure of the set $S$ as stated in the theorem.
Conversely, suppose that for each $\varepsilon>0$ there exists an element $c\in R$ of the form $c=\sum_jp_jf_j(a)$, where $p_j\in Z$ are projections with the sum $1$ and $f_j$ are Lipschitz functions with the Lipschitz constant $1$. Then for each pure state $\omega$ on $R$ and $x\in R$, $z\in Z$ the equality $\omega(zx)=\omega(z)\omega(x)$ holds [@KR 4.3.14]). In particular $\omega|Z$ is multiplicative, hence $\omega(p_{j_0})=1$ for one index $j_0$ and $\omega(p_j)=0$ if $j\ne j_0$. It follows now by a straightforward computation that ${D_{\omega}(c)}={D_{\omega}(f_{j_0}(a))}$, which is at most ${D_{\omega}(a)}$ by the same computation as in the last part of the proof of Theorem \[th3\]. Now, since $\|b-c\|<\varepsilon$, it follows from Lemma \[le1\] that ${D_{\omega}(b)}\leq{D_{\omega}(a)}$.
Is a derivation determined by the norms of its values?
======================================================
Given an operator $a\in{{\rm B}(\mathcal{H})}$, we will denote by $d_a$ the derivation on ${{\rm B}(\mathcal{H})}$ defined by $$d_a(x)=ax-xa.$$ For any vectors $\xi,\eta\in\h$ we denote by ${\xi\otimes\eta^*}$ the rank one operator on $\h$ defined by $({\xi\otimes\eta^*})(\zeta)={\langle \zeta,\eta\rangle}\xi.$ The following lemma enables us to interpret the results of the previous section in terms of derivations.
\[le41\]For each $\xi\in\h$ and $a\in{{\rm B}(\mathcal{H})}$ the equality $\|d_a({\xi\otimes\xi^*})\|^2={D_{\xi}(a)}$ holds.
Denote $x={\xi\otimes\xi^*}$. The square of the norm of $d_a(x)=a\xi\otimes\xi^*-\xi\otimes(a^*\xi)^*$ is equal to the spectral radius of the operator $T:=d_a(x)^*d_a(x)$, which is the largest eigenvalue of the restriction of $T$ to the span $\h_0$ of $\xi$ and $a^*\xi$. If $\xi$ and $a^*\xi$ are linearly independent, then the matrix of $T|\h_0$ in the basis $\{\xi,a^*\xi\}$ can easily be computed to be $$\left[\begin{array}{cc}
{D_{\xi}(a)}&{\langle \xi,a\xi\rangle}(\|a\xi\|^2-\|a^*\xi\|^2)\\
0&{D_{\xi}(a)}\end{array}\right].$$ Thus $\|d_a(x)\|^2={D_{\xi}(a)}$. By continuity (considering perturbations of $a$) we see that this equality holds even if $\xi$ and $a^*\xi$ are linearly dependent .
\[th42\]If $a,b\in{{\rm B}(\mathcal{H})}$ are such that $$\label{400}\|[b,x]\|=\|a,x]\|\ \ \mbox{for all}\ x\in{{\rm B}(\mathcal{H})},$$ then either $b=\sigma a+\lambda1$ for some scalars $\sigma,\lambda\in\bc$ with $|\sigma|=1$ or there exist a unitary $u$ and scalars $\alpha,\beta,\lambda,\mu$ in $\bc$ with $|\beta|=|\alpha|$ such that $a=\alpha u^*+\lambda1$ and $b=\beta u+\mu1$.
A variant of this theorem was proved in [@BMS 5.3, 5.4] in general C$^*$-algebras, but under the additional assumption that $a$ and $b$ are normal. The methods in [@BMS] are different from those we will use below. The author is not able to deduce Theorem \[th42\] as a direct consequence of the previous results; for a proof we will need two additional lemmas. We denote by $a^{(n)}$ the direct sum of $n$ copies of an operator $a\in{{\rm B}(\mathcal{H})}$, thus $a^{(n)}$ acts on $\h^n$. We will also use the usual notation $[x,y]:=xy-yx$, so that $d_a(x)=[a,x]$.
\[re42\]We will need the following, perhaps well-known, general fact: for any bounded linear operators $S,T:X\to Y$ between Banach spaces the inequality $$\label{200}\|Tx\|\leq\|Sx\|\ \ (x\in X)$$ implies $\|T^{\sharp\sharp}v\|
\leq\|S^{\sharp\sharp}v\|$ ($v\in X^{\sharp\sharp}$). This follows from [@JW 1.1, 1.3], but here is a slightly more direct proof. The inequality (\[200\]) simply means that there is a contraction $Q$ from the range of $S$ into the range of $T$ such that $T=QS$. But then $T^{\sharp\sharp}=Q^{\sharp\sharp}S^{\sharp\sharp}$, which clearly implies the desired conclusion.
The content of the following lemma was observed already by Kissin and Shulman in the proof of [@KS 3.3].
\[le42\][@KS] Let $a,b\in{{\rm B}(\mathcal{H})}$ and suppose that $$\label{41}\|[b,x]\|\leq\|[a,x]\|$$ for all $x\in{{\rm K}(\mathcal{H})}$. If $a$ is normal, then $\|[b^{(n)},x]\|\leq\|[a^{(n)},x]\|$ for all $x\in{{\rm M}_n({{\rm B}(\mathcal{H})})}$ ($n\times n$ matrices with the entries in ${{\rm B}(\mathcal{H})}$) and all $n\in\bn$.
Since $d_a={(d_a|{{\rm K}(\mathcal{H})})^{\sharp\sharp}}$ (the second adjoint in the Banach space sense), it follows from Remark \[re42\] that (\[41\]) holds for all $x\in{{\rm B}(\mathcal{H})}$.
Suppose now that $a$ is normal and note that ${(a)^{\prime}}$ is a C$^*$-algebra by the Fuglede-Putnam theorem. Since (\[41\]) holds for all $x\in{{\rm B}(\mathcal{H})}$, $b\in(a)^{\prime\prime}$. Further, by (\[41\]) the map $[a,x]\mapsto[b,x]$ is a contraction from $d_a({{\rm B}(\mathcal{H})})$ to $d_b({{\rm B}(\mathcal{H})})$. Clearly this map is a homomorphism of ${(a)^{\prime}}$-bimodules, hence by [@S 2.1, 2.2, 2.3] it is a complete contraction, which is equivalent to the conclusion of the lemma.
\[re400\] We will use below the following well-known fact. Given $c_j,e_j\in{{\rm B}(\mathcal{H})}$, [*an identity of the form ${\sum_{j=1}^{n}}c_jxe_j=0$, if it holds for all $x\in{{\rm B}(\mathcal{H})}$, implies that all $c_j$ must be $0$ if the $e_j$ are linearly independent. (See e. g. [@AM Theorem 5.1.7]).*]{}
We refer to [@BLM] or [@Pa] for the definition of the injective envelope of an operator space used in the following lemma.
\[le43\]Let ${\mathcal{R}}=d_a({{\rm B}(\mathcal{H})})$ and let ${\mathcal{S}}$ be the operator system $${\mathcal{S}}=\left\{\left[\begin{array}{ll}
\lambda&y\\
z^*&\mu\end{array}\right]: \ \lambda,\mu\in\bc\; \ y,z\in{\mathcal{R}}\right\}.$$ If $a$ does not satisfy any quadratic equation over $\bc$ then the C$^*$-algebra $C^*({\mathcal{S}})$ generated by ${\mathcal{S}}$ is irreducible and the injective envelope $I({\mathcal{S}})$ of ${\mathcal{S}}$ is ${{\rm M}_2({{\rm B}(\mathcal{H})})}$.
Since ${\mathcal{S}}$ contains the diagonal $2\times 2$ matrices with scalar entries, each element of ${S^{\prime}}$ (the commutant of ${\mathcal{S}}$) is a block diagonal matrix, that is, of the form $c\oplus e$, where $c,e\in{{\rm B}(\mathcal{H})}$. To prove the irreducibility of $C^*({\mathcal{S}})$ means to prove that each selfadjoint such element $c\oplus e$ is a scalar multiple of $1$. Since $c\oplus e$ commutes with elements of ${\mathcal{S}}$, we have that $cy=ye$ for all $y\in{\mathcal{R}}$. Setting $y=ax-xa$ in the last identity we obtain $$\label{401}cax-cxa-axe+xae=0\ \ \mbox{for all}\ x\in{{\rm B}(\mathcal{H})}.$$ Since in (\[401\]) the left coefficients $ca,-c,-a$ and $1$ are not all $0$, it follows that $1,a,e,ae$ are linearly dependent. Thus, if $1,a$ and $e$ are linearly independent, then $ae=\alpha1+\beta a+\gamma e$ for some scalars $\alpha,\beta,\gamma\in\bc$. Using this, we may rearrange (\[401\]) into $$\label{402}(ca+\alpha1)x+(\beta1-c)xa+(\gamma1-a)xe=0.$$ If $1,a$ and $e$ were linearly independent, then (\[402\]) would imply that $a=\gamma1$, but this would be in contradiction with the assumption about $a$. Hence $$\label{403}e=\delta1+\eta a\ \ \mbox{for some}\ \delta,\eta\in\bc.$$ Now (\[402\]) may be rewritten as $$[(\alpha+\gamma\delta)1+(c-\delta1)a]x+[(\beta+\gamma\eta)1-c-\eta a]xa=0,$$ which implies (since $1$ and $a$ are linearly independent) that $$(\alpha+\gamma\delta)1+(c-\delta1)a=0\ \ \mbox{and}\ \ (\beta+\gamma\eta)1-c-\eta a=0.$$ From these two equalities it follows (by putting into the first equality the expression for $c$ obtained from the second equality) that $a$ satisfies the quadratic equation $$\eta a^2+(\delta-\beta-\gamma\eta)a-(\alpha+\gamma\delta)1=0.$$ From the assumption about $a$ it follows now in particular that $\eta=0$, hence we see from (\[403\]) that $e=\delta1$. But then the identity $cy=ye$ ($y\in{\mathcal{R}}$) implies that $c=\delta1$, hence $c\oplus e$ is a scalar multiple of the identity.
It remains to consider the case when $1, a$ and $e$ are linearly dependent, say $e=\alpha1+\beta a$ ($\alpha,\beta\in\bc$). Then (\[401\]) can be rewritten as $$\label{404}(c-\alpha1)ax+(\alpha1-\beta a-c)xa+\beta xa^2=0.$$ Since $1,a$ and $a^2$ are linearly independent by assumption, we infer from (\[404\]) that $\beta=0$. But then $e=\alpha1$ and we conclude as above that $c\oplus e$ is also a scalar multiple of $1$. This proves the irreducibility of $C^*({\mathcal{S}})$.
Since ${\mathcal{S}}$ contains nonzero compact operators, the identity map on ${\mathcal{S}}$ has a unique completely positive extension to $C^*({\mathcal{S}})$ by the Arveson boundary theorem [@Ar1], which implies that $C^*({\mathcal{S}})\subseteq I({\mathcal{S}})$. (Otherwise a projection ${{\rm B}(\mathcal{H})}\to I({\mathcal{S}})$ restricted to $C^*({\mathcal{S}})$ would be a completely positive extension of $id_{{\mathcal{S}}}$, different from $id_{C^*({\mathcal{S}})}$.) But since $C^*({\mathcal{S}})$ is irreducible and contains nonzero compact operators, it follows that $C^*({\mathcal{S}})\supseteq{{\rm M}_2({{\rm K}(\mathcal{H})})}$, hence $I({\mathcal{S}})$ must contain the injective envelope $I({{\rm M}_2({{\rm K}(\mathcal{H})})})$, which is known to be ${{\rm M}_2({{\rm B}(\mathcal{H})})}$ [@BLM].
If $a$ (or $b$) is a scalar multiple of $1$ the proof is easy, so we assume from now on that this is not the case. If $a$ satisfies a quadratic equation of the form $$a^2+\beta a+\gamma1=0\ \ (\beta,\gamma\in\bc),$$ then each element of $(a)^{\prime\prime}$ is a polynomial in $a$ (this holds for any algebraic operator $a$ by [@Tu]), hence in particular $b$ is a linear polynomial in $a$, say $b=\sigma a+\lambda1$. Then the condition (\[400\]) obviously implies that $|\sigma|=1$. Hence we may assume that $a$ does not satisfy any quadratic equation over $\bc$. Further, if $b$ is not of the form $\sigma a+\lambda1$ (which we assume from now on), then by Theorem \[th1\] and Lemma \[le41\] $a$ is normal and (replacing $b$ by $\alpha b+\beta$ for suitable $\alpha,\beta\in\bc$) we may assume without loss of generality that $b=a^*$.
Denote by ${\mathcal{R}}_a$ and ${\mathcal{R}}_b$ the ranges of the derivations $d_a$ and $d_b$ and by ${\mathcal{S}}_a$ and ${\mathcal{S}}_b$ the corresponding operator systems (as in Lemma \[le43\]). Since $a$ is normal, by Lemma \[le42\] the map $$\phi:{\mathcal{R}}_a\to{\mathcal{R}}_b,\ \ \phi([a,x]):=[b,x]\ (x\in{{\rm B}(\mathcal{H})})$$ is completely contractive and the same holds for its inverse, hence $\phi$ is completely isometric and consequently the map $$\Phi:{\mathcal{S}}_a\to{\mathcal{S}}_b,\ \ \Phi\left(\left[\begin{array}{cc}
\alpha&y\\
z^*&\beta\end{array}\right]\right):=\left[\begin{array}{cc}
\alpha&\phi(y)\\
\phi(z)^*&\beta\end{array}\right]$$ is completely positive with completely positive inverse, hence also completely isometric (see [@Pa]). But then $\Phi$ extends to a complete isometry $\psi$ between the injective envelopes $I({\mathcal{S}}_a)$ and $I({\mathcal{S}}_b)$ (since both $\Phi$ and $\Phi^{-1}$ extend to complete contractions which must be each other’s inverse by rigidity). Since $a$ (and $b=a^*$) does not satisfy any quadratic equation over $\bc$, these injective envelopes are both ${{\rm M}_2({{\rm B}(\mathcal{H})})}$ by Lemma \[le43\], hence $\psi$ is a unital surjective complete isometry of ${{\rm M}_2({{\rm B}(\mathcal{H})})}={\rm B}(\h^2)$, thus by [@BLM 4.5.13] or [@KR Ex. 7.6.18] (and since all automorphisms of ${\rm B}(\h^2)$ are inner) $\psi$ is necessarily of the form $$\psi(y)=w^*yw\ \ (y\in{\rm B}(\h^2)),$$ where $w\in{\rm B}(\h^2)$ is unitary. Since by definition $\psi$ fixes the projections of $\h^2$ on the two summands, $w$ must commute with these two projections (by the multiplicative domain argument, see [@Pa p. 38]), consequently $w$ is of the form $w=u\oplus v$ for unitaries $u,v\in{{\rm B}(\mathcal{H})}$. It follows now from the definition of $\psi$ that $\phi$ is of the form $$\phi(y)=uyv\ \ (y\in{\mathcal{R}}_a),$$ that is $\phi([a,x])=u[a,x]v$. Hence $u[a,x]v=[b,x]$ for all $x\in{{\rm B}(\mathcal{H})}$, which can be rewritten as $$\label{405}uaxv-uxav-bx+xb=0\ \ (x\in{{\rm B}(\mathcal{H})}).$$ Thus by Remark \[re400\] we see from (\[405\]) that $v,av,1,$ and $b$ are linearly dependent. Hence, if $1$, $v$ and $b$ are linearly independent, then $av=\alpha1+\beta b+\gamma v$, where $\alpha,\beta\,\gamma\in\bc$, and (\[405\]) can be rewritten as $$(ua-\gamma u)xv-(\alpha u+b)x+(1-\beta u)xb=0.$$ But by Remark \[re400\] this implies in particular that $ua-\gamma u=0$, hence $a=\gamma 1$, a possibility which we have excluded in the first paragraph of this proof. So we may assume that $1,v$ and $b$ are linearly dependent. Then, since $b=a^*$ is not a scalar, $v$ can not to be a scalar. Hence $b=\alpha1+\beta v$ for suitable $\alpha,\beta\in\bc$. Since $v$ is unitary and $a=b^*$, this concludes the proof.
To extend Theorem \[th42\] to C$^*$-algebras we need a lemma.
\[le508\]Let $A\subseteq{{\rm B}(\mathcal{H})}$ be a C$^*$-algebra, $J$ a closed ideal in $A$, and let $a,b\in A$ satisfy $\|[b,x]\|\leq\|[a,x]\|$ for all $x\in A$. Then the same inequality holds for all $x\in{\overline{A}}$ and also for all cosets $\dot{x}\in A/J$.
The statement about the quotinet was observed already in [@BMS Proof of 5.4] and follows from the existence of a quasicentral approximate unit $(e_k)$ in $J$ [@Ar]. Namely, the conditions $\|[a,e_k]\|,\ \|[b,e_k]\to 0$ (from the definition of the quasicentral approximate unit) and the well-known propery that $\|\dot{y}\|=\lim_k\|y(1-e_k)\|$ ($y\in A$) imply that $$\|[\dot{b},\dot{x}]\|=\lim_k\|[b,x](1-e_k)\|=\lim_k\|[b,x(1-e_k)]\leq\lim_k\|[a,x(1-e_k)]\|=\|[\dot{a},\dot{x}]\|.$$
Let ${A^{\sharp\sharp}}$ be the universal von Neumann envelope of $A$ (= bidual of $A$) and regard $A$ as a subalgebra in ${A^{\sharp\sharp}}$ in the usual way. Since ${d_a^{\sharp\sharp}}$ is just the derivation induced by $a$ on ${A^{\sharp\sharp}}$, it follows from Remark \[re42\] that the condition $\|[b,x]\|\leq\|[a,x]\|$ holds for all $x\in{A^{\sharp\sharp}}$. Since ${\overline{A}}$ is a quotient of ${A^{\sharp\sharp}}$, it follows from the previous paragraph (applied to ${A^{\sharp\sharp}}$ instead of $A$) that the condition holds also in ${\overline{A}}$.
If $A$ is a C$^*$-algebra and $a,b\in A$ are such that $\|[b,x]\|=\|a,x]\|$ for all $x\in A,$ then there exist a projection $p$ in the center $Z$ of ${\overline{A}}$ and elements $s,d\in Zp$ with $s$ unitary, and $u,v,c,g,h\in Z{p^{\perp}}$ with $u,v$ unitary, such that $bp=sa+d$ and $a{p^{\perp}}=cu^*+g$, $b{p^{\perp}}=vcu+h$.
If $A$ is primitive the corollary follows immediately from Theorem \[th42\] and Lemma \[le508\] since ${\overline{A}}={{\rm B}(\mathcal{H})}$ if $A$ is irreducibly represented on $\h$. In general, Lemma \[le508\] reduces the proof to von Neumann algebras, where the arguments are similar as in the proof of Corollary \[co11\], so we will omit the details.
If $\|[b,x]\|\leq\|[a,x]\|$ for all $x\in A$ then ${D_{\omega}(b)}\leq{D_{\omega}(a)}$ for all pure states $\omega$ on $A$.
If $\pi:A\to{{\rm B}(\mathcal{H}_{\pi})}$ is the irreducible representation obtained from $\omega$ by the GNS construction, then ${\overline{\pi(A)}}={{\rm B}(\mathcal{H}_{\pi})}$, hence the corollary follows from Lemmas \[le508\] and \[le41\].
An inequality between norms of commutators
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In this section we study the inequality $$\label{50}\|[b,x]\|\leq\kappa\|[a,x]\|\ \ (\forall x\in{{\rm B}(\mathcal{H})}),$$ where $a,b\in{{\rm B}(\mathcal{H})}$ are fixed and $\kappa$ is a constant. For a normal $a$ it is proved in [@JW] that (\[50\]) holds (for some $\kappa$) if and only if
$$\label{51}d_b({{\rm B}(\mathcal{H})})\subseteq d_a({{\rm B}(\mathcal{H})}).$$
That for normal $a$ (\[50\]) implies (\[51\]) can be easily proved as follows. We have seen in the proof of Lemma \[le42\] that for normal $a$ the condition (\[50\]) is equivalent to the fact that the map $$d_a(x)\mapsto d_b(x)\ \ (x\in{{\rm B}(\mathcal{H})})$$ is a completely bounded homomorphism of ${(a)^{\prime}}$-bimodules $d_a({{\rm B}(\mathcal{H})})\to d_b({{\rm B}(\mathcal{H})})$. Then this map can be extended to a completely bounded ${(a)^{\prime}}$-bimodule endomorphism $\phi$ of ${{\rm B}(\mathcal{H})}$ by the Wittstock theorem (see [@BLM 3.6.2]), hence we have $$d_b(x)=\phi(d_a(x))=d_a(\phi(x))\ \ (x\in{{\rm B}(\mathcal{H})}).$$ This argument is perhaps easier than the one in [@JW], but the argument from [@JW] can be adapted to von Neumann algebras.
\[pr50\]Let $R\subseteq{{\rm B}(\mathcal{H})}$ be a von Neumann algebra, $a,b\in R$ and denote $(a)^c:=R\cap{(a)^{\prime}}$. If $\|[b,x]\leq\|[a,x]\|$ for all $x\in R$ and $a$ is normal, then there exists a normal bounded $(a)^c$-bimodule map $\psi$ on $R$ with $\|\psi\|\leq4\|b\|$ such that $d_b|R=\psi(d_a|R)=
(d_a|R)\psi$. Hence in particular $d_b(R)\subseteq d_a(R)$.
Since $(a)^c$ is an abelian von Neumann algebra there exists a projection $E_0$ from ${{\rm B}(\mathcal{H})}$ onto $(a)^c$ which is an $(a)^c$-bimodule map [@KR 8.3.12 and 8.7.24]. Let $E=E_0|R$ and ${E^{\perp}}=1-E$. Since ${E^{\perp}}(R)$ (the kernel of $E$) is a complementary subspace to $(a)^c$, the restriction $q_0:=q|{E^{\perp}}(R)$ of the quotient map $q:R\to R/(a)^c$ is an isomorphism of $(a)^c$-bimodules. Denote by $\tilde{d}_a:R/(a)^c\to{\overline{\overline{d_a(R)}}}$ the map induced by $d_a$ and let $\phi:{\overline{\overline{d_a(R)}}}\to{\overline{\overline{d_b(R)}}}$ be the continuous extension of the map $d_a(x)\mapsto d_b(x)$ ($x\in R$), regarded as a map into $R$. Then the composition $\psi:=\phi\tilde{d}_aq_0{E^{\perp}}$ is easily seen to be an $(a)^c$-bimodule map such that $\psi (d_a|R)=d_b|R$, hence $d_b|R=(d_a|R)\psi$ (since $a\in(a)^c$) and consequently $d_b(R)\subseteq d_a(R)$. The map $\psi$ is bounded with $\|\psi\|\leq\|\phi\tilde{d}_a\|\|q_0\|\|{E^{\perp}}\|
\leq2\|\phi\tilde{d}_a\|\leq4\|b\|$, but not automatically normal. However, if $\psi=\psi_n+\psi_s$ is the decomposition into the normal part $\psi_n$ and the singular part $\psi_s$ [@KR Section 10.1], then the identity $d_b|R-(d_a|R)\psi_n=(d_a|R)\psi_s$ implies that $(d_a|R)\psi_s=0$ (since the left side of the identity is normal, while the right side is singular). Similarly $\psi(d_a|R)=0$, hence we may replace $\psi$ by $\psi_n$.
We do not know if the range inclusion part of Proposition \[pr50\] can be extended to general C$^*$-algebras, but at least a somewhat weaker version holds.
\[co509\]In a C$^*$-algebra $A$, if elements $a,b$ satisfy $\|[b,x]\|\leq\|[a,x]\|$ for all $x\in A$ and $a$ is normal, then $d_b(A)\subseteq4\|b\|{\overline{\overline{d_a(B_A)}}}$, where $B_A$ denotes the closed unit ball of $A$. In particular $d_b(A)\subseteq{\overline{\overline{d_a(A)}}}$.
We may assume that $A\subseteq{{\rm B}(\mathcal{H})}$. Then by Lemma \[le508\] $\|[b,x]\|\leq\|[a,x]\|$ for all $x\in{\overline{A}}$, hence it follows from Proposition \[pr50\] that $d_b(B_{{\overline{A}}})
\subseteq 4\|b\| d_a(B_{{\overline{A}}})$. Since the Kaplansky density theorem implies that ${\overline{d_a(B_{{\overline{A}}})}}={\overline{d_a(B_A)}}$, it follows that $d_b(B_A)\subseteq 4\|b\|{\overline{d_a(B_A)}}\cap A=
4\|b\|{\overline{\overline{d_b(B_A)}}}$, where for the last equality we have used [@KR 10.1.4].
By [@KS 6.5], if $a$ is normal, (\[51\]) implies (\[50\]) in any C$^*$-algebra $A$. The proof uses the special case $A={{\rm B}(\mathcal{H})}$ proved earlier in [@JW]. We will sketch a somewhat simplified proof of this case, but first we need to recall a fact concerning operators in ${\rm B}(X,Y)$, the space of all bounded linear operators from $X$ into $Y$, where $X$ and $Y$ are Banch spaces. Denote by ${X^{\sharp}}$ the dual of $X$ and by ${T^{\sharp}}$ the adjoint of $T\in{\rm B}(X,Y)$. The following simple fact is well-known (see [@JW]).
\[le51\]Given $S,T\in {\rm B}(X,Y)$, the inclusion ${T^{\sharp}}({Y^{\sharp}})\subseteq{S^{\sharp}}({Y^{\sharp}})$ holds if and only if there exists a constant $\kappa$ such that $$\label{61}\|T\xi\|\leq\kappa\|S\xi\|$$ for all $\xi\in X$.
Since $d_a=-{(d_a|{{\rm T}(\mathcal{H})})^{\sharp}}$, where ${{\rm T}(\mathcal{H})}$ is the ideal in ${{\rm B}(\mathcal{H})}$ of trace class operators, the following is just a special case of Lemma \[le51\].
\[co52\]Let $a,b\in{{\rm B}(\mathcal{H})}$.
\(i) The inclusion $d_b({{\rm B}(\mathcal{H})})\subseteq d_a({{\rm B}(\mathcal{H})})$ holds if and only if there exists a constant $\kappa$ such that $\|d_b(t)\|\leq\kappa\|d_a(t)\|$ for all $t\in{{\rm T}(\mathcal{H})}$.
\(ii) The inclusion $d_b({{\rm T}(\mathcal{H})})\subseteq d_a({{\rm T}(\mathcal{H})})$ is equivalent to the existence of a constant $\kappa$ such that $\|d_b(x)\|\leq\kappa
\|d_a(x)\|$ for all $x\in{{\rm K}(\mathcal{H})}$ or (equivalently, by Lemma \[le508\]) for all $x\in{{\rm B}(\mathcal{H})}$.
\[le521\]Let $a\in{{\rm B}(\mathcal{H})}$ be a subnormal operator and $f$ a Lipschitz function on $\sigma(a)$. If $a$ is not normal, assume that $f$ is in the uniform closure of the set of rational functions with poles outside $\sigma(a)$, so that $b:=f(a)$ is defined. If $\|[b,x]\|\leq\kappa\|[a,x]\|$ for all $x\in{{\rm B}(\mathcal{H})}$, then for each sequence $(\lambda_i)\subseteq\sigma(a)$ the matrix $\Lambda(f;\lambda)$ defined by the right side of (\[54\]) is a Schur multiplier with the norm at most $2\kappa$. (That is, $f$ is a Schur function on $\sigma(a)$ as defined in the Introduction). Similarly, the condition $\|[b,x]\|_1\leq\kappa\|[a,x]\|_1$ for all $x\in{{\rm T}(\mathcal{H})}$ implies that $\Lambda(f;\lambda)$ is a Schur multiplier on ${{\rm T}(\mathcal{H})}$ with the norm $\leq2\kappa$.
First suppose that $(\lambda_i)_{i=1}^m$ is a finite subset of the boundary $\partial\sigma(a)$ of $\sigma(a)$. Then each $\lambda_i$ is an approximate eigenvalue of $a$ [@Co], hence there exists a sequence of unit vectors $\xi_{i,n}\in\h$ such that $\lim_n\|(a-\lambda_i1)\xi_{i,n}\|=0$. Since $a-\lambda_i1$ is hyponormal, $\|(a-\lambda_i1)^*\xi_{i,n}\|\leq\|(a-\lambda_i1)\xi_{i,n}\|$ and as $n\to\infty$ $$(\lambda_i-\lambda_j){\langle \xi_{i,n},\xi_{j,n}\rangle}={\langle \lambda_i\xi_{i,n},\xi_{j,n}\rangle}-
{\langle \xi_{i,n},\overline{\lambda}_j
\xi_{j,n}\rangle}$$ tends to $\lim_n({\langle a\xi_{i,n},\xi_{j,n}\rangle}-{\langle \xi_{i,n},a^*\xi_{j,n}\rangle}=0.$ Thus, if $i\ne j$, then $\lim{\langle \xi_{i,n},\xi_{j,n}\rangle}=0$, so the set $\{\xi_{1,n},\ldots,\xi_{m,n}\}$ is approximately orthonormal if $n$ is large. It follows that for each matrix $\alpha=[\alpha_{i,j}]\in{{\rm M}_m(\bc)}$ the norm of the operator $x:=\sum_{i,j=1}^m\alpha_{i,j}\xi_{i,n}\otimes\xi_{j,n}^*$ is approximately equal to the usual operator norm of $\alpha$. Further, for large $n$ we have approximate equalities $$d_a(x)=\sum_{i,j=1}^m\alpha_{i,j}(a\xi_{i,n}\otimes\xi_{j,n}^*-\xi_{i,n}\otimes(a^*\xi_{j,n})^*)\approx\sum_{i,j=1}^m\alpha_{i,j}(\lambda_i-\lambda_j)
\xi_{i,n}\otimes\xi_{j,n}^*$$ and $$d_b(x)\approx\sum_{i,j=1}^m\alpha_{i,j}(f(\lambda_i)-f(\lambda_j))\xi_{i,n}\otimes\xi_{j,n}^*,$$ hence it follows from the assumption $\|d_b(x)\|\leq\kappa\|d_a(x)\|$ that $$\label{201}\|[f(\lambda_i)-f(\lambda_j)\alpha_{i,j}]\|\leq\kappa\|[(\lambda_i-\lambda_j)\alpha_{i,j}]\|.$$ This estimate means that for a finite subset $\lambda=(\lambda_i)_{i=1}^m$ of $\partial\sigma(a)$ the matrix $\Lambda(f;\lambda)$ with the entries $$\label{54}\Lambda_{i,j}(f;\lambda)=\left\{\begin{array}{ll}\frac{f(\lambda_i)-f(\lambda_j)}{\lambda_i-\lambda_j},&\mbox{if}\ i\ne j\\
0,&\mbox{if}\ i=j\end{array}\right.$$ acts as a Schur multiplier with the norm at most $\kappa$ on the subspace $E_m\subseteq{{\rm M}_m(\bc)}$ of matrices of the form $[(\lambda_i-\lambda_j)\alpha_{i,j}]$. Since $E_m$ is just the set of all matrices with zero diagonal and the natural projection from ${{\rm M}_m(\bc)}$ onto the subspace $D_m$ of diagonal matrices has the Schur norm $1$ (and $\Lambda(f,\lambda)(D_m=0)$), it follows that the norm of $\Lambda(f;\lambda)$ as a Schur multiplier on ${{\rm M}_m(\bc)}$ is at most $2\kappa$.
Now let $\lambda_2,\ldots,\lambda_m$ be fixed elements of $\partial\sigma(a)$ and consider the function $$\lambda_1\mapsto\Lambda(f;\lambda_1,\lambda_1,\ldots,\lambda_m)$$ from $\sigma(a)$ into the Banach algebra ${{\rm M}_m(\bc)}$ equipped with the Schur norm. Since this function is holomorphic on the interior of $\sigma(a)$ and bounded on $\partial\sigma(a)$ (by $2\kappa$), it follows that the Schur norm of $\Lambda(f;\lambda_1,\lambda_2,\ldots,\lambda_m)$ is at most $2\kappa$ for all $\lambda_1\in\sigma(a)$. In the same way we show that the Schur norm of $\Lambda(f;\lambda_1,\ldots,\lambda_m)$ is at most $2\kappa$ for all $\lambda_i\in\sigma(a)$. Since the bound $2\kappa$ is the same for all $m$, the lemma is proved.
For a rank one operator $x$ the operators $d_a(x)$ and $d_b(x)$ have rank at most two and on such operators the trace class norm is equivalent to the usual operator norm. If $a$ is normal, we deduce now from Corollary \[co52\], Lemma \[le41\] and Theorem \[th3\] that each of the two range inclusions $d_b({{\rm B}(\mathcal{H})})\subseteq d_a({{\rm B}(\mathcal{H})})$ and $d_b({{\rm T}(\mathcal{H})})\subseteq d_a({{\rm T}(\mathcal{H})})$ implies that $b$ is of the form $b=f(a)$ for a Lipschitz function $f$ on $\sigma(a)$. Then $f$ is a Schur function by Corollary \[co52\] and Lemma \[le521\].
If $a$ is diagonal, the proof of Lemma \[le521\] shows that (\[50\]) and its analogue for the trace norm hold for any Schur function $f$ on $\sigma(a)$. Indeed, if $\lambda_i$ are the eigenvalues of $a$ and a matrix $[x_{i,j}]$ represents an operator $x$ relative to the ortonormal basis consisting of eigenvectors of $a$, the inequality (\[50\]) (and its analogue in the trace norm) assumes the form $$\label{533}\|[(f(\lambda_i)-f(\lambda_j))x_{i,j}]\|\leq\kappa\|[(\lambda_i-\lambda_j)x_{i,j}]\|.$$ Since the obvious projection from ${{\rm B}(\mathcal{H})}$ (or ${{\rm T}(\mathcal{H})}$) onto the subset of all diagonal matrices in ${{\rm B}(\mathcal{H})}$ (or in ${{\rm T}(\mathcal{H})}$) is contractive (in the Schur norm), we see that (\[533\]) and its trace analogue are equivalent to the requirements that the matrix $\Lambda(f;\lambda)$ defined as in (\[54\]) is a Schur multiplier on ${{\rm B}(\mathcal{H})}$ and ${{\rm T}(\mathcal{H})}$ (respectively). But it is well-known (and easy to see) that a matrix is a Schur multiplier on ${{\rm T}(\mathcal{H})}$ if and only if its transpose ia a Schur multiplier on ${{\rm B}(\mathcal{H})}$ and then the two have the same norm, hence the two conditions on $f$ are equivalent.
For a general normal $a$ and a Schur function $f$ on $\sigma(a)$, given $\varepsilon>0$, by the Weyl-von Neumann-Bergh theorem [@Co2 Corollary 39.6] there exists a diagonal $a_0$ such that $\sigma(a_0)\subseteq\sigma(a)$, $\|a-a_0\|<\varepsilon$ and (approximating $f$ by polynomials) $\|f(a)-f(a_0)\|<\varepsilon$. Then for each $x\in{{\rm B}(\mathcal{H})}$ with $\|x\|=1$ we have $\|[f(a),x]\|\leq\|[f(a_0),x]\|+2\varepsilon\leq\kappa\|[a_0,x]\|+2\varepsilon\leq\kappa\|[a,x]\|+2\kappa\varepsilon
+2\varepsilon$, so $\|[f(a),x]\|\leq\kappa\|[a,x]\|$. The same estimate holds also for the trace norm. Thus we may summarize the above discussion in the following theorem most of which was proved already by Johnson and Williams in [@JW] in a somewhat different way.
\[JW\][@JW] If $a\in{{\rm B}(\mathcal{H})}$ is normal, then for any $b\in{{\rm B}(\mathcal{H})}$ the inclusion $d_b({{\rm B}(\mathcal{H})})\subseteq d_a({{\rm B}(\mathcal{H})})$ holds if and only if there exists a constant $\kappa$ such that $\|d_b(x)\|\leq\kappa\|d_a(x)\|$ for all $x\in{{\rm B}(\mathcal{H})}$ and this is also equivalent to the condition that $b=f(a)$ for a Schur function $f$ on $\sigma(a)$.
If $a$ is not normal, then the range inclusion (\[51\]) does not necessarily imply that $b\in(a)^{\prime\prime}$ [@Ho], hence it does not imply (\[50\]). But we will prove that conversely (\[50\]) implies (\[51\]), if $a$ satisfies certain conditions which are much more general than normality.
\[pr53\]Denote ${\mathcal{R}}_a:=d_a({{\rm B}(\mathcal{H})})$. If ${\overline{{\mathcal{R}}_a}}+{(a)^{\prime}}={{\rm B}(\mathcal{H})}$, then for each $b\in{{\rm B}(\mathcal{H})}$ the condition (\[50\]) implies that ${\mathcal{R}}_b\subseteq{\mathcal{R}}_a$. Moreover, if ${\overline{{\mathcal{R}}_a}}={{\rm B}(\mathcal{H})}$, then there exists a weak\* continuous ${(a)^{\prime}}$-bimodule map $\phi$ on ${{\rm B}(\mathcal{H})}$ such that $d_b=\phi d_a=d_a\phi$.
By (\[50\]) the correspondence $d_a(x)\mapsto d_b(x)$ extends to a bounded map $\phi_0$ from ${\overline{\overline{{\mathcal{R}}_a}}}$ into ${\overline{\overline{{\mathcal{R}}_b}}}$ such that $\phi_0d_a=d_b$. Note that $\phi_0(d_a({{\rm K}(\mathcal{H})}))\subseteq
d_b({{\rm K}(\mathcal{H})})$. Identifying $d_a({{\rm K}(\mathcal{H})})^{\sharp\sharp}$ with ${\overline{d_a({{\rm K}(\mathcal{H})})}}$ inside ${{\rm K}(\mathcal{H})}^{\sharp\sharp}={{\rm B}(\mathcal{H})}$ in the usual way, it follows that $\phi:=\phi_0^{\sharp\sharp}$ is the weak\* continuous extension of $\phi_0$ to ${\overline{d_a({{\rm B}(\mathcal{H})})}}={\overline{d_a({{\rm K}(\mathcal{H})})}}$ satisfying $\phi d_a=d_b$. Since $\phi_0$ is an ${(a)^{\prime}}$-bimodule maps, so must be $\phi$ by continuity, hence in particular $$d_b(x)=\phi(d_a(x))=d_a\phi(x)\ \ \mbox{for all}\ x\in{\overline{{\mathcal{R}}_a}}={\overline{d_a({{\rm B}(\mathcal{H})})}}$$ and consequently $d_b({\overline{{\mathcal{R}}_a}})\subseteq {\mathcal{R}}_a$. Finally, to conclude the proof, note that the assumption ${\overline{{\mathcal{R}}_a}}+{(a)^{\prime}}={{\rm B}(\mathcal{H})}$ implies that ${\mathcal{R}}_b=d_b({\overline{{\mathcal{R}}_a)}}$ since from (\[50\]) ${(a)^{\prime}}\subseteq{(b)^{\prime}}=\ker d_b$.
By duality the condition ${\overline{d_a({{\rm B}(\mathcal{H})})}}={{\rm B}(\mathcal{H})}$ means that the kernel of $d_a|{{\rm T}(\mathcal{H})}$ is $0$, that is, ${(a)^{\prime}}\cap{{\rm T}(\mathcal{H})}=0$. There are many Hilbert space operators $a$ which do not even commute with any nonzero compact operator. This is so for example, if $a$ is normal and has no eigenvalues. (Namely, ${(a)^{\prime}}$ is a C$^*$-algebra and contains the spectral projection $p$ corresponding to any nonzero eigenvalue of each $h=h^*\in{(a)^{\prime}}$. Since $p$ is of finite rank, $ap$, and therefore also $a$, has eigenvalues if $h\ne0$.) For a general normal $a\in{{\rm B}(\mathcal{H})}$ we can decompose $\h$ into the orthogonal sum $\h=\h_1\oplus\h_2$, where $\h_1$ is the closed linear span of all eigenvectors of $a$ and $\h_2=\h_1^{\perp}$. Then $a$ also decomposes as $a_1\oplus a_2$, where ${(a_2)^{\prime}}$ contains non nonzero compact operators, while ${(a_1)^{\prime}}$ contains a net of finite rank projections converging strongly to the identity.
Another class of operators which admit the decomposition as in the previous paragraph are (rationally) cyclic subnormal operators. Such an operator can be decomposed into the direct sum of the normal and the pure part. An operator $c$ in the commutant of the pure part $a_0$ is subnormal (by Yoshino’s theorem [@Co3 5.4]), hence normal if compact [@Hal]. But then the (finite dimensional) eigenspace $\h_0$ of $c$ corresponding to a nonzero eigenvalue is invariant under $a_0$ and $a_0|\h_0$ is normal, contradicting the purity of $a_0$. (A general pure subnormal operator, however, can commute with a nonzero trace class operator; an example is in [@Wi 2.1].) By the Wold decomposition an isometry is a direct sum of a unitary and of copies of the unilateral shift, hence, the following theorem applies also to isometries.
\[th54\]Let $a\in{{\rm B}(\mathcal{H})}$ and suppose that $\h$ decomposes into the orthogonal sum $\h_1\oplus\h_2$ of two subspaces which are invariant under $a$, so that $a=a_1\oplus a_2$, where $a_i\in{\rm B}(\h_i)$. If ${(a_1)^{\prime}}\cap{{\rm T}(\mathcal{H}_1)}$ contains a bounded net $(e_k)$ converging to $1$ in the strong operator topology, while ${(a_2)^{\prime}}\cap{{\rm T}(\mathcal{H}_2)}=0$, then the condition $\|d_b(x)\|\leq\|d_a(x)\|$ ($\forall
x\in{{\rm B}(\mathcal{H})}$) implies that $d_b({{\rm B}(\mathcal{H})})\subseteq d_a({{\rm B}(\mathcal{H})})$.
Since $b\in(a)^{\prime\prime}$, $\h_1$ and $\h_2$ are invariant subspaces for $b$, so $b$ also decomposes as $b=b_1\oplus b_2$, where $b_i\in{\rm B}(\h_i)$. Relative to the same decomposition of $\h$ each $x\in{{\rm B}(\mathcal{H})}$ can be represented by a $2\times2$ operator matrix $x=[x_{i,j}]$ and $$d_b(x)=\left[\begin{array}{ll}
b_1x_{1,1}-x_{1,1}b_1&b_1x_{1,2}-x_{1,2}b_2\\
b_2x_{2,1}-x_{2,1}b_1&b_2x_{2,2}-x_{2,2}b_2
\end{array}\right].$$ Thus it suffices to show that for each pair $(i,j)$ of indexes and for each $x_{i,j}\in{\rm B}(\h_j,\h_i)$ the element $b_ix_{i,j}-x_{i,j}b_j$ is in the range of the map $d_{a_i,a_j}$ defined on ${\rm B}(\h_j,\h_i)$ by $d_{a_i,a_j}(y)=a_iy-ya_j$. In the case $i=2=j$ this follows from Proposition \[pr53\]. We will now consider the case $i=1$ and $j=2$, the remaining two cases are treated similarly.
From the norm inequality in the theorem we have in particular that $$\label{55}\|d_{b_2,b_1}(x)\|\leq\|d_{a_2,a_1}(x)\|\ \ (\forall x\in
{\rm B}(\h_1,\h_2)).$$ This implies that there exists a bounded ${(a_2)^{\prime}},{(a_1)^{\prime}}$-bimodule map $$\phi_0:{\overline{\overline{d_{a_2,a_1}({\rm K}(\h_1,\h_2))}}}\to{\rm K}(\h_1,\h_2)$$ such that $\phi_0d_{a_2,a_1}=d_{b_2,b_1}$. (To prove the bimodule property use that ${(a_i)^{\prime}}\subset{(b_i)^{\prime}}$, which follows from ${(a)^{\prime}}\subseteq{(b)^{\prime}}$.) Now observe that the closure $X$ of $d_{a_2,a_1}({\rm T}(\h_1,\h_2))$ in the trace norm is contained in ${\overline{\overline{d_{a_2,a_1}({\rm K}(\h_1,\h_2))}}}$ (the closure of $d_{a_2,a_1}({\rm K}(\h_1,\h_2))$ in the usual operator norm), and that $X\subseteq {\rm T}(\h_1,\h_2)$ is a nondegenerate right Banach module over the Banach algebra $A:={(a_1)^{\prime}}\cap{{\rm T}(\mathcal{H}_1)}$ since for each $t\in{\rm T}(\h_1,\h_2)$ the operators $te_k$ converge to $t$ in the trace norm. Moreover, $(e_k)$ is a bounded approximate identity for $A$, hence by the Cohen-Hewitt theorem [@CLM p. 108] each $t\in X$ can be factored as $t=sc$, where $s\in X$ and $c\in A$. Since $\phi_0$ is a homomorphism of right ${(a_1)^{\prime}}$-modules (hence also of right $A$-modules), it follows that $\phi_0(t)=\phi_0(s)c$, hence $\phi_0(t)\in{\rm T}(\h_1,\h_2)$ (since $c\in{{\rm T}(\mathcal{H}_1)}$). Thus $\phi_0$ maps $X$ into ${\rm T}(\h_1,\h_2)$. Moreover, it is easy to verify that the graph of the restriction $\psi:=\phi_0|X$ is closed, hence $\psi$ is bounded by the closed graph theorem. Now from $d_{b_2,b_1}|
{\rm T}(\h_1,\h_2)=\psi (d_{a_2,a_1}|{\rm T}(\h_1,\h_2))$ we infer (by taking the dual maps) that $d_{b_1,b_2}=\tilde{d}_{a_1,a_2}{\psi^{\sharp}}$, where $\tilde{d}_{a_1,a_2}:{\rm B}(\h_2,\h_1)/\ker{d_{a_1,a_2}}\to {\rm B}(\h_2,\h_1)$ is the map induced by $d_{a_1,a_2}$, hence the range of $d_{b_1,b_2}$ is indeed contained in the range of $d_{a_1,a_2}$.
As customary, ${\rm Rat}(K)$ denotes the algebra of all rational functions with poles outside a compact subset $K\subseteq\bc$ and, if $\mu$ is a positive Borel Measure on $K$, $R^2(K,\mu)$ is the closure in $L^2(\mu)$ of ${\rm Rat}(K)$. As before we denote by $\dot{a}$ the coset in the Calkin algebra ${{\rm C}(\mathcal{H})}$ of an operator $a\in{{\rm B}(\mathcal{H})}$.
\[pr6101\]Let $K$ be a compact subset of $\bc$, $a$ a subnormal operator with $\sigma(a)\subseteq K$ such that $a$ is cyclic for the algebra ${\rm Rat}(K)$ and let $c$ be the minimal normal extension of $a$. Assume that $\sigma(c)\subseteq \sigma(\dot{a})$, let $\mu$ be a scalar spectral measure for $c$ such that $a$ is the multiplication on $\h:=R^2(K,\mu)$ by the identity function $z$. Denote by $p$ the orthogonal projection from $\k:=L^2(\mu)$ onto $\h$ and assume that the only function $h\in C(\sigma(c))+(L^{\infty}(\mu)\cap R^2(K,\mu))$ for which the operator $T_h$ defined by $T_h(\xi):=p(f\xi)$ ($\xi\in\h$) is compact is $h=0$. Then for each $b\in{{\rm B}(\mathcal{H})}$ satisfying $\|[b,x]\|\leq\|[a,x]\|$ ($x\in{{\rm B}(\mathcal{H})}$) there exists a function $f\in C(\sigma(c))\cap R^2(K,\mu)$ such that $b=f(c)|\h$.
Moreover, if $K$ is the closure of a domain $G$ bounded by finitely many nonintersecting analytic Jordan curves and $a$ is the multiplication operator by $z$ on the Hardy space $H^2(G)$, $f$ can be extended to a Schur function on $K$.
It is well-known that a rationally cyclic subnormal operator $a$ can be represented as the multiplication on $R^2(K,\mu)$ by the independent variable $z$ [@Co3 p. 51] and that ${(a)^{\prime}}=R^2(K,\mu)\cap L^{\infty}(\mu)$ by Yoshino’s theorem [@Co3 p. 52]. Since $b\in{(a)^{\prime}}$, it follows that $b$ is the multiplication on $R^2(K,\mu)$ by a function $f\in R^2(K,\mu)\cap L^{\infty}(\mu)$.
On the other hand $a$ is essentially normal by the Berger-Show theorem [@Co3 p. 152], hence by Corollary \[co32\] and Lemma \[le41\] $\dot{b}=g(\dot{a})$ for a continuous function $g$ on $\sigma(\dot{a})$. Further, since $c$ is normal and $a$ is subnormal and essentially normal, an easy computation with $2\times 2$ operator matrices (relative to the decomposition $\k=\h\oplus\h^{\perp}$) shows that the operator ${p^{\perp}}c^*p$ is compact, hence (since also ${p^{\perp}}cp=0$) the map $h\mapsto \dot{T}_h$ from $C(\sigma(c))$ into the Calkin algebra is a $*$-homomorphism. Thus it must coincide with the $*$-homomorphism $h\mapsto h(\dot{c})$ since they coincide on the generator $c$. It follows in particular that $\dot{b}=g(\dot{a})=\dot{T_g}$, hence the operator $T_{g-f}=T_g-b$ is compact. But by the hypothesis this is possible only if $g-f=0$, hence $f$ is continuous.
In the case $a$ is the unilateral shift, $f$ is a continuous function on the circle and contained in the closure $P^2(\mu)$ of polynomials in $L^2(\mu)$, where $\mu$ is the normalized Lebesgue measure on the circle. It is well known that such a function can be holomorphically extended to disc $\bd$ such that the extension (denoted again by $f$) is continuous on ${\overline{\bd}}$. By Lemma \[le521\] $f$ is a Schur function on ${\overline{\bd}}$. Similar arguments apply to multiply connected domains by [@Ab 2.11, 1.1], [@Mu 4.3, 9.4]).
Perhaps, in general, (\[50\]) does not even imply that $d_b({{\rm B}(\mathcal{H})})\subseteq{\overline{\overline{d_a({{\rm B}(\mathcal{H})})}}}$, but a possible counterexample is not known to the author. Note, however, that by duality between ${{\rm T}(\mathcal{H})}$ and ${{\rm B}(\mathcal{H})}$ (\[50\]) implies that $d_b({{\rm B}(\mathcal{H})})\subseteq {\overline{d_a({{\rm B}(\mathcal{H})})}}$, hence in particular $d_b({{\rm K}(\mathcal{H})})\subseteq{\overline{\overline{d_a({{\rm K}(\mathcal{H})}}}}$ since the weak topology agrees on ${{\rm K}(\mathcal{H})}$ with the weak\* topology inhereted from ${{\rm B}(\mathcal{H})}$. More generally, we will see that the problem depends entirely on what happens in the Calkin algebra.
For a C$^*$-algebra $A$ and $a\in A$ note that a functional $\rho\in{A^{\sharp}}$ annihilates $d_a(A)$ if and only if $[a,\rho]=0$, where $[a,\rho]\in{A^{\sharp}}$ is defined by $([a,\rho])(x)=\rho(xa-ax)$. In other words, the annihilator in ${A^{\sharp}}$ of $d_a(A)$ is just the [*centralizer*]{} $C_a$ of $a$ in ${A^{\sharp}}$.
\[pr06\]If $a,b\in{{\rm B}(\mathcal{H})}$ satisfy $\|[b,x]\|\leq\|[a,x]\|$ for all $x\in{{\rm B}(\mathcal{H})}$, then $\|[\dot{b},\dot{x}]\|\leq\|[\dot{a},\dot{x}]\|$ in the Calkin algebra ${{\rm C}(\mathcal{H})}$. If this latter inequality implies that $C_{\dot{a}}\subseteq C_{\dot{b}}$, then $C_a\subseteq C_b$ also holds, hence $d_b({{\rm B}(\mathcal{H})})\subseteq
{\overline{\overline{d_a({{\rm B}(\mathcal{H})})}}}$.
The first statement follows from Lemma \[le508\]. To prove the rest of the proposition, first note that for any $a\in{{\rm B}(\mathcal{H})}$ and a functional $\rho\in C_a$ the normal part $\rho_n$ and the singular part $\rho_s$ are both in $C_a$. (Indeed, from $[a,\rho]=0$ we have $[a,\rho_n]=-[a,\rho_s]$, where the left side is normal and the right side is singular, hence both are $0$.) Further, since $\rho_n$ is given by a trace class operator $t$, $[a,t]=0$, hence the hypothesis of the proposition implies that $[b,t]=0$, so $[b,\rho_n]=0$. Since singular functionals annihilate ${{\rm K}(\mathcal{H})}$, they can be regarded as functionals on ${{\rm C}(\mathcal{H})}$. Thus, if the condition $\|[\dot{b},\dot{x}]\|\leq\|[\dot{a},\dot{x}]\|$ ($\dot{x}\in{{\rm C}(\mathcal{H})}$) implies that $C_{\dot{a}}\subseteq C_{\dot{b}}$, then we have $\rho_s\in C_{\dot{b}}$ and consequently also $\rho\in C_b$. This proves that $C_a\subseteq C_b$ and the Hahn-Banach theorem then implies that $d_b({{\rm B}(\mathcal{H})})\subseteq{\overline{\overline{d_a({{\rm B}(\mathcal{H})})}}}$.
\[co07\]Suppose $a,b\in{{\rm B}(\mathcal{H})}$ satisfy (\[50\]). If $a$ is essentially normal, then $d_b({{\rm B}(\mathcal{H})})\subseteq {\overline{\overline{d_a({{\rm B}(\mathcal{H})})}}}$.
By Lemma \[le508\] and Corollary \[co509\] $d_{\dot{b}}({{\rm C}(\mathcal{H})})\subseteq {\overline{\overline{d_{\dot{a}}({{\rm C}(\mathcal{H})})}}}$. Now Proposition \[pr06\] completes the proof.
Commutators and the completely bounded norm
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In this section we will study stronger variants of the condition $\|[b,x]\|\leq\|[a,x]\|$ ($x\in{{\rm B}(\mathcal{H})}$) in the context of completely bounded maps.
\[le61\]If $a,b\in{{\rm B}(\mathcal{H})}$ satisfy $$\label{6100}\|[b^{(n)},x]\|\leq\|[a^{(n)},x]\|\ \ \mbox{for all}\ x\in{{\rm M}_n({{\rm B}(\mathcal{H})})}\ \mbox{and all}\ n\in\bn,$$ then $$\label{62}\|[\pi(b),x]\|\leq\|[\pi(a),x]\|\ \mbox{for all}\ x\in{{\rm B}(\mathcal{H}_{\pi})}$$ for every unital $*$-representation $\pi:A\to{{\rm B}(\mathcal{H}_{\pi})}$ of the C$^*$-algebra $A$ generated by $1,a$ and $b$.
First assume that $\h_{\pi}$ is separable. Let $J={{\rm K}(\mathcal{H})}\cap A$, $\h_n=[\pi(J)\h_{\pi}]$, and let $\pi_n$ and $\pi_s$ be the representations of $A$ defined by $\pi_n(a)=\pi(a)|\h_n$ and $\pi_s(a)=\pi(a)|{\h_n^{\perp}}$ ($a\in A$), so that $\pi=\pi_n\oplus\pi_s$. By basic theory of representations of C$^*$-algebras of compact operators $\pi_n$ is a subrepresentation of a multiple $id^{(m)}$ of the identity representation. By Voiculescu’s theorem ([@Vo], [@Ar]) the representation $\pi\oplus id$ is approximately unitarily equivalent to $\pi_n\oplus id$, hence $\pi\oplus id$ is approximately unitarily equivalent to a subrepresentation of $id^{(m+1)}$. It follows easily from (\[6100\]) that (\[62\]) holds for any multiple of the identity representation in place of $\pi$, hence it must also hold for any subrepresentation $\rho$ of $id^{(m+1)}$ (to see this, just take in (\[62\]) for $x$ elements that live on the Hilbert space of $\rho$). But then it follows from the approximate equivalence that the condition (\[62\]) holds for $\pi\oplus id$ in place of $\pi$, hence also for $\pi$ itself.
In general, when $\h_{\pi}$ is not necessarily separable, $\h_{\pi}$ decomposes into an orthogonal sum $\oplus_{i\in\bi}\h_i$ of separable invariant subspaces for $\pi(A)$. For a fixed $x\in{{\rm B}(\mathcal{H}_{\pi})}$ there exists a countable subset $\bj$ of $\bi$ such that the norm of the operator $[\pi(b),x]$ is the same as the norm of its compression to $\l:=\oplus_{i\in\bj}\h_i$. Since $\l$ is separable, it follows from what we have already proved that $\|[\pi(b),x]\|\leq\|[\pi(a),x]\|$.
\[co610\]If $a,b\in{{\rm B}(\mathcal{H})}$ satisfy (\[6100\]) then $b$ is contained in the C$^*$-algebra $B$ generated by $a$ and $1$.
Let $\pi$ be the universal representation of $A=C^*(a,b,1)$ and $\h_{\pi}$ its Hilbert space. It follows from Lemma \[le61\] that $\pi(b)\in(\pi(a))^{\prime\prime}$, hence also $\pi(b)\in\pi(B)^{\prime\prime}$. But $\pi(B)^{\prime\prime}={\overline{\pi(B)}}$, thus $\pi(b)\in{\overline{\pi(B)}}\cap\pi(A)=\pi(B)$ by [@KR 10.1.4].
Let us say that a completely contractive Hilber module over an operator algebra $A$ (that is, a Hilbert space on which $A$ has a completely contractive representation) is a [*cogenerator*]{} if every completely contractive Hilbert $A$-module is contained (completely isomorphically) in a multiple of $\h$ (that is, in a direct sum of copies of $\h$). Here by an operator algebra we will always mean a norm complete algebra of operators on a Hilbert space.
\[th62\]If $a,b\in{{\rm B}(\mathcal{H})}$ satisfy (\[6100\]), where $\h$ is a cogenerator for the operator algebra $A_0$ generated by $a$ and $1$, then $b\in A_0$.
Let $\pi$ be the universal representation of the C$^*$-algebra $A$ generated by $1,a$ and $b$. Then $\h_{\pi}$ (the Hilbert space of $\pi$) is a cogenerator for $A_0$, hence by the Blecher-Solel bicommutation theorem (see [@BLM 3.2.14]) ${\overline{\pi(A_0)}}=\pi(A_0)^{\prime\prime}$. From Lemma \[le61\] $\pi(b)\in \pi(A_0)^{\prime\prime}$, hence $\pi(b)\in{\overline{\pi(A_0)}}\cap\pi(A)=\pi(A_0)$ by [@KR 10.1.4]).
\[re63\]Let $a,b\in{{\rm B}(\mathcal{H})}$, $A_0$ the norm closed operator algebra generated by $a$ and $1$, and $\h_0$ the direct sum of infinitely many copies of the Hilbert space of the universal representation of $C^*_{\rm max}(A_0)$ (the maximal C$^*$-cover of $A_0$, see [@BLM]). Then there exists a natural $*$-homomorphism $q$ from $C^*_{\rm max}(A_0)$ onto $C^*(A_0)$ (the C$^*$ subalgebra of ${{\rm B}(\mathcal{H})}$ generated by $A_0$). If (\[6100\]) holds, then $b\in C^*(A_0)$ by Corollary \[co610\], hence $b=q(b_0)$ for some $b_0\in C^*_{\rm max}(A_0)$. If $b_0$ can be chosen so that $$\label{622}\|[b_0,x\|\leq\|[a,x]\|\ \mbox{for all}\ x\in{\rm B}(\h_0),$$ then it follows by Proposition \[th62\] that $b_0$ can be approximated by polynomials in $a$ and $1$, hence so can be $b$. If in addition $a$ is subnormal, then we conclude that $b=f(a)$ for a function $f$ in the uniform closure of polynomials on $\widehat{\sigma(a)}$.
For a general subnormal operator $a$ the condition (\[6100\]) (in contrast to its more involved version (\[622\])) does not imply that $b$ is in the uniform closure of polynomials in $a$, however the author does not know if it implies that $b$ is of the form $b=f(a)$ for some more general function $f$.
If in (\[6100\]) $a$ is subnormal, is then $b$ necessarily of the form $b=f(a)$ for some function $f$? Is $b$ necessarily subnormal?
Commutators of functions of subnormal operators
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In this section we will show for a subnormal operator $a$ and a sufficiently regular function $f$ an estimate of the form $$\label{598}\|[f(a),x]\|\leq\kappa\|[a,x]\| \ \ \forall x\in{{\rm B}(\mathcal{H})}.$$ By Lemma \[le521\] such an estimate can hold only for Schur functions, but we are able to prove (\[598\]) for all Schur functions only if $\sigma(a)$ is nice (Theorem \[th621\]). It follows from the proof in [@JW Theorem 4.1] that a Schur function $f$ is complex differentiable in the sense that the limit $f^{\prime}(\zeta_0)=\lim_{\zeta\to\zeta_0,\ \zeta\in\sigma(a)}(f(\zeta)-f(\zeta_0))/(\zeta-\zeta_0)$ exists at each non-isolated point of $\sigma(a)$. Moreover, from the Lipschitz condition on $f$ we see that $f^{\prime}$ is bounded. However, the boundedness of $f^{\prime}$ is not sufficient for $f$ to be a Schur function. When $a$ is selfadjoint it is proved in [@JW 5.1] that (\[598\]) holds if $f^{(3)}$ is continuous. We will prove (\[598\]) for subnormal $a$ under a much milder condition on $f$ (for example, $f^{\prime}$ Lipschitz suffices), but perhaps when $a$ is normal our condition on $f$ is more restrictive than Peller’s condition that $f$ is a restriction of a function from the appropriate Besov space (see [@Pe] and [@APPS]).
We will start from the special case of the Cauchy-Green formula $$\label{599}g(\lambda)=-\frac{1}{\pi}\int_{\bc}\frac{{\overline{\partial}}g(\zeta)}{\zeta-\lambda}\, dm(\zeta),$$ which holds for a compactly supported differentiable function $g$ such that ${\overline{\partial}}g$ is bounded. Here $m$ denotes the planar Lebesgue measure and ${\overline{\partial}}g=(1/2)(\frac{\partial g}{\partial x}+i\frac{\partial g}{\partial y})$. (The proof in [@Ru 20.3] is valid for functions with the properties just stated.) We note that an operator calculus based on the Cauchy-Green formula was developed by Dynkin [@Dy], however we will need rather different results, specific to subnormal operators.
\[le610\]If $a\in{{\rm B}(\mathcal{H})}$ is a subnormal operator and $g:\bc\to\bc$ is a differentiable function with compact support such that ${\overline{\partial}}g$ is bounded and ${\overline{\partial}}g|\sigma(a)=0$, then $$\label{610}{\langle g(a)\eta,\xi\rangle}=-\frac{1}{\pi}\int_{\bc\setminus\sigma(a)}{\overline{\partial}}g(\zeta)
{\langle (\zeta1-a)^{-1}\eta,\xi\rangle}\, dm(\zeta),\ \ (\xi,\eta\in\h).$$
Let $c\in{{\rm B}(\mathcal{K})}$ be the minimal normal extension of $a$, $e$ the projection valued spectral measure of $c$, $K=\sigma(a)$ and $H=\{\zeta:\, {\overline{\partial}}g(\zeta)\ne0\}$. For $\eta\in\h$ and $\xi\in\k$ denote by $\mu$ the measure ${\langle e(\cdot)\eta,\xi\rangle}$. Then by the spectral theorem $g(c)=\int_Kg(\lambda)\, de(\lambda)$ and $(\zeta1-c)^{-1}=\int_K(\zeta-\lambda)^{-1}\,de(\lambda)$ for each $\zeta\in\bc\setminus K$ (in particular for $\zeta\in H$ since $H\cap K=0$ because of ${\overline{\partial}}|K=0$), hence by (\[599\]) $$\begin{aligned}
{\langle g(c)\eta,\xi\rangle}=\int_Kg(\lambda)\, d\mu(\lambda)=-\frac{1}{\pi}\int_K\int_H
{\overline{\partial}}g(\zeta)(\zeta-\lambda)^{-1}\, dm(\zeta)\,
d\mu(\lambda)\\=-\frac{1}{\pi}\int_H{\overline{\partial}}g(\zeta)\int_K
(\zeta-\lambda)^{-1}\, d\mu(\lambda)\, dm(\zeta)\\=-\frac{1}{\pi}\int_H
{\overline{\partial}}g(\zeta){\langle (\zeta1-c)^{-1}\eta,\xi\rangle}\,dm(\zeta)=-\frac{1}{\pi}\int_{\bc\setminus\sigma(a)}{\overline{\partial}}g(\zeta)
{\langle (\zeta1-a)^{-1}\eta,\xi\rangle}\,dm(\zeta).\end{aligned}$$ For all $\xi\in{\h^{\perp}}$ the last integrand is $0$ since $(\zeta1-a)^{-1}\eta\in\h$, hence $g(c)\eta\in\h$. Thus $\h$ is an invariant subspace for $g(c)$ and the usual definition of $g(a)$, namely $g(a):=g(c)|\h$ (see [@Co3 p. 85]), is compatible with (\[610\]). To justify the interchange of order of integration in the above computation, let $M=\sup_{\zeta\in\bc}|{\overline{\partial}}g(\zeta)|$ and let $R$ be a constant larger than the diameter of the set $H-K$, so that for each $\lambda\in K$ the disc $D(\lambda,R)$ with the center $\lambda$ and radius $R$ contains $H$. Introduce the polar coordinates by $\zeta=\lambda+re^{i\phi}$. Then by Fubini’s theorem $$\begin{aligned}
\int_H\int_K|{\overline{\partial}}g(\zeta)||\zeta-\lambda|^{-1}|\,d|\mu|(\lambda)\, dm(\zeta)
\leq M\int_H\int_K|\zeta-\lambda|^{-1}\, d|\mu|(\lambda)\, dm(\zeta)\\=
M\int_K\int_H|\zeta-\lambda|^{-1}dm\,(\zeta)\, d\mu(\lambda)
\leq M\int_K\int_{D(\lambda,R)}|\zeta-\lambda|^{-1}\,dm(\zeta)\, d|\mu|(\lambda)\\=M\int_K\int_0^{2\pi}\int_0^R\, dr\,d\phi\,
d|\mu|=2\pi MR|\mu|(K)<\infty.\end{aligned}$$
Now, if $a$ and $g$ are as in Lemma \[le610\] and if $b=g(a)$, we may compute formally for each $x\in{{\rm B}(\mathcal{H})}$ $$\begin{aligned}
=[g(a),x]=-\frac{1}{\pi}\int_{\bc\setminus\sigma(a)}{\overline{\partial}}g(\zeta)[(\zeta1-a)^{-1},x]\, dm(\zeta)\\=
-\frac{1}{\pi}\int_{\bc\setminus\sigma(a)}{\overline{\partial}}g(\zeta)(\zeta1-a)^{-1}[a,x](\zeta1-a)^{-1}\, dm(\zeta)=
[a,T_{a,f}(x)],\end{aligned}$$ where $$\label{611}T_{a,f}(x):=-\frac{1}{\pi}\int_{\bc\setminus\sigma(a)}{\overline{\partial}}g(\zeta)(\zeta1-a)^{-1}x(\zeta1-a)^{-1}\, dm(\zeta).$$ The problem here is, of course, the existence of the integral in (\[611\]). We have to show that the map $$\label{612}(\eta,\xi)\mapsto
-\frac{1}{\pi}\int_{\bc\setminus\sigma(a)}{\overline{\partial}}g(\zeta){\langle x(\zeta1-a)^{-1}\eta,(\overline{\zeta}1-a^*)^{-1}\xi\rangle}\, dm(\zeta)$$ is a bounded sesquilinear form on $\h$. The following lemma will be helpful.
\[le6111\]Let $a$, $g$ and $K:=\sigma(a)$ be as in Lemma \[le610\]. If $$\label{6111}\kappa:=\sup_{\lambda\in K}\int_{\bc\setminus K}|{\overline{\partial}}g||\zeta-\lambda|^{-2}\,dm(\zeta)
<\infty,$$ then the sesquilinear form defined by (\[612\]) is bounded by $\frac{2}{\pi}\|x\|\kappa$.
For any $t>0$, using first the Schwarz inequality and then the inequality $\alpha\beta\leq\frac{1}{2}(t^2\alpha^2+t^{-2}\beta^2)$ ($\alpha,\beta\geq0$) to estimate the inner product in the integral in (\[612\]), we see that the integral in (\[612\]) is dominated by $$\begin{aligned}
\|x\|\int_{K^c}|{\overline{\partial}}g(\zeta)|\|(\zeta1-a)^{-1}\eta\|\|(\overline{\zeta}1-a^*)^{-1}\xi\|\, dm(\zeta)\leq\\
\|x\|\frac{1}{2}[t^2\int_{K^c}|{\overline{\partial}}g(\zeta)|
\|(\zeta1-a)^{-1}\eta\|^2\, dm(\zeta)+t^{-2}\int_{K^c}|{\overline{\partial}}g(\zeta)|\|(\zeta1-a)^{-1}\xi\|^2\, dm(\zeta)].\end{aligned}$$ Using the notation from the proof of Lemma \[le610\] (with $\mu(\cdot):={\langle e(\cdot)\xi,\xi\rangle}$) and (\[6111\]), we have $$\begin{aligned}
\int_{\bc\setminus K}|{\overline{\partial}}g(\zeta)|\|(\zeta1-a)^{-1}\xi\|^2\, dm(\zeta)
=\int_H|{\overline{\partial}}g(\zeta)|\int_K|\zeta-\lambda|^{-2}\, d\mu(\lambda)\, dm(\zeta)\\
=\int_K\int_H|{\overline{\partial}}g(\zeta)||\zeta-\lambda|^{-2}\,dm(\zeta)\,d\mu(\lambda)\leq\kappa\mu(K)=\kappa\|\xi\|^2.\end{aligned}$$ Since a similar estimate holds with $\eta$ in place of $\xi$, it follows that $$\int_H|{\overline{\partial}}g(\zeta)|\|(\zeta1-a)^{-1}\eta\|\|(\overline{\zeta}1-a^*)^{-1}\xi\|\, dm(\zeta)\leq
\kappa(t^2\|\eta\|^2+t^{-2}\|\xi\|^2).$$ Taking the infimum over all $t>0$ we get $$\frac{1}{\pi}\int_H|{\overline{\partial}}g(\zeta)|\|t(\zeta1-a)^{-1}\eta\|\|t^{-1}(\overline{\zeta}1-a^*)^{-1}\xi\|\,
dm(\zeta)\leq\frac{2}{\pi}\kappa\|\eta\|\|\xi\|.$$
\[61111\]Lemma \[le6111\] applies, for example, if ${\overline{\partial}}g$ is a Lipschitz function of order $\alpha$, that is $|{\overline{\partial}}g(\zeta)-{\overline{\partial}}g(\zeta_0)|\leq\beta|\zeta-\zeta_0|^{\alpha}$ ($\zeta,\zeta_0\in\bc$) for some positive constants $\alpha$ and $\beta$, with ${\overline{\partial}}g|K=0$. In this case the integral (\[6111\]) may be estimated by noting that the Lipschitz condition (together with ${\overline{\partial}}g|K=0$) implies that $|{\overline{\partial}}g(\zeta)|\leq
\beta\delta(\zeta,K)^{\alpha}$, where $\delta(\zeta,K)$ is the distance from $\zeta$ to $K$. Let $R>0$ be so large that for each $\lambda\in K$ the closed dics $D(\lambda,R)$ with the center $\lambda$ and radius $R$ contains $H$. Introducing the polar coordinates by $\zeta=\lambda+re^{i\phi}$, for each $\lambda\in K$ we have $$\begin{aligned}
\int_{\bc\setminus K}|{\overline{\partial}}g(\zeta)||\zeta-\lambda|^{-2}\,dm(\zeta)
\leq\beta\int_H\frac{\delta(\zeta,K)^{\alpha}}{|\zeta-\lambda|^2}\, dm(\zeta)
\leq\beta\int_{D(\lambda,R)}|\zeta-\lambda|^{\alpha-2}\, dm(\zeta)\\
=2\pi\beta\alpha^{-1} R^{\alpha}.\end{aligned}$$
A function $f$ on a compact subset $K\subseteq\bc$ is in the [*class ${\rm L}(1+\alpha,K)$*]{} (where $\alpha\in(0,1]$) if the limit $$\label{63}f^{\prime}(\zeta_0)=\lim_{\zeta\to\zeta_0,\ \zeta\in\sigma(a)}\frac{f(\zeta)-f(\zeta_0)}{\zeta-\zeta_0}$$ exists for each (nonisolated) $\zeta_0\in K$ and if there exists a constant $\kappa>0$ such that $$\label{64}|f(\zeta)-f(\zeta_0)-f^{\prime}(\zeta_0)(\zeta-\zeta_0)|\leq\kappa|\zeta-\zeta_0|^{1+\alpha}$$ and $$\label{65}|f^{\prime}(\zeta)-f^{\prime}(\zeta_0)|\leq\kappa|\zeta-\zeta_0|^{\alpha}$$ for all $\zeta,\zeta_0\in K$.
We need the following consequence of the Whitney extension theorem.
\[le64\]Each $f\in{\rm L}(1+\alpha,K)$ can be extended to a continuously differentiable function $g$ with compact support such that ${\overline{\partial}}g$ is a Lipschitz function of order $\alpha$ and ${\overline{\partial}}g(\zeta)=0$ if $\zeta\in K$ (even though $K$ may have empty interior).
It suffices to extend $f$ to a differentiable function $g$ with Lipschitz ${\overline{\partial}}g$ and $\partial g$, for then we simply multiply $g$ by a smooth function with a compact support which is equal to $1$ on $K$. Let $\zeta=x+iy$, $f=f_1+if_2$ and $f^{\prime}(\zeta)=h_1(\zeta)+ih_2(\zeta)$, where $f_1, f_2$ and $h_1,h_2$ are real valued functions on $K$. It follows from (\[64\]) and (\[65\]) that for any $\zeta,\zeta_0\in K$ $$f_1(\zeta)=f_1(\zeta_0)+h_1(\zeta_0)(x-x_0)-h_2(\zeta_0)(y-y_0)+R(\zeta,\zeta_1)$$ and $$h_j(\zeta)=h_j(\zeta_0)+R_j(\zeta,\zeta_0)\ (j=1,2),$$ where $R$ and $R_j$ are function satisfying $|R(\zeta,\zeta_1)|\leq \kappa|\zeta-\zeta_0|^{1+\alpha}$ and $|R_j(\zeta,\zeta_0)|\leq\kappa|\zeta-\zeta_0|^{\alpha}$. By the Whitney extension theorem [@St p. 177] $f_1$ can be extended to a differentiable function $g_1$ on $\bc$ such that the partial derivatives of $g_1$ are Lipschitz of order $\alpha$ and $$\label{66}\frac{\partial g_1}{\partial x}=h_1,\ \ \frac{\partial g_1}{\partial y}=-h_2\ \mbox{on}\ K.$$ Similarly $f_2$ can be extended to an appropriate function $g_2$ such that $$\label{67}\frac{\partial g_2}{\partial x}=h_2,\ \ \frac{\partial g_2}{\partial y}=h_1\ \mbox{on}\ K.$$ Then $g:=g_1+ig_2$ is a required extension of $f$ since (\[66\]) and (\[67\]) imply that ${\overline{\partial}}g=0$ on $K$.
In all of the above discussion in this section we may replace $a$ by $a^{(\infty)}$ acting on $\h^{\infty}$, which implies that the map $T_{a,f}$ defined by (\[611\]) is completely bounded and, taking in (\[612\]) $\xi$ and $\eta$ to be in $\h^{\infty}$, we see that $$\label{107}{\langle T_{a,f}(x),\rho\rangle}=-\frac{1}{\pi}\int_{\bc\setminus\sigma(a)}{\overline{\partial}}g(\zeta)
{\langle x,(\zeta1-a)^{-1}\rho(\zeta1-a)^{-1}\rangle}\, dm(\zeta)={\langle x,(T_{a,f})_{\sharp}(\rho)\rangle}$$ for each $\rho=\eta\otimes\xi^*$ in the predual of ${{\rm B}(\mathcal{H})}$, where $$\begin{aligned}
(T_{a,f})_{\sharp}(\rho)=-\frac{1}{\pi}\int_{\bc\setminus\sigma(a)}{\overline{\partial}}g(\zeta)(\zeta1-a)^{-1}\rho(\zeta1-a)^{-1}\,
dm(\zeta)\\
=-\frac{1}{\pi}\int_{\bc\setminus\sigma(a)}{\overline{\partial}}g(\zeta)
(\zeta1-a)^{-1}\eta\otimes((\zeta1-a)^{-1})\xi)^*\, dm(\zeta).\end{aligned}$$ A similar computation as in the proof of Lemma \[le6111\] shows that the last integral exists and that $\|(T_{a,f})_{\sharp}(\rho)\|\leq{\rm const.}\|\rho\|$. Therefore we conclude that $T_{a,f}$ is weak\* continuous. Further, if $S$ is any weak\* continuous ${(a)^{\prime}}$-bimodule endomorphism of ${{\rm B}(\mathcal{H})}$, then $S$ commutes in particular with multiplications by $(\zeta1-a)^{-1}$ and using (\[107\]) it follows that $S$ commutes with $T_{a,f}$. Collecting all the above results, we have proved the following theorem.
\[th65\]For a subnormal operator $a\in{{\rm B}(\mathcal{H})}$ and a function $f\in{\rm L}(1+\alpha,\sigma(a))$ ($\alpha\in(0,1]$) the map $T_{a,f}$ defined by (\[611\]) is a central element in the algebra of all normal completely bounded ${(a)^{\prime}}$-bimodule endomorphisms of ${{\rm B}(\mathcal{H})}$ such that $[f(a),x]=[a,T_{a,f}(x)]=T_{a,f}([a,x])$ for all $x\in{{\rm B}(\mathcal{H})}$. (In particular the range of $d_{f(a)}$ is contained in the range of $d_a$ and (\[598\]) holds.)
Now we are going to show that if $\sigma(a)$ is nice enough, then the Lipschitz type condition on $f$ in Theorem \[th65\] can be relaxed: $f$ only needs to be a Schur function. First suppose that $\sigma(a)$ is the closed unit disc ${\overline{\bd}}$. For each $r\in(0,1)$ let $f_r(\zeta)=f(r\zeta)$. Thus each $f_r$ is a holomorphic function on a neighborhood $\Omega_r$ of ${\overline{\bd}}$ and $f_r(a)$ can be expressed as $f(a)=\frac{1}{2\pi i}\int_{\Gamma_r}f_r(\zeta)(\zeta1-a)^{-1}\,
d\zeta$, where $\Gamma_r$ is a contour in $\Omega_r$ surrounding $\sigma(a)$ once in a positive direction. Then for each $x\in{{\rm B}(\mathcal{H})}$ we have $$\label{100}[f_r(a),x]=\frac{1}{2\pi i}\int_{\Gamma_r}f(r\zeta)[(\zeta-a)^{-1},x]\, d\zeta=
\frac{1}{2\pi i}\int_{\Gamma_r}f(r\zeta)(\zeta-a)^{-1}[a,x](\zeta-a)^{-1}\, d\zeta$$ and also $$\label{101}[f_r(a),x]=[a,T_r(x)],\ \mbox{where}\ T_r(x)=\frac{1}{2\pi i}\int_{\Gamma_r}f(r\zeta)
(\zeta-a)^{-1}x(\zeta-a)^{-1}\,d\zeta.$$ If the set of completely bounded maps $T_r$ on ${{\rm B}(\mathcal{H})}$ ($0<r<1$) is bounded, then it has a limit point, say $T$, in the weak\* topology (which the space of all completely bounded maps on ${{\rm B}(\mathcal{H})}$ carries as a dual space, see e.g. [@BLM 1.5.14 (4)]). $T$ commutes with left and right multiplications by elements of ${(a)^{\prime}}$ (since all $T_r$ do). From (\[100\]) and (\[101\]) we see that $$[f(a),x]=T([a,x])=[a,Tx]\ \ (x\in{{\rm B}(\mathcal{H})}).$$ The same holds for $a^{(n)}$ in place of $a$ and $x\in{{\rm M}_n({{\rm B}(\mathcal{H})})}$, hence in particular (\[598\]) holds. To estimate the norms of the maps $T_r$, let $c$ on $\k\supseteq\h$ be the unitary power dilation of $a$ (so that $a^n=pc^n|\h$ for all $n\in\bn$, where $p$ is the orthogonal projection from $\k$ onto $\h$, see e.g. [@Hal] or [@Pa]). Let $S_r$ be the map on ${{\rm B}(\mathcal{K})}$ defined in the same way as $T_r$, that is, in the second formula in (\[101\]) we replace $a$ by $c$. Then (since $f_r$ can be approximated uniformly by polynomials) $T_r(x)=pS_r(x)|\h$ for each $x\in{{\rm B}(\mathcal{H})}$, where $x$ is regarded as an operator on $\k$ by setting $x|\h^{\perp}=0$. Hence $\|T_r\|\leq
\|S_r\|$. In the special case when $c$ is diagonal (relative to some orthonormal basis of $\k$) with eigenvalues $\lambda_i$ and $x=[x_{i,j}]$, a simple computation shows that $S_r(x)$ is represented by the matrix $[\frac{f(\lambda_i)-f(\lambda_j)}{\lambda_i-\lambda_j}x_{i,j}]$ (where the quotient is taken to be $0$ if $\lambda_i=
\lambda_j$). Hence in this case $\|S_r\|\leq\kappa$ since $f$ is a Schur function. Since any normal perator $c$ can be approximated uniformly by diagonal operators, the same estimate must hold for all such $c$ with $\sigma(c)\subseteq\bd$. A similar reasoning applies also to the completely bounded norm, hence it follows that $\sup_{0<r<1}\|T_r\|_{\rm cb}<
\infty$.
Let us now consider the case when $\sigma(a)$ is the closure of its interior $U$ and $U$ is simply connected. Let $h$ be a conformal map from $\bd$ onto $U$. If the boundary $\partial{\sigma(a)}$ of ${\sigma(a)}$ is sufficiently nice, say a Jordan curve of class $C^3$, then $h$ can be extended to a bijection, denoted again by $h$, from ${\overline{\bd}}$ onto ${\overline{U}}=\sigma(a)$, such that $h$ and $h^{-1}$ are in the class $C^2$ [@Kr 5.2.4]. Then by Theorem \[th65\] and Lemma \[le521\] $h$ and $h^{-1}$ are Schur functions. Let $a_0=h^{-1}(a)$. Note that $\{a_0,1\}$ generates the same Banach subalgebra as $\{a,1\}$ since $h$ and $h^{-1}$ can both be uniformly approximated by polynomials (by Mergelyan’s theorem). For any Schur function $f$ on $\sigma(a)$ the composition $f_0:=f\circ h$ is a Schur function on ${\overline{\bd}}$. (To see this, note that for any $\lambda\ne\mu$ in ${\overline{\bd}}$ we may write $\frac{f(h(\lambda))-f(h(\mu))}{\lambda-\mu}=\frac{f(h(\lambda))-f(h(\mu))}{h(\lambda)-h(\mu)}
\frac{h(\lambda)-h(\mu)}{\lambda-\mu}$ if $h(\lambda)\ne h(\mu)$ and that the inequality $\|[x_{i,j}y_{i,j}]\|_S\leq\|[x_{i,j}]\|_S\|[y_{i,j}]\|_S$ holds for the Shur norm of the Schur product of two matrices.) Note that $f(a)=f_0(a_0)$ and ${(a_0)^{\prime}}={(a)^{\prime}}$. By the previous paragraph there exists a completely bounded ${(a_0)^{\prime}}$-bimodule map $T$ on ${{\rm B}(\mathcal{H})}$ such that $$\label{102}[f_0(a_0),x]=[a_0,Tx]=T([a_0,x])\ \ \mbox{for all}\ x\in{{\rm B}(\mathcal{H})},$$ hence $[f(a),x]=[h^{-1}(a),Tx]=T([h^{-1}(a),x])$. Now the map $T$ is not a priori normal, but it can be replaced by its normal part $T_n$ in (\[102\]), hence we may achieve that $T$ is normal. By Theorem \[th65\] there exists a completely bounded ${(a)^{\prime}}$-bimodule map $S$ on ${{\rm B}(\mathcal{H})}$ such that $[h^{-1}(a),y]=[a,Sy]=S([a,y])$ for all $y\in{{\rm B}(\mathcal{H})}$ and $S$ commutes with all normal ${(a)^{\prime}}$-bimodule maps on ${{\rm B}(\mathcal{H})}$ (in particular with $T$). Hence we have now $[f(a),x]=[h^{-1}(a),Tx]=[a,STx]$ and $[f(a),x]=T([h^{-1}(a),x])=TS([a,x])$ for all $x\in{{\rm B}(\mathcal{H})}$. Denoting $T_{a,f}=TS=ST$, we have deduced the following theorem.
\[th621\]For a subnormal $a\in{{\rm B}(\mathcal{H})}$ suppose that $\sigma(a)$ is the closure of a simply connected domain bounded by a Jordan curve of class $C^3$. Then for each Schur function $f$ on $\sigma(a)$ there exists a (normal) completely bounded ${(a)^{\prime}}$-bimodule map $T_{a,f}$ on ${{\rm B}(\mathcal{H})}$ such that $[f(a),x]=[a,T_{a,f}(x)]=T_{a,f}([a,x])$ for all $x\in{{\rm B}(\mathcal{H})}$.
In general, the Lipschitz type condition in Theorem \[th65\] can be replaced by a similar, but less restrictive condition, which involves a regular modulus of continuity $\omega$ in the sense of [@St p. 175] (instead of just $\omega(t)=t^{\alpha}$) such that $\int_0^1\omega(r)/r\, dr<\infty$. (There exists an appropriate version of Whitney’s extension theorem [@St p. 194].) But probably even this is to restrictive, for we do not need any requirements about $\partial g$ of the extension $g$ (only requirements about ${\overline{\partial}}g$). Thus, at least in cases when $K$ has a nice boundary, it is worthwhile to try a more direct way of extending functions. We will only consider the case $K\subseteq\br$, from which the result can be generalized to sets bounded by sufficiently regular Jordan curves.
\[le81\]If $f$ is a continuously differentiable function on $\br$ with compact support such that $$\label{106}\sup_{\lambda\in\br}\int_{\br}\int_{0}^1|f^{\prime}(x+\lambda)-f^{\prime}(x+\lambda-h)|
\frac{{\rm ln}(1+\frac{x^2}{h^2})}{x^2}\,dh\,dx<\infty,$$ then $f$ can be extended to a differentiable function $g$ on $\bc$ with a compact support such that ${\overline{\partial}}g$ is bounded, ${\overline{\partial}}g|\br=0$ and $$\sup_{\lambda\in \br}\int_{\bc}|{\overline{\partial}}g(\zeta)||\zeta-\lambda|^{-2}\,dm(\zeta)<\infty.$$ The condition (\[106\]) holds in particular if $$\label{654}\int_{0}^1\frac{\|\Delta_hf^{\prime}\|_{\infty}}{h}\,dh<\infty,$$ where $$\|\Delta_hf^{\prime}\|_{\infty}=\sup_{x\in\br}|\Delta_hf^{\prime}|\ \mbox{and}\
(\Delta_hf^{\prime})(x)=f^{\prime}(x-h)-f^{\prime}(x).$$
An extension $g$ of $f$ satisfying ${\overline{\partial}}g|\br=0$ is given by $g(\zeta)=f(x)+iyf^{\prime}(x)$, where $\zeta=x+iy$. However, for this $g$ to be differentiable we must assume that $f$ is twice differentiable. To avoid this additional assumption on $f$, let $\phi:\br\to\br$ be a smooth function with support in $[-1,1]$ such that $\int_{\br} \phi(x)\, dx=1$, $\phi_y(x):=y^{-1}\phi(y^{-1}x)$, and let $$\label{68}g_0(\zeta)=\left\{\begin{array}{ll}f(x)+iy(\phi_{|y|}*f^{\prime})(x),&\mbox{if}\ y\ne0\\
f(x),&\mbox{if}\ y=0.
\end{array}\right.$$ Since $f^{\prime}$ is continuous, $\phi_{|y|}*f^{\prime}$ converges to $f^{\prime}$ pointwise and it follows by a simple computation that ${\overline{\partial}}g_0|\br=0$. If $y>0$, $$g_0(\zeta)=f(x)+i\int_{\br}\phi(\frac{x-t}{y})f^{\prime}(t)\, dt$$ and $$\begin{aligned}
2{\overline{\partial}}g_0(\zeta)&=\frac{\partial g_0}{\partial x}(x,y)+i\frac{\partial g_0}{\partial y}(x,y)\\
&=f^{\prime}(x)+\frac{i}{y}\int_{\br}\phi^{\prime}(\frac{x-t}{y})f^{\prime}(t)\, dt+\frac{1}{y}\int_{\br}\phi^{\prime}
(\frac{x-t}{y})\frac{x-t}{y}f^{\prime}(t)\, dt\\
&=f^{\prime}(x)+i\int_{\br}\phi^{\prime}(s)f^{\prime}(x-sy)\, ds+\int_{\br}\phi^{\prime}(s)sf^{\prime}(x-sy)\, ds.\end{aligned}$$ Denoting $\psi(s)=s\phi(s)$ and noting that $\int_{\br}\phi^{\prime}(s)\,ds=0=\int_{\br}\psi^{\prime}(s)\,ds$ (since $\phi$ and $\psi$ have compact support) and $\int_{\br}\phi(s)\,ds=1$, $2{\overline{\partial}}g_0(\zeta)$ can be rewritten as $$\begin{aligned}
2{\overline{\partial}}g_0(\zeta)&=i\int_{\br}\phi^{\prime}(s)f^{\prime}(x-sy)\, ds+f^{\prime}(x)+\int_{\br}(\psi^{\prime}
(s)-\phi(s))f^{\prime}(x-sy)\,ds\\
&=i\int_{\br}\phi^{\prime}(s)(f^{\prime}(x-sy)-f^{\prime}(x))\,ds-\int_{\br}\phi(s)(f^{\prime}(x-sy)-f^{\prime}(x))\,ds\\
&+\int_{\br}\psi^{\prime}(s)(f^{\prime}(x-sy)-f^{\prime}(x))\,ds.\end{aligned}$$ A similar computation is possible also for $y<0$ and we thus obtain $$\label{69}2{\overline{\partial}}g_0(\zeta)=\int_{-1}^1\theta(s)(f^{\prime}(x-s|y|)-f^{\prime}(x))\,ds \ (y\ne0),$$ where $\theta(s):=(i+s)\phi^{\prime}(s)$. Let $\delta>0$ and $\chi:\bc\to[0,1]$ a smooth function depending only on the variable $y$ which is equal to $1$ on the strip $-\delta\leq y\leq\delta$, and equal to $0$ if $|y|\geq1$. Set $g=\chi g_0$. Then $g$ has a compact support, say $H$, and $$\label{90}\begin{array}{l}\int_H|{\overline{\partial}}g(\zeta)||\zeta-\lambda|^{-2}\,dm(\zeta)=\\
\int_H|{\overline{\partial}}\chi(\zeta)||g_0(\zeta)|
|\zeta-\lambda|^{-2}\,dm(\zeta)+\int_H|\chi(\zeta)||{\overline{\partial}}g_0(\zeta)||\zeta-\lambda|^{-2}\,dm(\zeta).\end{array}$$ Since ${\overline{\partial}}\chi=0$ on a neighborhood of the real line, the first integral on the right side in (\[90\]) is bounded uniformly for $\lambda\in\br$. Since the support of $\chi$ is contained in $\br\times[-1,1]$, the second integral on the right side of (\[90\]) is dominated by $$\int_{\br\times[-1,1]}|{\overline{\partial}}g(\zeta+\lambda)||\zeta|^{-2}\,dm(\zeta).$$ Using (\[69\]) (and the boundedness of $\theta$) we see that the last integral is dominated by a constant multiple of $$\begin{aligned}
\int_{\br}\int_{0}^1\int_{-1}^1\frac{|f^{\prime}(x+\lambda))-f^{\prime}(x+\lambda-s|y|)|}{x^2+y^2}\,ds\, dy\,dx.\end{aligned}$$ Interchanging the order of integration over $y$ and $s$ and introducing new variable $h=sy$ instead of $y$, the last expression transforms into $$\label{652}\int_{\br}\int_{-1}^1\int_{0}^{|s|}
\frac{|f^{\prime}(x+\lambda)-f^{\prime}(x+\lambda-h)|}{x^2s^2+h^2}|s|\,
dh\,ds\,dx.$$ Interchanging the order of integration over $s$ and $h$ and noting that $$\int_{|h|}^{1}\frac{s}{s^2x^2+h^2}\,ds=\frac{{\rm ln}(1+\frac{x^2}{h^2})}{2x^2}-\frac{{\rm ln}(x^2+1)}{2x^2},$$ we transform (\[652\]) into $$\label{653}\begin{array}{r}2\int_{\br}\int_{0}^1|f^{\prime}(x+\lambda)-f^{\prime}(x+\lambda-h)|
\frac{{\rm ln}(1+\frac{x^2}{h^2})}{x^2}\,dh\,dx\\
-2\int_{\br}\int_{0}^1|f^{\prime}(x+\lambda)-f^{\prime}(x+\lambda-h)|
\frac{{\rm ln}(1+x^2)}{x^2}\,dh\,dx.\end{array}$$ Since the function $x^{-2}{\rm ln}(1+x^2)$ is bounded (by $1$) and $f$ has compact support, the second double integral in (\[653\]) is bounded uniformly in $\lambda$. The first double integral in (\[653\]) is dominated by $$\int_{0}^1\|\Delta_hf^{\prime}\|_{\infty}\int_{\br}x^{-2}{\rm ln}(1+\frac{x^2}{h^2}))\,dx
=\pi\int_{0}^1\frac{\|\Delta_hf^{\prime}\|_{\infty}}{|h|}\,dh=2\pi\int_0^1\frac{\|\Delta_hf^{\prime}\|_{\infty}}{h}\,dh.$$
We note that the continuity of $f^{\prime}$ in Proposition \[le81\] is not essential since the area of $\br$ is $0$ and in the proof of Lemma \[le610\] it suffices to require that ${\overline{\partial}}|\sigma(a)=0$ $m$-almost everywhere. (Also (\[599\]) still holds if $g$ fails to be differentiable on one line only.) Further, the condition (\[654\]) for a compactly supported function $f$ implies that $\int_0^1\frac{\|\Delta_h^2f\|_{\infty}}{h^2}\,dh<\infty$ (to see this, apply Lagrange’s theorem to the function $\Delta_hf$), which means that $f$ is in the Besov space $B^1_{\infty,1}(\br)$. Since for selfadjoint $a$ the condition $f\in B^1_{\infty,1}(\br)$ suffices for $\|[f(a),x]\|\leq\kappa
\|[a,x]\|$ ($x\in{{\rm B}(\mathcal{H})}_h$) by [@Pe], it would be interesting to know if the condition (\[106\]) allows compactly supported functions which are not in $B^1_{\infty,1}(\br)$.
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[^1]: Acknowledgment. I am grateful to to Miran Černe for discussions concerning complex anaysis topics, to Matej Brešar and Špela Špenko for conversations from which some of the questions studied in this paper have emerged, and to Victor Shulman for the correspondence concerning a question about Besov spaces.
[^2]: The author was supported in part by the Ministry of Science and Education of Slovenia.
|
{
"pile_set_name": "ArXiv"
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= 1.5ex ‘=11
Marco Matone
Dipartimento di Fisica “G. Galilei”
Istituto Nazionale di Fisica Nucleare
Università di Padova, Via Marzolo, 8 – 35131 Padova, Italy
We show that a suitable rescaling of the matrix model coupling constant makes manifest the duality group of the $N=2$ SYM theory with gauge group $SU(2)$. This is done by first identifying the possible modifications of the SYM moduli preserving the monodromy group. Then we show that in matrix models there is a simple rescaling of the pair $(S_D,S)$ which makes them dual variables with $\Gamma(2)$ monodromy. We then show that, thanks to a crucial scaling property of the free energy derived perturbatively by Dijkgraaf, Gukov, Kazakov and Vafa, this redefinition corresponds to a rescaling of the free energy which in turn fixes the rescaling of the coupling constant. Next, we show that in terms of the rescaled free energy one obtains a nonperturbative relation which is the matrix model counterpart of the relation between the $u$–modulus and the prepotential of $N=2$ SYM. This suggests considering a dual formulation of the matrix model in which the expansion of the prepotential in the strong coupling region, whose QFT derivation is still unknown, should follow from perturbation theory. The investigation concerns the $SU(2)$ gauge group and can be generalized to higher rank groups.
Recently Dijkgraaf and Vafa derived crucial relations between matrix models and SYM theories [@1; @2; @3]. Subsequently, in [@DGKV] Dijkgraaf, Gukov, Kazakov and Vafa provided the explicit relationship between the $N=2$ SYM theory [@SW] and matrix models. The original proposal was based on geometrical engineering analysis in string theory, while in [@n; @n+1] it has been argued that there exists a QFT proof of the Dijkgraaf–Vafa formulation. In these derivations a crucial role is played by holomorphy. This is a crucial issue as, for example, holomorphy and symmetries are at the basis of $N=2$ SYM duality. Therefore, a basic question in considering the matrix model formulation, is to identify the duality structure which is the essence of Seiberg–Witten theory [@SW]. There are several reasons which suggest introducing the powerful tool of duality directly in the matrix model formulation. For example, an interesting question would be to understand the analogous of the nonperturbative relation between the $u$–modulus and the prepotential [@rela]. This should be useful for a proof of the relationship between matrix model and $N=2$ SYM along the lines of [@proof]. We also note that this relation, which has been useful in investigating related issues [@rela2], should help in deriving possible exact results in matrix models. Furthermore, this duality may help in understanding what is the QFT formulation of $N=2$ SYM in the strong coupling region. In this context, one should expect that the expansion of the $N=2$ SYM prepotential in the strong coupling region should be obtained by means of a perturbative calculation in a dual matrix model formulation.
The aim of this paper is to introduce such a duality in matrix models. We will start by showing that, on general grounds, in order to preserve the Seiberg–Witten duality, only a class of redefinitions of the moduli $(a_D,a)$ is allowed. This is based on a mathematical general observation which involves the Picard–Fuchs equation.[^1] In particular, it is shown that if $$\tau(a)={\partial{\cal S}_D\over\partial{\cal S}}={\partial
a_D\over\partial a},$$ with $\tau$ the $N=2$ effective coupling constant, then $({\cal
S}_D,{\cal S})$ have the same monodromy of $(a_D,a)$ on the $u$–plane if $${\cal S}_D=fa_D+4(u^2-\Lambda_{SW}^4)f'a_D' ,\qquad {\cal
S}=fa+4(u^2-\Lambda_{SW}^4)f'a', \label{introsx}$$ with $f$ an arbitrary singlevalued function of $u$ (note that a possible additional $Z_2$ monodromy leaves $\tau$ invariant).
Next, we identify the explicit relationship between the matrix model variables $(S_D,S)$ and $(a_D,a)$. It turns out that $(S_D,S)$ cannot have $\Gamma(2)$–monodromy. Nevertheless, remarkably, the simple rescaling $${\cal S}=\left({\Lambda_{SW}\over 2^{3/2} u^{1/2}}\right)^3S,$$ restores duality, that is $${\cal S}={\Lambda_{SW}^{3}\over 3\cdot
2^6}\left[ u^{-1/2}a-2(u^2-\Lambda_{SW}^4)u^{-3/2}a'\right],$$ satisfies (\[introsx\]) with $f={\Lambda_{SW}^3\over \sqrt 2
\cdot 48} u^{-1/2}$. On the other hand, this fixes $S^D$ to be $${\cal S}_D={\Lambda_{SW}^3\over3\cdot 2^6}\left[
u^{-1/2}a_D-2(u^2-\Lambda_{SW}^4)u^{-3/2}a_D'\right],$$ which, in turn, defines ${\cal F}_0$ by $${\cal S}_D={\partial {\cal F}_0\over \partial {\cal S}}.$$ It then follows that the new pair has $\Gamma(2)$ monodromy $$\left(\begin{array}{c} \tilde {\cal S}_D\\ \tilde {\cal S}
\end{array}\right)= \left(\begin{array}{c}A\\C
\end{array}\begin{array}{cc}B\\D\end{array}\right)\left(\begin{array}{c}
{\cal S}_D\\ {\cal S}
\end{array}\right).
\label{2x}$$ We then show that thanks to a remarkable scaling property of the genus zero free energy ${\cal F}_{0}(
S_k,\Delta,\Lambda)$, passing to the new variables is equivalent to a simple rescaling, that is the change of variables ($\Lambda=2^{-1/2}\Lambda_{SW}$) $$S_k\longrightarrow {\cal S}_k=\left({\Lambda\over \Delta}\right)^3
S_k, \qquad \Delta\longrightarrow {\Lambda\over \Delta}
\Delta=\Lambda, \qquad \Lambda\longrightarrow {\Lambda\over
\Delta}\Lambda= {\Lambda^2\over \Delta},$$ induces the scaling transformation $${\cal F}_{0}\longrightarrow {\cal F}_0\left({\cal
S}_k,\Lambda,{\Lambda^2\over\Delta}\right)=
\left({\Lambda\over\Delta}\right)^6{\cal F}_{0}(
S_k,\Delta,\Lambda),$$ which has no effect in deriving the critical values since the factor $\left({\Lambda\over\Delta}\right)^6$ cancels the one from the Jacobian in $\tau_{ij}$, that is $\partial^2\mu^6{\cal
F}_{0}/\partial {\cal S}_i\partial {\cal S}_j=\partial^2{\cal
F}_{0}/\partial S_i\partial S_j$. As a result, even if the partition function remains invariant, we have the same rescaling for both the potential and the matrix coupling constant $$g_S\longrightarrow g_{\cal S}=\left({\Lambda\over\Delta}\right)^3
g_S,$$ $$W\longrightarrow{\cal
W}(\Phi)=\left({\Lambda\over\Delta}\right)^3 W(\Phi).$$ As a consequence the scaling generalizes to $${\cal F}_{g}\longrightarrow {\cal F}_g\left({\cal
S}_k,\Lambda,{\Lambda^2\over\Delta}\right)=
\left({\Lambda\over\Delta}\right)^{3(2-2g)}
{\cal F}_{g}( S_k,\Delta,\Lambda).$$ We then show that the new prepotential, which is obtained by integrating with respect to ${\cal S}$, the function $\tau$ at the extremum, satisfies the nonperturbative relation $$\left({\Lambda\over\Delta}\right)^4={48\pi
i\over\Lambda^6}\left({\cal F}_0-{{\cal S}\over 2}{\partial {\cal
F}_0\over\partial {\cal S}}\right).$$ Introducing duality then leads to consider a dual formulation of the matrix model that we propose should correspond to introduce the Legendre transform of the free energy $${\cal F}_{Dg}={\cal
F}_g-\sum_{i=1,2}{\cal S}_i{\partial {\cal F}_g\over\partial {\cal
S}_i},$$ where now ${\cal F}_g\equiv{\cal F}_g\left({\cal
S}_k,\Lambda,{\Lambda^2\over\Delta}\right)$.
Let us start by recalling that in matrix model the effective coupling constant of $N=2$ SYM has the form [@DGKV] $$\tau(a)={\partial^2 {\cal F}_0(S)\over \partial S^2},
\label{1}$$ which should be compared with $$\tau(a)={\partial^2 {\cal F}(a)\over \partial a^2}.
\label{2}$$ The problem is to find the relationship between ${\cal F}_0(S)$ and ${\cal F}(a)$. Let us introduce the dual $$S_D={\partial {\cal F}_0(S)\over \partial S},
\label{3}$$ so that $$\tau(a)={\partial_u S_D\over \partial_u S}={\partial_u a_D\over
\partial_u a}. \label{5}$$ It is clear that the dual pairs $(S_D,S)$ and $(a_D,a)$ should have the same monodromy on the $u$–plane. As observed in [@BIM] in considering a similar problem, we may use the differential equation [@EQ; @rela] $$\left(\partial_u^2+\frac{1}{4(u^2-\Lambda_{SW}^4)}\right)
\pmatrix{a_D\cr a}=0, \label{embhe}$$ to investigate the structure of the possible solutions of (\[5\]). Generalizing the analysis in [@BIM] we set[^2] ($'\equiv\partial_u$) $${\cal S}_D=f_Da_D+g_Da_D' ,\qquad {\cal S}=fa+ga' ,
\label{wodihw}$$ where the two dual pairs $(f_D,f)$ and $(g_D,g)$ are functions of $u$. Note that if these functions are singlevalued with respect to $u$, then $({\cal S}_D,{\cal S})$ would have the $\Gamma(2)$ monodromy of $(a_D,a)$. However, since a possible additional $Z_2$ monodromy of $({\cal S}_D,{\cal S})$ with respect to $(a_D,a)$ does not change the polymorphicity of ${\cal S}_D'/{\cal S}'$, the functions $(f_D,f)$ and $(g_D,g)$ should be singlevalued on the $u$–space except for a possible minus sign they may get winding around some point.
We now show that if the functions $(f_D,f)$ and $(g_D,g)$ solve a differential equation, then Eq.(\[5\]) is satisfied. By (\[embhe\]) and (\[wodihw\]) we have $${\cal S}_D'=\tilde f_Da_D+\tilde g_Da_D',\qquad {\cal S}'= \tilde
fa+\tilde ga',$$ where $$\tilde f_D=f_D'-\frac{1}{4(u^2-\Lambda_{SW}^4)}g_D, \qquad \tilde
g_D=f_D+g_D',$$ $$\tilde f=f'-\frac{1}{4(u^2-\Lambda_{SW}^4)}g, \qquad \tilde
g=f+g'.$$ Imposing $\tilde f_D=0=\tilde f$ $$g_D=4(u^2-\Lambda_{SW}^4)f_D',\qquad g=4(u^2-\Lambda_{SW}^4)f',
\label{eiuoh}$$ and $\tilde g_D=\tilde g$, that is $$\tilde g_D=f_D+8uf_D'+4(u^2-\Lambda_{SW}^4)f_D''=
f+8uf'+4(u^2-\Lambda_{SW}^4)f''=\tilde g,
\label{oiuce2}$$ we obtain $${\cal S}_D'=h a_D',\qquad {\cal S}'=h a',$$ where $h\equiv\tilde
g_D=\tilde g$. Since $f_D$ and $f$ satisfy the same differential equation (\[oiuce2\]), it follows that once either $f_D$ or $f$ is given, say $f$, besides the choice $f_D=f$ (which would imply $g_D=g$), one can also choose $f_D$ to be any other solution of (\[oiuce2\]). Summarizing, from (\[wodihw\]) and (\[eiuoh\]) we have $${\cal S}_D=f_Da_D+4(u^2-\Lambda_{SW}^4)f_D'a_D' ,\qquad {\cal
S}=fa+4(u^2-\Lambda_{SW}^4)f'a', \label{wodihwn}$$ and ${\cal S}_D'/{\cal S}'=a_D'/a' =\tau$.
Let us start considering the relationship between the $N=2$ SYM and matrix model variables. We first set $$\Lambda=2^{-1/2}\Lambda_{SW},\qquad \Delta^2=4u,$$ in the loop expansion of $S$ [@DGKV] $${S\over 2^3 u^{3/2}}={1\over 2^6}\left({\Lambda_{SW}^2\over u
}\right)^2+ {3\over2^{11}} \left({\Lambda_{SW}^2\over
u}\right)^4+{35\over 2^{16}} \left({\Lambda_{SW}^2\over u
}\right)^6+\ldots . \label{laserie}$$ We now show that rather than $S$ itself, it is the right hand side of (\[laserie\]) that matches with the expansion of ${\cal S}$ in (\[wodihwn\]) with $$f={1\over \sqrt 2 \cdot 48} u^{-1/2}.
\label{effediu}$$ Therefore, while ${S\over 2^3
u^{3/2}}$ is of the form that preserves duality, this is not the case for $S$ itself. As we will see, this will lead to a natural rescaling of the matrix model coupling constant which will make Seiberg–Witten duality manifest. In particular, we will see that one has to rescale $S$ to $$\left({\Lambda_{SW}\over 2^{3/2}u^{1/2}}\right)^3S
={\Lambda_{SW}^3u^{-3/2}\over3\cdot 2^6}\left[
ua-2(u^2-\Lambda_{SW}^4)a'\right].
\label{almostesse}$$ In order to compare (\[laserie\]) and (\[almostesse\]) we expand $a$ for $u\to\infty$ $$a(u)={\sqrt 2\over \pi}\int_{-\Lambda_{SW}^2}^{\Lambda_{SW}^2}
dx{\sqrt{x-u}\over \sqrt{x^2-\Lambda_{SW}^4}}$$ $$=\sqrt {2u}\left(1- {1\over 2^4}\left({\Lambda_{SW}^2\over
u}\right)^2-{15\over2^{10}} \left({\Lambda_{SW}^2\over
u}\right)^4-{105\over2^{14}} \left({\Lambda_{SW}^2\over
u}\right)^6 +\ldots \right),$$ that substituted in (\[almostesse\]) exactly reproduces (\[laserie\]). Substituting (\[almostesse\]) in (\[5\]) and using $$S'={1\over \sqrt2 \cdot
4}(a-2ua'),\label{Sprimo}$$ we see that the relation between $(S_D',S')$ and $(a_D',a')$ is rather involved $$S_D'={1\over \sqrt 2\cdot 4}(a-2ua'){a_D'\over a'}.$$ This is not only a formal question since $S_D'$ and $S'$ cannot have simultaneously $\Gamma(2)$ monodromy. Even if this is implicit in the above construction, it is instructive to illustrate it explicitly. In particular, if $S$ has $\Gamma(2)$ monodromy, this cannot be the case for $S_D$. Since the monodromy commutes with the derivative, we show this for $S'_D$ and $S'$. Under the action of $\Gamma(2)$ we have $$S'\longrightarrow \gamma(S')={1\over \sqrt 2\cdot 4}C(a_D-2ua_D')+
{1\over \sqrt 2\cdot 4}D(a-2ua'),$$ so $S'$ has $\Gamma(2)$ monodromy iff we consider as its dual $$\hat S_D'={1\over \sqrt 2\cdot 4}(a_D-2ua_D')\neq S_D',$$ so that $$\gamma(S')=C \hat S_D'+D S'.$$ Of course, as follows by the previous analysis, even if $\hat
S_D'$ and $S'$ have $\Gamma(2)$ monodromy, their ratio cannot correspond to $\tau$.
A similar reasoning holds for $S_D'$. Actually, since under $\Gamma(2)$ $${A\tau+B\over C\tau+D}={AS_D'+BS'\over
CS_D'+DS'}={\gamma(S_D')\over \gamma(S')},$$ we see that $$\gamma(S'_D)={(AS_D'+BS')(C\hat S_D'+DS')\over CS_D'+DS'},$$ which cannot correspond to the $\Gamma(2)$ monodromy, that is $$\gamma(S_D')\neq AS'_D+BS'.$$ Note that $$S_D={1\over \sqrt2\cdot 2^2}\int^u_{u_0} d\tilde u (\tau a
-2\tilde u\partial_{\tilde u}a_D)+S_D(u_0),\qquad S= {1\over
\sqrt2\cdot 6} (ua-2(u^2-\Lambda_{SW}^4)a'),
\label{oiiuxh}$$ and by (\[almostesse\]) and (\[Sprimo\]) $$a={\sqrt2\cdot 2\over\Lambda_{SW}^4}
[3uS-2(u^2-\Lambda_{SW}^4)S'].$$ The fact that the Seiberg–Witten duality is not manifest with the pair $(S_D,S)$ can be also seen by noticing that $S$ solves the differential equation $$\left(\partial_u^2-{3\over 4(u^2-\Lambda^4_{SW})}\right)S=0,
\label{odcwo}$$ which is not satisfied by $S_D$, indicating once again that they cannot have the same monodromy on the $u$–plane. Inverting Eq.(\[odcwo\]) we obtain $$4\left({\cal G}^2-\Lambda_{SW}^4\right){\partial^2 {\cal G}\over
\partial S^2}+3{\cal S}\left({\partial {\cal G}\over\partial
S}\right)^3=0, \label{oidhw}$$ where $$u={\cal G}(S).$$
To select a dual pair with $\Gamma(2)$ monodromy and whose ratio corresponds to $\tau$ is essential to recognize the underlying geometry of $N=2$ SYM. In particular, winding around the $u$–moduli space, the pair $(S_D,S)$ will not preserve the analogous relations satisfied by $(a_D,a)$. In order to restore manifest duality we rescale $S$ and define $${\cal S}=\left({\Lambda_{SW}\over 2^{3/2} u^{1/2}}\right)^3S,
\label{riscalato}$$ that is $${\cal S}={\Lambda_{SW}^{3}\over3\cdot 2^6}\left[
u^{-1/2}a-2(u^2-\Lambda_{SW}^4)u^{-3/2}a'\right],
\label{essecorsiva}$$ where the term $\Lambda_{SW}^3$ has been introduced to make $S$ and ${\cal S}$ of the same dimension. We now choose $f_D=f$, so that by (\[wodihwn\]) $${\cal S}_D={\Lambda_{SW}^3\over3\cdot 2^6}\left[
u^{-1/2}a_D-2(u^2-\Lambda_{SW}^4)u^{-3/2}a_D'\right],
\label{essecorsivaduale}$$ which, in turn, defines ${\cal F}_0$ by $${\cal S}_D={\partial {\cal F}_0\over \partial {\cal S}}.$$ By construction the pair $({\cal S}_D,{\cal S})$ has the same monodromy of $(a_D,a)$ on the $u$–plane except for a minus sign they get winding around $u=0$, as observed this does not change the polymorphicity properties of $\tau$.
We can now use the method introduced in [@rela] to derive the exact relation between the prepotential and the modular invariant. In this case, by means of $({\cal S}_D,{\cal S})$ we may construct the modular invariant $$v={2^{13}\cdot 3\pi i\over\Lambda_{SW}^6}\left({\cal F}_0-{{\cal
S}\over 2}{\partial {\cal F}_0\over\partial {\cal S}}\right),
\label{v}$$ which implies that the pair $({\cal
S}_D,{\cal S})$ satisfies the differential equation $$\left(\partial_v^2+\frac{1}{2}\{\sigma,v\}\right) \pmatrix{{\cal
S}_D\cr {\cal S}}=0, \label{embhe2}$$ where $\{g(x),x\}$ denotes the Schwarzian derivative $g'''/g'-{3\over2}(g''/g')^2$ and $\sigma$ is an arbitrary Möbius transformation of the ratio ${\cal S}_D/{\cal S}$. Later we will see that a simple redefinition of the matrix model coupling constant precisely leads to the above duality structure. Furthermore, we will see that $v=\Lambda_{SW}^4/u^2$ and will find the explicit expression of ${\cal S}(v)$ and ${\cal S}_D(v)$.
We now show that thanks to a scaling property of ${\cal F}_0$, it is possible to identify the right variables to make Seiberg–Witten duality in Dijkgraaf–Vafa theory manifest. First, we note that, by an overall rescaling, the loop expansion of the genus zero free energy in matrix model [@DGKV] reduces by one the number of variables ($\Lambda=2^{-1/2}\Lambda_{SW}$) $$\Delta^{-6}
{\cal F}_{0}(S_k,\Delta,\Lambda)=$$ $${1 \over 2} \sum_{i=1,2}
\left({S_i\over\Delta^3}\right)^2\left[\log \left( {S_i \over
\Delta^3} \right)-{3\over2}\right]-\left({S_1\over\Delta^3}
+{S_2\over\Delta^3}\right)^2 \log \left( {\Lambda \over \Delta}
\right)+\sum_{n\geq3}\sum_{i=0}^n c_{n,i}
\left({S_1\over\Delta^3}\right)^{n-i}\left({S_2\over\Delta^3}\right)^i,
\label{dacompararesotto}$$ where $$c_{n,i}=(-1)^nc_{n,n-i}, \qquad c_{n,i}=(-1)^i|c_{n,i}|,
\label{ojh}$$ so that, except for the first term, ${\cal F}_0$ is symmetric in $S_1$ and $-S_2$. By Euler theorem we have $$\sum_{i=1,2}S_i{\partial {\cal F}_0\over\partial
S_i}+{\Delta\over3}{\partial {\cal F}_0\over\partial\Delta}+
{\Lambda\over3}{\partial {\cal F}_0\over\partial\Lambda}=2{\cal
F}_0. \label{opicjas}$$ Eq.(\[dacompararesotto\]) would suggest that the natural variables are $S_k/\Delta^3$ rather than $S_k$. However, note that this would change the dimensional properties, so we should select $\Lambda^3S_k/\Delta^3$. Furthermore, we should also choose the scale $\Lambda$ as independent variable. So we should express ${\cal F}_0$ as a function of $${\cal S}_k=\left({\Lambda\over\Delta}\right)^3 S_k,\qquad
\mu=?,\qquad \Lambda.$$ It remains to find $\mu$ which, of course, should depend on $\Delta$ and possibly on $\Lambda$. A closer look to (\[dacompararesotto\]) fixes it. Actually, Eq.(\[dacompararesotto\]) suggests considering a natural rescaling of all dimensional quantities of the arguments of ${\cal
F}_0$, by the dimensionless factor ${\Lambda\over \Delta}$. In particular, if $[x]=[\Lambda]^n$, then $x\to\left({\Lambda\over
\Delta}\right)^n x$, that is $$S_k\longrightarrow \left({\Lambda\over \Delta}\right)^3 S_k,
\qquad \Delta\longrightarrow {\Lambda\over \Delta} \Delta=\Lambda,
\qquad \Lambda\longrightarrow {\Lambda\over \Delta}\Lambda=
{\Lambda^2\over \Delta},$$ and the map we define is $${\cal F}_{0}(S_k,\Delta,\Lambda)\qquad\longrightarrow \qquad{\cal
F}_{0}\left(\left({\Lambda\over \Delta}\right)^3
S_k,\Lambda,{\Lambda^2\over \Delta}\right),$$ showing that $S_k$, $\Delta$ and $\Lambda$ combine in such a way that the natural variables for ${\cal F}_0$ are $${\cal S}_1=\left({\Lambda\over \Delta}\right)^3 S_1,\qquad {\cal
S}_2= \left({\Lambda\over \Delta}\right)^3 S_2, \qquad
\mu={\Lambda\over \Delta},\qquad \Lambda.$$ This also follows by the scaling law which is crucial for us $${\cal F}_{0}(\mu^3S_k,\mu\Delta,\mu\Lambda)=\mu^6{\cal
F}_{0}(S_k,\Delta,\Lambda), \label{preoknco}$$ that we rewrite as $${\cal F}_{0}({\cal S}_k,\Lambda,\mu\Lambda)=\mu^6{\cal
F}_{0}(S_k,\Delta,\Lambda). \label{oknco}$$ Note that $${\cal F}_{0}({\cal S}_k,\Lambda,\mu\Lambda)=$$ $$\Lambda^6\left\{{1 \over 2} \sum_{i=1,2} \left({{\cal
S}_i\over\Lambda^3}\right)^2 \left[\log \left( {{\cal S}_i \over
\Lambda^3}\right)-{3\over2}\right]-\left({{\cal
S}_1\over\Lambda^3}+{{\cal
S}_2\over\Lambda^3}\right)^2\log\mu+\sum_{n\geq3}\sum_{i=0}^n
c_{n,i} \left({{\cal S}_1\over\Lambda^3}\right)^{n-i}\left({{\cal
S}_2\over\Lambda^3}\right)^i\right\},
\label{dacompararesopra}$$ that differs from ${\cal
F}_{0}({\cal S}_k,\Lambda,\Delta)$, which, we stress, is the original function with $\Lambda$ and $\Delta$ interchanged and $S_k$ replaced by ${\cal S}_k$, by the sign of the term $({\cal
S}_1+{\cal S}_2)^2 \log\mu$.
Since ${\cal F}_{0}({\cal S}_k,\Lambda,\mu\Lambda)$ is a function of ${\cal S}_k$, $\mu$, and $\Lambda$, it follows by (\[oknco\]) that this is the case also for ${\cal F}_{0}(S_k,\Delta,\Lambda)$. Therefore, we consider the map $(S_k,\Delta,\Lambda)
\longrightarrow ({\cal S}_k,\mu,\Lambda)$, as change of variables for ${\cal F}_0(S_k,\Delta,\Lambda)$. The relationships between the derivatives in the old and new variables are
$${\partial{\cal F}_{0}\over\partial S_1}=\mu^3{\partial{\cal
F}_{0}\over\partial {\cal S}_1},\qquad {\partial{\cal
F}_{0}\over\partial S_2}=\mu^3{\partial{\cal F}_{0}\over\partial
{\cal S}_2}, \label{d1}$$
$${\partial{\cal F}_{0}\over\partial
\Delta}=-3{\mu\over\Lambda}{\cal S}_1 {\partial{\cal
F}_{0}\over\partial {\cal S}_1}-3{\mu\over\Lambda}{\cal
S}_2{\partial{\cal F}_{0}\over\partial {\cal
S}_2}-{\mu^2\over\Lambda}{\partial{\cal F}_{0}\over\partial \mu},
\label{d2}$$
$${\partial{\cal F}_{0}\over\partial \Lambda}= {\partial{\cal
F}_{0}\over\partial \Lambda}+3{{\cal
S}_1\over\Lambda}{\partial{\cal F}_{0}\over\partial {\cal
S}_1}+3{{\cal S}_2\over\Lambda}{\partial{\cal F}_{0}\over\partial
{\cal S}_2}+{\mu\over\Lambda}{\partial{\cal F}_{0}\over\partial
\mu}, \label{d3}$$
\
where in the left hand side the derivatives have been taken considering ${\cal F}_{0}$ as function of the old variables, while on the right hand side it is seen as function of $({\cal
S}_k,\mu,\Lambda)$. In the following we make an abuse of notation and drop a factor $\Lambda$, that is $${\cal F}_{0}({\cal S}_k,\Lambda,\mu)\equiv{\cal F}_{0}({\cal S}_k,
\Lambda,\mu\Lambda)=\mu^6{\cal F}_{0}(S_k,\Delta,\Lambda).
\label{okncopokerdassi}$$ Minimizing $$W_{eff}=\sum_{i=1,2}{\partial{\cal F}_0\over\partial S_i},$$ we obtain, by (\[d1\]) and (\[okncopokerdassi\]) $$\sum_{i=1,2}{\partial^2{\cal F}_0\over
\partial S_i\partial S_j}=
\sum_{i=1,2}\mu^6{\partial^2{\cal F}_0\over \partial {\cal
S}_i\partial {\cal S}_j}=\sum_{i=1,2}{\partial^2{\cal F}_0({\cal
S}_k,\Lambda,\mu)\over
\partial {\cal S}_i\partial {\cal S}_j}=0,$$ which gives ${\cal S}={\cal S}_1=-{\cal S}_2$, where [@DGKV] $${\cal S} =\Lambda^3(\mu^4
+6\mu^8+140\mu^{12}+4620\mu^{16}+\ldots).$$ The effective coupling constant of $N=2$ SYM with gauge group $SU(2)$ is given by $$\tau={\partial^2 {\cal F}_0\over
\partial S_1\partial S_2}\bigg|_{S_1=-S_2=S},$$ and by (\[d1\]) and (\[okncopokerdassi\]) $$\tau={\partial^2 {\cal F}_{0}({\cal S}_k,\Lambda,\mu)\over
\partial {\cal S}_1\partial {\cal S}_2}\bigg|_{{\cal S}_1=-{\cal
S}_2={\cal S}},$$ where here ${\cal F}_0$ is rescaled by $1/\pi i$ with respect to the one in (\[dacompararesotto\]). So, we have seen that, thanks to the scaling property (\[okncopokerdassi\]), one obtains the same effective coupling constant $\tau(a)$, if in the matrix model one considers as variables the old ones rescaled by $\mu^n=(\Lambda/\Delta)^n$, with $n$ defined by $[x]=[\Lambda]^n$. As a consequence the duality structure of $N=2$ SYM with gauge group $SU(2)$ is manifest. Before showing this explicitly we explain how the above rescaling of ${\cal F}_0$ simply amounts to a different choice of the matrix model coupling constant. Let us set $$g_{\cal S}=\mu^3 g_S, \qquad {\cal
W}(\Phi)=\mu^3W(\Phi),\label{newgS}$$ and note that $$Z={1\over {\rm vol(G)}}\int d\Phi \exp \left(-{1\over g_S} {\rm
tr}\, W(\Phi)\right)={1\over {\rm vol(G)}}\int d\Phi \exp
\left(-{1\over g_{\cal S}} {\rm tr}\, {\cal W}(\Phi)\right),
\label{Z}$$ so that $$Z=\exp\left(-\sum_{g\geq0}g_S^{2g-2}{\cal F}_g\right)=
\exp\left(-\sum_{g\geq0}g_{\cal S}^{2g-2}\tilde{\cal F}_g\right),
\label{expann}$$ where $$\tilde{\cal F}_g=\mu^{3(2-2g)}{\cal F}_g.
\label{iuc}$$ In particular, by (\[okncopokerdassi\]) we see that $\tilde{\cal F}_0=\mu^6{\cal
F}_0={\cal F}_0({\cal S}_k,\Lambda,\mu)$. This indicates that also the higher genus contributions should be considered as functions of the new variables, that is $$\tilde{\cal F}_g=\mu^{3(2-2g)}{\cal F}_g={\cal F}_g({\cal
S}_k,\Lambda,\mu), \label{scalareale}$$ so we rewrite $$Z=\exp\left(-\sum_{g\geq0}g_{\cal S}^{2g-2}{\cal F}_g\right),
\label{expannoliik}$$ where now ${\cal F}_g\equiv{\cal
F}_g({\cal S}_k,\Lambda,\mu)$.
Let us now derive the explicit expression for ${\cal S}_D$ and ${\cal S}$ and show how the rescaling leads to make the $N=2$ SYM duality manifest. The trick is to first consider the derivative of $v$ with respect to $u$. In particular, by (\[essecorsiva\]) and (\[essecorsivaduale\]) we have $${\cal S}_D'=-{\Lambda_{SW}^7\over64} u^{-5/2}a_D',\qquad {\cal
S}'=-{\Lambda_{SW}^7\over64} u^{-5/2}a',
\label{iprimati}$$ and by (\[v\]) $$v'={2^{10}\cdot3{\pi i}\over\Lambda_{SW}^6}({\cal S}_D{\cal
S}'-{\cal S}{\cal S}_D')=\pi i\Lambda_{SW}^4(a_D'a-a_Da')u^{-3}.
\label{oiajx}$$ On the other hand, since $aa_D'-a_Da'=2i/\pi$, we have $$v'=-2{\Lambda_{SW}^4} u^{-3}, \label{oiajx2}$$ that is $$v= \left({\Lambda_{SW}^2\over u}\right)^2,
\label{oiajx3}$$ where the additive constant, that corresponds to fix the additive constant of ${\cal F}_0$, has been set to zero. By construction we know that ${\cal S}$ satisfies a second order differential equation with respect to $v$ in which the first derivative term is absent. Actually, taking the second derivative of ${\cal S}$ with respect to $v$, we have $$\partial^2_v {\cal S}=-(\partial_u
v)^{-3}\partial^2_uv\partial_u{\cal
S}+(\partial_uv)^{-2}\partial_u^2{\cal S}={3u^4\over
16\Lambda_{SW}^4(u^2-\Lambda_{SW}^4)} {\cal S},
\label{vabene}$$ that is $({\cal S}_D,{\cal S})$ satisfy the second order differential equation $$\left(\partial_v^2+{3\over 16v(v-1)} \right) \pmatrix{{\cal
S}_D\cr {\cal S}}=0, \label{embhe23}$$ whose solutions can be obtained directly by (\[essecorsiva\]) and (\[essecorsivaduale\]) using $a_D(u(v))$ and $a(u(v))$ $${\cal S}_D={\Lambda_{SW}^3\sqrt v\over\sqrt2\cdot96
\pi}\int_{-1}^{1\over\sqrt v} dx {x-\sqrt v\over
\sqrt{x^2-1}\sqrt{\sqrt v x -1}},\qquad {\cal
S}={\Lambda_{SW}^3\sqrt v\over\sqrt2\cdot96 \pi}\int_{-1}^1 dx
{x-\sqrt v\over \sqrt{x^2-1}\sqrt{\sqrt v x -1}}.
\label{opidjc}$$ Inverting Eq.(\[embhe23\]) we obtain the differential equation for $v={\cal H}({\cal S})$ $$16{\cal H}(1-{\cal H}){\partial^2 {\cal H}\over\partial {\cal
S}^2}+3{\cal S}\left({\partial {\cal H}\over\partial {\cal
S}}\right)^3=0. \label{iowuxch}$$ Since $$\mu=\left({\Lambda_{SW}^2\over 2^3 u}\right)^{1/2}\longrightarrow
v=2^6\mu^4,$$ we have $${\cal
S}_D={\Lambda_{SW}^3\mu^2\over\sqrt2\cdot12\pi}\int_{-1}^{1\over
8\mu^2} dx {x-8\mu^2\over \sqrt{x^2-1}\sqrt{8\mu^2 x -1}},\qquad
{\cal S}={\Lambda_{SW}^3\mu^2\over\sqrt2\cdot12\pi}\int_{-1}^1 dx
{x-8\mu^2\over \sqrt{x^2-1}\sqrt{8\mu^2x -1}}.
\label{opidjcbisse}$$ In terms of $\mu$ the nonperturbative relation (\[v\]) reads $$\mu^4={3\cdot 2^7\pi i\over\Lambda_{SW}^6}\left({\cal F}_0-{{\cal
S}\over 2}{\partial {\cal F}_0\over\partial {\cal S}}\right),
\label{muuu}$$ which is the matrix model analog of the relation between the $u$–modulus and the Seiberg–Witten prepotential [@rela].
Introducing manifest duality has several interesting consequences. For example, one may investigate to what corresponds in matrix model the strong coupling region of $N=2$ SYM. In particular, the QFT meaning of the strong coupling expansion of the prepotential at the points $u=\pm\Lambda_{SW}^2$ is a crucial open question. While in the weak coupling region the expansion of the SW prepotential corresponds to a one–loop term and to infinitely many instanton contributions, no QFT meaning is known for its expansion at strong coupling. In $N=2$ SYM, this region is investigated by performing a $S$–duality transformation on the fields. This corresponds to a Legendre transform of the prepotential. On the matrix model side one should consider a dual formulation corresponding to this region. It would be interesting whether perturbation theory would reproduce also in this region the $N=2$ SYM theory. One should consider the Legendre transform $${\cal F}_{Dg}={\cal F}_g-\sum_{i=1,2}{\cal S}_i{\partial {\cal
F}_g\over\partial {\cal S}_i}, \label{FrankZappa}$$ where ${\cal F}_g\equiv{\cal F}_g({\cal S}_k,\Lambda,\mu)$, and $$Z_D= \exp\left(-\sum_{g\geq0}g_{{\cal S}_D}^{2g-2}{\cal
F}_{Dg}\right), \label{expanndual}$$ which should induce the definition of ${\cal W}_D$ $$Z_D={1\over {\rm vol(G)}}\int d\Phi_D \exp \left(-{1\over g_{{\cal
S}_D}} {\rm tr}\, {\cal
W}_D(\Phi_D)\right).\label{Z_D}$$
Before concluding, let us note that this approach should be related with the derivation of the structure of the instanton moduli space of $N=2$ SYM obtained from the recursion relations for the instanton contributions to the prepotential [@sfere]. In particular, it was shown how the analogs of the recursive structure of the Deligne–Knudsen–Mumford compactification of moduli space of Riemann surfaces and of the Wolpert restriction phenomenon, essentially determine the structure of the instanton moduli spaces. These techniques are strictly related to the geometry of matrix models considered in the framework of Liouville quantum gravity [@LG]. So, it would be interesting to investigate whether there is a possible link between the matrix model approach to the $N=2$ SYM and the geometrical approach considered in [@sfere].
Finally, we note that making duality manifest, which generalizes to higher rank groups [@rangoalto], may have possible relations with recent work on matrix models [@vari].
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[^1]: A suitable generalization of the method introduced here, suggests a possible application in investigating the Picard–Fuchs equations in the framework of the Mirror conjecture.
[^2]: We are using the notation $({{\cal{S}}_D},{\cal S})$ rather than $(S_D,S)$ since, as we will see, the pair $(S_D,S)$ defined in matrix model has not $\Gamma(2)$–monodromy.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Stephan’s Quintet (SQ, HCG92) was observed with the Far-Infrared Surveyor (FIS) aboard *AKARI* in four far-infrared (IR) bands at 65, 90, 140, and 160 $\mu$m. The *AKARI* four-band images of the SQ show far-IR emission in the intergalactic medium (IGM) of the SQ. In particular, the 160 $\mu$m band image shows single peak emission in addition to the structure extending in the North-South direction along the shock ridge as seen in the 140 $\mu$m band, H$_2$ emission and X-ray emission. Whereas most of the far-IR emission in the shocked region comes from the cold dust component, shock-powered \[\]158$\mu$m emission can significantly contribute to the emission in the 160 $\mu$m band that shows a single peak at the shocked region. In the shocked region, the observed gas-to-dust mass ratio is in agreement with the Galactic one. The color temperature of the cold dust component ($\sim$20 K) is lower than that in surrounding galaxies ($\sim$30 K). We discuss a possible origin of the intergalactic dust emission.'
author:
- 'Toyoaki Suzuki, Hidehiro Kaneda, Takashi Onaka,'
- Tetsu Kitayama
title: 'Far-infrared emission from intergalactic medium in Stephan’s Quintet revealed by *AKARI*'
---
Introduction
============
The track of the galactic scale transmigration of gas and dust is indispensable to understand the galaxy evolution. Stephan’s Quintet (SQ, HCG92) is the well studied compact group of galaxies [@hickson] with the disturbed intergalactic medium (IGM) as shown in Fig. \[SQ\]. The intruder galaxy NGC 7318b is currently colliding with the IGM at a relative velocity of $\sim1000$ km s$^{-1}$ and causes a large-scale shock front and IGM starbursts (SQ-A and SQ-B). The shock front was first discovered by [@allen1972] with radio, and subsequently detected in the X-ray emission from shock-heated ($\sim6\times 10^6$ K) gas [@pietsch; @trinchieri2003; @trinchieri2005; @osullivan]. [@appleton] found powerful H$_2$ rotational line emission from warm ($\sim 10^2$-$10^3$ K) molecular gas in the center of the shock ridge. The significant discovery is the extremely large equivalent width ($EW$) of the H$_2$ line emission. This may indicate that far-infrared (IR) fine-structure lines such as \[\]158$\mu$m line also show extremely large $EW$ by the shock and have a significant contribution to far-IR luminosity. However, the fact has yet to be revealed observationally. [@culver2010] made H$_2$ maps of the SQ and found that the spatial distribution of the H$_2$ emission was similar to those of radio and X-ray emission in the shocked region. To explain the co-existence of both hot and H$_2$ gas, [@guillard2009] proposed a model of the shock in the inhomogeneous gas medium. While H$_2$ molecules are likely to be produced on the grain surface [@gould1963], direct association of the H$_2$ gas with dust has yet to be found in the shock ridge. Recently, [@natale2010] reported dust emission in the SQ with *Spitzer* observations. While the map at 160 $\mu$m shows the significant presence of cold dust in the shock ridge, it is difficult to make direct comparison of the diffuse 160 $\mu$m map with the H$_2$ emission map because of the effect of bright emission from the surrounding galaxies.
The Far-Infrared Surveyor (FIS) [@kawada] aboard *AKARI* [@murakami] has four far-IR bands with the central wavelengths of 65, 90, 140, and 160 $\mu$m, and achieves high sensitivity with relatively high spatial resolution. Finer allocation of *AKARI*/FIS 4 bands can provide the spectral information to constrain both the dust temperature and the \[\]158$\mu$m line emission and is the distinctive advantage compared to *Spitzer*/MIPS observations. In this paper we report the observation of the SQ with the FIS, and discuss the origin of the far-IR emission in the IGM.
Observations and Data reduction
===============================
The SQ was observed in part of the *AKARI* mission program “ISM in our Galaxy and Nearby Galaxies” [ISMGN; @kaneda2009] on 2007 Jun 18 using the FIS01 observation mode (Observation ID: 1402238-001). Details of the FIS instrument and its in-orbit performance/calibration are described in @kawada and @shirahata. The FIS was operated in the photometry mode with the four bands: N60 (65 $\mu$m), WIDE-S (90 $\mu$m), WIDE-L (140 $\mu$m), and N160 (160 $\mu$m). The four-band data were simultaneously obtained in a pointing observation for an area of about $10\arcmin\times20\arcmin$ around the SQ. The FIS data were processed with the *AKARI* official pipeline modules (version 20070914). Finally, four-band images were created with grid sizes of $12.5\arcsec$ for the WIDE-L and N160 bands and $7.5\arcsec$ for the WIDE-S and N60 bands. The widths (FWHM) of the point spread functions (PSFs) are $\sim45\arcsec$ for the WIDE-L and N160 bands and $\sim30\arcsec$ for the WIDE-S and N60 bands [@shirahata]. The four-band images are presented in Fig. \[4image\]. The images are smoothed with boxcar kernels with a width of $25\arcsec$ for the WIDE-L and N160 bands, and $15\arcsec$ for the WIDE-S and N60 bands.
To derive the spatial variation in color temperature, the spatial resolutions of the original WIDE-S and N60 images are reduced to match those of the WIDE-L and N160 images by convolving the former images with a Gaussian kernel. The images are then resized with the common spatial scale among the four bands, $12.5\arcsec$/pixel. The four-band flux densities at each image bin are derived after subtracting the sky background. The background levels were estimated and subtracted using nearby regions of the blank sky, which were observed in the beginning and at the end of the scan. Color corrections were also applied for the obtained flux densities by assuming a modified blackbody spectrum with the emissivity power-law index of unity and the temperatures of 20 K and 30 K for the shocked region and galaxies, respectively. Aperture corrections were not applied to the flux densities because we only consider diffuse emission components. At the centers of the shocked region, NGC 7319, and NGC 7320, we derive the flux densities by integrating the surface brightness within circular apertures. Figure \[sed\](a) shows the resulting spectral energy distributions (SEDs) at these positions. Note that the fluxes in Fig. \[sed\](a) are not the fluxes for the whole areas of the shocked region and the galaxies. The systematic errors for the flux densities are estimated to be $\sim$10 % for the WIDE-S and N60 bands, $\sim$20 % for the WIDE-L band, and $\sim$25 % for N160 band, which include the uncertainty due to radiation effects. The relative uncertainties for the flux densities are estimated to be below 5 % for the four bands.
Results
=======
In Fig. \[4image\], N60 and WIDE-S images show emission from NGC 7319 and NGC 7320, and from the SQ-A and SQ-B (Fig. \[4image\](c)). Although the WIDE-L image shows a structure extending in the North-South direction along the shock ridge, the N160 image clearly shows single peak emission in addition to the structure, but does not indicate any features associated with the individual galaxies. In the shocked region, the background-subtracted surface brightness in the N160 band is larger than 9 MJy sr$^{-1}$, which is comparable to 5$\sigma$ sky background noise. For the first time, *AKARI* observations reveal the dramatical change in the spatial distribution of far-IR emission at wavelengths around 140-160 $\mu$m. The difference in images between the N160 band and the MIPS160 band [@natale2010] may come from the fact that the spectral response of the N160 band covers longer wavelengths than that of the MIPS160 band; wavelength coverage at 10 % of the peak responsivity is $140$-$196$ $\mu$m for the N160 band (AKARI FIS Data Users Manual ver.1.3) and $129$-$184$ $\mu$m for the MIPS160 band (MIPS Instrument Handbook). Figure \[4image\](f) shows that the spatial distribution of 160 $\mu$m emission is quite similar to that of X-ray emission in the shocked region. In fact, X-ray, radio, and H$_2$ emission in the shocked region all show structures associated with the shock ridge and the bridge extending eastward from the shock ridge to NGC 7319. Our result clearly shows the spatial correlation of the far-IR emission with these structures.
Difference in images between N160 and WIDE-L bands
--------------------------------------------------
The cold dust temperatures in the shock and surrounding galaxies are about 20-30 K. Because the flux ratio between the N160 and WIDE-L bands is not very sensitive to the dust temperature, the difference in the spatial distribution of far-IR emission between the two bands is hard to explain only by the presence of the cold dust component. An alternative possibility is a contribution from the shock-powered \[\]158$\mu$m line emission to the N160 band. For the SQ, the wavelength of \[\]158$\mu$m line is redshifted to 161 $\mu$m [@hickson1992]. Thus, its contribution to the N160 band is larger than to the WIDE-L band. To estimate the contribution from the \[\] line, the WIDE-L flux density $F_\mathrm{WL}$ is simply subtracted from the N160 flux density $F_\mathrm{N160}$ by assuming that the dust emission is constant in units of $F_\nu$ over the two bands (Fig. \[sed\](b)). By using the same aperture as that in Fig. \[4image\](f), the subtracted flux density $F_\mathrm{sub}$ is estimated to be 40$^{+18}_{-22}$ mJy. The error includes the uncertainty that $F_\mathrm{WL}$ to be subtracted must be increased by 38 %, if the SED of the dust emission is considered as the modified blackbody spectrum whose temperature and amplitude are determined from the data at 90 and 140 $\mu$m. The uncertainty is comparable to the subtracted sky background noise. The \[\] luminosity surface density $\Sigma_{L_{\mathrm{[CII]}}}$ \[erg sec$^{-1}$ kpc$^{-2}$\] is given by $$\Sigma_{L_{\mathrm{[CII]}}} = 4\pi D^2
\frac{F_\mathrm{sub}}{A}\left(\frac{R_\mathrm{N160}(161\mu
\mathrm{m})}{\Delta\nu_{\mathrm{N160}}}-\frac{R_\mathrm{WL}(161\mu \mathrm{m})}{\Delta\nu_{\mathrm{WL}}}\right)^{-1},
\label{eq5}$$ where $D$ is the distance to the SQ of 94 Mpc, $A$ is the aperture area at the shocked region (260 kpc$^2$), $R_\mathrm{N160}$ and $R_\mathrm{WL}$ are the relative responses of the N160 and WIDE-L bands at 161 $\mu$m, respectively, and $\Delta\nu_{\mathrm{N160}}$ and $\Delta\nu_{\mathrm{WL}}$ are the effective bandwidths of the N160 and WIDE-L, respectively. From [@kawada], $R_\mathrm{N160}$, $R_\mathrm{WL}$, $\Delta\nu_{\mathrm{N160}}$, and $\Delta\nu_{\mathrm{WL}}$ are taken as 0.96, 0.59, 0.4 THz, and 0.8 THz, respectively. Thus, $\Sigma_{L_{\mathrm{[CII]}}}$ is estimated to be $(1.0^{+0.4}_{-0.5})\times10^{39}$ erg sec$^{-1}$ kpc$^{-2}$, which is comparable to the H$_2$ line luminosity surface density $\Sigma_{L_\mathrm{H_2}}$ of $2\times10^{39}$ erg sec$^{-1}$ kpc$^{-2}$[@culver2010]. By using the dust emission at 160 $\mu$m, the $EW$ of the \[\] line is estimated to be $\sim$10 $\mu$m. The $EW$ is about 10 times larger than that in nearby galaxies (the mean of 0.7 $\mu$m with the dispersion of 0.2-3.0 $\mu$m, [@boselli2002]). We calculated the contribution of the \[\] line emission to the fluxes in the WIDE-L and N160 bands by considering the constant dust emission in units of $F_\nu$ over the two bands. We obtained 8 % to WIDE-L band and 23 % to N160 band.
Properties of far-IR dust emission
----------------------------------
To estimate the far-IR luminosity $L_\mathrm{FIR}$ in the shocked region, the best-fit modified blackbody (the dash dotted line in Fig \[sed\](a)) was integrated between 3 and 3000 $\mu$m to yield $L_\mathrm{FIR}$ of $(5 \pm 2)\times 10^{42}$ erg sec$^{-1}$. By dividing $L_\mathrm{FIR}$ by the aperture area $A$, the luminosity surface density $\Sigma_{L_\mathrm{FIR}}$ is estimated to be $(2.0 \pm
0.6)\times 10^{40}$ erg sec$^{-1}$ kpc$^{-2}$, which is in agreement with the far-IR luminosity surface density ($1.6\times 10^{40}$ erg sec$^{-1}$ kpc$^{-2}$) of *ISO* observations [@xu2003]. In the shocked region, we calculate the dust mass according to Eq. (4) of [@hildebrand] with the emissivity power-law index of unity. The dust mass surface density $\Sigma_{M_\mathrm{d}}$ is given by $$\Sigma_{M_\mathrm{d}} = 1.1\times10^4\left(
\frac{\Sigma_{L_\mathrm{FIR}}}{2.0\times10^{40}\
\mathrm{erg\ sec^{-1}}\ \mathrm{kpc}^{-2}}\right)
\left(\frac{T_{\rm d}}{22\ \mathrm{K}}\right)^{-5}\ M_\sun\ \mathrm{ kpc^{-2}}.
\label{eq6}$$ The dust temperature $T_\mathrm{d}$ is set to be 22 K, which is derived from the SED fitting in Fig \[sed\](a). From Eq. (\[eq6\]), $\Sigma_{M_\mathrm{d}}$ is estimated to be $(1.1\pm0.3)\times10^4$ $M_\sun$ kpc$^{-2}$. If we consider the contribution of the \[\] line emission to $F_\mathrm{N160}$, $\Sigma_{M_\mathrm{d}}$ is reduced by $\sim$20 %. [@culver2010] estimated the mass surface density of warm H$_2$ gas in the shocked region as $(10\pm2)\times10^5$ $M_\sun$ kpc$^{-2}$. CO observations of the shocked region suggest that the mass ratio of the cold H$_2$ ($<$ 50 K) to warm H$_2$ gas masses is 2 with the CO-to-H$_2$ conversion factor of our Galaxy [@guillard2010b]. [@guillard2009] estimated an upper limit of the mass surface density of gas in the shocked region as $5\times10^5~ M_\sun$ kpc$^{-2}$. Thus, the gas-to-dust mass ratio is 170-210, which is in agreement with that of our Galaxy [@sodroski1997]. The estimated properties of the dust in the SQ are summarized in Table \[prop\].
Discussion
==========
Shock-powered \[\]158$\mu$m line emission from the shocked region
-----------------------------------------------------------------
To investigate the possibility of the luminous \[\] line emission from the shocked region, the C$^+$ abundance per hydrogen atom $X_\mathrm{C^+}$ is estimated with assumptions that the \[\] line emission comes from the warm H$_2$ gas and is optically thin. Within these assumptions, $X_\mathrm{C^+}$ is given as $$X_\mathrm{C^+} = 1.1\times10^{-4}\left(
\frac{\Sigma_{L_{\mathrm{[CII]}}}\
L_\sun\mathrm{kpc}^{-2}}{\Sigma_{\mathrm{M_{H_2}}}\
M_\sun\mathrm{kpc}^{-2}}\right)
\left(\frac{1+2\mathrm{exp}(-91\
\mathrm{K}/T)+n^\mathrm{crit}_\mathrm{H}/n_\mathrm{H}}{2\mathrm{exp}(-91\ \mathrm{K}/T)}\right),
\label{eq7}$$ where $T$ is the kinetic temperature of H$_2$ gas, $n^\mathrm{crit}_\mathrm{H}$ is the critical density of the \[\] line emission for collisions, and $n_\mathrm{H}$ is the number density of . At the shocked region, $T$ and $n_\mathrm{H}$ are 158 K and $>10^3$ cm$^{-3}$, respectively [@culver2010]. $n^\mathrm{crit}_\mathrm{H}$ is $3.0\times10^3$ cm$^{-3}$ at 158 K [@langer2010]. Thus, $X_\mathrm{C^+}$ is estimated to be $\sim1\times10^{-4}$. Assuming that the carbon in the \[CII\] line emitting region is in singly ionized form, the carbon abundance in the shocked region is in agreement with that in an interstellar gas-phase ($1.4\times10^{-4}$, [@cardelli1996]). Therefore, the luminous \[\] line emission from the shocked region is physically plausible provided that C$^+$ is the main carbon form in the warm H$_2$ gas.
Origin of the cold dust emission from the shocked region
--------------------------------------------------------
The clear spatial correlations among the cold dust, H$_2$, and X-ray emissions suggest two possibilities: radiative heating of dust grains in a low interstellar radiation field (ISRF) and collisional heating of dust grains in a hot plasma. To explain the formation of H$_2$ gas in the shocked region, [@guillard2009] introduced a model of the shock in an inhomogeneous gas phase. In their model, there are two critical assumptions to explain the formation of H$_2$ gas in the collision age: dust survival in the shock and the Galactic gas-to-dust mass ratio. The spatial correlation between the maps of the cold dust emission and H$_2$ rotational line strongly supports H$_2$ formation on dust grains. In addition, the observed gas-to-dust mass ratio in the shocked region is in agreement with the Galactic one. These facts indicate that the H$_2$ formation time scale should be shorter than the collision age. In the case of the multiphase medium, the cold dust emission comes from molecular gas clouds. [@guillard2010] modeled SEDs of dust emission associated with diffuse (the hydrogen column density $N_\mathrm{H}\sim2\times10^{20}$ cm$^{-2}$) or clumpy molecular gas ($N_\mathrm{H}\sim7\times10^{21}$ cm$^{-2}$) in radiation fields. As discussed in Section 4.1, the \[\] line emission can come from molecular gas clouds. The optical depth of the \[\] line $\tau_\mathrm{[CII]}$ is expressed in Eq. (A4) of [@crawford1985]. By using $X_\mathrm{C^+}$ of $1\times10^{-4}$ and the velocity dispersion of 870 km s$^{-1}$ [@culver2010], $N_\mathrm{H}$ is estimated to be $2\times10^{24}$ cm$^{-2}$ when $\tau_\mathrm{[CII]}$ becomes unity. Thus, the \[\] line emission is optically thin for both cloud models. To estimate color temperatures of cold dust for both cloud models, flux densities at 90 and 160 $\mu$m are obtained from Fig. 7 in [@guillard2009]. Assuming the modified blackbody spectrum with the emissivity power-law index of unity, the color temperature is estimated to be 24 K for the diffuse molecular gas model and 20 K for the clumpy one. We investigate the spatial distribution of the color temperature in the SQ (see Fig. \[color\]) by using the flux densities at 90 and 160 $\mu$m that are above the $3\sigma$ sky background noise. Errors in the color temperature are estimated as $2.0$-$2.5$ K. The color temperature in the shocked region is about 20 K, which favors the temperature for the clumpy molecular gas model rather than the diffuse one and significantly lower than that in the surrounding galaxies. Thus, it is possible that cold dust in the shocked region is radiatively heated by the ISRF.
The spatial correlation between the maps of the cold dust emission and X-ray suggests an alternative possibility for the origin of cold dust emission in the shocked region. If dust grains are exposed to the hot plasma for the collision age ($\sim 5\times 10^{6}$ yr), grains smaller than 0.1 $\mu$m in radius must have been destroyed in $\sim 5\times 10^{6}$ yr [@guillard2009]. We calculated the temperatures of dust grains with radii $a>0.1$ $\mu$m heated collisionally by ambient plasma electrons, based on the model of [@dwek1986; @dwek1987]. We assumed the size distribution $\mbox{d}n/\mbox{d}a \propto a^{-\alpha}$ with $\alpha$ = 2.5-3.5 over the range $0.1 < a < 1\ \mu
\mbox{m}$ for either silicate or graphite grains. The density and temperature of the electrons are taken to be $n_e=0.02$ cm$^{-3}$ and $T_e=6 \times 10^6$ K, respectively [@trinchieri2003]. As a result, we derive the dust temperature of 20-22 K, which is insensitive to a specific choice of $\alpha$ or grain composition. The derived temperature is in a range similar to that observed in the shocked region. Thus we cannot distinguish between radiative heating and collisional heating of dust grains solely from the dust temperature.
However, the similarity in the spatial distribution between the far-IR (cold dust and \[\] line) emission and H$_2$ emission strongly suggests that H$_2$ molecules form on dust grains and coexists with C$^{+}$. Moreover, [@natale2010] showed that the collisionally heated dust emission should be minor contribution to explain the X-ray temperature map. Therefore, it is likely that the far-IR dust emission arises mostly from radiative heating of cold dust in clumpy molecular gas. These facts support the multiphase medium model for the SQ.
Summary
=======
We observed the SQ with the FIS aboard *AKARI* in four far-IR bands. The N160 band image shows single peak emission in addition to the structure extending in the North-South direction along the shock ridge as seen in the WIDE-L band, H$_2$ emission and X-ray emission. Whereas most of the far-IR emission in the shocked region comes from the cold dust component, the \[\] line emission whose luminosity is comparable to that of the warm H$_2$ gas can significantly contribute to the single peak emission in the N160 band. It can be explained that the \[\] line emission comes from the warm H$_2$ gas. In the shocked region, the observed gas-to-dust mass ratio is in agreement with the Galactic one. These indicate that H$_2$ molecules are produced on the grain surface and their formation timescale should be shorter than the collision age. The color temperature in the shocked region ($\sim 20$ K) is significantly lower than that in the surrounding galaxies ($\sim 30$ K). It is considered that the cold dust emission comes mostly from radiative heating in dense and clumpy gas clouds. Our results support the scenario of the multiphase medium of the pre-shocked gas in the IGM.
We are grateful to the referee for providing us very important comments and corrections on our calculation. The present work is based on observations with *AKARI*, a JAXA project with the participation of ESA. The Digitized Sky Surveys were produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions. This work was supported in part by Grant-in-Aid for Young Scientists by MEXT (21740139).
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![Composite image of the Stephan’s Quintet (HCG92): *AKARI* 7 $\mu$m (red), Optical (green), and X-ray (blue). Optical and X-ray images are retrieved from Digitized Sky Survey and XMM-Newton Science Archive (observation ID: 0021140201, observation date: 2001-12-07), respectively. The SQ is composed of five galaxies (NGC 7317, NGC 7318a, NGC 7318b, NGC 7319, and NGC 7320c). NGC 7320 is an unrelated foreground object [@moles1997]. The box shows the boundary of the maps in Fig. \[4image\]. []{data-label="SQ"}](./fig1.eps)
![Four-band images of the SQ in the N60 (top-left), WIDE-S (top-middle), WIDE-L (bottom-left), and N160 (bottom-middle) bands. The center wavelengths of the four bands are 65 $\mu$m for N60, 90 $\mu$m for WIDE-S, 140 $\mu$m for WIDE-L, and 160 $\mu$m for N160. The contours are linearly spaced from 10 % to 99 % of the peak brightness with a step of 3 %. The peak brightness that includes the sky background is 22 MJy sr$^{-1}$ (N60), 17 MJy sr$^{-1}$ (WIDE-S), 22 MJy sr$^{-1}$ (WIDE-L), and 27 MJy sr$^{-1}$ (N160). In each image, the PSF size in FWHM is shown in the lower left corner. The upper-right and bottom-right panels show the images of WIDE-S and N160 bands overlaid on *AKARI* 7 $\mu$m and XMM/Newton X-ray contours, respectively. The three apertures for photometry of the shocked region ($r=20''$), NGC 7319 ($r=30''$), and NGC 7320 ($r=30''$) are shown in the panels of (c) and (f). As a reference, the shock ridge is shown by the solid white line in the panel (f).[]{data-label="4image"}](./fig2a.ps "fig:"){width="18cm"} ![Four-band images of the SQ in the N60 (top-left), WIDE-S (top-middle), WIDE-L (bottom-left), and N160 (bottom-middle) bands. The center wavelengths of the four bands are 65 $\mu$m for N60, 90 $\mu$m for WIDE-S, 140 $\mu$m for WIDE-L, and 160 $\mu$m for N160. The contours are linearly spaced from 10 % to 99 % of the peak brightness with a step of 3 %. The peak brightness that includes the sky background is 22 MJy sr$^{-1}$ (N60), 17 MJy sr$^{-1}$ (WIDE-S), 22 MJy sr$^{-1}$ (WIDE-L), and 27 MJy sr$^{-1}$ (N160). In each image, the PSF size in FWHM is shown in the lower left corner. The upper-right and bottom-right panels show the images of WIDE-S and N160 bands overlaid on *AKARI* 7 $\mu$m and XMM/Newton X-ray contours, respectively. The three apertures for photometry of the shocked region ($r=20''$), NGC 7319 ($r=30''$), and NGC 7320 ($r=30''$) are shown in the panels of (c) and (f). As a reference, the shock ridge is shown by the solid white line in the panel (f).[]{data-label="4image"}](./fig2b.ps "fig:"){width="18cm"}
.[]{data-label="color"}](./fig4.ps)
[cccc]{} $T_\mathrm{C}$ & $\Sigma_{L_\mathrm{FIR}}$ & $\Sigma_{M_\mathrm{d}}$ & Gas-to-dust\
(K) & (erg sec$^{-1}$ kpc$^{-2}$) & ($M_\sun$ kpc$^{-2}$) & mass ratio\
22$\pm$1 & $(2.0 \pm 0.6)\times 10^{40}$ & $(1.1\pm0.3)\times10^4$ & 170$-$210\
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a general model allowing “quantum simulation” of one-dimensional Dirac models with 2- and 4-component spinors using ultracold atoms in driven 1D tilted optical latices. The resulting Dirac physics is illustrated by one of its well-known manifestations, *Zitterbewegung*. This general model can be extended and applied with great flexibility to more complex situations.'
author:
- Jean Claude Garreau
- Véronique Zehnlé
title: Simulating Dirac models with ultracold atoms in optical lattices
---
\[sec:Intro\]Introduction
=========================
The Dirac theory of the electron (with its quantum-electrodynamical corrections) is the most complete, precise, and experimentally well-tested theory in physics. It combines quantum mechanics and relativistic covariance in a general frame, automatically including the spin degree of freedom, and predicting the existence of the positron. However, in atomic physics, and *a fortiori* in cold-atom physics, Dirac theory has played a relatively restricted role, because, experimentally, its domain of application ($v\sim c$) is not often attained (except for inner-shell electrons of heavy atoms) and, theoretically, many of its important results (e.g. fine structure) can be calculated with a good precision in the simpler frame of Pauli theory (that is, Schrödinger equation plus spin 1/2), at least for light atoms.
Recently, *quantum simulation* [@Georgescu:QuantumSimulation:RMP14] became a mainstream in ultracold-atom physics [@Bloch:QuantumSimulationsUltracoldGases:NP14]. The basic idea, inspired by early Feynman insights [@Feynman:SimulatingPhysics:IJTP82], is to generate the physical behavior corresponding to some model, e.g. condensed matter’s Hubbard Hamiltonians, by “artificially” creating a corresponding Hamiltonian in more controlled conditions, e.g. ultracold atoms in optical lattices [@Bloch:ManyBodyUltracold:RMP08]. This “Hamiltonian engineering” has been pushed quite far, with the introduction of artificial gauge fields [@Dalibard:ArtificialGaugePotentials:RMP11], spin-orbit couplings and Dirac equation simulations [@Gerritsma:QuantumSimulationDirac:N10; @Witthaut:EffectiveDiracDynamicsBichrOptLatt:PRA11; @Salger:KleinTunnelingBECOptLatt:PRL11; @Galitski:SpinOrbitCouplingQuantumGases:N13], quantum magnetism of neutral atoms [@Lin:SyntheticMagneticFieldsForUltracold:N09; @Struck:TunableGaugePotentialDrivenLattices:PRL12], and the physics of disordered systems [@Chabe:Anderson:PRL08; @Billy:AndersonBEC1D:N08; @Roati:AubryAndreBEC1D:N08; @Kondov:ThreeDimensionalAnderson:S11; @Manai:Anderson2DKR:PRL15].
Quantum simulation of Dirac physics has benefit of a large interest in recent years. This can be done in condensed matter systems by taking advantage of the flexible concept of quasi-particles, where in particular the Weyl semimetal [@Wang:QuantumTransportDiracAndWeylReview:17] is a pertinent concept, and recently the existence of “type-II” Weyl particles (that is a Weyl particle breaking Lorentz isotropy) [@Soluyanov:TypeIIWeyl:N17] has been suggested. Dirac quantum simulators using ion traps have also been proposed [@Lamata:RelativisticQuantumMechanicsTrappedIons:NJP11]. Another popular way of quantum-simulating Dirac physics is by using ultracold atoms in optical lattices, pioneered by Gerritsma *et al*. [@Gerritsma:QuantumSimulationDirac:N10; @Gerritsma:QuantumSimulationKleinParadox:PRL11], who studied the phenomenon of Klein tunneling, also studied in refs. [@Salger:KleinTunnelingBECOptLatt:PRL11; @Witthaut:EffectiveDiracDynamicsBichrOptLatt:PRA11; @Suchet:AnalogSimulationWeylParticles:EPL16]. Without trying to be exhaustive, a wealth of interesting related phenomena can also be studied: topological insulators, Dirac cones, spin-orbit coupling, and even cyclotron dynamics [@Mazza:OpticalLatticeBasedQuantumSimulator:NJP12; @Kolovsky:WannierStarkStatesAndBlochOsc:PRA13; @Tarruell:MergingDiracPtsHoneycombLattice:N12; @Lopez-Gonzalez:EffectiveDiracEquationOptLatt:PRA14; @JimenezGarcia:TunableSpinOrbitCouplingDrivingUltracold:PRL15; @Zhang:RelativisticQuantumEffectsOfDiracUltracold:FPH12; @Kolovsky:SimulatingCyclotronBlochDyn:FPH12].
The present work combines these two driving forces in the ultracold-atom field. We propose a general method for simulating Dirac physics in a “tilted” one-dimensional optical lattice, a system that has been very useful since the early days of the quantum simulation (even before the term *quantum simulation* was introduced), for example for the observation of Bloch oscillations or the (equivalent) Wannier-Stark ladder [@BenDahan:BlochOsc:PRL96; @Niu:LandauZennerWS:PRL96; @Kolovsky:BECsOnTiltedLattices:PRA10; @Kolovsky:BlochOscillationsBECDynamicalInst:Quantum:PRA09; @Korsch:FractStab:EL00; @Korsch:LifetimeWS:PRL99; @Korsh:WSREV:IDiew:PREP02]. The realization of such a system can be obtained by applying a far-detuned laser standing wave that ultracold atoms see as a sinusoidal potential acting on their center of mass variables [@Cohen-TannoudjiDGO:AdvancesInAtomicPhysics::11]. If the atom’s de Broglie wavelength is comparable to the lattice constant $\mathsf{a}=\lambda_{L}/2$, where $\lambda_{L}=2\pi/k_{L}$ is the radiation wavelength (we use sans serif symbols for dimensioned quantities), the system is in the quantum regime, a condition easily realized for temperatures of the order of a few $\mu$K. In order to obtain a tilted potential, one can simply chirp one of the beams forming the standing wave: A linear shift of the frequency produces a quadratic displacement of the nodes of the standing wave; in the rest frame with respect to the nodes, an inertial constant force creates a tilt, that is, a potential of the form $\mathsf{V}_{ws}(\mathsf{x})=-\mathsf{V}_{1}\cos(2k_{L}\mathit{\mathsf{x}})+\mathsf{Fx}$, with $\mathsf{V}_{1}$ proportional to the radiation intensity and $\mathsf{F}$ (constant) proportional to the frequency chirp. This kind of setup is by now quite common in cold atom physics. In what follows, we shall use dimensionless units such that spatial coordinate $x=\mathsf{x}/\mathsf{a}$ is measured in units of the lattice potential step $\mathsf{a}$, energy in units of the so-called “recoil energy” $\mathsf{E}_{R}=\hbar^{2}k_{L}^{2}/2M$ ($M$ is the mass of the atom), time in units of $\hbar/\mathsf{E}_{R}$; $m^{*}=\pi^{2}/2$ is a reduced mass, and $\hbar=1$ is the reduced Planck constant [@Thommen:WannierStark:PRA02]. This defines the (dimensionless) *Wannier-Stark* Hamiltonian $$\begin{aligned}
H_{0} & = & \frac{p_{x}^{2}}{2m^{*}}-V_{1}\cos(2\pi x)+Fx,\label{eq:H0}\end{aligned}$$ with $F\equiv\mathsf{Fa}/\mathsf{E}_{R}$ and $V_{1}=\mathsf{V_{1}}/\mathsf{E}_{R}$ . A given well (labeled by its position $x=n$) may, depending on $V_{1}$ and $F$, host a number of bound eigenstates, called Wannier-Stark (WS) states [@Nenciu:WS:RMP91]. We note $\varphi_{n}^{\ell}(x)$ the $\ell^{\mathrm{th}}$ bounded state of well $n$ [^1] (see Fig. \[fig:WSstates\]), with the corresponding eigenenergy $E_{n}^{\ell}$. The WS potential $V_{ws}$$=-V_{1}\cos(2\pi x)+Fx$ of Eq. (\[eq:H0\]) is invariant under a *simultaneous* spatial translation by an integer multiple $m$ of the lattice constant $a=1$ *and* an energy shift of $mF$, implying that $\varphi_{n+m}^{\ell}(x)=\varphi_{n}^{\ell}(x-m)$ and $E_{n+m}^{\ell}=E_{n}^{\ell}+m\omega_{B}$. These eigenenergies form the so-called *Wannier-Stark ladder* of step $\omega_{B}=F$, called *Bloch frequency* ($=|\mathsf{F}|\mathsf{a}/\hbar$ in dimensioned units). In the present work we shall consider at most two such ladders: The ground ladder $\ell=g$ of lowest energy and the first excited ladder $\ell=e$.
A perturbation (for example a temporal or spatial modulation of $V_{1}$ or $\mathit{F}$), creates couplings between WS states and may generate interesting dynamics [@Thommen:WannierStark:PRA02; @Thommen:DirectedWavePacket2D:PRA11; @Zenesini:LandauZener:PRL09; @Kolovsky:BECsOnTiltedLattices:PRA10; @Goldman:PeriodicallyDrivenQuantumMatter:PRA15]. The aim of the present work is to take advantage of these possibilities to quantum-simulate Dirac dynamics. By an adequate choice of these temporal modulations one can obtain either a spinor-2 model or a spinor-4 Dirac equation.
After a brief summary of the Dirac equation in sec. \[sec:The-Dirac-equation\], sec. \[sec:GeneralModel\] introduces the general frame of our study; the spinor-2 model and spinor-4 models are described in sec. \[sec:Spinor-2\] and in sec. \[sec:Spinor-4\] respectively. Section \[sec:ExperimentalRelization\] discusses the experimental feasibility of our theoretical proposals and Sec. \[sec:Conclusion\] draws general conclusions of this work.
![\[fig:WSstates\]The Wannier-Stark system. Red energy levels $E_{n}^{g}$ and amplitude $c_{n}$ form the “ground” Wannier-Stark ladder, the corresponding spatial probability distribution $\left|\varphi_{n}^{g}(x)\right|^{2}$ (for site $n$) is shown as the bottom red curve. Blue levels of energy $E_{n}^{e}$ and amplitude $d_{n}$ form the “excited” WS ladder and the corresponding eigenstate $\left|\varphi_{n}^{e}(x)\right|^{2}$ is shown as the top blue curve. Levels in the same well are separated by an energy $\Delta$ and levels in the same ladder are separated by $\omega_{B}$, the Bloch frequency. The parameters used in this work are $V_{1}=6$, $F=1$, for which one finds numerically $\Delta=5.66$.](ws_states){width="0.9\columnwidth"}
Compared to other works demonstrating ways to simulate Dirac physics, an advantage of our method is its simplicity both from the experimental and the theoretical point of view. We use simple 1D optical lattices modulated in time, for which analytic calculations can be pushed quite far. The system is realizable experimentally with state-of-the-art techniques (see Sec. \[sec:ExperimentalRelization\]). In particular, no Raman or Zeeman transitions are necessary. Moreover, the approach developed here is general and can be easily adapted to different situations, as it will be seen below (and in future works).
\[sec:The-Dirac-equation\]The Dirac equation in a nutshell
==========================================================
The Dirac equation governs massive spin-1/2 particles [@Dirac:QuantumTheoryElectron:PRSLA28; @Pal:DiracMajoranaWeylFermions:AJP11]. As shown by Dirac, the requirement for relativistic invariance leads to the existence of spin and antiparticles; the theory deals with a *spinor-4*, that is, a 4-component state vector whose components are themselves wave functions: $$\boldsymbol{\text{\ensuremath{\psi}}}=\left(\begin{array}{c}
\psi_{1}(x,t)\\
\psi_{2}(x,t)\\
\psi_{3}(x,t)\\
\psi_{4}(x,t)
\end{array}\right).$$ A possible representation for the Dirac equation for free particles of mass $m$ is $H\boldsymbol{\text{\ensuremath{\psi}}}=i\partial_{t}\boldsymbol{\text{\ensuremath{\psi}}}$, with the Dirac Hamiltonian $$H=\left(\boldsymbol{\alpha}\cdot\boldsymbol{p}c+\beta mc^{2}\right)\label{eq:H_Dirac}$$ where $\alpha_{j}$ ($j=x,y,z$) and $\beta$ are Dirac matrices $$\alpha_{j}=\left(\begin{array}{cc}
0 & \sigma_{j}\\
\sigma_{j} & 0
\end{array}\right),\qquad\beta=\left(\begin{array}{cc}
\mathbf{1} & 0\\
0 & -\mathbf{1}
\end{array}\right)$$ with $\sigma_{j}$ the Pauli matrices, $\mathbf{1}$ the $2\times2$ identity matrix, $p_{j}=-i\partial/\partial x_{j}$ ($x_{j}=x,y,z$) the momentum operator, $c$ the velocity of light, and $\hbar=1$. For massive particles, in the rest frame of reference, the two upper components of the spinor-4 can be identified with the spin components of the (positive rest energy state) “particle” and the two bottom components with the spin of the “antiparticle” (negative rest energy state), but in a frame in which the particle is in motion, the components are mixed and no such distinction is possible; a spinor-4 description is necessary. However this “contamination” is small if $p\ll mc$. The general eigenvalues of the Dirac Hamiltonian are $\pm\left(p^{2}c^{2}+m^{2}c^{4}\right)^{1/2}$, the distinction between positive and negative eigenstates thus subsists (for a free particle) in all cases.
For a massive free particle, if the momentum is parallel to the spin, that is in the $z$ direction (the arbitrary quantization axis for the spin), then the Dirac equation couples $\psi_{1}$ to $\psi_{3}$ and $\psi_{2}$ to $\psi_{4}$. If the momentum is orthogonal to the spin (i.e. along the $x$- or the $y$-axis), it couples $\psi_{1}$ to $\psi_{4}$ and $\psi_{2}$ to $\psi_{3}$. Therefore, in both cases the quantum dynamics can be described by two spinor-2, obeying decoupled, equivalent equations. We can thus, for instance in the latter case, form the *spinor-2* $$\bar{\boldsymbol{\psi}}=\left(\begin{array}{c}
\psi_{2}\\
\psi_{3}
\end{array}\right)$$ which, from Eq. (\[eq:H\_Dirac\]), obeys the spinor-2 Dirac equation
$$i\partial_{t}\bar{\boldsymbol{\psi}}=c\sigma_{j}p_{j}\bar{\boldsymbol{\psi}}+mc^{2}\sigma_{z}\bar{\boldsymbol{\psi}}\label{eq:DiracSpinor2}$$
where $j=x$ or $j=y$. A similar equation holds for $(\psi_{1},\psi_{4})$. In presence of a magnetic field, however, the quantization axis is imposed by the field and for an arbitrary direction of the momentum $\boldsymbol{p}$, the four components are coupled and the particle is described by a true spinor-4.
Equation (\[eq:H\_Dirac\]) is the original Hamiltonian written by Dirac. This representation is well adapted to the case $p\ll mc$, where the first term is small compared to the second; if the first term is neglected, the Hamiltonian is diagonal. Other representations exist, e.g., the so-called *Weyl representation* corresponds to the Hamiltonian $$H_{W}=c\left(\begin{array}{cc}
\boldsymbol{\sigma}\cdot\boldsymbol{p} & 0\\
0 & -\boldsymbol{\sigma}\cdot\boldsymbol{p}
\end{array}\right)+\gamma_{0}mc^{2}.\label{eq:H_Weyl}$$ with
$$\gamma_{0}=\left(\begin{array}{cc}
0 & \mathbf{1}\\
\mathbf{1} & 0
\end{array}\right).$$ This representation is well suited for the ultra-relativistic limit $p\gg mc$, where the mass term $\gamma_{0}mc^{2}$ in Eq. (\[eq:H\_Weyl\]) becomes much smaller than the first one; neglecting the mass term leaves a diagonal form. For massless particles, the system separates into two subsets of equivalent equations, and can be described by a spinor-2, the so-called *Weyl fermion*. The above form implies that these particles are characterized by a well-defined projection of the spin along the particle’s momentum $\boldsymbol{\sigma}\cdot\boldsymbol{p}/|\boldsymbol{p}|$, a quantity called, as for photons, *helicity*.
\[sec:GeneralModel\]General model
=================================
In this section we introduce the general model leading from Wannier-Stark Hamiltonians of the form Eq. (\[eq:H0\]) to Dirac-like Hamiltonians. We shall consider a restricted state space of one or two ladders, i.e one or two WS states per potential well; the ground WS state (indexed by $\ell=g$) $\varphi_{n}^{g}(x)=\left\langle x\right.\left|\varphi_{n}^{g}\right\rangle $ in the well $n$, of energy $E_{n}^{g}=n\omega_{B}$, and the first excited WS state ($\ell=e$) $\varphi_{n}^{e}(x)$ of energy $E_{n}^{e}=E_{n}^{g}+\Delta$$=n\omega_{B}+\Delta$ of same well $n$ where $\varDelta$ is the energy offset between $g$ and $e$ levels in the same well (cf. Fig. \[fig:WSstates\]). We assume in the following that none of these eigenenergies are degenerate.
The general evolution of an arbitrary wave function can then be written in the form $$\begin{aligned}
\Psi(x,t)= & \sum_{n}\left[c_{n}(t)\exp\left(-iE_{n}^{g}t\right)\varphi_{n}^{g}(x)\right.\nonumber \\
& \left.+d_{n}(t)\exp\left(-iE_{n}^{e}t\right)\varphi_{n}^{e}(x)\right]\label{eq:Psi-general}\end{aligned}$$ with $c_{n}(0)=\left\langle \varphi_{n}^{g}\right.\left|\Psi(0)\right\rangle $ and $d_{n}(0)=\left\langle \varphi_{n}^{e}\right.\left|\Psi(0)\right\rangle $.
We introduce a perturbation $\bar{H}(t)$ so that our complete Hamiltonian becomes $H=H_{0}+\bar{H}(t)$, with $$\bar{H}(t)=-V_{1}\cos(2\pi x)f_{1}(t)+V_{2}\cos(\pi x)f_{2}(t)+V_{S}(x).\label{eq:Hoverbar}$$ A suitable choice of the frequencies present in $f_{1}(t)$ and $f_{2}(t)$ induces interactions between **states that are resonantly coupled,** as shown in Fig. \[fig:WScouplings\]. For example, the ground-ladder level $\left|\varphi_{n}^{g}\right\rangle $ is resonantly coupled to excited-ladder level $\left|\varphi_{n+1}^{e}\right\rangle $ by a modulation of frequency $\Delta+\omega_{B}$, and to $\left|\varphi_{n-1}^{e}\right\rangle $ by a modulation of frequency $\Delta-\omega_{B}$, and so on. The perturbation term $V_{2}(x,t)=V_{2}\cos(\pi x)f_{2}(t)$ has double spatial period, and $V_{S}(x)$ is a static contribution whose utility will appear below.
![\[fig:WScouplings\]Energy levels and couplings in the Wannier-Stark system. A modulation of frequency $\omega_{B}$ induces an *intra-ladder* coupling between adjacent wells. *Inter-ladder* couplings are induced by perturbation frequencies $\Delta-\omega_{B}$ ($n\rightarrow n-1$), $\Delta$ ($n\rightarrow n$), and $\Delta+\omega_{B}$ ($n\rightarrow n+1$).](couplings_ws1){width="0.9\columnwidth"}
Under the action of $\bar{H}$ the coupled equations of motion for the amplitudes $c_{n}$ and $d_{n}$ of Eq. (\[eq:Psi-general\]) are developed in App. \[sec:Derivation\] and have the form: $$\begin{aligned}
i\frac{d}{dt}c_{n}= & \sum_{r\in\mathbb{Z}}\left\{ \left\langle \varphi_{n}^{g}\right|\bar{H}\left|\varphi_{n+r}^{g}\right\rangle e^{-ir\omega_{B}t}c_{n+r}\right.\nonumber \\
& \left.+\left\langle \varphi_{n}^{g}\right|\bar{H}\left|\varphi_{n+r}^{e}\right\rangle e^{-ir\omega_{B}t}e^{-i\Delta t}d_{n+r}\right\} \nonumber \\
i\frac{d}{dt}d_{n}= & \sum_{r\in\mathbb{Z}}\left\{ \left\langle \varphi_{n}^{e}\right|\bar{H}\left|\varphi_{n+r}^{g}\right\rangle e^{-ir\omega_{B}t}e^{i\Delta t}c_{n+r}\right.\nonumber \\
& \left.+\left\langle \varphi_{n}^{e}\right|\bar{H}\left|\varphi_{n+r}^{e}\right\rangle e^{-ir\omega_{B}t}d_{n+r}\right\} .\label{eq:GeneralEvolutionEqs}\end{aligned}$$ The functions $f_{\alpha}(t)$ ($\alpha=1,2$) appearing in $\bar{H}$, contain modulation frequencies of the form $\omega_{j,q}=j\omega_{B}+q\triangle$ with $j\in\mathbb{Z}$ and $q=0,\pm1$ $$f_{\alpha}(t)=\sum_{j,q}\left(A_{j,q}^{(\alpha)}e^{ij\omega_{B}t}e^{iq\Delta t}\right)\label{eq:f(t)}$$ where the reality condition implies $A_{j,q}^{(\alpha)}=A_{-j,-q}^{(\alpha)*}$. A great advantage of the Wannier-Stark model, within the assumption that parameters are such that there are no intrinsically degenerated states, is that tuning the amplitudes $A_{j,q}^{(\alpha)}$ allows us to choose which pairs of states are coupled, providing a very flexible control of the dynamics. For instance, one sees that modulations with $q=0$ induce intra-ladder couplings ($g-g$ and $e-e$) and modulations with $q=\pm1$ induce inter-ladder couplings $e-g$; taking$j=0$ creates a coupling $g-e$ in the *same* well, whereas $j=1$ couples wells $n\rightarrow n+1$ and $j=-1$ couples $n\rightarrow n-1$.
In the resonant case, Eqs. (\[eq:GeneralEvolutionEqs\]) can be formally written as $$\begin{aligned}
i\frac{d}{dt}c_{n} & =\sum_{r}\left(T_{n,r}^{gg}c_{n+r}+T_{n,r}^{ge}d_{n+r}\right)\nonumber \\
i\frac{d}{dt}d_{n} & =\sum_{r}\left(T_{n,r}^{ee}d_{n+r}+T_{n,r}^{eg}c_{n+r}\right)\label{eq:general_discrete_model}\end{aligned}$$ (see App. \[sec:Derivation\]). The explicit form of coupling coefficients $T_{n,r}^{ab}$ ($a,b\in\left\{ e,g\right\} $) between the sites $n$ and $n+r$ depend on the overlap integrals, which, thanks to the properties of the WS states, are $$\left\langle \varphi_{n}^{g,e}\right|\cos(2\pi x)\left|\varphi_{n+r}^{g,e}\right\rangle =\left\langle \varphi_{0}^{g,e}\right|\cos(2\pi x)\left|\varphi_{r}^{g,e}\right\rangle ,$$ $$\left\langle \varphi_{n}^{g,e}\right|\cos(\pi x)\left|\varphi_{n+r}^{g,e}\right\rangle =(-1)^{n}\left\langle \varphi_{0}^{g,e}\right|\cos(\pi x)\left|\varphi_{r}^{g,e}\right\rangle .$$ One then obtains intra-ladder coupling as $$\begin{aligned}
T_{n,r}^{gg}=\left\langle \varphi_{n}^{g}\right|V_{S}\left|\varphi_{n}^{g}\right\rangle \delta_{r,0}- & V_{1}A_{r,0}^{(1)}\left\langle \varphi_{0}^{g}\right|\cos(2\pi x)\left|\varphi_{r}^{g}\right\rangle \nonumber \\
& +(-1)^{n}V_{2}A_{r,0}^{(2)}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{r}^{g}\right\rangle \nonumber \\
T_{n,r}^{ee}=\left\langle \varphi_{n}^{e}\right|V_{S}\left|\varphi_{n}^{e}\right\rangle \delta_{r,0}- & V_{1}A_{r,0}^{(1)}\left\langle \varphi_{0}^{e}\right|\cos(2\pi x)\left|\varphi_{r}^{e}\right\rangle \nonumber \\
& +(-1)^{n}V_{2}A_{r,0}^{(2)}\left\langle \varphi_{0}^{e}\right|\cos(\pi x)\left|\varphi_{r}^{e}\right\rangle .\label{eq:interLadderCoupls}\end{aligned}$$ and inter-ladder couplings $$\begin{aligned}
T_{n,r}^{ge}= & -V_{1}A_{r,1}^{(1)}\left\langle \varphi_{0}^{g}\right|\cos(2\pi x)\left|\varphi_{r}^{e}\right\rangle \nonumber \\
& +(-1)^{n}V_{2}A_{r,1}^{(2)}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{r}^{e}\right\rangle \nonumber \\
T_{n,r}^{eg}= & -V_{1}A_{r,-1}^{(1)}\left\langle \varphi_{0}^{e}\right|\cos(2\pi x)\left|\varphi_{r}^{g}\right\rangle \nonumber \\
& +(-1)^{n}V_{2}A_{r,-1}^{(2)}\left\langle \varphi_{0}^{e}\right|\cos(\pi x)\left|\varphi_{r}^{g}\right\rangle .\label{eq:intraLadderCoupls}\end{aligned}$$
This general model spans all cases we will consider in the present work. In Sec. \[sec:Spinor-2\] we show how to construct a quantum simulator for a spinor-2 Dirac equation, and in Sec. \[sec:Spinor-4\] we show how the full spinor-4 Dirac or Weyl equations can be synthesized.
\[sec:Spinor-2\]Spinor-2 model
==============================
Many interesting phenomena related to the Dirac equation can be illustrated with a simpler spinor-2. In order to construct a spinor-2 quantum simulator we restrict our system to the ground state ladder with “self” ($c_{n}$$\leftrightarrows c_{n}$) and nearest neighbors ($c_{n}\leftrightarrows c_{n\pm1}$) couplings. Inter-ladder transitions are set off by keeping only the $q=0$ term in Eq. (\[eq:f(t)\]), and we start with an initial condition $d_{n}(0)=0$ for all sites [^2], so that the excited ladder is never populated. We also set $V_{1}=V_{S}=0$ in Eq. (\[eq:Hoverbar\]). The perturbation thus contains only contributions of double spatial period $$\bar{H}=V_{2}f_{2}(t)\cos(\pi x)\label{eq:Spinor2H1}$$ with, in Eq. (\[eq:f(t)\]), $j=0,\pm1$, $q=0$, that is $$\begin{aligned}
f_{2} & (t)=A_{0,0}^{(2)}+A_{1,0}^{(2)}e^{i\omega_{B}t}+A_{-1,0}^{(2)}e^{-i\omega_{B}t}\nonumber \\
& =A_{0}+A_{1}e^{i\omega_{B}t}+A_{1}^{*}e^{-i\omega_{B}t},\label{eq:Dirac2_modulation}\end{aligned}$$ where, in the second line, we suppressed for simplicity the fixed indexes $q=0$ and $\alpha=2$. The remaining coupling parameters are then [\[]{}Eq. (\[eq:interLadderCoupls\])[\]]{} $$\begin{aligned}
T_{n,1}^{gg} & =(-1)^{n}V_{2}A_{1}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{1}^{g}\right\rangle \\
T_{n,0}^{gg} & =(-1)^{n}V_{2}A_{0}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{0}^{g}\right\rangle \\
T_{n,-1}^{gg} & =(-1)^{n}V_{2}A_{-1}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{-1}^{g}\right\rangle \\
& =-(-1)^{n}V_{2}A_{1}^{*}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{1}^{g}\right\rangle .\end{aligned}$$ Eqs. (\[eq:general\_discrete\_model\]) then imply
$$\begin{aligned}
i\frac{d}{dt}c_{n}= & (-1)^{n}V_{2}A_{0}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{0}^{g}\right\rangle c_{n}\nonumber \\
& (-1)^{n}V_{2}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{1}^{g}\right\rangle \left[A_{1}c_{n+1}-A_{1}^{*}c_{n-1}\right].\label{eq:Dirac2_discret}\end{aligned}$$
A key point for realizing a spinor-2 system is that the perturbation of double spatial period creates alternate sign couplings from site to site (see App. \[sec:Derivation\]). This has a dynamical effect that is clearly visible in the reciprocal space, where we define “spin” states as “odd site” and “even site” amplitudes $$\begin{aligned}
\bar{c}_{+}(k,t) & = & \sum_{n}e^{2ink}c_{2n}(t)\nonumber \\
\bar{c}_{-}(k,t) & = & \sum_{n}e^{i(2n+1)k}c_{2n+1}(t).\label{eq:creciproc}\end{aligned}$$ Taking, for simplicity, $A_{1}$ real in Eq. (\[eq:Dirac2\_discret\]), one obtains the following coupled set of equations $$\begin{aligned}
i\frac{d}{dt}\bar{c}_{+}(k,t) & = & E_{0}\bar{c}_{+}(k,t)-2i\Omega_{2}\sin k\:\bar{c}_{-}(k,t)\nonumber \\
i\frac{d}{dt}\bar{c}_{-}(k,t) & = & -E_{0}\bar{c}_{-}(k,t)+2i\Omega_{2}\sin k\:\bar{c}_{+}(k,t),\label{eq:discretModel}\end{aligned}$$ where we defined the frequency $\Omega_{2}=$$V_{2}A_{1}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{1}^{g}\right\rangle $ and the “self-energy” $E_{0}=$$V_{2}A_{0}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{0}^{g}\right\rangle $. These two equations show the emergence of an effective pseudo spinor-2 which in $k$-space is $$\bar{\boldsymbol{\psi}}=\left(\begin{array}{c}
c_{+}(k,t)\\
c_{-}(k,t)
\end{array}\right).$$ Looking for solutions in $\exp\left(-i\omega(k)t\right)$, the corresponding eigenenergies $\omega(k)$ are
$$\omega_{\pm}(k)=\pm\sqrt{E_{0}^{2}+4\Omega_{2}^{2}\sin^{2}k}.\label{eq:2bands}$$
For $E_{0}=0$, the positive and negative eigenenergies $\pm2\left|\Omega_{2}\sin k\right|$ are associated to the eigenspinor $$\bar{\boldsymbol{\psi}}_{\pm}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}
1\\
\pm i\mathrm{sgn}(\Omega_{2}k)
\end{array}\right),\label{eq:Spinor2Eigenstates}$$ where $\mathrm{sgn}(x)$ is the sign function. The linear, phonon-like, dispersion relation for $k\rightarrow0$ , $\omega_{\pm}(k)=\pm2\left|\Omega_{2}k\right|$, reproduces the spectrum of the relativistic massless spin-1/2 fermion. A “1D-conical intersection” occurs as the two branches coalesce at $k=0$, creating a so-called *Dirac point*.
In real space, if the even- $c_{2n}(x,t)$ and odd-site $c_{2n+1}(x,t)$ amplitudes vary slowly on the scale of the lattice step $a=1$, one can take the continuous limit of Eqs. (\[eq:Dirac2\_discret\]), and define the functions $c_{\pm}(x,t)$ as the spatial envelopes of the $c_{n}(x,t)$ (cf. App. \[sec:Derivation\]), leading to the spinor-2
$$\boldsymbol{\phi}=\left(\begin{array}{c}
c_{+}(x,t)\\
c_{-}(x,t)
\end{array}\right)$$ which obeys an equation $$i\partial_{t}\boldsymbol{\phi}=-2\Omega_{2}(-i\partial_{x})\sigma_{y}\boldsymbol{\phi}+E_{0}\sigma_{z}\boldsymbol{\phi}\label{eq:SyntheticDirac2}$$ of the same form as Eq. (\[eq:DiracSpinor2\]) if one sets $p_{y}=-i\partial_{x}$ (the labeling of the axes is obviously arbitrary). By comparing Eqs. (\[eq:SyntheticDirac2\]) and (\[eq:H\_Dirac\]) we can make the following identifications: $E_{0}=\overline{m}\overline{c}^{2}$ and $2\left|\Omega_{2}\right|=\overline{c}$, where $\overline{m}$ and $\overline{c}$ are the effective mass and speed of light which can be adjusted by changing the modulation amplitudes $A_{0}$ and $A_{1}$ in Eq. (\[eq:Dirac2\_modulation\]).
The validity of the model Eq. (\[eq:discretModel\]) can be numerically tested by comparison with the simulation of the exact Schrödinger equation corresponding to the Hamiltonian $H_{0}+\bar{H}$ with $\bar{H}$ given by Eq. (\[eq:Spinor2H1\]). We chose a broad initial wave packet, with amplitudes: $$\begin{aligned}
c_{2n} & =a_{+}G^{(k_{0})}(2n),\;c_{2n+1}\label{eq:CI}\\
& =a_{-}G^{(k_{0})}(2n+1),\end{aligned}$$ with $G^{(k_{0})}(n)=\left(2\pi/\sigma\right)^{1/2}$$\exp\left(-ink_{0}\right)$$\exp\left(-n^{2}/\sigma^{2}\right)$, $\sigma\gg1,$ with the normalization condition $\left|a_{+}\right|^{2}+\left|a_{-}\right|^{2}=1$. The initial spinor is thus $$\boldsymbol{\phi}_{0}=\left(\begin{array}{c}
a_{\text{+}}\\
a_{-}
\end{array}\right)G^{(k_{0})}(x)\leftrightarrow\left(\begin{array}{c}
a_{\text{+}}\\
a_{-}
\end{array}\right)\bar{G}^{(k_{0})}(k)\label{eq:CI_x}$$ where the first expression is in real and the second in momentum space, and $\bar{G}^{(k_{0})}(k)$ is a narrow Gaussian function centered at $k=k_{0}$.
The dashed lines in Fig. \[fig:dyn\_behavior\] show the dynamical behavior of a massless particle (setting $A_{0}=0$ in Eq. (\[eq:Dirac2\_modulation\]) leads to $E_{0}=0$) obtained from the above model [\[]{}cf. Eq. (\[eq:Dirac2\_discret\])[\]]{} at time $t=0,$ $150T_{B}$ and $300T_{B}$, where $T_{B}=2\pi/F$ is the Bloch-period. The initial spinor $(a_{+},a_{-})=(1,0)$ with $k_{0}\rightarrow0^{+}$, corresponds to a superposition of the positive energy eigenspinor $(1,i)/\sqrt{2}$ and the negative energy eigenspinor $(1,-i)/\sqrt{2}$ [\[]{}cf. Eq. (\[eq:Spinor2Eigenstates\])[\]]{} having opposite drift velocities $\pm v_{D}$ which from Eq. (\[eq:2bands\]) read $$v_{D}=\left|\frac{d\omega_{\pm}}{dk}\right|_{k_{0}}=2\left|\Omega_{2}\cos k_{0}\right|\simeq2\left|\Omega_{2}\right|.$$ The comparison with the solution of exact Schrödinger equation (full lines) shows a very good agreement up to $t=300T_{B}$. One can verify that splitting into two separate wave packets moving with opposite group velocities $\pm v_{D}$ which matches the expected theoretical value.
![Evolution of an initial wave packet with $k_{0}=0$, $\sigma=10$ and $(a_{+},a_{-})=(1,0)$ [\[]{}Eq. (\[eq:CI\])[\]]{} using the discrete model given by Eq. (\[eq:Dirac2\_discret\]) (dashed lines), at times $t=0$ (red bottom line), $150T_{B}$ (blue middle line) and $300T_{B}$ (top green line) and compared to the ö Parameters are $V_{1}=6$, $F=1,$ $\bar{H}(x,t)=0.5\cos\left(\pi x\right)\cos\left(\omega_{B}t\right)$ which give $\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{1}^{g}\right\rangle =-2.31\times10^{-2}$ (numerical value), and $\Omega_{2}=-5.4\times10^{-3}$. The numerical value of the drift velocity $v_{D}$ agrees with the theoretical value $v_{D}=2\left|\Omega_{2}\cos k_{0}\right|=1.1\times10^{-2}$[]{data-label="fig:dyn_behavior"}](valid_s2){width="7cm"}
One of the most characteristic effects associated to the Dirac equation for *massive* particles is the so-called *Zitterbewegung* (“trembling motion”), an interference effect between the positive and negative energy parts of the spinor resulting in a spatial jitter of the wave packet [@Vaishnav:ObservingZitterbewegungUltracold:PRL08]. Such an effect was recently observed in quantum simulators of the Dirac equation with trapped ions [@Gerritsma:QuantumSimulationDirac:N10], with ultracold atoms [@LeBlanc:ObservationZitterbewegungBEC:NJP13; @Qu:ObservationZitterbewegungBEC:PRA13], and in a photonic device [@Dreisow:ClassicalSimulationZitterbewegungPhotLatt:PRL10; @Longhi:PhotonicAnalogZitterbewegung:OL10]. Figure \[fig:ZBFalseColors\] shows the spatio-temporal behavior of a wave packet for a massive particle governed by Eqs. (\[eq:Dirac2\_discret\]), with a an initial spinor $(a_{+},a_{-})=2^{-1/2}(1,1)$ and $k_{0}=0$, corresponding to superposition of positive and negative energy eigenstates (as can be seen from Eq. (\[eq:discretModel\]) in the limit $k\rightarrow0$). In order to give a mass to the particle, we set $A_{0}\neq0$ in Eq. (\[eq:Dirac2\_modulation\]), so that $E_{0}\neq0$). We verified that the same spatio-temporal behavior is obtained from the exact Schrödinger equation.
![*Zitterbewegung* obtained from the discrete model Eq. (\[eq:Dirac2\_discret\]) with initial spinor $(a_{+},a_{-})=2^{-1/2}(1,1)$, $\sigma=10$ and $k_{0}=0$. The probability density is displayed in false colors. Potential parameters are the same as in Fig. \[fig:dyn\_behavior\] except $\bar{H}(x,t)=\cos\pi x\left(0.5\cos\omega_{B}t+0.005\right)$. The time-independent contribution $A_{0}=0.005$ in $\bar{H}$ leads to a mass term $E_{0}=4.6\times10^{-3}$.[]{data-label="fig:ZBFalseColors"}](zb_discr){width="0.9\columnwidth"}
From Eq. (\[eq:discretModel\]) one can obtain the evolution of the wave packet’s average position $$\begin{aligned}
\frac{d\left\langle x\right\rangle }{dt} & =\frac{1}{i\hbar}\left\langle \left[x,H\right]\right\rangle =-2\Omega_{2}\left\langle \sigma_{y}\right\rangle \\
& =2i\Omega_{2}\int dx\left(c_{+}^{*}(x,t)c_{-}(x,t)-\mathrm{c.c.}\right)\\
& =2i\Omega_{2}\int dk\left(\bar{c}_{+}^{*}(k,t)\bar{c}_{-}(k,t)-\mathrm{c.c.}\right).\end{aligned}$$ The fact that the oscillation depends on $c_{+}^{*}c_{-}$ (in real or momentum space) shows that the Zitterbewegung is due to the coherence between positive- and negative-energy states, confirming its physical interpretation as a quantum beat between odd- and even- site contributions (or positive and negative energy states in Dirac’s language). To the leading order in $k\approx0$ we find $$\frac{d\left\langle x\right\rangle }{dt}=\frac{i2\Omega_{2}}{\sqrt{1-iDt}}a_{+}^{*}a_{-}e^{2iE_{0}t}+\mathrm{c.c}\label{eq:x_moyen}$$ with $D=4\Omega_{2}^{2}/(E_{0}\sigma^{2})$. In this approximation, the amplitude of the oscillation is seen to be directly proportional to the initial coherence $a_{+}^{*}a_{-}$. The oscillation has frequency $2E_{0}$, as it is the case for the electron’s Zitterbewegung, and is slowly damped by diffusion effects with an effective coefficient $D$; note that the amplitude of the oscillation for $Dt\rightarrow0$, is $\left|\Omega_{2}\right|/E_{0}=(2\bar{m}\overline{c})^{-1}$, that is, half the dimensionless Compton wavelength, also in agreement with the Zitterbewegung of an electron. As shown in Fig. \[fig:ZB\_discret\], the numerical calculations of $\left\langle x(t)\right\rangle $ from the exact Schrödinger equation and from the discrete model are in excellent agreement and match the theoretical amplitude and period deduced from Eq. (\[eq:x\_moyen\]).
![Evolution of the average position $\left\langle x(t)\right\rangle $ for $0\leq t\leq5T_{ZB}$. Solid red line: Numerical result obtained from the exact Schrödinger equation. Blue circles: calculation from the discrete model Eq. (\[eq:Dirac2\_discret\]). Same parameters as in Fig. \[fig:ZBFalseColors\]. The Zitterbewegung period is in excellent agreement with the theoretical value, $T_{ZB}=2\pi/(2E_{0})=109T_{B}$, and its amplitude with the prediction $\left|\Omega_{2}\right|/E_{0}=1.17$. Due to diffusion, the amplitude is attenuated by a factor $(1+D^{2}t^{2})^{-1/2}\sim0.75$ at $t=5T_{ZB}$ as compared to its initial value. []{data-label="fig:ZB_discret"}](zb_avgx){width="0.9\columnwidth"}
The effective parameters $\overline{m}=E_{0}/4\Omega_{2}^{2}$ and $\overline{c}=2\left|\Omega_{2}\right|$ can be calculated from the parameters used in the above simulations. For atoms of mass $M$, they read, in dimensioned units, $\overline{\mathsf{c}}=2\left|\Omega_{2}\right|E_{R}d/\hbar$$=\left|\Omega_{2}\right|\left(2\pi^{2}\hbar/M\lambda_{L}\right)$ and $\overline{\mathsf{m}}=E_{0}E_{R}/c^{2}$$=\left(E_{0}/2\pi^{2}\Omega_{2}^{2}\right)M$. For cesium atoms and for potential parameters chosen in this section (*$E_{0}=4.6\times10^{-3}$, $\left|\Omega_{2}\right|=5.4\times10^{-3}$*) this leads to $\overline{\mathsf{m}}\sim9.3M$ and $\overline{\mathsf{c}}\sim1.33\times10^{-4}\left|\Omega_{2}\right|$$\sim7\times10^{-7}$m/s$\approx2\times10^{-3}v_{R}$, where $v_{R}=\sqrt{2E_{R}/M}$ is the atom recoil velocity .
\[sec:Spinor-4\]Spinor-4 model
==============================
We can also construct a full Dirac equation with a spinor-4. Using different coupling schemes we obtain either a Dirac-like equation in the standard representation or its analog in the Weyl representation. This beautifully illustrates the flexibility of the general model presented in Sec. \[sec:GeneralModel\].
Spinor-4 Dirac representation
------------------------------
In order to construct a spinor-4 in the Dirac representation, we consider both ground and excited WS ladders, nearest-neighbors inter-ladder couplings are set on and intra-ladder couplings are set off. The perturbation is thus of the form [\[]{}cf. Eq. (\[eq:Hoverbar\])[\]]{} $$\bar{H}=-V_{1}f_{1}(t)\cos(2\pi x)+V_{S}(x)$$ with the modulation function $$f_{1}(t)=A_{1,1}^{(1)}e^{i\omega_{B}t}e^{i\Delta t}+A_{1,-1}^{(1)}e^{i\omega_{B}t}e^{-i\Delta t}+\mathrm{c.c}.$$ From Eq. (\[eq:general\_discrete\_model\]) we obtain the equations of motion $$\begin{aligned}
i\frac{d}{dt}c_{n} & = & T_{n,0}^{gg}c_{n}+T_{n,1}^{ge}d_{n+1}+T_{n,-1}^{ge}d_{n-1}\\
i\frac{d}{dt}d_{n} & = & T_{n,0}^{ee}d_{n}+T_{n,1}^{eg}c_{n+1}+T_{n,-1}^{eg}c_{n-1}\end{aligned}$$ with $$T_{n,-1}^{eg}=\left(T_{n,1}^{ge}\right)^{*},\;T_{n,1}^{eg}=\left(T_{n,-1}^{g,e}\right)^{*}.$$ A Dirac-like equation is obtained if the coupling coefficients $T_{n,1}^{ge}=-T_{n,-1}^{ge}$ are imaginary and if $A_{1,1}^{(1)}\left\langle \varphi_{0}^{g}\right|\cos(2\pi x)\left|\varphi_{1}^{e}\right\rangle $$=-A_{1,-1}^{(1)}\left\langle \varphi_{1}^{g}\right|\cos(2\pi x)\left|\varphi_{0}^{e}\right\rangle $, a condition that is realized by tuning the modulation amplitudes $A_{1,\pm1}^{(1)}$ so that they exactly compensate for the difference in the overlap integrals. The static perturbation $V_{S}(x)$ is chosen to be translation-invariant with respect to the reference lattice constant $a=1$, so that $T_{n,0}^{gg}=\left\langle \varphi_{0}^{g}\right|V_{S}\left|\varphi_{0}^{g}\right\rangle \equiv V_{S}^{g}$ and $T_{n,0}^{ee}=\left\langle \varphi_{0}^{e}\right|V_{S}\left|\varphi_{0}^{e}\right\rangle \equiv V_{S}^{e}$ do not depend on $n$; the simple form used here is $V_{S}(x)\propto\cos\left(4\pi x\right)$. Thus $$\begin{aligned}
i\frac{d}{dt}c_{n} & = & E_{0}c_{n}+i\Omega_{1}\left(d_{n+1}-d_{n-1}\right)\nonumber \\
i\frac{d}{dt}d_{n} & = & -E_{0}d_{n}+i\Omega_{1}\left(c_{n+1}-c_{n-1}\right)\label{eq:Spinor4cndn}\end{aligned}$$ where the coupling $\Omega_{1}$ is given by $$i\Omega_{1}=T_{n,1}^{ge}=-V_{1}A_{1,1}^{(1)}\left\langle \varphi_{0}^{g}\right|\cos(2\pi x)\left|\varphi_{1}^{e}\right\rangle$$ (with $A_{1,1}^{(1)}$ imaginary) and the effective rest mass $E_{0}$, controlled by the static potential $V_{S}(x),$ is given by [^3] $$\begin{aligned}
E_{0} & =\frac{V_{S}^{g}-V_{S}^{e}}{2}.\label{eq:E_0_4spinor}\end{aligned}$$
The coupled equations (\[eq:Spinor4cndn\]) can be split into two independent sub-lattices corresponding to sites $c_{n}$ with $n$ even coupled to $d_{n}$ with $n$ odd and conversely. Hence, we can build a 4-component Wannier-Stark spinor $$\boldsymbol{\psi}=\left(\begin{array}{c}
c_{+}\\
c_{-}\\
d_{+}\\
d_{-}
\end{array}\right)\label{eq:spinor-4}$$ where $c_{\pm}(x,t)$ and $d_{\pm}(x,t)$ are the slowly varying envelopes of $c_{n}$ and $d_{n}$ for $n$ odd and $n$ even respectively (in close analogy with what has been done in the spinor-2 case, Sec. \[sec:Spinor-2\] and in App. \[sec:Derivation\]), giving $$i\partial_{t}\boldsymbol{\psi}=\left(E_{0}\beta-2\Omega_{1}\alpha_{x}p_{x}\right)\boldsymbol{\psi}\label{eq:Dirac-4}$$ which corresponds to the Dirac equation described by Eq. (\[eq:H\_Dirac\]). As stated in Sec. \[sec:The-Dirac-equation\], this equation can be decoupled into two equivalent sets $$i\frac{\partial}{\partial t}\left(\begin{array}{c}
c_{+}\\
d_{-}
\end{array}\right)=\left(E_{0}\sigma_{z}-2\Omega_{1}p_{x}\sigma_{x}\right)\left(\begin{array}{c}
c_{+}\\
d_{-}
\end{array}\right)$$ the other components $(c_{-},d_{+})$ following exactly the same equation. The corresponding dispersion relation is again $\omega_{\pm}(k)=\pm\left(E_{0}^{2}+4\Omega_{1}^{2}k^{2}\right)^{1/2}$, but each eigenvalue has now a double degeneracy. Note that this degeneracy can be lifted by adding other terms in $\bar{H}$ (for instance, terms proportional to $\cos(\pi x)$ which break translation invariance with respect to the lattice step $a=1$) and will be studied in a forthcoming paper.
The Zitterbewegung is described in the same way as for the spinor-2 case: $$\begin{aligned}
\frac{d\left\langle x\right\rangle }{dt}= & -2\Omega_{1}\left\langle \alpha_{x}\right\rangle \\
= & -2\Omega_{1}\intop dx\left[c_{+}^{*}(x,t)d_{-}(x,t)+\mathrm{c.c}\right.\\
& +\left.c_{-}^{*}(x,t)d(x,t)+\mathrm{c.c.}\right].\end{aligned}$$ In the simple case $p_{x}=0$ with a spatially broad initial wave packet $\boldsymbol{\psi}=$$2^{-1}(a_{+},a_{-},b_{+},b-)G^{(k_{0})}(x)$ one obtains $$\frac{d\left\langle x\right\rangle }{dt}=-2\Omega_{1}\left[\left(a_{+}^{*}b_{-}+a_{-}^{*}b_{+}\right)\,e^{2iE_{0}t}+\mathrm{c.c.}\right]$$ showing an oscillation amplitude proportional to $\Omega_{1}/E_{0}$, controlled by the initial coherence. The superposition of a “spin up particle” $(a_{\text{+}},a_{-})$ and a “spin down antiparticle” $(b_{\text{+}},b_{-})$ $\boldsymbol{\psi}=2^{-1}(1,1,1,1)$ [^4], leads to $\left\langle x(t)\right\rangle =-\left(\Omega_{1}/E_{0}\right)\sin(2E_{0}t)$. States with $\left(a_{+}^{*}b_{-}+a_{-}^{*}b_{+}\right)=0$, for instance $\boldsymbol{\psi}=2^{-1}(1,-1,1,1)$, display no Zitterbewegung. These results are illustrated in Fig. \[fig:ZB\_4spinor\]. The oscillations (blue line) obtained from the Schrödinger equation are in good agreement with the simulation of Eq. (\[eq:Spinor4cndn\]) displayed in red. On the time scale of a few Zitterbewegung periods $T_{ZB}=1/2E_{0}$, diffusion effects are here negligible (one finds $DT_{ZB}=4\Omega_{1}^{2}\pi/(E_{0}^{2}\sigma^{2})\sim10^{-2}$), but in contrast to the spinor-2 model displayed in Fig. \[fig:ZB\_discret\], the exact Schrödinger equation shows parasitic Landau-Zener tunneling into the continuum (due to the presence of populated excited states $d_{\pm}$), leading to a slow decrease in the spinor negative-energy amplitudes and thus to the oscillation amplitude. Fast, small-amplitude Rabi oscillations at frequency $\Omega_{1}\gg T_{ZB}^{-1}$ between ground and excited states are responsible for the apparent thickening of the blue line in Fig. \[fig:ZB\_4spinor\]: it is due to the asymmetry of the excited state $\varphi_{n}^{e}(x)$ with respect to the center of its well $n$ leading thus an average position which differs by a fraction of a lattice step as compared to the ground state average position. Note finally that the second initial condition spinor $\boldsymbol{\psi}=2^{-1}(1,-1,1,1)$ (green line) do not display Zitterbewegung, as expected.
![\[fig:ZB\_4spinor\]Evolution of the average position $\left\langle x(t)\right\rangle $ for $0\leq t\leq3T_{ZB}$. Exact Schrödinger equation (thick blue line) and discrete model Eq. (\[eq:Spinor4cndn\]) (red line) for an initial spinor $2^{-1}(1,1,1,1)$, $\sigma^{2}=500$ and $k_{0}=0$. The non-oscillating green line is the exact Schrödinger equation result for an initial spinor $2^{-1}(1,-1,1,1)$, which does not show Zitterbewegung. Potential parameters are $V_{1}=6$, $F=1$, $A_{1,1}^{(1)}=-5.0\times10^{-3}i$ and $A_{1,-1}^{(1)}=-7.5\times10^{-3}i$ giving $\Omega_{1}=-1.5\times10^{-3}$. The effective mass $E_{0}=1.6\times10^{-3}$ is generated by the potential $V_{S}(x)=5\times10^{-3}\cos(4\pi x)$. The Zitterbewegung period and amplitude agree with the theoretical values $T_{ZB}=2\pi/(2E_{0})=310T_{B}$ and $\left|\Omega_{1}\right|/E_{0}=0.93$. ](zb_4spinor){width="0.9\columnwidth"}
Spinor-4 Weyl representation
-----------------------------
A Dirac equation in the Weyl representation can be obtained with a different coupling scheme. The calculation follows the same lines as in the previous section, and we shall simply indicate the main steps below. We use the Hamiltonian $\bar{H}$ of Eq. (\[eq:Hoverbar\]) with $V_{S}(x)=0$ and with the modulations $$\begin{aligned}
f_{2}(t) & =A_{1,1}^{(2)}e^{i\omega_{B}t}e^{i\Delta t}+A_{1,-1}^{(2)}e^{i\omega_{B}t}e^{-i\Delta t}+\mathrm{c.c.}\\
f_{1}(t) & =A_{0,1}^{(1)}e^{i\Delta t}+\mathrm{c.c}.\end{aligned}$$ The general developments of Sec. \[sec:GeneralModel\] then lead to: $$\begin{aligned}
i\frac{d}{dt}c_{n} & =T_{n,1}^{ge}d_{n+1}+T_{n,-1}^{ge}d_{n-1}+T_{n,0}^{ge}d_{n}\\
i\frac{d}{dt}d_{n} & =T_{n,1}^{eg}c_{n+1}+T_{n,-1}^{eg}c_{n-1}+T_{n,0}^{eg}c_{n}.\end{aligned}$$ We then choose $A_{0,1}^{(1)}$ real and define the real parameter $$\begin{aligned}
E_{W} & =T_{n,0}^{ge}=T_{n,0}^{eg}\\
& =-A_{0,1}^{(1)}V_{1}\left\langle \varphi_{0}^{g}\left|\cos(2\pi x)\right|\varphi_{0}^{e}\right\rangle .\end{aligned}$$ Making the amplitudes $A_{1,\pm1}^{(2)}$ imaginary and tuning them in such a way that $A_{1,1}^{(2)}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{1}^{e}\right\rangle $$=A_{-1,1}^{(2)}\left\langle \varphi_{1}^{g}\right|\cos(\pi x)\left|\varphi_{0}^{e}\right\rangle $, one has $$\begin{aligned}
T_{n,1}^{ge} & =(-1)^{n}V_{2}A_{1,1}^{(2)}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{1}^{e}\right\rangle =i(-1)^{n}\Omega_{W}\\
T_{n,-1}^{ge} & =(-1)^{n}V_{2}A_{-1,1}^{(2)}\left\langle \varphi_{0}^{g}\right|\cos(\pi x)\left|\varphi_{-1}^{e}\right\rangle =-i(-1)^{n}\Omega_{W}\\
T_{n,1}^{eg} & =T_{n,1}^{ge}\\
T_{n,-1}^{eg} & =-T_{n,1}^{ge}\end{aligned}$$ and thus
$$\begin{aligned}
i\frac{d}{dt}c_{n}= & (-1)^{n} & i\Omega_{W}\left(d_{n+1}-d_{n-1}\right)+E_{W}d_{n}\nonumber \\
i\frac{d}{dt}d_{n}= & -(-1)^{n} & i\Omega_{W}\left(c_{n+1}-c_{n-1}\right)+E_{W}c_{n}.\label{eq:Spinor4cndn-1}\end{aligned}$$
The continuous limit of these two equations gives $$\begin{aligned}
i\partial_{t}c_{+}(x,t) & =-2\Omega_{W}p_{x}d_{-}+E_{W}d_{+}\\
i\partial_{t}d_{-}(x,t) & =-2\Omega_{W}p_{x}c_{+}+E_{W}c_{-}\\
i\partial_{t}d_{+}(x,t) & =2\Omega_{W}p_{x}c_{-}+E_{W}c_{+}\\
i\partial_{t}c_{-}(x,t) & =2\Omega_{W}p_{x}d_{+}+E_{W}d_{-}.\end{aligned}$$ The Weyl spinor-4 is thus defined as $\boldsymbol{\psi}_{W}(x,t)=\left(c_{+},d_{-},d_{+},c_{-}\right)$ and follows the equation $i\partial_{t}\boldsymbol{\psi}_{W}=H_{W}\boldsymbol{\psi}_{W}$ with $$H_{W}=-2\Omega_{W}\left(\begin{array}{cc}
\sigma_{x}p_{x} & 0\\
0 & -\sigma_{x}p_{x}
\end{array}\right)+E_{W}\left(\begin{array}{cc}
0 & 1\\
1 & 0
\end{array}\right)$$ which is the Dirac Hamiltonian in the Weyl representation, Eq. (\[eq:H\_Weyl\]), with $\boldsymbol{p}$ parallel to the $x$ axis.
Prospects for an experimental realization\[sec:ExperimentalRelization\]
=======================================================================
The present proposal of a quantum simulator of Dirac physics depends on techniques that are widely used experimentally. It is based on driving of ultracold atoms by modulations of a 1D optical lattice [@Cohen-TannoudjiDGO:AdvancesInAtomicPhysics::11; @Eckardt:AtomicQuantumGasesPeriodicallyDrivenOptLatt:RMP17], a technique that has been used from the early days of optical lattice physics, from the seminal experiments of observation of Bloch oscillations [@BenDahan:BlochOsc:PRL96] and the Wannier-Stark ladder [@Niu:LandauZennerWS:PRL96], dynamical localization and Anderson physics [@Moore:LDynFirst:PRL94; @Garreau:QuantumSimulationOfDisordered:CRP17], Landau-Zener tunneling [@Zenesini:LandauZener:PRL09; @Creffield:ExpansionOfMatterWavesDrivenOptLat:PRA10], to, more recently, the generation of artificial gauge fields [@Struck:TunableGaugePotentialDrivenLattices:PRL12; @Hauke:NonAbelianGaugeFieldsShakenOL:PRL12]. This makes our system particularly simple, both conceptually and experimentally, not involving, for example, Raman transitions or Zeeman-level manipulation. The main limitation of driven systems is the loss of atoms to the continuum via dynamic Landau-Zener coupling, which requires a careful optimization of the parameters. However, most effects described here survive to moderate losses, e.g. the Zitterbewegung, as it can be seen from Figs. \[fig:ZBFalseColors\] and \[fig:ZB\_4spinor\].
Several techniques have also been developed for atom detection, recently attaining single-site resolution thanks to the *quantum gas microscope* [@Greif:SiteResolvedImagingQuantumGas:S16; @Haller:SingleAtomImagingOfFermionsQuantumGasMicroscope:NP17] or near-field imaging [@Zimmermann:HighResolutionImagingUltracoldAtoms:NJP11]. For the particular situation studied here, a possible difficulty is the necessity of distinguishing the contribution of atoms located in even and odd sites. This can obviously be done site by site if single-site resolution is attained. Another, potentially more practical, way to do so is to *select* atoms from even/odd sites *before* detection. A possible strategy is the following: After the desired dynamics is studied (e.g. Zitterbewegung) the tilt of the potential is adiabatically tuned to zero, leaving only a flat trapping potential $\mathsf{V}_{a}\cos(2\mathsf{k}_{L}\mathit{\mathsf{x}})\exp(-\mathsf{y}^{2}/\mathsf{w}_{a}^{2})$, where we take into account the tranverse Gaussian profile of the laser beam. One then turns on adiabatically a transversely-shifted double-period potential $\mathsf{V}_{b}\cos(\mathsf{k}_{L}\mathit{x}+\varphi)\exp\left[-(\mathsf{y}-\mathsf{y}_{0})^{2}/\mathsf{w}_{b}^{2}\right]$; for $\varphi=0$ (resp. $\pi$) this potential will mostly affect even (resp. odd) sites. By adjusting the ratio $\mathsf{V}_{b}/\mathsf{V}_{a}$ and the shift $\mathsf{y}_{0}$ one can create a transverse “gutter” that induces losses in even (resp. odd) sites. One can then either detect the lost atoms, that is even- (resp. odd-)site population, or remaining atoms, i.e. odd- (resp. even-) site population. If the potential allows two Wannier-Stark ladders, one can adjust $\mathsf{V}_{a}$ before turning $\mathsf{V}_{b}$ on so as to induce losses in the excited WS ladder.
As a concrete example, consider the 4-spinor $\boldsymbol{\psi}=(c_{+},c_{-},d_{+},d_{-})$ Eq. (\[eq:spinor-4\]). In the particle-antiparticle context, the first component $c_{+}$ (for example) corresponds to the spin-up component for a particle at rest. In our quantum simulator it corresponds to the slowly varying envelope of the population of the ground ladder odd sites. Such quantity can be measured by first lowering the potential barrier (or increasing the slope) so that the atoms in the excited ladder escape, and then measuring the population $\left|c_{+}(x)\right|^{2}$ using the techniques described above. For the excited ladder components as $\left|d_{+}(x)\right|^{2}$ (odd sites), one can first remove even-site atoms using the method presented above, then lower the lattice depth allowing the excited-ladder atoms to escape while ground-ladder atoms remain trapped, and one detects the atoms that are leaking. The other components can be detected in a similar way.
\[sec:Conclusion\]Conclusion
============================
The present work introduces a general scheme based on the Wannier-Stark Hamiltonian, realizable with ultracold atoms in 1D optical lattices, allowing for the quantum simulation of Dirac physics, with a great flexibility in the choice of the parameters and of the properties of the resulting quantum simulator. One can control the effective mass, realize spinor-2 and spinor-4 Dirac equations both in the standard and in the Weyl representation. Our general model opens a large field of other possibilities which will be developed in forthcoming papers. For instance, the spinor-4 obtained as two degenerate spinor-2 systems can be studied in the case where the degeneracy is lifted, leading to flat bands or to spin $3/2$-like relativistic particles. The possibilities are even more exciting if one generalizes the above approach to higher dimensions. In dimension 2, one can use lattice temporal modulations to generate non-trivial artificial gauge fields [@Dalibard:ArtificialGaugePotentials:RMP11; @Lamata:RelativisticQuantumMechanicsTrappedIons:NJP11], and quantum simulate the Dirac particle interaction with electromagnetic fields (e.g. simulate the “gyromagnetic factor” of our “artificial electron”). If one uses interacting bosonic atoms in the mean-field limit described by the Gross-Pitaevskii equation, we can study Dirac physics in the presence of a nonlinearity, which can lead to quasiclassical “relativistic” chaos [@Thommen:ChaosBEC:PRL03]. All these possibilities put into evidence the power of ultracold atoms and optical potentials as quantum simulator for a rich variety of physical systems.
This work is supported by Agence Nationale de la Recherche (Grant K-BEC No. ANR-13-BS04-0001-01), the Labex CEMPI (Grant No. ANR-11-LABX-0007-01), as well as by the Ministry of Higher Education and Research, Hauts de France council and European Regional Development Fund (ERDF) through the Contrat de Projets Etat-Region (CPER Photonics for Society, P4S).
\[sec:Derivation\]Detailed derivation of the Dirac Hamiltonian
==============================================================
This Appendix presents in more detail the calculation leading to the coupled equations of Eqs. (\[eq:GeneralEvolutionEqs\]), and show how a a Dirac-like Hamiltonian can be obtained.
We consider here the wave packet of Eq. (\[eq:Psi-general\]) and project the Schrodinger equation, $id\Psi/dt=(H_{0}+\bar{H})\Psi$ on the WS states [\[]{}noting that $c_{n}=\left\langle \varphi_{n}^{g}\right.\left|\Psi\right\rangle \exp\left(i\omega_{B}t\right)$ and $d_{n}=\left\langle \varphi_{n}^{e}\right.\left|\Psi\right\rangle \exp\left(i\omega_{B}t+i\Delta t\right)$[\]]{}: $$\begin{aligned}
i\frac{d}{dt}c_{n}= & \sum_{r\in\mathbb{Z}}\left\{ \left\langle \varphi_{n}^{g}\right|\bar{H}\left|\varphi_{n+r}^{g}\right\rangle e^{-ir\omega_{B}t}c_{n+r}\right.\nonumber \\
& \left.+\left\langle \varphi_{n}^{g}\right|\bar{H}\left|\varphi_{n+r}^{e}\right\rangle e^{-ir\omega_{B}t}e^{-i\Delta t}d_{n+r}\right\} \nonumber \\
i\frac{d}{dt}d_{n}= & \sum_{r\in\mathbb{Z}}\left\{ \left\langle \varphi_{n}^{e}\right|\bar{H}\left|\varphi_{n+r}^{g}\right\rangle e^{-ir\omega_{B}t}c_{n+r}\right.\nonumber \\
& \left.+\left\langle \varphi_{n}^{e}\right|\bar{H}\left|\varphi_{n+r}^{e}\right\rangle e^{-ir\omega_{B}t}e^{-i\Delta t}d_{n+r}\right\} \label{eq:app_evolution}\end{aligned}$$ where the “free evolution” due to $H_{0}$ is canceled out. In the following, we take as an example the particular perturbation $$\bar{H}(t)=-V_{1}\cos(2\pi x)f_{1}(t)\label{eq:app_Hbar}$$ with $$f_{1}(t)=A_{1,1}^{(1)}e^{i\omega_{B}t}e^{i\Delta t}+A_{1,-1}^{(1)}e^{i\omega_{B}t}e^{-i\Delta t}+\mathrm{c.c}.\label{eq:app_modulation}$$ The results for any other choice of Hamiltonian can be obtained along the same lines.
From Eqs. (\[eq:app\_evolution\]), we then have: $$\begin{aligned}
i\frac{d}{dt}c_{n}=-V_{1} & \sum_{r\in\mathbb{Z}}\left\{ \left\langle \varphi_{n}^{g}\right|\cos2\pi x\left|\varphi_{n+r}^{g}\right\rangle f_{1}(t)e^{-ir\omega_{B}t}c_{n+r}\right.\nonumber \\
& \left.+\left\langle \varphi_{n}^{g}\right|\cos2\pi x\left|\varphi_{n+r}^{e}\right\rangle f_{1}(t)e^{-ir\omega_{B}t}e^{-i\Delta t}d_{n+r}\right\} \nonumber \\
i\frac{d}{dt}d_{n}=-V_{1} & \sum_{r\in\mathbb{Z}}\left\{ \left\langle \varphi_{n}^{e}\right|\cos2\pi x\left|\varphi_{n+r}^{g}\right\rangle f_{1}(t)e^{-ir\omega_{B}t}e^{i\Delta t}c_{n+r}\right.\nonumber \\
& \left.+\left\langle \varphi_{n}^{e}\right|\cos2\pi x\left|\varphi_{n+r}^{e}\right\rangle f_{1}(t)e^{-ir\omega_{B}t}d_{n+r}\right\} .\label{eq:GeneralEv}\end{aligned}$$ We now introduce two simplifying assumptions: (*i*) The overlap integrals between WS states rapidly shrink to zero for $\left|r\right|>1$ and we can thus consider only nearest neighbor couplings, and (*ii*) we neglect fast oscillations and keep only resonant contributions in Eq. (\[eq:GeneralEv\]), which eliminates intra-ladder couplings (assuming that $\Delta$ is far from $\omega_{B}$). We obtain: $$\begin{aligned}
i\frac{d}{dt}c_{n}= & -V_{1}A_{1,1}^{(1)}\left\langle \varphi_{0}^{g}\right|\cos2\pi x\left|\varphi_{1}^{e}\right\rangle d_{n+1}\nonumber \\
& -V_{1}A_{1,-1}^{(1)*}\left\langle \varphi_{0}^{g}\right|\cos2\pi x\left|\varphi_{-1}^{e}\right\rangle d_{n-1}\nonumber \\
i\frac{d}{dt}d_{n}= & -V_{1}A_{1,-1}^{(1)}\left\langle \varphi_{0}^{e}\right|\cos2\pi x\left|\varphi_{1}^{g}\right\rangle c_{n+1}\nonumber \\
& -V_{1}A_{1,1}^{(1)*}\left\langle \varphi_{0}^{e}\right|\cos2\pi x\left|\varphi_{-1}^{g}\right\rangle c_{n-1},\label{eq:app_coupled_discrete}\end{aligned}$$ that is, Eq. (\[eq:general\_discrete\_model\]) with intra-ladder couplings off and the inter-ladder couplings of Eq.(\[eq:intraLadderCoupls\]). In Eq. (\[eq:app\_coupled\_discrete\]), we took into account the reality condition of $f_{1}(t)$, $A_{1,-1}^{(1)*}=A_{-1,1}^{(1)}$, $A_{1,1}^{(1)*}=A_{-1,-1}^{(1)}$, and the properties of overlap integrals: $$\begin{aligned}
\left\langle \varphi_{n}^{g}\right|\cos2\pi x\left|\varphi_{n\pm1}^{e}\right\rangle & =\int\varphi_{n}^{g}(x)\varphi_{n\pm1}^{e}(x)\cos(2\pi x)dx\\
& =\int\varphi_{0}^{g}(x-n)\varphi_{\pm1}^{e}(x-n)\cos(2\pi x)dx\\
& =\left\langle \varphi_{0}^{g}\right|\cos2\pi x\left|\varphi_{\pm1}^{e}\right\rangle \end{aligned}$$ and $$\begin{aligned}
\left\langle \varphi_{n}^{e}\right|\cos2\pi x\left|\varphi_{n\pm1}^{g}\right\rangle & =\left\langle \varphi_{0}^{e}\right|\cos2\pi x\left|\varphi_{\pm1}^{g}\right\rangle \\
& =\left\langle \varphi_{0}^{g}\right|\cos2\pi x\left|\varphi_{\mp1}^{e}\right\rangle \end{aligned}$$ where the translational invariance of WS states was used.
In the general framework of Sec. \[sec:GeneralModel\], other contributions to the coupling coefficients may have to be considered in Eqs. (\[eq:interLadderCoupls\]) and (\[eq:intraLadderCoupls\]), and can be obtained in the same way. Note that if a perturbation component proportional to $\cos(\pi x)$ is present, the overlap integrals are $$\begin{aligned}
\left\langle \varphi_{n}^{g}\right|\cos\pi x\left|\varphi_{n\pm1}^{e}\right\rangle & =\int\varphi_{n}^{g}(x)\varphi_{n\pm1}^{e}(x)\cos(\pi x)dx\\
& =\int\varphi_{0}^{g}(x)\varphi_{\pm1}^{e}(x)\cos(\pi x+\pi n)dx\\
& =(-1)^{n}\left\langle \varphi_{0}^{g}\right|\cos\pi x\left|\varphi_{\pm1}^{e}\right\rangle \\
\left\langle \varphi_{n}^{e}\right|\cos\pi x\left|\varphi_{n\pm1}^{g}\right\rangle & =\pm(-1)^{n}\left\langle \varphi_{0}^{e}\right|\cos\pi x\left|\varphi_{\pm1}^{g}\right\rangle ,\end{aligned}$$ and thus depend on the even or odd character of the site label $n$.
A Dirac-like equation can be derived from Eq. (\[eq:app\_coupled\_discrete\]). If we tune the modulation coefficients such that $$A_{1,1}^{(1)}\left\langle \varphi_{0}^{g}\right|\cos2\pi x\left|\varphi_{1}^{e}\right\rangle =-A_{1,-1}^{(1)*}\left\langle \varphi_{0}^{g}\right|\cos2\pi x\left|\varphi_{-1}^{e}\right\rangle$$ we find $$\begin{aligned}
i\frac{d}{dt}c_{n}= & -V_{1}A_{1,1}^{(1)}\left\langle \varphi_{0}^{g}\right|\cos2\pi x\left|\varphi_{1}^{e}\right\rangle \left[d_{n+1}-d_{n-1}\right],\\
i\frac{d}{dt}d_{n}= & A_{1,1}^{(1)*}V_{1}\left\langle \varphi_{1}^{e}\right|\cos2\pi x\left|\varphi_{0}^{g}\right\rangle \left[c_{n+1}-c_{n-1}\right]\end{aligned}$$ and assuming imaginary amplitudes (i.e choosing the phase of the modulations suitably) gives $$\begin{aligned}
i\frac{d}{dt}c_{n}= & i\Omega\left[d_{n+1}-d_{n-1}\right]\nonumber \\
i\frac{d}{dt}d_{n}= & i\Omega\left[c_{n+1}-c_{n-1}\right]\label{eq:app_Dirac_discrete}\end{aligned}$$ where, $V_{1}A_{1,1}^{(1)}\left\langle \varphi_{0}^{g}\right|\cos2\pi x\left|\varphi_{1}^{e}\right\rangle =-i\Omega$. Note that these equations correspond to two independent sub-lattices, the amplitudes $c_{n}$ for $n$ odd being coupled to $d_{n}$ for $n$ even, and conversely.
We thus conclude that the “suitable” form of the potential corresponding to Eqs. (\[eq:app\_Hbar\]) and (\[eq:app\_modulation\]) leading to Eq. (\[eq:app\_Dirac\_discrete\]) is $$f_{1}(t)=2a_{1,1}^{(1)}\sin\left(\omega_{B}t+\Delta t\right)+2a_{1,-1}^{(1)}\sin\left(\omega_{B}t-\Delta\right)$$ where $a_{1,\pm1}=-iA_{1,\pm1}$ are real amplitudes with relative weight obeying $a_{1,1}^{(1)}\left\langle \varphi_{0}^{g}\right|\cos2\pi x\left|\varphi_{1}^{e}\right\rangle $$=a_{1,-1}\left\langle \varphi_{0}^{g}\right|\cos2\pi x\left|\varphi_{-1}^{e}\right\rangle $.
We can take the continuous limit of these equations assuming that the amplitudes $c_{n},$ $d_{n}$ are slowly varying at the scale of the lattice step. We can then introduce the smooth envelopes associated to each sub-lattice: $c_{n}(t)=c_{\pm}(x=n,t)$ (the sign $\pm$ corresponding to $n$ odd or even) and $d_{n}(t)=d_{\pm}(x=n,t)$ . We then get $$\begin{aligned}
i\partial_{t}c_{\pm}= & i2\Omega\frac{\partial d_{\mp}(x,t)}{\partial x}\\
i\partial_{t}d_{\pm}= & i2\Omega\frac{\partial c_{\mp}(x,t)}{\partial x}\end{aligned}$$ This last expression written as a Dirac equation for a massless particle corresponding to Eq. (\[eq:Dirac-4\]) with $E_{0}=0$.
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[^1]: Technically speaking, in infinite space, WS states are “resonances” metastable states [@Nenciu:WS:RMP91], but for our present purposes they can be considered as stationary states as long as the duration of the experiment is much shorter than their lifetime. We checked numerically the validity of this hypothesis throughout this work.
[^2]: Experimentally this can be done by trapping the atoms on a shallow optical lattice and increasing adiabatically the lattice amplitude to the desired level.
[^3]: We simply redefined $\left(c_{n},d_{n}\right)$ as $\left(c_{n},d_{n}\right)\exp\left[-i(V_{S}^{g}+V_{S}^{e})t/2\right]$.
[^4]: This distinction is meaningful only if $p\ll mc$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The overtaking collisions of ion-acoustic solitons (IASs) in presence of trapping effects of electrons are studied based on a fully kinetic simulation approach. The method is able to provide all the kinetic details of the process alongside the fluid-level quantities self consistently. Solitons are produced naturally by utilizing the chain formation phenomenon, then are arranged in a new simulation box to test different scenarios of overtaking collisions. Three achievements are reported here. Firstly, simulations prove the long-time life span of the ion-acoustic solitons in the presence of trapping effect of electrons (kinetic effects), which serves as the benchmark of the simulation code. Secondly, their stability against overtaking mutual collisions is established by creating collisions between solitons with different number and shapes of trapped electrons, i.e. different trapping parameter. Finally, details of solitons during collisions for both ions and electrons are provided on both fluid and kinetic levels. These results show that on the kinetic level, trapped electron population accompanying each of the solitons are exchanged between the solitons during the collision. Furthermore, the behavior of electron holes accompanying solitons contradicts the theory about the electron holes interaction developed based on kinetic theory. They also show behaviors much different from other electron holes witnessed in processes such as nonlinear Landau damping (Bernstein-Greene-Kruskal -BGK- modes) or beam-plasma interaction (like two-beam instability).'
author:
- 'S. M. Hosseini Jenab[^1]'
- 'F. Spanier [^2]'
title: |
Simulation study of overtaking of ion-acoustic solitons\
in the fully kinetic regime
---
Introduction {#Sec_Intro}
============
Solitons are defined as nonlinear localized solutions which are stable against mutual collisions, namely head-on and overtaking collisions[@Wadati2001841]. These structures emerge after the collisions without any changes in their physical features (such as velocity, height, width and shape in both velocity and spatial directions). Stability is defined as their physical features remaining the same before and after the collisions (and not during collisions). As far as physical features are considered, solitons act as pseudo-particles, hence the suffix “ton” in the word “soliton”[@Zabusky1965]. During collisions, however, features of two solitons participating in collision merge and overlap. Hence, a collision between two solitons is interpreted as their overlapping in spatial direction. Solitons can be regarded as a subclass of solitary waves, nonlinear localized structures propagating steadily in a dissipative medium. “Stability against mutual collisions” distinguishes the two concepts apart. Although in the context of plasma physics, especially for ion-acoustic solitons (IASs), these two terms have been used interchangeably.
Historically, solitons were discovered in context of plasma fluid simulation[@Zabusky1965], and since then theoretical and simulation approaches based on “fluid framework” have played a major role in studying different forms of solitons in plasma physics[@kuznetsov1986soliton; @shafranov2012reviews]. In the fluid framework, densities of plasma species are considered as the their physical features. Temporal evolutions of these densities, coupled by Poisson’s equation, produce the whole physical picture of solitons by providing the electric potential and field for any given time. However, densities are reduced forms of distribution functions and hence kinetic effects, related to distribution functions, stays beyond the scope of fluid framework. In the experimental studies, the kinetic details are either not reported or overlooked [@ikezi1970formation; @lonngren1983soliton; @tran1979ion]. Theoretical approaches such as the Sagdeev pseudo-potential[@Sagdeev] and the BGK method[@bernstein1957exact] supply a chance to incorporate kinetic effects into solitons studies. However, this comes at the price of losing the temporal evolution, which consequently means that stability against mutual collisions can not be studied. These methods are able to discover the nonlinear solutions, but can’t prove/disprove them as solitons.
To address these limitations in the studies of solitons, we have employed a fully kinetic simulation approach, which can encompass the kinetic effects and supply the temporal evolution of the physical features. In this simulation method the dynamics of the plasma species, e.g. electrons and ions, are followed based on solving the Vlasov equations. Therefore, the dynamics of distribution functions, kinetic details, are provided and the fluid-level quantities, i.e. densities of each species, are achieved self-consistently. This theoretical framework, removes the limitations faced in the previous studies such as small-amplitude limitation of reductive perturbation method i.e. KdV model, the absence of temporal evolution in the Sagdeev’s approach or the BGK method and the lack of kinetic effects in fluid model.
One of the major kinetic effects in the context of solitons in plasmas is called *trapping effect*. Particles in a certain range of energy resonate with the potential well imposed by a soliton, and oscillate inside the well, hence are trapped by the soliton. Schamel has utilized a self-developed version of the BGK method[@schamel_1] to integrate the trapping effect into the study of ion-acoustic solitons (IASs). Trapping effect (controlled by the trapping parameter $\beta$) introduces its own extra nonlinearity to the KdV equation. Comparing this new nonlinearity with usual nonlinearity of KdV equation results in three regimes[@schamel_3], each having their own fluid dynamical equations, namely KdV, Schamel-KdV and Schamel equations. However it remains inconclusive, if the solutions should be regarded as solitons, i.e. if they can survive mutual collisions, due to the absence of the temporal evolution in the BGK method. Different fluid-based simulation methods have been employed to respond this concern[@Kakad2013; @Sharma2015]. However, their results are limited since trapping effect can’t be comprehensively considered by fluid models specially in case of large-amplitude solitons[@Kakad20145589]. Moreover, Particle-in-cell (PIC) simulation methods suffer from their inherit noise level and can’t provide a clear view of the kinetic-level interactions[@Kakad20145589; @Qi20153815].
Here, our main focus is on the overtaking collisions of ion-acoustic solitons in presence of electron trapping in two-species plasmas. However, we need to show that simulation method is adjusted and fine-tunned for long-time nonlinear simulations. Therefore, firstly the results of IASs propagation for long-time runs are presented in Sec.\[SubSec\_propagation\]. This section stands as the benchmarking of the simulation code which can assure the reliability of the simulation results in the long-time nonlinear stage. Afterwards, in Sec.\[SubSec\_Stability\] the question about the stability of IASs against collisions (here overtaking collisions) is addressed. Collisions between solitons with different sizes and trapping parameter are presented covering different regimes proposed by Schamel. Note that interaction time of head-on collisions are shorter than of overtaking ones. Hence stability against overtaking collisions should be regarded as the ultimate test for the stability.
Furthermore, the details of a mutual overtaking collision between two IASs is presented in the phase space by showing the temporal evolution of distribution function of both species (Sec.\[SubSec\_During\]). In case of electron distribution function, kinetic details of two electron holes accompanying IASs during overtaking collisions are studied alongside other types of shapes. The interaction between the holes reveals two contradictions with the existing theories. Firstly, the results unveil the complexity of interaction of solitons on kinetic level, i.e. particle exchanging, in contrast to the simple comprehension provided by the fluid framework. Secondly, it shows that the electron holes don’t merge while exchanging trapped particles. Electron holes (trapped electrons population) has shown a tendency of merging in context of other phenomena in plasmas such as BGK modes, two stream instability. Our results show a very different tendency among the electron holes when they are accompanying IASs.
Basic Equations and Numerical Scheme {#Sec_Model}
====================================
Equations and quantities are normalized based on table \[table\_normalization\]. Hence the normalized Vlasov-Poisson set of equations reads as follow: $$\begin{gathered}
\frac{\partial f_s(x,v,t)}{\partial t}
+ v \frac{\partial f_s(x,v,t)}{\partial x}
\\ + \frac{q_s}{m_s} E(x,t) \frac{\partial f_s(x,v,t)}{\partial v}
= 0, \ \ \ s = i,e
\label{Vlasov}\end{gathered}$$ $$\frac{\partial^2 \phi(x,t)}{\partial x^2} = n_e(x,t) - n_i(x,t)
\label{Poisson}$$ where $s = i,e$ represents the corresponding species. They are coupled by density integrations for each species to form a closed set of equations: $$\begin{aligned}
n_s(x,t) &= n_{0s} N_s(x,t) \\
N_s(x,t) &= \int f_s(x,v,t) dv
\label{density}\end{aligned}$$ in which $N$ stands for the number density. Note that by this normalization, ion sound velocity and electron plasma frequency are $v_C = 8.06$ and $\omega_{pe} = 10.0$, respectively.
----------- -------- ---------------------- -------------------------------------------------------------------------------
Name formula
Time $\tau$ ion plasma frequency $\omega_{pi} = {\big(\frac{n_{i0} e^2}{m_i \epsilon_0}\big)^{\frac{1}{2}} }$
Length $L$ ion Debye length $\lambda_{Di} = \sqrt{ \frac{\epsilon_0 K_B T_i}{n_{i0} e^2} }$
Velocity $v$ ion thermal velocity $v_{th_i} = \sqrt{\frac{K_B T_i}{m_i}}$
Energy $E$ [——-]{} $K_B T_i$
Potential $\phi$ [——-]{} $\frac{K_B T_i}{e}$
Charge $q$ elementary charge $e$
Mass $m$ ion mass $m_i$
----------- -------- ---------------------- -------------------------------------------------------------------------------
: Normalization of quantities.
\[table\_normalization\]
The Schamel distribution function [@schamel_1] has been utilized as the initial distribution function to invoke a self-consistent hole in phase space and a localized compressional density profile in density. The normalized version of it reads as follow: @size[8.3]{}@mathfonts @@@\#1$$f_{s}(v) =
\left\{\begin{array}{lr}
A \ exp \Big[- \big(\sqrt{\frac{\xi_s}{2}} v_0 + \sqrt{E(v)} \big)^2 \Big] &\textrm{if}
\left\{\begin{array}{lr}
v<v_0 - \sqrt{\frac{2E_{\phi}}{m_s}}\\
v>v_0+\sqrt{\frac{2E_{\phi}}{m_s}}
\end{array}\right. \\
A \ exp \Big[- \big(\frac{\xi_s}{2} v_0^2 + \beta_s E(v) \big) \Big] &\textrm{if}
\left\{\begin{array}{lr}
v>v_0-\sqrt{\frac{2E_{\phi}}{m_s}} \\
v<v_0 + \sqrt{\frac{2E_{\phi}}{m_s}}
\end{array}\right.
\end{array}\right.
\label{Schamel_Dif}$$in which $A = \sqrt{ \frac{\xi_s}{2 \pi}} n_{0s}$, and $\xi_s = \frac{m_s}{T_s}$ are the amplitude and the normalization factor respectively. $E(v) = \frac{\xi_s}{2}(v-v_0)^2 + \phi\frac{1}{T_s q_s}$ represents the (normalized) energy of particles. $v_0 = 0$ stands for the velocity of the initial density perturbation (IDP).
$\beta$ is the so called *trapping parameter* which describes the distribution function of trapped particles around $v_0$. Based on $\beta$, Fig. \[DF\_Beta\] shows that the distribution function of trapped particles can take three different types of shapes, namely *hole* ($\beta<0$), *plateau* ($\beta = 0$) and *hump* ($\beta>0$).
The hole in phase space produces a compressional pulse in the number density which is in turn introduces a localized structure in the electric potential. Early in temporal evolution, the initial density perturbation (IDP) breaks into two oppositely moving density perturbations (MDP) due to the symmetry in the velocity direction of the distribution function [@jenab2016IASWs]. Then these MDPs split into number of IASs through the chain formation process. Note that the resulted IASs are not mathematical structures imposed to the system and are produced self-consistently. Then, these solitons are isolated and inserted into new simulation boxes to create different scenarios of overtaking collisions. In other words, the distribution function of these self-consistent solitons are inserted into certain places in the spatial direction, while the rest of the new simulation box is filled by the unperturbed Maxwellian distribution function: $ f_m = \sqrt{ \frac{1}{2 \pi} \exp \big( - \frac{v^2}{2} \big)}.$
The constant parameters which remain fixed through all of our simulations include: mass ratio $\frac{m_i}{m_e} = 100$, temperature ratio $\frac{T_e}{T_i} = 64$ and $L = 1024$, where L is the length of the simulation box. The periodic boundary condition is adopted on the spatial direction.
The kinetic simulation approach utilized here is based on the Vlasov-Hybrid Simulation (VHS) method in which a distribution function is modeled by phase points[@nunn1993novel; @kazeminezhad2003vlasov; @jenab2011preventing]. The arrangement of phase points in the phase space at each time step provides the distribution function, and hence all the kinetic momentums e.g. density, entropy and etc. The initial value of distribution function associated to each of the phase points stay intact during simulation which guarantees the positiveness of distribution function under any circumstances. Deviation of the conservation laws, e.g. conservation of entropy and energy, are closely monitored to stay below one percent.
Each time step routine of the simulation consists of three steps which are summarized below. First step includes integrating distribution functions to achieve number densities. Then, by plugging these densities into Poisson’s equation, electric potential are obtained. Poisson’s equation is solved here based on a parallelized multi-grid method. On the third step, the Vlasov equation for each species is solved based on characteristics method utilizing leap-frog scheme to find the new arrangement of the phase points in the phase space for the next step.
Results and Discussion {#Sec_Results}
======================
Long time propagation of IA solitons {#SubSec_propagation}
------------------------------------
As the first step, we have considered the long-time simulation of solitons propagation, aimed at examining the reliability of the simulation approach utilized here. Figs.\[Fig\_PlR\_ZsR\_num\] and \[Fig\_PlR\_ZsR\_PhaseSpace\] present the results of two solitons with almost the same speed propagating in a periodically bounded plasma. They have different sizes and values of trapping parameter, e.g. $\beta = -0.1, 0.2$.
Fig. \[Fig\_PlR\_ZsR\_num\] displays the fluid-level physical features, i.e. number densities, for both species. Since number density is the starting point for all the other quantities such as electric field and potential, their stabilities during propagation guarantee these other quantities stability. Fig. \[Fig\_PlR\_ZsR\_PhaseSpace\], on the other hands, deals with the kinetic details of the distribution function of both species during the solitons’ long-time propagation. The overall shape and internal structure stay the same up-to $\tau = 1000$. Combining these two figures, all the physical features (such as velocity, height and width in both spatial and velocity directions) of a soliton are preserved by this simulation method for a long time. Considering the ion distribution function around the solitons, trapping effect can’t be seen and hence the simulations here are in the regime without ion trapping effect.
The example shown here, provides a benchmarking test for the simulation code. Firstly, they prove the the process of inserting solitons distribution functions into a new simulation box has not resulted in any numerical instability. Secondly, it shows that the routines in time step of the simulation code is capable of handling a long-time simulation for the nonlinear stage and justifies the results of the next sections to be reliable. Note that solitons are extremely sensitive nonlinear structures, and any numerical errors in the process of simulation temporal evolution can easily cause them to become unstable and deform or disappear from the simulation. Finally, this example clearly indicates that the ion-acoustic solitons in the presence of the electron trapping can exist (for more details see Ref. 6). The deviation of the total energy and entropy up-to $\tau = 1000$ are $0.2\%$ and $0.05\%$, respectively.
[c c]{} & \
&\
&\
\
\
\
[cc]{} & \
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\
Stability of IASs in overtaking collisions {#SubSec_Stability}
-------------------------------------------
Two cases of overtaking collisions between solitons are presented here. In the first case (Figs. \[Fig\_NlR\_NsR\_num\] and \[Fig\_NlR\_NsR\_PhaseSpace\]) IASs with same trapping parameters $\beta = -0.1$ collide while the larger soliton overtaking the smaller one. Second case is dedicated to solitons with different trapping parameter ($\beta = -0.1$ and $\beta = 0$). The results prove the stability of IASs in presence of trapping effect of electrons against mutual overtaking collisions.
In case of $\beta = -0.1$, the overtaking collision of IASs happens while they are accompanied by two holes in electron distribution function. Fig. \[Fig\_NlR\_NsR\_num\] shows the number densities, fluid-level features, before and after the collision. On the fluid level, the stability against mutual collision can be witnessed, since solitons’ features such as hight, shape and width remain the same before and after the collision. Note that the collision is defined as the time interval of solitons when they are overlapping each other. Hence the times $\tau < 200$ and $\tau > 600$ are considered as before and after the collision, respectively. During collision/overlapping (for example at time $\tau = 400$ shown in Fig. \[Fig\_NlR\_NsR\_num\]), the two solitons lose their distinctive shape and merge to some extent. However, the concept of stability is defined for solitons features before and after collisions and not during them.
Furthermore, the kinetic details of both species distribution functions before and after collision is shown in Fig. \[Fig\_NlR\_NsR\_PhaseSpace\]. Although the overall shape, width in velocity and spatial directions remain the same, the internal structures differ. The contrast of colors clearly indicates that the after the collision, each of the electron holes has acquired some of the trapped electrons’ population of the opposite electron hole. Each trapped population can be recognized after the collision by their core (inner part). In other words, the core remains untouched during collision. However, the outer (parts) are exchanged between the two electron holes. In case of ions distribution function no change can be seen before and after the collision.
-- --
-- --
Figs.\[Fig\_NlR\_ZsR\_num\] and \[Fig\_NlR\_ZsR\_PhaseSpace\] display the results of an overtaking collision between two solitons with different trapping parameter, namely $\beta = 0$ and $\beta = -0.1$. Hence on the kinetic level of electron, the collision takes place between a plateau ($\beta = 0$) and a hole ($\beta = -0.1$) accompanying small and large solitons, respectively. The same characteristics as the collision between two electron holes can be witnessed such as exchanging outer layers of trapped populations and no change in ion distribution function. **Fig.\[Fig\_NlR\_ZsR\_cross\] clearly indicates the exchange of population between the two solitons. Furthermore it implies the conservation of the trapped partilces.**
-- --
-- --
Details during overtaking collisions {#SubSec_During}
------------------------------------
Fig. \[Fig\_NlR\_ZsR\_num\_details\] displays the details of temporal evolution of number densities of the two species during the overtaking collision. Three interesting phenomena can be witnessed in this figure.
Firstly, Fig. \[Fig\_NlR\_ZsR\_num\_details\] reveals that during the overtaking collision, the solitons don’t cross each other. The larger (faster) soliton reaches the smaller (slower) soliton, while losing its height, hence slowing down. Meanwhile, the smaller (slower) soliton increase its height and therefore its velocity. Note that there is a direct relationship between velocity and height of an IAS. As the time of $\tau = 425$, the two soliton have the same height and velocity. Afterwards, the process of losing/gaining height and velocity by larger/smaller soliton continues. Until the smaller/larger one morphs itself completely to the opposite soliton. Further on, due to the velocity difference they start to depart each other, however this time the larger soliton appears ahead of the smaller one.
Secondly, a shift in the trajectories of both solitons can be recognized which is a specific property of collision between solitons and have been reported in context of other field of physics for variety of solitons. This shift can be conceived as the by product of the two solitons exchanging their features during collision. The larger soliton disappears during collision and emerges later in the place of the smaller one and hence it doesn’t follow a continuous path. This bending of trajectory appears as a shift seen in Fig. \[Fig\_NlR\_ZsR\_num\_details\].
Finally, during collision, two solitons overlap and their facing tales disappears. This overlapping can be seen at $\tau = 425$ in Fig. \[Fig\_NlR\_ZsR\_num\_details\] at the midst of the collision.
Kinetic details of the overtaking collision are presented in Fig. \[Fig\_NlR\_ZsR\_PhaseSpace\_details\]. The same phenomenon as on the fluid level (Fig. \[Fig\_NlR\_ZsR\_num\_details\]) can be seen here as well.
-- --
-- --
Firstly, during the collision, the two trapped electron population (here a plateau and a hole) starts to overlap and lose their boundaries on the facing side. As collision continues, more layers of their outer part are exchanged between them. Hence, the larger/smaller soliton shrinks/grows on the velocity direction which results in losing/gaining velocity. In the midst of the collision, they have reached the same velocity as well as the same width in the velocity and spatial directions ($\tau = 425$). Therefore they can’t continue increasing their overlapping region since they are both traveling with the same velocity. As the processes continue, the growth/decline in velocity of smaller/larger solitons cause them to depart. Since during collision they don’t exchange their core part of trapped electrons, hence the smaller/larger soliton splits while having its own core with the outer layer of the larger/smaller soliton. We have carried out these types of simulations for solitons with different trapping parameter ($\beta$) overtaking each other and the same patterns have been witnessed.
Secondly, electron holes/plateaus/humps accompanying the solitons after the overlapping, split and continue propagating steadily accompanying their associated solitons. However this is in contrast of the behavior of electron holes seen and predicted in other phenomena such as beam-plasma interactions [@berk1970phase; @omura1996electron] and nonlinear Landau damping (Bernstein-Greene-Kruskal -BGK- modes)[@ghizzo1988stability]. In these case, it is observed that the electron holes merge in pair until the system reaches the stable state of one hole. In other words in a periodically bounded system of a plasma, electron holes tend to merge until one big hole remains. Our simulations show that electron holes accompanying IASs don’t show any tendency of merging, even-though they are overlapping during collision. Moreover, they split after the collision which has never been reported (to the best of our knowledge) in the context of electron holes study.
Furthermore, based on a theory developed by Krasovsky *et al.* [@krasovsky1999interaction; @krasovsky1999interaction_small; @krasovsky2003electrostatic] utilizing energy conservation principle, collision between two electron holes is a dissipative process. The internal energy of a hole (kinetic energy of the trapped electrons in the co-moving frame) grows during collision and hence holes warm up. This is an irreversible process and causes an effective friction in the energy balance. In other words any collision between two electron holes should be inelastic and they should ultimately merge into one hole. Our simulation results display a complete opposite process. Krasovsky *et al.* argue that for two holes to merge, the relative velocity of the holes should be slow enough so that the trapped electrons oscillate at least once during collision (condition of merging). Here despite the slow relative velocity the two holes **do not merge**.
By following the temporal evolution even more until the two soliton collide again, the same patterns can be witnessed. This time, they exchange their outer parts, and the resulted trapped population shows the same color as the cores. Hence it seems that they exchange the same population of the trapped electrons as of the first overtaking collision. (see Fig. \[Fig\_NlR\_ZsR\_PhaseSpace\_second\]).
\
\
\
Conclusions {#Sec_Conclusions}
===========
Three aspects of the ion-acoustic soliton in the presence of trapped electrons are discussed. Initially, it is shown that these nonlinear localized structures can propagate for a long time using the simulation method proposed here without losing their distinctive characteristics both in fluid and kinetic levels[@jenab2016IASWs]. This has been used as benchmarking test of the simulation code as well.
The main focus of the paper is to provide proof of the stability of these IASs against mutual overtaking collisions. The results on both fluid and kinetic levels are provided and show that the physical features such as width, hight, and shape in both spatial and velocity directions stay the same before and after the collisions. Simulations with different trapping parameter, hence different shape of trapped populations, prove the stability for a range of the trapping parameter covering negative to positive values. However the internal structures of the trapped populations of electrons differs before and after the collisions.
Furthermore, we have studied and presented the details of the collision on both fluid and kinetic level. The overall dynamics of overtaking collisions can be reduced into three steps:\
1) closing-in step\
2) mid-collision step\
3) departing step\
in the first step, the two solitons come close and start overlapping in the spatial direction. Meanwhile, the exchanging of trapped particles starts and this cause the fast soliton to lose velocity while the slower one gets faster. During this step, due to particle exchange, the amplitude of the larger one is reduced while the opposite happens to the smaller soliton. Solitons increase their overlapping area until the collision process hits the second step.
At the mid-collision step, which is the exact mid time of collision process, the size of the two solitons are equal as well as their velocities, hence zero relative velocity. So they can’t continue increasing their overlapping area. Finally, in the third step, the processes of the first step continue. The old small-slow soliton becomes new large-fast soliton and visa versa. Hence the relative velocity increases and the two solitons starts departing. Finally they move further enough that they can’t no longer exchange particles and therefor the collision process finishes, i.e. overlapping stops.
Note that the two solitons don’t overtake each other, they basically come close to each other and exchange particles and depart. During this process they exchange their fluid-level identity, i.e. number density profile, and this appears as a shift in their trajectory on the fluid level.
In Conclusion, ion-acoustic solitons physical features don not change on the fluid level before and after overtaking collisions. However, on the kinetic level, the internal dynamics of the electron trapped population differs. But this doe not affect the fluid-level properties, hence it is safe to assume ion-acoustic solitons (in presence of trapping effect of electrons) as solitons, structures which can survive collisions.
This work is based upon research supported by the National Research Foundation and Department of Science and Technology. Any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and therefore the NRF and DST do not accept any liability in regard thereto.
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[^1]: Email: [email protected]
[^2]: Email: [email protected]
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'This work addresses the problem of providing and evaluating recommendations in data markets. Since most of the research in recommender systems is focused on the bipartite relationship between users and items (e.g., movies), we extend this view to the tripartite relationship between users, datasets and services, which is present in data markets. Between these entities, we identify four use cases for recommendations: (i) recommendation of datasets for users, (ii) recommendation of services for users, (iii) recommendation of services for datasets, and (iv) recommendation of datasets for services. Using the open Meta Kaggle dataset, we evaluate the recommendation accuracy of a popularity-based as well as a collaborative filtering-based algorithm for these four use cases and find that the recommendation accuracy strongly depends on the given use case. The presented work contributes to the tripartite recommendation problem in general and to the under-researched portfolio of evaluating recommender systems for data markets in particular.'
author:
- Dominik Kowald
- Matthias Traub
- Dieter Theiler
- Heimo Gursch
- Emanuel Lacic
- Stefanie Lindstaedt
- Roman Kern
- Elisabeth Lex
title: Using the Open Meta Kaggle Dataset to Evaluate Tripartite Recommendations in Data Markets
---
Introduction
============
Data-driven services are becoming an increasingly important aspect of the modern economy, with data markets playing a key role as broker between the stakeholders of the data-driven ecosystem. Various initiatives have been started to research the requirements and dynamics of data markets. To name two examples, the “Data Market Austria” (DMA)[^1] [@traub2017] is a national project in Austria, while “A European AI On Demand Platform and Ecosystem” (AI4EU)[^2] aims at creating a market platform for data and artificial intelligence solutions on the European level.
For successful collaborations in data markets, the different entities need to collaborate with each other in order to create new solutions and to be able to provide innovative data products [@Cavanillas2016; @curry2016big].
[**Problem and objective of this work.**]{} Recommender services thereby play a crucial role in data markets, since their suggestions allow to discover potential new combinations between users, datasets and services [@damiani2015applying]. This results in a more complex tripartite relationship comprising users, datasets and services, as well as an increased number of use cases, in comparison with a traditional recommender setting. The tripartite structure and use cases are depicted in Figure \[fig:dma-data\].
However, most of the research in recommender systems is focused on settings consisting only of users and items, like recommending new movies to viewers. Hence, these settings can be categorized as bipartite relationships. The work of [@Godoy2016] points out the research need for recommendations in tripartite relationship scenarios such as the data markets scenario investigated in the work at hand. Another issue is the lack of an open dataset for the evaluation of tripartite recommendations in data markets. Therefore, we propose the use of the open Meta Kaggle dataset of the well-known data science portal Kaggle.
![The tripartite relationship in a data market is spun between users, datasets and services. We identify four use cases for recommendations between these three identities, namely recommendation of datasets for users (UC1), recommendation of services for users (UC2), recommendation of datasets for services (UC3) and recommendation of services for datasets (UC4).[]{data-label="fig:dma-data"}](DMA-data_v2.pdf){width="45.00000%"}
[**Contributions and findings.**]{} The contributions of our work are two-fold:
- We propose four use cases as well as a system architecture for recommendations in data markets (see Section \[s:approach\]).
- We provide evaluation results for a popularity-based as well as collaborative filtering-based algorithm for these four use cases using the open Meta Kaggle dataset (see Section \[s:eval\]).
{width="85.00000%"}
Our results show that the recommendation accuracy strongly depends on the given use case. For example, in settings in which we have a limited set of candidate entities to recommend, already the simple popularity-based approach (recommending the most popular (MP) entities) provides good results. However, in more complex settings, where it is required to link services and datasets, a personalized approach such as collaborative filtering (CF) should be favored.
Taken together, our work contributes to the under-researched portfolio of recommender systems for data markets and thus, should be of interest for both researchers and practitioners in this area.
Recommendations in Data Markets {#s:approach}
===============================
This section gives an a detailed overview of the four central data market use cases followed by the architecture of the proposed recommender system and all its components.
Use Cases {#s:use}
---------
As depicted in Figure \[fig:dma-data\], data markets create a tripartite relationship between their entities users, datasets and services, thus leading to more complex recommendation problems. We identify four use cases for recommendations in the setting of data markets, investigated in more detail in the remainder of this subsection.
[**UC1: Recommendation of datasets for users.**]{} In the first use case, we recommend datasets to users. Thus, this one reflects a rather classic item2user recommendation problem, in which we analyze past user interactions between the target user and datasets (e.g., clicks or purchases) in order to recommend other datasets that could be interesting for the user (e.g., by using CF).
[**UC2: Recommendation of services for users.**]{} The second use case also reflects a classic item2user recommendation problem but this time we aim to recommend services for users of the data market. Since typically there are more services than datasets available in a data market (see Section \[s:data\]), the set of potential candidate services is also larger, which makes this recommendation problem potentially harder than the one of UC1.
[**UC3: Recommendation of datasets for services.**]{} UC3 reflects a more complex recommendation problem, in which we aim to recommend datasets for services. As both entities are now item types, we do no longer have classic user interactions for CF as we have in UC1 and UC2. To overcome this, we could establish an indirect connection between a dataset and a service when a user has interacted with both, the dataset and the service (see Section \[s:data\]).
[**UC4: Recommendation of services for datasets.**]{} In the fourth and final use case, we recommend services for datasets. As mentioned in UC2, we typically have more services than datasets available in a data market, which makes this use cases more complex than UC3, where the set of candidate entities to recommend is smaller. Furthermore, in UC4, we want to link services and datasets, where we do not have direct user interactions available. Thus, we believe that this use case is the most complex one and therefore, we also expect the lowest recommendation accuracy for this one (see Section \[s:results\]).
System Architecture {#s:framework}
-------------------
The design of the system architecture of our recommender system for data markets is centred upon the scalable recommendation framework ScaR[^3] [@lacic2015scar; @lacic2014towards]. In Figure \[fig:dma-rec\], we illustrate our main modules and how they interact with each other as well as with users and administrators of a data market. Apache ZooKeeper[^4] is used for handling the communication between the modules and for load balancing (e.g., deploying multiple instances of a module).
[**Service Provider (SP).**]{} The SP acts as a proxy for data markets to interact with the recommender system. It provides REST-based Web services to enable users to query recommendations of datasets and services, and to add new data (e.g., user interactions, datasets or services) to the system.
[**Data Modification Layer (DML) & Apache Solr.**]{} The DML encapsulates all database-related CRUD operations (i.e., create, retrieve, update, delete) in one module and thus, enables easy access to the underlying data backend. As shown in Figure \[fig:dma-rec\], we utilize the high-performance search engine Apache Solr[^5]. This data backend solution not only guarantees scalability and (near) real-time recommendations but also the support of multiple entities like the users, datasets and services we encounter here.
[**Recommender Engine (RE).**]{} The RE is the main module of our recommender system for data markets as it is responsible for calculating recommendations. Here, we make use of Apache Solr’s build-in data structures for efficient similarity calculations. Currently, we focus on popularity and CF-based recommendation algorithms, but the RE module could be easily extended with further algorithms as well (e.g., content-based filtering).
[**Recommender Customizer (RC).**]{} The RC is used to change the parameters (e.g., the number of recommended entities $k$) of the recommendation approaches on the fly. Thus, it holds a so-called recommendation profile for each approach, accessible and changeable by the data market administrator. These changes are then broadcast to the RE to be aware of how a specific approach should be executed.
[**Recommender Evaluator (REV).**]{} The REV is responsible for evaluating the recommendation algorithms implemented in the RE. Hence, it can be executed to perform an offline evaluation with training/test set splits (see Section \[s:metric\]). In the future, it will also be possible to conduct online evaluations in data markets via A/B-tests.
Evaluation {#s:eval}
==========
In this section, we present our evaluation study, in which we compare popularity-based with CF-based recommendations for all four use cases defined in Section \[s:use\].
Data {#s:data}
----
For our evaluation, we use the open Meta Kaggle dataset[^6] (2017-11-15) of the well-known Kaggle data science portal in order to simulate a real-world data market. Here, we have 6,108 users and 45 datasets that are connected via 2,926 user/dataset interactions, where an interaction is given by a user writing about a dataset in a discussion thread. Furthermore, we have 3,334 services that are connected to the 6,108 users via 18,593 user interactions. These interactions are created by users voting for a service.
Finally, we establish a collaboration network between datasets and services (see e.g., [@hasani2018consensus]). Thus, we create a link between a dataset and a service when a user has interacted with both, the dataset and the service, which leads to 95,249 interactions. The full statistics of our dataset are summarized in Table \[tab:datasets\].
[l|c]{} Feature & \#\
Number of users & 6,108\
Number of datasets & 45\
Number of services & 3,334\
Number of user/dataset interactions & 2,962\
Number of user/service interactions & 18,593\
Number of dataset/service interactions & 95,249\
Evaluation Method {#s:metric}
-----------------
In this section, we describe the evaluation protocol, recommendation algorithms and evaluation metrics used for our study.
[**Evaluation protocol.**]{} For measuring the recommendation quality in the settings of the four use cases UC1 - UC4, we follow common practice in the area of recommender systems and split our Meta Kaggle dataset into training and test splits as suggested by [@herlocker2004evaluating].
Specifically, we extract all entities with at least eleven interactions from whom we withhold ten interactions for the test set and use the rest for training[^7] [@parra2009collaborative]. Thus, for UC1, this results into 17 users for whom we recommend datasets; for UC2, this results into 184 users for whom we recommend services; for UC3, this results into 2,338 services for whom we recommend datasets; for UC4, this results into 44 datasets for whom we recommend services.
[**Recommendation algorithms.**]{} We evaluate our four uses cases for recommendations in data markets with two algorithms, namely most popular (MP) and (ii) collaborative filtering (CF). The recommendations are calculated and evaluated using the recommender system presented in Section \[s:framework\].
MP is a non-personalized algorithm and is especially useful for new entities in a data market without any interactions so far, commonly referred as cold-start entities [@schein2002methods]. This approach recommends datasets or services, which are weighted and ranked by the number of interactions. As mentioned, the MP approach is non-personalized and thus, each entity will receive the same recommendations.
CF algorithms [@ricci2011introduction] analyze the interactions between users and entities, e.g., datasets and services alike. In CF methods two users are treated as similar if they have interacted with similar entities in the past. Hence, entities a similar user has interacted with in the past are candidates to recommend to a target user, who has not interacted with those entities yet. In the case of data markets, we do not only have interactions between users and entities but also between entities themselves when we consider UC3 and UC4. Here, we realize the CF approach in a similar way but instead of calculating user similarities, we calculate similarities between datasets and services, respectively.
[**Evaluation metrics.**]{} For measuring the accuracy of the recommendations in data markets, we use a rich set of metrics, namely Precision (P@$k$), F1-score (F1@$k$), Recall (R@$k$), Mean Reciprocal Rank (MRR@$k$), Mean Average Precision (MAP@$k$) and normalized Discounted Cumulative Gain (nDCG@$k$) [@jarvelin2002cumulated].
We report these metrics for different numbers of recommended entities (= $k$), i.e., P@1 for $k$ = 1, F1@5 for $k$ = 5[^8], R@10 for $k$ = 10, MRR@10 for $k$ = 10, MAP@10 for $k$ = 10 and nDCG@10 for $k$ = 10. Please note that we set the maximum number of $k$ to 10, which is a common value for the evaluation of recommender systems [@said2014comparative].
Results {#s:results}
-------
In this section, we present the results of our evaluation with respect to our four use cases. Table \[tab:results\] holds the resulting numbers achieved in our experiments.
[l|cccccc]{} Approach & P@1 & F1@5 & R@10 & MRR@10 & MAP@10 & nDCG@10\
UC1: MP & **0.823** & **0.470** & **0.717** & **0.217** & **0.597** & **0.729**\
UC1: CF & 0.705 & 0.431 & 0.611 & 0.192 & 0.484 & 0.635\
UC2: MP & 0.103 & 0.050 & 0.066 & 0.023 & 0.026 & 0.072\
UC2: CF & **0.137** & **0.086** & **0.114** & **0.037** & **0.054** & **0.121**\
UC3: MP & 1.000 & 0.411 & 0.707 & 0.232 & 0.580 & 0.750\
UC3: CF & **1.000** & **0.636** & **0.934** & **0.281** & **0.925** & **0.948**\
UC4: MP & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000\
UC4: CF & **0.022** & **0.006** & **0.006** & **0.003** & **0.004** & **0.009**\
[**UC1: Recommendation of datasets for users.**]{} This use case reflects the least complex one as we recommend from a quite limited set of candidate entities (i.e., 45 datasets) with a small number of connections to the target entities (i.e., 2,962 user interactions). This is also reflected in the recommendation accuracy results presented in Table \[tab:results\] as the unpersonalized MP approach provides better results than the personalized CF one. This behavior of MP outperforming CF can only be observed in this use case, which shows that personalized approaches are not always necessary.
[**UC2: Recommendation of services for users.**]{} When recommending services for users, we face a more complex problem since we have a much larger set of candidate entities (i.e., 3,334 services). Thus, the accuracy results in UC2 are much lower than the ones in UC1. Furthermore, in this case, the CF approach, which analyzes the 18,593 interactions between users and services in a personalized manner, provides better results than MP.
[**UC3: Recommendation of datasets for services.**]{} Similar to UC1, in UC3, we also recommend datasets but this time for services instead of users. For this use case, we also have a large set of 95,249 interactions between datasets and services available, leading to the overall best results for CF across all four use cases. Interestingly, both MP and CF provide a perfect score for P@1 of 1.000, which indicates that both algorithms rank a highly-connected dataset on the first position that is relevant for all 2,338 evaluated services.
[**UC4: Recommendation of services for datasets.**]{} UC4 reflects the most complex of our use cases since we have a large set of 3,334 candidates services available, which are linked via 95,249 interactions to a small set of 44 datasets being the evaluated entities. This is reflected in the results shown in Table \[tab:results\] as both algorithms, MP and CF, provide the worst results across all use cases. Here, the unpersonalized MP approach even reaches a recommendation accuracy of 0.000 for all metrics, thus not recommending a single relevant service.
Discussion {#s:disc}
----------
Our evaluation results show that there is no one-size-fits-all solution for recommendations in data markets. One particular finding of us is, that in cases having a limited set of candidate entities available like in UC1, popularity-based methods such as MP provide good results. Another finding is that personalized methods such as CF should be favored when the use cases get more complex, for example if we have a larger set of candidate entities as it is the case in UC2. The same holds for the recommendations of entities to other entities, like datasets to services in UC3.
However, our results also show that both MP and CF provide poor results for UC4 being the most complex use case. For such a setting, we need more sophisticated methods that incorporate also other data sources, e.g., content-based filtering (CBF) approaches [@lops2011content]. For overcoming sparsity problems, these approaches could also be combined with word embeddings [@kenter2015short; @mikolov2013word2vec].
Conclusion and Future Work {#s:conc}
==========================
In this paper, we presented our initial steps for providing and evaluating recommendations in data markets. Therefore, we first provided four potential use cases, which included recommendation of datasets for users (UC1), recommendation of services for users (UC2), recommendation of datasets for services (UC3), and recommendation of services for datasets (UC4). Then, we proposed a system architecture for a recommender system for data markets based on the scalable recommendation framework ScaR.
Finally, we provided an evaluation of these four uses using the Meta Kaggle dataset and our proposed recommender system. Here, we find that the unpersonalized most popular approach (MP) provides the best results for UC1 and the personalized collaborative filtering approach (CF) provides the best results for the more complex use cases UC2, UC3 and UC4.
[**Limitations and future Work.**]{} One limitation of our evaluation is that we have simulated a real-world data market using the Meta Kaggle dataset. Although, this dataset provides all relevant entities of data markets (i.e., users, datasets and services), we plan to also conduct evaluation studies in real-world data markets such as the ones created in the DMA and AI4EU initiatives.
Furthermore, so far, we have only evaluated the two algorithms MP and CF. Thus, we also plan to extend our study with more recommendation approaches such as content-based filtering (see Section \[s:disc\]).
[**Acknowledgments.**]{} This work was supported by the Know-Center GmbH, the FFG flagship project Data Market Austria (DMA) and the H2020 project AI4EU (GA: 825619). The Know-Center GmbH is funded within the Austrian COMET Program - Competence Centers for Excellent Technologies - under the auspices of the Austrian Ministry of Transport, Innovation and Technology, the Austrian Ministry of Economics and Labor and by the State of Styria. COMET is managed by the Austrian Research Promotion Agency (FFG).
[^1]: <https://datamarket.at/en/>
[^2]: <https://www.ai4eu.eu/>
[^3]: <http://scar.know-center.tugraz.at/>
[^4]: <https://zookeeper.apache.org/>
[^5]: <http://lucene.apache.org/solr/>
[^6]: <https://www.kaggle.com/kaggle/meta-kaggle>
[^7]: We only evaluate users with a minimum of eleven interactions to ensure that we have at least one interactions for training when using ten interactions for testing.
[^8]: For 10 recommended entities, Precision typically reaches its highest value for $k$ = 1 and F1 for $k$ = 5.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We analyze the geometrical properties of trace-preserving Pauli maps. Using the Choi-Jamio[ł]{}kowski isomorphism, we express the Hilbert-Schmidt line and volume elements in terms of the eigenvalues of the Pauli map. We analytically compute the volumes of positive, trace-preserving Pauli maps and Pauli channels. In particular, we find the relative volumes of the entanglement breaking Pauli channels, as well as the channels that can be generated by a time-local generator. Finally, we show what part of the Pauli channels are P and CP-divisible, which is related to the notion of Markovianity.'
author:
- 'Katarzyna Siudzi[ń]{}ska'
bibliography:
- 'C:/Users/cynda/OneDrive/Fizyka/bibliography.bib'
title: Geometry of the Pauli maps and Pauli channels
---
Introduction
============
Recently, much attention has been given to the geometrical properties of quantum states. To determine the distance between two states, one introduces the metric, which determines the geometry of the underlying space. For pure states, there exists a unique unitarily invariant metric, which is induced by the Fubini-Study line element [@Zyczkowski]. However, for mixed states, there are several possible choices of the metric. In [@Vitale], the authors derived a family of metrics for $N$-level quantum states from the relative entropy. Bures and Życzkowski used the Hilbert-Schmidt metric [@Sommers2] and the Bures metric [@Sommers] to compute the volume of mixed quantum states in an arbitrary finite dimension. In the infinite-dimensional case, it was shown that three different metrics produce the same volume of the Gaussian states with a fixed global purity [@Link2015].
Similarly, there are papers dedicated to analyze the geometry of quantum channels. Narang and Arvind [@Arvind] calculated the volume of the Pauli channels that can be simulated by a one-qubit mixed state environment. Analogical calculations were carried out for the generalized amplitude damping channels [@Jung]. Due to the duality between quantum states and quantum maps, one can use the geometrical formalism developed for quantum states to study quantum maps. The one-to-one correspondence between states and channels is established by the Choi-Jamio[ł]{}kowski isomorphism [@Choi; @JAMIOLKOWSKI1972275]. It has been applied by Lovas and Andai [@Lovas] to compute the volumes of the general and unital qubit channels using the Lebesque measure. Szarek et. al. [@Szarek2] derived bounds for the Hilbert-Schmidt volumes of positive, trace-preserving maps and the subsets of completely positive, decomposable, and superpositive qudit maps. Recently, it was shown that the volume of the positive but not completely positive, trace-preserving Pauli maps is twice as large as the volume of the Pauli channels [@Jagadish]. The geometry of Gaussian quantum channels with respect to the Bures-Fisher metric was considered by Monras and Illuminati [@Monras2010].
In this paper, we derive the volumes of the positive and completely positive, trace-preserving Pauli maps. In order to achieve this, we calculate the Hilbert-Schmidt line and volume elements for the Choi-Jamio[ł]{}kowski states that correspond to the Pauli maps. In the next step, we consider the volumes of the entanglement breaking channels and the channels that can be generated using the time-local generator. Finally, we also find the relative volumes of the CP and P-divisible Pauli channels. Our results can be interpreted as a probability that a randomly selected Pauli channel is entanglement breaking, divisible, or has positive eigenvalues.
Pauli maps and Pauli channels
=============================
Consider the Pauli map, which is the most general bistochastic qubit map [@King; @Landau] defined by $$\label{Pauli}
\Lambda[\rho]=\sum_{\alpha=0}^3p_\alpha\sigma_\alpha\rho\sigma_\alpha$$ with the Pauli matrices $\sigma_0=\mathbb{I}_2$ and $$\sigma_1=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},\quad
\sigma_2=\begin{pmatrix} 0 & - i \\ i & 0 \end{pmatrix},\quad
\sigma_3=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$ Let us characterize these maps in terms of their eigenvalues $\lambda_\alpha$. The eigenvalue equation reads $$\Lambda[\sigma_\alpha]=\lambda_\alpha\sigma_\alpha$$ with $\lambda_0=1$. There is a simple relation between $p_\alpha$ and $\lambda_\alpha$; namely, $$\lambda_\alpha=p_0+2p_\alpha-\sum_{\beta=1}^3p_\beta,\qquad\alpha=1,2,3.$$ To map qubits into qubits, the Pauli map has to be positive, which is the case if and only if $$\label{P}
|\lambda_\alpha|\leq 1.$$ Now, if the extended map $\oper_N\otimes\Lambda$ transforms quantum states into quantum states, where $\oper_N$ is the $N$-dimensional identity map, then $\Lambda$ is completely positive and is called the [*Pauli channel*]{}. Recall that a Pauli map is completely positive if and only if its eigenvalues satisfy the Fujiwara-Algoet conditions [@Fujiwara] $$\label{CP}
|1\pm\lambda_3|\geq|\lambda_1\pm\lambda_2|.$$
Geometry of the Pauli maps
==========================
To analyze the geometry of the Pauli maps, we use the Choi-Jamio[ł]{}kowski isomorphism [@Choi; @JAMIOLKOWSKI1972275] to express quantum channels as quantum states. Indeed, to the Pauli map $\Lambda$, there corresponds the unique Choi matrix $$\begin{split}
\rho_\Lambda:&=\frac{1}{4}\sum_{i,j=0}^1|i\>\<j|\otimes\Lambda[|i\>\<j|]\\&
=\frac 14 \begin{pmatrix}
1+\lambda_3 & 0 & 0 & \lambda_1+\lambda_2 \\
0 & 1-\lambda_3 & \lambda_1-\lambda_2 & 0 \\
0 & \lambda_1-\lambda_2 & 1-\lambda_3 & 0 \\
\lambda_1+\lambda_2 & 0 & 0 & 1+\lambda_3
\end{pmatrix},
\end{split}$$ whose eigenvalues are $p_\alpha$ from eq. (\[Pauli\]). We equip the space of quantum maps with the metric $g=\frac 14 \mathrm{diag}(1,1,1)$ induced by the Hilbert-Schmidt line element $$\label{HS}
{\mathrm{d}}s^2:={\mathrm{Tr}}({\mathrm{d}}\rho_\Lambda^2)=\frac 14 ({\mathrm{d}}\lambda_1^2+{\mathrm{d}}\lambda_2^2+{\mathrm{d}}\lambda_3^2).$$ The associated volume element has a simple form of $$\label{dV}
{\mathrm{d}}V:=\sqrt{g}{\mathrm{d}}\lambda_1{\mathrm{d}}\lambda_2{\mathrm{d}}\lambda_3=
\frac 18 {\mathrm{d}}\lambda_1{\mathrm{d}}\lambda_2{\mathrm{d}}\lambda_3$$ If we integrate the volume element ${\mathrm{d}}V$ from eq. (\[dV\]) over the regions $\mathcal{C}_{\rm PT}$ and $\mathcal{C}_{\rm CPT}$ given by ineq. (\[P\]) and (\[CP\]), respectively, we obtain the volumes $$V(\mathcal{C}_{\rm PT})=1,\qquad V(\mathcal{C}_{\rm CPT})=\frac 13.$$ of the positive, trace-preserving Pauli maps (PT) and the Pauli channels (CPT). Note that there are twice as many completely positive than positive but not completely positive maps. This reproduces the result from ref. [@Jagadish], where the authors proposed a measure to determine the volumes of PT and CPT Pauli maps.
An important subclass of the Pauli channels are the entanglement breaking channels (EBC), whose extension $\oper_N\otimes\Lambda$ maps any state into a separable one. It was shown that $\Lambda$ is entanglement breaking if and only if [@qubitEBC] $$\label{EBC}
\sum_{\alpha=1}^3|\lambda_\alpha|\leq 1.$$ Integrating the Pauli channels over the region $\mathcal{C}_{\rm EBC}$ defined by ineq. (\[EBC\]) leads to the following volume, $$V(\mathcal{C}_{\rm CPT}\cap\mathcal{C}_{\rm EBC})=\frac 16.$$ In Fig. \[CPT\_EBC\], we show the ranges of $\lambda_\alpha$ that correspond to the Pauli channels and the entanglement breaking Pauli channels, respectively. The complete positivity region is a tetrahedron (light grey), and the entanglement breaking region is an inscribed octahedron (dark grey). Observe that four of the octahedron’s faces are coplanar with the tetrahedron’s faces. The volume of the octahedron is half the volume of the tetrahedron.
![ A graphical representation of the range of eigenvalues $\lambda_1,\ \lambda_2,\ \lambda_3$ that correspond to the Pauli channels (light grey) and the entanglement breaking Pauli channels (dark grey).[]{data-label="CPT_EBC"}](CPT_EBC3.jpg){width="40.00000%"}
Among their many applications, quantum channels are used to describe the dynamics of open quantum systems. Indeed, continuous time-evolution is provided with the use of time-parametrized families of quantum channels referred to as [*dynamical maps*]{}. As the simplest example of a Pauli dynamical map, one usually considers the Markovian semigroup $\Lambda(t)$, which is the solution of the master equation $$\label{MS}
\dot{\Lambda}(t)=\mathcal{L}\Lambda(t),\qquad\Lambda(0)=\oper,$$ with the Gorini-Kossakowski-Sudarshan-Lindblad generator [@GKS; @L] $$\label{L}
\mathcal{L}[\rho]=\frac 12 \sum_{\alpha=1}^3\gamma_\alpha(\sigma_\alpha\rho \sigma_\alpha-\rho),$$ where $\gamma_\alpha\geq 0$. To include non-Markovian effects, one introduces the time-local generator $\mathcal{L}(t)$, which has the same form as $\mathcal{L}$ in eq. (\[L\]) but with time-dependent $\gamma_\alpha(t)$ that are not necessarily positive. Now, $\mathcal{L}(t)$ generates the dynamical map $\Lambda(t)$ with the eigenvalues $$\lambda_\alpha(t)=\exp[\Gamma_0(t)-\Gamma_\alpha(t)],$$ where $\Gamma_\alpha(t)=\int_0^t\gamma_\alpha(\tau){\mathrm{d}}\tau$ and $\gamma_0(t)=\sum_{\alpha=1}^3\gamma_\alpha(t)$. Another way to generalize Markovian semigroup master equation (\[MS\]) is through the memory kernel master equation $$\label{K}
\dot{\Lambda}(t)=\int_0^tK(t,\tau)\Lambda(\tau){\mathrm{d}}\tau,\qquad\Lambda(0)=\oper,$$ with the non-local memory kernel $K(t,\tau)$. Note that the solutions of eq. (\[K\]) can have negative eigenvalues $\lambda_\alpha(t)$.
At this point, we make an important observation. Any Pauli channel $\Lambda$ can be obtained from a Pauli dynamical map $\Lambda^\prime(t=t_\ast)$ if and only if its eigenvalues $\lambda_\alpha$ are positive. The condition $\lambda_\alpha\geq 0$ corresponds to the region of integration $\mathcal{C}_{\rm TLG}$. Such channels are reachable with time-local generators (TLG), and their relative volume with respect to all Pauli channels is equal to $$\frac{V(\mathcal{C}_{\rm CPT}\cap\mathcal{C}_{\rm TLG})}{V(\mathcal{C}_{\rm CPT})}=\frac{3}{16}.$$ Meanwhile, the total volume of the positive maps reachable with time-local generators is $$V(\mathcal{C}_{\rm PT}\cap\mathcal{C}_{\rm TLG})=\frac 18.$$ Fig. \[CPT\_TLG\] shows the ranges of $\lambda_\alpha$ that correspond to the Pauli channels and the Pauli channels reachable with time-local generators. The complete positivity region is a tetrahedron (light grey), and the region of channels with positive eigenvalues is an inscribed triangular bipyramid (dark grey). Observe that two of the biparymid’s faces are coplanar with the tetrahedron’s faces. The volume of the triangular biparymid is $3/16$-th of the volume of the tetrahedron. The remaining part, $$1-\frac{V(\mathcal{C}_{\rm CPT}\cap\mathcal{C}_{\rm TLG})}{V(\mathcal{C}_{\rm CPT})}=\frac{13}{16},$$ is the relative volume of the Pauli channels that are reachable only with the memory kernel $K(t,\tau)$.
![ A graphical representation of the range of the eigenvalues $\lambda_1,\ \lambda_2,\ \lambda_3$ that describe the Pauli channels (light grey) and their subclass that is reachable with time-local generators (dark grey).[]{data-label="CPT_TLG"}](CPT_TLG3.jpg){width="40.00000%"}
Additionally, one could ask what part of the Pauli channels reachable with time-local generators are entanglement breaking. It is straightforward to show that the corresponding volume ratio is $$\frac{V(\mathcal{C}_{\rm CPT}\cap\mathcal{C}_{\rm TLG}\cap\mathcal{C}_{\rm EBC})}{V(\mathcal{C}_{\rm CPT}\cap\mathcal{C}_{\rm TLG})}=\frac 13.$$ The associated ranges of $\lambda_\alpha$ are presented in Fig. \[TLG\_EBC\]. The triangular bipyramid (light grey) corresponds to the channels achievable with a time-local generator, and the inscribed tetrahedron (dark grey) corresponds to the entanglement breaking region. Three of the tetrahedron’s faces lie on the same planes as the triangular bipyramid’s faces. The volume of the tetrahedron is a third of the volume of the triangular biparymid. The figure that is a combination of Figs. 1–3 has recently been plotted in [@Davalos], where divisibility of qubit channels is considered.
![ A graphical representation of the range of the eigenvalues $\lambda_1,\ \lambda_2,\ \lambda_3$ that describe the Pauli channels achievable with a time-local generator (light grey) and their entanglement breaking subclass (dark grey).[]{data-label="TLG_EBC"}](TLG_EBC3.jpg){width="40.00000%"}
Let us present the main results of this section in one picture. In Fig. \[pie\_chart\], the rectangle composed of $8\times 6$ identical squares represents the total volume of all positive, trace-preserving Pauli maps. The white region corresponds to the positive but not completely positive Pauli maps, whereas the grey region is associated with the Pauli channels (completely positive, trace-preserving Pauli maps). The single-hatched regions are the volumes of the entanglement breaking Pauli channels and the positive Pauli maps that are reachable with time-local generators, respectively (consult the legend in Fig. \[pie\_chart\] for details). Note that some of the regions overlap.
![A quantitative representation of the volumes of various Pauli maps. One square covers the region of $1/48$.[]{data-label="pie_chart"}](pie_chart.jpg){width="45.00000%"}
P and CP-divisibility
=====================
A special property of a quantum channel $\Lambda$ is its divisibility. Namely, $\Lambda$ is P-divisible if and only if it can be decomposed into $$\Lambda=V\Lambda^\prime,$$ where $\Lambda^\prime$ is a quantum channel and $V$ denotes a (non-unitary) positive, trace-preserving map. If $V$ is instead a quantum channel, then $\Lambda$ is said to be CP-divisible. In open quantum systems, divisibility of quantum dynamical maps is used to characterize Markovianity. Indeed, if $\Lambda(t)=V(t,s)\Lambda(s)$ for $s\leq t$ is CP-divisible, then it represents the Markovian evolution. A dynamical map that is only P-divisible is called [*weakly non-Markovian*]{} [@witness2; @ChManiscalco]. Moreover, for invertible Pauli dynamical maps, weak non-Markovianity is equivalent to the lack of information backflow from the system to the environment [@Filip2].
In the case of Pauli channels, divisibility is fully-characterized by the eigenvalues $\lambda_\alpha$. In particular, the Pauli channel is P-divisible if and only if [@Cirac] $$\lambda_1\lambda_2\lambda_3\geq 0,$$ which defines the P-divisibility region $\mathcal{C}_{\rm P-div}$. The necessary and sufficient condition for CP-divisibility of the invertible Pauli channel reads [@Wolf] $$0<\lambda_1\lambda_2\lambda_3\leq(\min_\alpha\lambda_\alpha)^2.$$ We denote the associated region by $\mathcal{C}_{\rm CP-div}$. Note that non-invertible Pauli channels do not contribute to the volume due to ${\mathrm{d}}\lambda_\alpha=0$. The relative volumes of the P and CP-divisible Pauli channels with respect to all Pauli channels amount to $$\begin{aligned}
\frac{V(\mathcal{C}_{\rm CPT}\cap\mathcal{C}_{\rm P-div})}{V(\mathcal{C}_{\rm CPT})}&=\frac{3}{4},\\
\frac{V(\mathcal{C}_{\rm CPT}\cap\mathcal{C}_{\rm CP-div})}{V(\mathcal{C}_{\rm CPT})}&=\frac 38,\end{aligned}$$ respectively. Interestingly, $V(\mathcal{C}_{\rm CPT}\cap\mathcal{C}_{\rm P-div})/V(\mathcal{C}_{\rm CPT})$ has the same value as the relative volume of the Pauli channels simulable with a one-qubit mixed state environment [@Arvind]. Analogical calculations can be carried over for the Pauli channels that are reachable with time-local generators. The results are $$\begin{aligned}
\frac{V(\mathcal{C}_{\rm CPT}\cap\mathcal{C}_{\rm TLG}\cap\mathcal{C}_{\rm P-div})}{V(\mathcal{C}_{\rm CPT}\cap\mathcal{C}_{\rm TLG})}&=1,\\
\frac{V(\mathcal{C}_{\rm CPT}\cap\mathcal{C}_{\rm TLG}\cap\mathcal{C}_{\rm CP-div})}{V(\mathcal{C}_{\rm CPT}\cap\mathcal{C}_{\rm TLG})}&=\frac{1}{2},\label{CPD}\end{aligned}$$ where eq. (\[CPD\]) agrees with Ref. [@Filippov]. Observe that half of the P-divisible maps are CP-divisible, even if we restrict our attention to the Pauli channels with positive eigenvalues. The geometrical regions representing the P and CP-divisible Pauli channels have already been plotted in [@Davalos], and they are very involved.
Fig. \[pie\_chart2\] summarizes the results of this section. The rectangle composed of $8\times 4$ identical squares represents the volume of all Pauli channels. The unhatched region corresponds to the indivisible Pauli channels, whereas the grey region is associated with the Pauli channels that are reachable with time-local generators. The single and double-hatched regions are the relative volumes of the P and CP-divisible Pauli channels, respectively. More details can be found in the legend in Fig. \[pie\_chart2\]. Note that some regions overlap.
![A quantitative representation of the volumes of divisible Pauli channels. One square covers $1/32$ of the volume of the Pauli channels.[]{data-label="pie_chart2"}](pie_chart2A.jpg){width="45.00000%"}
Conclusions
===========
In this paper, we analytically calculate the volumes of the positive, trace-preserving Pauli maps and Pauli channels. We use the Choi-Jamio[ł]{}kowski isomorphism and the Hilbert-Schmidt line element to introduce the metric in the space of Pauli maps. The associated volume element depends only on the map’s eigenvalues. Next, we find the volumes of some special classes of the Pauli channels: entanglement breaking, reachable with time-local generators (having positive eigenvalues), as well as P and CP-divisible.
Our results allow one to check the probability that a randomly generated Pauli channel or positive map has desired properties. Also, they make us realize what a small percentage of Pauli channels is covered by the time-local generators and how important it is to further develop the memory kernel approach. Similarly, the ratio of CP and P-divisible channels with respect to indivisible Pauli channels can shed some light on the amount of dynamical maps describing Markovian and non-Markovian quantum evolutions.
There are several topics that require further study. It would be interesting to consider the volumes of more general quantum maps, like non-unital qubit maps or unital qudit maps. However, one is sure to encounter problems with determining the regions of integration. For example, the range of admissible eigenvalues for non-unital qubit channels or positive qudit channels are not known.
Recall that the geometry of a given space depends on the metric. An alternative choice to the metric induced by the Hilbert-Schmidt line element in eq. (\[HS\]) is the one following from the Fisher-Rao line element [@Fisher] $${\mathrm{d}}s^2_{FR}:={\mathrm{Tr}}({\mathrm{d}}\sqrt{\rho_\Lambda})^2.$$ Observe that the corresponding volume element $${\mathrm{d}}V_{FR}=\frac{1}{8\sqrt{p_0p_1p_2p_3}}
{\mathrm{d}}\lambda_1{\mathrm{d}}\lambda_2{\mathrm{d}}\lambda_3$$ does not coincide with ${\mathrm{d}}V$ from eq. (\[dV\]). Moreover, one can encounter technical issues with analytical integration of ${\mathrm{d}}V_{FR}$, so numerical analysis might be necessary.
Acknowledgements {#acknowledgements .unnumbered}
================
This paper was supported by the Polish National Science Centre project No. 2018/31/N/ST2/00250.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A macronova (kilonova) was discovered with a short gamma-ray burst, GRB 130603B, which is widely believed to be powered by the radioactivity of $r$-process elements synthesized in the ejecta of a neutron star binary merger. As an alternative, we propose that macronovae are energized by the central engine, i.e., a black hole or neutron star, and the injected energy is emitted after the adiabatic expansion of ejecta. This engine model is motivated by extended emission of short GRBs. In order to compare the theoretical models with observations, we develop analytical formulae for the light curves of macronovae. The engine model allows a wider parameter range, especially smaller ejecta mass, and better fit to observations than the $r$-process model. Future observations of electromagnetic counterparts of gravitational waves should distinguish energy sources and constrain the activity of central engine and the $r$-process nucleosynthesis.'
author:
- Shota Kisaka
- Kunihito Ioka
- Hajime Takami
title: Energy Sources and Light curves of Macronovae
---
INTRODUCTION {#interduction}
============
{width="150mm"}
Gravitational wave (GW) observations are expected to provide a new view of relativistic phenomena in the Universe. One of the most promising candidates for the direct detection of GWs is the merger of compact binaries such as binary neutron stars (NSs). The second generation of ground-based GW detectors, such as Advanced LIGO [@Aba+10], Advanced VIRGO [@Ace+14] and KAGRA [@Kur+10], will reach the sensitivity required to detect GWs from the inspiral and coalescence of compact binary systems including binary NSs within a few hundred Mpc. Statistical studies suggest that a few tens of merger events should be observed per year [@Aba+10b].
Electromagnetic counterparts of GW emitters have been recently focused on to maximize a scientific return from the expected detection of GWs [e.g., @MB12]. Follow-up observations of these electromagnetic counterparts are important to confirm a GW detection and to investigate progenitors and environments. The electromagnetic detection also improves the localization of GW sources because the localization accuracy by photons is much better than that by the ground-based GW detectors $\sim10-100$ deg$^2$ [e.g., @Ess+14].
Sophisticated simulations have revealed mass ejection associated with the mergers of binary NSs by several mechanisms. Significant mass is dynamically ejected by gravitational torques and hydrodynamical interactions during the mergers, called dynamical ejecta [e.g., @Ros+99; @RJ01; @Hot+13]. General relativistic simulations show that these ejecta distribute nearly isotropic compared to Newtonian simulations in the cases of binary NSs [@Hot+13], while they are anisotropic for NS-black hole (BH) mergers [@KIS13]. Mass may be also ejected through winds driven by neutrinos [@Des+09], magnetic fields of and/or amplified by the merged objects [@SSKI11; @KKS12; @KKSSW14], viscous heating and nuclear recombination [@FM13; @FKMQ14].
A traditional electromagnetic counterpart is short-hard gamma-ray bursts [GRBs; @NPP92]. Recent simulations have revealed that a hypermassive NS is formed from the merger of a NS binary [e.g., @Hot+13], which is believed to collapse into a BH at later time. Non-collapsed matter and some ejecta falling back to the BH form a torus around the BH [e.g., @Ros07]. Then, a relativistic jet may be launched from the BH-torus system, which is believed to be the central engine of short-hard GRBs. Another interesting possibility is a so-called macronova/kilonova, which is thermal emission from ejecta [e.g., @LP98; @Kul05; @BK13]. The radiative energy of a macronova is estimated between that of a classical nova and supernova. Ejecta can also produce non-thermal emission at later time similarly to supernova remnants [@NP11; @PNR13; @TKI14]. Ejecta may accompany an advanced relativistic part, producing early emission [$\sim$ hours; @KIS14; @Met+14]. Emission from macronovae and NS binary merger remnants is almost isotropic and hence different from that of short GRBs which depends on the directions of their relativistic jets. Moreover, macronovae are closer in time to mergers than emission from merger remnants and do not depend on the properties of circumburst environments. Therefore, macronovae are expected to play a crucial role to localize a large sample of GW events [@MB12].
Recently, a macronova candidate following GRB 130603B was discovered [@Tan+13; @BFC13]. This candidate is widely interpreted as the results of the radioactive decay of $r$-process elements produced in the ejecta of a compact binary merger [@Tan+13; @BFC13; @Hot+13b; @PKR14; @Gro+14]. We call this scenario [*an $r$-process model*]{} throughout this paper. The ejecta from a merger of binary NSs is primarily neutron-rich. Then, heavy radioactive elements (mass number $\gtrsim 130$) are expected to form through neutron-capture onto nuclei ($r$-process nucleosynthesis) [e.g., @LS74]. Although the $r$-process nucleosynthesis ends a few hundred millisecond after a merger, synthesized elements release energy due to nuclear fission and beta decays up to $\sim100$ days [e.g., @Wan+14]. A schematic picture for this model is shown in the left panel of figure \[figure:model\]. If this scenario is correct, the observations also give important insights into the enrichment of $r$-process elements in the galaxy evolution [e.g., @PKR14]. Although the $r$-process model explains the observed light curve of the macronova, it is based on the limited observational data and the nuclear heating rate with large uncertainties. Required mass of dynamical ejecta to explain the observations is relatively large compared with the simulation results [@Gro+14]. In addition, the occurrence of $r$-process nucleosynthesis needs the ejecta with low electron fraction ($Y_e\lesssim0.1$). However, relatively high electron fraction ($Y_e\sim0.2-0.5$) can be also realized, which has been discussed for neutrino-driven wind [e.g., @FM13]. It is worth considering other possibilities such as the scenarios of an external shock between ejecta and surrounding medium [@Jin+13], a supramassive magnetar [@Fan+13] and dust grains [@TNI14].
In this study, we consider another power source of macronovae, i.e., energy injection from the activity of the central engine, in addition to the radioactive decay of $r$-process elements. This is similar to the early evolution of core-collapse supernovae [e.g., @A80; @P93]. We call this model [*an engine model*]{} throughout this paper. There are several motivations to consider that the activity of the central engine contributes to the heating of ejecta. One observational motivation is the extended emission following the prompt emission of short GRBs. The origin of extended emission is considered to be the activity of the central engine [@Bar+05] because the sharp drop of its light curve is difficult to be reproduced by afterglow emission [@IKZ05]. After the merger, a stable NS or a BH is formed. In the case that a BH with a torus (or disk) is formed, the energy injection to the ejecta is expected as a form of the jet and/or disk wind [e.g., @Nak+13]. In the case that a NS with strong poloidal magnetic field is formed as a result of a merger, the wind of relativistic particles is ejected [@Dai+06; @MQT08; @YZG13; @WD13; @MP14]. Then, the wind collides with the ejecta, and about half of the wind energy converts to the internal energy by the shock-heating. A schematic picture is shown in the right-hand side of figure \[figure:model\].
The ejecta emission powered by a stable magnetar has already been discussed [@YZG13; @WD13; @MP14]. They suggest that the magnetar-powered ejecta emit the brighter optical and X-ray emissions than that of the $r$-process model. However, they did not show that the magnetar-powered ejecta explain the detected infrared excess in GRB 130603B [@Tan+13; @BFC13].
The engine model can provide energy enough to reproduce the detected macronova candidate, GRB 130603B. We do not specify the specific heating sources. Alternatively, to estimate the luminosity and temperature, we assume that the internal energy $E_{\rm int0}\sim10^{51}$ erg is injected to the ejecta at the time $t_{\rm inj}\sim10^2$ s after the merger. These values are consistent with typical isotropic energy $E_{\rm iso}\sim10^{50}-10^{51}$ erg and duration $t_{\rm dur}\sim10-10^2$ s of the extended emission [@Sak+11]. Using the velocity of the ejecta $v$, the temperature at $t_{\rm inj}$ is $T_0\sim[E_{\rm int0}/(av^3t_{\rm inj}^3)]^{1/4}$, where $a$ is the radiative constant. If we only consider the adiabatic cooling for the cooling process of the ejecta, the evolution of the internal energy $E_{\rm int}$ and temperature $T$ is scaled as $E_{\rm int}\propto t^{-1}$ and $T\propto t^{-1}$. The luminosity is described as $L\sim E_{\rm int}/t$. Adopting the ejecta velocity $v\sim10^{10}$cm s$^{-1}$ [@Hot+13], the luminosity $L$ and the temperature $T$ at $t\sim10^6$s are $$\begin{aligned}
\label{sec1:L}
L &\sim& \frac{E_{\rm int0}}{t}\left(\frac{t}{t_{\rm inj}}\right)^{-1} \nonumber \\
&\sim&10^{41}\left(\frac{E_{\rm int0}}{10^{51}{\rm erg}}\right)\left(\frac{t_{\rm inj}}{10^2{\rm s}}\right)\left(\frac{t}{10^6{\rm s}}\right)^{-2}~{\rm erg}~{\rm s}^{-1},\end{aligned}$$ and $$\begin{aligned}
\label{sec1:T}
T&\sim& T_0\left(\frac{t}{t_{\rm inj}}\right)^{-1} \nonumber \\
&\sim&2\times10^3\left(\frac{E_{\rm int0}}{10^{51}{\rm erg}}\right)^{1/4}\left(\frac{t_{\rm inj}}{10^2{\rm s}}\right)^{1/4}\nonumber \\
& &\times\left(\frac{v}{10^{10}{\rm cm}~{\rm s}^{-1}}\right)^{-3/4}\left(\frac{t}{10^6{\rm s}}\right)^{-1}~{\rm K}.\end{aligned}$$ The observations of macronova of GRB 130603B give J-band luminosity $\sim10^{41}$ erg s$^{-1}$ and the difference between J-band and B-band $\gtrsim2.5$ mag which corresponds to the temperature $\lesssim4\times10^3$ K at $t\sim7$ days after GRB 130603B in the source rest frame [@Tan+13; @BFC13]. Therefore, in this estimate, the luminosity and temperature for the engine model is consistent with the observation of the macronova following GRB130603B.
We model the evolution of luminosity and temperature of a macronova. Unlike the previous studies, we treat the model in an analytical manner and formulate a light curve including the early phase ($\sim10^3-10^5$ s), which is important for the search of electromagnetic counterparts of GW emitters. We consider shock-heating due to the activity of a central engine as a heating mechanism of the ejecta. For comparison, the $r$-process model, which has been discussed in most papers [e.g., @LP98], is also formulated. Then, we compare the results of our models with observations to constrain the model parameters such as the ejected mass and the velocity of the ejecta. Although our models are simplified, it is valuable to make comparison between two heating models. In section \[model\], we introduce our model assumptions. We describe the analytical models for the evolution of luminosity and temperature in section \[evolution\]. Then, we compare our results with observations in section \[discuss\]. Implications for the discrimination between two models are also discussed. We summarize our results in section \[summary\]. In appendix A, we summarize the formulae for the observed temperature and bolometric luminosity.
MODEL
=====
Significant mass of material $\sim10^{-3}-10^{-1}M_{\odot}$ is ejected during a binary merger. We model ejecta by following the results of the general relativistic simulations of NS-NS mergers in @Hot+13. The simulations show that ejecta expand in a nearly homologous manner [see also @Ros+14]. The morphology of the ejecta is quasi-spherical in the case of a merger of binary NSs. According to these results, we assume an isotropic and homologous expansion for the ejecta. Then, the velocity of ejecta $v$ is $$\begin{aligned}
\label{sec2:v}
v\sim r/t\end{aligned}$$ where the radius $r$ originates the central engine and the time $t$ is measured from the time when a compact binary merges.
Note that in the case of a merger of NS-BH binary, the ejected mass expands with significant anisotropy [@KOST11; @Fou+13; @Fou+14; @Lov+13; @Dea+13; @KIS13]. We do not consider such anisotropic ejecta in this work.
{width="150mm"}
Density Profile {#density}
---------------
@Nag+14 found that the profile of ejecta obtained from simulations by @Hot+13 can be well fitted by a power-law function $\rho\propto v^{-\beta}$. The power-law index of snapshot density $\beta$ is more or less independent on the dynamics of mergers, which is in the range of $\beta \sim 3$–4 for $v_{\min}\le v\le v_{\max}$, where $v_{\max}$ and $v_{\rm min}$ are the velocities of the outer and inner edges of the ejecta, respectively. We choose the middle of this range $\beta=3.5$ in this study. We also fix the maximum velocity $v_{\max}=0.4c$ from simulation results [@Hot+13]. The maximum velocity $v_{\max}$ is comparable with the escape velocity of the system. The minimum velocity $v_{\min}$ is mainly determined by complicated dynamics at the initial stage of the merger $t\ll10^2{\rm s}$. For the mass density profile at the front of the ejecta, we assume the discrete boundary and the mass density $\rho=0$ at the region $r>v_{\max}t$. Although this profile may be far from the actual one [^1], our main aim is to compare two models for energy sources, so that our conclusions are not affected. In section \[outer\], we discuss the dependence on the mass density profile at the outer region of the ejecta for the observed light curve. Here, we only consider the evolution after the initial stage of the merger ($t\gg t_{\rm inj}$) and treat $v_{\min}$ as a model parameter. Because of homologous expansion, the density decreases as $\rho \propto t^{-3}$. Then, the density profile is described by $$\begin{aligned}
\label{sec2-1:rho}
\rho (t,v)=\rho_0 \left(\frac{t}{t_0}\right)^{-3}\left(\frac{v}{v_{\min}}\right)^{-\beta}.\end{aligned}$$ where $\rho_0$ and $t_0$ are normalization factors. The factor $\rho_0t_0^3$ is related to the total mass of the ejecta $M_{\rm ej}$ as following, $$\begin{aligned}
\label{sec2-1:M_ej}
M_{\rm ej}&=&4\pi\int_{v_{\min}t_0}^{v_{\max}t_0}\rho(t_0,v)r^2dr \nonumber \\
&=&\frac{4\pi}{\beta-3}\rho_0(v_{\min}t_0)^3\left[1-\left(\frac{v_{\max}}{v_{\min}}\right)^{3-\beta}\right],\end{aligned}$$ where we use $dr(t=t_0)=t_0dv$ from equation (\[sec2:v\]). We also introduce the radius of ejecta outer edge $$\begin{aligned}
\label{sec2-1:r_out}
r_{\rm out}=v_{\max}t,\end{aligned}$$ and their inner edge $$\begin{aligned}
\label{sec2-1:r_in}
r_{\rm in}=v_{\min}t.\end{aligned}$$
Diffusion Radius {#diffusion}
----------------
The inner part of the ejecta is optically thick, and therefore the propagation of radiation in the ejecta can be regarded as a diffusion process. Photons can diffusively escape from the region which satisfies that the diffusion time, $t_{\rm diff}$, is smaller than the dynamical time $t$, $$\begin{aligned}
\label{sec2-1:diffuse}
t_{\rm diff}\le t.\end{aligned}$$ The medium in this region is called to be effectively thin [@RL79]. For convenience, we introduce a diffusion radius $r_{\rm diff}(t)$ which is the radius satisfying the condition $t=t_{\rm diff}$. Furthermore, we divide ejecta into two regions called the effectively thin ($r\ge r_{\rm diff}$) and effectively thick ($r < r_{\rm diff}$) regions. Near the diffusion radius, the optical depth is $\tau\gg1$. We consider random walk for photons so that the mean number of scatterings to propagate for the distance $\Delta r$ is $(\Delta r/l_{\rm mfp})^2$, where $l_{\rm mfp}$ is the mean free path for a photon. Hence, the diffusion time $t_{\rm diff}$ for the propagation distance $\Delta r$ is $$\begin{aligned}
\label{sec2-1:t_diff}
t_{\rm diff}\sim\frac{l_{\rm mfp}}{c}\left(\frac{\Delta r}{l_{\rm mfp}}\right)^2\sim\tau\frac{\Delta r}{c}.\end{aligned}$$ In the right hand of equation (\[sec2-1:t\_diff\]), we use $\tau\sim\Delta r/l_{\rm mfp}$.
We calculate the diffusion radius $r_{\rm diff}$ from the condition $t_{\rm diff}=t$. Since the mass density profile of the ejecta is described by a decreasing power-law function (equation \[sec2-1:rho\]), the diffusion time $t_{\rm diff}$ is negligible in an outer part. Thus, in order to calculate the diffusion radius $r_{\rm diff}$, it is a good approximation to only consider scatterings near $r_{\rm diff}$ ($\Delta r\sim r_{\rm diff}$). However, in the early phase, the distance from the outer edge of the ejecta $r_{\rm out}$ to the diffusion radius $r_{\rm diff}$ is smaller than the diffusion radius $r_{\rm out}-r_{\rm diff}<r_{\rm diff}$. Therefore, we should take the propagation distance as $$\begin{aligned}
\label{sec2-1:Deltar}
\Delta r\sim \left\{ \begin{array}{ll}
r_{\rm out}-r_{\rm diff} &~~(r_{\rm diff}>0.5r_{\rm out}) \\
& \\
r_{\rm diff} &~~(r_{\rm diff}\le 0.5r_{\rm out}). \\
\end{array} \right.\end{aligned}$$ We call the first the thin-diffusion phase and the second the thick-diffusion phase throughout this paper. We schematically show these two phases in figure \[figure:diffuse\]. Note that in the thin-diffusion phase, since the size of the effectively thin region is much smaller than the size of the ejecta ($r_{\rm out}-r_{\rm in}$), the calculation of the radiative transfer using Monte Carlo technique [e.g., @BK13; @TH13] requires a large number of realizations to follow the temporal evolution, which do not seem to have been considered properly so far.
To obtain the diffusion radius, we need to calculate the optical depth $\tau$ of photons which propagate a distance $\Delta r$. Using equations (\[sec2-1:rho\]) and (\[sec2-1:Deltar\]), the optical depth $\tau$ is described as, $$\begin{aligned}
\label{sec2-1:tau}
\tau&=&\int_{r_{\rm diff}}^{r_{\rm out}}\kappa\rho dr \nonumber \\
&=&\frac{(\beta-3)\kappa M_{\rm ej}}{4\pi(\beta-1)v_{\min}^2t^2}\left[1-\left(\frac{v_{\max}}{v_{\min}}\right)^{3-\beta}\right]^{-1} \nonumber \\
& &\times
{\displaystyle \left[\left(\frac{r_{\rm diff}}{v_{\min}t}\right)^{1-\beta}-\left(\frac{v_{\max}}{v_{\min}}\right)^{1-\beta}\right]},\end{aligned}$$ in the thin-diffusion phase, and $$\begin{aligned}
\label{sec2-1:tau2}
\tau&=&\int_{r_{\rm diff}}^{2r_{\rm diff}}\kappa\rho dr \nonumber \\
&=&\frac{(\beta-3)\kappa M_{\rm ej}}{4\pi(\beta-1)v_{\min}^2t^2}\left[1-\left(\frac{v_{\max}}{v_{\min}}\right)^{3-\beta}\right]^{-1} \nonumber \\
& &\times
{\displaystyle \left(\frac{r_{\rm diff}}{v_{\min}t}\right)^{1-\beta}(1-2^{1-\beta})},\end{aligned}$$ in the thick-diffusion phase, where $\kappa$ is the opacity of the ejecta. For simplicity, we use a grey approximation and a spatially uniform value of the opacity $\kappa$. From the results of @TH13 [see also @KBB13] which consider the contribution from all $r$-process elements to the opacity of merger ejecta, the evolution of the bolometric luminosity can be approximately described by the constant value of the opacity, $\kappa\sim3-30$ cm$^2$g$^{-1}$. Following their results, we use this value for the opacity of the ejecta. Note that the exact value of the opacity of the ejecta has some uncertainties in the production efficiency of $r$-process elements and its spatial distribution. Moreover, if the ejecta temperature is low enough for dust formation ($T\lesssim2000$ K), the opacity significantly increases [@TNI14]. From these reasons, we consider the dependence on $\kappa$ in section \[evolution\].
Our model is based on the formulation of the light curves of supernovae [e.g., @C92; @NS10; @RW11], but there are several differences. In the case of type II supernovae, the opacity is significantly reduced due to hydrogen recombination [e.g., @GNS14]. However, since the ionization potentials of the lanthanides included in the $r$-process elements are generally lower than that of hydrogen and the iron group, the opacity remains high at relatively low temperature [@KBB13]. Therefore, we do not consider the recombination effects for the opacity.
As far as we know, the supernova studies [e.g., @C92; @NS10; @RW11] have not taken into account the thin-diffusion phase, which is necessary for treating the thickness of the diffusion length appropriately and estimating the physical quantities by the values at the outer edge of the ejecta in the analytical formulae. This phase may be also important for the case of supernovae.
Some supernova studies consider the planar phase [@PCW10; @NS10] in which the evolution of the ejecta is approximately planar as long as its radius do not double. In the case of the NS-NS merger, since the initial length scale of the merger system is small $\sim10^6$ cm and the velocity of the merger ejecta is subrelativistic, the planar phase is irrelevant for the observations.
Heating Mechanisms
------------------
### Radioactivity {#radio}
One of the two heating mechanisms we consider is nuclear heating by $r$-process elements. Since the beta decay products of $r$-process elements produced in NS binary mergers naturally heat ejecta, this mechanism is considered to power the emission of a macronova [e.g., @LP98]. The nuclear heating rate is calculated in several works [@Met+10; @Rob+11; @Kor+12; @Ros+14; @Wan+14]. The derived heating rates per unit mass $\dot{\epsilon}(t)$ are described by the following formula $$\begin{aligned}
\label{sec2-1:dotepsilon}
\dot{\epsilon}=\dot{\epsilon}_0\left(\frac{t}{1\ {\rm day}}\right)^{-\alpha}.\end{aligned}$$ In this study, we use $\alpha=1.3$ and $\dot{\epsilon}_0=2\times10^{10}~{\rm erg~s}^{-1}{\rm g}^{-1}$ obtained by @Wan+14. The value of $\dot{\epsilon}$ has been obtained by simulations under some simplified assumptions with only limited parameter regions. Thus, we should note that the value of $\dot{\epsilon}_0$ has uncertainties.
The injected internal energy by the nuclear decay is $\propto t^{1-\alpha}$ in the region $r < r_{\rm diff}$. On the other hand, the injected energy in this region is decreased by adiabatic cooling. The time-evolution of internal energy due to the adiabatic cooling is proportional to $t^{-1}$. Comparing the two temporal evolution, the index of the adiabatic cooling is smaller than that of the increase of internal energy due to the nuclear decay for $\alpha < 2$. Since we use $\alpha=1.3$, we neglect the injected internal energy in the region $r < r_{\rm diff}$.
### Engine-driven shock {#shock}
Unlike the $r$-process model, energy injection occurs only within the time $t_{\rm inj}$ in the engine model. We only consider adiabatic cooling as a cooling process of ejecta after $t_{\rm inj}$, and therefore, the temperature distribution at time $t$ is, $$\begin{aligned}
\label{sec2-1:T}
T(t,v)=T_0\left(\frac{t}{t_{\rm inj}}\right)^{-1}\left(\frac{v}{v_{\rm min}}\right)^{-\xi},\end{aligned}$$ where the index $\xi$ is a parameter for a snapshot distribution and $T_0$ is a normalization factor described later. The time dependence of $t^{-1}$ is the effect of adiabatic expansion.
The normalized value $T_0$ is determined by using the relation of total injected internal energy $E_{\rm int0}$ as $$\begin{aligned}
\label{sec2-1:E_int}
E_{\rm int0}&=&4\pi\int_{v_{\min}t_{\rm inj}}^{v_{\max}t_{\rm inj}}aT^4(t_{\rm inj},v)r^2dr \nonumber \\
&=&\frac{4\pi}{3-4\xi}aT_0^4(v_{\min}t_{\rm inj})^3\left[\left(\frac{v_{\max}}{v_{\min}}\right)^{3-4\xi}-1\right],\end{aligned}$$ where we use $dr=t_{\rm inj}dv$. For the temperature index $\xi>0.75$, the innermost region of ejecta has dominant internal energy. As will be shown in Section \[evolution\], since the luminosity and temperature always depend on the product of $E_{\rm int0}$ and $t_{\rm inj}$, we treat $E_{\rm int0}t_{\rm inj}$ as a parameter. Thus, the engine model has two parameters, $\xi$ and $E_{\rm int0}t_{\rm inj}$ instead of $\dot{\epsilon}_0$ and $\alpha$ in the $r$-process model.
Energy injection is not always a single event and the shock does not always get through the whole ejecta. It is considered that the activity of the central engine accompanies violent time variability. In this case, multiple shocks propagate into the ejecta. Some of the shock may not catch up with the outer edge of the ejecta. Current general relativistic simulations cannot calculate the evolution of ejecta for such a long time after merger ($t_{\rm inj}\sim 10^2$ s), so that the index $\xi$ of temperature distribution is highly uncertain. Therefore, we treat the temperature index $\xi$ as a parameter.
Unlike the case of core-collapse supernova [@NS10], it is difficult to determine the temperature distribution of heated ejecta by the activity of a central engine. In the case that the activity of a central engine injects the energy into the ejecta, the radiation-dominated shock (where the internal energy behind the shock is dominated by radiation) is formed in the ejecta. The ejecta are heated during the propagation of the shock. This situation is similar to the initial phase of core-collapse supernovae [e.g., @A80; @P93]. In the cases of core-collapse supernovae, the kinetic energy of ejecta before the shock heating is much smaller than the injected internal energy. In such ejecta, the relation between velocity and mass density was obtained by @S60 (in the non-relativistic case for the velocity of the ejecta). Using Sakurai’s (1960) solution and the equipartition between the kinetic energy after the shock heating and the internal energy [@NS10], the distribution of the temperature distribution is derived. However, in the case of compact binary mergers, the merger ejecta have a large velocity ($\sim0.01-0.1c$) before the shock heating [@Hot+13]. Then, injected internal energy is not always larger than the kinetic energy of ejecta so that it is not clear whether we can use the equipartition to estimate the distribution of internal energy or not.
The kinetic energy of the ejecta $E_{\rm kin}$ is described as $$\begin{aligned}
\label{sec2-1:E_kin}
E_{\rm kin}&=&\frac{1}{2}\times4\pi\int_{v_{\min}}^{v_{\max}}\rho(t,v)v^4t^3dv \nonumber \\
&=&{\displaystyle \frac{1}{2}M_{\rm ej}v_{\min}^2\frac{(\beta-3)\left[\left(\frac{v_{\max}}{v_{\min}}\right)^{5-\beta}-1\right]}{(5-\beta)\left[1-\left(\frac{v_{\max}}{v_{\min}}\right)^{3-\beta}\right]}}.\end{aligned}$$ Note that if the injected internal energy $E_{\rm int0}$ is larger than the kinetic energy of the ejecta, it is expected that some of the internal energy converts to the kinetic energy of the ejecta. As a result, the internal energy and the kinetic energy are equal as in the case of core-collapse supernovae. Then, the mass density distribution and the maximum velocity of the ejecta derived from simulations may be changed because the injection time may be long $\sim10^2$s compared to that calculated by simulations $\lesssim0.1$s [@Hot+13]. For simplicity, we only consider the case $E_{\rm int0}\le E_{\rm kin}$.
Evolution of Luminosities and Temperatures {#evolution}
==========================================
[clccc]{}\
Symbol & & Fiducial model & Minimum mass model & Hot interior model\
$M_{\rm ej}$ & Ejecta mass & $0.10M_{\odot}$ & $0.022M_{\odot}$ & $0.08M_{\odot}$\
$v_{\min}$ & Minimum velocity & $0.15c$ & $0.13c$ & $0.18c$\
$v_{\max}$ & Maximum velocity & $0.40c$ & $0.40c$ & $0.40c$\
$\beta$ & Index of the density profile & 3.5 & 3.5 & 3.5\
$\kappa$ & Opacity & 10 cm$^2$ g$^{-1}$ & 30 cm$^2$ g$^{-1}$ & 10 cm$^2$ g$^{-1}$\
$\dot{\epsilon}_0$ & Nuclear heating rate at 1 day & $2\times10^{10}$ erg s$^{-1}$ g$^{-1}$ & $\cdots$ & $2\times10^{10}$ erg s$^{-1}$ g$^{-1}$\
$\alpha$ & Index of nuclear heating rate & 1.3 & $\cdots$ & 1.3\
$E_{\rm int0}$ & Internal energy at $t_{\rm inj}$ & $1.3\times10^{51}$ erg & $0.9\times10^{51}$ erg & $0.8\times10^{51}$ erg\
$t_{\rm inj}$ & Injection time & $10^2$ s & $10^2$ s & $10^2$ s\
$\xi$ & Index of the temperature profile & 1.6 & 1.1 & 2.7\
\[tab:parameter\]
In this section, we present the evolution of the observed temperature and luminosity of a macronova using our model introduced in the previous section. In sections \[thin\] – \[transparent\], we focus on the parameter dependence of the evolution using some approximations. In section \[fiducial\], we calculate the temperature and luminosity using the fiducial model with parameters summarized in the first column of table 1.
To calculate the luminosity and temperature, we assume that the emission is well described by the blackbody radiation [e.g., @BK13]. For simplicity, we assume that the observed temperature equals to the temperature at the diffusion radius $r_{\rm diff}$. We also assume that the temperature is not so different from the diffusion radius $r_{\rm diff}$ to $2r_{\rm diff}$ so that in the thick-diffusion phase ($r_{\rm out}>2r_{\rm diff}$), we only consider the emission from $r_{\rm diff}$ to $2r_{\rm diff}$ to calculate the observed luminosity for both the $r$-process and engine models. In some studies [e.g., @Met+14], the observed temperature is approximated by the temperature at the radius of the photosphere $r_{\rm ph}$ where the optical depth is unity. Since the velocity of the ejecta is near the light speed, the optical depths at the diffusion radius $r_{\rm diff}$ and its twice $2r_{\rm diff}$ are $\tau\sim1-10^2$. Therefore, our assumed temperature approximately equals to the temperature at the photosphere.
In section \[diffusion\], we introduced two phases, the thin- and thick-diffusion phases (figure \[figure:diffuse\]), depending on the size of the region where photons make the diffusion in the ejecta $\Delta r$. We also introduce another phase $r_{\rm diff}\le r_{\rm in}$, the transparent phase, in which photons can diffuse out from the entire of the ejecta. Thus, we divide the evolution into these three phases for the values of the diffusion radius $r_{\rm diff}$ as described below.
Thin-diffusion phase {#thin}
--------------------
The size of the effectively thin region gets larger with time. At the early phase of a macronova, the diffusion radius $r_{\rm diff}$, which is the inner radius of the effectively thin region, is near the outer edge of the ejecta $r_{\rm out}$. In this early phase, we take the propagation distance $\Delta r$ of a photon as $\Delta r\sim r_{\rm out}-r_{\rm diff} (<r_{\rm diff})$. Since we assume that the density is a homologous function of the velocity $\rho\propto v^{-\beta}$, the density can be approximated as $\rho\sim\rho(v_{\max})$ in the region $r_{\rm diff}\gg\Delta r$. Using the escaping condition for the diffusing photons $t\sim t_{\rm diff}$, equation (\[sec2-1:t\_diff\]) and approximation on the optical depth $\tau\sim\Delta r\kappa\rho(t,v_{\max})$, the propagation distance $\Delta r$ can be estimated as $$\begin{aligned}
\label{sec2-2-1:Deltar}
\Delta r&\sim&\sqrt{\frac{ct}{\kappa\rho(t,v_{\max})}} \nonumber \\
&\propto&\kappa^{-1/2}M_{\rm ej}^{-1/2}v_{\min}^{\frac{3-\beta}{2}}v_{\max}^{\beta/2}t^2.\end{aligned}$$ In the discussion of parameter dependence (sections \[thin\] – \[transparent\]), we only consider the dominant term. For example, we neglect the second term in the right-hand side of equation (\[sec2-1:M\_ej\]) to derive the parameter equation (\[sec2-2-1:Deltar\]) because the index of the mass density is $\beta>3$ in our model. In section \[fiducial\], we include the subdominant terms to calculate the light curves numerically.
First we consider the $r$-process model. The evolution of temperature $T_{\rm obs}$ is obtained by the internal energy density $\dot{\epsilon}t\rho$ at the radius $r=r_{\rm out}$. Using equations (\[sec2-1:rho\]) and (\[sec2-1:M\_ej\]), the parameter dependence of the density is $\rho(t,v_{\max})\propto M_{\rm ej}v_{\min}^{\beta-3}v_{\max}^{-\beta}t^{-3}$. The observed temperature is $$\begin{aligned}
\label{sec2-2-1:T_obs,nuc}
T_{\rm obs}&\sim&\left(\frac{\dot{\epsilon}t\rho(t,v_{\max})}{a}\right)^{1/4} \nonumber \\
&\propto&M_{\rm ej}^{1/4}v_{\min}^{\frac{\beta-3}{4}}v_{\max}^{-\beta/4}t^{-\frac{2+\alpha}{4}}.\end{aligned}$$ For $\alpha=1.3$, the observed temperature evolves as $T_{\rm obs}\propto t^{-0.875}$. This is because in the thin-diffusion phase the ejecta is effectively a single expanding shell with $\rho\sim\rho(t,v_{\max})$ and the injected energy $\dot{\epsilon}t\propto t^{-0.3}$ is almost constant so that the observed temperature approximately follow adiabatic cooling $T\propto t^{-1}$. Note that in this phase the observed temperature does not depend on the opacity. The bolometric luminosity $L_{\rm bol}$ for the radioactivity is described as the product of the mass within the thickness $\Delta r$ in equation (\[sec2-2-1:Deltar\]) and the nuclear heating rate $\dot{\epsilon}$ in equation (\[sec2-1:dotepsilon\]) so that $$\begin{aligned}
\label{sec2-2-1:L_bol,nuc}
L_{\rm bol}&\sim& 4\pi r_{\rm out}^2\Delta r\rho(t,v_{\max})\dot{\epsilon} \nonumber \\
&\propto&\kappa^{-1/2}M_{\rm ej}^{1/2}v_{\min}^{\frac{\beta-3}{2}}v_{\max}^{\frac{4-\beta}{2}}t^{1-\alpha}.\end{aligned}$$ For $\alpha=1.3$, the evolution of the bolometric luminosity is $L_{\rm bol}\propto t^{-0.3}$.
Next we consider the engine model. We should take into account the freedom of the temperature index $\xi$ in the temperature distribution (equation \[sec2-1:T\]). Since we only consider a dominant term in the right-hand side of equation (\[sec2-1:E\_int\]) (the first term for $\xi<0.75$ or the second term for $\xi > 0.75$) in this subsection, the parameter dependence of the temperature $T_0$ is described as $$\begin{aligned}
\label{sec2-2-1:T_0}
T_0\propto E_{\rm int0}^{1/4}t_{\rm inj}^{-3/4}\times\left\{ \begin{array}{ll}
v_{\min}^{-\xi}v_{\max}^{\frac{4\xi-3}{4}} & (\xi < 0.75) \\
& \\
v_{\min}^{-3/4} & (\xi > 0.75) \\
\end{array} \right. .\end{aligned}$$ Substituting $v=v_{\max}$ into equation (\[sec2-1:T\]), the observed temperature is described as $$\begin{aligned}
\label{sec2-2-1:T_obs,sh}
T_{\rm obs}&\sim&T_0\left(\frac{t}{t_{\rm inj}}\right)^{-1}\left(\frac{v_{\max}}{v_{\min}}\right)^{-\xi} \nonumber \\
&\propto&E_{\rm int0}^{1/4}t_{\rm inj}^{1/4}t^{-1}\times\left\{ \begin{array}{ll}
v_{\max}^{-3/4} & (\xi < 0.75) \\
& \\
v_{\min}^{\frac{4\xi-3}{4}}v_{\max}^{-\xi} & (\xi > 0.75). \\
\end{array} \right.\end{aligned}$$ Since the observed temperature $T_{\rm obs}$ approximately equals to the temperature at the outer edge of the ejecta, $T_{\rm obs}\sim T(t,v_{\max})$, the evolution of the observed temperature and the luminosity are also determined by the adiabatic cooling. The bolometric luminosity in the effectively thin region is equal to the total radiation created by thermal emission in this region [@RL79]. Using equation (\[sec2-2-1:T\_obs,sh\]), the bolometric luminosity is described as $$\begin{aligned}
\label{sec2-2-1:L_bol,sh}
L_{\rm bol}&\sim&4\pi r_{\rm out}^2\Delta r\frac{aT_{\rm obs}^4}{t} \nonumber \\
&\propto&\kappa^{-1/2}M_{\rm ej}^{-1/2}E_{\rm int0}t_{\rm inj}t^{-1} \nonumber \\
& &\times\left\{ \begin{array}{ll}
v_{\min}^{\frac{3-\beta}{2}}v_{\max}^{\frac{\beta-2}{2}} & (\xi < 0.75) \\
& \\
v_{\min}^{\frac{-3-\beta+8\xi}{2}}v_{\max}^{\frac{4+\beta-8\xi}{2}} & (\xi > 0.75). \\
\end{array} \right.\end{aligned}$$ The time evolution of bolometric luminosity for the engine model does not depend on the temperature index $\xi$ in the thin-diffusion phase.
Comparing the engine model with the $r$-process model in the thin-diffusion phase, the bolometric luminosity and the observed temperature decrease faster in the engine model than those in the $r$-process model. These time-dependence do not depend on the indices of the density and temperature.
Note that the light curve may depend on the detailed profile of the front of the ejecta in this thin-diffusion phase. The profile of the ejecta front is difficult to calculate by the numerical simulation due to its low density, and hence has large uncertain [@KIS14]. We discuss its dependence in section \[discuss\].
Thick-diffusion phase {#thick}
---------------------
We consider diffusion to evaluate the diffusion radius in the thick-diffusion phase. We take the propagation distance $\Delta r\sim r_{\rm diff}$ after the time when the difference between the radius of the outer edge of the ejecta $r_{\rm out}$ and the diffusion radius $r_{\rm diff}$ is larger than the diffusion radius, $r_{\rm out}-r_{\rm diff}>r_{\rm diff}$, since the optical depth of the outer part is negligible for the density profile in equation (\[sec2-1:rho\]). In this thick-diffusion phase, mass density significantly deviates from $\rho(v_{\max})$. Substituting equations (\[sec2-1:t\_diff\]) and (\[sec2-1:tau2\]) into $t=t_{\rm diff}$, the diffusion radius $r_{\rm diff}$ is calculated as $$\begin{aligned}
\label{sec2-2-2:r_diff}
r_{\rm diff}&\sim&\left[\frac{(\beta-3)\kappa M_{\rm ej}v_{\min}^{\beta-3}t^{\beta-4}}{4\pi(\beta-1)c}\right]^{\frac{1}{\beta-2}} \nonumber \\
&\propto&\kappa^{\frac{1}{\beta-2}}M_{\rm ej}^{\frac{1}{\beta-2}}v_{\min}^{\frac{\beta-3}{\beta-2}}t^{\frac{\beta-4}{\beta-2}},\end{aligned}$$ where we use the relations $\Delta r\sim r_{\rm diff}$ and $v\sim r_{\rm diff}/t$. The latter is obtained from the assumption of the homologous expansion. Regarding the optical depth $\tau$, the second term is neglected in the right-hand side of equation (\[sec2-1:tau\]) to focus only on the dominant term to study parameter dependence in sections \[thin\] – \[transparent\]. For $\beta=3.5$, the diffusion radius decreases with time ($r_{\rm diff}\propto t^{-1/3}$). Then, emission from the region with relatively high mass density can be observed progressively in this phase ($\rho\propto t^{-3}(r_{\rm diff}/t)^{-\beta}\propto t^{(4\beta-9)/3}=t^{1.667}$).
We introduce the transition time $t_{\times}$ between thin- and thick-diffusion phases, which satisfies the relation $r_{\rm diff}=0.5r_{\rm out}$. Substituting equation (\[sec2-2-2:r\_diff\]) and $r_{\rm out}=v_{\max}t$ into the relation $r_{\rm diff}=0.5r_{\rm out}$, we can obtain the transition time $t_{\times}$ as $$\begin{aligned}
\label{sec2-2-2:t_times}
t_{\times}&\sim&\sqrt{\frac{2^{\beta-4}(\beta-3)\kappa M_{\rm ej}}{\pi(\beta-1)cv_{\max}}\left(\frac{v_{\max}}{v_{\min}}\right)^{3-\beta}} \nonumber \\
&\sim&4.1~\kappa_{10}^{1/2}M_{\rm ej,0.1}^{1/2}v_{\min,0.1}^{\frac{\beta-3}{2}}v_{\max,0.4}^{\frac{2-\beta}{2}}~{\rm day},\end{aligned}$$ where $\kappa_{10}\equiv \kappa/10$ cm$^2$ g$^{-1}$, $M_{\rm ej,0.1}\equiv M_{\rm ej}/0.1M_{\odot}$, $v_{\min,0.1}\equiv v_{\min}/0.1c$ and $v_{\max,0.4}\equiv v_{\max}/0.4c$. As seen above, the transition time $t_{\times}$ is typically several days. This timescale is expected to allow the follow-up observations [@Aasi+14]. Thus, we should consider both phases to predict something useful for follow-up observations. If we fix $\kappa$ and $v_{\max}$ and use $\beta=3.5$, equation (\[sec2-2-2:t\_times\]) gives $t_{\times}\propto M_{\rm ej}^{1/2}v_{\min}^{1/4}$. If we increase the total mass of the ejecta $M_{\rm ej}$ and the velocity at the inner edge of the ejecta $v_{\min}$, the mass density of the ejecta $\rho$ and hence the optical depth are increased. As a result, the transition time $t_{\times}$ becomes large.
First, we consider the $r$-process model. Here, we introduce the velocity $v_{\rm diff}=r_{\rm diff}/t$ based on the homologous relation. Using the velocity $v_{\rm diff}$ and equation (\[sec2-2-2:r\_diff\]) for the mass density (equation \[sec2-1:rho\]), the evolution of temperature is $$\begin{aligned}
\label{sec2-2-2:T_obs,nuc}
T_{\rm obs}&\sim&\left(\frac{\dot{\epsilon}t\rho(t,v_{\rm diff})}{a}\right)^{1/4} \nonumber \\
&\propto&\kappa^{\frac{\beta}{4(2-\beta)}}M_{\rm ej}^{\frac{1}{2(2-\beta)}}v_{\min}^{\frac{\beta-3}{2(2-\beta)}}t^{\frac{1}{\beta-2}-\frac{\alpha}{4}}.\end{aligned}$$ For $\alpha=1.3$ and $\beta=3.5$, the evolution is described by $T_{\rm obs}\propto t^{0.341}$ so that the observed temperature increases with time. The bolometric luminosity is described as the product of the mass between $r_{\rm diff}$ and $2r_{\rm diff}$ and the nuclear heating rate $\dot{\epsilon}$. Using equations (\[sec2-1:dotepsilon\]) and (\[sec2-2-2:r\_diff\]), we obtain the bolometric luminosity as, $$\begin{aligned}
\label{sec2-2-2:L_bol,nuc}
L_{\rm bol}&\sim&4\pi r_{\rm diff}^3\rho(t,v_{\rm diff})\dot{\epsilon} \nonumber \\
&\propto&\kappa^{\frac{3-\beta}{\beta-2}}M_{\rm ej}^{\frac{1}{\beta-2}}v_{\min}^{\frac{\beta-3}{\beta-2}}t^{\frac{2(\beta-3)}{\beta-2}-\alpha}.\end{aligned}$$ For $\alpha=1.3$ and $\beta=3.5$, the evolution of the bolometric luminosity is $L_{\rm bol}\propto t^{-0.633}$.
Next, we consider the engine model. Using equations (\[sec2-1:T\]) and (\[sec2-2-2:r\_diff\]), the evolution of the observed temperature is described as $$\begin{aligned}
\label{sec2-2-2:T_obs,sh}
T_{\rm obs}&\sim&T_0\left(\frac{t}{t_{\rm inj}}\right)^{-1}\left(\frac{v_{\rm diff}}{v_{\min}}\right)^{-\xi} \nonumber \\
&\propto&\kappa^{-\frac{\xi}{\beta-2}}M_{\rm ej}^{-\frac{\xi}{\beta-2}}E_{\rm int0}^{1/4}t_{\rm inj}^{1/4}t^{\frac{-\beta+2\xi+2}{\beta-2}} \nonumber \\
& &\times\left\{ \begin{array}{ll}
v_{\min}^{\frac{\xi(\beta-3)}{2-\beta}}v_{\max}^{\frac{4\xi-3}{4}} & (\xi < 0.75) \\
& \\
v_{\min}^{\frac{3\beta-6-4\xi}{4(2-\beta)}} & (\xi > 0.75). \\
\end{array} \right.\end{aligned}$$ For $\beta=3.5$, the value $\xi=0.75$ is the boundary whether the observed temperature increases with time ($\xi>0.75$) or not ($\xi<0.75$). The evolution of the luminosity equals to the total radiation created by thermal emission in the sphere with radius $r_{\rm diff}$. Using the relation $v(r_{\rm diff})\sim r_{\rm diff}/t$ and equations (\[sec2-2-2:r\_diff\]) and (\[sec2-2-2:T\_obs,sh\]), we obtain $$\begin{aligned}
\label{sec2-2-2:L_bol,sh}
L_{\rm bol}&\sim&4\pi r_{\rm diff}^3\frac{aT_{\rm obs}^4}{t} \nonumber \\
&\propto&\kappa^{\frac{3-4\xi}{\beta-2}}M_{\rm ej}^{\frac{3-4\xi}{\beta-2}}E_{\rm int0}t_{\rm inj}t^{\frac{2(\beta+1-4\xi)}{2-\beta}} \nonumber \\
& &\times\left\{ \begin{array}{ll}
v_{\min}^{\frac{(3-4\xi)(\beta-3)}{\beta-2}}v_{\max}^{4\xi-3} & (\xi < 0.75) \\
& \\
v_{\min}^{\frac{3-4\xi}{2-\beta}} & (\xi > 0.75). \\
\end{array} \right.\end{aligned}$$ If we take $\beta=3.5$ and $\xi=1.0$, the evolution of the bolometric luminosity is $L_{\rm bol}\propto t^{-0.666}$. This is almost the same dependence as in the $r$-process model. Note that even if the inner part of the ejecta has the larger internal energy ($\xi>0.75$), the bolometric luminosity does not always increase with time. Using the relation $E_{\rm int}(v,t)\propto t^{-1}$, from the adiabatic cooling, the evolution of bolometric luminosity for a given mass shell with $v$ is $L_{\rm bol}\sim E_{\rm int}(v)t^{-1}\propto t^{-2}$, where $E_{\rm int}(v,t)$ is the total internal energy for the mass shell with a given expanding velocity $v$. Since $E_{\rm int}(v_{\rm difff})\propto v^{3-4\xi}$ and $v_{\rm diff}= r_{\rm diff}/t\propto t^{\frac{2}{2-\beta}}$, the bolometric luminosity increases with time for the value of the temperature index $\xi>(\beta+1)/4=1.125$.
Transparent phase {#transparent}
-----------------
Once the diffusion radius reaches the inner edge of the ejecta $(r_{\rm diff}=r_{\rm in})$, all photons emitted from the ejecta can diffuse out within dynamical timescale. If energy is not injected into the ejecta in this transparent phase, the internal energy in the ejecta runs out immediately. The transition time from the thick-diffusion phase to the transparent phase $t_{\rm tr}$ is described as $$\begin{aligned}
\label{sec2-2-3:t_tr}
t_{\rm tr}&\sim&\sqrt{\frac{(\beta-3)\kappa M_{\rm ej}}{4\pi(\beta-1)cv_{\min}}} \nonumber \\
&\sim&6.9~\kappa_{10}^{1/2}M_{\rm ej,0.1}^{1/2}v_{\min,0.1}^{-1/2}~{\rm day},\end{aligned}$$ where we use the diffusion radius $r_{\rm diff}= r_{\rm in}$.
First we consider the $r$-process model. The observed temperature equals to the temperature at the inner edge of the ejecta, $T_{\rm obs}\sim [\dot{\epsilon}t\rho(v_{\min})/a]^{1/4}$. Using equation (\[sec2-1:rho\]), we obtain $$\begin{aligned}
\label{sec2-2-3:T_obs,nuc}
T_{\rm obs}&\sim&\left(\frac{\dot{\epsilon}t\rho(t,v_{\min})}{a}\right)^{1/4} \nonumber \\
&\propto&M_{\rm ej}^{1/4}v_{\min}^{-3/4}t^{-\frac{2+\alpha}{4}}.\end{aligned}$$ Since the energy is continuously injected due to the nuclear heating in the $r$-process model, the bolometric luminosity from the entire ejecta is described as $L_{\rm bol}\sim M_{\rm ej}\dot{\epsilon}$. However, the outer part of the ejecta emits photons with lower temperature and/or X-rays and $\gamma$-rays produced directly in radioactive decays. Although such emission contributes to the bolometric luminosity, we here focus only on the optical and infrared emissions. In the thick-diffusion phase, the observed emission comes from the region between $\sim r_{\rm diff}$ and $\sim2r_{\rm diff}$. In the transparent phase, we assume that the time evolution of the diffusion radius $r_{\rm diff}$ is the same as the thick-diffusion phase until $2r_{\rm diff}=r_{\rm in}$ and the observed luminosity comes from the region from $r_{\rm in}$ to $2r_{\rm diff}$ for simplicity. Then, the bolometric luminosity is described as $$\begin{aligned}
\label{sec2-2-3:L_bol,nuc}
L_{\rm bol}&\sim&4\pi r_{\rm in}^3\rho(t,v_{\min})\dot{\epsilon} \nonumber \\
&\propto&M_{\rm ej}t^{-\alpha}.\end{aligned}$$ Although it appears that this time evolution directly reflects the nuclear decay rate, when we calculate the mass between $r_{\rm in}$ and $2r_{\rm diff}$ the evolution of the upper limit of the integration $2r_{\rm diff}$ makes the decrease of the luminosity faster than $\propto t^{-\alpha}$ (see a dashed line in the middle panel of figure \[figure:evolution\]). In addition, the evolution of $r_{\rm diff}$ depends on the index $\beta$ (see equation \[sec2-2-2:r\_diff\]), so that the mass between $r_{\rm in}$ and $2r_{\rm diff}$ also depends on the index $\beta$.
Next we consider the engine model. We assume that the internal energy is exhausted when the diffusion radius reaches $2r_{\rm diff}=r_{\rm in}$. For the observed temperature $T_{\rm obs}$, we assume the relation $T_{\rm obs}=T(t,v_{\min})$ and use equation (\[sec2-1:T\]), $$\begin{aligned}
\label{sec2-2-3:T_obs,sh}
T_{\rm obs}&\sim&T_0\left(\frac{t}{t_{\rm inj}}\right)^{-1} \nonumber \\
&\propto&E_{\rm int0}^{1/4}t_{\rm inj}^{1/4}t^{-1} \nonumber \\
& &\times\left\{ \begin{array}{ll}
v_{\min}^{-\xi}v_{\max}^{\frac{4\xi-3}{4}} & (\xi < 0.75) \\
& \\
v_{\min}^{-3/4} & (\xi > 0.75). \\
\end{array} \right.\end{aligned}$$ The bolometric luminosity is described as $$\begin{aligned}
\label{sec2-2-3:L_bol,sh}
L_{\rm bol}&\sim&4\pi \int_{r_{\rm in}}^{2r_{\rm diff}}\frac{aT_{\rm obs}^4}{t} \nonumber \\
&\propto&E_{\rm int0}t_{\rm inj}t^{-2} \nonumber \\
& &\times\left\{ \begin{array}{ll}
\kappa^{\frac{3-4\xi}{\beta-2}}M_{\rm ej}^{\frac{3-4\xi}{\beta-2}}v_{\min}^{\frac{(\beta-3)(3-4\xi)}{\beta-2}} & \\
~~\times v_{\max}^{4\xi-3}t^{\frac{2(3-4\xi)}{2-\beta}} & (\xi < 0.75) \\
& \\
1 & (\xi > 0.75). \\
\end{array} \right.\end{aligned}$$ Since the internal energy at the innermost region almost equals to the total internal energy $E_{\rm int}(v_{\min})\sim E_{\rm int0}(t/t_{\rm inj})^{-1}$ and determines the bolometric luminosity $L_{\rm bol}\sim E_{\rm int}(v_{\min})/t$ for the temperature index $\xi>0.75$, the bolometric luminosity does not depend on the mass $M_{\rm ej}$ and velocities $v_{\max}$ and $v_{\min}$. This luminosity always corresponds to the maximum luminosity for $\xi > 0.75$, so that we can impose the lower limit on the parameter $E_{\rm int0}t_{\rm inj}$.
Fiducial Model {#fiducial}
--------------
![ Temporal evolution of the diffusion radius (top), bolometric luminosities (middle) and observed temperatures (bottom) in the fiducial model (first column of table 1). Thick dashed and solid lines show the evolution for the $r$-process model and the engine model, respectively. For comparison, we also plot the bolometric luminosity from the whole ejecta for the $r$-process model after the transparent phase ($t>t_{\rm tr}$ in equation \[sec2-2-3:t\_tr\]) as a blue long-dashed line in the middle panel.[]{data-label="figure:evolution"}](fig3.ps){width="70mm"}
{width="120mm"}
We show the temporal evolution of the diffusion radius $r_{\rm diff}$, the bolometric luminosity $L_{\rm bol}$ and the observed temperature $T_{\rm obs}$ in figure \[figure:evolution\] under the fiducial parameter set. The parameters are summarized in the first column of table 1. Here, we do not use approximations $\rho(v)\sim\rho(v_{\rm max})$ and $T(v)\sim T(v_{\max})$ at the thin-diffusion phase as in section \[thin\]. Instead, the diffusion radius $r_{\rm diff}$ is calculated from equations (\[sec2-1:t\_diff\]) – (\[sec2-1:tau\]) without approximations. Using the obtained diffusion radius $r_{\rm diff}$ and the relation $v_{\rm diff}=r_{\rm diff}/t$, we calculate the observed temperatures in the thin- and thick-diffusion phases, $T_{\rm obs}\sim[\dot{\epsilon}t\rho(t,v_{\rm diff})/a]^{1/4}$ (equation \[sec2-2-2:T\_obs,nuc\]), and $T_{\rm obs}\sim T_0(t/t_{\rm inj})^{-1}(v_{\rm diff}/v_{\min})^{-\xi}$ (equation \[sec2-2-2:T\_obs,sh\]) for the $r$-process and the engine models, respectively. In the transparent phase, the temperature in equations (\[sec2-2-3:T\_obs,nuc\]) and (\[sec2-2-3:T\_obs,sh\]) are evaluated with $v=v_{\min}$. Equations on observed temperature and bolometric luminosity for both models are summarized in appendix. The set of parameters we choose here explains the observed optical and infrared light curves of GRB 130603B (see next section). The vertical dash-dotted lines in figure \[figure:evolution\] show the time $t=t_{\times}$ (equation \[sec2-2-2:t\_times\]). The diffusion radius is plotted only up to the transition time $t=t_{\rm tr}$ (equation \[sec2-2-3:t\_tr\]).
In the thick-diffusion phase, the diffusion radius ($r_{\rm diff}\propto t^{\frac{\beta-4}{\beta-2}}=t^{-1/3}$ for $\beta=3.5$) moves inward in the ejecta ($r\propto t$). Since the observed luminosity and temperature are determined at the diffusion radius $r_{\rm diff}$, the time evolution of luminosity and temperature strongly depends on the indices of the profile, $\beta$ and $\xi$. For the $r$-process model, the bolometric luminosity decreases with time ($L_{\rm bol}\propto t^{\frac{2(\beta-3)}{\beta-2}-\alpha}=t^{-0.633}$) in the thick-diffusion phase (see equation \[sec2-2-2:L\_bol,nuc\]), which is more rapid than that in the thin-diffusion phase ($L_{\rm bol}\propto t^{1-\alpha}=t^{-0.3}$, see equation \[sec2-2-1:L\_bol,nuc\]). Since the index of the mass density $\beta=3.5$ is close to 3, in which the mass of each shell with a certain size $\delta r$ is the same value in logarithmic scale, the mass between the diffusion radius $r_{\rm diff}$ and its doubled value $2r_{\rm diff}$ does not significantly change with time. The luminosity is mainly determined by that mass, so that the evolution of the luminosity is slow compared with the evolution of nuclear heating rate ($\propto t^{-\alpha}$) in the thick-diffusion phase. On the other hand, bolometric luminosity and observed temperature increase with time in the engine model with the parameter set of the fiducial model. These mainly reflect the profile of the temperature distribution ($\xi=1.6$). In fact, using equation (\[sec2-2-2:L\_bol,sh\]), the index of the time $t$ for the bolometric luminosity is $-2(\beta+1-4\xi)/(\beta-2)\sim2.53$ for the engine model.
After the transition time $t\ge t_{\rm tr}$, the luminosity and temperature are almost determined by the quantities at the inner edge of the ejecta. Then, the evolution of the luminosity and temperature does not significantly depend on the indices of profile $\beta$ and $\xi$ as in the case of the thin-diffusion phase (except for the case $\xi<0.75$ of the engine model, equation \[sec2-2-3:L\_bol,sh\]). Since our used profile of mass density has an artificially steep cut-off at the inner edge of the ejecta (figure \[figure:diffuse\]), bolometric luminosity in both models rapidly declines after the time $t\ge t_{\rm tr}$. In the bottom panel of figure \[figure:evolution\], the observed temperature in both models has a steep cutoff at $2r_{\rm diff}=r_{\rm in}$. For comparison, we also consider the time evolution of bolometric luminosity from the whole ejecta $L_{\rm bol}=M_{\rm ej}t^{-\alpha}$ in the $r$-process model. Time evolution is shown in the middle panel of figure \[figure:evolution\] as a blue long-dashed line. This luminosity evolution ($L_{\rm bol}\propto t^{-\alpha}=t^{-1.3}$) is significantly slower than that of the engine model in the transparent phase. In section \[implication\], we discuss the implication for discriminating the $r$-process model and the engine model using these temporal behaviors.
DISCUSSION {#discuss}
==========
![Range of parameter space in order to explain the observations of the macronova, GRB 130603B ([*blue*]{} for the engine model and [*red*]{} for the $r$-process model). Since two regions are overlapped, the color looks like purple for the $r$-process model. These regions are only a schematic view. We fix $v_{\max}=0.4c$, $\beta=3.5$, $\dot{\epsilon}_0=2\times10^{10}$ erg s$^{-1}$ g$^{-1}$ and $\alpha=1.3$. For the opacity $\kappa$, we use the range $\kappa=3-30$ cm$^2$ g$^{-1}$. For the engine model, we treat $E_{\rm int0}t_{\rm inj}$ and $\xi$ as free parameters to fit the light curve. The circle, square and triangle denote the case of the fiducial model, minimum mass model and hot interior model, respectively (see table 1).[]{data-label="figure:parameter"}](fig5.ps){width="100mm"}
{width="170mm"}
Comparison with GRB 130603B
---------------------------
We compare the results with the optical and infrared observations of short GRB 130603B in figure \[figure:observation\]. The fiducial parameter set in table 1 is adopted. The $r$-process model and the engine model result in similar light curves at the optical and infrared bands. Both of them satisfy the observational data of GRB 130603B. Note that the detection point at F606W band at $\sim10^5$ s is consistent with the afterglow of GRB 130603B modeled as a smoothly broken power law [blue dashed line, @Tan+13]. We regard this detected value as an upper limit for the luminosity of emission from the ejecta. The detection point at F160W band at $\sim10^6$ s exceeds the extrapolation of the afterglow emission [red dashed line, @Tan+13], so that we regard this detected emission as a thermal radiation from the ejecta.
The range of the model parameters $v_{\min}$ and $M_{\rm ej}$ to satisfy the constraints obtained from the observation of GRB 130603B is shown in figure \[figure:parameter\] as colored areas (red area for the $r$-process model and blue area for the engine model). Note that the red area has a completely overlap with the blue area. We fix the other model parameters $v_{\max}=0.4c$, $\beta=3.5$, $\dot{\epsilon}_0=2\times10^{10}$ erg s$^{-1}$ g$^{-1}$ and $\alpha=1.3$ as in the fiducial model. We take into account the uncertain range of the opacity, $\kappa=3-30$ cm$^2$ g$^{-1}$ to constrain the parameters, $v_{\min}$ and $M_{\rm ej}$. In the engine model, $\xi$ and $E_{\rm int0}t_{\rm inj}$ are additionally treated as free parameters to derive the allowed area in figure \[figure:parameter\].
### Limits on ejecta mass
In the $r$-process model, the luminosity becomes smaller for smaller ejecta mass $M_{\rm ej}$. The small ejected mass $M_{\rm ej}\lesssim0.07M_{\odot}$ cannot reproduce the infrared excess of GRB 130603B (figure \[figure:observation\]). The required ejecta mass is relatively large compared to the mass indicated by recent numerical simulations for a merger of binary NSs [e.g., @Hot+13; @Ros+14; @Jus+14]. Note that in @BFC13, $0.03 - 0.08 M_{\odot}$ is required to explain the observed infrared excess, which is a factor $\sim$2 smaller than our results. Their theoretical light curves are based on the study of @BK13. In @BK13, a broken power-law mass density profile with the index $-1$ for the inner layer and $-10$ for the outer layer of ejecta is adopted, in which the mass of the ejecta is efficiently concentrated at the transition point of the density index. Therefore, the luminosity of ejecta is evaluated as the heating rate multiplied by the total ejecta mass at the moment when the diffusion radius reaches the transition point. On the other hand, the index of our mass density profile of the ejecta is $\beta = 3.5$, which is indicated by general relativistic simulations by @Hot+13. This profile is quite different from the profile adopted in @BK13; the index is close to 3, in which the mass in each logarithmic radius is constant. Then, at the time $t = t_{\rm tr}$ (equation \[sec2-2-3:t\_tr\]), the mass contributing to the luminosity is about $\sim$60% of the total ejecta mass in the case $v_{\min}=0.1c$. This profile predicts luminosity dimmer than that other studies. In fact, @Hot+13b tried to explain the observed infrared excess using the mass profile which is almost the same with ours. In the case of a binary NS merger with ejecta mass $\sim$0.02 $M_{\odot}$, even if they use a larger nuclear heating rate (larger by a factor of 2), their predicted luminosity is slightly smaller than the observed infrared excess (in the left panel of their figure 3). This result is consistent with our model, i.e., our model requires larger mass than most of previous studies.
In the engine model, the injected internal energy which determines the luminosity does not depend on the ejecta mass (except for the limit in equation \[sec2-1:E\_int\]). However, the luminosity declines rapidly after the transition time $t\gtrsim t_{\rm tr}$ which depends on the ejecta mass as in equation (\[sec2-2-3:t\_tr\]). The condition $t_{\rm tr}\gtrsim10^6$ s in the observer frame is required to reproduce the excess observed from GRB 130603B in the near-infrared band. This condition gives the lower limit for the ejecta mass in the engine model, $M_{\rm ej}\gtrsim0.02M_{\odot}$ with the opacity $\kappa\sim30$ cm$^2$ g$^{-1}$.
Note that the observed upper limit on the infrared luminosity at $\sim3\times10^6$ s in the observer frame (figure \[figure:observation\]), which corresponds to $t_{\rm tr}\lesssim3\times10^6$ s, gives the upper limit on the ejecta mass for both models. However, this limit is not important for the range $M_{\rm ej}<0.2M_{\odot}$ in the range of the opacity $\kappa=3-30$ cm$^2$ g$^{-1}$.
### Limits on the minimum velocity
The smaller minimum velocity $v_{\min}$ gives the smaller bolometric luminosity at certain time in the $r$-process model (see equations \[sec2-2-1:L\_bol,nuc\] and \[sec2-2-2:L\_bol,nuc\]). The small minimum velocity enlarges the size of ejecta (when we fix the maximum velocity $v_{\max}$). Then, the diffusion time $t_{\rm diff}$ of photons emitted from the inner region of the ejecta becomes long for the small velocity $v_{\min}$ (equation \[sec2-2-3:t\_tr\]). The mass between $r_{\rm diff}$ and $2r_{\rm diff}$ (or $r_{\rm out}$) increases toward inner region of the ejecta (as long as $\beta>3$) so that the mass is reduced for the small minimum velocity $v_{\rm min}$ at certain time. In fact, the dependence of the mass on the minimum velocity is $4\pi r_{\rm diff}^3\rho(t,v_{\rm diff})\propto v_{\min}^{\frac{\beta-3}{\beta-2}}=v_{\min}^{1/3}$. As a result, smaller minimum velocity gives smaller luminosity to reproduce the observed infrared excess of GRB 130603B. Moreover, smaller minimum velocity gives larger temperature $T_{\rm obs}$ at certain time (equations \[sec2-2-2:T\_obs,nuc\] and \[sec2-2-3:T\_obs,nuc\]) because mass density at a shell with small velocity is large. The difference between the detected luminosity at F160W band and the upper limit on the luminosity at F606W band at $\sim10^6$ s in the observer frame gives the upper limit on the observed temperature ($T_{\rm obs}\lesssim4\times10^3$ K). To satisfy the observed upper limit on the temperature from GRB 130603B, a lower limit of $v_{\min}\gtrsim0.1c$ is obtained for $M_{\rm ej}\sim0.1M_{\odot}$.
The smaller minimum velocity $v_{\min}$ gives higher temperature $T_{\rm obs}$ in the engine model (equations \[sec2-2-2:T\_obs,sh\] and \[sec2-2-3:T\_obs,sh\]). The observational limit for the temperature at $\sim10^6$ s in the observer frame indicates that the range of the minimum velocity $v_{\rm min}$ is limited in the engine model ($v_{\min}\gtrsim0.06c$ for $M_{\rm ej}\sim0.1M_{\odot}$).
### Dependence on opacity
We discuss the dependence on the value of $\kappa$. As mentioned in section 2.2, we use the temperature-independent opacity $\kappa$ with the grey approximation. In general, the $r$-process line opacity depends on frequency and changes with temperature and ionization state of the ejecta [@KBB13; @TH13]. The indicated grey opacity is $\kappa=3-30$ cm$^2$ g$^{-1}$.
In the case of the $r$-process model, the luminosity significantly depends on opacity $\kappa$. The larger opacity causes larger diffusion time $t_{\rm diff}$, so that larger time is required to observe the inner region of the ejecta for given ejecta mass $M_{\rm ej}$ and minimum velocity $v_{\min}$. In fact, two transition times $t_{\times}$ and $t_{\rm tr}$ are proportional to $\kappa^{1/2}$ (equations \[sec2-2-2:t\_times\] and \[sec2-2-3:t\_tr\]). Then, the mass around the diffusion radius $r_{\rm diff}$ is small at certain time, so that the luminosity is reduced. As a result, in order to explain the infrared excess observed in GRB 130603B, larger mass $M_{\rm ej}$ is required for the larger value of opacity $\kappa$. For the opacity $\kappa>30$ cm$^2$ g$^{-1}$, total ejecta mass $M_{\rm ej}\gtrsim0.2M_{\odot}$ is required to reproduce the observed excess, which is much larger than the simulation results of mergers of binary NSs [e.g., @Hot+13]. On the other hand, the transition time $t_{\rm tr}$ is smaller for the smaller value of the opacity. Then, the luminosity significantly increases at $\sim10^5$ s. For the opacity $\kappa\lesssim3$ cm$^2$ g$^{-1}$, there is no parameter set which gives smaller luminosity than the detection at F606W band ($\sim10^5$ s in the observer frame) and the luminosity comparable to the observed excess at F160W band simultaneously in the $r$-process model.
In the case of the engine model, a larger value of opacity $\kappa$ reduces the lower limit for the mass $M_{\rm ej}$ to explain the observed excess. For certain temperature and luminosity, the opacity $\kappa$ and the ejecta mass $M_{\rm ej}$ always degenerate in the form $\kappa M_{\rm ej}$ (see equations \[sec2-2-1:T\_obs,sh\], \[sec2-2-1:L\_bol,sh\], \[sec2-2-2:T\_obs,sh\], \[sec2-2-2:L\_bol,sh\], \[sec2-2-3:T\_obs,sh\] and \[sec2-2-3:L\_bol,sh\]). This dependence comes from the optical depth (equation \[sec2-1:tau\]) because the internal energy in the ejecta does not depend on the opacity and the ejecta mass, contrary to the $r$-process model. We present a parameter set to give the minimum ejecta mass $M_{\rm ej}$ in table 1 as the minimum mass parameter set. We also plot the value of $M_{\rm ej}$ and $v_{\min}$ of this model in figure \[figure:parameter\] as a square. This ejecta mass is naturally realized in general relativistic simulations [e.g., @Hot+13]. Although the larger value of the opacity $\kappa$ reduces the lower limit for the ejecta mass $M_{\rm ej}$, the kinetic energy is $E_{\rm kin}\sim1.1\times10^{51}$ erg (equation \[sec2-1:E\_kin\]) which is close to the initial injected energy $E_{\rm int0}=0.9\times10^{51}$ erg for the minimum mass parameter set. The lower ejecta mass $M_{\rm ej}$ reduces the kinetic energy of the ejecta, $E_{\rm kin}(\propto M_{\rm ej}v_{\max}^{5-\beta}v_{\min}^{\beta-3})$ in equation (\[sec2-1:E\_kin\]), so that the required energy $E_{\rm int0}$ may exceed the kinetic energy of the ejecta for the larger opacity. For the small value of the opacity, larger mass and smaller minimum velocity is required to satisfy the condition $t_{\rm tr}\gtrsim10^6$ s ($t_{\rm tr}\propto\kappa^{1/2}M_{\rm ej}^{1/2}v_{\min}^{-1/2}$ in equation \[sec2-2-3:t\_tr\]). For the opacity $\kappa=3$ cm$^2$ g$^{-1}$, the condition corresponds to $(M_{\rm ej}/0.2M_{\odot})(v_{\min}/0.1c)^{-1}\gtrsim1$. The observational constraint for the temperature also requires a large value of the minimum velocity $v_{\min}$. Then, there is no solution to explain the observed excess within the parameter range shown in figure \[figure:parameter\] for the opacity $\kappa\le3$ cm$^2$ g$^{-1}$. Therefore, for the small opacity $\kappa\le 3$ cm$^2$ g$^{-1}$ the engine model cannot explain the observed excess.
### Dependence on engine parameters
Since the engine model has additional free parameters, $\xi$ and $E_{\rm int0}t_{\rm inj}$, the allowed region of the parameters is larger than that of the $r$-process model. We can impose the lower limit on the parameter $E_{\rm int0}t_{\rm inj}$ by regarding the infrared luminosity $\sim10^{41}$ erg s$^{-1}$ at $t\sim 7$ day as bolometric luminosity in the source rest frame with equation (\[sec2-2-3:L\_bol,sh\]). The derived limit is $(E_{\rm int0}/10^{51}{\rm erg})(t_{\rm inj}/10^2{\rm s})\gtrsim0.4$. To satisfy the optical upper limit at $\sim10^5$ s and the detected luminosity at $\sim10^6$ s in the observer frame (figure \[figure:observation\]), we find the lower limit on the index of the temperature profile $\xi\gtrsim1.0$. For a smaller value of the index $\xi$, emission from the ejecta with relatively high temperature can be observed at time $\sim10^5$ s, so that luminosity at F606W band is larger than the observed upper limit of GRB 130603B. In addition, the smaller value of $\xi$ decreases relative internal energy in the inner edge of the ejecta. To reproduce the luminosity at time $\sim 10^6$ s in the observer frame when observed emission comes from the inner ejecta, the smaller $\xi$ requires the larger initial internal energy $E_{\rm int0}$ which exceeds the kinetic energy of the ejecta $E_{\rm kin}$ in some cases.
Comparison with Other GRBs with Deep Optical Observations {#other}
---------------------------------------------------------
Several deep optical observations of short GRBs give stringent upper limits on the luminosity of macronovae [@Kann+11]. We compare the results with two deep optical observations of short GRBs, GRB 050509B and GRB 080905A. For the fiducial parameter set, the luminosity exceeds the observational upper limits on these two observations. In the engine model, we can reduce the luminosity in the early phase $\lesssim10^5$ s without reducing the luminosity in the late phase $\sim10^6$ s by utilizing the steep temperature profile (large $\xi$).
Here, we introduce the hot interior parameter set with larger value of index $\xi$ than that of the fiducial parameter set. Since emission from the inner part of the ejecta is observed at the later time, the luminosity at the early phase decreases and avoids the observational limits if most of the internal energy is injected to the inner part of the ejecta. We show the light curve of the hot interior parameter set in figure \[figure:observation2\]. We choose the parameters as $M_{\rm ej}=0.08M_{\odot}, v_{\min}=0.18c, t_{\rm inj}=10^2{\rm s}, E_{\rm int0}=0.8\times10^{51}{\rm erg}$ and $\xi=2.7$ (the right column of table 1). From figure \[figure:observation2\], the light curves are consistent with all three observations using the same model parameters. A possible scenario for the hot interior parameter set is that the shock produced by the activity of the central engine may not be able to catch up with the outer part of the ejecta because the velocity of the ejecta is close to the light speed ($v_{\max}=0.4c$). Then, only the inner part of the ejecta will be heated. For comparison, we also show the light curves in the $r$-process model with the parameter set of the hot interior in figure \[figure:observation2\] as dashed lines. The luminosity of the $r$-process model exceeds the observed upper limits in two observations, GRB 050509B and GRB 080905A (left and middle panels of figure \[figure:observation2\]) if we choose the parameter set of the hot interior model. We are not able to find any parameter set in the $r$-process model, which simultaneously satisfies the observed limits of the three observations. Note that we do not argue that the $r$-process model is excluded from these results because we need to take into account the variations of the model parameters for each event.
Note that the extended emission was not detected in three short GRBs. However, it is not unreasonable to miss the extended emission of these bursts. One possibility is a selection effect. Observationally, the fraction of short GRBs with extended emission is significantly larger at softer energy bands: $\sim$25% in the Swift BAT samples [$>$15 keV; @NGS10] and $\sim$7% in the BATSE samples [$>$20 keV; @BKG13]. This suggests that observations with a low energy threshold may dramatically increase short GRBs with extended emission [@Nak+13]. This will be further tested by future soft X-ray survey facilities such as Wide-Field MAXI (0.7-10 keV) [@Kaw+14]. The three referred short GRBs were detected by Swift BAT and therefore Swift BAT could not detect extended emission by chance. Alternatively, the outflow following the main short GRB jet could not breakout the ejecta. @Nag+14 and @Mur+14 investigated the propagation of jets in merger ejecta. They found the cases that relativistic jets can penetrate merger ejecta and produce the prompt emission of short GRBs, but in the late energy injection cases, outflow fails to breakout the ejecta. Therefore, some extended emission may not be observed, although the central engine works actively.
Outer Region of Mass Density Profile {#outer}
------------------------------------
![Dependence of theoretical light curves for the $r$-process model on the shape of the front of the ejecta. We plot the case of GRB 130603B. Thick-dashed lines denote the fiducial model. Thin-solid, thin-dashed, and thin-dot-dashed lines correspond to the exponential profiles of mass density (equation \[sec3:rho\]) with $v_{\max}' = 0.4c, 0.5c$ and $0.6c$, respectively. We also plot the observational data from @Tan+13 [@BFC13; @Cuc13; @de14].[]{data-label="figure:exp_comp"}](fig7.ps){width="80mm"}
In the thin-diffusion phase, the light curve strongly depends on the density profile of the ejecta surface. The density profile is determined by the complex merger dynamics [@Hot+13], so that the density profile of the ejecta cannot be analytically derived as mentioned in section 2.3. Since the outer part of the density profile is difficult to calculate precisely, little attention has paid on the mass profile at the outer region in current numerical simulations. In order to investigate the dependence of the light curve on the mass profiles in the thin-diffusion phase, we consider other forms of the mass profile and compare the light curve with that of equation (\[sec2-1:rho\]). We adopt an exponential profile $$\begin{aligned}
\label{sec3:rho}
\rho(t,v)&=&\rho_0\left(\frac{t}{t_0}\right)^{-3}\left(\frac{v}{v_{\rm min}}\right)^{-\beta} \nonumber \\
& & \times\exp\left(-\frac{v-0.5v_{\max}}{v_{\max}'-v}\right).\end{aligned}$$ We introduce an additional free parameter $v_{\max}'(\ge v_{\max})$ and the ejecta expand $v_{\min}\le v\le v_{\max}'$. The calculations are the same except for using $\Delta r\sim v_{\max}'t - r_{\rm diff}$. We fix the mass with velocity larger than $0.5v_{\max}$ and calculate three models for $v_{\max}'=0.4c, 0.5c, 0.6c$. For the other parameters, we adopt from the fiducial parameter set in table 1. We show the results in the $r$-process model in figure \[figure:exp\_comp\]. Since we fix the mass with velocity larger than $0.5v_{\max}$, the bolometric luminosities at the time $t\sim t_{\times}$ are almost the same values. In the thin-diffusion phase ($t\ll t_{\times}$), both the luminosity and the temperature are smaller than the fiducial model. This is because the density at the front of the ejecta is reduced in this mass profile. Since the maximum velocity effectively becomes large and the adiabatic cooling becomes efficient, these effects for luminosity and temperature should be also seen in the engine model. We conclude that the luminosities in the thin-diffusion phase ($t\ll t_{\times}$) have uncertainties at least with $\sim1-2$ mag, which originates from the uncertainty of the outermost mass profile.
Note that the emission from the ejecta with the mass profile discussed here reduces the tension between the light curve in the $r$-process model and the upper limits of the deep optical observations (GRB 050509 and GRB 080905A) as discussed in section \[other\]. Especially, the optical luminosity in the case $v_{\max}'=0.6c$ (thin-dot-dashed line) significantly decreases after $t\gtrsim10^5$ s. Therefore, the ejecta with relatively shallow mass distribution at the front of the ejecta is able to explain the current optical follow-up observations in the $r$-process model.
Implications to Discriminate Two Models {#implication}
---------------------------------------
In the fiducial parameter set, the light curves for two models in the optical and infrared bands are similar (figure \[figure:observation\]). In figure \[figure:parameter\], the allowed parameter region to explain the observation of GRB 130603B for both models are also overlapped. Recall that emission from ejecta is described as blackbody radiation for the two models, whose spectrum is narrow in bands. Therefore, no excess at other wavelengths is expected and also the prediction is consistent with radio observations [@Fong+14]. Therefore, it is difficult to discriminate two models from the currently available observational data.
In the $r$-process model, the light curve with mass profile $\rho\propto v^{-\beta}$ ($3\lesssim\beta\lesssim4$) and a parameter set which explains the infrared excess detected from GRB 130603B cannot explain the upper limits obtained from the deep optical observations of some short GRBs. Therefore, if both stringent optical upper limits at $\sim10^5$ s and bright infrared emission at $\sim10^6$ s are simultaneously obtained from a single event (with a difference larger than two magnitudes $M_{\rm optical}(\sim10^5~{\rm s}) - M_{\rm infrared}(\sim10^6~{\rm s})\gtrsim2$ mag ), the $r$-process model is significantly restricted. For the engine model, these observations give a constraint for the temperature distribution in the ejecta, which may give new insights into the activity of the central engine.
As shown in the middle panel of figure \[figure:evolution\], the bolometric luminosity in the $r$-process model from the whole ejecta (blue long-dashed line), including low temperature and/or X-rays and $\gamma$-rays produced directly in radioactive decays [@Chu14], declines more gradually than that for the engine model. This is because there is no energy injection after the time $t>t_{\rm inj}$ for the engine model. Then, the luminosity significantly decreases when photons at the inner edge of the ejecta begin to diffuse out (see the middle panel of figure \[figure:evolution\] and figure \[figure:observation\]). The luminosity from the whole ejecta can be described as $L_{\rm bol}\sim M_{\rm ej}\dot{\epsilon}$ in the transparent phase. The index of time $t$ is determined by the nuclear heating rate, $\alpha\sim1.3$. Therefore, the two models are distinguishable by observing the temporal evolution of bolometric luminosity from the whole ejecta in this phase.
SUMMARY
=======
We calculated the light curves of macronovae by developing analytical models. We modeled the ejecta based on the results of numerical simulations for a merger of binary NSs. In addition to the nuclear decay of $r$-process elements (the $r$-process model which is often discussed), we considered another heating mechanism for the ejecta, the engine-driven shock (engine model). We compared the results with the optical and infrared observations of the first macronova candidate associated with GRB 130603B, and showed that both models can explain the observations. In order to reproduce the observed light curve, the $r$-process model requires relatively large ejecta mass $M_{\rm ej}\gtrsim0.07M_{\odot}$ which is mainly determined by the observed infrared luminosity $\sim10^{41}$ erg s$^{-1}$ at $\sim10^6$ s. In the engine model, the internal energy of ejecta, which mainly determines the observed luminosity, does not depend on the ejecta mass. Then, unless the entire of the ejecta is effectively thin (the diffusion time is smaller than the dynamical time, $t_{\rm diff}<t$, at the inner edge of the ejecta) [^2], the required ejecta mass is $M_{\rm ej}\gtrsim0.02M_{\odot}$, which is comparable to the recent numerical simulation results. The initial internal energy $E_{\rm int0}$ and the injection time $t_{\rm inj}$ are required as $(E_{\rm int0}/10^{51}{\rm erg})(t_{\rm inj}/10^2{\rm s})\gtrsim 1$, which is consistent with the observed extended emission of short GRBs, $E_{\rm iso}\sim10^{50}-10^{51}$ erg and $t_{\rm dur}\sim10-10^2$ s. The required minimum velocity is about $v_{\min}\gtrsim0.05c$ for both models, which is mainly determined by the constraint for the observed temperature $\lesssim4\times10^3$ K at $\sim10^6$ s. For the range of the opacity $\kappa\lesssim3$ cm$^2$ g$^{-1}$, it is difficult for both models to explain the observations of macronova associated with GRB 130603B by the ejecta mass less than $M_{\rm ej}<0.2M_{\odot}$.
If macronovae are identical, the upper limits on the luminosity obtained in the deep optical observations of other short GRBs give stringent constraints on the $r$-process model. On the other hand, the engine model satisfies these constraints if the temperature profile is centrally concentrated in the ejecta (large $\xi$). Thus, if the difference between the optical magnitude at $\sim10^5$ s and the infrared magnitude at $\sim10^6$ s is larger than $\sim2$ mag in a single event, the $r$-process model is difficult to explain the observations unless the front of the ejecta has much shallow mass distribution. Another difference in the light curves between two models is the bolometric luminosity at the transparent phase when dynamical time is smaller than the diffusion time at the inner edge of the ejecta $r_{\rm in}$. Although the optical and infrared luminosities rapidly decrease in the transparent phase, the bolometric luminosity from the whole ejecta, including lower frequency than near-infrared band and/or X-rays and $\gamma$-rays produced directly in radioactive decays, is determined by the energy injection rate of nuclear decay, $\dot{\epsilon}\propto t^{-\alpha}~(\alpha\sim1.3)$. For the engine model, the bolometric luminosity decreases rapidly in this phase (faster than $t^{-2}$). Therefore, we expect that the light curve of the bolometric luminosity from the whole ejecta can distinguish between two heating mechanisms.
Our results show that early light curves depend on the density profile of the outermost edge of the ejecta. It is necessary to develop a method to calculate the low-density region of the ejecta in either the analytical or numerical ways in order to precisely predict the early light curves of macronovae.
We thank K. Asano, K. Kashiyama, K. Kiuchi, H. Nagakura, T. Nakamura, Y. Sekiguchi, M. Shibata for fruitful discussions. This work is supported by KAKENHI 24103006 (S.K., K.I.), 24.9375 (H.T.), 24000004, 26247042, 26287051 (K.I.).
APPENDIX A. Analytic formulae for macronova light curves {#appendix-a.-analytic-formulae-for-macronova-light-curves .unnumbered}
========================================================
We summarize the formula for the observed temperature and bolometric luminosity. The detailed derivation of equations in this section is described in sections 2 and 3.
Since we assume that the observed luminosity and temperature approximate the luminosity and temperature at the diffusion radius $r_{\rm diff}$ (section 3), we need to calculate the diffusion radius. For the dynamics of the ejecta, we assume an isotropic and homologous expansion. Then, the velocity of ejecta $v$ is described by equation (\[sec2:v\]) $$\begin{aligned}
v\sim r/t\end{aligned}$$ where the radius $r$ originates the central engine and the time $t$ is measured from the time when a compact binary merges. As in section \[fiducial\], we calculate the diffusion radius $r_{\rm diff}$ from the condition that the diffusion time equals the dynamical time, $t_{\rm diff}=t$. The diffusion time is described by equation (\[sec2-1:t\_diff\]) as $$\begin{aligned}
t_{\rm diff}\sim\tau\frac{\Delta r}{c},\end{aligned}$$ where $c$ is the speed of the light, $\tau$ is the optical depth described by equation (\[sec2-1:tau\]) as $$\begin{aligned}
\tau=\left\{ \begin{array}{ll}
{\displaystyle \int_{r_{\rm diff}}^{r_{\rm out}}\kappa\rho dr} &~~(r_{\rm diff} > 0.5r_{\rm out}) \\
& \\
{\displaystyle \int_{r_{\rm diff}}^{2r_{\rm diff}}\kappa\rho dr} &~~(r_{\rm diff} \le 0.5r_{\rm out} ), \\
\end{array} \right.\end{aligned}$$ and $\Delta r$ is the width of the diffusion region described by the equation (\[sec2-1:Deltar\]) as $$\begin{aligned}
\Delta r\sim \left\{ \begin{array}{ll}
{\displaystyle r_{\rm out}-r_{\rm diff}} &~~(r_{\rm diff}>0.5r_{\rm out}) \\
& \\
{\displaystyle r_{\rm diff}} &~~(r_{\rm diff}\le 0.5r_{\rm out}). \\
\end{array} \right.\end{aligned}$$ In the calculation of the optical depth $\tau$, we use the spatially uniform value of the optical depth $\kappa$ with grey approximation and the ejecta mass density $\rho(t,v)$ described by equation (\[sec2-1:rho\]) as, $$\begin{aligned}
\rho (t,v)=\rho_0 \left(\frac{t}{t_0}\right)^{-3}\left(\frac{v}{v_{\min}}\right)^{-\beta},\end{aligned}$$ where $\rho_0$ and $t_0$ are normalized factors, and $v_{\min}$ is the velocity at the inner edge of the ejecta. The radius $r_{\rm out}$ is the outer edge of the ejecta, described by equation (\[sec2-1:r\_out\]) as $$\begin{aligned}
r_{\rm out}=v_{\max}t,\end{aligned}$$ where the velocity $v_{\max}$ is at the outer edge of the ejecta. The radius at the inner edge of the ejecta $r_{\rm in}$ is described by equation (\[sec2-1:r\_in\]) as $$\begin{aligned}
r_{\rm in}=v_{\min}t.\end{aligned}$$ The normalization factor $\rho_0t_0^3$ in the profile of the mass density is determined by the ejecta mass $M_{\rm ej}$ (in equation \[sec2-1:M\_ej\]) as $$\begin{aligned}
M_{\rm ej}=4\pi\int_{v_{\min}t_0}^{v_{\max}t_0}\rho(t_0,v)r^2dr.\end{aligned}$$
We consider two heating sources of the ejecta. In the $r$-process model, the internal energy of the ejecta is determined by the nuclear heating rate of the $r$-process element described in equation (\[sec2-1:dotepsilon\]) as $$\begin{aligned}
\dot{\epsilon}=\dot{\epsilon}_0\left(\frac{t}{1{\rm day}}\right)^{-\alpha}.\end{aligned}$$ For the engine model, we assume the temperature profile of the ejecta $T(t,v)$ as a result of the activity of the central engine, described in equation (\[sec2-1:T\]) as $$\begin{aligned}
T(t,v)\sim T_0\left(\frac{t}{t_{\rm inj}}\right)^{-1}\left(\frac{v}{v_{\min}}\right)^{-\xi},\end{aligned}$$ where $T_0$ is the normalization factor. This factor is determined by injected internal energy $E_{\rm int0}$ at the time $t_{\rm inj}$, which is described in equation (\[sec2-1:E\_int\]) as $$\begin{aligned}
E_{\rm int0}=4\pi\int_{v_{\min}t_{\rm inj}}^{v_{\max}t_{\rm inj}}aT^4(t_{\rm inj},v)r^2dr.\end{aligned}$$
Observed temperatures in the $r$-process model and the engine model are given by $$\begin{aligned}
T_{\rm obs}\sim\left\{ \begin{array}{l}
{\displaystyle \left[\frac{\dot{\epsilon}t\rho(t,v_{\rm diff})}{a}\right]^{1/4}} \\
~~~~~~~~~~~~~~~(r_{\rm diff}>r_{\rm in}) \\
\\
{\displaystyle \left[\frac{\dot{\epsilon}t\rho(t,v_{\min})}{a}\right]^{1/4}} \\
~~~~~~~~~~~~~~~(0.5r_{\rm in} < r_{\rm diff} \le r_{\rm in}) \\
\\
{\displaystyle 0} \\
~~~~~~~~~~~~~~~(r_{\rm diff} \le 0.5r_{\rm in}), \\
\end{array} \right.\end{aligned}$$ and by $$\begin{aligned}
T_{\rm obs}\sim \left\{ \begin{array}{l}
{\displaystyle T_0\left(\frac{t}{t_{\rm inj}}\right)^{-1}\left(\frac{v_{\rm diff}}{v_{\min}}\right)^{-\xi}} \\
~~~~~~~~~~~~~~~~~(r_{\rm diff} > r_{\rm in}) \\
\\
{\displaystyle T_0\left(\frac{t}{t_{\rm inj}}\right)^{-1}} \\
~~~~~~~~~~~~~~~~~(0.5r_{\rm in} < r_{\rm diff} \le r_{\rm in}) \\
\\
0 \\
~~~~~~~~~~~~~~~~~(r_{\rm diff} \le 0.5r_{\rm in}), \\
\end{array} \right.\end{aligned}$$ respectively. Note that we do not use the approximation $\rho(v,t)\sim\rho(v_{\max},t)$ and $T(v,t)\sim T(v_{\max},t)$ in the thin-diffusion case. The bolometric luminosities for the $r$-process model and the engine model are given by $$\begin{aligned}
L_{\rm bol}\sim\left\{ \begin{array}{l}
{\displaystyle 4\pi \int_{r_{\rm diff}}^{r_{\rm out}}\rho(v,t)\dot{\epsilon}r^2dr} \\
~~~~~~~~~~~~~~~(r_{\rm diff} > 0.5r_{\rm out}) \\
\\
{\displaystyle 4\pi \int_{r_{\rm diff}}^{2r_{\rm diff}}\rho(v,t)\dot{\epsilon}r^2dr} \\
~~~~~~~~~~~~~~~(r_{\rm in} < r_{\rm diff} \le 0.5r_{\rm out}) \\
\\
{\displaystyle 4\pi \int_{r_{\rm in}}^{2r_{\rm diff}}\rho(v,t)\dot{\epsilon}r^2dr} \\
~~~~~~~~~~~~~~~(0.5r_{\rm in} < r_{\rm diff} \le r_{\rm in}) \\
\\
{\displaystyle 0} \\
~~~~~~~~~~~~~~~(r_{\rm diff} \le 0.5r_{\rm in}), \\
\end{array} \right.\end{aligned}$$ and $$\begin{aligned}
L_{\rm bol}\sim\left\{ \begin{array}{l}
{\displaystyle 4\pi\int_{r_{\rm diff}}^{r_{\rm out}}\frac{aT_{\rm obs}^4}{t}r^2dr} \\
~~~~~~~~~~~~~~~(r_{\rm diff} > 0.5r_{\rm out}) \\
\\
{\displaystyle 4\pi\int_{r_{\rm diff}}^{2r_{\rm diff}}\frac{aT_{\rm obs}^4}{t}r^2dr} \\
~~~~~~~~~~~~~~~(r_{\rm in} < r_{\rm diff} \le 0.5r_{\rm out}) \\
\\
{\displaystyle 4\pi\int_{r_{\rm in}}^{2r_{\rm diff}}\frac{aT_{\rm obs}^4}{t}r^2dr} \\
~~~~~~~~~~~~~~~(0.5r_{\rm in} < r_{\rm diff} \le r_{\rm in}) \\
\\
0 \\
~~~~~~~~~~~~~~~(r_{\rm diff} \le 0.5r_{\rm in}), \\
\end{array} \right.\end{aligned}$$ respectively. An example of the calculated result is shown in figure \[figure:evolution\].
We present the numerical values with the parameter dependence for later use. Unlike equations (\[sec2-2-2:t\_times\]) and (\[sec2-2-3:t\_tr\]), we include the contribution from subdominant terms to the numerical values when we integrate equations. Some of the subdominant terms include the ratio $v_{\max}/v_{\min}$. Hereafter, the value $v_{\max}/v_{\min}=4$ in subdominant terms are fixed and are not included in the parameter dependence. We introduce the normalized quantities $M_{\rm ej,0.1}\equiv M_{\rm ej}/0.1M_{\odot}$, $v_{\min, 0.1}\equiv v_{\min}/0.1c$, $v_{\max,0.4}\equiv v_{\max}/0.4c$, $\kappa_{10}\equiv\kappa/10$ cm$^2$ g$^{-1}$, $E_{\rm int0,51}\equiv E_{\rm int0}/10^{51}$ erg and $t_{\rm inj,2}\equiv t_{\rm inj}/10^2$ s. For other parameters, we fix the index of the mass density profile $\beta=3.5$ and the parameters of the nuclear heating rate $\dot{\epsilon}_0=2\times10^{10}$ erg s$^{-1}$ g$^{-1}$ and $\alpha=1.3$. We also introduce the normalized time $t_5\equiv t/10^5$ s and $t_6\equiv t/10^6$ s. The values of observed temperature and bolometric luminosity in the thin-diffusion phase are $$\begin{aligned}
\label{app:T_obs,thin}
T_{\rm obs}\sim\left\{ \begin{array}{l}
{\displaystyle 5.63\times10^3~~{\rm K}} \\
~~{\displaystyle \times M_{\rm ej,0.1}^{0.25}v_{\min,0.1}^{0.125}v_{\max,0.4}^{-0.875}t_5^{-0.825}} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~(r-{\rm process}) \\
\\
{\displaystyle 6.72\times10^3~~{\rm K}} \\
~~{\displaystyle \times E_{\rm int0,51}^{0.25}t_{\rm inj,2}^{0.25}v_{\min,0.1}^{0.25}v_{\max,0.4}^{-1}t_5^{-1}} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=1) \\
\\
{\displaystyle 2.34\times10^3~~{\rm K}} \\
~~{\displaystyle \times E_{\rm int0,51}^{0.25}t_{\rm inj,2}^{0.25}v_{\min,0.1}^{1.25}v_{\max,0.4}^{-2}t_5^{-1}} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=2) \\
\\
{\displaystyle 6.77\times10^2~~{\rm K}} \\
~~{\displaystyle \times E_{\rm int0,51}^{0.25}t_{\rm inj,2}^{0.25}v_{\min,0.1}^{2.25}v_{\max,0.4}^{-3}t_5^{-1}} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=3), \\
\end{array} \right.\end{aligned}$$ and $$\begin{aligned}
\label{app:L_bol,thin}
L_{\rm bol}\sim\left\{ \begin{array}{l}
{\displaystyle 3.51\times10^{41}~~{\rm erg~s}^{-1}} \\
~~{\displaystyle \times \kappa_{10}^{-0.5}M_{\rm ej,0.1}^{0.5}v_{\min,0.1}^{0.25}v_{\max,0.4}^{0.25}t_5^{-0.3}} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(r-{\rm process}) \\
\\
{\displaystyle 7.10\times10^{41}~~{\rm erg~s}^{-1}} \\
~~{\displaystyle \times \kappa_{10}^{-0.5}M_{\rm ej,0.1}^{-0.5}E_{\rm int0,51}t_{\rm inj,2}v_{\min,0.1}^{0.75}v_{\max,0.4}^{-0.25}t_5^{-1}} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=1) \\
\\
{\displaystyle 1.04\times10^{40}~~{\rm erg~s}^{-1}} \\
~~{\displaystyle \times \kappa_{10}^{-0.5}M_{\rm ej,0.1}^{-0.5}E_{\rm int0,51}t_{\rm inj,2}v_{\min,0.1}^{4.75}v_{\max,0.4}^{-4.25}t_5^{-1}} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=2) \\
\\
{\displaystyle 7.32\times10^{37}~~{\rm erg~s}^{-1}} \\
~~{\displaystyle \times \kappa_{10}^{-0.5}M_{\rm ej,0.1}^{-0.5}E_{\rm int0,51}t_{\rm inj,2}v_{\min,0.1}^{8.75}v_{\max,0.4}^{-8.25}t_5^{-1}} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=3), \\
\end{array} \right.\end{aligned}$$ respectively.
The transition time from the thin-diffusion phase to the thick-diffusion phase $t_{\times}$ is $$\begin{aligned}
\label{app:t_times}
t_{\times}&=&\sqrt{\frac{2^{\beta-4}(\beta-3)(1-2^{1-\beta})\kappa M_{\rm ej}}{\pi(\beta-1)[1-(v_{\max}/v_{\min})^{3-\beta}]cv_{\max}}\left(\frac{v_{\max}}{v_{\min}}\right)^{3-\beta}} \nonumber \\
\nonumber \\
&\sim&4.53\times10^5~\kappa_{10}^{0.5}M_{\rm ej,0.1}^{0.5}v_{\min,0.1}^{0.25}v_{\max,0.4}^{-0.75}~{\rm s}.\end{aligned}$$ The values of the observed temperature and bolometric luminosity in the thick-diffusion phase are $$\begin{aligned}
T_{\rm obs}\sim\left\{ \begin{array}{l}
{\displaystyle 3.89\times10^3~~{\rm K}} \\
~~{\displaystyle \times \kappa_{10}^{-0.583}M_{\rm ej,0.1}^{-0.333}v_{\min,0.1}^{-0.167}t_6^{0.342}} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(r-{\rm process}) \\
\\
{\displaystyle 3.86\times10^3~~{\rm K}} \\
~~{\displaystyle \times \kappa_{10}^{-0.667}M_{\rm ej,0.1}^{-0.667}E_{\rm int0,51}^{0.25}t_{\rm inj,2}^{0.25}v_{\min,0.1}^{-0.083}t_6^{0.333}} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=1) \\
\\
{\displaystyle 7.73\times10^3~~{\rm K}} \\
~~{\displaystyle \times \kappa_{10}^{-1.333}M_{\rm ej,0.1}^{-1.333}E_{\rm int0,51}^{0.25}t_{\rm inj,2}^{0.25}v_{\min,0.1}^{0.583}t_6^{1.667}} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=2) \\
\\
{\displaystyle 1.29\times10^4~~{\rm K}} \\
~~{\displaystyle \times \kappa_{10}^{-2}M_{\rm ej,0.1}^{-2}E_{\rm int0,51}^{0.25}t_{\rm inj,2}^{0.25}v_{\min,0.1}^{1.25}t_6^3} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=3), \\
\end{array} \right.\end{aligned}$$ and $$\begin{aligned}
L_{\rm bol}\sim\left\{ \begin{array}{l}
{\displaystyle 1.16\times10^{41}~~{\rm erg~s}^{-1}} \\
{\displaystyle ~~\times \kappa_{10}^{-0.333}M_{\rm ej,0.1}^{0.667}v_{\min,0.1}^{0.333}t_6^{-0.633}} \\
{\displaystyle ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(r-{\rm process})} \\
\\
{\displaystyle 9.59\times10^{40}~~{\rm erg~s}^{-1}} \\
{\displaystyle ~~\times \kappa_{10}^{-0.667}M_{\rm ej,0.1}^{-0.667}E_{\rm int0,51}t_{\rm inj,2}v_{\min,0.1}^{0.667}t_6^{-0.667}} \\
{\displaystyle ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=1)} \\
\\
{\displaystyle 5.96\times10^{41}~~{\rm erg~s}^{-1}} \\
{\displaystyle ~~\times \kappa_{10}^{-3.333}M_{\rm ej,0.1}^{-3.333}E_{\rm int0,51}t_{\rm inj,2}v_{\min,0.1}^{3.333}t_6^{4.667}} \\
{\displaystyle ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=2)} \\
\\
{\displaystyle 2.62\times10^{42}~~{\rm erg~s}^{-1}} \\
{\displaystyle ~~\times \kappa_{10}^{-6}M_{\rm ej,0.1}^{-6}E_{\rm int0,51}t_{\rm inj,2}v_{\min,0.1}^6t_6^{10}} \\
{\displaystyle ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=3),} \\
\end{array} \right.\end{aligned}$$ respectively. Note that since the diffusion radius $r_{\rm diff}$ cannot be analytically described in the thin-diffusion phase, we use the approximations $\rho(v,t)\sim\rho(v_{\max},t)$ and $T(v,t)\sim T(v_{\max},t)$ in equations (\[app:T\_obs,thin\]) and (\[app:L\_bol,thin\]). These approximations make discontinuity at the transition time $t_{\times}$. The ratios of the temperature in the thick-diffusion phase to the temperature in the thin-diffusion phase for the $r$-process $A_{\rm T,r}$ and the engine model $A_{\rm T,e}$ at the time $t_{\times}$ are $$\begin{aligned}
A_{\rm T,r}&=&2^{\beta/4} \nonumber \\
&\sim&1.83\end{aligned}$$ and $$\begin{aligned}
A_{\rm T,e}&=&2^{\xi} \nonumber \\
&\sim&\left\{ \begin{array}{ll}
2.00 &~~(\xi=1) \\
4.00 &~~(\xi=2) \\
8.00 &~~(\xi=3), \\
\end{array} \right.\end{aligned}$$ respectively. The ratios of the luminosity in the thick-diffusion phase to the luminosity in the thin-diffusion phase for the $r$-process model $A_{\rm L,r}$ and the engine model $A_{\rm L,e}$ at the time $t_{\times}$ are $$\begin{aligned}
A_{\rm L,r}&=&2^{\frac{\beta-4}{2}}\left(\frac{1-2^{3-\beta}}{\beta-3}\right)\sqrt{\frac{\beta-1}{1-2^{1-\beta}}} \nonumber \\
&\sim&0.858\end{aligned}$$ and $$\begin{aligned}
A_{\rm L,e}&=&2^{\frac{-\beta-4+8\xi}{2}}\left(\frac{1-2^{3-4\xi}}{4\xi-3}\right)\sqrt{\frac{\beta-1}{1-2^{1-\beta}}} \nonumber \\
&\sim&\left\{ \begin{array}{ll}
1.04 &~~(\xi=1) \\
6.42 &~~(\xi=2) \\
58.8 &~~(\xi=3), \\
\end{array} \right.\end{aligned}$$ respectively.
The transition time from the thick-diffusion phase to the transparent phase $t_{\rm tr}$ is $$\begin{aligned}
\label{app:t_tr}
t_{\rm tr}&=&\sqrt{\frac{(\beta-3)(1-2^{1-\beta})\kappa M_{\rm ej}}{4\pi(\beta-1)[1-(v_{\max}/v_{\min})^{3-\beta}]cv_{\min}}} \nonumber \\
\nonumber \\
&\sim&7.62\times10^5~\kappa_{10}^{0.5}M_{\rm ej,0.1}^{0.5}v_{\min,0.1}^{-0.5}~{\rm s}.\end{aligned}$$ We introduce another transition time $t_{\rm tr2}$ when the upper limit of the integral for the luminosity $2r_{\rm diff}$ reaches the inner edge of the ejecta $r_{\rm in}$, $$\begin{aligned}
t_{\rm tr2}&=&2^{\frac{\beta-2}{2}}t_{\rm tr} \nonumber \\
&\sim&1.28\times10^6~\kappa_{10}^{0.5}M_{\rm ej,0.1}^{0.5}v_{\min,0.1}^{-0.5}~{\rm s}.\end{aligned}$$ The values of the observed temperature and bolometric luminosity in the transparent phase ($t_{\rm tr}\le t<t_{\rm tr2}$) are $$\begin{aligned}
T_{\rm obs}\sim\left\{ \begin{array}{l}
{\displaystyle 2.83\times10^3~~{\rm K}} \\
~~{\displaystyle \times M_{\rm ej,0.1}^{0.25}v_{\min,0.1}^{-0.75}t_6^{-0.825}} \\
~~~~~~~~~~~~~~~~~~~~(r-{\rm process}) \\
\\
{\displaystyle 2.69\times10^3~~{\rm K}} \\
~~{\displaystyle \times E_{\rm int0,51}^{0.25}t_{\rm inj,2}^{0.25}v_{\min,0.1}^{-0.75}t_6^{-1}} \\
~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=1) \\
\\
{\displaystyle 3.74\times10^3~~{\rm K}} \\
~~{\displaystyle \times E_{\rm int0,51}^{0.25}t_{\rm inj,2}^{0.25}v_{\min,0.1}^{-0.75}t_6^{-1}} \\
~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=2) \\
\\
{\displaystyle 4.33\times10^3~~{\rm K}} \\
~~{\displaystyle \times E_{\rm int0,51}^{0.25}t_{\rm inj,2}^{0.25}v_{\min,0.1}^{-0.75}t_6^{-1}} \\
~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=3) \\
\end{array} \right.\end{aligned}$$ and $$\begin{aligned}
L_{\rm bol}\sim\left\{ \begin{array}{l}
{\displaystyle 3.30\times10^{41}~~{\rm erg~s}^{-1}} \\
~~{\displaystyle \times M_{\rm ej,0.1}t_6^{-1.3}\left[1-\left(\frac{t}{t_{\rm tr2}}\right)^{0.667}\right]} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(r-{\rm process}) \\
\\
{\displaystyle 1.33\times10^{41}~~{\rm erg~s}^{-1}} \\
~~{\displaystyle \times E_{\rm int0,51}t_{\rm inj,2}t_6^{-2}\left[1-\left(\frac{t}{t_{\rm tr2}}\right)^{1.333}\right]} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=1), \\
\\
{\displaystyle 1.00\times10^{41}~~{\rm erg~s}^{-1}} \\
~~{\displaystyle \times E_{\rm int0,51}t_{\rm inj,2}t_6^{-2}\left[1-\left(\frac{t}{t_{\rm tr2}}\right)^{6.667}\right]} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=2), \\
\\
{\displaystyle 1.00\times10^{41}~~{\rm erg~s}^{-1}} \\
~~{\displaystyle \times E_{\rm int0,51}t_{\rm inj,2}t_6^{-2}\left[1-\left(\frac{t}{t_{\rm tr2}}\right)^{12}\right]} \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm engine},~\xi=3), \\
\end{array} \right.\end{aligned}$$ respectively.
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[^1]: The outer ejecta may have a relativistic component [@KIS14] and/or an exponential or a power-law profile (see Sec. \[outer\]). Such a profile is difficult to calculate precisely with current numerical calculations.
[^2]: Since there is no energy injection after $t>t_{\rm inj}\sim10^2$ s in the engine model, the luminosity rapidly decreases after the time $t_{\rm tr}\propto M_{\rm ej}^{1/2}$ when photons can diffuse out from the inner edge of the ejecta (section \[transparent\]).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we consider rational cuspidal plane curves having exactly two cusps whose complements have logarithmic Kodaira dimension two. We classify such curves with the property that the strict transforms of them via the minimal embedded resolution of the cusps have maximal self-intersection number.'
author:
- Keita Tono
title: On a new class of rational cuspidal plane curves with two cusps
---
Introduction
============
Let $C$ be an algebraic curve on $\SP^2=\SP^2(\SC)$. A singular point of $C$ is said to be a *cusp* if it is a locally irreducible singular point. We say that $C$ is *cuspidal* (resp. *bicuspidal*) if $C$ has only cusps (resp. two cusps) as its singular points. For a cusp $P$ of $C$, we denote the *multiplicity sequence* of $(C,P)$ by $\ms_P(C)$, or simply by $\ms_P$. We usually omit the last 1’s in $\ms_P$. We use the abbreviation $m_k$ for a subsequence of $\ms_P$ consisting of $k$ consecutive $m$’s. For example, $(2_k)$ means an $A_{2k}$ singularity. The set of the multiplicity sequences of the cusps of a cuspidal plane curve $C$ will be called the *numerical data* of $C$. For example, the rational quartic with three cusps has the numerical data $\{(2),(2),(2)\}$. We denote by $\KB=\KB(\SP^2\setminus C)$ the logarithmic Kodaira dimension of the complement $\SP^2\setminus C$. Suppose that $C$ is rational and bicuspidal. By [@Wak; @Ts], we have $\KB\ge 1$. Let $C'$ denote the strict transform of $C$ via the minimal embedded resolution of the cusps of $C$. We characterize rational bicuspidal plane curves $C$ with $\KB=1$ by $(C')^2$ in the following way.
\[thm0\] If $C$ is a rational bicuspidal plane curve, then $(C')^2\le 0$. Moreover, $(C')^2=0$ if and only if $\KB=1$.
We next consider rational bicuspidal plane curves $C$ with $(C')^2=-1$.
\[thm1\] The numerical data of a rational bicuspidal plane curve $C$ with $(C')^2=-1$ coincides with one of those in the following table, where $a$ is a positive integer.
No.
----- -------------------------------------------------------------- ------------
1 $\{(ab+b-1,ab-1,b_{a-1},b-1),$ $((ab)_2,b_{a})\}$ ($b\ge 2$) $2ab+b-1$
2 $\{(ab+b,ab,b_{a}),$ $((ab+1)_2,b_{a})\}$ ($b\ge 2$) $2ab+b+1$
3 $\{(ab+1,ab-b+1,b_{a-1}),$ $((ab)_2,b_{a})\}$ ($b\ge 3$) $2ab+1$
4 $\{(ab+b,ab,b_{a}),$ $((ab+b-1)_2,b_a,b-1)\}$ ($b\ge 3$) $2ab+2b-1$
Conversely, for a given numerical data in the above table, there exists a rational cuspidal plane curve having that data.
In [@fe], many sequences of rational bicuspidal plane curves were constructed. The numerical data of the curves with $(C')^2=-1$ among them coincide with the data 1, 2 and 3 with $a=1$ in Theorem \[thm1\].
Preliminaries
=============
Let $D$ be a divisor on a smooth surface $V$, $\varphi:V'\rightarrow V$ a composite of successive blow-ups and $B\subset V'$ a divisor. We say that $\varphi$ *contracts* $B$ to $D$, or simply that $B$ *shrinks to* $D$ if $\varphi(\supp{B})=\supp{D}$ and each center of blow-ups of $\varphi$ is on $D$ or one of its preimages. Let $D_1,\ldots,D_r$ be the irreducible components of $D$. We call $D$ an *SNC-divisor* if $D$ is a reduced effective divisor, each $D_i$ is smooth, $D_iD_j\le 1$ for distinct $D_i,D_j$, and $D_i\cap D_j\cap D_k=\emptyset$ for distinct $D_i,D_j,D_k$. Assume that $D$ is an SNC-divisor and that each $D_i$ is projective. Let $\Gamma=\Gamma(D)$ denote the dual graph of $D$. We give the vertex corresponding to a component $D_i$ the weight $D_i^2$. We sometimes do not distinguish between $D$ and its weighted dual graph $\Gamma$. We use the following notation and terminology (cf. [@fu Section 3] and [@mits Chapter 1]). A blow-up at a point $P\in D$ is said to be *sprouting* (resp. *subdivisional*) *with respect to* $D$ if $P$ is a smooth point (resp. node) of $D$. We also use this terminology for the case in which $D$ is a point. By definition, the blow-up is subdivisional in this case. A component $D_i$ is called a *branching component* of $D$ if $D_i(D-D_i)\ge 3$. Assume that $\Gamma$ is connected and linear. In cases where $r>1$, the weighted linear graph $\Gamma$ together with a direction from an endpoint to the other is called a *linear chain*. By definition, the empty graph $\emptyset$ and a weighted graph consisting of a single vertex without edges are linear chains. If necessary, renumber $D_1,\ldots,D_r$ so that the direction of the linear chain $\Gamma$ is from $D_1$ to $D_r$ and $D_iD_{i+1}=1$ for $i=1,\ldots,r-1$. We denote $\Gamma$ by $[-D_1^2,\ldots,-D_r^2]$. We sometimes write $\Gamma$ as $[D_1,\ldots,D_r]$. The linear chain is called *rational* if every $D_i$ is rational. In this paper, we always assume that every linear chain is rational. The linear chain $\Gamma$ is called *admissible* if it is not empty and $D_i^2\le -2$ for each $i$. Set $\NV(\Gamma)=r$. We define the *discriminant* $d(\Gamma)$ of $\Gamma$ as the determinant of the $r\times r$ matrix $(-D_i D_j)$. We set $d(\emptyset)=1$. Let $A=[a_1,\ldots,a_r]$ be a linear chain. We use the following notation if $A\ne\emptyset$: $$\TP{A}:=[a_r,\ldots,a_1],\
\overline{A}:=[a_2,\ldots,a_r],\
\underline{A}:=[a_1,\ldots,a_{r-1}].$$ The discriminant $d(A)$ has the following properties ([@fu Lemma 3.6]).
\[lem:det1\] Let $A=[a_1,\ldots,a_r]$ be a linear chain.
1. If $r>1$, then $d(A)=a_1 d(\overline{A})-d(\overline{\overline{A}})=d(\TP{A})=a_r d(\underline{A})-d(\underline{\underline{A}})$.
2. If $r>1$, then $d(\overline{A})d(\underline{A})-d(A)d(\underline{\overline{A}})=1$.
3. If $A$ is admissible, then $\gcd(d(A),d(\overline{A}))=1$ and $d(A)>d(\overline{A})>0$.
Let $A=[a_1,\ldots,a_r]$ be an admissible linear chain. The rational number $e(A):=d(\overline{A})/d(A)$ is called the *inductance* of $A$. By [@fu Corollary 3.8], the function $e$ defines a one-to-one correspondence between the set of all the admissible linear chains and the set of rational numbers in the interval $(0,1)$. For a given admissible linear chain $A$, the admissible linear chain $A^{\ast}:=e^{-1}(1-e(\TP{A}))$ is called the *adjoint* of $A$ ([@fu 3.9]). Admissible linear chains and their adjoints have the following properties ([@fu Corollary 3.7, Proposition 4.7]).
\[lem:indf\] Let $A$ and $B$ be admissible linear chains.
1. If $e(A)+e(B)=1$, then $d(A)=d(B)$ and $e(\TP{A})+e(\TP{B})=1$.
2. We have $A^{\ast\ast}=A$, $\TP{(A^{\ast})}=(\TP{A})^{\ast}$ and $d(A)=d(A^{\ast})=d(\overline{A^{\ast}})+d(\underline{A})$.
3. The linear chain $[A,1,B]$ shrinks to $[0]$ if and only if $A=B^{\ast}$.
For integers $m$, $n$ with $n\ge 0$, we define $[m_n]=[\overbrace{m,\ldots,m}^n]$, $\TW{n}=[2_n]$. For non-empty linear chains $A=[a_1,\ldots,a_r]$, $B=[b_1,\ldots,b_s]$, we write $A\TA B=[\underline{A},a_r+b_1-1,\overline{B}]$, $A^{\ast n}=\overbrace{A\TA\cdots\TA A}^n$, where $n\ge 1$. We remark that $(A\TA B)\TA C=A\TA(B\TA C)$ for non-empty linear chains $A$, $B$ and $C$. By using Lemma \[lem:det1\] and Lemma \[lem:indf\], we can show the following lemma.
\[lem:adj\] Let $A=[a_1,\ldots,a_r]$ be an admissible linear chain.
1. For a positive integer $n$, we have $[A,n+1]^{\ast}=\TW{n}\TA A^{\ast}$.
2. We have $A^{\ast}=\TW{a_r-1}\TA\cdots\TA\TW{a_1-1}$.
3. If there exist positive integers $m$, $n$ such that $[A,m+1]=[n+1,A]$ (resp. $A\TA\TW{m}=\TW{n}\TA A$), then $m=n$, $a_1=\cdots=a_r=n+1$ (resp. $A=\TW{n}^{\ast\NV(A^{\ast})}$).
We will use the following lemma ([@to:orev Corollary 8]).
\[lem:bu\] Let $a$ be a positive integer and $A$ an admissible linear chain. Let $B$ be a linear chain which is empty or admissible. Assume that a composite $\pi$ of blow-downs contracts $[A,1,B]$ to $[a]$ and that $[a]$ is the image of $A$ under $\pi$.
1. The linear chain $[a]$ is the image of the first curve of $A$. There exits a positive integer $n$ such that $A^{\ast}=[B,n+1,\TW{a-1}]$. Moreover, $A=[a]\TA\TW{n}\TA B^{\ast}$ if $B\ne\emptyset$.
2. The first $n$ blow-ups of $\pi$ are sprouting and the remaining ones are subdivisional with respect to $[a]$ or its preimages. The composite of the subdivisional blow-ups contracts $[A,1,B]$ to $[[a]\TA\TW{n},1]$.
3. The exceptional curve of each blow-up of $\pi$ is a unique ($-1$)-curve in the preimage of $[a]$.
Conversely, $[[a]\TA\TW{n}\TA B^{\ast},1,B]$ shrinks to $[a]$ for given positive integers $a$, $n$ and an admissible linear chain $B$.
Resolution of a cusp {#sec:cres}
--------------------
Let $(C,P)$ be a curve germ on a smooth surface $V$. Suppose that $(C,P)$ is a cusp. Let $\sigma:V'\rightarrow V$ be the minimal embedded resolution of $(C,P)$. That is, $\sigma$ is the composite of the shortest sequence of blow-ups such that the strict transform $C'$ of $C$ intersects $\sigma^{-1}(P)$ transversally. Let $
V'=V_n\stackrel{\sigma_{n-1}}{\longrightarrow}V_{n-1}
\longrightarrow\cdots\longrightarrow
V_2\stackrel{\sigma_1}{\longrightarrow}
V_1\stackrel{\sigma_0}{\longrightarrow}V_0=V
$ be the blow-ups of $\sigma$. The following lemma follows from the assumptions that $(C,P)$ is a cusp and $\sigma$ is minimal.
\[lem:cres0\] For $i\ge1$, the strict transform of $C$ on $V_i$ intersects $(\sigma_0\circ\cdots\circ\sigma_{i-1})^{-1}(P)$ in one point, which is on the exceptional curve of $\sigma_{i-1}$. The point of intersection is the center of $\sigma_{i}$ if $i<n$.
Let $D_0$ denote the exceptional curve of the last blow-up of $\sigma$.
\[lem:cres\] The following assertions hold.
1. The dual graph of $\sigma^{-1}(C)$ has the following shape, where $g\ge1$ and $A_1$ contains the exceptional curve of $\sigma_0$ by definition.
We number the irreducible components $A_{i,1}, A_{i,2},\ldots$ of $A_i$ (resp. $B_{i,1}$, $B_{i,2},\ldots$ of $B_i$) from the left-hand side to the right (resp. the bottom to the top) in the above figure. With these directions and the weights $A_{i,1}^2,A_{i,2}^2,\ldots$, $B_{i,1}^2,B_{i,2}^2,\ldots$, we regard $A_i,B_i$ as linear chains.
2. The morphism $\sigma$ can be written as $\sigma=\sigma_0\circ\rho_1'\circ\rho_1''\circ\cdots\circ\rho_g'\circ\rho_g''$, where each $\rho_i'$ (resp. $\rho_i''$) consists of sprouting (resp. subdivisional) blow-ups of $\sigma$ with respect to preimages of $P$.
3. The morphisms $\rho_i:=\rho_i'\circ\rho_i''$ have the following properties.
1. For $j<i$, $\rho_i$ does not change the linear chains $A_j,B_j$.
2. For each $i$, $\rho_{i}\circ\dots\circ\rho_{g}$ maps $A_{i,1}$ to a ($-1$)-curve.
3. $\rho_g$ contracts the linear chain $A_g+D_0+B_g$ to the ($-1$)-curve $\rho_g(A_{g,1})$. For $i<g$, $\rho_{i}$ contracts the linear chain $(\rho_{i+1}\circ\dots\circ\rho_{g})(A_{i}+A_{i+1,1}+B_{i})$ to the ($-1$)-curve $(\rho_{i}\circ\dots\circ\rho_{g})(A_{i,1})$.
We regard $A_i$ and $B_i$ as linear chains in the same way as in Lemma \[lem:cres\] (i). By Lemma \[lem:cres0\], these linear chains are admissible. Let $\NS_i$ denote the number of the blow-ups in $\rho_i'$. The following proposition follows from Lemma \[lem:bu\].
\[prop:cres\] The following assertions hold for $i=1,\ldots,g$.
1. We have $A_i=\TW{\NS_i}\TA B_i^{\ast}$, $A_i^{\ast}=[B_i,\NS_i+1]$.
2. The linear chain $A_i$ contains an irreducible component $E$ with $E^2\le-3$.
The characteristic sequence of a cusp {#sec:char}
-------------------------------------
Let the notation be as in the previous subsection. Put $\alpha_0=\mult_P C$. We take local coordinates $(x,y)$ of $V$ around $P=(0,0)$ such that the germ $(C,P)$ has a local parameterization: $$x=t^{\alpha_0},\
y=\sum_{i=\alpha_1}^{\infty}c_i t^i
\quad
\textnormal{($c_{\alpha_1}\ne0$, $\alpha_1>\alpha_0$, $\alpha_1 \not\equiv 0 \pmod{\alpha_0}$)}.$$ The *characteristic sequence* of $(C,P)$, which is denoted by $\Ch_{P}=\Ch_{P}(C)$, is a sequence $(\alpha_0,\alpha_1,\ldots,\alpha_k)$ of positive integers defined by the following conditions.
1. $\gcd(\alpha_0,\ldots,\alpha_{k})=1$.
2. If $\gcd(\alpha_0,\ldots,\alpha_{i-1})>1$, then $\alpha_i$ is the smallest $j$ such that $c_j \ne 0$ and that $\gcd(\alpha_0,\ldots,\alpha_{i-1})>\gcd(\alpha_0,\ldots,\alpha_{i-1},j)$.
The multiplicity sequence of $P$ is determined by $\Ch_{P}$ as follows. Put $\gamma_i=\alpha_i-\alpha_{i-1}$ for $i=1,\ldots,k$. Perform the Euclidean algorithm for $i=1,\ldots,k$: $$\vbox{\ialign{\hfil$#$&$#$\hfil\quad&#\hfil\cr
\gamma_i&{}=a_{i,1}m_{i,1}+m_{i,2}&$(0<m_{i,2}<m_{i,1})$,\cr
m_{i,1}&{}=a_{i,2}m_{i,2}+m_{i,3}&$(0<m_{i,3}<m_{i,2})$,\cr
&\cdots&\hfil$\cdots$\cr
m_{i,n_i-2}&{}=a_{i,n_i-1}m_{i,n_i-1}+m_{i,n_i}&$(0<m_{i,n_i}<m_{i,n_i-1})$,\cr
m_{i,n_i-1}&{}=a_{i,n_i}m_{i,n_i},&\cr
}}$$ where $m_{1,1}=\alpha_0$ and $m_{i+1,1}=m_{i,n_i}$. Note that $a_{i,n_i}>1$, $n_i>1$, and that $a_{i,j}>0$ if $j>1$ but $a_{i,1}\ge 0$ for each $i$. The multiplicity sequence of $P$ is given by $$(\alpha_0,
\overbrace{m_{1,1},\ldots,m_{1,1}}^{a_{1,1}},\ldots,
\overbrace{m_{i,j},\ldots,m_{i,j}}^{a_{i,j}},\ldots,
\overbrace{1,\ldots,1}^{a_{k,n_k}}).$$ Conversely, $\Ch_{P}$ is determined from $\ms_P$ by the above relation. See [@bk p.516, Theorem 12] for details, where $\gamma_1$ is defined as $\gamma_1=\alpha_1$. We remark that the *Puiseux pairs* $(q_1,p_1),\ldots,(q_k,p_k)$ of $(C,P)$ are computed from $\Ch_P$ by the relations: $$\alpha_0=q_1\cdots q_k,\,\,
\frac{\alpha_i}{\alpha_0}=\frac{p_i}{q_1\cdots q_i},\,\,
\gcd(q_i,p_i)=1\text{ for $i=1,\ldots,k$.}$$ We next describe the relation between the multiplicity sequence determined by $\Ch_{P}$ and the linear chains $A_i$, $B_i$.
\[prop:ms\] We have the following relations between the multiplicity sequence $(m_{1,1},(m_{1,1})_{a_{1,1}},\ldots,(m_{k,n_k})_{a_{k,n_k}})$ and $A_1,B_1,\ldots,A_g,B_g$. In particular $g=k$.
1. If $n_i$ is an odd number, then $$\begin{aligned}
&A_i=\TW{a_{i,1}+1}\TA[a_{i,2}]\TA\cdots\TA\TW{a_{i,n_i-2}+1}\TA[a_{i,n_i-1}]\TA\TW{a_{i,n_i}},\\
&B_i=[a_{i,n_i}]\TA\TW{a_{i,n_i-1}+1}\TA\cdots\TA[a_{i,5}]\TA\TW{a_{i,4}+1}\TA[a_{i,3}]\TA\TW{a_{i,2}},\end{aligned}$$ where we interpret $A_i$, $B_i$ as $A_i=\TW{a_{i,1}+1}\TA[a_{i,2}]\TA\TW{a_{i,3}}$, $B_i=[a_{i,3}]\TA\TW{a_{i,2}}$ when $n_i=3$.
2. If $n_i$ is an even number, then $$\begin{aligned}
&A_i=\TW{a_{i,1}+1}\TA[a_{i,2}]\TA\cdots\TA\TW{a_{i,n_i-1}+1}\TA[a_{i,n_i}],\\
&B_i=\TW{a_{i,n_i}}\TA[a_{i,n_i-1}]\TA\TW{a_{i,n_i-2}+1}\TA\cdots\TA[a_{i,5}]\TA\TW{a_{i,4}+1}\TA[a_{i,3}]\TA\TW{a_{i,2}},\end{aligned}$$ where we interpret $A_i$, $B_i$ as $A_i=\TW{a_{i,1}+1}\TA[a_{i,2}]$, $B_i=\TW{a_{i,2}-1}$ when $n_i=2$.
We have the weighted dual graphs in Figure \[fig-ch\] of $A_i$ and $B_i$, where the vertices are ordered from the left-hand side to the right, and $\ast$ (resp. $\bullet$) denotes a $(-1)$-curve (resp. $(-2)$-curve).
![The weighted dual graphs of $A_i$ and $B_i$[]{data-label="fig-ch"}](fig-ms)
In order to prove Proposition \[prop:ms\], we need Lemma \[lem:ch1\] and Lemma \[lem:ch2\] below. Let $
V'=V_n\stackrel{\sigma_{n-1}}{\longrightarrow}V_{n-1}
\longrightarrow\cdots\longrightarrow
V_2\stackrel{\sigma_1}{\longrightarrow}
V_1\stackrel{\sigma_0}{\longrightarrow}V_0=V
$ be the blow-ups of the minimal embedded resolution $\sigma$ of the cusp $P$ as in the previous subsection. For $i>j$, put $\tau_{i,j}=\sigma_{j}\circ\sigma_{j+1}\circ\cdots\circ\sigma_{i-1}:V_i\rightarrow V_j$. Let $E_i$ denote the exceptional curve of $\sigma_{i-1}$. We use the same symbol to denote the strict transforms of $E_i$. Let $(C_i,P_i)$ denote the strict transform of the curve germ $(C,P)$ on $V_i$, where $C_i\cap E_i=\{P_i\}$. Write $\ms_{P}(C)$ as $\ms_{P}(C)=(m_0,m_1,\ldots)$.
\[lem:ch1\] Suppose $m_0=\cdots=m_{q-1}$.
1. $(C_q E_q)_{P_q}=m_0$ and $(C_q E_i)_{P_q}=0$ for each $i\ne q$.
2. The dual graph of $\tau_{q,0}^{-1}(P)$ is linear. We have $$\tau_{q,0}^{-1}(P)=[E_1, E_2, \ldots, E_q]=[\TW{q-1},1].$$
The assertion (i) follows from [@fz:dd2 Lemma 1.3]. We prove the assertion (ii) by induction on $q$. The assertion is clear if $q=1$. Assume $q>1$. We have $\ms_{P_1}=(m_1,m_2,\ldots)$. By the induction hypothesis, the dual graph of $\tau_{q,1}^{-1}(P_1)$ is linear and $\tau_{q,1}^{-1}(P_1)=[E_2,\ldots,E_q]=[\TW{q-2},1]$. By (i), the center of $\sigma_1$ is on $E_1$, while that of $\sigma_i$ is not for $i\ge 2$. This means that $E_1^2=-2$ on $V_q$ and that $E_1$ intersects only $E_2$ among $E_2,\ldots,E_q$.
\[lem:ch2\] Let $l$ be a projective curve on $V$ which is smooth at $P$. Let $l_i$ denote the strict transform of $l$ on $V_i$. Write $(C l)_P=q m_0+r$, where $1\le q$, $0\le r<m_0$.
1. We have $m_0=\cdots=m_{q-1}$. Moreover, $m_{q}=r$ if $r>0$.
2. We have $(C_q l_q)_{P_q}=r$, $l_q^2=l^2-q$, $\tau_{q,0}^{-1}(l)=E_1+\cdots+E_q+l_q$, $E_q l_q=1$ and $E_1 l_q=\cdots=E_{q-1} l_q=0$.
The assertion (i) follows from [@fz:dd2 Lemma 1.4]. We prove the assertion (ii) by induction on $q$. On $V_1$, we have $(C_1 l_1)_{P_1}=(q-1)m_0+r$, $l_1^2=l^2-1$ and $\sigma_0^{-1}(l)=E_1+l_1$. So the assertion is clear if $q=1$. Assume that $q>1$. We use the induction hypothesis on $V_1$. Since $(C_1l_1)_{P_1}=(q-1)m_1+r$, we have $(C_q l_q)_{P_q}=r$, $l_q^2=l_1^2-q+1=l^2-q$, $\tau_{q,1}^{-1}(l_1)=E_2+\cdots+E_q+l_q$, $E_q l_q=1$ and $E_2 l_q=\cdots=E_{q-1} l_q=0$. Since $E_1l_1=1$ on $V_1$, the curve $E_1$ does not intersect $l_q$.
We first show the assertion for $A_1$ and $B_1$ by induction on $n_1$. Put $b_i=1+\sum_{j=1}^{i}a_{1,j}$. By applying Lemma \[lem:ch1\] to $(C,P)$ with $q=b_1$, we have $\tau_{b_1,0}^{-1}(P)=[E_1,\ldots,E_{b_1-1},E_{b_1}]=[\TW{a_{1,1}},1]$ and $(C_{b_1} E_{b_1})_{P_{b_1}}=m_{1,1}$. We see $\ms_{P_{b_1}}(C_{b_1})=((m_{1,2})_{a_{1,2}},\ldots)$.
We next apply Lemma \[lem:ch1\] to $(C_{b_1},P_{b_1})$ with $q=a_{1,2}$. We have $\tau_{b_2,b_1}^{-1}(P_{b_1})=[E_{b_1+1},\ldots,E_{b_2-1},E_{b_2}]=[\TW{a_{1,2}},1]$ and $(C_{b_2} E_{b_2})_{P_{b_2}}=m_{1,2}$.
We then apply Lemma \[lem:ch2\] to $E_{b_1}$ and $(C_{b_1},P_{b_1})$. Because $(C_{b_1}E_{b_1})_{P_{b_1}}=a_{1,2}m_{1,2}$ ($n_1=2$) or $(C_{b_1}E_{b_1})_{P_{b_1}}=a_{1,2}m_{1,2}+m_{1,3}$ ($n_1>2$), it follows that $\tau_{b_2,b_1}^{-1}(E_{b_1})=[E_{b_1+1},\ldots,E_{b_2-1},E_{b_2},E_{b_1}]=[\TW{a_{1,2}-1},1,1+a_{1,2}]$ and that $(C_{b_2}E_{b_1})_{P_{b_2}}=0$ ($n_1=2$) or $(C_{b_2}E_{b_1})_{P_{b_2}}=m_{1,3}$ ($n_1>2$). Since $P_{b_1}\not\in E_i$ for $i<b_1$, we see $\tau_{b_2,0}^{-1}(P)=[E_1,\ldots,E_{b_1-1},E_{b_1},E_{b_2},E_{b_2-1},\ldots,E_{b_1+1}]=[\TW{a_{1,1}},1+a_{1,2},1,\TW{a_{1,2}-1}]$.
Suppose that $n_1=2$. Since $m_{1,1}=a_{1,2}m_{1,2}$, we have $P_{b_2}\not\in E_{i}$ for $i< b_2$. Thus the weighted dual graph of $\tau_{b_2,0}^{-1}(P)-E_{b_2}$ is unchanged by the remaining blow-ups. The vertex corresponding to $E_{b_2}$ is a branching component of the dual graph of $\sigma^{-1}(P)+C'$. Because $A_1$ contains $E_1$, we have $A_1=\TW{a_{1,1}+1}\TA[a_{1,2}]$, $B_1=\TW{a_{1,2}-1}$. Suppose that $n_1>2$. We have $\ms_{P_{b_1}}(C_{b_1})=((m_{1,2})_{a_{1,2}},m_{1,3},\ldots)$. Since $(C_{b_2}E_{b_1})_{P_{b_2}}=m_{1,3}=\mult_{P_{b_2}}(C_{b_2})$, we see $P_{b_2}\in E_{b_1}$ and $P_i\not\in E_{b_1}$ for $i>b_2$. It follows that $E_{b_1}^2=-a_{1,2}-2$ and that $E_{b_1}$ intersects $E_{b_2+1}$ on $V_i$ for $i>b_2$. We apply the induction hypothesis to $(C_{b_1},P_{b_1})$. Put $T=\tau_{b_{n_1},b_1}^{-1}(P_{b_1})$. We write it as $T=[A,1,B]$, where $A$ contains $E_{b_1+1}$. If $n_1$ is an odd number, then $$\begin{aligned}
&A=\TW{a_{1,2}}\TA[a_{1,3}]\TA\TW{a_{1,4}+1}\TA[a_{1,5}]\TA\cdots\TA\TW{a_{1,n_1-1}+1}\TA[a_{1,n_1}],\\
&B=\TW{a_{1,n_1}}\TA[a_{1,n_1-1}]\TA\TW{a_{1,n_1-2}+1}\TA\cdots\TA[a_{1,6}]\TA\TW{a_{1,5}+1}\TA[a_{1,4}]\TA\TW{a_{1,3}}.\end{aligned}$$ If $n_1$ is an even number, then $$\begin{aligned}
&A=\TW{a_{1,2}}\TA[a_{1,3}]\TA\TW{a_{1,4}+1}\TA[a_{1,5}]\TA\cdots\TA\TW{a_{1,n_1-2}+1}\TA[a_{1,n_1-1}]\TA\TW{a_{1,n_1}},\\
&B=[a_{1,n_1}]\TA\TW{a_{1,n_1-1}+1}\TA\cdots\TA[a_{1,6}]\TA\TW{a_{1,5}+1}\TA[a_{1,4}]\TA\TW{a_{1,3}}.\end{aligned}$$ The first curve of $A$ is $E_{b_1+1}$ by Lemma \[lem:cres\] (iii). It follows that $\tau_{b_{n_1},0}^{-1}(P)=[E_{1},\ldots,E_{b_1-1},E_{b_1},\TP{T}]$. By the induction hypothesis, $A$ and $B$ are unchanged by the remaining blow-ups. We infer that $\tau_{b_{n_1},0}^{-1}(P)-E_{b_{n_1}}$ is also unchanged by the remaining blow-ups. Hence $A_1=[\TW{a_{1,1}},a_{1,2}+2,\TP{B}]$, $B_1=\TP{A}$. We can prove the assertion for $A_i$ and $B_i$ with $i\ge 2$ by using the same arguments as above, where $(C_b,P_b)$ ($b=\sum_{j=1}^{i-1}\sum_{k=1}^{n_j}a_{j,k}$) plays the role of $(C,P)$.
Proof of Theorem \[thm0\] {#sec:bck1}
=========================
Let $C$ be a rational bicuspidal plane curve. Let $P_1,P_2$ denote the cusps of $C$. Let $\sigma:V\rightarrow\SP^2$ be the minimal embedded resolution of the cusps and $C'$ the strict transform of $C$ via $\sigma$. Put $D:=\sigma^{-1}(C)$. We may assume $\sigma=\sigma^{(1)}\circ\sigma^{(2)}$, where $\sigma^{(k)}$ consists of the blow-ups over $P_k$. We decompose the dual graph of $\sigma^{-1}(P_k)$ ($k=1,2$) into subgraphs $A^{(k)}_1,B^{(k)}_1,\ldots,A^{(k)}_{g_k},B^{(k)}_{g_k},D^{(k)}_{0}$ in the same way as in Lemma \[lem:cres\].
By definition, $A^{(k)}_1$ contains the exceptional curve of the first blow-up over $P_k$. We give the weighted graphs $A^{(k)}_1,\ldots,A^{(k)}_{g_k}$ (resp. $B^{(k)}_1,\ldots,B^{(k)}_{g_k}$) the direction from the left-hand side to the right (resp. from the bottom to the top) of the above figure. With these directions, we regard $A^{(k)}_i$ and $B^{(k)}_i$ as linear chains. Let $\sigma^{(k)}_0$ denote the first blow-up of $\sigma^{(k)}$. By Lemma \[lem:cres\], there exists a decomposition $\sigma^{(k)}=\sigma^{(k)}_0\circ\sigma^{(k)}_{1,1}\circ\sigma^{(k)}_{1,2}\circ\cdots\circ\sigma^{(k)}_{g_k,1}\circ\sigma^{(k)}_{g_k,2}$ such that each $\sigma^{(k)}_{i,1}$ (resp. $\sigma^{(k)}_{i,2}$) consists of sprouting (resp. subdivisional) blow-ups with respect to preimages of $P_k$. The morphism $\sigma^{(k)}_{i,1}\circ\sigma^{(k)}_{i,2}$ contracts $[A^{(k)}_i,1,B^{(k)}_i]$ to a ($-1$)-curve for $i\ge1$. Let $\NS^{(k)}_i$ denote the number of the blow-ups of $\sigma^{(k)}_{i,1}$. We first show the “if” part of Theorem \[thm0\]. Assume that $\KB(\SP^2\setminus C)=1$. Put $D_{1}^{(k)}=B_{g_k}^{(k)}$ and $D_{2}^{(k)}=D^{(k)}-(D_{0}^{(k)}+D_{1}^{(k)})$. The dual graph of $D$ has the following shape.
Following [@fz:def], we consider a strictly minimal model $(\widetilde{V},\widetilde{D})$ of $(V,D)$. We successively contract ($-1$)-curves $E$ satisfying one of the following conditions: (1) $E \subset D$ and $(D-E) E=0$, (2) $E \subset D$ and $(D-E) E=1$, (3) $E \subset D$ and $(D-E) E=2$, (4) $E \not\subset D$ and $D E=0$, (5) $E \not\subset D$ and $D E=1$. After a finite number of contractions, we have no $(-1)$-curves satisfying the above conditions. Let $\pi:V \rightarrow \widetilde{V}$ be the composite of the contractions.
\[lem:pi\] The morphism $\pi$ does not contract irreducible curves meeting with $C'$. In particular, $(C')^2=-1$ if and only if $C'$ is contracted by $\pi$.
Suppose that there exists an irreducible curve $E$ on $V$ which intersects $C'$ and is contracted by $\pi$. If $E$ is a component of $D$, then $E$ is either $D_{0}^{(1)}$ or $D_{0}^{(2)}$. Since $E$ is a ($-1$)-curve, we may assume that $\pi$ contracts $E$ first. But this contraction is not allowed, since $(D-E) E=3$. Thus $E\not\subset D$. Since $E$ is contracted by $\pi$, $E$ does not intersect any components of $D$ other than $C'$. This means that $\sigma(E)$ is a plane curve with $\sigma(E)^2 \le -1$, which is impossible.
For a divisor $E$ on $V$, we write $\widetilde{E}=\pi_{\ast}(E)$. It is clear that $\widetilde{D}$ is an SNC-divisor and $\KB(\widetilde{V}\setminus\widetilde{D})=1$.
There exists a fibration $\widetilde{p}:\widetilde{V}\rightarrow \SP^1$ whose general fiber $F$ is $\SP^1$ and $\widetilde{D} F=2$.
By [@ka:cls Theorem 2.3] and the fact that $\widetilde{V}\setminus \widetilde{D}$ is affine, there exists a fibration $\widetilde{p}:\widetilde{V}\rightarrow W$ over a smooth curve $W$ whose general fiber $F$ is $\SP^1$ and $\widetilde{D} F=2$. Since $q(\widetilde{V})=0$, the curve $W$ must be $\SP^1$.
The fibration $\widetilde{p}$ is obtained from a $\SP^1$-bundle $\hat{p}:\Sigma\rightarrow\SP^1$ by successive blow-ups $\widetilde{\pi}:\widetilde{V}\rightarrow\Sigma$. Putting $p=\widetilde{p}\circ\pi$, we have the following commutative diagram.
Following [@fz:def], we use the following terminology. The triple $(\widetilde{V},\widetilde{D},\widetilde{p})$ is called a *$\SC^{\ast}$-triple*. A component of $\widetilde{D}$ is called *horizontal* if the image of it under $\widetilde{p}$ is 1-dimensional. Let $\widetilde{H}$ be the sum of the horizontal components of $(\widetilde{V},\widetilde{D},\widetilde{p})$. The $\SC^{\ast}$-triple $(\widetilde{V},\widetilde{D},\widetilde{p})$ is called of *twisted type* if $\widetilde{H}$ is irreducible; otherwise it is called of *untwisted type*. A fiber of $\widetilde{p}$ is called a *full fiber* of $(\widetilde{V},\widetilde{D},\widetilde{p})$ if it is contained in $\widetilde{D}$. Let $f$ denote the number of the full fibers of $(\widetilde{V},\widetilde{D},\widetilde{p})$.
\[lem:a1\] The $\SC^{\ast}$-triple has the following properties.
1. The $\SC^{\ast}$-triple is of untwisted type.
2. We have $f\le 1$. The fibration $\widetilde{p}$ has at least two singular fibers.
3. The weighted dual graph of a singular fiber of $\widetilde{p}$ is a linear chain $[A,1,B]$, where $A$, $B$ are admissible and are connected components of $\widetilde{D}-\widetilde{H}$. The curve $\widetilde{H}$ intersects only the first vertex of $A$ and the last of $B$.
By [@kiz Theorem 3], the $\SC^{\ast}$-triple is of untwisted type. The assertions (ii), (iii) follow from [@fz:def Lemma 4.4, Theorem 5.8 and 5.11].
Lemma \[lem:pi\] and the assertion (i) of the following proposition show the “if” part of Theorem \[thm0\].
\[prop:unt\] The following assertions hold.
1. We have $\widetilde{H}=\widetilde{D}_{0}^{(1)}+\widetilde{D}_{0}^{(2)}$. The curve $\widetilde{C}'$ is a full fiber of $\widetilde{p}$.
2. The fibration $\widetilde{p}$ has exactly two singular fibers $\widetilde{F}_1=\widetilde{D}_{1}^{(1)}+\widetilde{E}_1+\widetilde{D}_{a}^{(2)}$, $\widetilde{F}_2=\widetilde{D}_{2}^{(1)}+\widetilde{E}_2+\widetilde{D}_{b}^{(2)}$, where $\{a,b\}=\{1,2\}$ and $\widetilde{E}_i$ is the ($-1$)-curve in $\widetilde{F}_i$.
We first show that $\pi$ does not contract $C'$. Assume the contrary. Since $(\widetilde{D}_{0}^{(1)})^2 \ge 0$, $\widetilde{D}_{0}^{(1)}$ is either a horizontal component or a full fiber. Assume $\widetilde{D}_{0}^{(1)}$ is a full fiber. Since $(\widetilde{D}-\widetilde{D}_{0}^{(1)})\widetilde{D}_{0}^{(1)}=2$, one of $D_{1}^{(1)}$ or $D_{2}^{(1)}$ is contracted by $\pi$ to a point on $\widetilde{D}_{0}^{(1)}$. Thus we have $(\widetilde{D}_{0}^{(1)})^2>0$, which is a contradiction. Similarly, $\widetilde{D}_{0}^{(2)}$ is not a full fiber. Thus $\widetilde{H}=\widetilde{D}_{0}^{(1)}+\widetilde{D}_{0}^{(2)}$. Let $\widetilde{F}$ be the fiber of $\widetilde{p}$ passing through the point of intersection of $\widetilde{D}_{0}^{(1)}$ and $\widetilde{D}_{0}^{(2)}$. The strict transform $F$ of $\widetilde{F}$ in $V$ intersects only $C'$ among the irreducible components of $D$. Hence $\sigma(F)$ is a plane curve with $\sigma(F)^2<0$, which is a contradiction. Thus $\pi$ does not contract $C'$. Since $(\widetilde{D}_{0}^{(1)})^2 \ge -1$, $\widetilde{D}_{0}^{(1)}$ is either a horizontal component or a full fiber. Suppose that $\widetilde{D}_{0}^{(1)}$ is a full fiber. Then $\widetilde{C}'$ must be a horizontal component. This means that $\widetilde{p}$ has at most one singular fiber, which contradicts Lemma \[lem:a1\]. Thus $\widetilde{D}_{0}^{(1)}$ is a horizontal component. Similarly, $\widetilde{D}_{0}^{(2)}$ is a horizontal component. Hence $\widetilde{C}'$ must be a full fiber of $\widetilde{p}$. The assertion (ii) follows from (i) and Lemma \[lem:a1\].
We prove the remaining assertions of Theorem \[thm0\]. Let $C$ be a rational bicuspidal plane curve. Suppose $(C')^2\ge 0$. Since $\dim |C'|=1+(C')^2$, it follows that $\SP^2\setminus C$ contains a surface $\SC^{\ast}\times B$, where $B$ is a curve. Hence we have $\KB(\SP^2\setminus C)\le 1$. By [@Wak], $\KB(\SP^2\setminus C)\ge 0$. By [@Ts Proposition 1], $\KB(\SP^2\setminus C)\ge 1$. See also [@ko; @or]. Hence we have $\KB(\SP^2\setminus C)=1$ and $(C')^2=0$.
Proof of Theorem \[thm1\] {#sec:str}
=========================
Let $C$ be a rational bicuspidal plane curve. Let $P_1,P_2$ denote the cusps of $C$. Let $\sigma:V\rightarrow\SP^2$ be the minimal embedded resolution of the cusps. Let $C'$, $D$, etc. have the same meaning as in the first paragraph of the previous section. Assume that $(C')^2=-1$. Put $F_0'=D^{(1)}_0$. Let $\sigma':V\rightarrow V'$ be the contraction of $C'$. Since $(F_0')^2=0$ on $V'$, there exists a $\SP^1$-fibration $p':V'\rightarrow\SP^1$ such that $F_0'$ is a nonsingular fiber. Put $p=p'\circ\sigma':V\rightarrow\SP^1$ and $F_0=F_0'+C'$.
Since $(D^{(2)}_0)^2=0$ on $V'$, there exists another $\SP^1$-fibration such that $D^{(2)}_0$ is a nonsingular fiber.
The surface $X=V\setminus D$ is a $\SQ$-homology plane. Namely $h^i(X,\SQ)=0$ for $i>0$. A general fiber of $p|_{X}$ is a curve $\SC^{\ast\ast}=\SP^1\setminus\{3\text{ points}\}$. Such fibrations have already been classified in [@misu]. We will use their result to prove our theorem. There exists a birational morphism $\varphi:V\rightarrow\Sigma_n$ from $V$ onto the Hirzebruch surface $\Sigma_n$ of degree $n$ for some $n$ such that $p\circ\varphi^{-1}:\Sigma_n\rightarrow\SP^1$ is a $\SP^1$-bundle. The morphism $\varphi$ is the composite of the successive contractions of the ($-1$)-curves in the singular fibers of $p$. Let $S_1$ and $S_3$ be the irreducible components of $A^{(1)}_{g_1}+B^{(1)}_{g_1}$ meeting with $D^{(1)}_0$. Put $S_2=D^{(2)}_0$. The curves $S_1$, $S_2$ and $S_3$ are 1-sections of $p$. The divisor $D$ contains no other sections of $p$.
\[lem:sec\] We may assume that $\varphi(S_1+S_2+S_3)$ is smooth. We have $\varphi(S_1)\sim \varphi(S_2)\sim \varphi(S_3)$ (linearly equivalent) and $n=\varphi(S_i)^2=0$ for each $i$.
We only prove the first assertion. Suppose $\varphi(S_1+S_2+S_3)$ has a singular point $P$. Let $\phi_1$ be the blow-up at $P$. Since $S_1+S_2+S_3$ is smooth on $V$, we can arrange the order of the blow-ups of $\varphi$ so that $\varphi=\phi_1\circ\varphi'$. Let $F'$ be the strict transform via $\phi_1$ of the fiber of $p\circ\varphi^{-1}$ passing through $P$. Let $\phi_2$ be the contraction of $F'$. Since $F'$ is an irreducible component of a singular fiber of $p\circ{\varphi'}^{-1}$, we can replace $\varphi$ with $\phi_2\circ\varphi'$. We infer that $P$ can be resolved by repeating the above process. Hence we may assume that $\varphi(S_1+S_2+S_3)$ is smooth.
Let $F_1,\ldots,F_l$ be all singular fibers of $p$ other than $F_0$. For $i=1,\ldots,l$, let $E_i$ be the sum of the irreducible components of $F_i$ which are not components of $D$. Since $D$ contains no loop, each $E_i$ is not empty. It follows that the base curve of the $\SC^{\ast\ast}$-fibration $p|_X$ is $\SC$. Because $\KB(V\setminus D)=2$, each irreducible component of $E_i$ meets with $D$ in at least two points by [@mits2 Main Theorem]. In [@misu Lemma 1.5], singular fibers of a $\SC^{\ast\ast}$-fibration with three 1-sections were classified into several types. Among them, only singular fibers of type ($\mathrm{I_1}$) and ($\mathrm{III_1}$) satisfy the conditions that each irreducible component of $E_i$ meets with $D$ in at least two points. From the fact that $D$ contains no loop, we infer that each $F_i$ is of type ($\mathrm{III_1}$). By [@misu Lemma 2.3], $p$ has at most two singular fibers other than $F_0$. Since $S_2$ meets with $D-S_2$ in three points, $p$ has exactly three singular fibers $F_0$, $F_1$ and $F_2$. For $i=1,2$, the dual graph of $F_i+S_1+S_2+S_3$ coincides with one of those in Figure \[figsf\], where $\ast$ denotes a ($-1$)-curve and $E_i=E_{i1}+E_{i2}$. The graph $T_{i,j}$ may be empty for each $j$.
![Candidates for the dual graph of $F_i$, $i=1,2$[]{data-label="figsf"}](figsf)
\[lem:fib\] We have $\varphi(F_i)=\varphi(F_i')$ for $i=0,1,2$. For $i=1,2$, the dual graph of $F_i+S_1+S_2+S_3$ must be the first one in Figure \[figsf\].
By Lemma \[lem:sec\], we have $\varphi(F_0)=\varphi(F_0')$. Suppose that $\varphi$ contracts $F_1'$. Let $F_1'$ intersect $S_j$. Write $\varphi$ as $\varphi=\varphi_3\circ\varphi_2\circ\varphi_1$, where $\varphi_2$ is the contraction of $F_1'$. The curve $\varphi_1(F_1')$ intersects three irreducible components of $D+E_1+E_2$. By Lemma \[lem:sec\], $\varphi_1(F_1')$ does not intersect the images under $\varphi_1$ of sections other than $S_j$. It follows that $\varphi_{2}(\varphi_1(S_j))\varphi_{2}(\varphi_1(F_1))>1$, which is absurd. Thus $\varphi$ does not contract $F_1'$. Similarly, $\varphi$ does not contract $F_2'$. If one of $F_1'$, $F_2'$ does not intersect $S_2$, then $\varphi(S_2)^2> 0$, which contradicts Lemma \[lem:sec\]. Thus $F_1'$ and $F_2'$ intersect $S_2$.
By Lemma \[lem:fib\], the dual graph of $D+E_1+E_2$ must coincide with that in Figure \[fig\].
Proof of Theorem \[thm1\] — continued
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Let the notation be as in the previous section. We infer $g_i\le 2$ for $i=1,2$. With the direction from the left-hand side to the right of Figure \[fig\], we regard $T_{ij}$’s as linear chains. Put $s_i=-S_i^2$ and $f_j=-(F_j')^2$ for $i\ne 2$ and $j=1,2$. We have $s_i\ge 2$ and $f_j\ge 2$.
\[lem:0\] The following assertions hold.
1. We may assume $B^{(1)}_{g_1}=[S_1,T_{11}]$. We have $T_{21}=\emptyset$. There exists a non-negative integer $l_{22}$ such that $T_{22}=\TW{l_{22}}$.
2. There exist positive integers $k_{12}$, $k_{34}$ such that $[S_1,T_{11}]^{\ast}=[T_{12},k_{12}+1,\TW{l_{22}}]$, $[F_1',T_{13}]^{\ast}=[T_{14},k_{34}+1,\TW{k_{12}-1}]$ and $[T_{24},S_3]^{\ast}=[\TW{k_{34}-1},f_2,T_{23}]$. We have $A^{(1)}_{g_1}=\TW{\NS^{(1)}_{g_1}}\TA[T_{12},k_{12}+1,\TW{l_{22}}]$.
\(i) We may assume $B^{(1)}_{g_1}=[S_1,T_{11}]$ because the dual graph of $D+E_1+E_2$ is symmetric about the line passing through $F_1'$, $S_2$ and $F_2'$ in Figure \[fig\], and the line passing through $S_1$, $S_2$ and $S_3$. We have $T_{21}=\emptyset$. If $T_{22}\ne\emptyset$, then $\varphi$ contracts $[E_{21},T_{22}]$ to a ($-1$)-curve. By Lemma \[lem:bu\], there exists a positive integer $l_{22}$ such that $T_{22}=\TW{l_{22}}$. We set $l_{22}=0$ if $T_{22}=\emptyset$.
\(ii) We may assume that $\varphi=\varphi_0\circ\varphi_{21}\circ\varphi_{11}\circ\varphi_{12}\circ\varphi_{22}$, where $\varphi_0$ contracts $C'$ and $\varphi_{ij}$ contracts $T_{i,2j-1}+E_{ij}+T_{i,2j}$ to a point. Since $[E_{21},T_{22}]=[1,\TW{l_{22}}]$ and $\varphi(S_1)^2=0$, $\varphi_{11}$ contracts $[S_1,T_{11},E_{11},T_{12}]$ to $[l_{22}+1]$ by Lemma \[lem:indf\] (iii). By Lemma \[lem:bu\], there exists a positive integer $k_{12}$ such that $[S_1,T_{11}]^{\ast}=[T_{12},k_{12}+1,\TW{l_{22}}]$. The composite of the subdivisional blow-ups of $\varphi_{11}$ with respect to the preimages of $S_1$ contracts $[T_{11},E_{11},T_{12}]$ to $[\TW{k_{12}-1},1]$. Since $\varphi(F_1')^2=0$, $\varphi_{12}$ contracts $[F_1',T_{13},E_{12},T_{14}]$ to $[k_{12}]$ by Lemma \[lem:indf\] (iii). By Lemma \[lem:bu\], there exists a positive integer $k_{34}$ such that $[F_1',T_{13}]^{\ast}=[T_{14},k_{34}+1,\TW{k_{12}-1}]$. Similarly, $\varphi_{22}$ contracts $[T_{23},E_{22},T_{24},S_3]$ to $[k_{34}]$. By Lemma \[lem:bu\], there exists a positive integer $k$ such that $[S_3,\TP{T}_{24}]^{\ast}=[\TP{T}_{23},k+1,\TW{k_{34}-1}]$. Since $\varphi(F_2')^2=0$, we have $0=-f_2+k+1$. The last assertion follows from Proposition \[prop:cres\].
Now we prove Theorem \[thm1\]. The linear chain $B^{(2)}_{g_2}$ coincides with one of $\TP{[T_{12},F_1']}$, $[F_1',T_{13}]$, $\TP{[T_{22},F_2']}$ or $[F_2',T_{23}]$.
![The dual graph of $D+E_1+E_2$[]{data-label="fig"}](figd)
$B^{(2)}_{g_2}=\protect\TP{[T_{12},F_1']}$ {#subsec:1}
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\[lem:1\] We have $T_{13}=\emptyset$, $k_{34}=1$, $T_{14}=\TW{f_1-k_{12}-1}$ and $[T_{24},S_3]^{\ast}=[f_2,T_{23}]$.
It is clear that $T_{13}=\emptyset$. By Lemma \[lem:0\], we see $[T_{14},k_{34}+1,\TW{k_{12}-1}]=\TW{f_1-1}$. This means that $k_{34}=1$, $T_{14}=\TW{f_1-k_{12}-1}$. By Lemma \[lem:0\], we have $[T_{24},S_3]^{\ast}=[f_2,T_{23}]$.
Case (1): $g_2=1$. Either $A^{(2)}_1=[T_{22},F_2']$ or $A^{(2)}_1=\TP{[F_2',T_{23}]}$. Suppose $A^{(2)}_1=[T_{22},F_2']$. We have $T_{23}=\emptyset$. By Lemma \[lem:1\], we get $[T_{24},S_3]=\TW{f_2-1}$. Thus $T_{14}+S_3+T_{24}$ consists of ($-2$)-curves and so does $A^{(1)}_{g_1}$, which contradicts Proposition \[prop:cres\]. Hence $A^{(2)}_1=\TP{[F_2',T_{23}]}$. It follows that $T_{22}=\emptyset$. By Lemma \[lem:1\] and Proposition \[prop:cres\], we obtain $[T_{24},S_3]=[\NS^{(2)}_1+1,T_{12},F_1']$. We have $T_{24}\ne\emptyset$. Suppose $T_{14}=\emptyset$. We have $f_1=k_{12}+1$, $g_1=1$ and $A^{(1)}_1=[T_{24},S_3]$. Since $\TW{l_{22}}=T_{22}=\emptyset$, we get $B^{(1)}_1=[T_{12},f_1]^{\ast}$ by Lemma \[lem:0\]. By Proposition \[prop:cres\], $[\NS^{(2)}_1+1,T_{12},f_1]=[T_{24},S_3]=\TW{\NS^{(1)}_1}\TA[T_{12},f_1]$. Thus $\NS^{(1)}_1=2$, $\NS^{(2)}_1=1$. Since $[T_{12},f_1]=\TW{1}\TA[T_{12},f_1]$, we infer $[T_{12},f_1]^{\ast}=[[T_{12},f_1]^{\ast},2]$, which is absurd. Hence $T_{14}\ne\emptyset$. If $g_1=1$, then $A^{(1)}_1=[T_{14},s_3]$ and $T_{24}=\emptyset$, which is a contradiction. Thus $g_1=2$. Since $A^{(1)}_2=S_3$, we have $T_{12}=\emptyset$, $\NS^{(1)}_2=1$ and $A^{(1)}_2=S_3=[k_{12}+2]$ by Lemma \[lem:0\]. By Proposition \[prop:cres\], $B^{(1)}_2=\TW{k_{12}}$. By Lemma \[lem:1\], $T_{14}$ consists of ($-2$)-curves. By Proposition \[prop:cres\], $B^{(1)}_1=\TP{T}_{14}$ and $A^{(1)}_1=T_{24}$. By Lemma \[lem:1\] and Proposition \[prop:cres\], we get $[f_2,T_{23}]=\TW{k_{12}+1}\TA[\TW{f_1-k_{12}-1},\NS^{(1)}_1+1]$. Since $\emptyset\ne T_{14}=\TW{f_1-k_{12}-1}$, we have $[f_2,T_{23}]=[\TW{k_{12}},3,\TW{f_1-k_{12}-2},\NS^{(1)}_1+1]$. On the other hand, $[f_2,T_{23}]=\TP{A}^{(2)}_1=\TW{f_1-1}\TA\TW{\NS^{(2)}_1}=[\TW{f_1-2},3,\TW{\NS^{(2)}_1-1}]$ by Proposition \[prop:cres\]. This means that $\NS^{(2)}_1=2$, $\NS^{(1)}_1=1$ and $f_1=k_{12}+2$. Write $k=k_{12}$. We have $A^{(1)}_2=[k+2]$, $B^{(1)}_2=\TW{k}$, $B^{(1)}_1=[2]$ and $B^{(2)}_1=[k+2]$. By Proposition \[prop:cres\], we see $A^{(1)}_1=[3]$ and $A^{(2)}_1=[2,3,\TW{k}]$. It follows from Proposition \[prop:ms\] that the numerical data of $C$ is equal to $\{(2(k+1),(k+1)_2),\;\;((k+2)_2,k+1)\}$, which coincides with the data 2 with $a=1$, $b=k+1$. Case (2): $g_2=2$. By Lemma \[lem:0\], $T_{22}$ consists of ($-2$)-curves. By Proposition \[prop:cres\], we have $B^{(2)}_1=\TP{T}_{22}$ and $A^{(2)}_1=\TP{T}_{23}$. Thus $A^{(2)}_1=\TW{\NS^{(2)}_1}\TA[l_{22}+1]$. Since $T_{22}\ne\emptyset$, we infer $\NV(A^{(1)}_{g_1})\ge2$ by Lemma \[lem:0\]. It follows that $g_1=1$. Either $A^{(1)}_1=[T_{14},S_3]$ or $A^{(1)}_1=[T_{24},S_3]$. By Lemma \[lem:1\], $T_{14}$ consists of ($-2$)-curves. By Lemma \[lem:0\], $A^{(1)}_1-S_3$ contains an irreducible component other than a ($-2$)-curve. Thus $A^{(1)}_1=[T_{24},S_3]$, $T_{14}=\emptyset$ and $f_1=k_{12}+1$. Because $A^{(2)}_2=F_2'$, we have $\TW{f_2-1}=[\NS^{(2)}_2+1,T_{12},f_1]$ by Proposition \[prop:cres\]. This shows that $\NS^{(2)}_2=1$, $f_1=2$ and $T_{12}=\TW{f_2-3}$. Thus $k_{12}=1$. By Lemma \[lem:0\], $[S_1,T_{11}]=\TW{f_2+l_{22}-2}^{\ast}=[f_2+l_{22}-1]$. By Proposition \[prop:cres\] and Lemma \[lem:1\], $[f_2,T_{23}]={A^{(1)}_1}^{\ast}=[S_1,T_{11},\NS^{(1)}_1+1]=[f_2+l_{22}-1,\NS^{(1)}_1+1]$. Hence $l_{22}=1$, $T_{23}=[\NS^{(1)}_1+1]$. We infer $\NS^{(2)}_1=1$ and $\NS^{(1)}_1=2$. Put $k=f_2-2$. We have $k\ge 1$, $B^{(1)}_1=[k+2]$, $A^{(2)}_2=[k+2]$, $B^{(2)}_1=[2]$ and $A^{(2)}_1=[3]$. By Proposition \[prop:cres\], we get $A^{(1)}_1=[2,3,\TW{k}]$ and $B^{(2)}_2=\TW{k}$. The numerical data of $C$ is equal to $\{(2(k+1),(k+1)_2),\;\;((k+2)_2,k+1)\}$, which coincides with the data 2 with $a=1$, $b=k+1$.
$B^{(2)}_{g_2}=[F_1',T_{13}]$
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If $T_{13}=\emptyset$, then this case is contained in the case \[subsec:1\]. Thus we may assume $T_{13}\ne\emptyset$. We have $T_{12}=\emptyset$.
\[lem:2\] We have $g_2=1$, $T_{22}=\emptyset$ and $A^{(2)}_1=\TP{[F_2',T_{23}]}$. Moreover, $[T_{24},S_3]^{\ast}=[\TW{k_{12}+k_{34}-2},k_{34}+1,\TP{T}_{14}]\TA\TW{\NS^{(2)}_1}$, $B^{(1)}_{g_1}=\TW{k_{12}}$.
Since $T_{12}=\emptyset$, $[S_1,T_{11}]=[l_{22}+2,\TW{k_{12}-1}]$ by Lemma \[lem:0\]. This means that $s_1=l_{22}+2$, $T_{11}=\TW{k_{12}-1}$. By Proposition \[prop:cres\] and Lemma \[lem:0\], $A^{(2)}_{g_2}=\TW{\NS^{(2)}_{g_2}}\TA[T_{14},k_{34}+1,\TW{k_{12}-1}]$. Suppose $g_2=2$. Since $A^{(2)}_2=F_2'$, we get $\NS^{(2)}_2=1$, $T_{14}=\emptyset$, $k_{12}=1$ and $f_2=k_{34}+2$. We have $g_1=1$ and $A^{(1)}_1=[T_{24},S_3]$. By Proposition \[prop:cres\], $[T_{24},S_3]^{\ast}=[l_{22}+2,\NS^{(1)}_1+1]$. By Lemma \[lem:0\], $T_{22}$ consists of ($-2$)-curves. By Proposition \[prop:cres\], we have $A^{(2)}_1=\TP{T}_{23}$, $B^{(2)}_1=\TP{T}_{22}$ and $T_{23}=[l_{22}+2,\TW{\NS^{(2)}_1-1}]$. By Lemma \[lem:0\], $[T_{24},S_3]^{\ast}=[\TW{k_{34}-1},k_{34}+2,l_{22}+2,\TW{\NS^{(2)}_1-1}]$. Thus $[l_{22}+2,\NS^{(1)}_1+1]=[\TW{k_{34}-1},k_{34}+2,l_{22}+2,\TW{\NS^{(2)}_1-1}]$. This shows $k_{34}=1$. We have $[F_1',T_{13}]=[2]$ by Lemma \[lem:0\], which contradicts $T_{13}\ne\emptyset$. Hence $g_2=1$. Suppose $T_{22}\ne\emptyset$. We have $T_{23}=\emptyset$ and $A^{(2)}_1=[T_{22},F_2']$. Since $A^{(2)}_{1}=\TW{\NS^{(2)}_{1}}\TA[T_{14},k_{34}+1,\TW{k_{12}-1}]$ and $T_{22}=\TW{l_{22}}$, we infer $T_{14}=\emptyset$, $k_{12}=1$ and $f_2=k_{34}+2$. We have $g_1=1$ and $A^{(1)}_1=[T_{24},S_3]$. By Proposition \[prop:cres\], $[T_{24},S_3]^{\ast}=[l_{22}+2,\NS^{(1)}_1+1]$. By Lemma \[lem:0\], $[\TW{k_{34}-1},k_{34}+2]=[l_{22}+2,\NS^{(1)}_1+1]$. This shows $k_{34}=2$ and $l_{22}=0$, which is absurd. Hence $T_{22}=\emptyset$. We get $A^{(2)}_1=\TP{[F_2',T_{23}]}$. By Lemma \[lem:0\], we have $[T_{24},S_3]^{\ast}=[\TW{k_{12}+k_{34}-2},k_{34}+1,\TP{T}_{14}]\TA\TW{\NS^{(2)}_1}$ and $B^{(1)}_{g_1}=\TW{k_{12}}$.
Case (1): $k_{34}=1$. Suppose $T_{14}=\emptyset$. We have $g_1=1$ and $A^{(1)}_1=[T_{24},S_3]$. By Lemma \[lem:0\], $[T_{24},S_3]=[\TW{\NS^{(1)}_1-1},k_{12}+2]$. On the other hand, $[T_{24},S_3]=[\NS^{(2)}_1+1,k_{12}+1]$ by Lemma \[lem:2\], which is impossible. Thus $T_{14}\ne\emptyset$. By Lemma \[lem:2\], $[T_{24},S_3]=[\NS^{(2)}_1+1,\TP{T}_{14}^{\ast}\TA\TW{1}^{\ast k_{12}}]$. Thus $T_{24}\ne\emptyset$. We have $g_1=2$ and $A^{(1)}_2=S_3$. By Lemma \[lem:0\], $\NS^{(1)}_2=1$ and $s_3=k_{12}+2$. Either $A^{(1)}_1=T_{14}$ or $A^{(1)}_1=T_{24}$. If $A^{(1)}_1=T_{14}$, then $T_{14}^{\ast}=[\TP{T}_{24},\NS^{(1)}_1+1$\] by Proposition \[prop:cres\]. We get $[T_{24},S_3]=[\NS^{(2)}_1+1,\NS^{(1)}_1+1,T_{24}\TA\TW{1}^{\ast k_{12}}]$, which is a contradiction. Hence $A^{(1)}_1=T_{24}$ and $B^{(1)}_1=\TP{T}_{14}$. By Proposition \[prop:cres\], $T_{24}=\TW{\NS^{(1)}_1}\TA\TP{T}_{14}^{\ast}$. Thus $[\TW{\NS^{(1)}_1}\TA\TP{T}_{14}^{\ast},S_3]=[\NS^{(2)}_1+1,\TP{T}_{14}^{\ast}\TA\TW{1}^{\ast k_{12}}]$. Hence $\NS^{(1)}_1=1$. We have $\TW{s_3-1}\TA[\TP{T}_{14},2]=[\TW{k_{12}},\TP{T}_{14}]\TA\TW{\NS^{(2)}_1}$. This shows $s_3=k_{12}+\NS^{(2)}_1$ and $\NS^{(2)}_1=2$. Thus $[\TP{T}_{14}^{\ast},2]=[2,\TP{T}_{14}^{\ast}]$. There exists a positive integer $l$ such that $T_{14}^{\ast}=\TW{l}$ by Lemma \[lem:adj\] (iii). Write $k=k_{12}$. We have $B^{(1)}_2=\TW{k}$, $A^{(1)}_2=[k+2]$ and $B^{(1)}_1=[l+1]$. By Lemma \[lem:0\], $B^{(2)}_1=[k+2,\TW{l-1}]$. By Proposition \[prop:cres\], we have $A^{(1)}_1=[3,\TW{l-1}]$ and $A^{(2)}_1=[2,l+2,\TW{k}]$. The numerical data of $C$ is equal to $\{(((l+1)(k+1),l(k+1),(k+1)_l),\;\;((l(k+1)+1)_2,(k+1)_l)\}$, which coincides with the data 2 with $a=l$, $b=k+1$.
Case (2): $k_{34}>1$, $T_{14}=\emptyset$. We have $g_1=1$ and $A^{(1)}_1=[T_{24},S_3]$. By Proposition \[prop:cres\] and Lemma \[lem:2\], $[T_{24},S_3]^{\ast}=[B^{(1)}_1,\NS^{(1)}_1+1]=[\TW{k_{12}},\NS^{(1)}_1+1]$. On the other hand, $[T_{24},S_3]^{\ast}=[\TW{k_{12}+k_{34}-2},k_{34}+2,\TW{\NS^{(2)}_1-1}]$ by Lemma \[lem:2\]. Hence $k_{34}=2$, $\NS^{(2)}_1=1$ and $\NS^{(1)}_1=3$. Write $k=k_{12}$. We have $B^{(1)}_1=\TW{k}$. By Lemma \[lem:0\], $B^{(2)}_1=[k+1,2]$. By Proposition \[prop:cres\], we have $A^{(1)}_1=[2,2,k+2]$ and $A^{(2)}_1=[4,\TW{k-1}]$. The numerical data of $C$ is equal to $\{((k+1)_3),\;\;(2k+1,k_2)\}$, which coincides with the data 1 with $a=1$, $b=k+1$.
Case (3): $k_{34}>1$, $T_{14}\ne\emptyset$. By Lemma \[lem:2\], $[T_{24},S_3]=[\NS^{(2)}_1+1,\TP{T}_{14}^{\ast}\TA\TW{k_{34}-1},k_{12}+k_{34}]$. We have $T_{24}=[\NS^{(2)}_1+1,\TP{T}_{14}^{\ast}\TA\TW{k_{34}-1}]$ and $s_3=k_{12}+k_{34}$. We infer $T_{24}\ne\emptyset$ and $g_1=2$. Since $A^{(1)}_2=S_3$, we get $\NS^{(1)}_2=1$ and $k_{34}=2$ by Lemma \[lem:0\]. Either $B^{(1)}_1=\TP{T}_{14}$ or $B^{(1)}_1=\TP{T}_{24}$. If $B^{(1)}_1=\TP{T}_{24}$, then $\TP{T}_{14}^{\ast}=[\NS^{(1)}_1+1,T_{24}]$ by Proposition \[prop:cres\]. Thus $T_{24}=[\NS^{(2)}_1+1,\NS^{(1)}_1+1,T_{24}\TA\TW{1}]$, which is absurd. Hence $B^{(1)}_1=\TP{T}_{14}$. By Proposition \[prop:cres\], $T_{24}=A^{(1)}_1=\TW{\NS^{(1)}_1}\TA\TP{T}_{14}^{\ast}$. This means that $\NS^{(1)}_1=2$, $\NS^{(2)}_1=1$ and $\TW{1}\TA\TP{T}_{14}^{\ast}=\TP{T}_{14}^{\ast}\TA\TW{1}$. Thus $[T_{14},2]=[2,T_{14}]$. There exists a positive integer $l$ such that $T_{14}=\TW{l}$. Write $k=k_{12}$. We have $B^{(1)}_2=\TW{k}$, $A^{(1)}_2=[k+2]$, $B^{(1)}_1=\TW{l}$. By Lemma \[lem:0\], $B^{(2)}_1=[k+1,l+2]$. By Proposition \[prop:cres\], we have $A^{(1)}_1=[2,l+2]$ and $A^{(2)}_1=[3,\TW{l-1},3,\TW{k-1}]$. The numerical data of $C$ is equal to $\{(((l+1)(k+1))_2,(k+1)_{l+1}),\;\;((l+1)(k+1)+k,l(k+1)+k,(k+1)_l,k)\}$, which coincides with the data 1 with $a=l+1$, $b=k+1$.
$B^{(2)}_{g_2}=\protect\TP{[T_{22},F_2']}$
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We have $T_{23}=\emptyset$. We may assume $T_{11}\ne\emptyset$ because this case is contained in the case \[subsec:1\] if $T_{11}=\emptyset$.
\[lem:3\] We have $g_1=1$, $T_{24}=\emptyset$, $A^{(1)}_1=[T_{14},S_3]$, $f_2=2$, $s_3=k_{34}+1$, $f_1=l_{22}+3$, and $[F_1',T_{13}]^{\ast}=\TW{\NS^{(1)}_1}\TA[T_{12},k_{12}+1,\TW{l_{22}+k_{12}-1}]$.
By Lemma \[lem:0\], we have $[T_{24},S_3]=[\TW{f_2-2},k_{34}+1]$. This shows that $T_{24}=\TW{f_2-2}$ and $s_3=k_{34}+1$. Suppose $g_1=2$. Since $A^{(1)}_2=S_3$, we have $[k_{34}+1]=\TW{\NS^{(1)}_2}\TA[T_{12},k_{12}+1,\TW{l_{22}}]$ by Lemma \[lem:0\]. This means that $\NS^{(1)}_2=1$, $T_{12}=\emptyset$, $l_{22}=0$ and $k_{34}=k_{12}+1$. We infer $g_2=1$ and $A^{(2)}_1=\TP{[F_1',T_{13}]}$. By Proposition \[prop:cres\], $[F_1',T_{13}]^{\ast}=\TP{A}^{(2)\ast}_1=[\NS^{(2)}_1+1,F_2']$. By Lemma \[lem:0\], $[\NS^{(2)}_1+1,F_2']=[T_{14},k_{12}+2,\TW{k_{12}-1}]$. Because $T_{14}\ne\emptyset$, we have $k_{12}=1$. By Lemma \[lem:0\], we obtain $[S_1,T_{11}]=[2]$, which contradicts $T_{11}\ne\emptyset$. Hence $g_1=1$. Suppose $T_{24}\ne\emptyset$. We have $T_{14}=\emptyset$ and $A^{(1)}_1=[T_{24},S_3]=[\TW{f_2-2},k_{34}+1]$. Thus $f_2>2$. By Lemma \[lem:0\], $[F_1',T_{13}]=[k_{12}+1,\TW{k_{34}-1}]$. This shows $f_1=k_{12}+1$. By Proposition \[prop:cres\], $A^{(2)}_{g_2}=[\TW{\NS^{(2)}_{g_2}-1},l_{22}+3,\TW{f_2-2}]$. Since $A^{(2)}_{g_2}=\TP{[F_1',\ldots]}$, we have $f_1=2$ and $k_{12}=1$. By Lemma \[lem:0\], $A^{(1)}_{1}=\TW{\NS^{(1)}_{1}}\TA[T_{12},\TW{l_{22}+1}]$. Thus $[\TW{f_2-2},k_{34}+1]=\TW{\NS^{(1)}_{1}}\TA[T_{12},\TW{l_{22}+1}]$. If $T_{12}\ne\emptyset$ or $l_{22}>0$, then $k_{34}=1$ and $\TW{f_2-2}=\TW{\NS^{(1)}_{1}}\TA[T_{12},\TW{l_{22}}]$, which is impossible. Thus $T_{12}=\emptyset$ and $l_{22}=0$. By Lemma \[lem:0\], $[S_1,T_{11}]=[2]$, which contradicts $T_{11}\ne\emptyset$. Hence $T_{24}=\emptyset$ and $A^{(1)}_1=[T_{14},S_3]$. We have $f_2=2$. By Proposition \[prop:cres\], $A^{(2)}_{g_2}=[\TW{\NS^{(2)}_{g_2}-1},l_{22}+3]$. This shows $f_1=l_{22}+3$. By Lemma \[lem:0\], $[F_1,T_{13}]^{\ast}=\TW{\NS^{(1)}_1}\TA[T_{12},k_{12}+1,\TW{l_{22}+k_{12}-1}]$.
Case (1): $g_2=1$. If $T_{13}=\emptyset$, then $\TW{f_1-1}=\TW{\NS^{(1)}_1}\TA[T_{12},k_{12}+1,\TW{l_{22}+k_{12}-1}]$ by Lemma \[lem:3\], which is absurd. Thus $T_{13}\ne\emptyset$. We have $T_{12}=\emptyset$ and $A^{(2)}_1=\TP{[F_1',T_{13}]}$. By Lemma \[lem:3\], $\TP{A}^{(2)\ast}_1=\TW{\NS^{(1)}_1}\TA[k_{12}+1,\TW{l_{22}+k_{12}-1}]$. By Proposition \[prop:cres\], $\TP{A}^{(2)\ast}_1=[\NS^{(2)}_1+1,\TW{l_{22}+1}]$. It follows that $\NS^{(1)}_1+k_{12}=3$. By Lemma \[lem:0\], $[S_1,T_{11}]=[l_{22}+2,\TW{k_{12}-1}]$. Since $T_{11}\ne\emptyset$, we see $k_{12}=2$, $\NS^{(1)}_1=1$ and $\NS^{(2)}_1=3$. Put $k=l_{22}+1$. We have $k\ge 1$, $B^{(1)}_1=[k+1,2]$ and $B^{(2)}_1=\TW{k}$. By Proposition \[prop:cres\], we get $A^{(1)}_1=[4,\TW{k-1}]$ and $A^{(2)}_1=[2,2,k+2]$. The numerical data of $C$ is equal to $\{(2k+1,k_2),\;\;((k+1)_3)\}$, which coincides with the data 1 with $a=1$, $b=k+1$. Case (2): $g_2=2$. We have $A^{(2)}_2=F_1'$, $T_{12}\ne\emptyset$ and $T_{13}\ne\emptyset$.
\[lem:3-2\] We have $B^{(2)}_1=\TP{T}_{12}$, $A^{(2)}_1=\TP{T}_{13}$, $[\NS^{(2)}_1+1,T_{12}\ast\TW{l_{22}+2}]=[\TW{\NS^{(1)}_1}\TA T_{12},k_{12}+1,\TW{l_{22}+k_{12}-1}]$ and $3=\NS^{(1)}_1+k_{12}$.
Either $B^{(2)}_1=\TP{T}_{12}$ or $B^{(2)}_1=T_{13}$. Suppose $B^{(2)}_1=T_{13}$. By Proposition \[prop:cres\], $T_{12}=A^{(2)}_1=\TW{\NS^{(2)}_1}\TA T_{13}^{\ast}$. By Lemma \[lem:3\], $[l_{22}+3,T_{13}]=[\TW{\NS^{(1)}_1}\TA\TW{\NS^{(2)}_1}\TA T_{13}^{\ast},k_{12}+1,\TW{l_{22}+k_{12}-1}]^{\ast}=\TW{1}^{\ast l_{22}+k_{12}-1}\TA\TW{k_{12}}\TA[T_{13},\NS^{(2)}_1+1,\NS^{(1)}_1+1]$, which is impossible. Thus $B^{(2)}_1=\TP{T}_{12}$ and $A^{(2)}_1=\TP{T}_{13}$. By Proposition \[prop:cres\], $T_{13}^{\ast}=\TP{A}^{(2)\ast}_{1}=[\NS^{(2)}_1+1,T_{12}]$. By Lemma \[lem:3\], $[\NS^{(2)}_1+1,T_{12}\ast\TW{l_{22}+2}]=[\TW{\NS^{(1)}_1}\TA T_{12},k_{12}+1,\TW{l_{22}+k_{12}-1}]$. Hence $3=\NS^{(1)}_1+k_{12}$.
Case (2–1): $k_{12}=1$. By Lemma \[lem:3-2\], we have $\NS^{(1)}_1=2$, $\NS^{(2)}_1=1$ and $T_{12}\TA\TW{1}=\TW{1}\TA T_{12}$. Thus $[2,T_{12}^{\ast}]=[T_{12}^{\ast},2]$. There exists a positive integer $l'$ such that $T_{12}^{\ast}=\TW{l'}$. Hence $T_{12}=[l'+1]$. Put $l=l'-1$ and $k=l_{22}+1$. By Lemma \[lem:0\], $B^{(1)}_1=[k+2,\TW{l}]$. Since $T_{11}\ne\emptyset$, we see $l\ge 1$. We have $B^{(2)}_2=\TW{k}$, $A^{(2)}_2=[k+2]$ and $B^{(2)}_1=[l+2]$. By Proposition \[prop:cres\], we see $A^{(1)}_1=[2,l+3,\TW{k}]$ and $A^{(2)}_1=[3,\TW{l}]$. The numerical data of $C$ is equal to $\{(((l+1)(k+1)+1)_2,(k+1)_{l+1}),\;\;((l+2)(k+1),(l+1)(k+1),(k+1)_{l+1})\}$, which coincides with the data 2 with $a=l+1$, $b=k+1$. Case (2–2): $k_{12}=2$. By Lemma \[lem:3-2\], $\NS^{(1)}_1=1$ and $[\NS^{(2)}_1,T_{12}]=[T_{12},2]$. We have $\NS^{(2)}_1\ge2$. Since $T_{12}^{\ast}\TA\TW{\NS^{(2)}_1-1}=\TW{1}\TA T_{12}^{\ast}$, we see $\NS^{(2)}_1=2$. There exists a positive integer $l$ such that $T_{12}=\TW{l}$. Put $k=l_{22}+1$. We have $B^{(2)}_2=\TW{k}$, $A^{(2)}_2=[k+2]$ and $B^{(2)}_1=\TW{l}$. By Lemma \[lem:0\], $B^{(1)}_1=[k+1,l+2]$. By Proposition \[prop:cres\], we have $A^{(1)}_1=[3,\TW{l-1},3,\TW{k-1}]$ and $A^{(2)}_1=[2,l+2]$. The numerical data of $C$ is equal to $\{((l+1)(k+1)+k,l(k+1)+k,(k+1)_{l},k),\;\;(((l+1)(k+1))_2,(k+1)_{l+1})\}$, which coincides with the data 1 with $a=l+1$, $b=k+1$.
$B^{(2)}_{g_2}=[F_2',T_{23}]$
-----------------------------
We have $T_{22}=\emptyset$. We may assume $T_{11}\ne\emptyset\ne T_{23}$; otherwise this case is contained in another case. Case (1): $g_2=1$. We show the following lemma.
\[lem:4-1\] We have $T_{12}=\emptyset$, $A_1^{(2)}=\TP{[F_1',T_{13}]}$, $f_2=2$, $B^{(1)}_{g_1}=\TW{k_{12}}$, $k_{12}\ge 2$ and $[T_{14},k_{34}+1,\TW{k_{12}-2}]=[\NS^{(2)}_1+1,\TP{T_{23}}]$.
Suppose $T_{12}\ne\emptyset$. We have $A^{(2)}_1=[T_{12},F_1']$ and $T_{13}=\emptyset$. By Lemma \[lem:0\], $[T_{14},k_{34}+1,\TW{k_{12}-1}]=\TW{f_1-1}$. This shows that $k_{34}=1$ and $T_{14}=\TW{f_1-k_{12}-1}$. By Lemma \[lem:0\], $A^{(1)}_{g_1}=\TW{\NS^{(1)}_{g_1}}\TA [T_{12},k_{12}+1]$. Thus $A^{(1)}_{g_1}$ contains at least two irreducible components. It follows that $g_1=1$. Either $A^{(1)}_1=[T_{14},S_3]$ or $A^{(1)}_1=[T_{24},S_3]$. Because $T_{14}$ consists of ($-2$)-curves, the latter case must occur. We infer $T_{24}=\TW{\NS^{(1)}_1}\TA T_{12}$. By Proposition \[prop:cres\] and Lemma \[lem:0\], $[T_{12},F_1']=A^{(2)}_1=\TW{\NS^{(2)}_1}\TA[F_2',T_{23}]^{\ast}=\TW{\NS^{(2)}_1}\TA[T_{24},S_3]$. We have $T_{12}=\TW{\NS^{(2)}_1}\TA T_{24}$. Thus $T_{24}=\TW{\NS^{(1)}_1}\TA\TW{\NS^{(2)}_1}\TA T_{24}$, which is impossible. Hence $T_{12}=\emptyset$. We have $A_1^{(2)}=\TP{[F_1',T_{13}]}$. By Lemma \[lem:0\], $[T_{14},k_{34}+1,\TW{k_{12}-1}]=\TP{A}^{(2)\ast}_1$ and $B^{(1)}_{g_1}=[S_1,T_{11}]=\TW{k_{12}}$. We infer $k_{12}\ge 2$. By Proposition \[prop:cres\], $[T_{14},k_{34}+1,\TW{k_{12}-1}]=[\NS^{(2)}_1+1,\TP{T_{23}},F_2']$. This shows that $f_2=2$ and $[T_{14},k_{34}+1,\TW{k_{12}-2}]=[\NS^{(2)}_1+1,\TP{T_{23}}]$.
Case (1–1): $g_1=1$. Suppose $T_{24}\ne\emptyset$. We have $T_{14}=\emptyset$ and $A^{(1)}_1=[T_{24},S_3]$. By Lemma \[lem:0\] and Lemma \[lem:4-1\], $[\TW{k_{34}},T_{23}]=A^{(1)\ast}_1$. By Proposition \[prop:cres\], we have $[\TW{k_{34}},T_{23}]=[\TW{k_{12}},\NS^{(1)}_1+1]$. By Lemma \[lem:4-1\], $[T_{23},\NS^{(2)}_1+1]=[\TW{k_{12}-2},k_{34}+1]$. We infer $[\TW{k_{12}},\NS^{(1)}_1+1,\NS^{(2)}_1+1]=[\TW{k_{12}+k_{34}-2},k_{34}+1]$. This means that $k_{34}=3$ and $\NS^{(1)}_1=1$. By Proposition \[prop:cres\], $[T_{24},S_3]=[k_{12}+2]$, which is absurd. Thus $T_{24}=\emptyset$. We have $A^{(1)}_1=[T_{14},S_3]$. By Lemma \[lem:0\], we get $[T_{14},S_3]=[\TW{\NS^{(1)}_1-1},k_{12}+2]$. Hence $s_3=k_{12}+2$ and $T_{14}=\TW{\NS^{(1)}_1-1}$. By Proposition \[prop:cres\] and Lemma \[lem:4-1\], $[F_2',T_{23},\NS^{(2)}_1+1]=A^{(2)\ast}_1=\TP{[F_1',T_{13}]}^{\ast}$. By Lemma \[lem:0\], $[F_2',T_{23},\NS^{(2)}_1+1]=[\TW{k_{12}-1},k_{34}+1,\TW{\NS^{(1)}_1-1}]$. Since $k_{12}\ge 2$, we see $[T_{23},\NS^{(2)}_1+1]=[\TW{k_{12}-2},k_{34}+1,\TW{\NS^{(1)}_1-1}]$. By Lemma \[lem:0\], $[\TW{k_{34}-1},F_2',T_{23}]=\TW{s_3-1}=\TW{k_{12}+1}$. Thus $T_{23}=\TW{k_{12}-k_{34}+1}$. Hence $[\TW{k_{12}-k_{34}+1},\NS^{(2)}_1+1]=[\TW{k_{12}-2},k_{34}+1,\TW{\NS^{(1)}_1-1}]$. We infer $4=k_{34}+\NS^{(1)}_1$. Case ($\text{1--1}_\text{a}$): $k_{34}=1$. We have $\NS^{(1)}_1=3$, $\NS^{(2)}_1=1$ and $[F_2',T_{23}]=\TW{k_{12}+1}$. Put $k=k_{12}-1$. By Lemma \[lem:4-1\], $k\ge 1$ and $B^{(1)}_1=\TW{k+1}$. We have $B^{(2)}_1=\TW{k+2}$. By Proposition \[prop:cres\], $A^{(1)}_1=[2,2,k+3]$ and $A^{(2)}_1=[k+4]$. The numerical data of $C$ is equal to $\{(k+3),\, ((k+2)_3)\}$, which coincides with the data 3 with $a=1$, $b=k+2$. Case ($\text{1--1}_\text{b}$): $k_{34}>1$. If $\NS^{(1)}_1=2$, then $[\TW{k_{12}-k_{34}+1},\NS^{(2)}_1+1]=[\TW{k_{12}-2},3,2]$, which is impossible. We have $\NS^{(1)}_1=1$, $k_{34}=3$ and $\NS^{(2)}_1=3$. Put $k=k_{12}-2$. Since $[F_2',T_{23}]=\TW{k+1}$, we see $k\ge 1$. We have $B^{(1)}_1=\TW{k+2}$ and $B^{(2)}_1=\TW{k+1}$. By Proposition \[prop:cres\], we obtain $A^{(1)}_1=[k+4]$ and $A^{(2)}_1=[2,2,k+3]$. The numerical data of $C$ is equal to $\{((k+2)_3),\, (k+3)\}$, which coincides with the data 3 with $a=1$, $b=k+2$. Case (1–2): $g_1=2$. We have $A^{(1)}_2=S_3$, $T_{14}\ne\emptyset$ and $T_{24}\ne\emptyset$.
\[lem:4-1-2\] We have $\NS^{(1)}_2=1$, $s_3=k_{12}+2$, $A^{(1)}_1=T_{14}$, $B^{(1)}_1=\TP{T}_{24}$, $[\TW{k_{12}+k_{34}-2},k_{34}+1,T_{24}^{\ast}\TA\TW{\NS^{(1)}_1}]=[\TW{k_{12}+1}\TA T_{24}^{\ast},\NS^{(2)}_1+1]$, $k_{34}+\NS^{(1)}_1=3$.
By Lemma \[lem:0\], we get $\NS^{(1)}_2=1$, $s_3=k_{12}+2$ and $[\TW{k_{34}},T_{23}]=\TW{k_{12}+1}\TA T_{24}^{\ast}$. Either $B^{(1)}_1=\TP{T}_{24}$ or $B^{(1)}_1=\TP{T}_{14}$. Suppose $B^{(1)}_1=\TP{T}_{14}$. We have $A^{(1)}_1=T_{24}$. By Proposition \[prop:cres\], $[\TW{k_{34}},T_{23}]=[\TW{k_{12}+1}\TA \TP{T}_{14},\NS^{(1)}_1+1]$. By Lemma \[lem:4-1\], we have $[T_{14},k_{34}+1,\TW{k_{12}+k_{34}-2}]=[\NS^{(2)}_1+1,\TP{T}_{23},\TW{k_{34}}]=[\NS^{(2)}_1+1,\NS^{(1)}_1+1,T_{14}\TA\TW{k_{12}+1}]$. Thus $k_{34}=3$. It follows that $[T_{14},4,2]=[\NS^{(2)}_1+1,\NS^{(1)}_1+1,T_{14}\TA\TW{1}]$, which is impossible. Hence $A^{(1)}_1=T_{14}$, $B^{(1)}_1=\TP{T}_{24}$. By Proposition \[prop:cres\], $T_{14}=\TW{\NS^{(1)}_1}\TA \TP{T}_{24}^{\ast}$. By Lemma \[lem:4-1\], $[T_{23},\NS^{(2)}_1+1]=[\TW{k_{12}-2},k_{34}+1,T_{24}^{\ast}\TA\TW{\NS^{(1)}_1}]$. Thus $[\TW{k_{12}+1}\TA T_{24}^{\ast},\NS^{(2)}_1+1]=[\TW{k_{12}+k_{34}-2},k_{34}+1,T_{24}^{\ast}\TA\TW{\NS^{(1)}_1}]$. Hence $3=k_{34}+\NS^{(1)}_1$.
Case (1–$2_\text{a}$): $k_{34}=1$. By Lemma \[lem:4-1-2\], we have $\NS^{(1)}_1=2$, $\NS^{(2)}_1=1$ and $[\TW{k_{12}+1}\TA T_{24}^{\ast},2]=[\TW{k_{12}},T_{24}^{\ast}\TA\TW{2}]$. Thus $\TW{1}\TA T_{24}^{\ast}=T_{24}^{\ast}\TA\TW{1}$. Hence $[T_{24},2]=[2,T_{24}]$. There exists a positive integer $l$ such that $T_{24}=\TW{l}$. Put $k=k_{12}-1$. By Lemma \[lem:4-1\], we have $k\ge1$, $[S_1,T_{11}]=B^{(1)}_{2}=\TW{k+1}$. By Lemma \[lem:4-1-2\], $A^{(1)}_2=[k+3]$. Since $B^{(1)}_1=\TW{l}$, we get $A^{(1)}_1=[2,l+2]$ by Proposition \[prop:cres\]. By Lemma \[lem:0\], we infer $B^{(2)}_1=[\TW{k+1},l+2]$. By Proposition \[prop:cres\], $A^{(2)}_1=[3,\TW{l-1},k+3]$. It follows that the numerical data of $C$ is equal to $\{(((l+1)(k+2))_2,(k+2)_{l+1}),\;\;((l+1)(k+2)+1,l(k+2)+1,(k+2)_l)\}$, which coincides with the data 3 with $a=l+1$, $b=k+2$. Case (1–$2_\text{b}$): $k_{34}=2$. By Lemma \[lem:4-1-2\], we have $\NS^{(1)}_1=1$ and $[\TW{k_{12}},3,T_{24}^{\ast}\TA\TW{1}]=[\TW{k_{12}+1}\TA T_{24}^{\ast},\NS^{(2)}_1+1]$. Thus $[2,T_{24}^{\ast}]=[T_{24}^{\ast},\NS^{(2)}_1]$. Hence $T_{24}\TA\TW{1}=\TW{\NS^{(2)}_1-1}\TA T_{24}$. This shows that $\NS^{(2)}_1=2$ and $[2,T_{24}^{\ast}]=[T_{24}^{\ast},2]$. There exists a positive integer $l$ such that $T_{24}^{\ast}=\TW{l}$. We have $T_{24}=[l+1]$. Put $k=k_{12}-1$. By Lemma \[lem:4-1\], we see $k\ge1$, $[S_1,T_{11}]=B^{(1)}_2=\TW{k+1}$. We have $A^{(1)}_2=[k+3]$ by Lemma \[lem:4-1-2\]. Since $B^{(1)}_1=[l+1]$, we get $A^{(1)}_1=[3,\TW{l-1}]$ by Proposition \[prop:cres\]. By Lemma \[lem:0\], we infer $B^{(2)}_1=[\TW{k},3,\TW{l-1}]$. By Proposition \[prop:cres\], $A^{(2)}_1=[2,l+2,k+2]$. It follows that the numerical data of $C$ is equal to $\{((l+1)(k+2),l(k+2),(k+2)_{l}),\;\;((l(k+2)+k+1)_2,(k+2)_l,k+1)\}$, which coincides with the data 4 with $a=l$, $b=k+2$. Case (2): $g_2=2$. We have $T_{12}\ne\emptyset\ne T_{13}$ and $A^{(2)}_2=F_1'$.
\[lem:4-2\] The following assertions hold.
1. $g_1=1$, $T_{24}=\emptyset$, $A^{(1)}_1=[T_{14},S_3]$, $B^{(2)}_1=\TP{T}_{12}$, $A^{(2)}_1=\TP{T}_{13}$.
2. $k_{12}\ge2$, $f_1\ge4$, $f_2=2$, $s_3=k_{12}+1=k_{34}+f_1-2$, $k_{34}+\NS^{(1)}_1=3$, $\NS^{(2)}_1=k_{34}$, $\NS^{(2)}_2=1$.
3. $T_{13}=T_{12}^{\ast}\TA\TW{\NS^{(2)}_1}$, $T_{14}=\TW{\NS^{(1)}_1}\TA T_{12}$, $T_{23}=\TW{f_1-3}$, $[F_1',T_{12}^{\ast}\TA\TW{\NS^{(2)}_1}]=[\TW{1}^{\ast k_{12}-1}\TA\TW{k_{34}}\TA T_{12}^{\ast},\NS^{(1)}_1+1]$.
By Proposition \[prop:cres\], $[F_2',T_{23},\NS^{(2)}_2+1]=\TW{f_1-1}$. This shows that $f_1\ge4$, $f_2=2$, $\NS^{(2)}_2=1$ and $T_{23}=\TW{f_1-3}$. By Lemma \[lem:0\], $[T_{24},S_3]=[k_{34}+f_1-2]$. We have $T_{24}=\emptyset$ and $s_3=k_{34}+f_1-2$. It follows that $g_1=1$ and $A^{(1)}_1=[T_{14},S_3]$. By Lemma \[lem:0\], $[T_{14},S_3]=\TW{\NS^{(1)}_1}\TA[T_{12},k_{12}+1]$. Since $T_{12}\ne\emptyset$, we obtain $\emptyset\ne\TW{\NS^{(1)}_1}\TA T_{12}=T_{14}$ and $k_{12}+1=s_3=k_{34}+f_1-2$. We have $k_{12}\ge2$ and $T_{14}^{\ast}=[T_{12}^{\ast},\NS^{(1)}_1+1]$. By Lemma \[lem:0\], $[F_1',T_{13}]=[\TW{1}^{\ast k_{12}-1}\TA\TW{k_{34}}\TA T_{12}^{\ast},\NS^{(1)}_1+1]$. Either $B^{(2)}_1=T_{13}$ or $B^{(2)}_1=\TP{T}_{12}$. If $B^{(2)}_1=T_{13}$, then $A^{(2)}_1=T_{12}$. By Proposition \[prop:cres\], $[T_{13},\NS^{(2)}_1+1]=T_{12}^{\ast}$. Hence $[F_1',T_{13}]=[\TW{1}^{\ast k_{12}-1}\TA\TW{k_{34}}\TA T_{13},\NS^{(2)}_1+1,\NS^{(1)}_1+1]$, which is impossible. Thus $B^{(2)}_1=\TP{T}_{12}$, $A^{(2)}_1=\TP{T}_{13}$. By Proposition \[prop:cres\], $T_{13}=T_{12}^{\ast}\TA\TW{\NS^{(2)}_1}$. We have $[F_1',T_{12}^{\ast}\TA\TW{\NS^{(2)}_1}]=[\TW{1}^{\ast k_{12}-1}\TA\TW{k_{34}}\TA T_{12}^{\ast},\NS^{(1)}_1+1]$. This means that $\NS^{(2)}_1=k_{34}$ and $[\NS^{(2)}_1+1,T_{12}\TA\TW{f_1-1}]=[\TW{\NS^{(1)}_1}\TA T_{12},k_{34}+1,\TW{k_{12}-1}]$. We have $f_1=\NS^{(1)}_1+k_{12}$. Hence $k_{34}+\NS^{(1)}_1=3$.
Case (2–1): $k_{34}=1$. By Lemma \[lem:4-2\], $k_{12}=f_1-2$, $\NS^{(1)}_1=2$, $\NS^{(2)}_1=1$, $[F_1',T_{12}^{\ast}\TA\TW{1}]=[\TW{1}^{\ast k_{12}}\TA T_{12}^{\ast},3]$. We have $[2,T_{12}\TA\TW{f_1-1}]=[\TW{2}\TA T_{12},\TW{f_1-2}]$. This shows $T_{12}\TA\TW{1}=\TW{1}\TA T_{12}$. There exists a positive integer $l$ such that $T_{12}=[l+1]$. Put $k=k_{12}-1$. By Lemma \[lem:4-2\], we have $k\ge1$, $B^{(2)}_1=\TP{T}_{12}=[l+1]$. Furthermore, we see $A^{(2)}_1=\TP{T}_{13}=[3,\TW{l-1}]$, $A^{(2)}_2=[k+3]$, $A^{(1)}_1=[T_{14},S_3]=[2,l+2,k+2]$, $B^{(2)}_2=[F_2',T_{23}]=\TW{k+1}$. By Proposition \[prop:cres\], we obtain $[B^{(1)}_1,3]=A^{(1)\ast}_1=\TW{k+1}\TA\TW{l+1}\TA\TW{1}$. Hence $B^{(1)}_1=[\TW{k},3,\TW{l-1}]$. The numerical data of $C$ is equal to $\{((l(k+2)+k+1)_2,(k+2)_l,k+1),\;\;((l+1)(k+2),l(k+2),(k+2)_l)\}$, which coincides with the data 4 with $a=l$, $b=k+2$. Case (2–2): $k_{34}>1$. By Lemma \[lem:4-2\], we get $k_{34}=2$, $\NS^{(1)}_1=1$, $\NS^{(2)}_1=2$, $s_3=k_{12}+1=f_1\ge4$, $[F_1',T_{12}^{\ast}\TA\TW{2}]=[k_{12}+1,\TW{1}\TA T_{12}^{\ast},2]$. We have $T_{12}^{\ast}\TA\TW{1}=\TW{1}\TA T_{12}^{\ast}$. There exists a positive integer $l$ such that $T_{12}^{\ast}=[l+1]$. Put $k=k_{12}-2$. We have $k\ge 1$, $B^{(2)}_1=\TP{T}_{12}=\TW{l}$. Moreover, we see $A^{(2)}_1=\TP{T}_{13}=[2,l+2]$, $A^{(2)}_2=[k+3]$, $A^{(1)}_1=[T_{14},S_3]=[3,\TW{l-1},k+3]$, $B^{(2)}_2=[F_2',T_{23}]=\TW{k+1}$. By Proposition \[prop:cres\], we obtain $[B^{(1)}_1,2]=A^{(1)\ast}_1=\TW{k+2}\TA[l+1,2]$. Hence $B^{(1)}_1=[\TW{k+1},l+2]$. The numerical data of $C$ is equal to $\{((l+1)(k+2)+1,l(k+2)+1,(k+2)_{l}),\;\;(((l+1)(k+2))_2,(k+2)_{l+1})\}$, which coincides with the data 3 with $a=l+1$, $b=k+2$.
![The dual graph of $D+E_1+E_2$[]{data-label="figf"}](figf)
![The dual graph of $D+E_1+E_2$ — continued[]{data-label="figf2"}](figf2)
![The dual graph of $D+E_1+E_2$ — continued[]{data-label="figf3"}](figf3)
![The dual graph of $D+E_1+E_2$ — continued[]{data-label="figf4"}](figf4)
We list the dual graphs of $D+E_1+E_2$ in Figure \[figf\]. We prove the converse assertion of Theorem \[thm1\]. Let $\Gamma$ be one of the weighted dual graphs in Figure \[figf\]. It follows from [@fu Proposition 4.7] that the sub-graphs $F_0$, $F_1$ and $F_2$ of $\Gamma$ can be contracted to three disjoint $0$-curves. After the contraction, $S_1$, $S_2$ and $S_3$ become disjoint $0$-curves and meet with each curve $F_i$ transversally. Thus $\Gamma$ can be realized by blow-ups over three sections and fibers of $\Sigma_0$. By Lemma \[lem:bu\], $\Gamma-E_1-E_2-C'$ can be contracted to two points of $\SP^2$. Hence all the numerical data in Theorem \[thm1\] can be realized as those of rational cuspidal plane curves.
The author would like to express his thanks to Professor Fumio Sakai for his helpful advice. The author was supported by the Fuujukai Foundation.
[MaSa]{} Brieskorn, E., Knörrer, H.: Plane algebraic curves. Basel, Boston, Stuttgart: Birkhäuser 1986. Fenske, T.: Rational 1- and 2-cuspidal plane curves, Beiträge zur Algebra und Geometrie **40**, (1999), 309–329. Fujita, T.: On the topology of non-complete algebraic surfaces, J. Fac. Sci. Univ. Tokyo **29**, (1982), 503–566. Flenner, H., Zaidenberg, M.: $\SQ$-acyclic surfaces and their deformations, Contemp. Math. **162**, (1994), 143–208. Flenner, H., Zaidenberg, M.: On a class of rational cuspidal plane curves, Manuscripta Math. **89**, (1996), 439–459. Kawamata, Y.: On the classification of non-complete algebraic surfaces, Proc. Copenhagen, Lecture Notes in Math. **732**, (1979), 215–232. Kizuka, T.: Rational functions of $\SC^{\ast}$-type on the two-dimensional complex projective space, Tôhoku Math. J. **38**, (1986), 123–178. Kojima, H.: Complements of plane curves with logarithmic Kodaira dimension zero, J. Math. Soc. Japan **52**, (2000), 793–806. Matsuoka, T., Sakai, F.: The degree of rational cuspidal curves, Math. Ann. **285**, (1989), 233–247. Miyanishi, M., Sugie, T.: $\SQ$-homology planes with $\SC^{\ast\ast}$-fibrations, Osaka J. Math. **28**, (1991), 1–26. Miyanishi, M., Tsunoda, S.: Non-complete algebraic surfaces with logarithmic Kodaira dimension $-\infty$ and with non-connected boundaries at infinity, Japan. J. Math. **10**, No. 2, (1984), 195–242. Miyanishi, M., Tsunoda, S.: Absence of the affine lines on the homology planes of general type, J. Math. Kyoto Univ. **32**, (1992), 443–450. Orevkov, S. Yu.: On rational cuspidal curves I. Sharp estimate for degree via multiplicities, Math. Ann. **324**, (2002), 657–673. Tono, K.: On Orevkov’s rational cuspidal plane curves, J. Math. Soc. Japan **64**, (2012), 365–385. Tsunoda, S.: The complements of projective plane curves, RIMS-Kôkyûroku **446**, (1981), 48–56. Wakabayashi, I.: On the logarithmic Kodaira dimension of the complement of a curve in $\SP^2$, Proc. Japan Acad. **54**, Ser. A, (1978), 157–162.
\
<span style="font-variant:small-caps;">[Department of Mathematics, Graduate School of Science and Engineering, Saitama University,\
Saitama-City, Saitama 338–8570, Japan.]{}</span>\
[*E-mail address*: `[email protected]` ]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Combining the WKB expansion at large distances and Perturbation Theory at small distances it is constructed a compact uniform approximation for eight low-lying eigenfunctions: with the quantum numbers $(n,m,{\Lambda},\pm)$ , where $n=m=0$ at ${\Lambda}=0,1,2$, and $n=1$, $m=0$ at ${\Lambda}=0$. For any of these states this approximation provides the relative accuracy $\lesssim 10^{-5}$ (not less than 5 s.d.) for any real $x$ in eigenfunctions and for total energy $E(R)$ it gives 10-11 s.d. for internuclear distances $R \in [0,50]$. Corrections to proposed approximation are evaluated systematically in a framework of the convergent perturbation theory. Separation constants are found with 8 s.d. The oscillator strength for the electric dipole transitions are accurately calculated and compared with existing data with coincidence on the level of 7 s.d. The magnetic dipole and electric quadrupole transitions are accurately calculated for the first time with not less than 7 s.d.'
author:
- 'H. Olivares-Pilón'
- 'A.V. Turbiner'
date: 'January 30, 2013'
title: 'The H$_2^+$ molecular ion: low-lying states'
---
**INTRODUCTION**
The simplest molecular system which appears in Nature is the H$_2^+$ molecular ion. Needless to say that this system plays a fundamental role in different physical sciences, in particular, in atomic-molecular physics, in laser and plasma physics being also a traditional example of two-center Coulomb system $(Z, Z, e)$ which enters to all QM textbooks (see e.g. [@LL]). Due to the fact that protons are much more heavy than electron a standard consideration of the problem is made in the so-called static approximation (or, in other words, in the Bohr-Oppenheimer approximation of the zero order). In this approximation the protons are simply assumed to be infinitely heavy. It can be immediately checked that the projection of the angular momentum to the molecular axis (the line connecting the proton positions) $L_{\phi}$ is preserved, $[L_{\phi}, {\cal H}]=0$, where ${\cal H}$ is the Hamiltonian. Thus, the angular variable $\phi$ can be separated out. Hence, the problem is reduced to two-dimensional, which admits itself the separation of variables in elliptic coordinates. It reflects a unique property of general two-center Coulomb problem $(Z_1, Z_2, e)$ of the complete separation of variables (in prolate ellipsoidal coordinates).
In general, two-center Coulomb problem $(Z, Z, e)$ is non-solvable exactly, it can be solved in approximate way only. Thus, we need to introduce a natural definition of [*solvability*]{} of non-solvable spectral problem: for any eigenfunction $\Psi$ we can indicate constructively an uniform approximation $\Psi_{app}$ such that $$\label{delta}
\vert \frac{\Psi(x) - \Psi_{app}(x)}{\Psi_{app}(x)} \vert \lesssim 10^{-{\delta}} \ ,$$ in the coordinate space. In vicinity of the nodal surface, $\Psi_{app}(x)=0$ the absolute deviation $$\label{delta}
\vert {\Psi(x) - \Psi_{app}(x)} \vert \lesssim 10^{-{\delta}} \ ,$$ where the parameter ${\delta}> 0$ characterizes a number of significant digits (s.d.), which the approximation reproduces exactly. It implies that any observable, any matrix element can be found with accuracy not less than ${\delta}$.
A simple idea we are going to employ in order to construct an approximation is to combine WKB-expansion at large distances with perturbation theory at small distances near extremum the potential for the phase of wavefunction in a single interpolation. In the case of excited states this interpolation was complemented by a polynomial factor which carried the information about nodes. This idea was realized successfully for quartic anharmonic oscillator [@Turbiner:2005] and double-well potential [@Turbiner:2010]. In both cases for the lowest states it was constructed two-three parametric uniform approximations of the phase of eigenfunction leading to 10 s.d. in energies and with ${\delta}\sim {5 - 6}$ for any value of the coupling constant and size of the barrier. Recently, we announced the results of the similar quality for two lowest (and the most important) states $1s{\sigma}_{g}$ and $2p{\sigma}_{u}$ of the H$^+_2$ molecular ion [@Turbiner:2011]. A few parametric approximation leading to ${\delta}\sim 5 - 6$ was found. The goal of this paper is to extend and profound this analysis constructing approximations with ${\delta}\sim 5 - 6$ for eight low lying states of the H$^+_2$ molecular ion, including two above mentioned states. In order to check accuracy of obtained approximations a special convergent perturbation theory (PT) is developed. This PT allows us to evaluate a local deviation of the approximation from the exact eigenfunction. Eventually, we calculate systematically separation constants and the oscillator strength for the electric dipole and quadrupole, and magnetic dipole transitions.
It is worth mentioning that a study of the wavefunctions of the H$^+_2$ molecular ion in a form of expansion in some basis was initiated by Hylleraas [@Hylleraas:1931] and was successfully realized in the remarkable paper [@Bates:1953] (see also [@Montgomery:1977; @Bishop:1978]). Attempts to find bases leading to fast convergence are still continuing. At present, the basis of pure exponential functions seems the most fast convergent (see e.g. [@Korobov:2000] and references therein). Let us notice that following the analysis of classical mechanics of the H$_2^+$ system and its subsequent semiclassical quantization it was attempted to build some uniform approximations of wavefunctions of low lying electronic states [@Strand:1979]. Local accuracies of these approximations are unclear whilst eigenvalues are found with a few significant digits.
Generalities
============
The Schrödinger equation, which describes the electron in the field of two fixed centers of the charges $Z_1,
Z_2$ at the distance $R$, is of the form $$\label{Sch}
\left(-{\Delta}- \frac{2 Z_1}{r_1}- \frac{2 Z_2}{r_2}\right)\Psi \ =\ E' \Psi\ ,\
\Psi \in L^2 ({\bf R^3})\ ,$$ where $E'=(E - \frac{2 Z_1 Z_2 }{R})$ and the total energy $E$ are in Rydbergs, $r_{1,2}$ are the distances from electron to first (second) center, respectively. Following [@LL] let us introduce the dimensionless 2D elliptic coordinates and azimuthal angle $\varphi$ with respect to the molecular axis [^1]: $$\label{ell}
\xi = \frac{r_1+r_2}{R}\ ,\quad \eta = \frac{r_2-r_1}{R}\ ,\quad 1 \leq \xi \leq \infty\ ,\quad -1 \leq \eta \leq 1\ .$$ In these coordinates the Coulomb singularities are situated at $$\xi=1\quad ,\quad \eta = \pm 1\ ,$$ being at the boundaries of the configuration space. The Jacobian is $\propto (\xi^2-\eta^2)$. The equation (\[Sch\]) admits separation of variables in (\[ell\]). Since the projection of the angular momentum to the molecular axis $\hat L_{\phi}$ commutes with the Hamiltonian [^2] the eigenstate has a definite magnetic quantum number ${\Lambda}$. If $Z_1=Z_2$ the Hamiltonian is permutationally-symmetric $r_1 {\leftrightarrow}r_2$, or, equivalently, $\eta {\rightarrow}-\eta$, hence, any eigenfunction is of a definite parity ($\pm$). As a result, it can be represented in a form $$\label{psi}
\Psi \ =\ X(\xi) (\xi^2-1)^{{\Lambda}/2} Y(\eta) (1-\eta^2)^{{\Lambda}/2} e^{ \pm i {\Lambda}\phi}\ ,\ {\Lambda}=0,1,2,\ldots$$ where $Y(\eta)$ is of definite parity. Following the analysis we introduce the notation for a state as $(n,m,{\Lambda},\pm)$ where $n,m=0,1,\ldots$ are the quantum numbers in $\xi$ and $\eta$ coordinates, respectively, they have a meaning of number of nodes in $\xi$ and $\eta$, ${\Lambda}$ is a magnetic quantum number and $\pm$ is parity. It is easy to check that the ground state with the lowest total energy is $(0,0,0,+)$.
The factors $(\xi^2-1)^{{\Lambda}/2}$ and $(1-\eta^2)^{{\Lambda}/2}$ are introduced (\[psi\]) to take into account a singular behavior of the eigenfunction near Coulomb singularities. After substitution of the representation (\[psi\]) into (\[Sch\]) we arrive at the equations for $X(\xi)$ and $Y(\eta)$, $$\label{X_L}
{\partial}_{\xi} [(\xi^2-1) {\partial}_{\xi} X] + 2 {\Lambda}\xi {\partial}_{\xi} X +\left[-p^2\xi^2 + 2R \xi +A\right] X\ =\ 0\ ,\
X \in L^2(\xi \in [1,\infty))\ ,$$ $$\label{Y_L}
{\partial}_{\eta} [(\eta^2-1) {\partial}_{\eta} Y] + 2 {\Lambda}\eta {\partial}_{\eta} Y + \left[-p^2\eta^2 + A \right] Y\ =\ 0\ ,\
Y \in L^2(\eta \in [-1,1])\ ,$$ respectively, where following [@Bates:1953] we denote, $$\label{p}
p^2\ =\ - \frac{E'R^2}{4}\ ,$$ and $A$ is a separation constant. Equations (\[X\_L\]), (\[Y\_L\]) define a bispectral problem with $E,A$ as spectral parameters for any given $R$. Square-integrability of the function $\Psi$ (\[psi\]) implies a non-singular behavior of $X$ at $\xi {\rightarrow}1$ and decay at $\xi {\rightarrow}\infty$ as well as non-singular behavior of $Y$ at $\eta {\rightarrow}\pm 1$. Such a non-singular solution $X$ can be continued from the interval $[1, +\infty)$ to the whole line $(-\infty, +\infty)$. It implies searching a solution of the spectral problem (\[X\_L\]) which grows at $\xi {\rightarrow}-\infty$, decays at $\xi {\rightarrow}+\infty$ being a constant at $\xi=1$. A non-singular solution $Y(\eta)$ at $\eta=\pm 1$ can be unambiguously continued in $\eta$ beyond the interval $[-1,1]$ to $(-\infty, +\infty)$, it corresponds to growing (non-decaying) at $|\eta| {\rightarrow}\infty$.
The equation (\[X\_L\]) formally coincides with equation (\[Y\_L\]) at $R=0$ (united atom limit). It is evident that a domain for (\[Y\_L\]) is extended to $[1,\infty)$ it has no $L^2$ solutions since there is no degeneracy at any $R$ with $R=0$. Hence, at $E,A$ the solution found at the equation (\[X\_L\]) should be non-normalizable solution. Since at $R=0$ the problem becomes one-center Coulomb problem and can be solved exactly, the above consideration can be checked explicitly. It is also in agreement with large-$\eta$ behavior of the Hund-Mulliken function (it mimics the incoherent interaction of electron with charged centers) for both $1s{\sigma}_g$ (parity +) and $2p{\sigma}_u$ (parity -) states $$\label{HM}
\Psi^{(\pm)}_{HM}\ =\ e^{- 2{\alpha}_2 r_1} \pm e^{- 2{\alpha}_2 r_2}\
=\ 2 e^{-{\alpha}_2 R \xi}
\left[
\begin{array}{c}
\cosh ({\alpha}_2 R \eta) \\
\sinh ({\alpha}_2 R \eta)
\end{array}
\right]
\ ,$$ which describes large $R$ behavior. Similarly, for the Guillemin-Zener function (it mimics the coherent interaction of electron with charged centers) we get $$\label{GZ}
\Psi^{(\pm)}_{GZ}\ =\ e^{- 2{\alpha}_3 r_1 - 2{\alpha}_4 r_2} \pm e^{- 2{\alpha}_3 r_2 - 2{\alpha}_4 r_1}\
=\ 2 e^{-({\alpha}_3+{\alpha}_4) R \xi}
\left[
\begin{array}{c}
\cosh (({\alpha}_3-{\alpha}_4) R \eta) \\
\sinh (({\alpha}_3-{\alpha}_4) R \eta)
\end{array}
\right]
\ ,$$ which has to describe small $R$ behavior.
*Asymptotics.*
--------------
If we assume a representation $X=e^{-\varphi}$, then the WKB-expansion of phase at $\xi {\rightarrow}\infty$, $$\label{X-inf}
\varphi\ =\ p\xi - \left( \frac{R}{p}-{\Lambda}-1 \right) \log \xi + \left[\frac{A+(\frac{R}{p}-{\Lambda}-1)(\frac{R}{p}+{\Lambda})}{p} - p\right]\frac{1}{2\xi}
+\ldots\ ,$$ while at $\xi {\rightarrow}0$, $$\label{X-small}
\varphi\ =\ -\frac{A}{2}\xi^2 - \frac{R}{3}\xi^3 + \frac{(p^2+A^2-A(2{\Lambda}+3))}{12} \xi^4\
+\ \ldots \ .$$ Similarly to $X$ if we put $Y = e^{-\varrho}$, then at $\eta {\rightarrow}\infty$, $$\label{Y-inf}
\varrho\ =\ -p\eta + ({\Lambda}+1)\log \eta - \left(\frac{A-{\Lambda}({\Lambda}+1)}{p} - p\right)\frac{1}{2\eta}
+\ldots\ ,$$ when at $\eta {\rightarrow}0$, $$\label{Y-small}
\varrho\ =\ -\frac{A}{2}\eta^2 + \frac{(p^2+A^2-A(2{\Lambda}+3))}{12} \eta^4 + \ldots\ .$$ The important property of the expansions (\[X-inf\]) and (\[Y-inf\]) is that the coefficients in front of the growing terms at large distances (linear and logarithmic) are found explicitly, since they do not depend on the separation constant $A$.
*Approximation*
---------------
Making interpolation between WKB-expansion (\[X-inf\]) and the perturbation theory (\[X-small\]) for $X$, (\[Y-inf\]) and (\[Y-small\]) for $Y$, correspondingly, and taking into account that the $Z_2$-symmetry of $\Psi$: $\eta {\rightarrow}-\eta$ is realized through use of $\cosh(\sinh)$-function (cf. (\[HM\]) and (\[GZ\])) we arrive at the following expression [@Turbiner:2011] $$\label{appr}
\Psi^{(\pm)}_{n,m,{\Lambda}} = \frac{(\xi^2-1)^{{\Lambda}/2}P_n (\xi)}{(\gamma + \xi)^{1+n+{\Lambda}-\frac{R}{p}}}
e^{-\xi \frac{{\alpha}+ p \xi}{\gamma + \xi}}
\frac{(1-\eta^2)^{{\Lambda}/2}Q_m(\eta^2)}{(1 + b_2 \eta^2 + b_3 \eta^4)^{\frac{1+2m+{\Lambda}}{4}}}
\left[
\begin{array}{c}
\cosh \\
\sinh
\end{array}
\left(\eta \frac{a_1 + p a_2 \eta^2 + p b_3 \eta^4}
{1 + b_2\eta^2 + b_3 \eta^4}\right) \right] e^{\pm i {\Lambda}\phi} \ ,$$ for the eigenfunction of the state with the quantum numbers $(n,m,{\Lambda},\pm)$. Here ${\alpha},\gamma$ and $a_{1,2}, b_{2,3}$ are parameters (see below), $P_n (\xi)$ and $Q_m(\eta^2)$ are some polynomials of degrees $n$ and $m$ with real coefficients with $n$ and $m$ real roots in the intervals $[1,\infty)$ and $[0,1]$, respectively. One can choose these polynomials to ensure their orthogonality to all states with lower total energies.
Results
=======
Ground state of positive/negative parity
----------------------------------------
Let us consider two lowest states - one of positive and one of negative parity, $1s {\sigma}_g\ (0,0,0,+)$ and $2p {\sigma}_u\ (0,0,0,-)$, respectively, following the consideration [@Turbiner:2011]. Corresponding approximations have the form $$\Psi^{(\pm)}_{0,0,0} = \frac{1}{(\gamma + \xi)^{1-\frac{R}{p}}}
e^{-\xi \frac{{\alpha}+ p \xi}{\gamma + \xi}}
\frac{1}{(1 + b_2 \eta^2 + b_3 \eta^4)^{1/4}}
\left[
\begin{array}{c}
\cosh \\
\sinh
\end{array}
\left(\eta \frac{a_1 + p a_2 \eta^2 + p b_3 \eta^4}
{1 + b_2\eta^2 + b_3 \eta^4}\right) \right]$$ $$\label{appr-0}
\equiv X_0(\xi) Y_0^{(\pm)}(\eta) \ ,$$ (cf.(\[appr\])), respectively, and each of them depends on six parameters ${\alpha},\gamma$ and $a_{1,2}, b_{2,3}$. The easiest way to find these parameters is to make a variational calculation taking (\[appr-0\]) as a trial function for $R$ fixed and with $p$ as an extra variational parameter. Immediate striking result of the variational study is that for all $R \in [1,50]$ the optimal value of the parameter $p$ coincides with the exact value of $p$ (see (\[p\])) with extremely high accuracy for both $1s {\sigma}_g$ and $2p {\sigma}_u$ states. It implies a very high quality of the trial function - the variational optimization wants to reproduce with very high accuracy a domain where the eigenfunction is exponentially small, hence, the domain which gives a very small contribution to the energy functional. In Tables \[Ten1ssg\] and \[Ten2psu\] the results for the total energy (as well as for sensitive $p$) vs $R$ of $1s {\sigma}_g$ and $2p {\sigma}_u$ states are shown as well as their comparison with ones obtained by Montgomery [@Montgomery:1977] in highly-accurate realization of the approach by Bates et al [@Bates:1953], and also with the results we obtained in the Lagrange mesh method based on Vincke-Baye approach [@Baye:2006] (details will be given elsewhere). For all studied values of $R$ for both $1s {\sigma}_g$ and $2p {\sigma}_u$ states our variational energy turns out to be in agreement on the level of 10 s.d. with these two alternative calculations. Variational parameters are smooth slow-changing functions of $R$, see Tables \[table3\]-\[table4\]. All calculations were implemented in double precision arithmetics and checked in quadruple precision one. It is worth noting that the number of optimization parameters can be reduced putting $a_2=b_2=0$, however, in this case the accuracy in energy drops from 10-11 to 5-6 significant digits.
Hence, our relatively-simple, few parametric functions (\[appr-0\]) taken as trial functions in a variational study provide very high accuracy in energy in comparison with highly-accurate alternative calculations. Two naturally related questions occur: (i) can we estimate the accuracy of variationally obtained energies without making a comparison with other calculations and (ii) how close locally our functions to the exact ones in configuration space. In order to answer these questions we develop a perturbation theory for the Schroedinger equation (\[Sch\]) taking a trial function (\[appr-0\]) as zero approximation.
Let us choose $X_0, Y_0$ (\[appr-0\]) with parameters fixed variationally (see above) as zero approximation in perturbation theory (\[PTx\]), (\[PTy\]) (see Appendix). It is evident that by construction of $X_0, Y_0$ the emerging perturbation theory has to be convergent. Assuming the condition (\[An\]) is fulfilled for the first corrections, namely, $A_{1,\xi}\ =\ A_{1,\eta}=A_1$, we find the first corrections $\varphi_1(\xi)$ and $\varrho_1(\eta)$ as functions of $A_1$. Then we modify the trial function (\[appr-0\]), $$\label{appr-1}
\Psi^{(\pm)}_{0,0,0} {\rightarrow}X_0(\xi) Y_0^{(\pm)}(\eta)\ e^{-\varphi_1(\xi)-\varrho_1(\eta)}$$ and make the variational calculation with this trial function minimizing with respect to parameter $p$. The (expected) result is that the optimal value of parameter $p$ remained unchanged with respect to the value obtained for the trial function (\[appr-0\]) with extremely high accuracy - within 10 s.d.! It indicates that the condition (\[An\]) is fulfilled with high accuracy. The variational energy is changed beyond the 10 s.d. Therefore, the our energies presented in Tables \[Ten1ssg\], \[Ten2psu\] are correct in all digits. The separation parameters $A_{1,\xi}, A_{1,\eta}$ are presented in Table \[Aval\] together with those corresponding to other states (see below) . It allows us to find explicitly $\varphi_1(\xi)$ and $\varrho_1(\eta)$. As an illustration in Figs. \[fig-1ssg-X\], \[fig-1ssg-Y\], \[fig-2psu-X\] and \[fig-2psu-Y\] the functions $X_0(\xi)$, $Y_0^{(\pm)}(\eta)$ and the first correction to the phases are shown for $R=2$ a.u. Similar behavior appears for other values of $R$.
[ccc]{} R\[a.u.\] & $E_t$\[Ry\] (Present/[@Montgomery:1977]/Mesh) &$p$\
&-0.90357262676 &0.8519936\
&-0.90357262676 &\
&-0.90357262676 &\
------------------------------------------------------------------------
& -1.20526923821 &1.483403\
& – &\
& -1.20526923821 &\
------------------------------------------------------------------------
&-1.20526842899 &1.485015\
&-1.20526842899 &\
&-1.20526842899 &\
------------------------------------------------------------------------
&-1.0239380968 &3.49506\
&-1.0239380969 &\
&-1.0239380969 &\
------------------------------------------------------------------------
&-1.0011574578 &5.47987\
&-1.0011574579 &\
&-1.0011574579 &\
------------------------------------------------------------------------
&-1.0002611115 &6.73221\
&—– &\
&-1.0002611116 &\
------------------------------------------------------------------------
&-1.0000055815 & 15.492\
&—– &\
&-1.0000055815 &\
------------------------------------------------------------------------
&-1.0000017622 & 20.4939\
&—– &\
&-1.0000017622 &\
------------------------------------------------------------------------
&-1.0000007211 & 25.49511\
&—– &\
&-1.0000007211 &\
[ccc]{} R \[a.u.\] & $E_t$ (Present/[@Montgomery:1977]/Mesh) \[Ry\] & $p$\
&0.8703727499 & 0.5314196\
&0.8703727498 &\
&0.8703727498 &\
------------------------------------------------------------------------
&-0.3332800331 & 1.1536645\
&—– &\
&-0.33328003316 &\
------------------------------------------------------------------------
&-0.3350687844 & 1.155452\
&-0.3350687844 &\
&-0.3350687844 &\
------------------------------------------------------------------------
&-0.8911012787 & 2.3589\
&-0.8911012787 &\
&-0.8911012787 &\
------------------------------------------------------------------------
&-0.9998021372 &5.47678\
&-0.9998021372 &\
&-0.9998021372 &\
------------------------------------------------------------------------
&-1.0001215811 & 6.75434\
&— &\
&-1.0001215811 &\
------------------------------------------------------------------------
&-1.0000283953 & 10.4882\
&-1.0000283953 &\
&-1.0000283953 &\
------------------------------------------------------------------------
&-1.0000055815 & 15.492\
&— &\
&-1.0000055815 &\
------------------------------------------------------------------------
&-1.0000017622 & 20.4939\
&— &\
&-1.0000017622 &\
[c|c|c|c]{} & $R_{eq}$=1.997193 a.u. & $R$=6.0 a.u. & $R$=20.0 a.u.\
------------------------------------------------------------------------
${\alpha}$ & 1.48407 & 3.32381 & 10.0453\
$p$ & 1.483403 & 3.49506 & 10.4882\
$\gamma$ & 1.0299 & 0.96357 & 0.95774\
------------------------------------------------------------------------
$a_1$ & 0.9164 & 2.597355 & 9.8775\
$a_2$ & 0.05384 & 0.53443 & 6.8392\
$b_2$ & 0.06 & 0.588072 & 6.9016\
$b_3$ & 0.00011 & 0.00552 & 1.352\
[c|c|c|c]{} & $R$=6.0 a.u. & $R_{min}=$12.54525 a.u. & R=20.0 a.u.\
------------------------------------------------------------------------
${\alpha}$ & 3.24715 & 6.5275 & 10.7397\
$p$ & 3.43971 & 6.75434 & 10.4882\
$\gamma$ & 0.95706 & 0.97045 & 1.03027\
------------------------------------------------------------------------
$a_1$ & 2.84566 & 6.075 & 9.8077\
$a_2$ & 0.22098 & 1.46757 & 2.3784\
$b_2$ & 0.23611 & 1.5349 & 2.43705\
$b_3$ &-0.0027 & 0.1675 & 0.367\
$(0,0,{\Lambda},\pm)$ states with ${\Lambda}=1,2$
-------------------------------------------------
As the further check the quality of the approximation [(\[appr\])]{} proposed in [@Turbiner:2011], we considered the states with magnetic quantum number ${\Lambda}=1,2$ and both parities $(\pm)$. These four states $(0,0,1,+)$, $(0,0,1,-)$, $(0,0,2,+)$ and $(0,0,2,-)$ correspond to the states $2p\pi_u$, $3d\pi_g$, $3d{\delta}_g$ and $4f{\delta}_u$ in the united atom nomenclature, respectively. The approximation takes the form $$\label{apprL}
\Psi^{(\pm)}_{0,0,{\Lambda}}=\frac{(\xi^2-1)^{{\Lambda}/2}e^{-\xi\frac{{\alpha}+p\xi}{\gamma+\xi}}}
{(\gamma+\xi)^{1+{\Lambda}-\frac{R}{p}}}
\frac{(1-\eta^2)^{{\Lambda}/2}}{(1 + b_2 \eta^2 + b_3 \eta^4)^{\frac{1+{\Lambda}}{4}}}
\left[
\begin{array}{c}
\cosh\\
\sinh
\end{array}
\left(\eta\frac{a_1+p a_2\eta^2 +p b_3\eta^4}{1+b_2\eta^2+b_3\eta^4}\right)\right]e^{\pm i {\Lambda}\phi} \ ,$$ for positive and negative parity, respectively; it depends on six free parameters ${\alpha},{\gamma}$ and $a_{1,2}$, $b_{2,3}$ as well as $p$ which can be taken as an extra variational parameter. Due to the presence of the last factor in $\Psi^{(\pm)}_{0,0,{\Lambda}}$ the function (\[apprL\]) is orthogonal to (\[appr-0\]). Taking [(\[apprL\])]{} as a trial function and using the variational method, the optimized values of these parameters are obtained for each fixed value of the internuclear distance $R$. The results for the total energy and the value of p for the states with ${\Lambda}=1, 2$ and both parities $(\pm)$ as a function of the internuclear distance $R$ are presented in Table \[tb00Lpm\]. For each $R$-value, the second line are the results presented by Madsen and Peek [@Mar:1971]. In general, the agreement is on the level of $10$ s.d. except for a few values of $R$ where the agreement is on $8-9$ s.d. On Figs. \[fig-2ppu-X\], \[fig-2ppu-Y\], \[fig-3dpg-X\], \[fig-3dpg-Y\],\[fig-3ddg-X\], \[fig-3ddg-Y\],\[fig-4fdu-X\] and \[fig-4fdu-Y\] the trial functions $X_0(\xi)$, $Y_0^{(\pm)}(\eta)$ and the first corrections to the phases $\varphi_1(\xi)$ and $\varrho_1(\eta)$ for $R=2$ a.u. are present. We must emphasize that the variational parameter $p$ in Table \[tb00Lpm\] coincides with the value of $p$ found from the variational energy, (\[p\]) on the level of 5 -9 s.d. It indicates a very high quality of the function (\[apprL\]).
R(a.u.) $E_t(0,0,1,+)$ $p(0,0,1,+)$ $E_t(0,0,1,-)$ $p(0,0,1,-)$ $E_t(0,0,2,+)$ $p(0,0,2,+)$ $E_t(0,0,2,-)$ $p(0,0,2,-)$
--------- -------------------- -------------- ------------------- -------------- -------------------- -------------- ------------------- --------------
1.0 1.05178408748(77) 0.486882 1.55288688583 0.3343325 1.560917409665 0.331316537 1.7500049256 0.249997537
1.0517840874746 1.5528868858238 1.5609174096654 1.7500049255960
2.0 0.14245636021(08) 0.926037 0.54660074672 0.673349 0.574534636379 0.652277061 0.7500749141 0.499925080
0.14245636020826 0.5466007467126 0.5745346363784 0.7500749141264
4.0 -0.20164928823 1.67529 0.03809311538 1.3592746 0.111109971587 1.247220956 0.2509887461 0.998020549
-0.2016492882302 0.0380931153803 0.11110997158626 0.2509887460990
6.0 -0.26064979131 2.31211 -0.12174444493 2.023784 -0.019437228851 1.781834745 0.0871067833 1.488636608
-0.2606497913114 -0.12174444495100 -0.019437228851128 0.08710678324228
8.0 -0.26902126254(37) 2.881725 -0.18878303657 2.649628 -0.071453569562 2.267875022 0.0085759370 1.965396909
-0.2690212625382 -0.18878303658772 -0.071453569562040 0.00857593687662
10.0 -0.26543258028 3.41113 -0.21983374901 3.23973 -0.095093601175 2.716125923 -0.0351710338 2.424721809
-0.2654325802914 -0.2198337490582 -0.09509360117488 -0.03517103400198
14.0 -0.25539654598 4.417514 -0.24231908611 4.3444 -0.111495667118 3.530338183 -0.0780596930 3.290125373
-0.2553965462922 -0.2423190861426 -0.11149566711852 -0.07805969328034
20.0 -0.25016709781 5.9175 -0.24875292671 5.905531 -0.113758110521 4.623398216 -0.1006238789 4.479105705
-0.2501670989774 -0.248752926741 -0.1137581106214 -0.10062387914096
30.0 -0.24975590514 8.43772 -0.24973461946 8.4374 -0.110960223739 6.321870831 -0.1091161952 6.288970023
-0.2497559054846 -0.2497346194714 -0.11096022579684 -0.10911619534154
40.0 -0.24987285888 10.9521 -0.24987261006 10.9521 -0.110588155306 8.014690427 -0.1104296208 8.010733322
-0.2498728599708 -0.2498726100936 -0.1105881565852 -0.11042962089928
50.0 -0.24992875005 13.4613 -0.24992874750 13.46126 -0.110756575302 9.706846056 -0.1107458300 9.706500079
-0.2499287500956 -0.2499287475080 -0.11075657655914 -0.11074583006112
Ellipsoidal nodal surfaces: the $(1,0,0,\pm)$ states
----------------------------------------------------
The proposed approximation [(\[appr\])]{} [@Turbiner:2011] allows us to study the $n$th excited state in $\xi$ direction with $n$ nodes in the $\xi$ variable. Let us consider the simplest case, $n=1$ and ${\Lambda}=0$ of the parity $(\pm)$, $(1,0,0,\pm)$ or, differently, $2s{\sigma}_g$ and $3p{\sigma}_u$, respectively. The main difference with the approximation for the ground state [(\[appr-0\])]{} comes due to the presence of a monomial factor $(\xi-\xi_0)$ in the expression for $X_0(\xi)$, while the $Y_0(\eta)$ remains functionally the same, $$\label{apprNX}
X_{0} = \frac{(\xi-\xi_0)}{(\gamma + \xi)^{2-\frac{R}{p}}}
e^{-\xi \frac{{\alpha}+ p \xi}{\gamma + \xi}}\ .$$ Here $\xi_0$ defines the position of the node and it can be fixed by imposing the orthogonality condition between these states the ($\pm$ parity) and the lowest states, [*i.e.*]{} $\langle(0,0,0,\pm)|(1,0,0,\pm)\rangle = 0$. The orthogonality with the states $(0,0,{\Lambda},\pm)$ for any ${\Lambda}$ is always fulfilled. Eventually, the approximation $\Psi^{(\pm)}_{1,0,0}$ contains six free parameters which are obtained using the variational method. Results are presented in Table \[tb10Lpm\] for the two states $2s{\sigma}_g$ $(1,0,0,+)$ and $3p{\sigma}_u$ $(1,0,0,-)$ as a function of the internuclear distance $R$. Comparison the variational energy with previous, highly accurate results [@Mar:1971] (given on the second line) for each $R$-value is presented. The agreement is on the level of $10$ s.d. For each state the variational value of $p$ (when $p$ is taken as a variational parameter in (\[appr\])) as well as the node position $\xi^{\pm}_0$ are given. In both cases $(1,0,0,\pm)$ the node position is a decreasing function of the internuclear distance having a finite value for small $R$ and conversely approaching to the lower limit in $\xi$-coordinate, $\xi = 1$ at large $R$, roughly as $\sim 1/R$. At the point $\xi^{\pm}_0$, the wave function (\[apprNX\]) vanishes. In the configuration space it corresponds to a nodal surface which is a prolate spheroid of eccentricity $\varepsilon = 1/\xi^{\pm}_0$.
Functions $X_0(\xi)$, $Y_0^{(\pm)}(\eta)$ and the first corrections to the phases are shown in Figs. \[fig-2ssg-X\], \[fig-2ssg-Y\], \[fig-3psu-X\] and \[fig-3psu-Y\] for $R=2$ a.u. as an illustration.
R(a.u.) $E_t(1,0,0,+)$ $p(1,0,0,+)$ $\xi^{+}_0$ $E_t(1,0,0,-)$ $p(1,0,0,-)$ $\xi^{-}_0$
--------- ------------------- -------------- ------------- ------------------- -------------- -------------
1.0 1.1541508226 0.459850295 2.782853311 1.521369039285 0.345916 5.360475264
1.154150823003 1.5213690392720
2.0 0.278270249325 0.849546791 1.907869613 0.489173669829 0.714721 2.532742379
0.2782702493234 0.4891736698286
4.0 -0.0770297349135 1.519249466 1.477672193 0.00978089990 1.40031296 1.589362953
-0.07702973491498 0.009780899904368
6.0 -0.161775845624 2.110919849 1.330973187 -0.1215317623 2.02331 1.364704127
-0.16177584562974 -0.12153176233782
8.0 -0.193554665734 2.663995993 1.254298836 -0.17496728916 2.60758 1.265974957
-0.19355466573518 -0.17496728919184
10.0 -0.20942125179 3.199301689 1.206019531 -0.20117150595 3.16691 1.210160770
-0.2094212518184 -0.201171506037
20.0 -0.23699860692 5.805158111 1.103266490 -0.236904750195 5.80435 1.103289607
-0.2369986069452 -0.2369047502114
30.0 -0.24389262263 8.359177003 1.068352565 -0.24389177096 8.35916 1.068352748
-0.2438926229736 -0.2438917709742
40.0 -0.24647865989 10.889970797 1.051017992 -0.24647865270 10.88997 1.051017930
-0.2464786599118 -0.2464786527404
50.0 -0.247714222867 13.409749785 1.040679396 -0.24771422280 13.40975 1.040679432
-0.2477142228738 -0.2477142228160
: Total energy $E_t(R)$ for the $2s{\sigma}_g$ $(1,0,0,+)$ and $3p{\sigma}_u$ $(1,0,0,-)$ states of the H$_2^+$ molecular ion (the first line) compared to [@Mar:1971] (the second line).[]{data-label="tb10Lpm"}
Separation constant $A$
-----------------------
In developed perturbation theory so as to estimate the accuracy of the approximation [(\[appr\])]{} for $X_0(\xi)$ and $Y_0(\eta)$, two expressions, one for each variable, for the separation constant are obtained $A_{n,\xi}\ $ and $ A_{n,\eta}$ (see Appendix and Eqs. [(\[xA\_n\])]{} and [(\[yA\_n\])]{}). However, the condition of consistency $A_{n, \xi}\ =\ A_{n,\eta}$ should be imposed. Table \[Aval\] presents the separation constant for all considered states. For each $R$-value the first/second line correspond to $A_{\xi}$ /$A_{\eta}$ calculated with [(\[xA\_n\])]{} / [(\[yA\_n\])]{} compared to Marcela et al [@Mar:1971] (third row). It turns out that as a result of variational calculations the condition $A_{n,\xi}\ =\ A_{n,\eta}$ is fulfilled automatically, up to $\sim 8$ significant digits which is in agreement with those presented by Marcela et al [@Mar:1971]. Hence, there is no need to impose the equality condition. It is a reflection of the outstanding accuracy of the approximation (15).
------ ----------------- ----------------- ---------------- ------------------ ---------------- ----------------- ---------------- -----------------
$(0 0 0 +)$ $( 0 0 0 -)$ $( 0 0 1 +)$ $(0 0 1 -)$ $( 0 0 2 +)$ $(0 0 2 -)$ $( 1 0 0 +)$ $( 1 0 0 -)$
R $1s\sigma_g$ $2p\sigma_u$ $2p\pi_u$ $3d\pi_g$ $3d\delta_g$ $4f\delta_u$ $2s\sigma_g$ $3p\sigma_u$
1.0 0.2499462430 -1.8300104198 0.0476692616 -3.9520464219 0.0157049965 -5.9791583275 0.0711543055 -1.9281072878
0.2499462409 -1.8300104197 0.0476693150 -3.9520464344 0.0157049889 -5.9791583064 0.0711543140 -1.9281072817
0.2499462406113 -1.830010419730 0.047669315711 -3.952046434393 0.015704988875 -5.979158306119 0.071154314127 -1.928107280448
2.0 0.8117295877 -1.1868893947 0.1749484742 -3.8048856116 0.0611354153 -5.9165512457 0.2484661667 -1.6917231809
0.8117295852 -1.1868893929 0.1749484725 -3.8048856050 0.0611354010 -5.9165512311 0.2484661712 -1.6917231733
0.8117295846248 -1.186889392359 0.174948472433 -3.804885604702 0.061135400906 -5.916551230876 0.248466171440 -1.691723172798
4.0 2.7995887561 1.5384644804 0.6001486772 -3.1948053489 0.2270652065 -5.6657454590 0.8535318015 -0.7976034401
2.7995887582 1.5384644803 0.6001486748 -3.1948053506 0.2270652107 -5.6657454689 0.8535318003 -0.7976034382
2.799588759471 1.538464480300 0.600148674671 -3.194805350518 0.227065210827 -5.665745469006 0.853531800197 -0.797603437898
6.0 6.4536037434 5.9279301781 1.2199716980 -2.1786687874 0.4743694112 -5.2501595578 1.8115068883 0.5663869192
6.4536037423 5.9279301759 1.2199717011 -2.1786687836 0.4743694166 -5.2501595612 1.8115068932 0.5663869192
6.453603742887 5.927930173726 1.219971701568 -2.178668782566 0.474369416805 -5.250159561131 1.811506894227 0.566386919545
8.0 12.2261746132 12.0646853402 2.0537173294 -0.7961022597 0.7914989890 -4.6781903409 3.2069680505 2.3733521986
12.2261746118 12.0646853394 2.0537173246 -0.7961022613 0.7914989805 -4.6781903532 3.2069680527 2.3733521972
12.22617461542 12.06468533824 2.053717323829 -0.7961022613695 0.791498980083 -4.678190353126 3.206968053370 2.373352197778
10.0 20.1333096527 20.0921239053 3.1610270665 0.9355443423 1.1760019683 -3.9601419353 5.1293596287 4.6288376336
20.1333042259 20.0921157054 3.1610270649 0.9355443394 1.1760019677 -3.9601419604 5.1293596245 4.6288376291
20.13329317839 20.09209890008 3.161027064845 0.9355443386850 1.176001967652 -3.960141960690 5.129359623687 4.628837627894
20.0 90.0528911866 90.0528775638 15.6431425753 15.4372141472 4.4202357771 1.6768434995 23.1467951638 23.1310108444
90.0528911837 90.0528775637 15.6431424784 15.4372141468 4.4202357567 1.6768434549 23.1467951625 23.1310108423
90.05289119141 90.05287756706 15.64314256883 15.43721414965 4.420235762270 1.676843453846 23.14679516399 23.13101084191
30.0 210.0345966014 210.0345966014 41.5927047072 41.5865009061 11.8536327107 11.1439910435 54.1918175098 54.1915412139
210.0345965987 210.0345965997 41.5927046648 41.5865009042 11.8536321491 11.1439910147 54.1918174666 54.1915412094
210.0345965903 210.0345965883 41.59270470411 41.58650090379 11.85363268535 11.14399101596 54.19181751174 54.19154120499
40.0 380.0257071902 380.0257071902 80.2475884726 80.2474668189 25.5692520539 25.4727279329 97.8369229167 97.8369191379
380.0257071899 380.0257071899 80.2475883011 80.2474668173 25.5692515860 25.4727279007 97.8369229125 97.8369191305
380.0257071871 380.0257071871 80.24758848264 80.24746682685 25.56925202708 25.47272792120 97.83692292343 97.83691912308
50.0 600.0204520196 600.0204516482 131.4445904451 131.4445885530 45.2845813578 25.4727279329 154.0220957323 154.0220957009
600.0204519899 600.0204516470 131.4445904398 131.4445885530 45.2845807868 25.4727279007 154.0220957308 154.0220956952
600.0204516331 600.0204516331 131.4445904563 131.4445885619 45.28458134150 45.27511009129 154.0220957319 154.0220956865
------ ----------------- ----------------- ---------------- ------------------ ---------------- ----------------- ---------------- -----------------
Transitions
===========
Knowledge of wave functions with high local relative accuracy $\lesssim 10^{-5} - 10^{-6}$ gives us a chance to calculate matrix elements with controlled relative accuracy $\lesssim 10^{-5} - 10^{-6}$. As a demonstration we calculate E1, E2 and B1 Oscillator Strength as a function of interproton distance for the permitted radiative transitions from excited states to the ground state $1s{\sigma}_g\,(0,0,0,+)$ .
E1 Oscillator Strength
----------------------
Following Bates [@BDHS:1953; @BDHS:1954], with the energy given in Rydbergs, the electric dipole oscillator strength from a lower electronic (initial) state $\Psi_i$ to an upper electronic (final) state $\Psi_f$, is given by $$\label{OSE1}
f_{i\rightarrow f}^{(E1)}(R) = \frac{1}{3}\,G\,(E_f(R)-E_i(R))\,{\bf S}^{(1)}_{if}\ ,$$ where $G$ is the orbital degeneracy factor, ${\bf S}^{(1)}_{if}(R)$ is the square of the matrix element $${\bf S}^{(1)}_{if}(R)=|\langle \Psi_i(R)| {\bf r} |\Psi_f(R)\rangle|^2\ ,$$ and ${\bf r}$ is the vector of the electron position measured from the interproton midpoint. The involved excited states for permitted electric dipole transitions from the ground state $1s{\sigma}_g$ are the states $2p{\sigma}_u$, $2p\pi_u$ and $3p{\sigma}_u$. In Table \[Tose1\] the E1 oscillator strength is presented for two transitions: $1s{\sigma}_g-2p\pi_u$ and $1s{\sigma}_g-3p{\sigma}_u$. The transition $1s{\sigma}_g-2p{\sigma}_u$ was calculated and discussed in [@Turbiner:2011] and we won’t present here the results. The orbital degeneracy factor is $G=2$ for $f_{1s{\sigma}_g-2p\pi_u}$ and $G=1$ for $f_{1s{\sigma}_g-3p{\sigma}_u}$. It is assumed this calculation should provide at least 5 s.d. correctly. As a result for all internuclear distances they coincide in 6 s.d. with Tsogbayar et al, [@ts:2010] for $1s{\sigma}_g-2p\pi_u$ (with an exception at $R$=1 a.u. where it deviates in one unit at the 6th digit) which increases up to 7 figures for intermediate $R$. The E1 oscillator strength $f_{1s{\sigma}_g-3p{\sigma}_u}$ is compared with Bates et al [@BDHS:1954] only for two values of $R=2,4 a.u.$ and the agreement is in 2 s.d. We also confirm the striking qualitative result by Bates et al that the E1 oscillator strength increases in $\sim$20 times coming from $R=2 a.u.$ to 4 a.u.
------ --------------- ---------------- --------------------------------- ------------------------
$R$ Present [@ts:2010] Present [@BDHS:1954]
1.0 3.934 370 22 3.934 381 322 2.203 421 96
2.0 4.601 871 35 4.601 869 855 8.249 067 00 $\times 10^{-2}$ 8.24 $\times 10^{-2}$
4.0 4.655 236 69 4.655 237 278 1.614 379 52 1.61
6.0 3.841 069 41 3.841 069 351 4.145 987 07
8.0 3.035 614 09 3.035 614 626 5.567 828 37
10.0 2.617 504 44 2.617 505 281 6.106 005 25
20.0 2.717 465 95 2.717 469 205 6.503 362 27
30.0 2.774 375 78 6.610 830 83
40.0 2.775 806 83 6.673 398 05
50.0 2.775 499 12 6.715 284 57
------ --------------- ---------------- --------------------------------- ------------------------
: Electric dipole oscillator strength for transition $1s{\sigma}_g-2p\pi_u$ and $1s{\sigma}_g-3p{\sigma}_u$ vs $R$ compared to Tsogbayar et al [@ts:2010] and Bates et al [@BDHS:1954] (rounded).[]{data-label="Tose1"}
B1 Oscillator Strength
----------------------
It is known that the magnetic dipole transitions are much smaller than the electric dipole transition. The magnetic dipole B1 Oscillator Strength, with the energy in Rydbergs, is given by $$\label{OSB1}
f_{i\rightarrow f}^{(B1)}(R) = \frac{1}{3}\,(E_f(R)-E_i(R))|{\bf S}(R)|^2\ ,$$ where ${\bf S}(R)$ is the matrix element $${\bf S}(R)=-\mu_B\langle \Psi_i(R)| {\bf L} |\Psi_f(R)\rangle\ ,$$ ${\bf L}$ is the angular momentum operator and $\mu_B$ is the Bohr magneton. Between the states we consider at present article, there is only one permitted magnetic dipole transition from the ground state to $f_{1s{\sigma}_g-3d\pi_g}$. This B1 Oscillator strength is presented in Table \[TB1OS\]. Comparison is made with previously known results by Dalgarno et al. [@DaMc:1953] at $R=2,4 a.u.$ only with 3 s.d. We confirm the striking qualitative observation that the B1 oscillator strength increases in $\sim$10 times coming from $R=2 a.u.$ to 4 a.u.
------ ------------------- --------------
$R$ Present [@DaMc:1953]
1.0 1.050 606 8 E-08
2.0 1.666 176 0 E-07 1.67 E-07
4.0 2.008 469 0 E-06 2.01 E-06
6.0 6.251 641 6 E-06
8.0 1.129 235 3 E-05
10.0 1.633 841 0 E-05
20.0 5.260 000 6 E-05
30.0 1.169 171 1 E-04
40.0 2.078 469 5 E-04
50.0 3.247 423 8 E-04
------ ------------------- --------------
: Magnetic dipole oscillator strength for transition $1s{\sigma}_g-3d\pi_g$ vs $R$ compared to Dalgarno et al. [@DaMc:1953].[]{data-label="TB1OS"}
E2 Oscillator Strength
----------------------
It is known that the electric quadrupole transitions are much smaller than the electric dipole transition but comparable with magnetic dipole transitions. For the first time we calculate electric quadrupole transitions in $H_2^+$ molecular ion for transitions ${1s{\sigma}_g-3d\pi_g}$, ${1s{\sigma}_g-3d{\delta}_g}$ and ${1s{\sigma}_g-2s{\sigma}_g}$.
The electric quadrupole E2 oscillator strength with the energy in Rydbergs is given by $$\label{OSE2}
f_{i\rightarrow f}^{(E2)}(R) = \frac{\alpha^2}{240}\,G\,(E_f(R)-E_i(R))^3 {\bf S}^{(2)}_{if}(R)\ ,$$ where ${\bf S}^{(2)}_{if}(R)$ is the square of the matrix element of the electric quadrupole moment and ${\alpha}$ is the fine structure constant. The orbital degeneracy factor is $G=2$ for $f_{1s{\sigma}_g-3d\pi_g}$ and $f_{1s{\sigma}_g-3d{\delta}_g}$ and $G=1$ for $f_{1s{\sigma}_g-2s{\sigma}_g}$. It is assumed this calculation should provide at least 5 s.d. correctly. Results are presented in Table \[Tose2\]. Comparing the electric dipole transition $f_{1s{\sigma}_g-2p\pi_u}$, see Table \[Tose1\] with the magnetic dipole transition $f_{1s{\sigma}_g-3d\pi_g}$, see Table \[TB1OS\], and electric quadrupole transition $f_{1s{\sigma}_g-3d\pi_g}$, see Table \[Tose2\] oscillator strengths, one can see that at $R=2 a.u.$ the E1 oscillator strength is six orders of magnitude larger than E2 oscillator strength and seven order of magnitude larger than B1.
$R$ $f_{1s{\sigma}_g-3d\pi_g}$ $f_{1s{\sigma}_g-3d{\delta}_g}$ $f_{1s{\sigma}_g-2s{\sigma}_g}$
------- ----------------------------- --------------------------------- --------------------------------- --
1.0 1.500 688 2 E-06 1.240 332 8 E-06 1.386 514 0 E-09
2.0 2.608 637 5 E-06 1.557 357 3 E-06 1.378 379 7 E-08
4.0 4.539 815 8 E-06 1.436 910 5 E-06 1.372 684 3 E-07
6.0 6.122 024 1 E-06 9.655 519 1 E-07 5.240 450 5 E-07
8.0 7.884 701 2 E-06 5.901 755 7 E-07 1.222 375 2 E-06
10.0 1.010 476 3 E-05 3.817 367 8 E-07 2.179 700 3 E-06
20.0 3.114 396 7 E-05 1.558 008 9 E-07 9.918 089 5 E-06
30.0 6.984 147 5 E-05 1.735 371 6 E-07 2.253 646 5 E-05
40.0 1.244 746 2 E-04 1.864 752 8 E-07 4.027 405 2 E-05
50.0 1.946 579 9 E-04 1.879 848 1 E-07 6.317 650 4 E-05
: Quadrupole oscillator strength $f$ for transitions ${1s{\sigma}_g-3d\pi_g}$, ${1s{\sigma}_g-3d{\delta}_g}$ and ${1s{\sigma}_g-2s{\sigma}_g}$ [*vs*]{} $R$.[]{data-label="Tose2"}
H$_2^+$ molecular ion in the united atomic ion He$^+$ limit
-----------------------------------------------------------
When for H$_2^+$ molecular ion the internuclear distance tends to zero, $R \rightarrow 0$, we arrive at one-electron atomic system with nuclear charge $Z=2$, [*i.e.*]{} the He$^+$ ion. In practice, at $R \rightarrow 0$ we have $$\begin{aligned}
\label{ellx}
\lim_{R\rightarrow 0}R\,\xi\ =\ & 2 r\ ,\quad 0\leq r\leq \infty \ ,\\
\lim_{R\rightarrow 0}\eta \ =\ & \cos{\theta}\ ,\quad 0 \leq \theta \leq \pi \ ,\\
\lim_{R\rightarrow 0}\phi \ =\ & \phi \ ,\quad 0 \leq \phi \leq 2\pi \ ,\end{aligned}$$ where $(r,\theta,\phi)$ are the spherical coordinates. However, although in this limit the parameter $p \rightarrow 0$, the ratio $$\lim_{R\rightarrow 0} \frac{R}{p} = \frac{2}{\sqrt{-E}}= \frac{2{\mathtt n}}{\mathcal{Z}}\Bigg|_{\mathcal{Z}=2}={\mathtt n}\,,$$ (cf. [(\[p\])]{}), takes a finite value; here $E=-\mathcal{Z}^2/ \mathtt{n}^2$ is the total energy of the hydrogen-like atom of $\mathcal{Z}$-charge ($\mathcal{Z}=2$) with principal quantum number $\mathtt{n}$. Now taking the variational parameters ${\alpha}\rightarrow 0$, $\gamma\sim$ [*const*]{}, $a_{1} \rightarrow 0$ $b_{2}=b_{3}\rightarrow0$, the limit of approximation [(\[appr\])]{} at $R\rightarrow 0$ (up to a normalization factor) is $$\Psi^{(\pm)}_{n,m,{\Lambda};\mathtt{n}} \propto r^{\mathtt{n}-n-1}P_n(r)
e^{-\frac{2}{\mathtt{n}}r} \sin^{\Lambda}{{\theta}}\,\,Q_m(\cos^2{{\theta}})
\left[
\begin{array}{c}
1 \\
\cos{{\theta}}
\end{array}
\right] e^{\pm i {\Lambda}\phi} \ .
\label{plim}$$ This formulas realizes the correspondence between the states of the molecular ion H$_2^+$ and ones of the atomic ion He$^+$. The examples of this correspondence are displayed in Table \[Rto0\]. The first column presents the molecular orbital $(n,m,{\Lambda},\pm)$ approximated by [(\[appr\])]{}. Its united atom nomenclature is given in the second column. In the limit $R \rightarrow 0$ this approximation takes the form [(\[plim\])]{} (third column). Clearly, these functions coincide to the exact wavefunctions of the atomic ion He$^+$ (up to normalization factor), when the constant $c$ in the polynomial $P_n(r)$ (when present) takes a certain value (see the fourth column). Hence, the molecular orbital $(n,m,{\Lambda},\pm)$ in approximation ([(\[appr\])]{}) in the limit $R {\rightarrow}0$ corresponds to the exact atomic orbital $\mathtt{(n,l,m)}$ with appropriate value of $l$, see the last column Table \[Rto0\].
----------------------- ---------------- -------------------------------------------------------------- ------- --------------------
Molecular Orbital United Atom Atomic Orbital
$(n,m,{\Lambda},\pm)$ Designation [(\[plim\])]{} $c$ $\mathtt{(n,l,m)}$
$(0,0,0,+)$ $1s{\sigma}_g$ $e^{-2r}$ $\mathtt{(1,0,0)}$
$(0,0,0,-)$ $2p{\sigma}_u$ $re^{-r}\cos{\theta}$ $\mathtt{(2,1,0)}$
$(0,0,1,+)$ $2p\pi_u$ $re^{-r}\sin{\theta}e^{i\phi}$ $\mathtt{(2,1,1)}$
$(0,0,1,-)$ $3d\pi_g$ $r^2e^{-\frac{2}{3}r}\sin{\theta}\cos{\theta}\,e^{i\phi}$ $\mathtt{(3,2,1)}$
$(0,0,2,+)$ $3d{\delta}_g$ $r^2e^{-\frac{2}{3}r}\sin^2{\theta}\,e^{2i\phi}$ $\mathtt{(3,2,2)}$
$(0,0,2,-)$ $4f{\delta}_u$ $r^3e^{-\frac{1}{2}r}\sin^2{\theta}\cos{\theta}\,e^{2i\phi}$ $\mathtt{(4,3,2)}$
$(1,0,0,+)$ $2s{\sigma}_g$ $(r-c)e^{-2r}$ 2 $\mathtt{(2,0,0)}$
$(1,0,0,-)$ $3p{\sigma}_u$ $r(r-c)e^{-\frac{2}{3}r}\,\cos{\theta}$ 3 $\mathtt{(3,1,0)}$
$(0,1,0,+)$ $3d{\sigma}_g$ $r^2e^{-\frac{2}{3}r}(\cos^2{\theta}-c)$ 1/3 $\mathtt{(3,2,0)}$
$(0,1,0,-)$ $4f{\sigma}_u$ $r^3e^{-\frac{1}{2}r}(\cos^2{\theta}-c)\,\cos{\theta}$ 3/5 $\mathtt{(4,3,0)}$
----------------------- ---------------- -------------------------------------------------------------- ------- --------------------
: Correspondence between the molecular orbital $(n,m,\Lambda,\pm)$ and the atomic orbital $\mathtt{(n,l,m)}$ in the limit $R \rightarrow 0$. The molecular approximation [(\[appr\])]{} takes the form [(\[plim\])]{}, where $c$ is the constant term in the polynomial.[]{data-label="Rto0"}
Conclusions
===========
Summarizing we want to state that a simple uniform approximation of the eigenfunctions for the H$_2^+$ molecular ion is presented. It allows us to calculate any expectation value or matrix element with guaranteed accuracy. It manifests the approximate solution of the problem of spectra of the H$_2^+$ molecular ion. In a quite straightforward way similar approximations can be constructed for general two-center, one-electron system $(Z_a,Z_b,e)$, in particular, for $(\rm HeH)^{++}$. It will be done elsewhere.
The key element of the procedure is to construct an interpolation between the WKB expansion at large distances and perturbation series at small distances for the phase of the wavefunction. Or, in other words, to find an approximate solution for the corresponding eikonal equation. Separation of variables allowed us to solve this problem. In the case of non-separability of variables the WKB expansion of a solution of the eikonal equation can not be constructed in unified way, since all depends on the way to approach to infinity. However, a reasonable approximation of the first growing terms of the WKB expansion seems sufficient to construct the interpolation between large and small distances giving high accuracy results. This program was realized for the problem of the hydrogen atom in a magnetic field and will be published elsewhere.
It is worth mentioning a curious fact that the problem (\[Sch\]) possesses the hidden algebra $sl(2)\oplus sl(2)$. It can be immediately seen - making the gauge rotation of the operators in r.h.s. of the equations (\[X\_L\]) and (\[Y\_L\]) with gauge factors $e^{-p\xi}$ and $e^{p\eta}$, respectively. We obtain the operators which are in the universal enveloping algebra of $sl(2)$ (see e.g. [@Turbiner:1988]). The dimension of the representation is $-{\Lambda}$ and $-{\Lambda}+\frac{R}{p}$, respectively. For non-physical values of ${\Lambda}$ and integer ratio $\frac{R}{p}$ the algebras $sl(2)$ appear in the finite-dimensional representation realized in action on polynomials in $\xi, \eta$. It explains a mystery sometimes observed of the existence of polynomial solutions for non-physical values of ${\Lambda}$ in the problem (2) (details will be given elsewhere).
***Acknowledgements***.
The research is supported in part by PAPIIT grant [**IN115709**]{} and CONACyT grant [**166189**]{} (Mexico). H.O.P. is grateful to Universit[é]{} Libre de Bruxelles (Belgium) and Instituto de Ciencias Nucleares, UNAM (Mexico) for a kind hospitality extended to him where a certain stages of a present work were carried out. A.V.T. thanks the University Program FENOMEC (UNAM, Mexico) for partial support.
Appendix {#ap1 .unnumbered}
========
The easiest way to calculate a deviation of the approximation from the exact eigenfunction is to develop a perturbation theory in framework of the so-called [*non-linearization procedure*]{} [@Turbiner:1984]: for a chosen approximation $\psi_0$ a corresponding potential $V_0=\frac{{\Delta}\psi_0}{\psi_0}$ is found with $E_0=0$, for which $\psi_0$ is the exact eigensolution. Then the potential is written in the form $V=V_0 + {\lambda}V_1$, then it is looked for energy and the eigenfunction in the form of power series in the parameter ${\lambda}$, $E=\sum {\lambda}^n E_n$ and $\Psi=\Psi_0 \exp (-\sum {\lambda}^n \varphi_n)$, respectively. Eventually, ${\lambda}$ is placed equal to one.
Due to specifics of (1) because of the separation of variables the procedure can be developed for both functions $X$ and $Y$ (see (\[psi\])) separately as well as for the separation parameter $A$, while keeping the energy $E$ fixed. It can be done for the system of equations (\[X\_L\]), (\[Y\_L\]). As a first step let us transform (\[X\_L\]), (\[Y\_L\]) into the Riccati form by introducing $X=fe^{-\varphi}$ and $Y=ge^{-\varrho}$, respectively, $$\label{X_L-phi}
(\xi^2-1)[f(x'- x^2)+2f'x-f''] + 2 ({\Lambda}+1)\xi[fx-f'] =\
[A - V(\xi)]f \ ,\quad x=\varphi'_{\xi}$$ where the “potential” $V(\xi) = p^2\xi^2 - 2R \xi$, and $$\label{Y_L-rho}
(\eta^2-1)[g(y' - y^2)+2g'y-g''] + 2 ({\Lambda}+1)\eta[gy-g'] =\
[A - W(\eta)]g \ ,\quad y = \varrho'_{\eta}$$ where the “potential” $W(\eta) = p^2\eta^2$.
Let us choose some $x_0(\xi)=\varphi_0'(\xi)$, then substitute it to the l.h.s. of (\[X\_L-phi\]) and call the result as unperturbed “potential” $V_0(\xi)$ putting without loss of generality $A_0=0$. The difference between the original $V(\xi)$ and generated $V_0(\xi)$ is the perturbation, $V_1(\xi)=V(\xi)-V_0(\xi)$. For a sake of convenience we can insert a parameter ${\lambda}$ in front of $V_1$ and develop the perturbation theory in powers of it. The perturbation theory is also developed for node states where a node position is also looked for the form of power expansion in ${\lambda}$. $$\label{PTx}
x=\sum {\lambda}^n x_n\ , \ f=\sum {\lambda}^n f_{n,\xi}\ , \ A=\sum {\lambda}^n A_{n,\xi}\ .$$ The equation for $n$th correction has a form, $$\label{x_n}
\left\{(\xi^2-1)^{{\Lambda}+1}X_0^2\left[x_n-\left(\frac{f_{n,\xi}}{f_{0,\xi}}\right)'\right]\right\}'
=(\xi^2-1)^{{\Lambda}}X_0^2[A_{n,\xi} - Q_n],$$ where $Q_1=V_1$ and $$\begin{aligned}
Q_n &=& -(\xi^2-1)\sum_{i=1}^{n-1}x_ix_{n-i}\nonumber\\
& & -\frac{1}{f_{0,\xi}}\left[\sum_{k=1}^{n-1}f_{k,\xi}\left((\xi^2-1)\sum_{i=0}^{n-k}x_ix_{n-k-i}-\frac{((\xi^2-1)^{{\Lambda}+1}x_{n-k})'}{(\xi^2-1)^{{\Lambda}}} +A_{n-k,\xi}-V_{n-k}\right)\right.\nonumber\\
& & -\left.2(\xi^2-1)\sum_{k=1}^{n-1}x_kf'_{n-k,\xi}\right],\end{aligned}$$ for $n>1$. Integrating [(\[x\_n\])]{} we obtain $$\label{x_n-solu}
x_n\ =\left(\frac{f_{n,\xi}}{f_{0,\xi}}\right)'+\frac{1}{(\xi^2-1)^{{\Lambda}+1}X_0}\int_1^{\xi} (A_{n,\xi} - Q_n) (\xi^2-1)^{{\Lambda}}X_0^2 \,d\xi\ ,$$ where $f_{n,\xi}$ and $A_{n,\xi}$ are obtained in the same way. These are $$\label{xf_n}
f_{n,\xi}(\xi_0)\ =\frac{1}{(\xi_0^2-1)^{{\Lambda}+1}e^{-2 \varphi_0}f'_{0,\xi}(\xi_0)} \int_1^{\xi_0}(A_{n,\xi} - Q_n) (\xi^2-1)^{{\Lambda}}X_0^2\,d\xi,$$ and $$\label{xA_n}
A_{n,\xi}\ =\ \frac{\int_1^{\infty} Q_n (\xi^2-1)^{{\Lambda}}X_0^2d\xi}
{\int_1^{\infty} (\xi^2-1)^{{\Lambda}}X_0^2\,d\xi}.$$ In a similar way by choosing $y_0(\eta)=\varrho_0'(\eta)$, building the unperturbed “potential” $W_0(\eta)$ and putting $A_0=0$ as zero approximation one can develop perturbation theory in the equation (\[Y\_L-rho\]) $$\label{PTy}
y=\sum {\lambda}^n y_n\ , \ g=\sum {\lambda}^n g_{n,\eta}\, \ A=\sum {\lambda}^n A_{n,\eta}\ .$$ The equation for $n$th correction has a form similar to (\[x\_n\]), $$\label{y_n}
\left\{(\eta^2-1)^{{\Lambda}+1}Y_0^2\left[y_n-\left(\frac{g_{n,\eta}}{g_{0,\eta}}\right)'\right]\right\}'
=(\eta^2-1)^{{\Lambda}}Y_0^2[A_{n,\eta} - Q_n],$$ where $Q_1=W_1$ and $$\begin{aligned}
Q_n &=& -(\eta^2-1)\sum_{i=1}^{n-1}y_iy_{n-i}\nonumber\\
& & -\frac{1}{g_{0,\eta}}\left[\sum_{k=1}^{n-1}g_{k,\eta}\left((\eta^2-1)\sum_{i=0}^{n-k}y_iy_{n-k-i}-\frac{((\eta^2-1)^{{\Lambda}+1}y_{n-k})'}{(\eta^2-1)^{{\Lambda}}} +A_{n-k,\eta}-V_{n-k}\right)\right.\nonumber\\
& & -\left.2(\eta^2-1)\sum_{k=1}^{n-1}y_{k}g'_{n-k,\eta}\right],\end{aligned}$$ for $n>1$. Its solution is given by (cf.(\[x\_n-solu\])) $$\label{y_n-solu}
y_n\ =\left(\frac{g_{n,\eta}}{g_{0,\eta}}\right)'+\frac{1}{(\eta^2-1)^{{\Lambda}+1}Y_0}\int_{-1}^{\eta} (A_{n,\eta} - Q_n) (\eta^2-1)^{{\Lambda}}Y_0^2 \,d\eta\ ,$$ where $g_{n,\eta}$ and $A_{n,\eta}$ are obtained in the same way. These are (cf.(\[xf\_n\]) and [(\[xA\_n\])]{}) $$\label{yg_n}
g_{n,\eta}(\eta_0)\ =\frac{1}{(\eta_0^2-1)^{{\Lambda}+1}e^{-2 \varrho_0}g'_{0,\eta}(\eta_0)} \int_1^{\eta_0}(A_{n,\eta} - Q_n) (\eta^2-1)^{{\Lambda}}Y_0^2\,d\eta,$$ and $$\label{yA_n}
A_{n,\eta}\ =\ \frac{\int_{-1}^{1} Q_n (\eta^2-1)^{{\Lambda}}Y_0^2d\eta}
{\int_{-1}^{1} (\eta^2-1)^{{\Lambda}}Y_0^2\,d\eta}.$$
In order to realize this perturbation theory a condition of consistency should be imposed $$\label{An}
A_{n,\xi}\ =\ A_{n,\eta}\ .$$ This condition allows us to find the parameter $p$ and, hence, the energy $E'$ and $E$ (see (\[p\])).
Sufficient condition for such a perturbation theory to be convergent is to require a perturbation “potential” to be bounded, $$\label{PerPot}
|V_1(\xi)| \leq C_{\xi}\ ,\ |W_1(\eta)| \leq C_{\eta}\ ,$$ where $C_{\xi}, C_{\eta}$ are constants. Obviously, that the rate of convergence gets faster with smaller values of $C_{\xi}, C_{\eta}$. It is evident that the perturbations $V_1(\xi)$ and $W_1(\eta)$ get bounded if $\varphi_0(\xi)$ and $\varrho_0(\eta)$ are smooth functions vanishing at the origin but reproduce exactly the growing terms at $|\xi|, |\eta|$ tending to infinity in (\[X-inf\]), (\[Y-inf\]), respectively.
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[99]{}
L.D. Landau and E.M. Lifshitz,\
[*Quantum Mechanics, Non-relativistic Theory [(]{}Course of Theoretical Physics [vol 3)]{}*]{}, [3rd edn (Oxford:Pergamon Press)]{}, 1977
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[*Lett.Math.Phys. **74***]{}, 169-180 (2005)
A.V. Turbiner,\
*Double well potential: perturbation theory, tunneling, WKB (beyond instantons)*,\
[*Int.Journ.Mod.Phys. **A25***]{}, 647-658 (2010)
A.V. Turbiner and H. Olivares-Pilon,\
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[*Journ. of Phys. **B44***]{} (2011) 101002 (7 pp)
E.A. Hylleraas,\
[*Z. Physik **71***]{} (1931) 739
D.R. Bates, K. Ledsham and A.D. Stewart,\
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H.E. Montgomery Jr.,\
*One-electron wavefunctions. Accurate expectation values*,\
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D.M. Bishop and L.M. Cheung,\
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V.I. Korobov,\
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[*Phys. Rev. **A 61***]{} (2000) 064503
M.P. Strand and W.P. Reinhardt,\
*Semiclassical quantization of the low lying electronic states of H$_2^+$*,\
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H. A. Erikson and E. L. Hill,\
*A note about one-electron states of diatomic molecules*,\
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C.A. Coulson and A. Joseph,\
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M. Vincke and D. Baye,\
*Hydrogen molecular ion in an aligned strong magnetic field by the Lagrange-mesh method*,\
[*J. Phys. **B 39***]{}, 2605-2618 (2006)
A.V. Turbiner, *On Perturbation Theory and Variational Methods in Quantum Mechanics*,\
[*ZhETF **79***]{}, 1719 (1980); [*Soviet Phys.-JETP **52***]{}, 868 (1980) (English Translation);\
*The Problem of Spectra in Quantum Mechanics and the ‘Non-Linearization’ Procedure*,\
[*Usp. Fiz. Nauk. **144***]{}, 35 (1984); [*Sov. Phys. – Uspekhi **27***]{}, 668 (1984) (English Translation)
T.C. Scott, M. Aubert-Frecon and J. Grotendorst,\
*New Approach for the Electronic Energies of the Hydrogen Molecular Ion*,\
[*J. Chem. Physics **324***]{}, 323-338 (2006)
Ts. Tsogbayar and Ts. Banzragch,\
[*The Oscillator Strengths of H$_2^{+}$, 1$s{\sigma}_g$-2$p{\sigma}_u$, 1$s{\sigma}_g$-2$p\pi_u$*]{},\
arXiv:physics.atom-ph/1007.4354v1 (2010)
A.V. Turbiner,\
*Quasi-Exactly-Solvable Problems and the $SL(2,R)$ Group*,\
[*Comm.Math.Phys. **118***]{}, 467-474 (1988)
Marcela M. Madsen and James M. Peek,\
[*Eigenparameters for the lowest twenty electronic states of the Hydrogen molecular ion*]{},\
[*Atomic Data, [**2**]{}*]{}, 171-204 (1971) A. Dalgarno and R. McCarroll,\
[*Properties of the Hydrogen Molecular Ion VII: Magnetic Dipole Oscillator Strengths of the $1s\sigma_g-3d\pi_g$ Transition*]{},\
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[^1]: From $3D$ point of view they are [*prolate spheroidal*]{}.
[^2]: Due to complete separation of variables one more integral in a form of the second order polynomial in momentum exists [@Erikson:1949], it is closely related to Runge-Lenz vector [@Coulson:1967] and commutes with $\hat L_{\phi}$; hence, the H$_2^+$ ion in adiabatic (Born-Oppenheimer) approximation is completely-integrable system.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Hongxia Guo[^1]'
- 'Changfeng Gui[^2]'
- 'Ping Lin[^3]'
- 'Mingfeng Zhao[^4]'
title: 'Multiple solutions and their asymptotics for laminar flows through a porous channel with different permeabilities[^5]'
---
[^1]: Department of Applied Mathematics, University of Science and Technology, Beijing, 100083, China ().
[^2]: Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA; Institute of Mathematics, Hunan University, Changsha, 410082, China ().
[^3]: Division of Mathematics, University of Dundee, Dundee, DD1 4HN, United Kingdom; Department of Applied Mathematics, University of Science and Technology, Beijing, 100083, China ().
[^4]: Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China ().
[^5]:
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present alternative definitions of the first-order stable model semantics and its extension to incorporate generalized quantifiers by referring to the familiar notion of a reduct instead of referring to the $\sm$ operator in the original definitions. Also, we extend the FLP stable model semantics to allow generalized quantifiers by referring to an operator that is similar to the $\sm$ operator. For a reasonable syntactic class of logic programs, we show that the two stable model semantics of generalized quantifiers are interchangeable.'
author:
- Joohyung Lee and Yunsong Meng
title: Two New Definitions of Stable Models of Logic Programs with Generalized Quantifiers
---
Introduction
============
Most versions of the stable model semantics involve grounding. For instance, according to the FLP semantics from [@fab04; @faber11semantics], assuming that the domain is $\{-1,1,2\}$, program p(2) & & x:p(x) -1\
p(1) && p(-1) is identified with its ground instance w.r.t the domain: p(2) & & {-1:p(-1), 1:p(1), 2:p(2)} -1\
p(1) && p(-1) . As described in [@fab04], it is straightforward to extend the definition of satisfaction to ground aggregate expressions. For instance, set $\{p(-1),p(1)\}$ does not satisfy the body of the first rule of , but satisfies the bodies of the other rules. The FLP reduct of program relative to $\{p(-1),p(1)\}$ consists of the last two rules, and $\{p(-1), p(1)\}$ is its minimal model. Indeed, $\{p(-1), p(1)\}$ is the only FLP answer set of program .
On the other hand, according to the semantics from [@fer05], program is identified with some complex propositional formula containing nested implications: $$\ba l
\Big(\neg \big((p(2)\!\rar\! p(-1)\!\lor\! p(1))
\land (p(1)\!\land\! p(2)\!\rar\! p(-1))
\land (p(-1)\!\land\! p(1)\!\land\! p(2)\!\rar\!\bot)\big)\rar p(2)\Big) \\
\land\ \Big(\big(p(-1)\!\rar\! p(1)\!\lor\! p(2)\big)\rar p(-1)\Big) \\
\land\ \Big(p(-1)\rar p(1)\Big)\ .
\ea$$ Under the stable model semantics of propositional formulas [@fer05], this formula has two answer sets: $\{p(-1),p(1)\}$ and $\{p(-1),p(1),p(2)\}$. The relationship between the FLP and the Ferraris semantics was studied in [@lee09; @bartholomew11first-order].
Unlike the FLP semantics, the definition from [@fer05] is not applicable when the domain is infinite because it would require the representation of an aggregate expression to involve “infinite” conjunctions and disjunctions. This limitation was overcome in the semantics presented in [@lee09; @ferr10], which extends the first-order stable model semantics from [@fer07a; @ferraris11stable] to incorporate aggregate expressions. Recently, it was further extended to formulas involving generalized quantifiers [@lee12stable1], which provides a unifying framework of various extensions of the stable model semantics, including programs with aggregates, programs with abstract constraint atoms [@mare04], and programs with nonmonotonic dl-atoms [@eiter08combining].
In this paper, we revisit the first-order stable model semantics and its extension to incorporate generalized quantifiers. We provide an alternative, equivalent definition of a stable model by referring to grounding and reduct instead of the $\sm$ operator. Our work is inspired by the work of Truszczynski \[[-@truszczynski12connecting]\], who introduces infinite conjunctions and disjunctions to account for grounding quantified sentences. Our definition of a stable model can be viewed as a reformulation and a further generalization of his definition to incorporate generalized quantifiers. We define grounding in the same way as done in the FLP semantics, but define a reduct differently so that the semantics agrees with the one by Ferraris \[[-@fer05]\]. As we explain in Section \[sec:reduct\], our reduct of program relative to $\{p(-1),p(1)\}$ is & &\
p(-1) && {-1:p(-1), 1:p(1), 2:)} >-1\
p(1) && p(-1) , which is the program obtained from (\[ex1-ground\]) by replacing each maximal subformula that is not satisfied by $\{p(-1),p(1)\}$ with $\bot$. Set $\{p(-1), p(1)\}$ is an answer set of program (\[ex1\]) as it is a minimal model of the reduct. Likewise the reduct relative to $\{p(-1),p(1),p(2)\}$ is $$\ba {rcl}
p(2) &\ \ar\ & \top \\
p(-1) &\ar& \sum\{\langle -1\!:\!p(-1), 1\!:\!p(1), 2\!:\!p(2)\rangle\} \!>\!-1 \\
p(1) &\ar& p(-1)
\ea$$ and $\{p(-1),p(1),p(2)\}$ is a minimal model of the program. The semantics is more direct than the one from [@fer05] as it does not involve the complex translation into a propositional formula.
While the FLP semantics in [@fab04] was defined in the context of logic programs with aggregates, it can be straightforwardly extended to allow other “complex atoms.” Indeed, the FLP reduct is the basis of the semantics of HEX programs [@eite05]. In [@fink10alogical], the FLP reduct was applied to provide a semantics of nonmonotonic dl-programs [@eiter08combining]. In [@bartholomew11first-order], the FLP semantics of logic programs with aggregates was generalized to the first-order level. That semantics is defined in terms of the $\flp$ operator, which is similar to the $\sm$ operator. This paper further extends the definition to allow generalized quantifiers. By providing an alternative definition in the way that the other semantics was defined, this paper provides a useful insight into the relationship between the first-order stable model semantics and the FLP stable model semantics for programs with generalized quantifiers. While the two semantics behave differently in the general case, we show that they coincide on some reasonable syntactic class of logic programs. This implies that an implementation of one of the semantics can be viewed as an implementation of the other semantics if we limit attention to that class of logic programs.
The paper is organized as follows. Section \[sec:fosm\] reviews the first-order stable model semantics and its equivalent definition in terms of grounding and reduct, and Section \[sec:smgq\] extends that definition to incorporate generalized quantifiers. Section \[sec:flp\] provides an alternative definition of the FLP semantics with generalized quantifiers via a translation into second-order formulas. Section \[sec:comparison\] compares the FLP semantics and the first-order stable model semantics in the general context of programs with generalized quantifiers.
First-Order Stable Model Semantics {#sec:fosm}
==================================
Review of First-Order Stable Model Semantics {#ssec:review-fosm}
--------------------------------------------
This review follows [@ferraris11stable], a journal version of [@fer07a], which distinguishes between intensional and non-intensional predicates.
A [*formula*]{} is defined the same as in first-order logic. A [ *signature*]{} consists of [*function constants*]{} and [ *predicate constants*]{}. Function constants of arity $0$ are also called [*object constants*]{}. We assume the following set of primitive propositional connectives and quantifiers: $$\bot, \top,\ \land,\ \lor,\ \rar,\ \forall,\ \exists\ .$$ $\neg F$ is an abbreviation of $F\rar\bot$, and $F\lrar G$ stands for $(F\rar G)\land(G\rar F)$. We distinguish between atoms and atomic formulas as follows: an [*atom*]{} of a signature $\sigma$ is an $n$-ary predicate constant followed by a list of $n$ terms that can be formed from function constants in $\sigma$ and object variables; [*atomic formulas*]{} of $\sigma$ are atoms of $\sigma$, equalities between terms of $\sigma$, and the 0-place connectives $\bot$ and $\top$.
The stable models of $F$ relative to a list of predicates ${\bf p} =
(p_1,\dots,p_n)$ are defined via the [*stable model operator with the intensional predicates ${\bf p}$*]{}, denoted by $\sm[F; {\bf
p}]$.[^1] Let ${\bf u}$ be a list of distinct predicate variables $u_1,\dots,u_n$. By ${\bf u}={\bf p}$ we denote the conjunction of the formulas $\forall {\bf x}(u_i({\bf x})\lrar p_i({\bf x}))$, where ${\bf x}$ is a list of distinct object variables of the same length as the arity of $p_i$, for all $i=1,\dots, n$. By ${\bf u}\leq{\bf p}$ we denote the conjunction of the formulas $\forall {\bf x}(u_i({\bf x})\rar p_i({\bf x}))$ for all $i=1,\dots, n$, and ${\bf u}<{\bf p}$ stands for $({\bf u}\leq{\bf p})\land\neg({\bf u}={\bf p})$. For any first-order sentence $F$, expression $\sm[F;{\bf p}]$ stands for the second-order sentence $$F \land \neg \exists {\bf u} (({\bf u}<{\bf p}) \land F^*({\bf u})),$$ where $F^*({\bf u})$ is defined recursively:
- $p_i({\bf t})^* = u_i({\bf t})$ for any list ${\bf t}$ of terms;
- $F^* = F$ for any atomic formula $F$ that does not contain members of ${\bf p}$;
- $(F\land G)^* = F^* \land G^*$;
- $(F\lor G)^* = F^* \lor G^*$;
- $(F\rar G)^* = (F^* \rar G^*)\land (F \rar G)$;
- $(\forall xF)^* = \forall xF^*$;
- $(\exists xF)^* = \exists xF^*$.
A model of a sentence $F$ (in the sense of first-order logic) is called [*${\bf p}$-stable*]{} if it satisfies $\sm[F; {\bf p}]$.
\[ex:2\] Let $F$ be sentence $\forall x (\neg p(x)\rar q(x))$, and let $I$ be an interpretation whose universe is the set of all nonnegative integers ${\bf N}$, and $p^I(n) = \false$, $q^I(n)=\true$ for all $n\in {\bf
N}$. Section 2.4 of [@ferraris11stable] tells us that $I$ satisfies $\sm[F; pq]$.
Alternative Definition of First-Order Stable Models via Reduct {#ssec:reduct-fosm}
--------------------------------------------------------------
For any signature $\sigma$ and its interpretation $I$, by $\sigma^{I}$ we mean the signature obtained from $\sigma$ by adding new object constants $\xi^\dia$, called [*object names*]{}, for every element $\xi$ in the universe of $I$. We identify an interpretation $I$ of $\sigma$ with its extension to $\sigma^{I}$ defined by $I(\xi^\dia) = \xi$.
In order to facilitate defining a reduct, we provide a reformulation of the standard semantics of first-order logic via “a ground formula w.r.t. an interpretation.”
\[def:gr-formula\] For any interpretation $I$ of a signature $\sigma$, a [*ground formula w.r.t. $I$*]{} is defined recursively as follows.
- $p(\xi_1^\dia,\dots,\xi_n^\dia)$, where $p$ is a predicate constant of $\sigma$ and $\xi_i^\dia$ are object names of $\sigma^I$, is a ground formula w.r.t. $I$;
- $\top$ and $\bot$ are ground formulas w.r.t. $I$;
- If $F$ and $G$ are ground formulas w.r.t. $I$, then $F\land G$, $F\lor G$, $F\rar G$ are ground formulas w.r.t. $I$;
- If $S$ is a set of pairs of the form $\xi^\dia\!\!:\!F$ where $\xi^\dia$ is an object name in $\sigma^I$ and $F$ is a ground formula w.r.t. $I$, then $\forall (S)$ and $\exists (S)$ are ground formulas w.r.t. $I$.
The following definition describes a process that turns any first-order sentence into a ground formula w.r.t. an interpretation:
\[def:gr-fo\] Let $F$ be any first-order sentence of a signature $\sigma$, and let $I$ be an interpretation of $\sigma$ whose universe is $U$. By $gr_I[F]$ we denote the ground formula w.r.t. $I$, which is obtained by the following process:
- $gr_I[p(t_1,\dots,t_n)] = p((t_1^I)^\dia, \dots, (t_n^I)^\dia)$;
- $gr_I[t_1=t_2] =
\begin{cases}
\top & \text{ if $t_1^I=t_2^I$, and} \\
\bot & \text{otherwise};
\end{cases}
$
- $gr_I[\top] = \top$; $gr_I[\bot]=\bot$;
- $gr_I[F\odot G]= gr_I[F]\odot gr_I[G]\ \ \ \
(\odot\in\{\land,\lor,\rar\})$;
- $gr_I[Qx F(x)] = Q(\{\xi^\dia\!\!:\!gr_I[F(\xi^\dia)] \mid \xi\in U\})$ ($Q\in\{\forall, \exists\}$).
\[def:sat-fo\] For any interpretation $I$ and any ground formula $F$ w.r.t. $I$, the truth value of $F$ under $I$, denoted by $F^I$, is defined recursively as follows.
- $p(\xi_1^\dia,\dots,\xi_n^\dia)^I = p^I(\xi_1,\dots,\xi_n)$;
- $\top^I=\true$; $\bot^I=\false$;
- $(F\land G)^I=\true$ iff $F^I=\true$ and $G^I=\true$;
- $(F\lor G)^I=\true$ iff $F^I=\true$ or $G^I=\true$;
- $(F\rar G)^I=\true$ iff $G^I=\true$ whenever $F^I=\true$;
- $\forall (S)^I=\true$ iff the set $\{\xi \mid \xi^\dia\!\!:\!F(\xi^\dia)\in S \text{ and }
F(\xi^\dia)^I=\true\}$ is the same as the universe of $I$;
- $\exists (S)^I=\true$ iff the set $\{\xi \mid \xi^\dia\!\!:\!F(\xi^\dia)\in S \text{ and }
F(\xi^\dia)^I=\true\}$ is not empty.
We say that $I$ [*satisfies*]{} $F$, denoted $I\models F$, if $F^I=\true$.
[**Example \[ex:2\] continued (I)**]{}.
An interpretation $I$ of a signature $\sigma$ can be represented as a pair $\langle I^\mi{func},I^\mi{pred}\rangle$, where $I^\mi{func}$ is the restriction of $I$ to the function constants of $\sigma$, and $I^\mi{pred}$ is the set of atoms, formed using predicate constants from $\sigma$ and the object names from $\sigma^I$, which are satisfied by $I$. For example, interpretation $I$ in Example \[ex:2\] can be represented as $\langle I^\mi{func},\ \{q(n^\dia)\mid n\in {\bf N}\}\rangle$, where $I^\mi{func}$ maps each integer to itself.
The following proposition is immediate from the definitions:
\[fo-sat\] Let $\sigma$ be a signature that contains finitely many predicate constants, let $\sigma^\mi{pred}$ be the set of predicate constants in $\sigma$, let $I=\langle I^\mi{func},I^\mi{pred}\rangle$ be an interpretation of $\sigma$, and let $F$ be a first-order sentence of $\sigma$. Then $I\models F$ iff $I^\mi{pred}\models gr_I[F]$.
The introduction of the intermediate form of a ground formula w.r.t. an interpretation helps us define a reduct.
\[def:reduct-fosm\] For any ground formula $F$ w.r.t. $I$, the [*reduct*]{} of $F$ relative to $I$, denoted by $F^\mu{I}$, is obtained by replacing each maximal subformula that is not satisfied by $I$ with $\bot$. It can also be defined recursively as follows.
- $
(p(\xi_1^\dia,\dots,\xi_n^\dia))^\mu{I} =
\begin{cases}
p(\xi_1^\dia,\dots,\xi_n^\dia) & \text{if $I\models
p(\xi_1^\dia,\dots,\xi_n^\dia)$}, \\
\bot & \text{otherwise;}
\end{cases}
$
- $\top^\mu{I} = \top;\ \ \ \ \bot^\mu{I}=\bot$;
- $(F\odot G)^\mu{I} =
\begin{cases}
F^\mu{I}\odot G^\mu{I} & \text{if $I\models F\odot G$} \
\ \ \ (\odot\in\{\land,\lor,\rar\}), \\
\bot & \text{otherwise;}
\end{cases}
$
- $Q (S)^\mu{I} =
\begin{cases}
Q (\{\xi^\dia\!\!:\!(F(\xi^\dia))^\mu{I} \mid \xi^\dia\!\!:\!F(\xi^\dia)\in S\}) &
\text{if $I\models Q (S)$}
\ \ \ \ (Q\in\{\forall,\exists\}), \\
\bot & \text{otherwise.}
\end{cases}\\
$
The following theorem tells us how first-order stable models can be characterized in terms of grounding and reduct.
\[prop:ground-fosm\] Let $\sigma$ be a signature that contains finitely many predicate constants, let $\sigma^\mi{pred}$ be the set of predicate constants in $\sigma$, let $I=\langle I^\mi{func},I^\mi{pred}\rangle$ be an interpretation of $\sigma$, and let $F$ be a first-order sentence of $\sigma$. $I$ satisfies $\sm[F;\sigma^\mi{pred}]$ iff $I^\mi{pred}$ is a minimal set of atoms that satisfies $(\i{gr}_{I}[F])^\mu{I}$.
[**Example \[ex:2\] continued (II)**]{}.
Relation to Infinitary Formulas by Truszczynski
-----------------------------------------------
The definitions of grounding and a reduct in the previous section are inspired by the work of Truszczynski \[[-@truszczynski12connecting]\], where he introduces infinite conjunctions and disjunctions to account for the result of grounding $\forall$ and $\exists$ w.r.t. a given interpretation. Differences between the two approaches are illustrated in the following example:
Consider the formula $F=\forall x\ p(x)$ and the interpretation $I$ whose universe is the set of all nonnegative integers ${\bf
N}$. According to [@truszczynski12connecting], grounding of $F$ w.r.t. $I$ results in the infinitary propositional formula $$\{ p(n^\dia)\mid n\in N \}^{\land}\ . % \bigwedge_{i\in N} p(i^\dia).$$ On the other hand, formula $gr_I[F]$ is $$\forall (\{n^\dia\!\!:\!p(n^\dia)\mid n\in N\}).$$
Our definition of a reduct is essentially equivalent to the one defined in [@truszczynski12connecting]. In the next section, we extend our definition to incorporate generalized quantifiers.
Stable Models of Formulas with Generalized Quantifiers {#sec:smgq}
======================================================
Review: Formulas with Generalized Quantifiers
---------------------------------------------
We follow the definition of a formula with generalized quantifiers from [@wes08 Section 5] (that is to say, with Lindstr[ö]{}m quantifiers [@lindstrom66first] without the isomorphism closure condition).
We assume a set ${\bf Q}$ of symbols for generalized quantifiers. Each symbol in ${\bf Q}$ is associated with a tuple of nonnegative integers $\langle n_1,\dots,n_k \rangle$ ($k\ge 0$, and each $n_i$ is ), called the [*type*]{}. A [*(GQ-)formula (with the set ${\bf Q}$ of generalized quantifiers)*]{} is defined in a recursive way:
- an atomic formula (in the sense of first-order logic) is a GQ-formula;
- if $F_1,\dots, F_k$ ($k\ge 0$) are GQ-formulas and $Q$ is a generalized quantifier of type $\langle n_1,\dots,n_k\rangle$ in ${\bf Q}$, then Q\[\_1\]…\[\_k\] (F\_1(\_1),…,F\_k(\_k)) is a GQ-formula, where each $\bX_i$ ($1\le i\le k$) is a list of distinct object variables whose length is $n_i$.
We say that an occurrence of a variable $x$ in a GQ-formula $F$ is [*bound*]{} if it belongs to a subformula of $F$ that has the form $Q[\bX_1]\dots[\bX_k] (F_1(\bX_1),\dots,F_k(\bX_k))$ such that $x$ is in some $\bX_i$. Otherwise the occurrence is [*free*]{}. We say that $x$ is [*free*]{} in $F$ if $F$ contains a free occurrence of $x$. A [*(GQ-)sentence*]{} is a GQ-formula with no free variables.
We assume that ${\bf Q}$ contains type $\langle\rangle$ quantifiers $Q_\bot$ and $Q_\top$, type $\langle 0,0\rangle$ quantifiers $Q_\land,Q_\lor,Q_\rar$, and type $\langle 1\rangle$ quantifiers $Q_\forall,Q_\exists$. Each of them corresponds to the standard logical connectives and quantifiers — $\bot,\top,\land,\lor,\rar,\forall,\exists$. These generalized quantifiers will often be written in the familiar form. For example, we write $F\land G$ in place of $Q_\land[][](F,G)$, and write $\forall xF(x)$ in place of $Q_\forall[x] (F(x))$.
As in first-order logic, an interpretation $I$ consists of the universe $U$ and the evaluation of predicate constants and function constants. For each generalized quantifier $Q$ of type $\langle
n_1,\ldots,n_k\rangle$, $Q^U$ is a function from ${\cal P}(U^{n_1})\times\dots\times {\cal P}(U^{n_k})$ to $\{\true, \false\}$, where ${\cal P}(U^{n_i})$ denotes the power set of $U^{n_i}$.
\[e-1\] Besides the standard connectives and quantifiers, the following are some examples of generalized quantifiers.
- type $\langle 1\rangle$ quantifier $ Q_{\le 2} $ such that $Q_{\le 2}^{U}(R)=\true$ iff $|R|\le
2$; [^2]
- type $\langle 1\rangle $ quantifier $Q_{majority}$ such that $Q_{majority}^{U}(R)=\true$ iff $|R|> |U\setminus R|$;
- type $\langle 1,1\rangle$ quantifier $Q_{(\sum, <)}$ such that $Q_{(\sum, <)}^{U}(R_1,R_2) = \true$ iff
- $\sum(R_1)$ is defined,
- $R_2=\{b\}$, where $b$ is an integer, and
- $\sum(R_1)< b$.
Given a sentence $F$ of $\sigma^I$, $F^I$ is defined recursively as follows:
- $p(t_1,\dots,t_n)^I=p^I(t_1^I,\dots,t_n^I)$,
- $(t_1=t_2)^I= (t_1^I=t_2^I)$,
- For a generalized quantifier $Q$ of type $\langle n_1,\dots,n_k\rangle$, $$\ba l
(Q[\bX_1]\dots[\bX_k](F_1(\bX_1),\dots,F_k(\bX_k)))^I
= Q^{U}((\bX_1\!:\!F_1(\bX_1))^I,\dots,(\bX_k\!:\!F_k(\bX_k))^I),
\ea$$ where $(\bX_i\!:\!F_i(\bX_i))^I=\{\bfxi\in U^{n_i} \mid
(F_i(\bfxi^\dia))^I=\true \}$.
We assume that, for the standard logical connectives and quantifiers $Q$, functions $Q^{U}$ have the standard meaning:
[2]{}
- $Q_\forall^{U}(R)=\true$ iff $R=U$;
- $Q_\exists^{U}(R)=\true$ iff $R\cap U\ne\emptyset$;
- $Q_\land^{U}(R_1,R_2)=\true$ iff $R_1=R_2=\{\epsilon\}$;[^3]
- $Q_\lor^{U}(R_1,R_2) =\true$ iff $R_1=\{\epsilon\}$ or $R_2=\{\epsilon\}$;
- $Q_\rar^{U}(R_1,R_2) =\true$ iff $R_1$ is $\emptyset$ or $R_2$ is $\{\epsilon\}$;
- $Q_\bot^{U}() =\false$;
- $Q_\top^{U}() =\true$.
We say that an interpretation $I$ [*satisfies*]{} a GQ-sentence $F$, or is a [*model*]{} of $F$, and write $I\models F$, if $F^I = \true$. A GQ-sentence $F$ is [*logically valid*]{} if every interpretation satisfies $F$. A GQ-formula with free variables is said to be [ *logically valid*]{} if its universal closure is logically valid.
\[ex:main\] Program (\[ex1\]) in the introduction is identified with the following GQ-formula $F_1$: $$\ba l
(\neg Q_{(\sum, <)} [x][y] (p(x),\ y\mvis 2) \rar p(2))\\
\land ~(Q_{(\sum, >)} [x][y] (p(x),\ y\mvis -1) \rar p(-1))\\
\land ~( p(-1)\rar p(1)) \ .
\ea$$ Consider two Herbrand interpretations of the universe $U=\{-1,1,2\}$: $I_1=\{p(-1),p(1)\}$ and $I_2=\{p(-1),p(1), p(2)\}$. We have $(Q_{(\sum, <)} [x][y] (p(x),\ y=2))^{I_1}=\true$ since
- $(x:p(x))^{I_1}=\{-1,1\}$ and $(y: y\mvis 2)^{I_1}=\{2\}$;
- $Q_{(\sum, <)}^U(\{-1,1\}, \{2\})=\true$.
Similarly, $(Q_{(\sum, >)} [x][y] (p(x),\ y\mvis
-1))^{I_2}=\true$ since
- $(x:p(x))^{I_2}=\{-1,1,2\}$ and $(y:y\mvis -1)^{I_2}=\{-1\}$;
- $Q_{(\sum, >)}^U(\{-1,1,2\}, \{-1\})=\true$.
Consequently, both $I_1$ and $I_2$ satisfy $F_1$.
Review: $\sm$-Based Definition of Stable Models of GQ-Formulas {#ssec:smgq}
--------------------------------------------------------------
For any GQ-formula $F$ and any list of predicates ${\bf p}=(p_1,\dots,p_n)$, formula $\sm[F; {\bf p}]$ is defined as $$%beq
F\land\neg\exists {\bf u} (({\bf u}<{\bf p}) \land F^*({\bf u})),$$ where $F^*({\bf u})$ is defined recursively:
- $p_i({\bf t})^* = u_i({\bf t})$ for any list ${\bf t}$ of terms;
- $F^* = F$ for any atomic formula $F$ that does not contain members of ${\bf p}$;
- $$%\beq
\ba l
(Q[\bX_1]\dots[\bX_k] (F_1(\bX_1),\dots,F_k(\bX_k)))^* = \\
\hspace{3em} Q[\bX_1]\dots[\bX_k] (F_1^*(\bX_1),\dots,F_k^*(\bX_k))
\land\ Q[\bX_1]\dots[\bX_k] (F_1(\bX_1),\dots,F_k(\bX_k)).
\ea$$
When $F$ is a sentence, the models of $\sm[F; {\bf p}]$ are called the [*${\bf p}$-stable*]{} models of $F$: they are the models of $F$ that are “stable” on ${\bf p}$. We often simply write $\sm[F]$ in place of $\sm[F;{\bf p}]$ when ${\bf p}$ is the list of all predicate constants occurring in $F$, and call ${\bf p}$-stable models simply stable models.
As explained in [@lee12stable], this definition of a stable model is a proper generalization of the first-order stable model semantics.
[**Example \[ex:main\] continued (I)**]{}.
Reduct-Based Definition of Stable Models of GQ-Formulas {#sec:reduct}
-------------------------------------------------------
The reduct-based definition of stable models presented in Section \[ssec:reduct-fosm\] can be extended to GQ-formulas as follows.
Let $I$ be an interpretation of a signature $\sigma$. As before, we assume a set ${\bf Q}$ of generalized quantifiers, which contains all propositional connectives and standard quantifiers.
\[def:gr-gq-formula\] A ground GQ-formula w.r.t. $I$ is defined recursively as follows:
- $p(\xi_1^\dia,\dots,\xi_n^\dia)$, where $p$ is a predicate constant of $\sigma$ and $\xi_i^\dia$ are object names of $\sigma^I$, is a ground GQ-formula w.r.t. $I$;
- for any $Q\in {\bf Q}$ of type $\langle n_1,\ldots,n_k\rangle$, if each $S_i$ is a set of pairs of the form $\bfxi^\dia\!\!:\!F$ where $\bfxi^\dia$ is a list of object names from $\sigma^I$ whose length is $n_i$ and $F$ is a ground GQ-formula w.r.t. $I$, then $$Q (S_1,\dots,S_k)$$ is a ground GQ-formula w.r.t. $I$.
The following definition of grounding turns any GQ-sentence into a ground GQ-formula w.r.t. an interpretation:
\[def:gr-gq\] Let $F$ be a GQ-sentence of a signature $\sigma$, and let $I$ be an interpretation of $\sigma$. By $gr_I[F]$ we denote the ground GQ-formula w.r.t. $I$ that is obtained by the process similar to the one in Definition \[def:gr-fo\] except that the last two clauses are replaced by the following single clause:
- $
gr_I[Q[\bX_1]\dots[\bX_k] (F_1(\bX_1),\dots,F_k(\bX_k))] = Q (S_1,\dots,S_k)
$
where $S_i=\{\bfxi^\dia\!\!:\!gr_I[F_i(\bfxi^\dia)]\mid \bfxi^\dia \text{ is a list of
object names from $\sigma^I$ whose length is $n_i$} \}$.
For any interpretation $I$ and any ground GQ-formula $F$ w.r.t. $I$, the satisfaction relation $I\models F$ is defined recursively as follows.
\[def:sat-gq\] For any interpretation $I$ and any ground GQ-formula $F$ w.r.t. $I$, the satisfaction relation $I\models F$ is defined similar to Definition \[def:sat-fo\] except that the last five clauses are replaced by the following single clause:
- $Q(S_1,\dots,S_k)^I = Q^U(S_1^I,\dots,S_k^I)$ where $S_i^I=\{\bfxi \mid \bfxi^\dia\!\!:\!F(\bfxi^\dia)\in S_i,\ F(\bfxi^\dia)^I =\true \}$.
[**Example \[ex:main\] continued (II)**]{}.
\[prop:gq-sat\] Let $\sigma$ be a signature that contains finitely many predicate constants, let $\sigma^\mi{pred}$ be the set of predicate constants in $\sigma$, let $I=\langle I^\mi{func},I^\mi{pred}\rangle$ be an interpretation of $\sigma$, and let $F$ be a GQ-sentence of $\sigma$. Then $I\models F$ iff $I^\mi{pred}\models gr_{I}[F]$.
For any GQ-formula $F$ w.r.t. $I$, the reduct of $F$ relative to $I$, denoted by $F^\mu{I}$, is defined in the same way as in Definition \[def:reduct-fosm\] by replacing the last two clauses with the following single clause:
- $(Q (S_1,\dots,S_k))^\mu{I} =
\begin{cases}
Q (S_1^\mu{I},\dots,S_k^\mu{I}) & \text{if
$I\models Q (S_1,\dots,S_k)$}, \\
\bot & \text{otherwise;}
\end{cases}\\
$ where $S_i^\mu{I}=\{\bfxi^\dia\!\!:\!(F(\bfxi^\dia))^\mu{I}\mid
\bfxi^\dia\!\!:\!F(\bfxi^\dia)\in S_i\}$.
\[prop:ground-smgq\] Let $\sigma$ be a signature that contains finitely many predicate constants, let $\sigma^\mi{pred}$ be the set of predicate constants in $\sigma$, let $I=\langle I^\mi{func},I^\mi{pred}\rangle$ be an interpretation of $\sigma$, and let $F$ be a GQ-sentence of $\sigma$. $I\models \sm[F;\sigma^\mi{pred}]$ iff $I^\mi{pred}$ is a minimal set of atoms that satisfies $(\i{gr}_{I}[F])^\mu{I}$ .
[**Example \[ex:main\] continued (III).**]{}
Extending Theorem \[prop:ground-smgq\] to allow an arbitrary list of intensional predicates, rather than $\sigma^\mi{pred}$, is straightforward in view of Proposition 1 from [@lee12reformulating].
FLP Semantics of Programs with Generalized Quantifiers {#sec:flp}
======================================================
The FLP stable model semantics [@fab04] is an alternative way to define stable models. It is the basis of HEX programs, an extension of the stable model semantics with higher-order and external atoms, which is implemented in system [dlv-hex]{}. The first-order generalization of the FLP stable model semantics for programs with aggregates was given in [@bartholomew11first-order], using the $\flp$ operator that is similar to the $\sm$ operator. In this section we show how it can be extended to allow generalized quantifiers.
FLP Semantics of Programs with Generalized Quantifiers {#flp-semantics-of-programs-with-generalized-quantifiers}
------------------------------------------------------
A [*(general) rule*]{} is of the form HB where $H$ and $B$ are arbitrary GQ-formulas. A [*(general) program*]{} is a finite set of rules.
Let ${\bf p}$ be a list of distinct predicate constants $p_1,\dots,p_n$, and let ${\bf u}$ be a list of distinct predicate variables $u_1,\dots,u_n$. For any formula $G$, formula $G({\bf u})$ is obtained from $G$ by replacing all occurrences of predicates from ${\bf p}$ with the corresponding predicate variables from ${\bf u}$.
Let $\Pi$ be a finite program whose rules have the form (\[rule-f\]). The [*GQ-representation*]{} $\Pi^{GQ}$ of $\Pi$ is the conjunction of the universal closures of $B\rar H$ for all rules (\[rule-f\]) in $\Pi$. By $\flp[\Pi; {\bf p}]$ we denote the second-order formula $$\Pi^{GQ}\land\neg\exists {\bf u}({\bf u}<{\bf p}\land
\Pi^\tri({\bf u}))
% \bigwedge_{H\!\ar\! B\in \Pi}
% \Big(\neg B\lor (B({\bf u})\!\rar\! H({\bf u}))\Big).
% \Pi^\mathit{FOL}\land\neg\exists {\bf u}(({\bf u}<{\bf p})\land F^\tri({\bf u}))$$ where $\Pi^\tri({\bf u})$ is defined as the conjunction of the universal closures of $$B\land B({\bf u})\!\rar\! H({\bf u})$$ for all rules $H\ar B$ in $\Pi$.
We will often simply write $\flp[\Pi]$ instead of $\flp[\Pi;{\bf p}]$ when ${\bf p}$ is the list of all predicate constants occurring in $\Pi$, and call a model of $\flp[\Pi]$ an [*FLP-stable*]{} model of $\Pi$.
[**Example \[ex:main\] continued (IV).**]{}
Comparing the FLP Semantics and the First-Order Stable Model Semantics {#sec:comparison}
======================================================================
In this section, we show a class of programs with GQs for which the FLP semantics and the first-order stable model semantics coincide.
The following definition is from [@lee12stable]. We say that a generalized quantifier $Q$ is [ *monote in the $i$-th argument position*]{} if the following holds for any universe $U$: if $Q^{U}(R_1,\dots,R_k)=\true$ and $R_i\subseteq R_i' \subseteq U^{n_i}$, then $$Q^{U}(R_1,\dots,R_{i-1},R_i',R_{i+1},\dots,R_k)=\true.$$
Consider a program $\Pi$ consisting of rules of the form $$A_1;\dots; A_l \ar\ E_1,\dots,E_m, \no\ E_{m+1},\dots,\no\ E_n$$ ($l\ge 0$; $n\ge m\ge 0$), where each $A_i$ is an atomic formula and each $E_i$ is an atomic formula or a GQ-formula such that all $F_1(\bX_1),\dots,F_k(\bX_k)$ are atomic formulas. Furthermore we require that, for every GQ-formula in one of $E_{m+1},\dots, E_n$, $Q$ is monotone in all its argument positions.
\[cor:flp-sm-hex\] Let $\Pi$ be a program whose syntax is described as above, and let $F$ be the GQ-representation of $\Pi$. Then $\flp[\Pi; {\bf p}]$ is equivalent to $\sm[F; {\bf p}]$.
Consider the following one-rule program: p(a) Q\_[0]{} \[x\] p(x) . This program does not belong to the syntactic class of programs stated in Proposition \[cor:flp-sm-hex\] since $Q_{\le 0} [x]\ p(x)$ is not monotone in $\{1\}$. Indeed, both $\emptyset$ and $\{p(a)\}$ satisfy $\sm[\Pi; p]$, but only $\emptyset$ satisfies $\flp[\Pi; p]$.
Conditions under which the FLP semantics coincides with the first-order stable model semantics has been studied in [@lee09; @bartholomew11first-order] in the context of logic programs with aggregates.
Conclusion
==========
We introduced two definitions of a stable model. One is a reformulation of the first-order stable model semantics and its extension to allow generalized quantifiers by referring to grounding and reduct, and the other is a reformulation of the FLP semantics and its extension to allow generalized quantifiers by referring to a translation into second-order logic. These new definitions help us understand the relationship between the FLP semantics and the first-order stable model semantics, and their extensions. For the class of programs where the two semantics coincide, system [dlv-hex]{} can be viewed as an implementation of the stable model semantics of GQ-formulas; A recent extension of system [ f2lp]{} [@lee09a] to allow “complex” atoms may be considered as a front-end to [dlv-hex]{} to implement the generalized FLP semantics.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to Vladimir Lifschitz for useful discussions related to this paper. We are also grateful to Joseph Babb and the anonymous referees for their useful comments. This work was partially supported by the National Science Foundation under Grant IIS-0916116 and by the South Korea IT R&D program MKE/KIAT 2010-TD-300404-001.
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[^1]: The intensional predicates ${\bf p}$ are the predicates that we “intend to characterize” by $F$.
[^2]: It is clear from the type of the quantifier that $R$ is any subset of $U$. We will skip such explanation.
[^3]: $\epsilon$ denotes the empty tuple. For any interpretation $I$, $U^0=\{\epsilon\}$. For $I$ to satisfy $Q_{\land}[][](F, G)$, both $(\epsilon\!:\!F)^I$ and $(\epsilon\!:\!G)^I$ have to be $\{\epsilon\}$, which means that $F^I=G^I=\true$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Let $R$ be a regular local ring, containing [**a finite field**]{}. Let ${\mathbf G}$ be a reductive group scheme over $R$. We prove that a principal ${\mathbf G}$-bundle over $R$ is trivial, if it is trivial over the fraction field of $R$. In other words, if $K$ is the fraction field of $R$, then the map of non-abelian cohomology pointed sets $$H^1_{\text{\'et}}(R,{\mathbf G})\to H^1_{\text{\'et}}(K,{\mathbf G}),$$ induced by the inclusion of $R$ into $K$, has a trivial kernel.
Certain arguments used in the present preprint do not work if the ring $R$ contains a characteristic zero field. In that case and, more generally, in the case when the regular local ring $R$ contains [**an infinite field**]{} this result is proved in [@FP].
address: 'Steklov Institute of Mathematics at St.-Petersburg, Fontanka 27, St.-Petersburg 191023, Russia'
author:
- Ivan Panin
title: 'Proof of Grothendieck-Serre conjecture on principal bundles over regular local rings containing a finite field'
---
Introduction
============
Assume that $U$ is a regular scheme, ${\mathbf G}$ is a reductive $U$-group scheme. Recall that a $U$-scheme ${\mathcal G}$ with an action of ${\mathbf G}$ is called *a principal ${\mathbf G}$-bundle over $U$*, if ${\mathcal G}$ is faithfully flat and quasi-compact over $U$ and the action is simple transitive, that is, the natural morphism ${\mathbf G}\times_U{\mathcal G}\to{\mathcal G}\times_U{\mathcal G}$ is an isomorphism, see [@FGA Section 6]. It is well known that such a bundle is trivial locally in étale topology but in general not in Zariski topology. Grothendieck and Serre conjectured that ${\mathcal G}$ is trivial locally in Zariski topology, if it is trivial generically. More precisely
Let $R$ be a regular local ring, let $K$ be its field of fractions. Let ${\mathbf G}$ be a reductive group scheme over $U:=\operatorname{Spec}R$, let ${\mathcal G}$ be a principal ${\mathbf G}$-bundle. If ${\mathcal G}$ is trivial over $\operatorname{Spec}K$, then it is trivial. Equivalently, the map of non-abelian cohomology pointed sets $$H^1_{\text{\'et}}(R,{\mathbf G})\to H^1_{\text{\'et}}(K,{\mathbf G}),$$ induced by the inclusion of $R$ into $K$, has a trivial kernel.
The main result of this paper is a proof of this conjecture for regular semi-local domains $R$, containing [**a finite field**]{}. Our proof was inspired by the preprint \[FP\], where the conjecture is proven for semi-local regular domains containing an [**infinite**]{} field. [*Thus, the conjecture holds for semi-local regular domains containing a field*]{}.
The proof in the present preprint uses [@Pan1 Thm.1.1], [@Pan2 Thm.1.0.1], the key ideas of the paper [@FP] and a Bertini type theorem from [@Poo]. Our result implies that two principal ${\mathbf G}$-bundles over $U$ are isomorphic, if they are isomorphic over $\operatorname{Spec}K$ as proved in the next section. This result is new even for constant group schemes (that is, for group schemes coming from the ground field).
Recall that a part of the Gersten conjecture asserts that the natural homomorphism of $\mathrm K$-groups $\mathrm K_q(R)\to\mathrm K_q(K)$ is injective. Very roughly speaking, the Grothendieck–Serre conjecture is a non-abelian version of this part of the Gersten conjecture.
History of the topic
--------------------
Here is a list of known results in the same vein, corroborating the Grothendieck–Serre conjecture.
$\bullet$ The case, where the group scheme ${\mathbf G}$ comes from an infinite ground field, is completely solved by J.-L. Colliot-Thélène, M. Ojanguren, and M. S. Raghunatan in [@C-TO] and [@R1; @R2]; O. Gabber announced a proof for group schemes coming from arbitrary ground fields.
$\bullet$ The case of an arbitrary reductive group scheme over a discrete valuation ring or over a henselian ring is completely solved by Y. Nisnevich in [@Ni1]. He also proved the conjecture for two-dimensional local rings in the case, when ${\mathbf G}$ is quasi-split in [@Ni2].
$\bullet$ The case, where ${\mathbf G}$ is an arbitrary reductive group scheme over a regular semi-local domain containing an infinite field, was settled by R. Fedorov and I. Panin in [@FP].
$\bullet$ The case, where ${\mathbf G}$ is an arbitrary torus over a regular local ring, was settled by J.-L. Colliot-Thélène and J.-J. Sansuc in [@C-T-S].
$\bullet$ For some simple group schemes of classical series the conjecture is solved in works of the author, A. Suslin, M. Ojanguren, and K. Zainoulline; see [@Oj1], [@Oj2], [@PS], [@OP2], [@Z], [@OPZ].
$\bullet$ Under an isotropy condition on ${\mathbf G}$ and assuming that the ring contains an infinite field the conjecture is proved in a series of preprints [@PSV] and [@Pa2].
$\bullet$ The case of strongly inner simple adjoint group schemes of the types $E_6$ and $E_7$ is done by the second author, V. Petrov, and A. Stavrova in [@PPS]. No isotropy condition is imposed there, however it is supposed that the ring contains an infinite field.
$\bullet$ The case, when ${\mathbf G}$ is of the type $F_4$ with trivial $g_3$-invariant and the field is of characteristic zero, is settled by V. Chernousov in [@Chernous]; the case, when ${\mathbf G}$ is of the type $F_4$ with trivial $f_3$-invariant and the field is infinite and perfect, is settled by V. Petrov and A. Stavrova in [@PetrovStavrova].
Acknowledgments
---------------
The author thanks A.Suslin for his interest to the topic of the present preprint.
Main results {#Introduction}
============
Let $R$ be a commutative unital ring. Recall that an $R$-group scheme ${\mathbf G}$ is called *reductive*, if it is affine and smooth as an $R$-scheme and if, moreover, for each algebraically closed field $\Omega$ and for each ring homomorphism $R\to\Omega$ the scalar extension ${\mathbf G}_\Omega$ is a connected reductive algebraic group over $\Omega$. This definition of a reductive $R$-group scheme coincides with [@SGA3-3 Exp. XIX, Definition 2.7]. A well-known conjecture due to J.-P. Serre and A. Grothendieck (see [@Se Remarque, p.31], [@Gr1 Remarque 3, p.26-27], and [@Gr2 Remarque 1.11.a]) asserts that given a regular local ring $R$ and its field of fractions $K$ and given a reductive group scheme ${\mathbf G}$ over $R$, the map $$H^1_{\text{\'et}}(R,{\mathbf G})\to H^1_{\text{\'et}}(K,{\mathbf G}),$$ induced by the inclusion of $R$ into $K$, has a trivial kernel. The following theorem, which is the main result of the present paper, asserts that this conjecture holds, provided that $R$ contains [**a finite field**]{}. If $R$ contains an infinite field, then the conjecture is proved in \[FP\].
\[MainThm1\] Let $R$ be a regular semi-local domain containing [**a finite field**]{}, and let $K$ be its field of fractions. Let ${\mathbf G}$ be a reductive group scheme over $R$. Then the map $$H^1_{\text{\'et}}(R,{\mathbf G})\to H^1_{\text{\'et}}(K,{\mathbf G}),$$ induced by the inclusion of $R$ into $K$, has a trivial kernel. In other words, under the above assumptions on $R$ and ${\mathbf G}$, each principal ${\mathbf G}$-bundle over $R$ having a $K$-rational point is trivial.
Theorem \[MainThm1\] has the following
Under the hypothesis of Theorem \[MainThm1\], the map $$H^1_{\text{\'et}}(R,{\mathbf G})\to H^1_{\text{\'et}}(K,{\mathbf G}),$$ induced by the inclusion of $R$ into $K$, is injective. Equivalently, if ${\mathcal G}_1$ and ${\mathcal G}_2$ are two principal bundles isomorphic over $\operatorname{Spec}K$, then they are isomorphic.
Let ${\mathcal G}_1$ and ${\mathcal G}_2$ be two principal ${\mathbf G}$-bundles isomorphic over $\operatorname{Spec}K$. Let $\operatorname{Iso}({\mathcal G}_1,{\mathcal G}_2)$ be the scheme of isomorphisms. This scheme is a principal $\operatorname{Aut}{\mathcal G}_2$-bundle. By Theorem \[MainThm1\] it is trivial, and we see that ${\mathcal G}_1\cong{\mathcal G}_2$.
Note that, while Theorem \[MainThm1\] was previously known for reductive group schemes ${\mathbf G}$ coming from the ground field (an unpublished result due to O.Gabber), in many cases the corollary is a new result even for such group schemes.
For a scheme $U$ we denote by ${\mathbb A}^1_U$ the affine line over $U$ and by ${\mathbb P}^1_U$ the projective line over $U$. Let $T$ be a $U$-scheme. By a principal ${\mathbf G}$-bundle over $T$ we understand a principal ${\mathbf G}\times_UT$-bundle.
In Section \[sect:redtopsv\] we deduce Theorem \[MainThm1\] from the following result of independent interest (cf. [@PSV Thm.1.3]).
\[th:psv\] Let $R$ be the semi-local ring of finitely many closed points on an irreducible smooth affine variety over [**a finite field**]{} $k$, set $U=\operatorname{Spec}R$. Let ${\mathbf G}$ be a simple simply-connected group scheme over $U$ (see [@SGA3-3 Exp. XXIV, Sect. 5.3] for the definition). Let ${\mathcal E}_t$ be a principal ${\mathbf G}$-bundle over the affine line ${\mathbb A}^1_U=\operatorname{Spec}R[t]$, and let $h(t)\in R[t]$ be a monic polynomial. Denote by $({\mathbb A}^1_U)_h$ the open subscheme in ${\mathbb A}^1_U$ given by $h(t)\ne0$ and assume that the restriction of ${\mathcal E}_t$ to $({\mathbb A}^1_U)_h$ is a trivial principal ${\mathbf G}$-bundle. Then for each section $s:U\to{\mathbb A}^1_U$ of the projection ${\mathbb A}^1_U\to U$ the ${\mathbf G}$-bundle $s^*{\mathcal E}_t$ over $U$ is trivial.
The derivation of Theorem \[MainThm1\] from Theorem \[th:psv\] is based on [@Pan2 Thm.1.0.1] and [@Pan1 Thm.1.1].
Let $Y$ be a semi-local scheme. We will call a simple $Y$-group scheme quasi-split if its restriction to each connected component of $Y$ contains [**a Borel subgroup scheme**]{}.
\[MainThm2\] Let $R$, $U$, and ${\mathbf G}$ be as in Theorem \[th:psv\]. Let $Z\subset{\mathbb P}^1_U$ be a closed subscheme finite over $U$. Let $Y\subset{\mathbb P}^1_U$ be a closed subscheme finite and étale over $U$ and such that\
(i) ${\mathbf G}_Y:={\mathbf G}\times_UY$ is quasi-split,\
(ii) $Y\cap Z={\varnothing}$ and $Y \cap \{\infty\}\times U={\varnothing}= Z \cap \{\infty\}\times U$,\
(iii) for any closed point $u \in U$ one has $Pic({\mathbb P}^1_u - Y_u)=0$, where $Y_u:={\mathbb P}^1_u\cap Y$.\
Let ${\mathcal G}$ be a principal ${\mathbf G}$-bundle over ${\mathbb P}^1_U$ such that its restriction to ${\mathbb P}^1_U- Z$ is trivial. Then the restriction of ${\mathcal G}$ to ${\mathbb P}^1_U-Y$ is also trivial.\
In particular, the principal ${\mathbf G}$-bundle ${\mathcal G}$ is trivial locally for the Zarisky topology.
The proof of this result is inspired by [@FP Thm.3].
Organization of the paper
-------------------------
In Section \[sect:redtopsv\], we reduce Theorem \[MainThm1\] to Theorem \[th:psv\]. In Section \[sect:reducing\], we reduce Theorem \[th:psv\] to Theorem \[MainThm2\]. This reduction is based on [@Pan2 Thm.1.0.1], [@Pan1 Thm.1.1], on a theorem of D. Popescu [@P] and on Proposition \[SchemeY\]. The latter proposition is a new ingredient comparing with respecting arguments from [@FP Section 4].
In Section \[sect:proof2\] we prove Theorem \[MainThm2\]. We give an outline of the proof in Section \[sect:outline\]. We use the technique of henselization. In Section \[sect:application\] we give an application of Theorem \[MainThm1\].
In the Appendix we recall the definition of henselization from [@Gabber Section 0].
Reducing Theorem \[MainThm1\] to Theorem \[th:psv\] {#sect:redtopsv}
===================================================
In what follows “${\mathbf G}$-bundle” always means “principal ${\mathbf G}$-bundle”. Now we assume that Theorem \[th:psv\] holds. We start with the following particular case of Theorem \[MainThm1\].
\[pr:geometric\] Let $R$, $U=\operatorname{Spec}R$, and ${\mathbf G}$ be as in Theorem \[th:psv\]. Let ${\mathcal E}$ be a principal ${\mathbf G}$-bundle over $U$, trivial at the generic point of $U$. Then ${\mathcal E}$ is trivial.
Under the hypothesis of the proposition, the following data are constructed in [@Pan1 Thm.1.1]:\
[[)]{} ]{}a principal ${\mathbf G}$-bundle ${\mathcal E}_t$ over ${\mathbb A}^1_U$;\
[[)]{} ]{}a monic polynomial $h(t)\in R[t]$.\
Moreover these data satisfies the following conditions:\
(1) the restriction of ${\mathcal E}_t$ to $({\mathbb A}^1_U)_h$ is a trivial principal ${\mathbf G}$-bundle;\
(2) there is a section $s:U\to{\mathbb A}^1_U$ such that $s^*{\mathcal E}_t={\mathcal E}$.
Now it follows from Theorem \[th:psv\] that ${\mathcal E}$ is trivial.
\[pr:reductivegeometric\] Let $U$ be as in Theorem \[th:psv\]. Let ${\mathbf G}$ be a reductive group scheme over $U$. Let ${\mathcal E}$ be a principal ${\mathbf G}$-bundle over $U$ trivial at the generic point of $U$. Then ${\mathcal E}$ is trivial.
Firstly, using [@Pan2 Thm.1.0.1], we can assume that ${\mathbf G}$ is semi-simple and simply-connected. Secondly, standard arguments (see for instance [@PSV Section 9]) show that we can assume that ${\mathbf G}$ is simple and simply-connected. (Note that for this reduction it is necessary to work with semi-local rings.) Now the proposition is reduced to Proposition \[pr:geometric\].
Let us prove a general statement first. Let $k'$ be [**a finite field**]{}, $X$ be a $k'$-smooth irreducible affine variety, ${\mathbf H}$ be a reductive group scheme over $X$. Denote by $k'[X]$ the ring of regular functions on $X$ and by $k'(X)$ the field of rational functions on $X$. Let ${\mathcal H}$ be a principal ${\mathbf H}$-bundle over $X$ trivial over $k'(X)$. Let $\mathfrak p_1,\dots,\mathfrak p_n$ be prime ideals in $k'[X]$, and let ${\mathcal O}_{\mathfrak p_1,\dots,\mathfrak p_n}$ be the corresponding semi-local ring.
\[lm:primemax\] The principal ${\mathbf H}$-bundle ${\mathcal H}$ is trivial over ${\mathcal O}_{\mathfrak p_1,\dots,\mathfrak p_n}$.
For each $i=1,2,\ldots,n$ choose a maximal ideal $\mathfrak m_i\subset k'[X]$ containing $\mathfrak p_i$. One has inclusions of $k'$-algebras $${\mathcal O}_{\mathfrak m_1,\dots,\mathfrak m_n}\subset{\mathcal O}_{\mathfrak p_1,\dots,\mathfrak p_n}\subset k'(X).$$ By Proposition \[pr:reductivegeometric\] the principal ${\mathbf H}$-bundle ${\mathcal H}$ is trivial over ${\mathcal O}_{\mathfrak m_1,\dots,\mathfrak m_n}$. Thus it is trivial over ${\mathcal O}_{\mathfrak p_1,\dots,\mathfrak p_n}$.
Let us return to our situation. Let ${\mathfrak m}_1,\ldots,{\mathfrak m}_n$ be all the maximal ideals of $R$. Let ${\mathcal E}$ be a ${\mathbf G}$-bundle over $R$ trivial over the fraction field of $R$. Clearly, there is a non-zero $f\in R$ such that ${\mathcal E}$ is trivial over $R_f$. Let $k$ be the prime field of $R$. Note that $k$ is perfect. It follows from Popescu’s theorem ([@P; @Sw]) that $R$ is a filtered inductive limit of smooth $k$-algebras $R_\alpha$. Modifying the inductive system $R_\alpha$ if necessary, we can assume that each $R_\alpha$ is integral. There are an index $\alpha$, a reductive group scheme ${\mathbf G}_{\alpha}$ over $R_{\alpha}$, a principal ${\mathbf G}_{\alpha}$-bundle ${\mathcal E}_{\alpha}$ over $R_{\alpha}$, and an element $f_{\alpha }\in R_{\alpha}$ such that ${\mathbf G}={\mathbf G}_{\alpha}\times_{\operatorname{Spec}R_{\alpha}}\operatorname{Spec}R$, ${\mathcal E}$ is isomorphic to ${\mathcal E}_{\alpha}\times_{\operatorname{Spec}R_{\alpha}}\operatorname{Spec}R$ as principal ${\mathbf G}$-bundle, $f$ is the image of $f_{\alpha}$ under the homomorphism ${\varphi}_{\alpha}: R_{\alpha}\to R$, ${\mathcal E}_{\alpha}$ is trivial over $(R_{\alpha})_{f_{\alpha}}$.
For each maximal ideal $\mathfrak m_i$ in $R$ ($i=1,\dots, n$) set $\mathfrak p_i={\varphi}_{\alpha}^{-1}(\mathfrak m_i)$. The homomorphism ${\varphi}_\alpha$ induces a homomorphism of semi-local rings $(R_{\alpha})_{\mathfrak p_1,\dots,\mathfrak p_n}\to R$. By Lemma \[lm:primemax\] the principal ${\mathbf G}_{\alpha}$-bundle ${\mathcal E}_{\alpha}$ is trivial over $(R_{\alpha})_{\mathfrak p_1,\dots,\mathfrak p_n}$. Whence the ${\mathbf G}$-bundle ${\mathcal E}$ is trivial over $R$.
Reducing Theorem \[th:psv\] to Theorem \[MainThm2\] {#sect:reducing}
===================================================
Now we assume that Theorem \[MainThm2\] is true. Let $k$, $U$ and ${\mathbf G}$ be as in Theorem \[th:psv\]. Let $u_1,\ldots,u_n$ be all the closed points of $U$. Let $k(u_i)$ be the residue field of $u_i$. Consider the reduced closed subscheme ${\mathbf u}$ of $U$, whose points are $u_1$, …, $u_n$. Thus $${\mathbf u}\cong\coprod_i\operatorname{Spec}k(u_i).$$ Set ${\mathbf G}_{\mathbf u}={\mathbf G}\times_U{\mathbf u}$. By ${\mathbf G}_{u_i}$ we denote the fiber of ${\mathbf G}$ over $u_i$; it is a simple simply-connected algebraic group over $k(u_i)$.
\[SchemeY\] Let $Z\subset{\mathbb A}^1_U$ be a closed subscheme finite over $U$. There is a closed subscheme $Y\subset{\mathbb A}^1_U$ which is étale and finite over $U$ and such that\
(i) ${\mathbf G}_Y:={\mathbf G}\times_UY$ is quasi-split,\
(ii) $Y\cap Z={\varnothing}$,\
(iii) for any closed point $u \in U$ one has $Pic({\mathbb P}^1_u - Y_u)=0$, where $Y_u:={\mathbb P}^1_u\cap Y$.\
(Note that $Y$ and $Z$ are closed in ${\mathbb P}^1_U$ since they are finite over $U$).
For every $u_i$ in ${\mathbf u}$ choose a Borel subgroup ${\mathbf B}_{u_i}$ in ${\mathbf G}_{u_i}$. [*The laller is possible since the fields $k(u_i)$ are finite.*]{} Let ${\mathcal B}$ be the $U$-scheme of Borel subgroup schemes of ${\mathbf G}$. It is a smooth projective $U$-scheme (see [@SGA3-3 Cor. 3.5, Exp. XXVI]). The subgroup ${\mathbf B}_{u_i}$ in ${\mathbf G}_{u_i}$ is a $k(u_i)$-rational point $b_i$ in the fibre of ${\mathcal B}$ over the point $u_i$. Using a variant of Bertini theorem ([**see [@Poo Thm.1.2]**]{}), we can find a closed subscheme $Y^{\prime}$ of ${\mathcal B}$ such that $Y^{\prime}$ is étale over $U$ and all the $b_i$’s are in $Y$ (take an embedding of ${\mathcal B}$ into a projective space ${\mathbb P}^N_U$ and intersect ${\mathcal B}$ with appropriately chosen family of hypersurfaces containing the points $b_i$. Arguing as in the proof of [@OP2 Lemma 7.2], we get a scheme $Y^{\prime}$ finite and étale over $U$). For any closed point $u_i$ in $U$ the fibre $Y^{\prime}_{u_i}$ of $Y^{\prime}$ over $u_i$ contains a $k(u_i)$-rational point (it is the point $b_i$).
To continue the proof of the Proposition we need the following
\[F1F2\] Let $U$ be as in the Proposition. Let $Z\subset{\mathbb A}^1_U$ be a closed subscheme finite over $U$. Let $Y^{\prime} \to U$ be a finite étale morphism such that for any closed point $u_i$ in $U$ the fibre $Y^{\prime}_{u_i}$ of $Y^{\prime}$ over $u_i$ contains a $k(u_i)$-rational point. Then there are finite field extensions $k_1$ and $k_2$ of the finite field $k$ such that\
(i) the degrees $[k_1: k]$ and $[k_2: k]$ are coprime,\
(ii) $k(u_i) \otimes_k k_r$ is a field for $r=1$ and $r=2$,\
(iii) the degrees $[k_1: k]$ and $[k_2: k]$ are strictly greater than any of the degrees $[k(z): k]$ , where $z$ runs over all closed points of $Z$,\
(iv) there is a closed embedding of $U$-schemes $Y^{\prime\prime}=((Y^{\prime}\otimes_k k_1) \coprod (Y^{\prime}\otimes_k k_2)) \xrightarrow{i} {\mathbb A}^1_U$,\
(v) for $Y=i(Y^{\prime\prime})$ one has $Y \cap Z = {\varnothing}$,\
(vi) for any closed point $u_i$ in $U$ one has $Pic({\mathbb P}^1_{u_i}-Y_{u_i})=0$.
To prove this Lemma note that it’s easy to find field extensions $k_1$ and $k_2$ subjecting (i) to (iii). To satisfy (iv) it suffices to require that for any closed point $u_i$ in $U$ and for $r=1$ and $r=2$ the number of closed points in $Y^{\prime}_{u_i}\otimes_k k_r$ is the same as the number of closed points in $Y^{\prime}_{u_i}$, and to require that for any integer $n>0$ and any closed point $u_i$ in $U$ the number of points $y \in Y^{\prime\prime}_{u_i}$ with $[k(y): k(u_i)]=n$ is not more than the number of points $x \in {\mathbb A}^1_{u_i}$ with $[k(x): k(u_i)]=n$. Clearly, these requirements can be satisfied, which proves the item (iv).
The condition (v) holds for any closed $U$-embedding $i: Y^{\prime\prime} \hookrightarrow {\mathbb A}^1_U$ from item (iv), since the property (iii). The condition (vi) holds since the property (i).
Now complete the proof of Proposition \[SchemeY\]. Take the $U$-scheme $Y^{\prime} \subset {\mathbf B}$ as in the beginning of the proof. This $U$-scheme $Y^{\prime}$ satisfies the assumption of Lemma \[F1F2\]. Take the closed subscheme $Y$ of ${\mathbb A}^1_U$ as in the item (v) of the Lemma. For this $Y$ the conditions (ii) and (iii) of the Proposition are obviously satisfied. The condition (i) is satisfied too, since already it is satisfied for the $U$-scheme $Y^{\prime}$. The Proposition follows.
Set $Z:=\{h=0\}\cup s(U)\subset{\mathbb A}^1_U$. Clearly, $Z$ is finite over $U$. Since the principal ${\mathbf G}$-bundle ${\mathcal E}_t$ is trivial over $({\mathbb A}^1_U)_h$ it is trivial over ${\mathbb A}^1_U-Z$. Note that $\{h=0\}$ is closed in ${\mathbb P}^1_U$ and finite over $U$ because $h$ is monic. Further, $s(U)$ is also closed in ${\mathbb P}^1_U$ and finite over $U$ because it is a zero set of a degree one monic polynomial. Thus $Z\subset{\mathbb P}^1_U$ is closed and finite over $U$. Since the principal ${\mathbf G}$-bundle ${\mathcal E}_t$ is trivial over $({\mathbb A}^1_U)_h$, and ${\mathbf G}$-bundles can be glued in Zariski topology, there exists a principal ${\mathbf G}$-bundle ${\mathcal G}$ over ${\mathbb P}^1_U$ such that\
(i) its restriction to ${\mathbb A}^1_U$ coincides with ${\mathcal E}_t$;\
(ii) its restriction to ${\mathbb P}^1_U-Z$ is trivial.
Now choose $Y$ in ${\mathbb A}^1_U$ as in Proposition \[SchemeY\]. Clearly, $Y$ is finite étale over $U$ and closed in ${\mathbb P}^1_U$. Moreover, $Y \cap \{\infty\}\times U={\varnothing}= Z \cap \{\infty\}\times U$ and $Y\cap Z= {\varnothing}$. Applying Theorem \[MainThm2\] with this choice of $Y$ and $Z$, we see that the restriction of ${\mathcal G}$ to ${\mathbb P}^1_U-Y$ is a trivial ${\mathbf G}$-bundle. Since $s(U)$ is in ${\mathbb A}^1_U-Y$ and ${\mathcal G}|_{{\mathbb A}^1_U}$ coincides with ${\mathcal E}_t$, we conclude that $s^*{\mathcal E}_t$ is a trivial principal ${\mathbf G}$-bundle over $U$.
Proof of Theorem \[MainThm2\] {#sect:proof2}
=============================
We will be using notation from Theorem \[MainThm2\]. Let ${\mathbf u}$ be as in Section \[sect:reducing\]. For $u\in{\mathbf u}$ set ${\mathbf G}_u={\mathbf G}|_u$.
\[pr:trivclsdfbr\] Let ${\mathcal E}$ be a ${\mathbf G}$-bundle over ${\mathbb P}^1_U$ such that ${\mathcal E}|_{{\mathbb P}^1_u}$ is a trivial ${\mathbf G}_u$-bundle for all $u\in{\mathbf u}$. Assume that there exists a closed subscheme $T$ of ${\mathbb P}^1_U$ finite over $U$ such that the restriction of ${\mathcal E}$ to ${\mathbb P}^1_U-T$ is trivial and $(\infty \times U) \cap T = {\varnothing}$. Then ${\mathcal E}$ is trivial.
This follows from Theorem 9.6 of [@PSV], since ${\mathcal E}|_{(\infty \times U)}$ is a trivial ${\mathbf G}$-bundle.
An outline of the proof of Theorem \[MainThm2\] {#sect:outline}
-----------------------------------------------
Our proof of this Theorem almost literally coincides with the proof of [@FP Thm.3]. Our arguments are simpler at certain points. An outline of the proof. Denote by $Y^h$ the henselization of the pair $({\mathbb A}_U^1,Y)$, it is a scheme over ${\mathbb A}_U^1$. Let $s:Y\to Y^h$ be the canonical closed embedding, see Section \[sect:distinguishedLimit\] for more details. Set $\dot Y^h:=Y^h-s(Y)$. Let ${\mathcal G}'$ be a ${\mathbf G}$-bundle over ${\mathbb P}^1_U-Y$. Denote by $\operatorname{Gl}({\mathcal G}',{\varphi})$ the ${\mathbf G}$-bundle over ${\mathbb P}_U^1$ obtained by gluing ${\mathcal G}'$ with the trivial ${\mathbf G}$-bundle ${\mathbf G}\times_U Y^h$ via a ${\mathbf G}$-bundle isomorphism ${\varphi}:{\mathbf G}\times_U\dot Y^h\to{\mathcal G}'|_{\dot Y^h}$.
Note that the ${\mathbf G}$-bundle ${\mathcal G}$ can be presented in the form $\operatorname{Gl}({\mathcal G}',{\varphi})$, where ${\mathcal G}'= {\mathcal G}|_{{\mathbb P}^1_U-Y}$. The idea is to show that
($*$) **
If we find $\alpha$ satisfying condition ($*$), then Proposition \[pr:trivclsdfbr\], applied to $T=Y\cup Z$, shows that the ${\mathbf G}$-bundle $\operatorname{Gl}({\mathcal G}',{\varphi}\circ\alpha)$ is trivial over ${\mathbb P}^1_U$. On the other hand, its restriction to ${\mathbb P}^1_U-Y$ coincides with the ${\mathbf G}$-bundle ${\mathcal G}'={\mathcal G}|_{{\mathbb P}^1_U-Y}$. *Thus ${\mathcal G}|_{{\mathbb P}^1_U-Y}$ is a trivial ${\mathbf G}$-bundle*. To prove ($*$) it suffices to show that\
(i) the bundle ${\mathcal G}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}}$ is trivial;\
(ii) each element $\gamma_{\mathbf u}\in{\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)$ can be written in the form $$\alpha|_{\dot Y_{\mathbf u}^h}\cdot\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h}$$ for certain elements $\alpha\in{\mathbf G}(\dot Y^h)$ and $\beta_{\mathbf u}\in{\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h)$.
A realization of this plan in details is given below in the paper.
Henselization of affine pairs {#sect:distinguishedLimit}
-----------------------------
We will use the theory of henselian pairs and, in particular, a notion of a henselization $A^h_I$ of a commutative ring $A$ at an ideal $I$ (see Appendix and [@Gabber Section 0]). We refer to [@FP subsection 5.2] for the geometric counterpart. Let $S=\operatorname{Spec}A$ be a scheme and $T=\operatorname{Spec}(A/I)$ be a closed subscheme. Let $(T^h,\pi: T^h \to S,s: T \to T^h)$ be *the henselization of the pair $(S,T)$* (cf. Definition \[def:henzelisation\]). By definition the scheme $T^h$ is [*affine*]{} and the composite morphism $\pi \circ s: T \to S$ is the closed embedding $T \hookrightarrow S$. [*Recall*]{} that the pair $(T^h,s(T))$ is henselian, which means that for any affine étale morphism $\pi:Z\to T^h$, any section $\sigma$ of $\pi$ over $s(T)$ uniquely extends to a section of $\pi$ over $T^h$. It is known that $\pi^{-1}(T)=s(T)$.
In the notation of [@Gabber Section 0] we have $T^h=\operatorname{Spec}A^h_I$, $\pi:T^h\to S$ is induced by the structure of $A$-algebra on $A^h_I$.
[*Recall three properties of henselization of affine pairs*]{}\
(i) Let $T$ be a semi-local scheme. Then the henselization commutes with restriction to closed subschemes. In more details, if $S'\subset S$ is a closed subscheme, then there is a natural morphism $(T\times_SS')^h\to T^h\times_SS'$. This morphism is an isomorphism and the canonical section $s':T\times_SS'\to(T\times_SS')^h$ coincides under this identification with $$s\times_S\operatorname{Id}_{S'}:T\times_SS'\to T^h\times_SS'.$$ (ii) If $T=\coprod_i T_i$ is a disjoint union, then $T^h=\coprod_i T_i^h$.\
(iii) If we replace in a pair $(S,T)$ the scheme $S$ by an étale affine neighborhood of $T$, then the $(T^h,\pi,s)$ [*remains the same*]{}. In more details, given a pair $(S,T)$ as above we write temporarily $(S^{hen}_T, \pi_{S,T}, s_{S,T})$ for $(T^h,\pi,s)$. If $p: W \to S$ is an étale morphism and $t: T \hookrightarrow W$ is such that $p \circ t: T \hookrightarrow S$ coincides with the closed embedding $T$ into $S$, then there is a canonical [*isomorphism*]{} $\rho: W^{hen}_T \to S^{hen}_T$ of the $S$-schemes $(W^{hen}_T, \pi_{W,T})$ and $(S^{hen}_T, \pi_{S,T})$ such that $\rho \circ s_{W,T}=s_{S,T}$.
Gluing principal ${\mathbf G}$-bundles
--------------------------------------
Recall that $U=\operatorname{Spec}R$, where $R$ is the semi-local ring of finitely many closed points on an irreducible $k$-smooth affine variety over a finite field $k$. Also, ${\mathbf G}$ is a simple simply-connected group scheme over $U$, and $Y$ is a closed subscheme of ${\mathbb P}_U^1$ finite and étale over $U$.
[*We will assume below in the preprint that*]{} $Y\subset{\mathbb A}_U^1$ (as in the hypotheses of Theorem \[MainThm2\]). Let $(Y^h,\pi,s)$ be the henselization of the pair $({\mathbb A}_U^1,Y)$ and let $\dot Y^h=Y^h-s(Y)$ and let $in: {\mathbb A}_U^1 \hookrightarrow {\mathbb P}_U^1$ be the open inclusion.
[@FP] \[pr:affine\]The schemes $Y^h$ and $\dot Y^h$ are affine.
Let us make a general remark. Let ${\mathcal F}$ be a ${\mathbf G}$-bundle over a $U$-scheme $T$. By definition, a trivialization of ${\mathcal F}$ is a ${\mathbf G}$-equivariant isomorphism ${\mathbf G}\times_UT\to{\mathcal F}$. Equivalently, it is a section of the projection ${\mathcal F}\to T$. If ${\varphi}$ is such a trivialization and $f:T'\to T$ is a $U$-morphism, we get a trivialization $f^*{\varphi}$ of $f^*{\mathcal F}$. Sometimes we denote this trivialization by ${\varphi}|_{T'}$. We also sometimes call a trivialization of $f^*{\mathcal F}$ *a trivialization of ${\mathcal F}$ on $T'$*.
The main cartesian square we will work with is $$\begin{CD}\label{eq:distinguishedLimit}
\dot Y^h @>>> Y^h\\
@VVV @VV{in\circ\pi}V\\
{\mathbb P}^1_U - Y @>>>{\mathbb P}^1_U.
\end{CD}$$ Let ${\mathcal A}$ be the category of pairs $({\mathcal E},\psi)$, where ${\mathcal E}$ is a ${\mathbf G}$-bundle on ${\mathbb P}_U^1$, $\psi$ is a trivialization of ${\mathcal E}|_{Y^h}:=(in\circ\pi)^*{\mathcal E}$. A morphism between $({\mathcal E},\psi)$ and $({\mathcal E}',\psi')$ is an isomorphism ${\mathcal E}\to{\mathcal E}'$ compatible with trivializations.
Similarly, let ${\mathcal B}$ be the category of pairs $({\mathcal E},\psi)$, where ${\mathcal E}$ is a ${\mathbf G}$-bundle on ${\mathbb P}_U^1-Y$, $\psi$ is a trivialization of ${\mathcal E}|_{\dot Y^h}$.
Consider the restriction functor $\Psi:{\mathcal A}\to{\mathcal B}$.
[@FP] \[pr:gluing2\] The functor $\Psi$ is an equivalence of categories.
[@FP] \[equally\_well\] By Proposition \[pr:gluing2\] we can choose a functor quasi-inverse to $\Psi$. Fix such a functor $\Theta$. Let $\Lambda$ be the forgetful functor from ${\mathcal A}$ to the category of ${\mathbf G}$-bundles over ${\mathbb P}_U^1$. For $({\mathcal E},\psi)\in{\mathcal B}$ set $$\operatorname{Gl}({\mathcal E},\psi)=\Lambda(\Theta({\mathcal E},\psi)).$$ Note that $\operatorname{Gl}({\mathcal E},\psi)$ comes with a canonical trivialization over $Y^h$.
Conversely, if ${\mathcal E}$ is a principal ${\mathbf G}$-bundle over ${\mathbb P}_U^1$ such that its restriction to $Y^h$ is trivial, then ${\mathcal E}$ can be represented as $\operatorname{Gl}({\mathcal E}',\psi)$, where ${\mathcal E}'={\mathcal E}|_{{\mathbb P}_U^1-Y}$, $\psi$ is a trivialization of ${\mathcal E}'$ on $\dot Y^h$.
Let ${\mathbf u}$ be as in Section \[sect:reducing\], $Y_{\mathbf u}:=Y\times_U{\mathbf u}$. Let $(Y_{\mathbf u}^h,\pi_{\mathbf u},s_{\mathbf u})$ be the henselization of $({\mathbb A}^1_{\mathbf u},Y_{\mathbf u})$. Using property (i) of henselization, we get $Y_{\mathbf u}^h=Y^h\times_U{\mathbf u}$. Thus we have a natural closed embedding $Y_{\mathbf u}^h\to Y^h$. Set $\dot Y_{\mathbf u}^h=Y_{\mathbf u}^h-s_{\mathbf u}(Y_{\mathbf u})$. We get a closed embedding $$\label{dotYdotYh}
\dot Y_{\mathbf u}^h\hookrightarrow \dot Y^h.$$ Thus the pull-back of the cartesian square (\[eq:distinguishedLimit\]) by means of the closed embedding ${\mathbf u}\hookrightarrow U$ has the form $$\begin{CD}
\dot Y_{\mathbf u}^h @>>>Y_{\mathbf u}^h\\
@VVV @VV in_{\mathbf u}\circ\pi_{\mathbf u}V\\
{\mathbb P}^1_{\mathbf u}- Y_{\mathbf u}@>>>{\mathbb P}^1_{\mathbf u},
\end{CD}$$ where $in_{\mathbf u}:{\mathbb A}_{\mathbf u}^1\to{\mathbb P}_{\mathbf u}^1$.
Similarly to the above, we can define categories ${\mathcal A}_{\mathbf u}$ and ${\mathcal B}_{\mathbf u}$ and an equivalence of categories $\Psi_{\mathbf u}:{\mathcal A}_{\mathbf u}\to{\mathcal B}_{\mathbf u}$. Let $\Theta_{\mathbf u}$ be a functor quasi-inverse to $\Psi_{\mathbf u}$ and $\Lambda_{\mathbf u}$ be the forgetful functor from ${\mathcal A}_{\mathbf u}$ to the category of ${\mathbf G}_{\mathbf u}$-bundles over ${\mathbb P}_{\mathbf u}^1$. Let ${\mathcal E}_{\mathbf u}$ be a principal ${\mathbf G}_{\mathbf u}$-bundle over ${\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}$ and $\psi_{\mathbf u}$ be a trivialization of ${\mathbf G}_{\mathbf u}$ on $\dot Y_{\mathbf u}^h$. Set $$\operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u})=\Lambda_{\mathbf u}(\Theta_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u})).$$
[@FP] \[basechange\_limit\] Let $({\mathcal E},\psi)\in{\mathcal B}$, and let $\operatorname{Gl}({\mathcal E},\psi)$ be the ${\mathbf G}$-bundle obtained by Construction \[equally\_well\]. Then $$\operatorname{Gl}_{\mathbf u}({\mathcal E}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},\psi|_{\dot Y_{\mathbf u}^h}) \ \text{and} \ \operatorname{Gl}({\mathcal E},\psi)|_{{\mathbb P}^1_{\mathbf u}}$$ are isomorphic as ${\mathbf G}_{\mathbf u}$-bundles over ${\mathbb P}^1_{\mathbf u}$.
[@FP] \[coboundary\_limit\] For any $({\mathcal E}_{\mathbf u},\psi_{\mathbf u})\in{\mathcal B}_{\mathbf u}$ and any $\beta_{\mathbf u}\in{\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h)$ the ${\mathbf G}_{\mathbf u}$-bundles $$\operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u})\ \text{and} \ \operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u}\circ\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h})$$ are isomorphic (here $\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h}$ is regarded as an automorphism of the ${\mathbf G}_{\mathbf u}$-bundle ${\mathbf G}_{\mathbf u}\times_{\mathbf u}\dot Y_{\mathbf u}^h$ given by the right translation by $\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h}$).
Proof of Theorem \[MainThm2\]: presentation of ${\mathcal G}$ in the form $\operatorname{Gl}({\mathcal G}',{\varphi})$ {#sect:presentation}
----------------------------------------------------------------------------------------------------------------------
Let $U$, ${\mathbf G}$, $Z$, $Y$ and ${\mathcal G}$ be as in Theorem \[MainThm2\].
[@FP] \[presentation\] The ${\mathbf G}$-bundle ${\mathcal G}$ over ${\mathbb P}^1_U$ is of the form $\operatorname{Gl}({\mathcal G}',{\varphi})$ for the ${\mathbf G}$-bundle ${\mathcal G}':={\mathcal G}|_{{\mathbb P}^1_U-Y}$ and a trivialization ${\varphi}$ of ${\mathcal G}'$ over $\dot Y^h$.
In view of Construction \[equally\_well\], it is enough to prove that the restriction of the principal ${\mathbf G}$-bundle ${\mathcal G}$ to $Y^h$ is trivial. Let us choose a closed subscheme $Z'\subset{\mathbb A}^1_U$ such that $Z'$ contains $Z$, $Z'\cap Y={\varnothing}$, and ${\mathbb A}^1_U-Z'$ is affine. Then ${\mathbb A}^1_U-Z'$ is an affine neighborhood of $Y$. By the property (iii) from subsection \[sect:distinguishedLimit\] the henselization of the pair $({\mathbb A}^1_U-Z',Y)$ coincides with the henselization of the pair $({\mathbb A}^1_U,Y)$. Since ${\mathcal G}$ is trivial over ${\mathbb A}^1_U-Z'$, its pull-back to $Y^h$ is trivial too. The proposition is proved.
*Our aim is to modify the trivialization ${\varphi}$ via an element $$\alpha\in{\mathbf G}(\dot Y^h)$$ so that the ${\mathbf G}$-bundle $\operatorname{Gl}({\mathcal G}',{\varphi}\circ\alpha)$ becomes trivial over* ${\mathbb P}^1_U$.
Proof of Theorem \[MainThm2\]: proof of property (i) from the outline {#sect:properties_i}
---------------------------------------------------------------------
Now we are able to prove property (i) from the outline of the proof. In fact, we will prove the following modification of [@FP Lemma 5.11].
\[podpravka\] Let $\operatorname{Gl}({\mathcal G}',{\varphi})$ be the presentation of the ${\mathbf G}$-bundle ${\mathcal G}$ over ${\mathbb P}^1_U$ given in Proposition \[presentation\]. Set ${\varphi}_{\mathbf u}:={\varphi}|_{\dot Y_{\mathbf u}^h}$. Then there is $\gamma_{\mathbf u}\in{\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)$ such that the ${\mathbf G}_{\mathbf u}$-bundle $\operatorname{Gl}_{\mathbf u}({\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},{\varphi}_{\mathbf u}\circ\gamma_{\mathbf u})$ is trivial.
We show first that ${\mathcal G}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}}$ is trivial. One has $${\mathbb P}^1_{\mathbf u}= \coprod_{u\in{\mathbf u}}{\mathbb P}^1_u
$$ For $u\in{\mathbf u}$ set $Y_u:=Y\times_Uu$, ${\mathbf G}_u:={\mathbf G}\times_Uu$, and ${\mathcal G}_u:={\mathcal G}\times_Uu$.
Take $u\in{\mathbf u}$. By our assumption on $Y$, $Pic({\mathbb P}^1_u-Y_u)=0$. The ${\mathbf G}_u$-bundle ${\mathcal G}_u$ is trivial over ${\mathbb A}^1_u-Z_u$. Thus, by [@GilleTorseurs Corollary 3.10(a)], it is trivial over ${\mathbb P}^1_u-Y_u$.
We see that ${\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}}={\mathcal G}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}}$ is trivial. Choosing a trivialization, we may identify ${\varphi}_{\mathbf u}$ with an element of ${\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)$. Set $\gamma_{\mathbf u}={\varphi}_{\mathbf u}^{-1}$. By the very choice of $\gamma_{\mathbf u}$ the ${\mathbf G}_{\mathbf u}$-bundle $\operatorname{Gl}_{\mathbf u}({\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},{\varphi}_{\mathbf u}\circ\gamma_{\mathbf u})$ is trivial.
Proof of Theorem \[MainThm2\]: reduction to property (ii) from the outline {#sect:properties_ii}
--------------------------------------------------------------------------
The aim of this section is to deduce Theorem \[MainThm2\] from the following
[@FP] \[alpha\] Each element $\gamma_{\mathbf u}\in{\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)$ can be written in the form $$\alpha|_{\dot Y_{\mathbf u}^h}\cdot\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h}$$ for certain elements $\alpha\in{\mathbf G}(\dot Y^h)$ and $\beta_{\mathbf u}\in{\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h)$.
[@FP] Let $\operatorname{Gl}({\mathcal G}',{\varphi})$ be the presentation of the ${\mathbf G}$-bundle ${\mathcal G}$ from Proposition \[presentation\]. Let $\gamma_{\mathbf u}\in{\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)$ be the element from Lemma \[podpravka\]. Let $\alpha\in{\mathbf G}(\dot Y^h)$ and $\beta_{\mathbf u}\in{\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h)$ be the elements from Proposition \[alpha\]. Set $${\mathcal G}^{new}=\operatorname{Gl}({\mathcal G}',{\varphi}\circ\alpha).$$ *Claim.* The ${\mathbf G}$-bundle ${\mathcal G}^{new}$ is trivial over ${\mathbb P}^1_U$.\
Indeed, by Lemmas \[basechange\_limit\] and \[coboundary\_limit\] one has a chain of isomorphisms of ${\mathbf G}_{\mathbf u}$-bundles $$\begin{gathered}
{\mathcal G}^{new}|_{{\mathbb P}^1_{\mathbf u}}\cong
\operatorname{Gl}_{\mathbf u}({\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},{\varphi}_{\mathbf u}\circ \alpha|_{\dot Y_{\mathbf u}^h})
\cong\\
\operatorname{Gl}_{\mathbf u}({\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},{\varphi}_{\mathbf u}\circ
\alpha|_{\dot Y_{\mathbf u}^h}\circ\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h})=
\operatorname{Gl}_{\mathbf u}({\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}}, {\varphi}_{\mathbf u}\circ \gamma_{\mathbf u}),\end{gathered}$$ which is trivial by the choice of $\gamma_{\mathbf u}$. The ${\mathbf G}$-bundles ${\mathcal G}|_{{\mathbb P}^1_U-Y}$ and ${\mathcal G}^{new}|_{{\mathbb P}^1_U-Y}$ coincide by the very construction of ${\mathcal G}^{new}$. By Proposition \[pr:trivclsdfbr\], applied to $T=Z\cup Y$, the ${\mathbf G}$-bundle ${\mathcal G}^{new}$ is trivial. Whence the claim.
The claim above implies that the ${\mathbf G}$-bundle ${\mathcal G}|_{{\mathbb P}^1_U-Y}= {\mathcal G}^{new}|_{{\mathbb P}^1_U-Y}$ is trivial. Theorem \[MainThm2\] is proved.
End of proof of Theorem \[MainThm2\]: proof of property (ii) from the outline
-----------------------------------------------------------------------------
*In the remaining part of Section \[sect:proof2\] we will prove Proposition \[alpha\]. This will complete the proof of Theorem \[MainThm2\]*.
By our assumption on $Y$, the group scheme ${\mathbf G}_Y={\mathbf G}\times_UY$ is [**quasi-split**]{}. Thus we can and will choose a Borel subgroup scheme ${\mathbf B}^+$ in ${\mathbf G}_Y$. Since $Y$ is an affine scheme, by [@SGA3-3 Exp. XXVI, Cor. 2.3, Th 4.3.2(a)] there is an opposite to ${\mathbf B}^+$ Borel subgroup scheme ${\mathbf B}^-$ in ${\mathbf G}_Y$. Let ${\mathbf U}^+$ be the unipotent radical of ${\mathbf B}^+$, and let ${\mathbf U}^-$ be the unipotent radical of ${\mathbf B}^-$.
\[EYi\] We will write ${\mathbf E}$ for the functor, sending a $Y$-scheme $T$ to the subgroup ${\mathbf E}(T)$ of the group ${\mathbf G}_Y(T)={\mathbf G}(T)$ generated by the subgroups ${\mathbf U}^+(T)$ and ${\mathbf U}^-(T)$ of the group ${\mathbf G}_Y(T)={\mathbf G}(T)$.
\[lm:surjectivity\] The functor ${\mathbf E}$ has the property that for every closed subscheme $S$ in an affine $Y$-scheme $T$ the induced map ${\mathbf E}(T)\to{\mathbf E}(S)$ is surjective.
The restriction maps ${\mathbf U}^\pm(T)\to{\mathbf U}^\pm(S)$ are surjective, since ${\mathbf U}^\pm$ are isomorphic to vector bundles as $Y$-schemes (see [@SGA3-3 Exp. XXVI, Cor. 2.5]).
Recall that $(Y^h,\pi,s)$ is the henselization of the pair $({\mathbb A}_U^1,Y)$. Also, $in:{\mathbb A}_U^1\to{\mathbb P}_U^1$ is the embedding. Denote the projection ${\mathbb A}_U^1\to U$ by $pr$ and the projection ${\mathbb A}_Y^1\to Y$ by $pr_Y$.
[@FP] \[lm:retraction\] There is a morphism $r:Y^h\to Y$ making the following diagram commutative $$\label{eq:retraction}
\begin{CD}
Y^h @>r>> Y\\
@V{in\circ\pi}VV @VV{pr|_{Y}}V\\
{\mathbb P}^1_U @>pr>> U
\end{CD}$$ and such that $r\circ s=\operatorname{Id}_Y$.
We view $Y^h$ as a $Y$-scheme via $r$. Thus various subschemes of $Y^h$ also become $Y$-schemes. In particular, $\dot Y^h$ and $\dot Y_{\mathbf u}^h$ are $Y$-schemes, and we can consider $${\mathbf E}(\dot Y^h)\subset{\mathbf G}(\dot Y^h) \ \text{ and } \ {\mathbf E}(\dot Y_{\mathbf u}^h)\subset{\mathbf G}(\dot Y_{\mathbf u}^h)={\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h).$$
\[nastia\] $${\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)={\mathbf E}(\dot Y_{\mathbf u}^h){\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h).$$
Firstly, one has $Y_{\mathbf u}=\coprod_{u\in{\mathbf u}}\coprod_{y\in Y_u}y$. (Note that $Y_u$ is a finite scheme.) Thus by property (ii) of henselization, we have $$Y_{\mathbf u}^h=\coprod_{u\in{\mathbf u}}\coprod_{y\in Y_u}y^h,\qquad
\dot Y_{\mathbf u}^h=\coprod_{u\in{\mathbf u}}\coprod_{y\in Y_u}\dot y^h,$$ where $(y^h,\pi_y,s_y)$ is the henselization of the pair $({\mathbb A}_u^1,y)$, $\dot y^h:=y^h-s_y(y)$. We see that $y^h$ and $\dot y^h$ are subschemes of $Y^h$, so we can view them as $Y$-schemes, and ${\mathbf G}_{y^h}:={\mathbf G}_Y\times_Y{y^h}$ is [**quasi-split**]{}. Also, ${\mathbf E}(\dot y^h)$ makes sense as a subgroup of ${\mathbf G}(\dot y^h)={\mathbf G}_u(\dot y^h)={\mathbf G}_{y^h}(\dot y^h)$.
One has $$\begin{split}
{\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)&=\prod_{u\in{\mathbf u}}\prod_{y\in Y_u}{\mathbf G}_u(\dot y^h)=\prod_{u\in{\mathbf u}}\prod_{y\in Y_u}{\mathbf G}_{y^h}(\dot y^h),\\
{\mathbf E}(\dot Y_{\mathbf u}^h)&=\prod_{u\in{\mathbf u}}\prod_{y\in Y_u}{\mathbf E}(\dot y^h),\\
{\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h)&=\prod_{u\in{\mathbf u}}\prod_{y\in Y_u}{\mathbf G}_u(y^h)=\prod_{u\in{\mathbf u}}\prod_{y\in Y_u}{\mathbf G}_{y^h}(y^h).
\end{split}$$ Thus it suffices for each $u\in{\mathbf u}$ and each $y\in Y_u$ to check the equality $${\mathbf G}_{y^h}(\dot y^h)={\mathbf E}(\dot y^h){\mathbf G}_{y^h}(y^h).$$ This equality holds by Fait 4.3 and Lemma 4.5 of [@Gille:BourbakiTalk]. In fact, $y^h=\operatorname{Spec}\mathcal O$, where $\mathcal O=k(u)[t]_{\mathfrak m_y}^h$ is a henselian discrete valuation ring, and $\mathfrak m_y\subset k(u)[t]$ is the maximal ideal defining the point $y\in{\mathbb A}^1_u$. Further, $\dot y^h=\operatorname{Spec}L$, where $L$ is the fraction field of $\mathcal O$. The lemma is proved.
We have the closed embedding (\[dotYdotYh\]) and the scheme $\dot Y^h$ is affine by Proposition \[pr:affine\]. Recall that we regard $\dot Y^h$ as a $Y$-scheme via the morphism $r|_{\dot Y^h}$. Thus by Lemma \[lm:surjectivity\] the restriction map ${\mathbf E}(\dot Y^h)\to{\mathbf E}(\dot Y_{\mathbf u}^h)$ is surjective. Since ${\mathbf E}(\dot Y^h)\subset{\mathbf G}(\dot Y^h)$, the proposition \[alpha\] follows. *This completes the proof of Theorem \[MainThm2\]*.
An application {#sect:application}
==============
The following result is a straightforward consequence of Theorem \[MainThm1\] and an exact sequence for étale cohomology. Recall that by our definition a reductive group scheme has geometrically connected fibres.
\[Norms\] Let $R$ be as in Theorem \[MainThm1\] and ${\mathbf G}$ be a reductive $R$-group scheme. Let $\mu:{\mathbf G}\to{\mathbf T}$ be a group scheme morphism to an $R$-torus ${\mathbf T}$ such that $\mu$ is locally in the étale topology on $\operatorname{Spec}R$ surjective. Assume further that the $R$-group scheme ${\mathbf H}:=\operatorname{Ker}(\mu)$ is reductive. Let $K$ be the fraction field of $R$. Then the group homomorphism $${\mathbf T}(R)/\mu({\mathbf G}(R))\to{\mathbf T}(K)/\mu({\mathbf G}(K))$$ is injective.
This theorem extends all the known results of this form proven in [@C-TO], [@PS], [@Z], [@OPZ].
[@FP]
=====
For a commutative ring $A$ we denote by $\operatorname{Rad}(A)$ its Jacobson ideal. The following definition one can find in [@Gabber Section 0].
If $I$ is an ideal in a commutative ring $A$, then the pair $(A,I)$ is called *henselian*, if $I\subset\operatorname{Rad}(A)$ and for every two relatively prime monic polynomials $\bar g, \bar h\in\bar A[t]$, where $\bar A=A/I$, and monic lifting $f\in A[t]$ of $\bar g\bar h$, there exist monic liftings $g,h\in A[t]$ such that $f=gh$. (Two polynomials are called relatively prime, if they generate the unit ideal.)
[@FP] \[basechhens\] Let $(A,I)$ be a henselian pair with a semi-local ring $A$ and $J\subset A$ be an ideal. Then the pair $(A/J,(I+J)/J)$ is henselian.
The following definition one can find in [@Gabber Section 0].
\[def:henzelisation\] The henselization of any pair $(A,I)$ is the pair $(A_I^h,I^h)$ (over $(A,I)$) defined as follows $$(A_I^h,I^h):=\text{the filtered inductive limit over the category $\mathcal N$ of }(A',\operatorname{Ker}(\sigma)),$$ where $\mathcal N$ is the filtered category of pairs $(A',\sigma)$ such that $A'$ is an étale $A$-algebra and $\sigma\in\text{Hom}_{A-alg}(A',A/I)$.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We define the local trace function for subspaces of ${L^{2}\left(\mathbb{R}^n\right)}$ which are invariant under integer translation. Our trace function contains the dimension function and the spectral function defined in [@BoRz] and completely characterizes the given translation invariant subspace. It has properties such as positivity, additivity, monotony and some form of continuity. It behaves nicely under dilations and modulations. We use the local trace function to deduce, using short and simple arguments, some fundamental facts about wavelets such as the characterizing equations, the equality between the dimension function and the multiplicity function and some new relations between scaling functions and wavelets.'
address: |
Department of Mathematics\
The University of Iowa\
14 MacLean Hall\
Iowa City, IA 52242-1419\
U.S.A.
author:
- Dorin Ervin Dutkay
title: The Local Trace Function of Shift Invariant Subspaces
---
[^1]
\[Intro\]Introduction
=====================
Shift invariant spaces are closed subspaces of ${L^{2}\left(\mathbb{R}^n\right)}$ that are invariant under all integer translations (also called shifts). They lie at the very heart of several areas such as the theory of wavelets, spline systems, Gabor systems or approximation theory. Thus, a good understanding of the shift invariant spaces can prove itself fruitful and give results in each of these areas. The local trace function is a new instrument for the analysis of shift invariant spaces. It is associated to a shift invariant subspace of ${L^{2}\left(\mathbb{R}^n\right)}$ and a positive operator on ${l^{2}\left(\mathbb{Z}^n\right)}$. And it confirms our expectations: when applied to a specific case it gives results. Several fundamental facts about wavelets can be deduced quite easily with the aid of the local trace function ( we mention some of these facts: the equations that characterize wavelets (remark \[rem5\_2\]), the equality between the dimension function and the multiplicity function (remark \[rem5\_3\]) and some new equations that relate multiscaling functions to multiwavelets (theorem \[th5\_4\] and corrolary \[cor5\_5\])). Another nice thing about the local trace function is that it includes, as special cases, the dimension function and the spectral function introduced by M.Bownik and Z. Rzeszotnik in [@BoRz] ( their paper was the main source of inspiration for us, several of the theorems and proofs presented here are just extensions of the theorems and proofs from [@BoRz]). To get the dimension function, just compute the local trace function using the operator $I$-the identity on ${l^{2}\left(\mathbb{Z}^n\right)}$. The spectral function is just the local trace function associated to the projection $P_{\delta_0}$ onto the $0$-th component (see proposition \[prop3\_10\]).
The local trace function is based on two main concepts: the range function introduced by Helson ([@H]) for shift invariant subspaces of ${L^{2}\left(\mathbb{R}^n\right)}$ and the trace function for positive operators on ${l^{2}\left(\mathbb{Z}^n\right)}$. The definition of the local trace functions combines these concepts (see definition \[def3\_1\]). Another key fact is a simple observation regarding the link between the trace function and normalized tight frames, namely that the trace function can be computed as in its definition but using a normalized tight frame instead of an orthonormal basis (see proposition \[prop21\_2\]). Moreover, the normalized tight frames are the only families of vectors that can be used to compute the trace (see proposition \[prop21\_4\_1\] and theorem \[th21\_4\_2\]).
We recall that a subset $\{e_i\,|\,i\in I\}$ of a Hilbert space $H$ is called a frame with positive constants $A$ and $B$ if $$A\|f\|^2\leq\sum_{i\in I}|{\left\langle f\,|\,e_i\right\rangle}|^2\leq B\|f\|^2,\quad(f\in H).$$ If $A=B=1$ then it is called a normalized tight frame (or shortly NTF).
In this paper we define the local trace function, investigate some of its properties and apply it to wavelets to obtain, using only short arguments, some known and some new results. In section \[Therange\] and section \[Thetrace\] we recall some facts about the range function and the trace function, respectively. We establish the main properties of these functions, properties that we need for section \[Thelocal\] where we define and study the main character of our paper: the local trace function. Many properties shared by the range function and the trace function are inherited by the local trace function: additivity (proposition \[prop3\_5\] and \[prop3\_6\]), monotony (proposition \[prop3\_8\]), nice behavior with respect to modulation and dilation (proposition \[prop3\_8\] and \[prop3\_9\]).
Also, there is a strong connection between our trace function and the Gramian introduced and effectively used by A.Ron and Z.Shen in [@RS1],[@RS2], [@RS3] (see the remarks \[rem21\_4\_4\] and \[rem3\_3\_3\]).
Another fact that should be noticed is that the local trace function completely determines the shift invariant subspace: two shift invariant subspaces are equal if and only if their local trace functions are equal (proposition \[prop3\_7\]).
Section \[convergence\] contains some results about the behavior of the local trace function with respect to limits. That is, when a sequence of shift invariant spaces has a limit (which is also shift invariant), in some cases the local trace function of the limit space is the limit of the sequence of local trace functions. One of the conditions is that the convergence is in the strong operator topology, but we put restrictions on the operator (theorem \[th4\_2\]); the other theorem is a monotone convergence theorem (theorem \[th4\_3\]) and it works for any positive operator on ${l^{2}\left(\mathbb{Z}^n\right)}$.
In section \[wavelets\], we apply the local trace function to wavelets. Some very simple arguments lead to important results: the equations that characterize wavelets (\[eq5\_2\_1\]), (\[eq5\_2\_2\]), the equality between the multiplicity function and the dimension function (remark \[rem5\_3\]) and the relation between scaling functions and wavelets (relation (\[eq5\_4\_2\])-(\[eq5\_5\_2\])) are obtained just by writing some local trace function in two ways.
Before we engage in the analysis of the local trace function, we have to recall some definitions and theorems.
The Fourier transform is given by $$\widehat{f}(\xi)=\int_{{\mathbb{R}^n}}f(x)e^{-i{\left\langle x\,|\,\xi\right\rangle}}\,dx,\quad(\xi\in{\mathbb{R}^n}).$$ If $V$ is closed subspace of a Hilbert space $H$ and $f\in H$ we denote by $P_V$ the projection onto $V$ and by $P_f$ the operator defined by: $$P_f(v)={\left\langle v\,|\,f\right\rangle}f,\quad(v\in H).$$
\[def1\_1\] A closed subspace $V$ of ${L^{2}\left(\mathbb{R}\right)}$ is called shift invariant (or shortly SI) if $$T_kV=V,\quad(k\in{\mathbb{Z}^n}),$$ where $T_k$ is the translation by $k$ on ${L^{2}\left(\mathbb{R}^n\right)}$: $$(T_kf)(\xi)=f(\xi-k),\quad(\xi\in{\mathbb{R}^n},f\in{L^{2}\left(\mathbb{R}^n\right)}).$$ If $\mathcal{A}$ is a subset of ${L^{2}\left(\mathbb{R}^n\right)}$ then we denote by $S(\mathcal{A})$ the shift invariant space generated by $\mathcal{A}$, $$S(\mathcal{A})=\overline{\operatorname*{span}}\{T_k\varphi\,|\,k\in{\mathbb{Z}^n},\varphi\in\mathcal{A}\}.$$
\[def1\_2\] Let $V$ be a shift invariant subspace of ${L^{2}\left(\mathbb{R}^n\right)}$. A subset $\phi$ of $V$ is called a normalized tight frame generator (or NTF generator) for $V$ if $$\{T_k\varphi\,|\,k\in{\mathbb{Z}^n},\varphi\in\phi\}$$ is a NTF for $V$.
We use also the notation $S(\varphi):=S(\{\varphi\})$. $\varphi$ is called a quasi-orthogonal generator for $S(\varphi)$ if $$\{T_k\varphi\,|\,k\in{\mathbb{Z}^n}\}$$ is a NTF for $S(\varphi)$ and for all $\xi\in{\mathbb{R}^n}$, $$\operatorname*{Per}|\widehat{\varphi}|^2(\xi):=\sum_{k\in{\mathbb{Z}^n}}|\widehat{\varphi}|^2(\xi+2k\pi)\in\{0,1\}.$$ (actually, the second condition is a consequence of the first but we include it anyway).
Several proofs are available in the literature for the next theorem which guarantees the existence of NTF generators (see [@Bo1], [@B], [@BMM]).
\[th1\_3\] Suppose $V$ is a SI subspace of ${L^{2}\left(\mathbb{R}^n\right)}$. Then $V$ can be decomposed as an orthogonal sum $$V=\bigoplus_{i\in\mathbb{N}}S(\varphi_i),$$ where $\varphi_i$ is a quasi-orthogonal generator of $S(\varphi_i)$. Moreover $\{\varphi_i\,|\, i\in\mathbb{N}\}$ is a NTF generator for $V$.
In fact, it is possible to derive a choice for the generators $\varphi_i$ in theorem \[th1\_3\] directly from the Stone-spectral multiplicity theorem, as explained in [@BMM].
\[Therange\]The range function
==============================
In this section we define the range function and state some of its properties. Some preliminary notations are needed.
The Hilbert space of square integrable vector functions $L^2\left([-\pi,\pi]^n,{l^{2}\left(\mathbb{Z}^n\right)}\right)$ consists of all vector valued measurable functions $\phi: [-\pi,\pi]^n\rightarrow{l^{2}\left(\mathbb{Z}^n\right)}$ with the norm $$\|\phi\|=\left(\int_{[-\pi,\pi]^n}\|\phi(\xi)\|_{{l^{2}\left(\mathbb{Z}^n\right)}}^2\,d\xi\right)^{1/2}<\infty.$$ The scalar product is given by $${\left\langle \phi\,|\,\psi\right\rangle}:=\int_{[-\pi,\pi]^n}{\left\langle \phi(\xi)\,|\,\psi(\xi)\right\rangle}_{{l^{2}\left(\mathbb{Z}^n\right)}}\,d\xi.$$
The map $\mathcal{T}:{L^{2}\left(\mathbb{R}^n\right)}\rightarrow L^2\left([-\pi,\pi]^n,{l^{2}\left(\mathbb{Z}^n\right)}\right)$ defined for $f\in{L^{2}\left(\mathbb{R}^n\right)}$ by $$\mathcal{T}f:[-\pi,\pi]^n\rightarrow{l^{2}\left(\mathbb{Z}^n\right)},\quad\mathcal{T}f(\xi)=(\widehat{f}(\xi+2k\pi))_{k\in{\mathbb{Z}^n}},\quad(\xi\in
[-\pi,\pi]^n)$$ is an isometric isomorphism (up to multiplication by $1/(2\pi)^{n/2}$) between ${L^{2}\left(\mathbb{R}^n\right)}$ and $L^2\left([-\pi,\pi]^n,{l^{2}\left(\mathbb{Z}^n\right)}\right)$.
We will need some variations of these maps because they will make the formulas nicer. We define $L_{per}^2\left({\mathbb{R}^n},{l^{2}\left(\mathbb{Z}^n\right)}\right)$ to be the space of measurable vector valued functions $\phi:{\mathbb{R}^n}\rightarrow{l^{2}\left(\mathbb{Z}^n\right)}$ with the property that $\phi|_{[-\pi,\pi]^n}$ belongs to $L^2\left([-\pi,\pi]^n,{l^{2}\left(\mathbb{Z}^n\right)}\right)$ and they are periodic in the following sense: $$\phi(\xi+2k\pi)=\lambda(k)^*(\phi(\xi)),\quad(\xi\in{\mathbb{R}^n}),$$ where, for every $k\in{\mathbb{Z}^n}$, $\lambda(k)$ is the shift by $k$ on ${l^{2}\left(\mathbb{Z}^n\right)}$ that is $$(\lambda(k)\alpha)(l)=\alpha(l-k),\quad(l\in{\mathbb{Z}^n},\alpha\in{l^{2}\left(\mathbb{Z}^n\right)}).$$ The scalar product is defined by the same formula, the integral being taken over $[-\pi,\pi]^n$.
The map ${\mathcal{T}_{per}}:{L^{2}\left(\mathbb{R}^n\right)}\rightarrow L_{per}^2\left({\mathbb{R}^n},{l^{2}\left(\mathbb{Z}^n\right)}\right)$ is defined by the same formula as $\mathcal{T}$, $${\mathcal{T}_{per}}f:{\mathbb{R}^n}\rightarrow{l^{2}\left(\mathbb{Z}^n\right)},\,{\mathcal{T}_{per}}f(\xi)=(\widehat{f}(\xi+2k\pi))_{k\in{\mathbb{Z}^n}},\quad (\xi\in{\mathbb{R}^n})$$ but notice that $\xi$ is now in ${\mathbb{R}^n}$. Also, observe the periodicity property of ${\mathcal{T}_{per}}$: $${\mathcal{T}_{per}}f(\xi+2s\pi)=\lambda(s)^*({\mathcal{T}_{per}}f(\xi)),\quad(\xi\in{\mathbb{R}^n},s\in{\mathbb{Z}^n}).$$
\[def2\_0\] A range function is a measurable mapping $$J:[-\pi,\pi]^n\rightarrow\{\mbox{ closed subspaces of }{l^{2}\left(\mathbb{Z}^n\right)}\}.$$ Measurable means weakly operator measurable, i.e., $\xi\mapsto{\left\langle P_{J(\xi)}a\,|\,b\right\rangle}$ is measurable for any choice of vectors $a,b\in{l^{2}\left(\mathbb{Z}^n\right)}$.
A periodic range function is a measurable function $$J_{per}:{\mathbb{R}^n}\rightarrow\{\mbox{ closed subspaces of }{l^{2}\left(\mathbb{Z}^n\right)}\},$$ with the periodicity property: $$J_{per}(\xi+2k\pi)=\lambda(k)^*\left(J_{per}(\xi)\right),\quad(k\in{\mathbb{Z}^n},\xi\in{\mathbb{R}^n}).$$
Sometimes we will use the same letter to denote the subspace $J_{per}(\xi)$ and the projection onto $J_{per}(\xi)$. In terms of projections, the periodicity can be written as: $$J_{per}(\xi+2k\pi)=\lambda(k)^*J_{per}(\xi)\lambda(k),\quad(k\in{\mathbb{Z}^n},\xi\in{\mathbb{R}^n}).$$
The next theorem due to Helson [@H] establishes the fundamental connection between shift invariant spaces and the range function. The theorem appears in this form in [@Bo1], proposition 1.5, the only modification needed is to work with the whole ${\mathbb{R}^n}$ instead of $[-\pi,\pi]^n$ and use the periodicity.
\[th2\_1\] A closed subspace $V$ of ${L^{2}\left(\mathbb{R}^n\right)}$ is shift invariant if and only if $$V=\{f\in{L^{2}\left(\mathbb{R}^n\right)}\,|\, {\mathcal{T}_{per}}f(\xi)\in J_{per}(\xi)\mbox{ for a.e. }\xi\in\mathbb{R}^n\},$$ for some measurable periodic range function $J_{per}$. The correspondence between $V$ and $J_{per}$ is bijective under the convention that range functions are identified if they are equal a.e. Furthermore, if $V=S(\mathcal{A})$ for some countable $\mathcal{A}\subset{L^{2}\left(\mathbb{R}^n\right)}$, then $$J_{per}(\xi)=\overline{\operatorname*{span}}\{{\mathcal{T}_{per}}\varphi(\xi)\,|\,\varphi\in\mathcal{A}\},\quad \mbox{for
a.e. }\xi\in{\mathbb{R}^n}.$$
\[def2\_1\_1\] If $V$ is a SI subspace of ${L^{2}\left(\mathbb{R}^n\right)}$ then $J_{per}$ associated to $V$ as in theorem $\ref{th2_1}$ is called the periodic range function of $V$.
The first elementary property of the range function that we will need is additivity. The range function is also unitary in the sense that it preserves orthogonality of subspaces. The precise formulation of these properties in given below.
\[prop2\_2\] Let $(V_i)_{i\in I}$ be a countable family of mutually orthogonal SI subspaces of ${L^{2}\left(\mathbb{R}^n\right)}$ and denote by $J_{V_i}$ the periodic range function of $V_i$. If $$V=\bigoplus_{i\in I}V_i$$ and $J_V$ is its periodic range function then $$J_V(\xi)=\bigoplus_{i\in I}J_{V_i}(\xi),\quad\mbox{for a.e. }\xi\in{\mathbb{R}^n},$$ where the sum is an orthogonal one.
Pick some countable $\phi_i\subset V_i$ such that $\{T_k\varphi\,|\, k\in{\mathbb{Z}^n},\varphi\in\phi_i\}$ spans $V_i$. Then it is clear that $\{T_k\varphi\,|\, k\in{\mathbb{Z}^n},\varphi\in\phi\}$ spans $V$, with the notation $\phi=\cup_{i\in I}\phi_i$. Using Helson’s theorem \[th2\_1\] we can determine the periodic range functions: $$J_{V_i}(\xi)=\overline{\operatorname*{span}}\{{\mathcal{T}_{per}}\varphi(\xi)\,|\,\varphi\in\phi_i\},\quad(i\in
I),$$ $$J_{V}(\xi)=\overline{\operatorname*{span}}\{{\mathcal{T}_{per}}\varphi(\xi)\,|\,\varphi\in\phi\},$$ for almost every point $\xi\in{\mathbb{R}^n}$. This shows that, if we check the orthogonality of the subspaces $J_{V_i}(\xi)$, then we are done.
To check that these subspaces of ${l^{2}\left(\mathbb{Z}^n\right)}$ are mutually orthogonal, take two arbitrary $i\neq j\in
I$ and $\varphi_1\in\phi_i$,$\varphi_2\in\phi_j$. Then, since we are dealing with SI spaces, $T_k\varphi_1$ is perpendicular to $\varphi_2$ for any choice of $k\in{\mathbb{Z}^n}$. Rewriting this in terms of the Fourier transform, we obtain $$0=\int_{{\mathbb{R}^n}}e^{-i{\left\langle k\,|\,\xi\right\rangle}}\widehat{\varphi}_1(\xi)\overline{\widehat{\varphi}_2}(\xi)\,d\xi=
\int_{[-\pi,\pi]^n}e^{-i{\left\langle k\,|\,\xi\right\rangle}}{\left\langle \mathcal{T}\varphi_1(\xi)\,|\,\mathcal{T}\varphi_2(\xi)\right\rangle}\,d\xi.$$ For the second equality we applied a periodization. But this shows that all Fourier coefficients of the map $\xi\mapsto{\left\langle \mathcal{T}\varphi_1(\xi)\,|\,\mathcal{T}\varphi_2(\xi)\right\rangle}$ are 0 so the map itself is 0 which implies that ${\mathcal{T}_{per}}\varphi_1(\xi)$ is perpendicular to ${\mathcal{T}_{per}}\varphi_2(\xi)$ for a.e. $\xi\in[-\pi,\pi]^n$ and therefore, because of the periodicity, for a.e. $\xi\in{\mathbb{R}^n}$. Consequently, the subspaces are mutually orthogonal and the proposition is proved.
An easy consequence of the additivity of the range function is the following monotony property:
\[prop2\_3\] Let $(V_j)_{j\in\mathbb{N}}$ be an increasing sequence of SI subspaces of ${L^{2}\left(\mathbb{R}^n\right)}$. Denote by $$V:=\overline{\bigcup_{j\in\mathbb{N}}V_j}.$$ Then for a.e. $\xi\in{\mathbb{R}^n}$, $(J_{V_j}(\xi))_{j\in\mathbb{N}}$ is increasing and $$J_V(\xi)=\overline{\bigcup_{j\in\mathbb{N}}J_{V_j}(\xi)}.$$
Let $W_j$ be the orthogonal complement of $V_j$ in $V_{j+1}$, ($j\in\mathbb{N}$). Then $W_j$ are shift invariant too, $V_{j+1}=V_{j}\oplus W_j$ for all $j\in\mathbb{N}$ and $$V=V_0\oplus\bigoplus_{j\in\mathbb{N}}W_j,\quad
V_l=V_0\oplus\bigoplus_{j=0}^{l-1}W_j,\quad(l\in\mathbb{N}).$$ The proposition then follows from the additivity property stated in proposition \[prop2\_2\].
The next proposition can also be found in [@H] and [@Bo1]. Again the only modification is that we use the periodic extension of the range function.
\[prop2\_4\] Let $V$ be a SI subspace of ${L^{2}\left(\mathbb{R}^n\right)}$ and $J_{per}$ its periodic range function. Then $${\mathcal{T}_{per}}(P_Vf)(\xi)=J_{per}(\xi)({\mathcal{T}_{per}}f(\xi)),\quad\mbox{ for a.e. }\xi\in{\mathbb{R}^n}.$$
\[def6\_1\] A subset $\{e_i\,|\,i\in I\}$ of a Hilbert space $H$ is called a Bessel sequence with constant $B>0$ if $$\sum_{i\in I}|{\left\langle e_i\,|\,f\right\rangle}|^2\leq B\|f\|^2,\quad(f\in H).$$
The next theorem will be extensively used in this paper. Its proof can be found in [@Bo1].
\[th2\_5\] Let $V$ be a SI subspace of ${L^{2}\left(\mathbb{R}^n\right)}$, $J_{per}$ its periodic range function and $\phi$ a countable subset of $V$. $\{T_k\varphi\,|\,k\in{\mathbb{Z}^n},\varphi\in\phi\}$ is a frame with constants $A$ and $B$ for $V$ (Bessel family with constant $B$) if and only if $\{{\mathcal{T}_{per}}\varphi(\xi)\,|\,\varphi\in\phi\}$ is a frame with constants $A$ and $B$ for $J_{per}(\xi)$ (Bessel sequence with constant $B$) for almost every $\xi\in{\mathbb{R}^n}$.
\[Thetrace\]The trace function
==============================
In this section we recall the definition of the trace function and gather some of its properties, the main ones and also some others that we will need in the sequel.
\[def21\_1\] Let $H$ be a Hilbert space and $T$ a positive operator on $H$. Then the trace of the operator $T$ is the positive number (can be also $\infty$) defined by $${\operatorname*{Trace}}(T)=\sum_{i\in I}{\left\langle Te_i\,|\,e_i\right\rangle},$$ where $\{e_i\,|\,i\in I\}$ is an orthonormal basis for $H$.
The next proposition shows that the trace is well defined, that is it doesn’t depend on the choice of the orthonormal basis, and moreover, it can be computed with the same formula using a normalized tight frame.
\[prop21\_2\] Let $H$ be a Hilbert space, $T$ a positive operator on $H$ and $\{f_j\,|\,j\in J\}$ a normalized tight frame for $H$. Then $${\operatorname*{Trace}}(T)=\sum_{j\in J}{\left\langle Tf_j\,|\,f_j\right\rangle}.$$
Let $\{e_i\,|\,i\in I\}$ be an orthonormal basis for $H$. $$\begin{aligned}
{\operatorname*{Trace}}(T)&=\sum_{i\in I}{\left\langle Te_i\,|\,e_i\right\rangle}=\sum_{i\in I}{\left\langle T^{1/2}e_i\,|\,T^{1/2}e_i\right\rangle}=\sum_{i\in I}\|T^{1/2}e_i\|^2\\
&=\sum_{i\in I}\sum_{j\in J}|{\left\langle T^{1/2}e_i\,|\,f_j\right\rangle}|^2=\sum_{j\in J}\sum_{i\in I}|{\left\langle T^{1/2}e_i\,|\,f_j\right\rangle}|^2\\
&=\sum_{j\in J}\sum_{i\in I}|{\left\langle e_i\,|\,T^{1/2}f_j\right\rangle}|^2=\sum_{j\in J}\|T^{1/2}f_j\|^2\\
&=\sum_{j\in J}{\left\langle Tf_j\,|\,f_j\right\rangle}.\end{aligned}$$
The following proposition enumerates some of the elementary properties of the trace. For a proof look in any basic book on operator theory (e.g. [@StZs]).
\[prop21\_3\] The trace has the following properties: for all $a,b$ positive operators and $\lambda\geq0$:
1. ${\operatorname*{Trace}}(a+b)={\operatorname*{Trace}}(a)+{\operatorname*{Trace}}(b)$;
2. ${\operatorname*{Trace}}(\lambda a)=\lambda{\operatorname*{Trace}}(a)$;
3. If $a\leq b$ then ${\operatorname*{Trace}}(a)\leq{\operatorname*{Trace}}(b)$;
4. ${\operatorname*{Trace}}(v^*av)\leq{\operatorname*{Trace}}(a)$ whenever $v$ is a partial isometry;
5. ${\operatorname*{Trace}}(u^*au)={\operatorname*{Trace}}(a)$ for all unitary $u$;
6. ${\operatorname*{Trace}}(x^*x)={\operatorname*{Trace}}(xx^*)$ for every operator $x$ on $H$;
7. ${\operatorname*{Trace}}(|x|)={\operatorname*{Trace}}(|x^*|)$ for every operator $x$ on $H$ ( $|x|:=(x^*x)^{1/2}$);
8. ${\operatorname*{Trace}}(P)=\operatorname*{dim}PH$ for any projection $P$ in $H$.
\[prop21\_4\] Let $P$ be a projection and $\{e_i\,|\,i\in I\}$ a NTF for its range. Then, for any positive operator $T$, and any vector $f\in H$. $${\operatorname*{Trace}}(TP)=\sum_{i\in I}{\left\langle Te_i\,|\,e_i\right\rangle},$$ $${\operatorname*{Trace}}(P_fP)=\sum_{i\in I}|{\left\langle f\,|\,e_i\right\rangle}|^2=\|Pf\|^2.$$
Let $\{f_j\,|\,j\in J\}$ be an orthonormal basis for the orthogonal complement of the range of $P$. Then $\{e_i\,|\,i\in I\}\cup\{f_j\,|\,j\in J\}$ is a NTF for the entire Hilbert space. Using proposition \[prop21\_2\], the formula for ${\operatorname*{Trace}}(TP)$ follows.
Particularize the formula for $T=P_f$: $${\operatorname*{Trace}}(P_fP)=\sum_{i\in I}{\left\langle {\left\langle e_i\,|\,f\right\rangle}f\,|\,e_i\right\rangle}=\sum_{iu\in I}|{\left\langle f\,|\,e_i\right\rangle}|^2=\|Pf\|^2.$$
In fact even more is true: the equations of proposition \[prop21\_4\] characterize the normalized tight frames for the range of $P$. This is described in the next statement.
\[prop21\_4\_1\] Let $H$ be a Hilbert space, $H_0$ a closed subspace and $\{e_i\,|\,i\in I\}$ a family of vectors from $H$. The following affirmations are equivalent:
1. $\{e_i\,|\,i\in I\}$ is a NTF for $H_0$;
2. For every positive operator $T$ on $H$, $${\operatorname*{Trace}}(TP_{H_0})=\sum_{i\in I}{\left\langle Te_i\,|\,e_i\right\rangle};$$
3. For every vector $v\in H$, $${\operatorname*{Trace}}(P_vP_{H_0})=\sum_{i\in I}|{\left\langle v\,|\,e_i\right\rangle}|^2.$$
(Note that we do not require in (ii) and (iii) that the $e_i$’s be in $H_0$. This will follow from the formulas).
\(i) implies (ii) and (ii) implies (iii) according to proposition \[prop21\_4\_1\] and its proof. So we only have to worry about the implication from (iii) to (i). The hypotheses imply that, if $v\perp H_0$, then $P_vP_{H_0}=0$ so $v$ is perpendicular to $e_i$ for every $i\in I$.
Also, for $v\in H_0$, $$\sum_{i\in I}|{\left\langle v\,|\,e_i\right\rangle}|^2=\|v\|^2.$$ This shows that all $e_i$’s are in $H_0$ and they form a NTF for it.
We can weaken the condition (iii) in proposition \[prop21\_4\_1\]. The family of vectors is a NTF for the subspace if it satisfies the NTF condition just for some special vectors as shown below:
\[th21\_4\_2\] Let $H$ be a Hilbert space and $\{\delta_k\,|\,k\in K\}$ a total subset of $H$ (i.e. its closed linear span is $H$). Let $H_0$ be a closed subspace for $H$ and $\{e_i\,|\,i\in I\}$ a family of vectors in $H$. The following affirmations are equivalent:
1. $\{e_i\,|\,i\in I\}$ is a NTF for $H_0$;
2. For every $r\neq s\in K$ and $\lambda\in\{0,1,i=\sqrt{-1}\}$ $$\label{eq21_4_2_1}
\sum_{i\in I}|{\left\langle \delta_r+\lambda\delta_s\,|\,e_i\right\rangle}|^2=\|P_{H_0}(\delta_r+\lambda\delta_s)\|^2.$$
\(i) implies (ii) clearly.
Assume (ii). Take $\{f_j\,|\,j\in J\}$ a NTF for $H_0$. Then $$\label{eq21_4_2_2}
\|P_{H_0}(v)\|^2=\sum_{j\in J}|{\left\langle v\,|\,f_j\right\rangle}|^2,\quad(v\in H).$$ Take $\lambda=0$ and use (\[eq21\_4\_2\_1\]) and (\[eq21\_4\_2\_2\]): $$\label{eq21_4_2_3}
\sum_{i\in I}|{\left\langle \delta_r\,|\,e_i\right\rangle}|^2=\sum_{j\in J}|{\left\langle \delta_r\,|\,f_j\right\rangle}|^2,\quad(v\in H).$$ Now take $r\neq s$ and $\lambda\in\{1,i\}$. From (\[eq21\_4\_2\_1\]) and (\[eq21\_4\_2\_2\]): $$\sum_{i\in I}(|{\left\langle \delta_r\,|\,e_i\right\rangle}|^2+{\left\langle \delta_r\,|\,e_i\right\rangle}\overline{\lambda}\overline{{\left\langle \delta_s\,|\,e_i\right\rangle}}+
\overline{{\left\langle \delta_r\,|\,e_i\right\rangle}}\lambda{\left\langle \delta_s\,|\,e_i\right\rangle}+|\lambda|^2|{\left\langle \delta_s\,|\,e_i\right\rangle}|^2)=$$ $$\sum_{j\in J}(|{\left\langle \delta_r\,|\,f_j\right\rangle}|^2+{\left\langle \delta_r\,|\,f_j\right\rangle}\overline{\lambda}\overline{{\left\langle \delta_s\,|\,f_j\right\rangle}}+
\overline{{\left\langle \delta_r\,|\,f_j\right\rangle}}\lambda{\left\langle \delta_s\,|\,f_j\right\rangle}+|\lambda|^2|{\left\langle \delta_s\,|\,f_j\right\rangle}|^2).$$ With (\[eq21\_4\_2\_3\]) we can reduce this to $$\overline{\lambda}\sum_{i\in I}{\left\langle \delta_r\,|\,e_i\right\rangle}\overline{{\left\langle \delta_s\,|\,e_i\right\rangle}}+\lambda\sum_{i\in
I}\overline{{\left\langle \delta_r\,|\,e_i\right\rangle}}{\left\langle \delta_s\,|\,e_i\right\rangle}=$$ $$=\overline{\lambda}\sum_{j\in J}{\left\langle \delta_r\,|\,f_j\right\rangle}\overline{{\left\langle \delta_s\,|\,f_j\right\rangle}}+\lambda\sum_{j\in
J}\overline{{\left\langle \delta_r\,|\,f_j\right\rangle}}{\left\langle \delta_s\,|\,f_j\right\rangle}$$ Now take $\lambda=1$ and $\lambda=i$ and the two resulting equation will yield: $$\label{eq21_4_2_4}
\sum_{i\in I}{\left\langle \delta_r\,|\,e_i\right\rangle}\overline{{\left\langle \delta_s\,|\,e_i\right\rangle}}=\sum_{j\in J}{\left\langle \delta_r\,|\,f_j\right\rangle}\overline{{\left\langle \delta_s\,|\,f_j\right\rangle}}$$ Next we prove that $$\sum_{i\in I}|{\left\langle v\,|\,e_i\right\rangle}|^2=\sum_{j\in J}|{\left\langle v\,|\,f_j\right\rangle}|^2$$ holds for all $v\in S$ where $$S:=\{\sum_{k\in K_0}v_k\delta_k\,|\,K_0\subset K\mbox{ finite }\}.$$ For this take an arbitrary $v=\sum_{k\in K_0}v_k\delta_k$, with $K_0$ finite. Then $$\begin{aligned}
\sum_{i\in I}|{\left\langle v\,|\,e_i\right\rangle}|^2&=\sum_{i\in I}|\sum_{k\in K_0}v_k{\left\langle \delta_k\,|\,e_i\right\rangle}|^2\\
&=\sum_{i\in I}\sum_{k,k'\in K_0}v_k\overline{v}_{k'}{\left\langle \delta_k\,|\,e_i\right\rangle}\overline{{\left\langle \delta_{k'}\,|\,e_i\right\rangle}}\\
&=\sum_{k,k'\in K_0}v_k\overline{v}_{k'}\sum_{i\in I}{\left\langle \delta_k\,|\,e_i\right\rangle}\overline{{\left\langle \delta_{k'}\,|\,e_i\right\rangle}}\\
&=\sum_{k,k'\in K_0}v_k\overline{v}_{k'}\sum_{j\in J}{\left\langle \delta_k\,|\,f_j\right\rangle}\overline{{\left\langle \delta_{k'}\,|\,f_j\right\rangle}}\quad(\mbox{with (\ref{eq21_4_2_4})})\\
&=\sum_{j\in J}|{\left\langle v\,|\,f_j\right\rangle}|^2.\end{aligned}$$ To prove the relation for arbitrary $v\in H$, define $\tilde{T}_1:S\rightarrow l^2(I)$, $T_2:H\rightarrow l^2(J)$ by $$\tilde{T}_1v=({\left\langle v\,|\,e_i\right\rangle})_{i\in I},\quad(v\in S),\quad T_2v=({\left\langle v\,|\,f_j\right\rangle})_{j\in J}.$$ Then $\tilde{T}_1$ is a well defined linear operator, $\|T_2v\|^2=\|P_{H_0}v\|^2$ for $v\in H$ and $\|\tilde{T}_1v\|=\|T_2v\|$ for $v\in S$. This shows that $\|\tilde{T}_1v\|^2\leq\|v\|^2$ for $v\in S$. Hence, as $S$ is a dense subspace of $H$, we can extend $\tilde{T}_1$ to a linear operator $T_1$ on $H$ with $$\|T_1v\|^2\leq\|v\|^2,\quad\mbox{for all }v\in H.$$ We claim that $T_1v=({\left\langle v\,|\,e_i\right\rangle})_{i\in I}$ for any $v\in H$. Indeed, $v$ can be approximated by vectors $v_n$ in $S$. Then, for each $i\in I$, $$(T_1v)_i=\lim_{n\rightarrow\infty}(T_1v_n)_i=\lim_{n\rightarrow\infty}{\left\langle v_n\,|\,e_i\right\rangle}={\left\langle v\,|\,e_i\right\rangle}.$$ Also, taking the limit, $$\|T_1v\|^2=\|T_2v\|^2=\|P_{H_0}v\|^2,\quad(v\in H),$$ and this implies (i).
\[th21\_4\_3\] Let $H$ be a Hilbert space $H_0$ a closed subspace , $\{\delta_k\,|\,k\in K\}$ a total set for $H$, and $\{f_j\,|\,j\in J\}$ a NTF for $H_0$. Let $\{e_i\,|\,i\in I\}$ be a family of vectors in $H$. The following affirmations are equivalent:
1. $\{e_i\,|\,i\in I\}$ is a NTF for $H_0$;
2. For all $r,s\in H_0$, $$\sum_{i\in I}{\left\langle \delta_r\,|\,e_i\right\rangle}\overline{{\left\langle \delta_s\,|\,e_i\right\rangle}}=\sum_{j\in J}{\left\langle \delta_r\,|\,f_j\right\rangle}\overline{{\left\langle \delta_s\,|\,f_j\right\rangle}}.$$
Just examine the proof of theorem \[th21\_4\_2\].
\[rem21\_4\_4\] Theorem \[th21\_4\_3\] shows that $\{e_i\,|\, i\in I\}$ is a NTF for $H_0$ if and only if it has the same Gramian as $\{f_j\,|\,j\in J\}$. We recall briefly this notion. The Gramian was introduced by A.Ron and Z.Shen in a series of papers and they used it to analize the structure of shift invariant spaces ([@RS1], [@RS2], [@RS3]).
For a given countable family of vectors $\{e_i\,|\,i\in I\}$ define the operator $K:H\rightarrow l^2(I)$, initially on sequences $f=(f_i)_{i\in I}$ with compact support, by $$\label{eq6_1}
K(f)=\sum_{i\in I}f_ie_i.$$ If $K$ extends to a bounded operator (this is the case when $\{e_i\,|\,i\in I\}$ forms a Bessel family), then its adjoint $K^*:l^2(I)\rightarrow H$ is given by $$\label{eq6_2}
K^*(v)=({\left\langle v\,|\,e_i\right\rangle})_{i\in I},\quad(v\in{l^{2}\left(\mathbb{Z}^n\right)}).$$
Suppose $\{e_i\,|\,i\in I\}$ is a subset of $H$ and $K$ is defined as in (\[eq6\_1\]). The Gramian of the system $\{e_i\,|\,i\in
I\}$ is $G:l^2(I)\rightarrow l^2(I)$ defined by $G=K^*K$. The dual Gramian of the system $\{e_i\,|\,i\in I\}$ is $\tilde{G}:H\rightarrow H$ defined by $\tilde{G}=KK^*$.
Note that $$\label{eq6_2_1}
{\left\langle \tilde{G}f\,|\,g\right\rangle}={\left\langle K^*f\,|\,K^*g\right\rangle}=\sum_{i\in I}{\left\langle f\,|\,e_i\right\rangle}\overline{{\left\langle g\,|\,e_i\right\rangle}}=:B(f,g).$$ The trace function involves expressions of the form $$\sum_{i\in I}|{\left\langle f\,|\,e_i\right\rangle}|^2=B(f,f)={\left\langle \tilde{G}f\,|\,f\right\rangle}.$$ We can recuperate the dual Gramian if these expressions are given, because by polarization $$B(f,g)=\frac{1}{4}\sum_{k=0}^3i^kB(f+i^kg,f+i^kg).$$ $B$, as a sesquilinear form gives rise to an operator and this operator has to be the dual Gramian $\tilde{G}$. Another fact that is worth noticing is that, if $\{e_i\,|\,i\in I\}$ is a NTF for some subspace $H_0$ of ${l^{2}\left(\mathbb{Z}^n\right)}$, then the dual Gramian $\tilde{G}$ is the projection onto $H_0$, $P_{H_0}$. This is because $P_{H_0}$ verifies the equation (\[eq6\_2\_1\]).
\[prop21\_5\] Let $(P_j)_{j\in J}$ be a family of mutually orthogonal projections in a Hilbert space $H$. Then $${\operatorname*{Trace}}(T\sum_{j\in J}P_j)=\sum_{j\in J}{\operatorname*{Trace}}(TP_j),$$ for any positive operator $T$ on $H$.
Consider $\{e_i\,|\,i\in I_j\}$ an orthonormal basis for the range of $P_j$, ($j\in J$). Then, $\{e_i\,|\,i\in\cup
I_j\}$ is an orthonormal basis for the range of $\sum_{j\in J}P_j$. The additivity property is now clear, if we use proposition \[prop21\_4\].
The next proposition is an easy consequence of the additivity of the trace mentioned in proposition \[prop21\_5\].
\[prop21\_6\] Let $(P_j)_{j\in\mathbb{N}}$ be an increasing sequence of projections in some Hilbert space $H$, $P=\sup_{j\in\mathbb{N}}P_j$ and $T$ a positive operator on $H$. Then\
$({\operatorname*{Trace}}(TP_j))_{j\in\mathbb{N}}$ increases to ${\operatorname*{Trace}}(TP)$.
\[prop21\_6\_1\] Let $T$ be a positive operator on a Hilbert space $H$, $P$ a projection on $H$ and $U$ a unitary on $H$. Then $${\operatorname*{Trace}}(UTPU^*)={\operatorname*{Trace}}(TP).$$
Let $\{e_i\,|\,i\in I\}$ be an orthonormal basis for $H$ such that $\{e_i\,|\, i\in I_0\}$ is an orthonormal basis for $UPH$, with $I_0\subset I$. Then $\{U^*e_i\,|\,i\in I_0\}$ is an orthonormal basis for $PH$ and $PU^*e_i=0$ for $i\in I\setminus I_0$; so $${\operatorname*{Trace}}(UTPU^*)=\sum_{i\in I_0}{\left\langle UTPU^*e_i\,|\,e_i\right\rangle}=\sum_{i\in
I_0}{\left\langle TPU^*e_i\,|\,U^*e_i\right\rangle}={\operatorname*{Trace}}(TP).$$
\[lem21\_8\_1\] If $H$ is a Hilbert space, $H_0$ a closed subspace and $f\in H$, then ${\operatorname*{Trace}}(P_fP_{H_0})\leq\|f\|^2$ with equality if and only if $f\in H_0$.
Let $\{e_i\,|\, i\in I\}$ be an orthonormal basis for $H$ such that $\{e_i\,|\, i\in I_0\}$ is an orthonormal basis for $H_0$ with $I_0\subset I$. Then $$\begin{aligned}
\|f\|^2&=\sum_{i\in I_0}|{\left\langle f\,|\,e_i\right\rangle}|^2+\sum_{i\in I\setminus I_0}|{\left\langle f\,|\,e_i\right\rangle}|^2\\
&=\sum_{i\in I_0}{\left\langle P_fe_i\,|\,e_i\right\rangle}+\sum_{i\in I\setminus I_0}|{\left\langle f\,|\,e_i\right\rangle}|^2\\
&={\operatorname*{Trace}}(P_fP_{H_0})+\sum_{i\in I\setminus I_0}|{\left\langle f\,|\,e_i\right\rangle}|^2\end{aligned}$$ Thus the inequality holds and ${\operatorname*{Trace}}(P_fP_{H_0})=\|f\|^2$ iff ${\left\langle f\,|\,e_i\right\rangle}=0$ for all $i\in I\setminus I_0$ which is equivalent to $f\in H_0$.
\[lem21\_9\_1\] Let $H$ be a Hilbert space, $H_1$ and $H_2$ two closed subspaces. Then $H_1\subset H_2$ if and only if ${\operatorname*{Trace}}(P_fP_{H_1})\leq
{\operatorname*{Trace}}(P_fP_{H_2})$ for all $f\in H$.
If $H_1$ is contained in $H_2$ then, using proposition \[prop21\_5\], the inequality between the traces is immediate.
For the converse, assume $f\in H_1$. Then, by lemma \[lem21\_8\_1\], ${\operatorname*{Trace}}(P_fP_{H_1})=\|f\|^2$ so $\|f\|^2\leq{\operatorname*{Trace}}(P_fP_{H_2})$. But then, with the same lemma, ${\operatorname*{Trace}}(P_fP_{H_2})=\|f\|^2$ which implies that $f\in H_2$. As $f$ was arbitrary, the inclusion is proved.
\[Thelocal\]The local trace function
====================================
\[def3\_1\] Let $V$ be a SI subspace of ${L^{2}\left(\mathbb{R}^n\right)}$, $T$ a positive operator on ${l^{2}\left(\mathbb{Z}^n\right)}$ and let $J_{per}$ be the range function associated to $V$. We define the local trace function associated to $V$ and $T$ as the map from $\mathbb{R}^n$ to $[0,\infty]$ given by the formula $$\tau_{V,T}(\xi)={\operatorname*{Trace}}\left(TJ_{per}(\xi)\right),\quad(\xi\in\mathbb{R}).$$ We define the restricted local trace function associated to $V$ and a vector $f$ in ${l^{2}\left(\mathbb{Z}^n\right)}$ by $$\tau_{V,f}(\xi)={\operatorname*{Trace}}\left(P_fJ_{per}(\xi)\right)(=\tau_{V,P_f}(\xi)),\quad(\xi\in\mathbb{R}^n),$$ where $P_f$ is the operator on ${l^{2}\left(\mathbb{Z}^n\right)}$ defined by $P_f(v)={\left\langle v\,|\,f\right\rangle}f$.
\[prop3\_1\] For all $f\in{l^{2}\left(\mathbb{Z}^n\right)}$, $$\tau_{V,f}(\xi)=\|J_{per}(\xi)(f)\|^2,\quad(\mbox{ for a.e. }\xi\in\mathbb{R}^n).$$
Use the proposition \[prop21\_4\].
\[th3\_3\] Let $V$ be a SI subspace of ${L^{2}\left(\mathbb{R}^n\right)}$ and $\phi\subset V$ a NTF generator for $V$. Then for every positive operator $T$ on ${l^{2}\left(\mathbb{Z}^n\right)}$ and any $f\in{l^{2}\left(\mathbb{Z}^n\right)}$, $$\label{eq3_3_1}
\tau_{V,T}(\xi)=\sum_{\varphi\in\phi}{\left\langle T{\mathcal{T}_{per}}\varphi(\xi)\,|\,{\mathcal{T}_{per}}\varphi(\xi)\right\rangle},\quad(\mbox{
for a.e. }\xi\in\mathbb{R}^n);$$ $$\label{eq3_3_2}
\tau_{V,f}(\xi)=\sum_{\varphi\in\phi}|{\left\langle f\,|\,{\mathcal{T}_{per}}\varphi(\xi)\right\rangle}|^2,\quad \mbox{ for a.e. }\xi\in\mathbb{R}^n).$$
According to theorem \[th2\_5\], $\{{\mathcal{T}_{per}}\varphi(\xi)\,|\,\varphi\in\phi\}$ is a NTF for $J_{per}(\xi)$ for a.e. $\xi\in\mathbb{R}^n$. With the proposition \[prop21\_4\] we obtain (\[eq3\_3\_1\]). Take $T=P_f$ and (\[eq3\_3\_1\]) becomes (\[eq3\_3\_2\]).
\[th3\_3\_1\] Let $V$ be a SI subspace of ${L^{2}\left(\mathbb{R}^n\right)}$, $J_{per}$ its periodic range function and $\phi$ a countable subset of ${L^{2}\left(\mathbb{R}^n\right)}$. Then following affirmations are equivalent:
1. $\phi\subset V$ and $\phi$ is a NTF generator for $V$;
2. For every $f\in{l^{2}\left(\mathbb{Z}^n\right)}$ $$\label{eq3_3_1_1}
\sum_{\varphi\in\phi}|{\left\langle f\,|\,{\mathcal{T}_{per}}\varphi(\xi)\right\rangle}|^2=\|J_{per}(\xi)(f)\|^2,\quad\mbox{for a.e. }\xi\in{\mathbb{R}^n}$$
3. For every $0\neq l\in{\mathbb{Z}^n}$ and $\alpha\in\{0,1,i\}$, $$\label{eq3_3_1_2}
\sum_{\varphi\in\phi}|\widehat{\varphi}(\xi)+\overline{\alpha}\widehat{\varphi}(\xi+2l\pi)|^2=\|J_{per}(\xi)(\delta_0+\alpha\delta_l)\|^2,\quad\mbox{for
a.e. }\xi\in{\mathbb{R}^n}.$$
\(i) implies (ii) just as an application of theorem \[th3\_3\] and proposition \[prop3\_1\]. (iii) is just a particular case of (ii), namely $f=\delta_0+\alpha\delta_l$. So assume (iii) holds. Then (\[eq3\_3\_1\_2\]) can be rewritten as: $$\sum_{\varphi\in\phi}{{\left\langle \delta_0+\alpha\delta_l\,|\,{\mathcal{T}_{per}}(\xi)\right\rangle}|^2=\|J_{per}(\delta_0+\alpha\delta_l)}\|^2.$$ Now take $r\neq s\in{\mathbb{Z}^n}$ and apply this equation to $l=s-r$, $\xi=\xi+2\pi r$. A short computation, that uses the periodicity of ${\mathcal{T}_{per}}$ and $J_{per}$ and the fact that $\lambda(r)$ is unitary, will lead to $$\sum_{\varphi\in\phi}|{\left\langle \delta_r+\alpha\delta_s\,|\,{\mathcal{T}_{per}}(\xi)\right\rangle}|^2=\|J_{per}(\xi)(\delta_r+\alpha\delta_s)\|^2,\quad\mbox{
for a.e. }\xi\in{\mathbb{R}^n}.$$ Now we can use theorem \[th21\_4\_2\] to conclude that, for a.e. $\xi\in{\mathbb{R}^n}$, $$\{{\mathcal{T}_{per}}\varphi(\xi)\,|\,\varphi\in\phi\}$$ is a NTF for $J_{per}(\xi)$. And, with theorem \[th2\_5\], (i) is obtained.
Applying this theorem to $V={L^{2}\left(\mathbb{R}^n\right)}$ we deduce the next corollary which can be found also in [@RS1]:
\[cor3\_3\_1\] A countable subset $\varphi$ of ${L^{2}\left(\mathbb{R}^n\right)}$ is a NTF generator for ${L^{2}\left(\mathbb{R}^n\right)}$ if and only if the following equations hold for a.e. $\xi\in{\mathbb{R}^n}$: $$\sum_{\varphi\in\phi}|\widehat{\varphi}(\xi)|^2=1,$$ $$\sum_{\varphi\in\phi}\widehat{\varphi}(\xi)\overline{\widehat{\varphi}}(\xi+2l\pi)=0,(l\in{\mathbb{Z}^n},l\neq 0).$$
\[th3\_3\_2\] Let $V$ be a SI subspace of ${L^{2}\left(\mathbb{R}^n\right)}$, $\phi_1$ a NTF generator for $V$ and $\phi_2$ a countable family of vectors from ${L^{2}\left(\mathbb{R}^n\right)}$. The following affirmations are equivalent:
1. $\phi_2\subset V$ and $\phi_2$ is a NTF generator for $V$;
2. For every $l\in{\mathbb{Z}^n}$, $$\sum_{\varphi\in\phi_2}\widehat{\varphi}(\xi)\overline{\widehat{\varphi}}(\xi+2l\pi)=
\sum_{\varphi\in\phi_1}\widehat{\varphi}(\xi)\overline{\widehat{\varphi}}(\xi+2l\pi),\quad\mbox{for a.e. }\xi\in{\mathbb{R}^n}.$$
By theorem \[th2\_5\], (i) is equivalent to $$\label{eq3_3_2_1}
\{{\mathcal{T}_{per}}\varphi(\xi)\,|\,\varphi\in\phi_2\}\mbox{ is a NTF for }J_{per}(\xi)\mbox{ for a.e. }\xi\in{\mathbb{R}^n}.$$ But we know that $\{{\mathcal{T}_{per}}\varphi(\xi)\,|\,\varphi\in\phi_1\}$ is a NTF for $J_{per}(\xi)$ (again, by theorem \[th2\_5\]). Therefore, using theorem \[th21\_4\_3\], (\[eq3\_3\_2\_1\]) is equivalent to:
For every $r,s\in{\mathbb{Z}^n}$ and a.e. $\xi\in{\mathbb{R}^n}$, $$\sum_{\varphi\in\phi_2}{\left\langle \delta_r\,|\,{\mathcal{T}_{per}}\varphi(\xi)\right\rangle}\overline{{\left\langle \delta_s\,|\,{\mathcal{T}_{per}}\varphi(\xi)\right\rangle}}
=\sum_{\varphi\in\phi_1}{\left\langle \delta_r\,|\,{\mathcal{T}_{per}}\varphi(\xi)\right\rangle}\overline{{\left\langle \delta_s\,|\,{\mathcal{T}_{per}}\varphi(\xi)\right\rangle}},$$ which is exactly $$\sum_{\varphi\in\phi_2}\widehat{\varphi}(\xi+2r\pi)\overline{\widehat{\varphi}}(\xi+2s\pi)=
\sum_{\varphi\in\phi_1}\widehat{\varphi}(\xi+2r\pi)\overline{\widehat{\varphi}}(\xi+2s\pi),\quad(r,s\in{\mathbb{Z}^n},\xi\in{\mathbb{R}^n}).$$ This implies (ii) and (ii) implies this, because one can take $l=s-r$, $\xi=\xi+2r\pi$.
\[rem3\_3\_3\] Theorem \[th3\_3\_2\] shows that $\phi_2$ is a NTF generator for $V$ iff it has the same dual Gramian as $\phi_1$. Recall, that the dual Gramian of a countable subset $\phi$ of ${L^{2}\left(\mathbb{R}^n\right)}$ is defined as the function which assigns to each $\xi\in{\mathbb{R}^n}$ the dual Gramian of the set $\{{\mathcal{T}_{per}}\varphi(\xi)\,|\,\varphi\in\phi\}$ (see remark \[rem21\_4\_4\] and [@RS1], [@RS2], [@RS3], [@Bo1] for details). The dual Gramian satisfies the equations: $${\left\langle \tilde{G}(\xi)\delta_r\,|\,\delta_s\right\rangle}=\sum_{\varphi\in\phi}\widehat{\varphi}(\xi+2\pi r)\overline{\widehat{\varphi}}(\xi+2\pi s).$$
\[prop3\_4\] [**\[Periodicity\]**]{} Let $V$ be a SI subspace, $T$ a positive operator on ${l^{2}\left(\mathbb{Z}^n\right)}$, $f\in{l^{2}\left(\mathbb{Z}^n\right)}$. Then, for $k\in\mathbb{Z}^n$ $$\label{eq3_4_1}
\tau_{V,T}(\xi+2k\pi)=\tau_{V,\lambda(k)T\lambda(k)^*}(\xi),\quad
(\mbox{ for a.e. }\xi\in\mathbb{R}^n);$$ $$\label{eq3_4_2}
\tau_{V,f}(\xi+2k\pi)=\tau_{V,\lambda(k)f}(\xi),\quad (\mbox{ for a.e. }\in\mathbb{R}^n),$$ where $\lambda(k)$ is the unitary operator on ${l^{2}\left(\mathbb{Z}^n\right)}$ defined by $(\lambda(k)\alpha)(l)=\alpha(l-k)$ for all $l\in\mathbb{Z}^n$, $\alpha\in{l^{2}\left(\mathbb{Z}^n\right)}$.
The periodicity of the local trace function is a consequence of the periodicity of the range function. Indeed, by definition \[def2\_0\], we know that for a.e. $\xi\in\mathbb{R}^n$ and every $k\in\mathbb{Z}^n$, $$J_{per}(\xi+2k\pi)=\lambda(k)^*J_{per}(\xi)\lambda(k).$$ Apply this in the definition of the local trace function and use proposition \[prop21\_6\_1\]: $$\tau_{V,T}(\xi+2k\pi)={\operatorname*{Trace}}(TJ_{per}(\xi+2k\pi))={\operatorname*{Trace}}(T\lambda(k)^*J_{per}(\xi)\lambda(k))$$ $$={\operatorname*{Trace}}(\lambda(k)T\lambda(k)^*J_{per}(\xi))=\tau_{V,\lambda(k)T\lambda(k)^*}(\xi).$$ Since $\lambda(k)P_f\lambda(k)^*=P_{\lambda(k)f}$, (\[eq3\_4\_2\]) follows from (\[eq3\_4\_1\]).
\[prop3\_5\] Let $V$ be a SI space, $T,S$ positive operators on ${l^{2}\left(\mathbb{Z}^n\right)}$, $f\in{l^{2}\left(\mathbb{Z}^n\right)}$
1. $0\leq\tau_{V,T}(\xi)\leq\infty$, $0\leq\tau_{V,f}\leq\|f\|^2$;
2. $\tau_{V,T+S}=\tau_{V,T}+\tau_{V,S}$;
3. $\tau_{V,\lambda T}=\lambda\tau_{V,T},\quad (\lambda>0)$;
4. $\tau_{V,\lambda f}=|\lambda|^2\tau_{V,f},\quad(\lambda\in\mathbb{C}).$
Everything follows from section \[Thetrace\].
\[prop3\_6\] [**\[Additivity\]**]{} Suppose $(V_i)_{i\in I}$ are mutually orthogonal SI subspaces ($I$ countable) and let $V=\oplus_{i\in I}V_i$. Then, for every positive operator $T$ on ${l^{2}\left(\mathbb{Z}^n\right)}$ and every $f\in{l^{2}\left(\mathbb{Z}^n\right)}$: $$\label{eq3_6_1}
\tau_{V,T}=\sum_{i\in I}\tau_{V_i,T},\quad\mbox{a.e. on }\mathbb{R}^n;$$ $$\label{eq3_6_2}
\tau_{V,f}=\sum_{i\in I}\tau_{V_i,f},\quad\mbox{a.e. on }\mathbb{R}^n.$$
Let $J_i$ be the periodic range function of $V_i$, ($i\in I$) and $J$ the periodic range function of $V$. The range function is additive (proposition \[prop2\_2\]) so $$J(\xi)=\sum_{i\in I}J_i(\xi),\quad\mbox{ a.e. on }\mathbb{R}^n.$$ Also the trace has additive properties (proposition \[prop21\_5\]) and these imply the additivity of the local trace function expressed in (\[eq3\_6\_1\]). Again (\[eq3\_6\_2\]) is just a particular case of (\[eq3\_6\_1\]).
\[prop3\_7\] [**\[Monotony and injectivity\]**]{} Let $V,W$ be SI subspaces.
\(i) $V\subset W$ iff $\tau_{V,T}\leq\tau_{W,T}$ a.e. for all positive operators $T$ iff $\tau_{V,f}\leq\tau_{W,f}$ a.e. for all $f\in{l^{2}\left(\mathbb{Z}^n\right)}$.
\(ii) $V=W$ iff $\tau_{V,T}=\tau_{W,T}$ a.e. for all positive operators $T$ iff $\tau_{V,f}=\tau_{W,f}$ a.e for all $f\in{l^{2}\left(\mathbb{Z}^n\right)}$.
\(i) It is clear that the first statement implies the second; just use the additivity property for the SI spaces $V$ and $W\ominus V$. The third statement is just a particular case of the second one. So the only interesting implication is from the third statement to the first one.
Let $J_V$ and $J_W$ be the corresponding periodic range functions for $V$ and $W$. The hypothesis implies, according to lemma \[lem21\_9\_1\], that $$J_V(\xi)\subset J_W(\xi),\quad(\mbox{ for a.e. }\xi\in\mathbb{R}^n),$$ and this implies in turn that $V\subset W$ (just look at theorem \[th2\_1\]).
\(ii) is a consequence of (i) by a double inclusion argument.
For $a\in\mathbb{R}^n$ we define the modulation of $f\in{L^{2}\left(\mathbb{R}^n\right)}$ by $$M_a(f)(x)=e^{i{\left\langle a\,|\,x\right\rangle}}f(x),\quad(x\in\mathbb{R}^n).$$ The local trace function behaves nicely under modulation. This is expressed in the next proposition.
\[prop3\_8\] [**\[Modulation\]**]{} Let $V$ be a SI subspace and $a\in\mathbb{R}^n$. Then $M_aV$ is a SI subspace and for all positive operators $T$ on ${l^{2}\left(\mathbb{Z}^n\right)}$ and all vectors $f\in{l^{2}\left(\mathbb{Z}^n\right)}$: $$\label{eq3_8_1}
\tau_{M_aV,T}(\xi)=\tau_{V,T}(\xi-a),\quad\mbox{ for a.e. }\xi\in\mathbb{R}^n;$$ $$\label{eq3_8_2}
\tau_{M_aV,f}(\xi)=\tau_{V,f}(\xi-a),\quad\mbox{ for a.e. }\xi\in\mathbb{R}^n.$$
The modulation and the translations satisfy a commutation relation: for $k\in\mathbb{Z}^n$ and $\varphi\in{L^{2}\left(\mathbb{R}^n\right)}$, $$T_kM_a\varphi(x)=e^{i{\left\langle a\,|\,x-k\right\rangle}}\varphi(x-k)=e^{-i{\left\langle a\,|\,k\right\rangle}}M_aT_k\varphi(x),\quad(x\in\mathbb{R}^n).$$ This relation shows that $M_aV$ is shift invariant.
Now take a NTF generator $\phi$ for $V$ (it exists by theorem \[th1\_3\]). Then, as $M_a$ is unitary, $$\{M_aT_k\varphi\,|\,\varphi\in\phi\}$$ is a NTF for $M_aV$. Using the commutation relation and the fact that $e^{-i{\left\langle a\,|\,k\right\rangle}}$ are just constants of modulus 1, we see that $$\{T_kM_a\varphi\,|\,\varphi\in\phi\}$$ is a NTF for $M_aV$. Therefore we can safely use theorem \[th3\_3\] and compute: $$\tau_{M_aV,T}(\xi)=\sum_{\varphi\in\phi}{\left\langle T{\mathcal{T}_{per}}(M_a\varphi)(\xi)\,|\,{\mathcal{T}_{per}}(M_a\varphi)(\xi)\right\rangle}.$$ But $${\mathcal{T}_{per}}M_a\varphi(\xi)=\left(\widehat{M_a\varphi}(\xi+2k\pi)\right)_{k\in\mathbb{Z}^n}=\left(\widehat{\varphi}(\xi-a+2k\pi)\right)_{k\in\mathbb{Z}^n}$$ $$={\mathcal{T}_{per}}\varphi(\xi-a),$$ and (\[eq3\_8\_1\]) follows with (\[eq3\_8\_2\]) as its consequence.
The dilation by an $n\times n$ non-singular matrix $A$ is the unitary operator on ${L^{2}\left(\mathbb{R}\right)}$ defined by $$D_Af(x)=|\operatorname*{det}A|^{\frac{1}{2}}f(Ax),\quad(x\in\mathbb{R}^n,f\in{L^{2}\left(\mathbb{R}^n\right)}).$$ We will consider only matrices $A$ which preserve the lattice $\mathbb{Z}^n$, because in this case $D_AV$ is shift invariant whenever $V$ is.
\[prop3\_9\] [**\[Dilation\]**]{} Let $V$ be a SI subspace and $A$ an $n\times n$ integer matrix with $\operatorname*{det}A\neq 0$. Then $D_AV$ is shift invariant and, for every positive operator $T$ on ${l^{2}\left(\mathbb{Z}^n\right)}$ and every vector $f\in{l^{2}\left(\mathbb{Z}^n\right)}$: $$\label{eq3_9_1}
\tau_{D_AV,T}(\xi)=\sum_{d\in\mathcal{D}}\tau_{V,D_d^*TD_d}\left(\left(A^*\right)^{-1}(\xi+2d\pi)\right),\quad\mbox{ for a.e.
}\xi\in\mathbb{R}^n,$$ $$\label{eq3_9_2}
\tau_{D_AV,f}(\xi)=\sum_{d\in\mathcal{D}}\tau_{V,D_d^*f}\left(\left(A^*\right)^{-1}(\xi+2d\pi)\right),\quad\mbox{
for
a.e. }\xi\in\mathbb{R}^n,$$ where $\mathcal{D}$ is a complete set of representatives of the cosets $\mathbb{Z}^n/A^*\mathbb{Z}^n$ and $D_d$ is the linear operator on ${l^{2}\left(\mathbb{Z}^n\right)}$ defined by $$(D_d\alpha)(k)=\left\{\begin{array}{ccc}
\alpha(l),&\mbox{if}&k=d+A^*l\\
0,& & otherwise
\end{array}
\right.,\quad(k\in\mathbb{Z}^n,\alpha\in{l^{2}\left(\mathbb{Z}^n\right)}).$$
The dilation and the translation satisfy the following commutation relation which can be easily verified: $$T_kD_A=D_AT_{Ak},\quad(k\in\mathbb{Z}^n).$$ This shows that $D_AV$ is shift invariant.
We can decompose $V$ as the orthogonal sum $V=\oplus_{i\in I}S(\varphi_i)$, where $\varphi_i$ is a quasi-orthogonal generator of $S(\varphi_i)$ (see theorem \[th1\_3\]). Since $D_A$ is unitary, $D_AV=\oplus_{i\in I}D_AS(\varphi_i)$ and, using the additivity property of the local trace function (proposition \[prop3\_6\]), it suffices to prove the formula (\[eq3\_9\_1\]) just for the case when $V=S(\varphi)$ with $\varphi$ quasi-orthogonal generator for $S(\varphi)$. We will assume this is the case.
From the commutation relation we see that, if $\mathcal{L}$ is a complete set of $|\operatorname*{det}A|$ representatives of the cosets $\mathbb{Z}^n/A\mathbb{Z}^n$, then $$\{D_AT_l\varphi\,|\, l\in\mathcal{L}\}$$ will span $D_AV$ by translations.
For $l\in\mathcal{L}$, consider $$\begin{aligned}
\phi_l(\xi)&={\mathcal{T}_{per}}(D_AT_l\varphi)(\xi)\\
&=\left(|\operatorname*{det}
A|^{-\frac{1}{2}}\widehat{\varphi}\left(\left(A^*\right)^{-1}(\xi+2k\pi)\right)
e^{-i{\left\langle (A^*)^{-1}(\xi+2k\pi)\,\,|\,\,l\right\rangle}}\right)_{k\in\mathbb{Z}^n}.\end{aligned}$$ For $d\in\mathcal{D}$ define $\psi_d\in L^2_{per}({\mathbb{R}^n},{l^{2}\left(\mathbb{Z}^n\right)})$ by $$\begin{aligned}
\psi_d(\xi)(k)&=\left\{\begin{array}{ccc}
\widehat{\varphi}((A^*)^{-1}(\xi+2k\pi)),&\mbox{if}&k\in d+A^*{\mathbb{Z}^n}\\
0,& &\mbox{otherwise}
\end{array}
\right.\\
&=\left\{\begin{array}{ccc}
\widehat{\varphi}((A^*)^{-1}(\xi+2d\pi)+2\pi l),&\mbox{if}&k= d+A^*l,\mbox{ with }l\in{\mathbb{Z}^n}\\
0,& &\mbox{otherwise}
\end{array}
\right.\\
&=D_d\left({\mathcal{T}_{per}}\varphi((A^*)^{-1}(\xi+2\pi d))\right).\end{aligned}$$ Then $\phi_l(\xi)$ and $\psi_d(\xi)$ are related by the following linear equations: $$\phi_l(\xi)=e^{-i{\left\langle (A^*)^{-1}\xi\,\,|\,\,l\right\rangle}}|\operatorname*{det}A|^{-1/2}
\sum_{d\in\mathcal{D}}e^{-i{\left\langle (A^*)^{-1}2\pi d\,\,|\,\,l\right\rangle}}\psi_d(\xi).$$ Since the $|\operatorname*{det}A|\times |\operatorname*{det}A|$ matrix $$\left( |\operatorname{det}A|^{-1/2}e^{-i{\left\langle (A^*)^{-1}2\pi
d\,\,|\,\,l\right\rangle}}\right)_{d\in\mathcal{D},l\in\mathcal{L}}$$ is unitary, it follows that $\{\psi_d(\xi)\,|\, d\in\mathcal{D}\}$ and $\{\phi_l(\xi)\,|\, l\in\mathcal{L}\}$ span the same subspace of ${l^{2}\left(\mathbb{Z}^n\right)}$, namely $J_{D_AV}(\xi)$ (use theorem \[th2\_1\]).
$\varphi$ is a quasi-orthogonal generator so $\operatorname*{Per}|\widehat{\varphi}|^2$ is a characteristic function that is $\|{\mathcal{T}_{per}}\varphi(\xi)\|_{{l^{2}\left(\mathbb{Z}^n\right)}}$ is either 0 or 1, and as $D_d$ is an isometry, $\|\psi_d(\xi)\|_{{l^{2}\left(\mathbb{Z}^n\right)}}\in\{0,1\}$. Also $\psi_d(\xi)$ and $\psi_{d'}(\xi)$ are perpendicular when $d\neq d'$ and, in conclusion $\{\psi_d(\xi)\,|\, d\in\mathcal{D}\}$ is a NTF for $J_{D_AV}(\xi)$. Therefore we can use these vectors to compute the local trace function (proposition \[prop21\_4\]): $$\begin{aligned}
\tau_{D_AV,T}(\xi)&=\sum_{d\in\mathcal{D}}{\left\langle T\psi_d(\xi)\,|\,\psi_d(\xi)\right\rangle}\\
&=\sum_{d\in\mathcal{D}}{\left\langle TD_d\left({\mathcal{T}_{per}}\varphi((A^*)^{-1}(\xi+2\pi
d))\right)\,|\,D_d\left({\mathcal{T}_{per}}\varphi((A^*)^{-1}(\xi+2\pi d))\right)\right\rangle}\\
&=\sum_{d\in\mathcal{D}}{\left\langle D_d^*TD_d\left({\mathcal{T}_{per}}\varphi((A^*)^{-1}(\xi+2\pi
d))\right)\,|\,{\mathcal{T}_{per}}\varphi((A^*)^{-1}(\xi+2\pi d))\right\rangle}\\
&=\sum_{d\in\mathcal{D}}\tau_{V,D_d^*TD_d}((A^*)^{-1}(\xi+2\pi d))\end{aligned}$$ This proves (\[eq3\_9\_1\]). (\[eq3\_9\_2\]) follows from (\[eq3\_9\_1\]) because $D_d^*P_fD_d=P_{D_d^*f}$.
As we promised, the local trace function incorporates the dimension function and the spectral function defined by M.Bownik and Z.Rzeszotnik in [@BoRz].
Recall that for a shift invariant subspace $V$ of ${L^{2}\left(\mathbb{R}^n\right)}$, its dimension function is defined as $${\operatorname*{dim}}_V(\xi)=\operatorname*{dim}J_{per}(\xi),\quad(\xi\in{\mathbb{R}^n}),$$ where $J_{per}$ is the periodic function associated to $V$.
The spectral function introduced in [@BoRz] is defined by $$\sigma_V(\xi+2k\pi)=\|J_{per}(\xi)\delta_k\|^2,\quad(\xi\in[-\pi,\pi]^n,k\in{\mathbb{Z}^n}),$$ where $\delta_k\in{l^{2}\left(\mathbb{Z}^n\right)}$, $\delta_k(l)=\left\{\begin{array}{ccc}
1,&\mbox{if}&k=l\\
0,& &\mbox{otherwise}
\end{array}
\right.$ $(k\in\mathbb{Z}^n)$.
For a similar treatment of the dimension function, using the Gramian the reader can also consult [@RS4].
\[prop3\_10\] Let $V$ be a SI space.\
(i) $$\label{eq3_10_1}
\tau_{V,I}={\operatorname*{dim}}_V.$$ (ii) $$\label{eq3_10_2}
\tau_{V,\delta_0}=\sigma_V.$$
Let $J_{per}$ be the periodic range function of $V$. $$\tau_{V,I}(\xi)={\operatorname*{Trace}}(IJ_{per}(\xi))=\operatorname*{dim}J_{per}(\xi)=\dim_V(\xi).$$
For $\xi\in [-\pi,\pi)^n$, by proposition \[prop3\_1\], $$\tau_{V,\delta_0}(\xi)=\|J_{per}(\xi)\delta_0\|^2=\sigma_V(\xi).$$ If $k\in\mathbb{Z}^n$ then, using the periodicity of the local trace function stated in proposition \[prop3\_4\], $$\tau_{V,\delta_0}(\xi+2k\pi)=\tau_{V,\lambda(k)\delta_0}(\xi)=\tau_{V,\delta_k}(\xi)=
\|J_{per}(\xi)\delta_k\|^2=\sigma_V(\xi).$$
\[convergence\]Convergence theorems
===================================
In this section we study the behavior of the local trace function with respect to limits. More precisely, we consider the following question: if $(V_j)_{j\in\mathbb{N}}$ is a sequence of SI subspaces of ${L^{2}\left(\mathbb{R}^n\right)}$ such that $P_{V_j}$ converges in the strong operator topology to $P_V$ for some SI subspace $V$, what can be said about the convergence of the local trace functions $\tau_{V_i,T}$?
We begin with a lemma and then give some useful partial answers to this question.
\[le4\_1\] Suppose $(V_j)_{j\in\mathbb{N}}$ and $V$ are SI subspaces of ${L^{2}\left(\mathbb{R}^n\right)}$ such that $P_{V_j}$ converges in the strong operator topology to $P_V$. Then for any $f\in{L^{2}\left(\mathbb{R}^n\right)}$ and any $A>0$ $$\int_{[-A,A]^n}\|J_{V_j}(\xi)({\mathcal{T}_{per}}f(\xi))-J_V(\xi)({\mathcal{T}_{per}}f(\xi))\|^2\,d\xi\mbox{ converges to
}0\mbox{ as }j\rightarrow\infty,$$ where $J_{V_j}$ and $J_{V}$ are the corresponding periodic range functions.
We can consider $A=\pi$ because then the result is obtained using the periodicity of the range function. With proposition \[prop2\_4\], $$J_{V_j}(\xi)({\mathcal{T}_{per}}f(\xi))={\mathcal{T}_{per}}(P_{V_j}f)(\xi),\quad (\xi\in\mathbb{R}^n).$$ And then $$\int_{[-\pi,\pi]^n}\|J_{V_j}(\xi)({\mathcal{T}_{per}}f(\xi))-J_V(\xi)({\mathcal{T}_{per}}f(\xi))\|^2\,d\xi$$ $$=\int_{[-\pi,\pi]^n}\|{\mathcal{T}_{per}}(P_{V_j}f)(\xi)-{\mathcal{T}_{per}}(P_Vf)(\xi)\|^2\,d\xi$$ $$=(2\pi)^n\|P_{V_j}f-P_Vf\|^2_{{L^{2}\left(\mathbb{R}^n\right)}}\rightarrow 0,\mbox{ as }j\rightarrow\infty.$$
\[th4\_2\] Let $(V_j)_{j\in\mathbb{N}}$ and $V$ be SI subspaces such that $P_{V_j}$ converges to $P_V$ in the strong operator topology. Then, for any bounded measurable set $E$ and every $f\in{l^{2}\left(\mathbb{Z}^n\right)}$ $$\int_E|\tau_{V_j,f}(\xi)-\tau_{V,f}(\xi)|\,d\xi\rightarrow 0,\mbox{ as
}j\rightarrow\infty.$$
Let $J_{V_j}$ and $J_V$ be the corresponding periodic range function. We can assume $E=[-\pi,\pi]^n$ because then the result is a consequence of the periodicity of the local trace function (proposition \[prop3\_4\]) and we can assume also $\|f\|=1$. Using lemma \[le4\_1\] for a $g\in{L^{2}\left(\mathbb{R}^n\right)}$ with $\mathcal{T}g(\xi)=f$ for all $\xi\in[-\pi,\pi]^n$, we have $$\begin{aligned}
0&=\lim_{j\rightarrow\infty}\int_{[-\pi,\pi]^n}\|J_{V_j}(\xi)f-J_V(\xi)f\|^2\,d\xi\\
&\geq\limsup_{j\rightarrow\infty}\int_{[-\pi,\pi]^n}\left|\|J_{V_j}(\xi)f\|-\|J_V(\xi)f\|\right|^2\,d\xi\\
&=\limsup_{j\rightarrow\infty}\int_{[-\pi,\pi]^n}|\tau_{V_j,f}^{1/2}(\xi)-\tau_{V,f}^{1/2}(\xi)|^2\,d\xi\\
&\geq\frac{1}{4}\limsup_{j\rightarrow\infty}\int_{[-\pi,\pi]^n}|\tau_{V_j,f}(\xi)-\tau_{V,f}(\xi)|^2\,d\xi\\
&\geq\frac{1}{4}(2\pi)^n\limsup_{j\rightarrow\infty}\int_{[-\pi,\pi]^n}|\tau_{V_j,f}(\xi)-\tau_{V,f}(\xi)|\,d\xi,\end{aligned}$$ where for the third line we used proposition \[prop3\_1\], for the fourth we used the fact that $\tau_{V,f},\tau_{V_j,f}$ are less then $\|f\|^2=1$ (proposition \[prop3\_5\] (i)) and for the last one we used Holder’s inequality.
\[th4\_3\] Let $(V_j)_{j\in\mathbb{N}}$ be an increasing sequence of SI subspaces of ${L^{2}\left(\mathbb{R}^n\right)}$, $$V:=\overline{\bigcup_{j\in\mathbb{N}}V_j},$$ $T$ a positive operator on ${l^{2}\left(\mathbb{Z}^n\right)}$ and $f\in{l^{2}\left(\mathbb{Z}^n\right)}$. Then, for a.e. $\xi\in\mathbb{R}^n$, $\tau_{V_j,T}(\xi)$ increases to $\tau_{V,T}(\xi)$ and $\tau_{V_j,f}(\xi)$ increases to $\tau_{V,f}(\xi)$.
The monotony is a consequence of proposition \[prop3\_7\].
Again, we denote by $J_{V_j}$ and $J_V$ the corresponding periodic range function. By proposition \[prop2\_3\], $(J_{V_j}(\xi))_{j\in\mathbb{N}}$ is increasing and $$J_V(\xi)=\overline{\bigcup_{j\in\mathbb{N}}J_{V_j}(\xi)},$$ for almost every $\xi\in\mathbb{R}^n$.
Apply now proposition \[prop21\_6\] to our situation to conclude that $$\tau_{V,T}(\xi)={\operatorname*{Trace}}(TJ_V(\xi))=\lim_{j\rightarrow\infty}{\operatorname*{Trace}}(TJ_{V_j}(\xi))=\lim_{j\rightarrow\infty}\tau_{V_j,T}(\xi).$$ The second statement is a particular case of the first.
\[wavelets\]Wavelets and the local trace function
=================================================
We reserved this section for applications of the local trace function to wavelets. Throughout this section $A$ will be an $n\times n$ dilation matrix (i.e all eigenvalues $\lambda$ have $|\lambda|>1$), and $A$ preserves the lattice ${\mathbb{Z}^n}$ (i.e. $A{\mathbb{Z}^n}\subset {\mathbb{Z}^n}$).
\[rem5\_2\] The local trace function can be used to obtain the characterization of wavelets (see [@Bo2] or [@Ca]), namely:
A set $\Psi=\{\psi^1,...,\psi^L\}$ is a NTF wavelet (i.e. the set $$X(\Psi):=\{D_A^jT_k\psi^l\,|\,j\in\mathbb{Z},k\in{\mathbb{Z}^n},l\in\{1,...,L\}\}$$ is a NTF for ${L^{2}\left(\mathbb{R}^n\right)}$) if and only if the following equations hold for a.e. $\xi\in{\mathbb{R}^n}$: $$\label{eq5_2_1}
\sum_{\psi\in\Psi}\sum_{j=-\infty}^\infty|\widehat{\psi}|^2((A^*)^j\xi)=1,$$ $$\label{eq5_2_2}
\sum_{\psi\in\Psi}\sum_{j\geq0}\widehat{\psi}((A^*)^j\xi)\overline{\widehat{\psi}}((A^*)^j(\xi+2s\pi))=0,\quad(s\in{\mathbb{Z}^n}\setminus A^*{\mathbb{Z}^n}).$$
The argument used here will be the same as the one used in [@Bo2], the only difference is that, instead of the Gramian we employ the local trace function. Here is a sketch of the proof: It is known (see [@CSS]) that $X(\Psi)$ is an affine NTF for ${L^{2}\left(\mathbb{R}^n\right)}$ if and only if the quasi-affine system $X^q(\Psi)$ is a NTF for ${L^{2}\left(\mathbb{R}^n\right)}$. Recall that $$X^q(\Psi):=\{\tilde{\psi}^l_{j,k}\,|j\in\mathbb{Z},k\in{\mathbb{Z}^n},l\in\{1,...,L\}\},$$ with the convention $$\tilde{\psi}_{j,k}(x)=\left\{\begin{array}{ccc}
D_A^jT_k\psi(x),&\mbox{if}&j\geq0,k\in{\mathbb{Z}^n}\\
|\operatorname*{det} A|^{j/2}T_kD_A^j\psi(x)&\mbox{if}&j<0,k\in{\mathbb{Z}^n}.
\end{array}\right.$$ This can be reformulated as $$\{\tilde{\psi}^l_{j,0}\,|\, j<0\}\cup\{\tilde{\psi}_{j,r}^l\,|\,j\geq0,r\in\mathcal{L}_j\}$$ is a NTF generator for ${L^{2}\left(\mathbb{R}^n\right)}$ (here $\mathcal{L}_j$ is a complete set of representatives of ${\mathbb{Z}^n}/A^j{\mathbb{Z}^n}$.
Now use corollary \[cor3\_3\_1\], and all one has to do is a computation. The computation is done in lemma 2.3 in [@Bo2] and one obtains, for a.e. $\xi\in{\mathbb{R}^n}$, $$\sum_{\psi\in\Psi}\sum_{j=-\infty}^\infty|\widehat{\psi}|^2((A^*)^j(\xi)=1,$$ $$t_{(A^*)^{-m}p}((A^*)^{-m}(\xi))=0,\quad(p\in{\mathbb{Z}^n},p\neq0),$$ where $$t_s(\xi):=\sum_{\psi\in\Psi}\sum_{j\geq0}\widehat{\psi}((A^*)^j\xi)\overline{\widehat{\psi}}((A^*)^j(\xi+2s\pi)),\quad(\xi\in{\mathbb{R}^n},s\in{\mathbb{Z}^n}\setminus
A^*{\mathbb{Z}^n})$$ and $m=\max\{j\in\mathbb{Z}\,|\,(A^*)^{-j}p\in{\mathbb{Z}^n}\}$. This leads immediately to the equivalence to (\[eq5\_2\_1\]) and (\[eq5\_2\_2\]).
Another observation is that, if we know that the equivalence “ $X(\psi)$ is a NTF for ${L^{2}\left(\mathbb{R}^n\right)}$ iff (\[eq5\_2\_1\]) and (\[eq5\_2\_2\]) hold”, then the argument above also shows that the affine system $X(\psi)$ is a NTF for ${L^{2}\left(\mathbb{R}^n\right)}$ iff $X^q(\psi)$ is a NTF for ${L^{2}\left(\mathbb{R}^n\right)}$.
\[th5\_1\] Let $\Psi=\{\psi^1,...,\psi^L\}$ be a semiorthogonal wavelet, i.e. the affine system $$\{D_{A}^jT_k\psi\,|\, j\in\mathbb{Z}, k\in\mathbb{Z}^n,\psi\in\Psi\}$$ is a NTF for ${L^{2}\left(\mathbb{R}^n\right)}$ and $W_i\perp W_j$ for $i\neq j$, where $$W_j=\overline{\operatorname*{span}}\{D_{A}^jT_k\psi\,|\,k\in\mathbb{Z}^n,\psi\in\Psi\}=D_{A}^j(S(\Psi)),\quad(j\in\mathbb{Z}),$$ $$V_j=\bigoplus_{i<j}W_i.$$ Then\
(i) The set $$\{|\operatorname*{det}A|^{j/2}T_kD_{A}^j\psi\,|\,j<0,k\in\mathbb{Z}^n\}$$ is a NTF for $V_0$.\
(ii) For $\psi\in\Psi$, $j\geq 1$ and $\xi\in\mathbb{R}^n$, denote by $f_{\psi}^j(\xi)$ the vector in ${l^{2}\left(\mathbb{Z}^n\right)}$ defined by $$f_{\psi}^j(\xi)(k)=\widehat{\psi}((A^*)^j(\xi+2k\pi)),\quad(k\in{\mathbb{Z}^n}).$$ Then, for almost every $\xi\in\mathbb{R}^n$, the set $$\{f_{\psi}^j(\xi)\,|\, j\geq 1,\psi\in\Psi\}$$ is a NTF for $J_{V_0}(\xi)$, where $J_{V_0}$ is the periodic range function of $V_0$.\
(iii) For every positive operator $T$ on ${l^{2}\left(\mathbb{Z}^n\right)}$, every $f\in{l^{2}\left(\mathbb{Z}^n\right)}$ and almost every $\xi\in\mathbb{R}^n$, $$\label{eq5_1_1}
\tau_{V_0,f}(\xi)=\sum_{\psi\in\Psi}\sum_{j=1}^\infty{\left\langle Tf_{\psi}^j(\xi)\,|\,f_{\psi}^j(\xi)\right\rangle},$$ $$\label{eq5_1_2}
\tau_{V_0,f}(\xi)=\sum_{\psi\in\Psi}\sum_{j=1}^\infty|\sum_{k\in\mathbb{Z}^n}\widehat{\psi}((A^*)^j(\xi+2k\pi))\overline{f}_k|^2.$$
\(i) follows, as we mentioned before, from the equivalence between affine frames and quasi-affine frames (see [@CSS] or theorem 1.4 in [@Bo2]) and the orthogonality relations given in the hypothesis.
\(ii) If we compute the Fourier transform of $|\operatorname*{det}A|^{-j/2}D_{A}^{-j}\psi$ for $j>0$ we get $\widehat{\psi}((A^*)^j\xi)$ and (ii) is obtained from (i) and theorem \[th2\_5\].
\(iii) The fact asserted in (ii) shows that we can use the vectors $f_{\psi}^j$ to compute the trace and the resulting formulas are exactly (\[eq5\_1\_1\]) and (\[eq5\_1\_2\]).
\[rem5\_3\] Another fundamental fact from the theory of wavelets can be obtained with the aid of the local trace function: the dimension function equals the trace function (see [@Web]). Let’s recall some notions. Consider $\Psi:=\{\psi^1,...,\psi^L\}$ a semi-orthogonal wavelet (as in theorem \[th5\_1\]). We keep the notations for $V_j$ and $W_j$. The dimension function associated to $\Psi$ is $$D_{\Psi}(\xi)=\sum_{k\in{\mathbb{Z}^n}}\sum_{\psi\in\Psi}\sum_{j\geq
1}|\widehat{\psi}|^2\left((A^*)^j(\xi+2k\pi)\right).$$
The definition of the multiplicity function requires a little bit of harmonic analysis (for details look in [@BMM] or [@B]). The subspace $V_0$ in invariant under translations by integers (shift invariant). So one has a unitary representation of the locally compact abelian group ${\mathbb{Z}^n}$ on $V_0$ by translations. The Stone-Mackey theory shows that this representation is determined by a projection-valued measure which in turn is determined by a positive measure class on the dual group $\widehat{{\mathbb{Z}^n}}$ and a measurable multiplicity function $m_{V_0}:\widehat{{\mathbb{Z}^n}}\rightarrow\{0,1...,\infty\}$. $\widehat{{\mathbb{Z}^n}}$ can be identified with $[-\pi,\pi]^n$. In our case, the measure is the Lebesgue measure, so the determinant is the multiplicity function $m_{V_0}$. The beautiful result is $$\label{eq5_3_1}
D_{\Psi}(\xi)=m_{V_0}(\xi),\quad(\xi\in[-\pi,\pi]^n).$$ We use the local trace function to prove it. First, if we use the equation (\[eq5\_1\_1\]) with $T=I$, the identity on ${l^{2}\left(\mathbb{Z}^n\right)}$, we obtain $$\label{eq5_3_2}
\dim_{V_0}(\xi)=\tau_{V_0,I}(\xi)=D_{\Psi}(\xi),\quad(\xi\in[-\pi,\pi]^n).$$ To equate the local trace function with the multiplicity function we use a result from [@BM]: if $$S_j:=\{\xi\in[-\pi,\pi]^n\,|\,m_{V_0}(\xi)\geq j\},\quad(j\in\mathbb{N},j\geq 1)$$ then there exist a NTF generator $\{\varphi_j\,|\,j\geq 1\}$ for $V_0$ such that $$\label{eq5_3_2_1}
\sum_{k\in{\mathbb{Z}^n}}\widehat{\varphi}_i\overline{\widehat{\varphi}}_j(\xi+2k\pi)=
\left\{\begin{array}{ccc}
\chi_{S_i}(\xi),&\mbox{if}&i=j\\
0,&\mbox{if}&i\neq j.
\end{array}
\right.$$ We can use theorem \[th3\_3\] for $T=I$ and we have for a.e. $\xi$: $$\tau_{V_0,I}(\xi)=\sum_{j\geq 1}{\left\langle {\mathcal{T}_{per}}\varphi_j(\xi)\,|\,{\mathcal{T}_{per}}\varphi_j(\xi)
\right\rangle}=\sum_{j\geq
1}\sum_{k\in{\mathbb{Z}^n}}|\widehat{\varphi}_j|^2(\xi)=\sum_{j\geq 1}\chi_{S_j}(\xi).$$ Therefore $$\label{eq5_3_3}
\dim_{V_0}(\xi)=\tau_{V_0,I}(\xi)=m_{V_0}(\xi),\quad(\xi\in[-\pi,\pi]^n).$$ Consequently the multiplicity function and the dimension function are just two disguises of the local trace function at $T=I$.
The next theorem gives the equations that relates scaling functions (i.e NTF generators for $V_0$) wavelets.
\[th5\_4\] Let $\Psi=\{\psi^1,...,\psi^L\}$ be a semi-orthogonal wavelet as in theorem \[th5\_1\]. Let $\Phi$ be a countable subset of ${L^{2}\left(\mathbb{R}^n\right)}$. The following affirmations are equivalent:
1. $\Phi$ is contained in $V_0$ and is a NTF generator for $V_0$;
2. The following equations hold: for every $s\in{\mathbb{Z}^n}$, $$\label{eq5_4_2}
\sum_{\psi\in\Psi}\sum_{j\geq 1}\widehat{\psi}((A^*)^j\xi)\overline{\widehat{\psi}}((A^*)^j(\xi+2s\pi))=
\sum_{\varphi\in\Phi}\widehat{\varphi}(\xi)\overline{\widehat{\varphi}}(\xi+2s\pi).$$ for a.e. $\xi\in{\mathbb{R}^n}$.
Use theorem \[th3\_3\_2\] and theorem \[th5\_1\].
\[cor5\_5\] Let $\Psi=\{\psi^1,...,\psi^L\}$ be a semi-orthogonal wavelet as in theorem \[th5\_1\] and $\Phi$ a NTF generator for $V_0$. Then $$\label{eq5_5_1}
\sum_{\psi\in\Psi}|\widehat{\psi}|^2(\xi)=\sum_{\varphi\in\Phi}|\widehat{\varphi}|^2((A^*)^{-1}\xi)-
\sum_{\varphi\in\Phi}|\widehat{\varphi}|^2(\xi),\quad\mbox{a.e. on }{\mathbb{R}^n};$$ For all $s\in{\mathbb{Z}^n}\setminus A^*{\mathbb{Z}^n}$, $$\label{eq5_5_2}
\sum_{\psi\in\Psi}\widehat{\psi}(\xi)\overline{\widehat{\psi}}(\xi+2s\pi)=
-\sum_{\varphi\in\Phi}\widehat{\varphi}(\xi)\overline{\widehat{\varphi}}(\xi+2s\pi),\quad\mbox{a.e. on }{\mathbb{R}^n}.$$
For (\[eq5\_5\_1\]), take $s=0$, write the equation (\[eq5\_4\_2\]) for $\xi$ and $(A^*)^{-1}\xi$, and substract the first from the second. For (\[eq5\_5\_2\]), substract equation (\[eq5\_4\_2\]) from equation (\[eq5\_2\_2\]).
[abcdef]{} L. Baggett, [*[A]{}n abstract interpretation of the wavelet dimension function using group representations*]{}, J. Funct. Anal. [**173**]{} (2000) 1-20 L. Baggett,K. Merrill, [*[A]{}bstract harmonic analysis and wavelets in ${\mathbb{R}^n}$*]{}, Contemporary Mathematics, The functional and harmonic analysis of wavelets and frames (1999) 17-29. L. Baggett,H. Medina,K. Merrill, [*[G]{}eneralized multiresolution analyses and a construction procedure for all wavelet sets in ${\mathbb{R}^n}$*]{}, J. Four. Anal. Appl. [**5**]{}, (1999) 563-573 M. Bownik, [*[T]{}he structure of shift invariant subspaces of ${L^{2}\left(\mathbb{R}^n\right)}$*]{}, J. Funct. Anal., [**177**]{} (2000),282-309 M. Bownik, [*[O]{}n characterizations of multiwavelets in ${L^{2}\left(\mathbb{R}^n\right)}$*]{}, Proc. Amer, Math. Soc. , [**129**]{} (2001) 3265-3274 M. Bownik,Z. Rzeszotnik, [*[T]{}he spectral function of shift-invariant spaces*]{}, to appear. A. Calgolero, [*[A]{} characterization of wavelets on general lattices*]{}, J. Geom. Anal. C.K. Chui,X. Shi,J.Stoeckler, [*[A]{}ffine frames, quasi-affine frames, and their duals*]{}, Adv. Comput. Math,[**8**]{} (1998),1-17 H. Helson, [*[L]{}ectures on Invariant Subspaces*]{}, Academic Press,1964 A. Ron,Z. Shen, [*[F]{}rames and stable bases for shift-invariant subspaces of ${L^{2}\left(\mathbb{R}^n\right)}$*]{}, Canad. J. Math.,[**47**]{}, (1995) 1051-1094 A. Ron,Z. Shen, [*[A]{}ffine systems in ${L^{2}\left(\mathbb{R}^n\right)}$: the analysis of the analysis operator*]{}, J. Funct. Anal.[**148**]{}, (1997), 408-447 A. Ron,Z. Shen, [*[A]{}ffine systems in ${L^{2}\left(\mathbb{R}^n\right)}$. II. Dual systems*]{}, J. Fourier Anal. Appl.[**3**]{} (1997), 617-637 A. Ron,Z. Shen, [*[T]{}he wavelet dimension function is the trace function of a shift-invariant system*]{}, to appear. S. Stratila,L. Zsido, [*[L]{}ectures on von Neumann algebras*]{}, Taylor and Francis, Inc. 1979 E. Weber, [*[A]{}pplications of the wavelet multiplicity function*]{}, in The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999) Amer. Math. Soc., 1999, 297-306
[^1]:
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Recently, several simple quantum mechanical toy models of black hole evaporation have appeared in the literature attempting to illuminate the black hole information paradox. We present a general class of models that is large enough to describe both unitary and nonunitary evaporation, and study a few specific examples to clarify some potential confusions regarding recent results. We also generalize Mathur’s bound on small corrections to black hole dynamics. Conclusions are then drawn about the requirements for unitary evaporation of black holes in this class of models. We present a one-parameter family of models that continuously deforms nonunitary Hawking evaporation into a unitary process. The required deformation is large.'
author:
- 'Steven G. Avery'
bibliography:
- 'qmbh.bib'
title: Qubit Models of Black Hole Evaporation
---
Introduction
============
Hawking’s calculation of black hole evaporation [@Hawking:1974sw] leads to a direct conflict between general relativity and the unitary evolution of quantum mechanics [@Hawking:1976ra]. This prompted the suggestion that the requirement of unitary evolution should be relaxed [@Hawking:1982dj; @Page:1979tc], and pure states allowed to evolve into mixed states. Such nonunitary evolution, however, seems problematic [@Banks:1983by].[^1]
Moreover, since black hole evaporation is a very slow process involving a large number of emitted particles, and since one only expects to start recovering information after about one half of the radiation has been emitted [@Page:1993df], one might imagine that unitarity is restored by the accumulation of small (say, nonperturbative) corrections over the course of the entire evolution process, cf. [@Page:1993wv]. To evaluate this claim, Mathur [@Mathur:2009hf] introduces a qubit model of black hole evaporation and then derives bounds on the entanglement entropy of the emitted radiation, which show that this scenario is *not possible*. Below we generalize his result to consider more general deformations of the pair creation dynamics. In particular, for small changes in the evaporation process the entanglement entropy continues to increase, therefore (barring remnants and variants thereof) the evolution is not unitary.
Thus, for unitarity to be restored, one needs to make *large* corrections to the semi-classical evaporation process described by Hawking. Recently, several specific unitary models have been introduced [@Czech:2011wy; @Giddings:2011ks; @Mathur:2011wg] as proposed alternatives to Hawking’s semiclassical evolution. As discussed in [@Mathur:2011uj], models of this kind could be called “burning paper” models that involve large corrections to the Hawking evolution. While these models are unitary, they are typically written in a way that makes it difficult to compare to the semiclassical evolution and to see how Mathur’s bound operates. The difficulty arises because the bound is derived in terms of dynamics in an ever-enlarging Hilbert space, whereas unitary models are typically written as dynamics in a fixed-dimensional Hilbert space.
The primary goal of this paper is to clarify the meaning of Mathur’s bound in [@Mathur:2009hf]. In particular, until one writes unitary models as (large) corrections to nonunitary, semiclassical evolution, the meaning of Mathur’s result remains obscure. Previous investigations of corrections in this context either directly introduce unitary models that are difficult to compare to Hawking evaporation [@Czech:2011wy; @Giddings:2011ks; @Mathur:2011wg]; or consider small corrections to Hawking evaporation that *do not produce unitary evolution when made sufficiently large [@Mathur:2011wg; @Mathur:2010kx]*, and thus are unconvincing illustrations of the result in [@Mathur:2009hf]. A second goal is to characterize what kinds of corrections produce the desired unitary evolution. That the corrections need to be large is a necessary condition derived in [@Mathur:2009hf], but sufficient conditions have not been discussed in the same way.
In order to address the above points, it is necessary to introduce a general model space that provides a uniform language to discuss both unitary and nonunitary black hole evaporation. This allows us, for example, to continuously deform the semiclassical Hawking evolution to unitary evolution. One can then explicitly see that the deformation is large in an appropriate sense, and therefore in agreement with a suitable generalization of Mathur’s argument. Let us emphasize that it is not our intention here to advocate for nonunitary evolution, only to demonstrate that unitarity demands there be a significant alteration of the traditional semiclassical evolution. To restrict ourselves to unitary evolution at this stage would be to beg the question.
In Section \[sec:gen\], we introduce a very general framework for qubit models of black hole evaporation that is appropriate for both unitary and nonunitary evolution. Most of the discussion focuses on how to interpret the models, and their connection to ideas in quantum information theory. In Section \[sec:models\], we apply the formalism to a number of sample models; some of the models were chosen because they were discussed previously in the literature, and others because they illustrate some interesting issues. In Section \[sec:bound\], we briefly review Mathur’s argument against small corrections restoring unitary evolution, and generalize the main entanglement entropy bound to allow arbitary deformations. Mathur’s original result [@Mathur:2009hf] only explicitly considers one kind of perturbation. In Section \[sec:unitarity\], using observations from Section \[sec:models\], we discuss what conditions ensure unitary evolution. In Section \[sec:one-par\], we present a one-parameter family of models that continuously connects Hawking evaporation to a unitary model; one sees clearly that the deformation required is large. In Section \[sec:conc\], we conclude with some brief comments.
The General Model {#sec:gen}
=================
Before presenting our model, we make a few preliminary comments on nonunitary evolution in Section \[sec:nonunitary\]. Then in Section \[sec:quops\], using some results from quantum information theory, we explain how to describe nonunitary evolution that still has a good probabilistic interpretation. Along the way, we clarify some potential confusions regarding previous work. Finally, we present our general class of models in Section \[sec:themodel\], discussing the physical interpretation in Section \[sec:physics\].
Nonunitarity {#sec:nonunitary}
------------
When we model the evolution of a closed quantum system, the state of the system is given by a ket $\ket{\psi(t)}$ that satisfies $$\label{eq:unit-evol-1}
\ket{\psi(t)} = U(t)\ket{\psi(0)}$$ for a unitary time evolution operator $U(t)$. We may equivalently write the state of the system as a density matrix $\rho(t) =
\ket{\psi(t)}\bra{\psi(t)}$ where $\rho(t)$ satisfies $$\label{eq:unit-evol-rho}
\rho(t) = U\rho(0)U^{\dagger}.$$ Unitary evolution satisfies several nice conditions, namely,[^2]
1. \[it:lin\] Linearity
2. \[it:norm\] Preservation of the norm: unit norm states evolve to unit norm states, ensuring that a probabilistic interpretation makes sense. For density matrices, the desired condition is that unit-trace, completely positive density matrices evolve to unit-trace, completely positive density matrices.
3. \[it:inv\] Invertibility: previous states can be found from the current state.
4. \[it:pure\] Purity: pure states evolve to pure states.
However, it has sometimes been suggested [@Hawking:1982dj; @Hawking:1976ra; @Page:1979tc] that theories of quantum gravity (especially in the presence of black holes) will not be unitary. Let us note that the negation of unitary is ambiguous, since it is not clear which of the above conditions is relaxed. Let us consider three illustrative mappings.
1. This evolution does not conserve probability—the norm is not conserved; however, pure states still evolve to pure states, and the evolution is invertible: $$\ket{\psi} \mapsto \frac{3}{4}\ket{\psi}.$$ This kind of evolution is sometimes useful in modelling a system that decays into something that is outside the model. For instance, in modeling alpha decay. An equivalent way to describe this evolution is to say that the system has energies with an imaginary part. For a fundamental description that includes all degrees of freedom, however, the nonconservation of probability is nonsensical.
2. This evolution preserves the norm, is invertible, but evolves pure states to mixed states: $$\ket{\psi_1} \mapsto \rho_1 = \frac{1}{2}\ket{\psi_1}\bra{\psi_1}
+ \frac{1}{2}\ket{\phi_1}\bra{\phi_1}\qquad
\ket{\psi_2} \mapsto \rho_2 = \frac{1}{2}\ket{\psi_2}\bra{\psi_2}
+ \frac{1}{2}\ket{\phi_2}\bra{\phi_2}\qquad
\dots$$ While this model evolves pure states to mixed states, information is still preserved (in a weak sense),[^3] and probability is conserved.
3. This evolution conserves probability, evolves pure states to pure states, but is not invertible. $$\ket{\psi_1} \mapsto \ket{\psi_0}\qquad
\ket{\psi_2} \mapsto \ket{\psi_0}\qquad
\dots$$ In this model one cannot reconstruct the past from the current state, and therefore information is not preserved. In Section \[sec:G1-broken\], we introduce some models of this type.
In the black hole information paradox, one considers some initial configuration of matter $\ket{\psi_m}$ that collapses into a black hole, and then completely evaporates to radiation in a thermal mixed state $\rho \sim e^{-\beta H}$. Since the final state is both mixed and independent of the initial state, the evolution *both* fails to be invertibile and pure in the above senses.
Suppose for the nonce that Hawking’s original argument is correct [@Hawking:1976ra], and the fundamental theory of quantum gravity is not unitary. We can no longer write evolution as in . We restrict our considerations to evolution that satisfies conditions \[it:lin\] and \[it:norm\], but not necessarily condition \[it:inv\] and \[it:pure\]. Then, assuming some very basic conditions that ensure a good probabilistic interpretation, we can write the most general possible evolution in the operator-sum representation (cf. [@nielsen]) $$\label{eq:op-sum-rep}
\rho(t) = \sum_k E_k \rho(0) E_k^{\dagger}$$ for some set of operator $E_k$ that satisfy the completeness relation $$\sum_k E_k(t)^{\dagger}E_k(t) = I.$$ This is one way to write the evolution of an open quantum system, and the transformation from $\rho(0)$ to $\rho(f)$ is called a “quantum operation” in the quantum information theory literature [@nielsen]. The operators $E_k$ determine the evolution of the density matrix. When there is only one $E_k$, the evolution is unitary.[^4]
Quantum Operations {#sec:quops}
------------------
As in [@Mathur:2009hf; @Giddings:2011ks; @Mathur:2011uj; @Mathur:2010kx; @Mathur:2011wg], we model the Hawking evaporation process as a discrete set of mappings on qubits. In the initial state, the system consists entirely of matter in a pure state. There have been some suggestions [@Braunstein:2009my] in this context that the entanglement between the initial black hole-forming matter and the outside matter plays an important role; we do not address these issues at this time. The initial state is modeled as a set of $n$ “matter qubits”: $$\rho_0 = \ket{\psi_0}\bra{\psi_0} \qquad \ket{\psi_0} \in\operatorname{\mathrm{span}}\left\{\ket{\hat{q}_1\hat{q}_2\cdots\hat{q}_{n}}\right\},$$ where each $\hat{q}$ is a qubit, a quantum state labeled by $0$ or $1$. After a sequence of intermediate steps, the end state consists entirely of radiation (again, we are assuming no remnants), modeled as a (possibly mixed) density matrix acting on $n$ “radiation” qubits, $\rho_f$. Throughout the evolution, we keep the total dimension of the Hilbert space fixed. This is certainly true for the unitary evolution of closed systems, but here we put it in as a reasonable assumption. We are motivated in part by the black hole’s entropy. The black hole initially has $S\sim M^2$, which entirely radiates away on the time scale $\sim M^3$ with an emission every $\sim M$. This is consistent with a model having a fixed number of physical qubits.
Following [@Giddings:2011ks], we use hats to distinguish the internal black hole qubits from the external radiation qubits. The hatted qubits represent all degrees of freedom that are inaccessible outside the black hole; unlike [@Mathur:2009hf], we do not distinguish between degrees of freedom from the initial matter, from gravitational interactions, or any that arise during the evaporation process. We write basis elements for the final state as $$\{\ket{q_nq_{n-1}\cdots q_1}\},$$ where we have put the labels on the qubits in reverse order for reasons which should become clear. At the $i$th step, we have a density matrix acting on $n-i$ black hole qubits and $i$ radiation qubits, so that the total dimension of the Hilbert space is fixed. The evaporation concludes on the $n$th step when there is only radiation: $$\rho_0 \to \rho_1 \to \cdots \to \rho_{n-1} \to \rho_n$$
In general the mapping from $\rho_0$ to $\rho_n = \rho_f$ should be a quantum operation, which is the composition of $n$ quantum operations (one for each emission). Therefore the total evolution from $\rho_0$ to $\rho_n$ (and each intermediate step) may be written in terms of some $E_k$s, like in Equation . Quantum operations, however, may be written in an equivalent, alternative form that is more natural when discussing black hole evaporation. This form connects more directly with the discussion in [@Mathur:2009hf].
We motivate this alternate form, by noting that any mixed density matrix may be “purified” by enlarging the Hilbert space. For example, a density matrix of the form $$\rho = p_1 \ket{A}\bra{A} + p_2 \ket{B}\bra{B}$$ with orthonormal $\{\ket{A}, \ket{B}\}$ can be purified by introducing the orthonormal states $\{\ket{\alpha},\ket{\beta}\}$, and defining $$\ket{\Psi} = \sqrt{p_1}\ket{A}\otimes\ket{\alpha} + \sqrt{p_2}\ket{B}\otimes\ket{\beta}.$$ Then, one sees that $\rho$ is the reduced density matrix found by tracing out the new degrees of freedom. Let us emphasize *purification is a formal mathematical operation, and not a dynamical process*. In particular, for us, the new kets that we direct producted into the Hilbert space do not correspond to any physical degrees of freedom.[^5] Roughly speaking, then, we can imagine purifying each of the $\rho_i$s by enlarging the Hilbert space, and then the evolution in this enlarged Hilbert space would be unitary.
As it turns out, any quantum operation from say $\rho_0$ to $\rho_n$ may be written in the following way: $$\rho_n = \operatorname{\mathrm{tr}}_{\text{aux}}[U (\rho_{\text{aux}}\otimes \rho_0)U^{\dagger}],$$ for a unitary transformation $U$ acting on some auxillary degrees of freedom as well as the physical degrees of freedom. In particular, if $\rho_0$ acts on a $d$-dimensional Hilbert space, then we need introduce at most a $d^2$-dimensional auxillary Hilbert space to write the most general quantum operation in this form [@nielsen]. Thus for our $n$-qubit system, we need only introduce $2n$ auxillary qubits to capture the most general evolution of density matrices.
In this language, we can think about the semiclassical Hawking evolution in the following way. We start with $n$ black hole qubits initially in a pure state. We imagine that $n$ might be roughly given by the entropy of the black hole.[^6] At the first time step, a pair of qubits is created at the horizon in an entangled state, $$\frac{1}{\sqrt{2}}(\ket{\hat{0}}\ket{0} + \ket{\hat{1}}\ket{1}).$$ The zero represents no particle and the one represents a particle. We refer the reader to [@Mathur:2009hf; @Mathur:2010kx] for a thorough discussion on the origin of this description; see also [@Giddings:2011ks]. We have now added two new qubits to the system, increasing the size of our Hilbert space. At each time step a new entangled pair is produced in the above state, and the Hilbert space keeps increasing in size. By the end of the evaporation process, on the $n$th step, we have added $2n$ qubits to the initial $n$ qubits for a total $3n$ qubits. Since the black hole has completely evaporated and there are only the $n$ physical qubits of radiation, the remaining $2n$ hatted qubits should be interpreted as auxillary degrees of freedom as in the above discussion. Since presumeably the total number of physical degrees of freedom should remain fixed at $n$ qubits, at the $i$th step we have $2i$ auxillary (hatted) qubits, but the semiclassical analysis does not make any clear identification of the auxillary qubits at intermediate stages in the evaporation. It *is* clear, however, that by the end of the evaporation process all of the black hole (hatted) qubits must be auxillary. Specifying the auxillary subspace (in combination with giving the internal dynamics) corresponds to taking into account back reaction on the geometry.
The Hilbert space is illustrated in Figure \[fig:H\], where the intermediate state is shown with some highlighted auxillary degrees of freedom. In the figure, the highlighted region contains some of the inital matter qubits (circles) and some of the new infalling qubits (squares); this represents one possibility. One could also consider cases where for the first $n/2$ steps only the inital matter is auxillary, for example. The parameterization of the auxillary degrees of freedom at intermediate steps should be considered part of a model, so that one may trace out the auxillary degrees of freedom to arive at a fixed-dimensional Hilbert space description. Since we are mostly interested in the final state, where the auxillary space is unambiguous, we do not always specify the auxillary degrees of freedom. In the cases where the evolution is unitary, it should be clear what the auxillary degrees of freedom are. Although nothing profoundly new has been said here, we hope this discussion may help clarify potential confusions[^7] regarding [@Mathur:2009hf; @Mathur:2010kx], and other investigations [@Mathur:2011wg; @Giddings:2011ks; @Czech:2011wy].
The General Model {#sec:themodel}
-----------------
We are now ready to present the general class of models that we consider. We start with $n$ hatted qubits, and at each step we add a hatted qubit and an unhatted qubit. The number of qubits at each step is summarized in Table \[tab:no-qubits\].
step no. BH qubits no. rad. qubits total no. qubits no. aux. qubits state
------ --------------- ----------------- ------------------ ----------------- ----------------
$0$ $n$ $0$ $n$ 0 $\ket{\psi_0}$
$1$ $n+1$ $1$ $n+2$ 2 $\ket{\psi_1}$
$2$ $n+2$ $2$ $n+4$ 4 $\ket{\psi_2}$
$i$ $n+i$ $i$ $n+2i$ $2i$ $\ket{\psi_i}$
$n$ $2n$ $n$ $3n$ $2n$ $\ket{\psi_n}$
: Here we outline the discrete steps in our models. At the $0$th or initial step there are $n$ black hole (BH) qubits and no radiation qubits. At each step in the evolution, the state is given by the ket $\ket{\psi_i}$ in an enlarging Hilbert space.[]{data-label="tab:no-qubits"}
We model the evolution in two steps: a creation step effected by operators $C_i$; and an internal evolution step effected by $\hat{U}_i$ acting on the hatted qubits and $U_i$ acting on the unhatted radiation qubits. Basis vectors at each step look like $$\begin{gathered}
\left\{\ket{\hat{q}_1\hat{q}_2\cdots\hat{q}_{n+i}}\ket{q_iq_{i-1}\cdots q_1}\right\}
\xrightarrow{C_i}
\left\{\ket{\hat{q}_1\hat{q}_2\cdots\hat{q}_{n+i}\hat{q}_{n+i+1}}\ket{q_{i+1}q_iq_{i-1}\cdots q_1}\right\}\\
\xrightarrow{\hat{U}_i\otimes U_i}
\left\{\hat{U}\ket{\hat{q}_1\hat{q}_2\cdots\hat{q}_{n+i}\hat{q}_{n+i+1}}U\ket{q_{i+1}q_iq_{i-1}\cdots q_1}\right\}.\end{gathered}$$ Of course, one can combine $C_i$ and $\hat{U}_i\otimes U_i$ into a single operator, but it is useful to break up the evolution in this way. Also, there is some physical motivation for thinking about the evolution in this way, since the pair creation time scale is roughly $\sim M$, the black hole mass, while there are some conjectures that the internal dynamics of the black hole should be as fast [@Hayden:2007cs; @Sekino:2008he]. (For a 3+1 dimensional Schwarzschild black hole, the scrambling time is speculated to be $\sim M\log M$, but the evolution time step is $\sim M$.)
For the majority of the discussion, we focus on the $C_i$ and are content to set $\hat{U}_i = U_i = I$. What properties should the $C_i$ satisfy? We want $C_i$ to preserve the norm and be linear, which means that we should require $$\label{eq:CdgC}
(C_i)^{\dagger}C_i = I,$$ where there is no sum on $i$. Note that this is not the same as $C_i(C_i)^{\dagger}$ since the $C_i$ have *non*square matrix representations; the above requirement makes $C_i$ an isometric, but nonunitary mapping. We also assume that the $C_i$ act only on the hatted black hole qubits and not on the unhatted radiation qubits which are far away from the pair creation site.
We can write the $C_i$ in the following form $$C_i = \ket{\varphi_1}\otimes \hat{P}_1+\ket{\varphi_2}\otimes \hat{P}_2+\ket{\varphi_3}\otimes \hat{P}_3 + \ket{\varphi_4}\otimes \hat{P}_4,$$ where $\ket{\varphi_j}$ are an orthonormal basis for the created pair qubits, and the $\hat{P}$’s are linear operators which act on the hatted qubits (with implicit $i$ dependence). Following [@Mathur:2011wg; @Giddings:2011ks], we use the basis $$\begin{aligned}
\ket{\varphi^i_1} &= \frac{1}{\sqrt{2}}\big(\ket{\hat{0}_{n+i+1}}\ket{0_{i+1}}
+ \ket{\hat{1}_{n+i+1}}\ket{1_{i+1}}\big)\\
\ket{\varphi^i_2} &= \frac{1}{\sqrt{2}}\big(\ket{\hat{0}_{n+i+1}}\ket{0_{i+1}}
- \ket{\hat{1}_{n+i+1}}\ket{1_{i+1}}\big)\\
\ket{\varphi^i_3} &= \ket{\hat{0}_{n+i+1}}\ket{1_{i+1}} \\
\ket{\varphi^i_4} &= \ket{\hat{1}_{n+i+1}}\ket{0_{i+1}},
\end{aligned}$$ for the newly created pair. The constraint in Equation implies the following condition on the $\hat{P}$s: $$\label{eq:completeness}
(C_i)^{\dagger}C_i = \hat{P}_1^{\dagger}\hat{P}_1 + \hat{P}_2^{\dagger}\hat{P}_2
+ \hat{P}_3^{\dagger}\hat{P}_3 + \hat{P}_4^{\dagger}\hat{P}_4 = \hat{I}.$$ Note that this defines the $\hat{P}$s as a set of generalized measurement operators acting on the black hole Hilbert space.
A fully specified model, then, entails
1. A set of $\hat{P}$s at each step $i$ that satisfy the completeness relation .
2. The unitary operators $\hat{U}_i$ and $U_i$ for each $i$.
3. A clear delineation of the auxillary subspace at each step $i$.
The last item is frequently omitted in our discussion; it should be clear for unitary models, and it does not make significant differences for the nonunitary models. If one wants to acquire the fixed-dimensional Hilbert space description, however, then one must trace out the auxillary degrees of freedom at each step. This gives a very general model space that makes it easy to compare and contrast different models of evolution.
Physical Motivations {#sec:physics}
--------------------
At this point, the class of models introduced above may seem fairly abstract with little contact with the original black hole problem. Let us review the physical motivations for this type of model as laid out in [@Mathur:2009hf] and in [@Giddings:2011ks].
We consider an initial configuration of spherically symmetric matter that forms a black hole, which we expect should be well described by the Schwarzschild solution. The model is based on the semiclassical evolution of fields in the background of such a Schwarzschild black hole. In order to give a Hilbert space description of evolution, it is necessary to specify a spacelike slicing of the geometry so that we can specify the quantum state of fields on each slice. It is important to the arguments advanced in [@Mathur:2009hf] that there exists a “nice slicing” of the black hole geometry [@Lowe:1995ac]. This slicing avoids the geometry’s strong curvature, has sub-Planck scale extrinsic and intrinsic curvature, and yet cuts through the initial matter, horizon, and outgoing Hawking radiation in a smooth way [@Lowe:1995ac; @Mathur:2009hf]. Thus, all quantum gravity effects seem to be under control.
Our model should be considered an effective description of the dynamics on the slicing. As is well known, in the presence of curved backgrounds the quantum field theory notion of particle becomes observer dependent. If we expand our fields on the slice into modes inside the horizon and outside the horizon, then one finds that pairs of particles are created inside and outside of the horizon. More explicitly there is a Bogoliubov transformation such that the in vacuum evolves to a state of the form $\exp(\gamma a_\text{inside}^{\dagger}a_\text{outside}^{\dagger})$ acting on the out vacuum. We reduce our problem to an essentially two-dimensional one by expanding the modes in spherical harmonics. From a two-dimensional perspective, each harmonic corresponds to a different field. Then following [@Hawking:1974sw; @Giddings:1992ff], as emphasized in this context in [@Giddings:2011ks], we can use a set of modes that are localized wavepackets so that we can talk about locality. This is implicit in the discussion of [@Mathur:2009hf]. Moreover, we can truncate the Fock space to occupation numbers zero or one. We then are effectively left with a discussion of qubits, with $\ket{0}$ representing no excitation and $\ket{1}$ representing an excitation.
In this description, a pair of particles are created roughly every $M$ in Planck units, with the outgoing particles traveling freely outward on the slices and the ingoing particles traveling freely inward toward the initial matter that is very far away on each slice. The pair of particles are created entangled, as should be clear from the above exponential. This all suggests an effective, discrete time evolution with [@Mathur:2009hf; @Mathur:2010kx; @Mathur:2011wg; @Mathur:2011uj; @Giddings:2011ks] $$\hat{P}_1 = \hat{I} \qquad \hat{P}_{2,3,4} = 0\qquad
\hat{U}=U = I.$$ Because the particles are well-separated on the slice, we do not expect interparticle interactions to be significant which is represented by the choice $\hat{U}=U=I$. We call this point in model space, the Hawking model. Location of the particles on the slice can then be read in the following way from the states. Consider, for illustrative purposes, $n=3$ with a state of the form $$\ket{\hat{q}_1\hat{q}_2\hat{q}_3}_\text{initial}\ket{\hat{1}_4\hat{0}_5}_\text{infalling}\ket{0_2 1_1}_\text{outgoing}.$$ The first three hatted qubits represent the initial infalling matter. Note that in the Hawking model this matter plays no role in the evolution. In general, we imagine that we find some qubit description of the initial matter, the details of which are irrelevant to our concerns here. The next two hatted qubits represent the infalling Hawking radiation as it travels inward on the slices. We see that on the first time step, a particle was emitted but not on the second time step. The two unhatted bits represent the outgoing radiation, so the above implies an outgoing particle was emitted on the first time step and then no particle on the second time step. We have written the qubits in the above order so that reading from left to right loosely corresponds to traveling outward on the slice. This allows us to talk about a coarse form of locality [@Mathur:2009hf; @Giddings:2011ks]. Note that in the Hawking model the above state would be superposed with several other direct product states.
As discussed in the Introduction and for this context in [@Mathur:2009hf; @Mathur:2011uj; @Mathur:2010kx; @Giddings:2011ks], the above semiclassical description of Hawking evaporation is incomplete. In particular, one expects quantum gravity effects, backreaction, and interactions to play a role. Because of the nature of the nice slicing and the low curvature at the horizon, however, one generally expects all of these corrections to be small. That is to say, on the pair creation time scale, one expects from naive estimates that the dynamics be $\epsilon$ away from the above. This expectation is in considerable tension with the expectation that the dynamics are unitary. For instance, we might introduce a set of $\hat{P}$s that act on the last emitted ingoing particle. This would suggest that the horizon is not effectively the vacuum and still “remembers” the previous emission. One might also allow some mild nonlocal interactions inside the black hole via some nearest neighbor $\hat{U}_i$s. Or, motivated by holography and fast scrambling [@Hayden:2007cs; @Sekino:2008he; @Susskind:2011ap], one might consider general $\hat{U}_i$s, in which case one gives up all notions of locality on the slice inside the black hole. The distinction between the initial matter and the infalling particles is also lost. Allowing general internal dynamics does not affect the argument of [@Mathur:2009hf] and its generalization in Section \[sec:bound\], which only relies on the pair creation, $\hat{P}_i$s, being close to the Hawking model. We will always use $U=I$, since there is no physical motivation to consider strong interactions among the outgoing radiation. (Such corrections would also be irrelevant to Mathur’s bound on the entanglement entropy.) This should become clear after we examine some examples.
Examples {#sec:models}
========
In this section, we highlight some special points in model space that may be of interest and/or were discussed in the recent literature. One of the main results of this paper is the one-parameter family of models presented in Section \[sec:one-par\] that continuously deforms the Hawking pair production in Section \[sec:hawking\] into the unitary evolution in Section \[sec:G2\]; however, it is also useful to write the various models in a common form, so that the similarities and differences are manifest. This is especially true when one wants to compare unitary models to nonunitary models. We start with the canonical Hawking evaporation model. This should be thought of as the baseline model to which all other models should be compared.
The Hawking Model {#sec:hawking}
-----------------
The standard Hawking evaporation corresponds to creating a new pair in the state $\ket{\varphi_1}$ irrespective of the state of the system, as discussed at length in Section \[sec:physics\]. Thus, it can be written as $$\label{eq:hawking}
C^{H}_i = \ket{\varphi_1}\otimes \hat{I}\qquad \hat{U}=U = I,$$ and so we can write the $\hat{P}$s as $$\hat{P}_1 = \hat{I} \qquad \hat{P}_{2,3,4} = 0.$$ In [@Mathur:2009hf], Mathur showed that if the created pair is at most $\epsilon$ away from $\ket{\varphi_1}$, then the entanglement entropy of the radiation continues to grow with each step and thus the final state is mixed and unitarity is lost. In this language, the bound shows that if the $\hat{P}$s are small deformations from the above, then the final state will be mixed and the evolution will not be unitary. In the sequel, we demonstrate this quite explicitly.
A Burning Paper Model {#sec:paper}
---------------------
Here, we present a unitary “burning paper” model that is equivalent to the one given in [@Mathur:2011wg]:[^8] $$\begin{gathered}\label{eq:paper}
\hat{P}_1 = \hat{P}_2 = \left[\frac{1}{\sqrt{2}}\ket{\hat{0}\hat{0}}\bra{\hat{0}\hat{0}}
+ \frac{1}{2}\ket{\hat{1}\hat{0}}\bra{\hat{1}\hat{0}}
- \frac{1}{2}\ket{\hat{1}\hat{0}}\bra{\hat{0}\hat{1}}\right]_{n+i-1, n+i}\\
\hat{P}_3 = \left[\ket{\hat{1}\hat{0}}\bra{\hat{1}\hat{1}}
+ \frac{1}{\sqrt{2}}\ket{\hat{0}\hat{0}}\bra{\hat{1}\hat{0}}
+ \frac{1}{\sqrt{2}}\ket{\hat{0}\hat{0}}\bra{\hat{0}\hat{1}}\right]_{n+i-1,n+i}\\
\hat{P}_4 = 0,
\end{gathered}$$ where the subscript on the brackets indicates that the qubits referred to are the $(n+i-1)$th and the $(n+i)$th qubits. It should be clear that this model is quite far from the Hawking model. This models the creation step as $$\begin{gathered}
C_i = \ket{\hat{0}_{n+i+1}}\ket{1_i}\otimes
\left[\ket{\hat{1}\hat{0}}\bra{\hat{1}\hat{1}}
+ \frac{1}{\sqrt{2}}\ket{\hat{0}\hat{0}}\bra{\hat{1}\hat{0}}
+ \frac{1}{\sqrt{2}}\ket{\hat{0}\hat{0}}\bra{\hat{0}\hat{1}}\right]_{n+i-1,n+i}\\
+ \ket{\hat{0}_{n+i+1}}\ket{0_i}\otimes
\left[\ket{\hat{0}\hat{0}}\bra{\hat{0}\hat{0}}
+ \frac{1}{\sqrt{2}}\ket{\hat{1}\hat{0}}\bra{\hat{1}\hat{0}}
- \frac{1}{\sqrt{2}}\ket{\hat{1}\hat{0}}\bra{\hat{0}\hat{1}}\right]_{n+i-1,n+i}.\end{gathered}$$ The important property of the evolution to note is that the $(n+i+1)$th black hole qubit is always $\hat{0}$, as is the $(n+i)$th qubit. Thus these two qubits are “zeroed,” and effectively deactivated. We use the word zeroed in this sense, even if the qubit under discussion is deactivated to a different value. (It could even be something like $i\,\mathrm{mod}\,2$. In this situation, the information is sometimes said to be “bleached” out of the state.) It is clear, then, that these two qubits should be thought of as the auxillary qubits at intermediate steps. We’ll discuss this a bit more in Section \[sec:unitarity\].
We also introduce some interesting internal dynamics. First, we need to move the auxillary qubits out of the way, so that they don’t affect the next radiation step. So we first cyclically shift all of the qubits 2 positions to the right, thus shoving the two $\hat{0}$s to the two leftmost positions, $1$ and $2$. Then, we introduce some dynamics for the physical degrees of freedom. We cyclically shift *only* the nonauxillary qubits to the right by one unit. This defines $\hat{U}$ so that the model agrees with the burning paper model studied in [@Mathur:2011wg]. If we chop off the zeroed qubits, we recover the model exactly.
The first model in [@Giddings:2011ks] is in the same class of models. It too zeroes two qubits, one of which is the newly created black hole qubit. The main difference being that instead of the radiation being determined by the two rightmost hatted qubits, it is instead determined by the leftmost qubit. This model may be written as $$\begin{gathered}\label{eq:G1}
\hat{P}_1 = \hat{P}_2
= \frac{1}{\sqrt{2}}\ket{\hat{0}_{i+1}}\bra{\hat{0}_{i+1}}\otimes \hat{u}\\
\hat{P}_3 = \ket{\hat{0}_{i+1}}\bra{\hat{1}_{i+1}}\otimes \hat{u}',
\end{gathered}$$ where $\hat{u}$ and $\hat{u}$’ are unitary operators acting on the remaining hatted qubits. While it is not stated in [@Giddings:2011ks], we should require that $\hat{u}$ and $\hat{u}$’ do not mix the first $i+1$ or the last $i$ auxillary hatted qubits with the remaining physical qubits. The simplest case is to take $\hat{u}=\hat{u}' = \hat{I}$. In this model, the entanglement entropy of the radiation is always zero, which contrasts with our expectations from [@Page:1993df].
“Nonlocal” Unitary Evolution {#sec:G2}
----------------------------
In [@Giddings:2011ks], Giddings presents three unitary models of evolution. We focus on the second model that he presents. This second model can be written in our notation as $$\begin{aligned}\label{eq:G2}
\hat{P}_1 &= \ket{\hat{0}_{2i+1}\hat{0}_{2i+2}}\bra{\hat{0}_{2i+1}\hat{0}_{2i+2}}\\
\hat{P}_2 &= \ket{\hat{0}\hat{0}}\bra{\hat{1}\hat{1}}\\
\hat{P}_3 &= \ket{\hat{0}\hat{0}}\bra{\hat{0}\hat{1}}\\
\hat{P}_4 &= \ket{\hat{0}\hat{0}}\bra{\hat{1}\hat{0}}
\end{aligned}\qquad \hat{U} = \hat{I},$$ where we have suppressed the $(2i+1)$ and $(2i+2)$ subscripts in all but the first $\hat{P}$. One sees that as in the models presented in Section \[sec:paper\], two hatted qubits are zeroed at each step. In this case, they are the $(2i+1)$th and $(2i+2)$th qubits. Thus as the evolution progresses the hatted qubits are gradually put into a fiducial form. By the $i$th step, the first $2i$ qubits are zeroed, and should be thought of as auxillary. Note that the above evolution rule breaks down on the penultimate step, when there is only one nonauxillary qubit left. By then, we expect the black hole to be on the Planck scale, and so we can just emit the last qubit freely. This model corresponds to $\theta=\frac{\pi}{2}$ in the model presented in Section \[sec:one-par\].
This model is nonlocal when one considers the model in the original nice slicing of the black hole. In this context, the $(2i+1)$th and $(2i+2)$th qubit are very far from the pair creation site at the horizon. Note that this property is shared by the model in Equation , and to a lesser extent the model in Equation . The difference being how far away the zeroed qubits are. One can either interpret these unitary models as nonlocal interactions transmitting information far down the nice slice to the horizon [@Giddings:2011ks], or in terms of fuzzball microstates altering the state at the horizon, or as burning paper; these information theoretic models are too crude to distinguish. This point is discussed futher in the Conclusion.
A Pure, but Not Invertible Model {#sec:G1-broken}
--------------------------------
There are several of models of this kind. The simplest to consider is $$\hat{P}_{1,2,4} = 0\qquad
\hat{P}_3 = \hat{I}\qquad \hat{U}=\hat{I}.$$ In this model, regardless of the state of the system, the new pair is created in the state $\ket{\varphi_3}$. The state $\ket{\varphi_3}$ is not entangled, and thus we can think of this model as zeroing the new black hole qubit *and* the new radiation qubit. Instead of putting the internal qubits into a fiducial form, we put the radiation into a fiducial form. We hope that this convincingly demonstrates that purity of the final state does not ensure unitarity.
Another interesting example of (almost) pure but not invertible evolution is the model in Equation , with $$\label{eq:G1-broken}
\hat{u} = \hat{u}' = \hat{S}^{i+2, n+i}_1,$$ where $\hat{S}^{i+2,n+1}_1$ is the operator that cyclically shifts the $(i+2)$th through $(n+i)$th qubits to the left. For example, consider the evolution: $$\begin{aligned}\label{eq:samp-run}
&\ket{\hat{0}\hat{q}_1\hat{q}_2\cdots\hat{q}_{n-2}\hat{0}}\\
\xrightarrow{C_0}
&\ket{\hat{0}\hat{0}\hat{q}_1\hat{q}_2\cdots\hat{q}_{n-2}\hat{0}}\ket{0}\\
\xrightarrow{C_1}
&\ket{\hat{0}\hat{0}\hat{0}\hat{q}_1\hat{q}_2\cdots\hat{q}_{n-2}\hat{0}}\ket{00}\\
\xrightarrow{C_2}
&\ket{\hat{0}\hat{0}\hat{0}\hat{0}\hat{q}_1\hat{q}_2\cdots\hat{q}_{n-2}\hat{0}}\ket{000}\\
\vdots
\end{aligned}$$ In the above example, the radiation is never entangled with the black hole degrees of freedom, but its state is only determined by the first and last qubit of the initial state. Thus, the evolution is not unitary. This is a potential problem with the model even as it is defined in [@Giddings:2011ks]. What happened? In essence, the model keeps “reading” and zeroing the same qubit, which has the net effect of zeroing the radiation qubits as well as the new black hole qubits. This illustrates that unitary evolution can be lost when auxillary qubits mix with nonauxillary qubits.
This model is not quite pure for arbitrary initial states, since $$\begin{aligned}
&\ket{\hat{1}\hat{q}_1\hat{q}_2\cdots\hat{q}_{n-2}\hat{1}}\\
\xrightarrow{C_0}
&\ket{\hat{0}\hat{1}\hat{q}_1\hat{q}_2\cdots\hat{q}_{n-2}\hat{0}}\ket{1}\\
\xrightarrow{C_1}
&\ket{\hat{0}\hat{0}\hat{0}\hat{q}_1\hat{q}_2\cdots\hat{q}_{n-2}\hat{0}}\ket{11}\\
\xrightarrow{C_2}
&\ket{\hat{0}\hat{0}\hat{0}\hat{0}\hat{q}_1\hat{q}_2\cdots\hat{q}_{n-2}\hat{0}}\ket{011}\\
\vdots
\end{aligned},$$ which gives a pure final state, but if one considers a nontrivial superposition of the above initial state and the one in then the final state is mixed. Note that the radiation only carries two qubits of information about the initial state, and so the entropy of the final state is very small although nonvanishing on some initial states. It is in this sense that we call the evolution almost pure.
An Impure Model
---------------
A generic model that one constructs leads to impure evolution. When thinking about the different forms that $\hat{P}$s can take, a particularly natural variation on the Section \[sec:G2\] model to consider might be $$\begin{aligned}
\hat{P}_1 &= \ket{\hat{0}\hat{0}}\bra{\hat{0}\hat{0}}\\
\hat{P}_2 &= \ket{\hat{1}\hat{1}}\bra{\hat{1}\hat{1}}\\
\hat{P}_3 &= \ket{\hat{0}\hat{1}}\bra{\hat{0}\hat{1}}\\
\hat{P}_4 &= \ket{\hat{1}\hat{0}}\bra{\hat{1}\hat{0}},
\end{aligned}$$ where as in the above operators act on the $(2i+1)$th and $(2i+2)$th qubits. It should be clear that this model leads to mixed states. Note that it is also a large deformation from the Hawking model. It is easy to see that this model is not invertible either, by considering $\ket{\hat{0}\hat{0}}$ and $\ket{\hat{1}\hat{1}}$ as initial states.
Mathur–Plumberg Shift–Anti-Shift Models {#sec:shift}
---------------------------------------
In [@Mathur:2011wg], Mathur and Plumberg present several models. We can write their “Model A,” in the following way. Let $\hat{T}_j$ be the operator that cyclically shifts *only* the newly created (not the first $n$) hatted qubits to the right by $j$. Then, the model is of the form $$\hat{P}_1 = \lambda_1\hat{T}_1\qquad \hat{P}_2 = \lambda_2\hat{T}_{-1}
\qquad \hat{P}_{3,4}=0,$$ where the completeness relation requires $$\label{eq:model-A-constraint}
|\lambda_1|^2 + |\lambda_2|^2 = 1.$$ More generally, we can replace $\hat{T}_1$ and $\hat{T}_{-1}$ by other unitary transformations that act on the hatted qubits. The smallness of the corrections to Hawking is loosely determined by $\lambda_2$; see [@Mathur:2011wg] for more details and numerical results.
The “Model B” of [@Mathur:2011wg] may be written as $$\hat{P}_1 = \lambda_1\,\hat{T}_1\otimes \ket{\hat{1}_{i+1}}\bra{\hat{1}_{i+1}}
+ \hat{I}\otimes\ket{\hat{0}_{i+1}}\bra{\hat{0}_{i+1}}
\quad \hat{P}_2 = \lambda_2\,\hat{T}_{-1}\otimes\ket{\hat{1}_{i+1}}\bra{\hat{1}_{i+1}}\quad \hat{P}_{3,4}=0,$$ where one can confirm that the completeness relation imposes the same constraint .
Note that neither of the above models zeroes any qubits, and so one does not expect information about the initial matter to be transmitted out in the radiation.
Mathur “Ising” Model
--------------------
In [@Mathur:2010kx], Mathur presents a model that can be mapped onto the one-dimensional Ising model and thus solved analytically. The model can be written in the form $$\hat{P}_1 = \lambda_1 \left[\ket{\hat{0}}\bra{\hat{0}}
+ \ket{\hat{1}}\bra{\hat{1}}\right]_{n+i}\qquad
\hat{P}_2 = \lambda_2 \left[\ket{\hat{0}}\bra{\hat{0}}
- \ket{\hat{1}}\bra{\hat{1}}\right]_{n+i}\qquad
\hat{P}_{3,4}=0,$$ where $\lambda_1$ and $\lambda_2$ are related to the parameters $a$ and $b$ in [@Mathur:2010kx] via $$\lambda_1 = \frac{e^a + e^b}{2}\qquad \lambda_2 = \frac{e^a-e^b}{2}.$$ The above rule is not valid for the first step, for which we use the Hawking model ($\lambda_1 =1$ and $\lambda_2 = 0$). The parameters $\lambda_1$ and $\lambda_2$ are subject to the same constraint as before $$|\lambda_1|^2 + |\lambda_2|^2 = 1.$$ In fact, this model is qualitatively the same as the model in Section \[sec:shift\]: both set $\hat{P}_1$ and $\hat{P}_2$ to unitary transformations, and $\hat{P}_3=\hat{P}_4=0$. The expression for the final state entropy can be written in the form [@Mathur:2010kx] $$S = n\log 2 - (n-1)\left[ae^{2a} + be^{2b}\right].$$ Let us note that for no value of $a$ and $b$ that solves the completeness relation does this vanish. Thus, no matter how large the corrections are in this model, unitarity is lost; however, one *can* set $\lambda_1 = \lambda_2$ in which case the entanglement entropy of the radiation is $\log 2$ for all time. In this case, the pair creation (after the first step) is given by $$C_i = \ket{\hat{0}_{n+i+1}}\ket{0_i}\otimes\ket{\hat{0}_{n+i}}\bra{\hat{0}_{n+i}}
+\ket{\hat{1}_{n+i+1}}\ket{1_i}\otimes\ket{\hat{1}_{n+i}}\bra{\hat{1}_{n+i}}.$$ One sees that after the first step, no further entanglement between the hatted and unhatted qubits is generated. One might think there would be more entanglement generated when the above acts on qubits which are in a superposition of $1$ and $0$, but since the above only acts on previously created pairs we do not have to worry about that issue. One can imagine that the model breaks down when the black hole becomes very small and the last qubit is emitted freely, so this model is effectively pure in this case. Let us note that keeping the entanglement entropy of the radiation constant is in stark contrast with expectations we have from Page [@Page:1993df]; there is no characteristic rise and fall of the entanglement entropy of the radiation. Furthermore, the final state consisting entirely of radiation is *completely* independent of the initial matter that formed the black hole; the evolution is (almost) pure but far from invertible like the model discussed in Section \[sec:G1-broken\]. This model, while interesting to consider, never leads to unitary evolution, even when one considers arbitrarily large deformations from the Hawking point in model space.
Review and Generalization of Mathur’s Argument {#sec:bound}
==============================================
In [@Mathur:2009hf], Mathur argues that small corrections to the pair creation process are insufficient to restore unitarity. More specifically, he demonstrates that small corrections don’t accumulate. Let us define $S_i$ as the entanglement entropy of the radiation with the rest of the system at step $i$. If the black hole evaporation process is unitary, then in the limit of large $n$ we expect $S_i$ to rise linearly with $i$ until about the halfway point, $i=n/2$, and then rapidly turn over and fall linearly to zero on the final step [@Page:1993wv]. On the other hand, if one uses the Hawking model of evolution $C^H$, then one sees that $S_i$ increases by $\log
2$ at each time step: $$S^{\text{Hawking}}_i = i \log 2;$$ there is no turnover. This is also the maximum entropy that is possible for the $i$ radiation qubits, which indicates that the radiation carries no information about the initial state.
The central insight in Mathur’s argument [@Mathur:2009hf] is that the marginal increase in entanglement entropy, $$\Delta S_i = S_{i+1}-S_i,$$ varies smoothly with small deformations away from the Hawking model, and thus a large deformation is needed to make $\Delta S$ negative if one starts from the Hawking model’s $\log 2$. In [@Mathur:2009hf], only a $\hat{P}_2$-type deformation to the Hawking model was explicitly considered; however, we demonstrate below that the the argument generalizes to all deformations in the model space.
In particular, we claim in the class of models discussed in this paper, if $$\|C_i - C_i^H\|<{\varepsilon}< 1,$$ then $$\label{eq:my-bound}
\Delta S_i \geq \log 2 - k_{\varepsilon},$$ where $k_{\varepsilon}$ is parametrically small, positive, and vanishes as ${\varepsilon}$ goes to zero. In fact, we demonstrate below that $k_{\varepsilon}$ behaves as no worse than $k_{\varepsilon}\sim -9{\varepsilon}\log{\varepsilon}$ as ${\varepsilon}$ approaches zero. Above, $\|\cdot\|$ is the operator norm. For an operator $O$, $\|O\|$ is the square root of the largest eigenvalue of $O^{\dagger}O$.[^9]
The proof follows that in [@Mathur:2009hf] with some minor modifications. To begin, we use strong subadditivity of (von Neumann) entanglement entropy. Throughout, without loss of generality, we set $\hat{U}=U=I$ since they do not affect the entanglement entropies. Let us consider the $(i+1)$th state $$\ket{\psi_{i+1}} = C_i\ket{\psi_i}$$ Let $R_i$ denote the first $i$ emitted radiation qubits, $r$ denote the $(i+1)$th emitted radiation qubit, $B_i$ denote the first $n+i$ black hole qubits, and $b$ denote the $(n+i+1)$th black hole qubit. Then, strong subadditivity implies $$S(R_i\cup r)+S(r\cup b)\geq S(R_i) + S(b).$$ By definition, $S(R_i\cup r)$ is simply $S_{i+1}$, whereas since the $C_i$ acts trivially on emitted radiation $S(R_i) = S_i$. Thus, we can write the above as $$\Delta S_i \geq S(b) - S(r\cup b).$$ Note that for the Hawking model $S(b) = \log 2$ and $S(r\cup b) = 0$, and the bound is saturated.
Now, we need to use the condition on $C_i$ to place bounds on $S(b)$ and $S(r\cup b)$. The operator norm is compatible with the Hilbert space norm, which means $$\|(C_i - C^H)\ket{\lambda}\|\leq \|C_i - C^H\|\,\|\ket{\lambda}\|$$ for all $\ket{\lambda}$. Applying this to $\ket{\psi_i}$ gives the condition $$\label{eq:ket-bound}
\|\ket{\psi_{i+1}} - C_i^H\ket{\psi_i}\| < {\varepsilon}.$$ We may write the two kets in the form $$\ket{\psi_{i+1}} = \alpha_1 \ket{\varphi_1}\ket{\Lambda_1}
+ \alpha_2 \ket{\varphi_2}\ket{\Lambda_2}
+ \alpha_3 \ket{\varphi_3}\ket{\Lambda_3}
+ \alpha_4 \ket{\varphi_4}\ket{\Lambda_4},\qquad
C_i^H\ket{\psi_i} = \ket{\varphi_1}\ket{\Lambda_0},$$ where the $\ket{\Lambda_i}$ are normalized, but not necessarily orthogonal kets in the $n+2i$ qubit space $R_i\cup B_i$. Of course normalization demands that $$|\alpha_1|^2 + |\alpha_2|^2+|\alpha_3|^2+|\alpha_4|^2 = 1.$$ One can show that the condition implies that $${\operatorname{Re}}\big(\alpha_1 \braket{\Lambda_0|\Lambda_1}\big) > 1-\frac{{\varepsilon}^2}{2},$$ and since $|\braket{\Lambda_0|\Lambda_1}|\leq 1$, we see $$1-\frac{{\varepsilon}^2}{2}<|\alpha_1|\leq 1.$$ This, in turn, implies that $$\label{eq:delta-def}
1-\delta^2 <|\alpha_1|^2<1,\qquad |\alpha_2|,|\alpha_3|,|\alpha_4|<\delta = {\varepsilon}\sqrt{1-\frac{{\varepsilon}^2}{4}},$$ where we have defined $\delta$ for convenience. Note that $\delta$ is what should be directly compared with $\epsilon$ in [@Mathur:2009hf].
Let us use the above to place an upper bound on $S(r\cup b)$. The corresponding reduced density matrix is given by $$\rho_{rb} = \begin{pmatrix}
|\alpha_1|^2 & \alpha_1\alpha_2^*\braket{\Lambda_2|\Lambda_1}
& \alpha_1\alpha_3^*\braket{\Lambda_3|\Lambda_1}
& \alpha_1\alpha_4^*\braket{\Lambda_4|\Lambda_1}\\
\alpha_1^*\alpha_2\braket{\Lambda_1|\Lambda_2} & |\alpha_2|^2
& \alpha_2\alpha_3^*\braket{\Lambda_3|\Lambda_2}
& \alpha_2\alpha_4^*\braket{\Lambda_4|\Lambda_2}\\
\alpha_1^*\alpha_3\braket{\Lambda_1|\Lambda_3}
& \alpha_2^*\alpha_3\braket{\Lambda_2|\Lambda_3} & |\alpha_3|^2
& \alpha_3\alpha_4^*\braket{\Lambda_4|\Lambda_3}\\
\alpha_1^*\alpha_4\braket{\Lambda_1|\Lambda_4}
& \alpha_2^*\alpha_4\braket{\Lambda_2|\Lambda_4}
& \alpha_3^*\alpha_4\braket{\Lambda_3|\Lambda_4} & |\alpha_4|^2
\end{pmatrix}.$$ We can now use Audenaert’s optimal generalization [@audenaert] of Fannes’ inequality, which places a bound on the difference in entropy of two $d$-dimensional density matrices $\rho$ and $\sigma$.[^10] Let $T$ be the trace distance between $\rho$ and $\sigma$, then [@audenaert] $$\label{eq:fannes}
|S(\rho) - S(\sigma)|\leq T\log (d-1) - T\log T-(1-T)\log(1-T),$$ where the trace distance is defined as $$T = \frac{1}{2}\operatorname{\mathrm{tr}}\left[\sqrt{(\rho-\sigma)^{\dagger}(\rho-\sigma)}\right],$$ or one-half the sum of the absolute value of the eigenvalues of $\rho-\sigma$. Note that the above definition of the trace distance differs by a factor of $2$ from some references. With the above normalization $0\leq T\leq 1$ for all unit-trace density matrices $\rho$ and $\sigma$. In this case we consider the $\sigma$ to be the density matrix with $|\alpha_1| = 1$, and $\rho$ to be $\rho_{rb}$. Next, we may use Gershgorin’s circle theorem to bound the eigenvalues, $\lambda$, and therefore the trace distance, $T$. Gershgorin’s theorem tells us all the eigenvalues must lie in the union of discs in the complex plane centered on the diagonal entries with radii given by the sum of the absolute value of off-diagonal entries for each row. For instance the first row gives a disc $$D_1:\quad |\lambda -(|\alpha_1|^2-1)| \leq
|\alpha_1|\,|\alpha_2|\, |\braket{\Lambda_2|\Lambda_1}|
+ |\alpha_1|\,|\alpha_3|\,|\braket{\Lambda_3|\Lambda_1}|
+ |\alpha_1|\,|\alpha_4|\,|\braket{\Lambda_4|\Lambda_1}|$$ Applying our inequalities on the components and the hermiticity of $\rho-\sigma$ (and therefore reality of its spectrum), we can find an interval that must include any eigenvalues in the above disc: $$I_1= (-3\delta-\delta^2, 3\delta).$$ One finds the remaining three rows can be encompassed by the interval $$I_2 = (-\delta-2\delta^2, \delta+3\delta^2),$$ and so all eigenvalues must satisfy $$|\lambda| < 3\delta+\delta^2.$$ Thus, we may conclude that the trace distance must satisfy $$T < 2(3 \delta + \delta^2),$$ and therefore $$\label{eq:x1}
S(r\cup b) \leq -x_1
\log\left(\tfrac{1}{3}x_1\right)
- (1-x_1)\log(1-x_1)\qquad
x_1 =\min\big(\tfrac{3}{4},\, 2(3 \delta + \delta^2)\big).$$ The $3/4$ comes by finding critical point of the right-hand side of as a function of $T$.
The bound on $S(b)$ can be derived in analogous fashion. The state $\ket{\psi_{i+1}}$ may be written out as $$\begin{gathered}
\ket{\psi_{i+1}} = \ket{\hat{0}}\ket{\chi_0}+\ket{\hat{1}}\ket{\chi_1}\\
\ket{\chi_0} = \frac{\alpha_1}{\sqrt{2}}\ket{0}\ket{\Lambda_1}
+\frac{\alpha_2}{\sqrt{2}}\ket{0}\ket{\Lambda_2}
+\alpha_3\ket{1}\ket{\Lambda_3}\\
\ket{\chi_1} = \frac{\alpha_1}{\sqrt{2}}\ket{1}\ket{\Lambda_1}
-\frac{\alpha_2}{\sqrt{2}}\ket{1}\ket{\Lambda_2}
+\alpha_4\ket{0}\ket{\Lambda_4}.
\end{gathered}$$ The reduced density matrix can be written as $$\rho_b = \begin{pmatrix}
\braket{\chi_0|\chi_0} & \braket{\chi_1|\chi_0}\\
\braket{\chi_0|\chi_1} & \braket{\chi_1|\chi_1}
\end{pmatrix}.$$ As before we can place bounds on the above components $$\begin{aligned}
\left|\braket{\chi_0|\chi_0}-\frac{1}{2}\right|&<\delta + \frac{\delta^2}{2}\\
\left|\braket{\chi_1|\chi_1}-\frac{1}{2}\right|&<\delta + \frac{\delta^2}{2}\\
|\braket{\chi_1|\chi_0}|&<\sqrt{2}(\delta + \delta^2).
\end{aligned}$$ One can once again use the Fannes–Audenaert inequality along with Gershgorin’s theorem to bound $S(b)$, where $\sigma$ is the $\alpha_1=1$ density matrix, $I/2$. One finds the trace distance satisfies $$T < (1+\sqrt{2})\delta + (\tfrac{1}{2}+\sqrt{2})\delta^2,$$ and this gives $$\label{eq:x2}
S(b) \geq\log 2 + x_2\log x_2 + (1-x_2)\log(1-x_2)\qquad
x_2 = \min\big(\tfrac{1}{2},\,(1+\sqrt{2})\delta + (\tfrac{1}{2}+\sqrt{2})\delta^2\big).$$ Finally, this allows us to write $$\label{eq:k}
k_{\varepsilon}= -x_1\log (\tfrac{1}{3}x_1) - x_2\log x_2 -(1-x_1)\log(1-x_1)-(1-x_2)\log(1-x_2),$$ where recall $x_1$ is defined in Equation , $x_2$ in Equation , and $\delta$ in Equation . For asymptotically small ${\varepsilon}$, $k_{\varepsilon}\sim
-(7+\sqrt{2}){\varepsilon}\log{\varepsilon}$; and in fact for small but finite ${\varepsilon}$, $k_{\varepsilon}< -9{\varepsilon}\log{\varepsilon}$. Numerically, one finds that $k_{\varepsilon}$ first surpasses $\log 2$, thus allowing the entanglement entropy to decrease for ${\varepsilon}\approx .02$. Furthermore, one finds $k_{\varepsilon}$ reaches $2\log 2$, thus allowing the maximal marginal decrease of entanglement entropy for ${\varepsilon}\approx .05$. For larger ${\varepsilon}$, the inequality with $k_{\varepsilon}$ given in places no restriction on the marginal change in entanglement.
Since we are not especially interested in making the tightest possible bound or even the above numerical values, we may as well write the bound in slightly less unwieldy form, $$\Delta S_i \geq \log 2 + 9{\varepsilon}\log{\varepsilon}\qquad {\varepsilon}\ll 1.$$ This establishes the claim. Let us note that the above bound’s asymptotic behavior is weaker than the inequality derived in [@Mathur:2009hf][^11] as a consequence of using more general arguments to include arbitrary perturbations. On the other hand, the result presented here is stronger in the sense that [@Mathur:2009hf] finds only a leading order result valid to order $O(\epsilon^2)$ whereas the bound with $k_{\varepsilon}$ given in is valid for finite ${\varepsilon}\in(0,1)$. The above bounds could possibly be strengthened with more work;[^12] however, that is irrelevant to the basic claim that small corrections to the low energy pair creation process cannot restore unitarity.
Requirements for Unitarity {#sec:unitarity}
==========================
While it is interesting to think about the different kinds of evolution that one could have, perhaps the most interesting question to ask is what kinds of models are unitary or, equivalently what sorts of corrections to the Hawking evolution can restore unitarity. Above we see that small corrections to the evolution *cannot* restore unitarity; this gives a necessary condition that the corrections are *large*. It would be nice to also have some sufficient conditions, since it is clear that not every large correction one could consider leads to unitary evolution.
In order for the evolution to be pure, we need the final state (including the auxillary qubits) to be a direct product of the form $$\ket{\psi_n} = \ket{\hat{\phi}}\otimes\ket{\chi},$$ where $\ket{\hat{\phi}}$ is a state in the $2n$-qubit auxillary space and $\ket{\chi}$ is the state of the physical radiation qubits. Our first observation is that $\ket{\hat{\phi}}$ should be independent of the initial state. Suppose that this were not true: $$\begin{aligned}
\ket{\phi_0^{(1)}} &\mapsto \ket{\hat{\phi}^{(1)}}\otimes\ket{\chi^{(1)}}\\
\ket{\phi_0^{(2)}} &\mapsto \ket{\hat{\phi}^{(2)}}\otimes\ket{\chi^{(2)}}
\end{aligned},$$ which seems fine until one considers an initial state which is a superposition of the above two states; the final state is then mixed when one traces out the hatted qubits. (This argument assumes that the $\ket{\chi}$s are linearly independent so that the evolution is invertible.) This is basically a variant of the no-cloning theorem.
The next observation is that we need the final radiation state $\ket{\chi}$ to be a unitary transformation of the initial state $\ket{\psi_0}$. Let $F$ be the total map from initial state to the final state, then $F$ is a linear, isometric (norm-preserving) mapping from $n$ hatted qubits to $2n$ hatted plus $n$ unhatted qubits. From the above, unitarity demands that $$\label{eq:Funit}
F_\text{unitary}: \ket{\hat{\psi}}\mapsto \ket{\hat{\phi}}\otimes \ket{\chi}\qquad
\ket{\chi} = U\ket{\hat{\psi}},$$ where $U$ in the above is a unitary transformation from the initial $n$ hatted qubits to the final $n$ unhatted qubits, and $\ket{\hat{\phi}}$ is fixed. The total map $F$ is just the product of all the $C_i$s and $\hat{U}_i\otimes U_i$s. All of the hatted qubits have to be zeroed or bleached, and the information stored in the initial matter transferred to the radiation.
Let us think about how we can zero or bleach the hatted qubits. We have $n$ steps to project $2n$ qubits to a unique state with the $\hat{P}^i$s; the unitary $\hat{U}$s clearly cannot zero qubits. At each step, we can zero *at most two qubits*. If $C$ bleaches some subspace to a state $\ket{\hat{\alpha}}$, then it may be written as $$C = \ket{\hat{\alpha}}\otimes O,$$ for an unspecified operator $O$. If $\ket{\hat{\alpha}}$ is a $p$-qubit subspace, then $O$ maps $n+i$ qubits to $n+i+2-p$ qubits and must satisfy $$O^{\dagger}O = I;$$ this is only possible if $n+i+2-p\geq n+i$, immediately implying $p\leq 2$.
Our key observation is that the desire to zero the hatted qubits is in tension with the completeness relation . Since we only have four $\hat{P}$s, at any given step the *best* we can do is project out a four-dimensional subspace, or two qubits. It is this tension that connects the need to zero qubits with the requirement to have large corrections to the Hawking model. This might help elucidate the results in [@Mathur:2009hf]. Note that this is very much in agreement with the picture presented in Figure \[fig:H\], wherein at each stage there are two new auxillary qubits. For the state to be pure, these auxillary qubits must be zeroed.
Let us note that there are two different ways to zero two qubits at each step, although the distinction is not actually that sharp when one considers the full $C_i$s. In the burning paper model of Section \[sec:paper\] we use only three $\hat{P}$s, which zero one qubit. The three $\hat{P}$s were chosen to ensure that the newly created $\hat{q}$ is also zeroed. The second way is illustrated in Section \[sec:G2\], in which all four $\hat{P}$s are used to zero two old $\hat{q}$s. One can consider various unitary transformations, however, these are the only two qualitative kinds of models that lead to a pure radiation final state. Remember that which $\hat{P}$s get used tell us which pair state is created at the horizon. It is impossible to preferentially use only $\hat{P}_1$ and simultaneously have unitary evolution.
As we saw in Section \[sec:G1-broken\], it is possible for the evolution to be pure, but not invertible. In the model in Equation , qubits that were zeroed in previous steps mixed with nonzeroed qubits. When we zero the qubits, we are then thinking of them as auxillary degrees of freedom that should be erased in the operator-sum description . Thus, it does not make physical sense to allow mixing with the auxillary degrees of freedom if one wants unitary evolution.
The requirements outlined above for purity and invertibilty should ensure unitary evolution.
A One-Parameter Interpolating Model {#sec:one-par}
===================================
We can interpolate between the Hawking model in Section \[sec:hawking\] and the unitary model in Section \[sec:G2\] via $$\begin{aligned}\label{eq:theta-model}
\hat{P}_1 &= \cos\theta\,\hat{I}
+ (1-\cos\theta)\ket{\hat{0}_{2i+1}\hat{0}_{2i+2}}\bra{\hat{0}_{2i+1}\hat{0}_{2i+2}}\\
\hat{P}_2 &= \sin\theta\ket{\hat{0}\hat{0}}\bra{\hat{1}\hat{1}}\\
\hat{P}_3 &= \sin\theta\ket{\hat{0}\hat{0}}\bra{\hat{0}\hat{1}}\\
\hat{P}_4 &= \sin\theta\ket{\hat{0}\hat{0}}\bra{\hat{1}\hat{0}}
\end{aligned}\qquad \hat{U}=\hat{I},$$ with $\theta=0$ giving Hawking’s evolution and $\theta=\frac{\pi}{2}$ giving the unitary evolution in Equation . (Once again we suppressed subscripts on the qubits after the first line.) We may write $$(C - C^H)^{\dagger}(C - C^H) = 2 \hat{I} - \hat{P}_1-\hat{P}_1^{\dagger}= 2(1-\cos\theta)\big(\hat{I}-\ket{\hat{0}\hat{0}}\bra{\hat{0}\hat{0}}\big),$$ so that one finds $$\|C - C^H\| = 2|\sin\tfrac{\theta}{2}|.$$ We clearly see that this is in accord with Mathur’s argument and its generalization in Section \[sec:bound\].
One of the main results of this paper is the above model, which continuously connects the Hawking model to a unitary model, clearly illustrating that they are far apart in model space. Previous efforts to illuminate Mathur’s bound [@Mathur:2011wg; @Mathur:2010kx], considered different types of small corrections and showed that they did not significantly affect the entropy of the final state; however, they did not consider corrections that when made *large* would give a unitary “burning paper” type model. The above model fills this gap. In Figure \[fig:theta\], we plot the second Rényi entanglement entropies of the radiation as a function of $\theta$ for three different initial states. Recall that the second Rényi entropy is defined as $$S_2(\rho_\text{red}) = - \log\operatorname{\mathrm{tr}}(\rho^2_{\text{red}}),$$ and is a positive-definite measure of entanglement that vanishes if and only if $\rho_{\text{red}}$ is pure. For computational purposes, however, it is a bit more convenient than the traditional von Neumann entropy. It should also be noted that $S_2$ gives a lower bound for the von Neumann entropy. Moreover, when one examines the von Neumann entropy it behaves qualitatively similarly.
In Figure \[fig:theta\], one sees that at $\theta=0$ (Hawking model) the entropy rises by one at each step, except for the final step where it drops down by one. This is an artifact of the way we end the evolution, since the model breaks down on the penultimate step. We chose to just emit the last qubit freely, which makes sense on physical grounds since the black hole should be quite small by that point; one can easily imagine large corrections in the final stage(s) of black hole evaporation. At $\theta=\frac{\pi}{2}$, we have the unitary model, and the entanglement entropy of the radiation has the expected rise and fall. For no $\theta$ close to $\theta=0$, however, does the entanglement entropy of the radiation fail to increase (excluding the final step). Of course, a $5$ qubit initial state is not realistic for the macroscopically large black holes we are thinking about; however, it suffices to demonstrate the qualitative behaviour which should not change as $n$ is increased. The author was limited by the computational power required, which grows quite rapidly with $n$.
Conclusion {#sec:conc}
==========
We have presented a very general framework that provides a natural, unifying language to compare information-theoretic models of black hole evaporation. The framework involves describing the dynamics of pure state vectors $\ket{\psi}$ in an ever enlarging Hilbert space. The only constraint on the models is the completeness relation . In Section \[sec:quops\], we explain how to interpret this model in terms of potentially mixed evolution in a fixed dimensional Hilbert space: one must trace out some auxillary degrees of freedom to arrive at a dynamical equation like in Equation . Part of a full model, then, involves specifying the auxillary degrees of freedom. While at intermediate steps this can be ambiguous, by the end of the evolution we are left only with radiation and thus any nonradiation degrees of freedom are by default auxillary. This excludes remnants or other scenarios where one can identify physical degrees of freedom that the Hawking radiation is entangled with at the end of the evaporation.
In Section \[sec:models\], we show how to write a number of interesting models in our unifying notation. Many of them had been introduced and studied previously in the literature. These models illustrate some of the key obstructions and requirements to have unitary evolution.
In Section \[sec:unitarity\], we discuss the requirements for unitarity, and summarize a set of sufficient conditions. A key backdrop to our discussion, and indeed the whole paper, is a recent theorem [@Mathur:2009hf] showing that small corrections to the Hawking model *cannot* give unitary evolution. A corollary is that *large* corrections are necessary to have unitary evolution. As we show, this is, however, not a sufficient condition; unsurprisingly, there are many large corrections that fail to give unitary evolution. One interesting observation is how the unitary requirement that internal qubits be zeroed becomes connected to corrections to the pair creation via the completeness relation . This may help elucidate the results in [@Mathur:2009hf].
Finally, in Section \[sec:one-par\], we give a one-parameter family of models that continuously interpolates between the Hawking model and a unitary model. In terms of this parameter, one can clearly see that the unitary model is far from the Hawking model, thus illustrating the theorem in [@Mathur:2009hf].
One of the key points emphasized in [@Mathur:2009hf; @Mathur:2011uj] is that the nice slicing of the Schwarzschild solution implies at best small corrections to the Hawking model in Equation , and therefore a loss of unitarity evolution. To restore unitarity, large corrections are required of the form discussed here; however, one must show why these corrections arise in the black hole *and not* in all of our earth-based experiments and observations. It seems quite difficult to do this, since in the nice slice construction no geometric quantity is large. The only quantity that seems to be large is the number of degrees of freedom or number of particles required to form the black hole, but this is not a basic geometric quantity.
Let us further note that for our discussion there is a factorization of the internal dynamics and the pair creation dynamics. The internal black hole dynamics can be as nonlocal, or scramble as rapidly as one wants, but whether the evaporation process is unitary or not is (modulo a few caveats mentioned in Sections \[sec:models\] and \[sec:unitarity\]) entirely determined by the pair creation process. Thus, the discussion in [@Hayden:2007cs; @Sekino:2008he; @Susskind:2011ap; @Lashkari:2011yi] is not directly relevant to our concerns here, although it is important for better understanding black holes. The pair creation process is localized near the horizon, where for a large black hole the geometry suggests one can trust the semiclassical approximation even if one might doubt its validity deep within the black hole. On the pair creation time scale, however, it is precisely this physics that needs an order unity correction [@Mathur:2009hf].
One other issue that one may wish to raise in our discussion is the issue of conservation laws [@Czech:2011wy; @Braunstein:2011gz]. While we hope we have sufficiently addressed the issue of an expanding Hilbert space as raised in [@Czech:2011wy], one may still be concerned that we haven’t discussed conservation of energy (or angular momentum, electric charge, etc.). These issues are rebutted in [@Mathur:2011uj]. Let us note, however, by not discussing the original spacetime physics and the resulting at most small corrections, the fundamental issue has been totally elided. While the results presented in [@Czech:2011wy; @Braunstein:2011gz] are interesting unto themselves, they do not provide a plausible physical mechanism to modify the pair creation process to get the dynamics they suggest.
If, as suggested by string theory, or from other considerations, we think black hole evaporation is a unitary process; then, the pair creation process must not strictly adhere to the causal structure on the Schwarzschild nice slicing. There are two obvious frameworks (ignoring the possibility of remnants) to discuss this deviation: fuzzballs or nonlocality.
The fuzzball proposal (see [@Skenderis:2008qn; @Bena:2007kg; @Balasubramanian:2008da; @Mathur:2005ai; @Mathur:2005zp] for reviews) suggests that the black hole metric is only an effective geometry that approximates $e^{S_\text{BH}}$ microstates. The microstates differ from each other on the horizon scale, thus large corrections to the Hawking evolution are anticipated and information is transmitted from local excitations. How the fuzzball proposal relates to these qubit models is discussed in [@Mathur:2010kx; @Mathur:2011uj; @Mathur:2009hf; @Mathur:2011wg]. The main point being that since the geometry at the would-be horizon depends on the internal state, their is a physical mechanism to get large corrections to the pair creation process. Since the fuzzball’s interior geometry (and the whole causal structure) is quite different from the original black hole solution in which the nice slices were constructed, one should probably not interpret them with the original notion of locality for the internal qubits discussed in Section \[sec:physics\]. Moreover, let us note that by adding nontrivial dynamics of the internal fuzzball structure via a $\hat{U}$ to the model in , one can effectively change which qubits get emitted. This is the point referred to at the end of Section \[sec:G2\].
There is one explicit family of (non-extremal) fuzzball microstates [@Jejjala:2005yu] for which one can understand the bulk Hawking emission process. As first suggested in [@Chowdhury:2007jx], the geometry’s ergoregion instability [@Cardoso:2007ws] can be interpreted as a Bose enhanced version of the Hawking instability for the corresponding black hole. This explanation was justified by comparing gravitational emission to the dual CFT emission process for both ergoregion emission from the fuzzballs and Hawking radiation from the corresponding black hole [@Chowdhury:2007jx; @Chowdhury:2008bd; @Chowdhury:2008uj; @Avery:2009tu; @Avery:2009xr]. In [@Chowdhury:2007jx], a toy model was presented for the ergoregion emission, based on the CFT description. The toy model consists of a set of two-level atoms that can spontaneously emit or absorb photons. In the geometric description, the de-exciting atoms correspond to accumulating particles in the ergoregion that decrease the geometry’s mass and angular momentum. Loosely, in our language, the toy model is in the class discussed in Section \[sec:paper\]. To properly capture the Bose enhancement, however, is a bit trickier.
In [@Giddings:2011ks], several unitary models of evolution (some of which were discussed here) were presented, motivated by proposed nonlocal physics on the Schwarzschild background. As mentioned in [@Giddings:2009ae], it is unclear what sets the scale of the proposed nonlocality, so as to ensure it operates in the black hole background but not in everyday low-energy experiments. While the sorts of models discussed here remain too crude to distinguish between nonlocal physics or fuzzball microstates, their utility lies in their generality, which serves to sharpen our information theoretic understanding of black hole evaporation. One obvious task that remains is to translate Mathur’s bound and its generalization in Section \[sec:bound\] into a sharper, quantitative statement about the breakdown of the semiclassical limit of quantum gravity. Since the entire discussion has been in a Hamiltonian framework, it would be especially nice to have analogous bounds on the path integral.
The author is grateful for comments and correspondence related to this work from C. Asplund, S. Ghosh, S. Giddings, and S. Mathur. The material here especially benefitted from discussions with S. Kalyana Rama.
[^1]: See [@Unruh:1995gn; @Unruh:2012vd], however, for criticisms of the arguments made in [@Banks:1983by].
[^2]: We do not claim that these are completely independent.
[^3]: Provided large ensembles of identical copies of the system, one can with some confidence distinguish distinct density matrices; however, this usage of the phrase “information preservation” is not canonical, and is problematic if one starts considering density matrices which are very close to each other. This would mean in an experiment repeated many times with identical initial conditions, one could reconstruct some information about the initial state from the final state. In a technical sense, however, quantum information is lost. We make the distinction here, since the semiclassical description implies that information is not preserved even in this weak sense.
[^4]: Note that we model evolution with discrete time evolution, and do not discuss continuous evolution, which might be governed by the Lindblad equation.
[^5]: If one does treat the new degrees of freedom as physical in the final state, then one is considering a remnant scenario.
[^6]: If we want the initial state to model the matter just before it collapses into a black hole this may not be a good assumption, since in a suitably fine-grained description the number of degrees of freedom available to ordinary matter is parametrically smaller than the entropy of the black hole [@Giddings:2009gj; @'tHooft:1993gx]. We do not concern ourselves with this issue here, but the author is grateful to S. Giddings for pointing this out.
[^7]: As a specific example, Reference [@Czech:2011wy] raises the issue of an ever-enlarging Hilbert space as potential issue in the analysis of [@Mathur:2009hf].
[^8]: Throughout the discussion, the reader may assume that the identity acts on any subspaces which are not explicitly shown.
[^9]: In fact, the result is unchanged by using any other norm that is compatible with the Hilbert space norm.
[^10]: One could instead use Fannes’ inequality for a weaker bound with stronger restrictions on ${\varepsilon}$.
[^11]: An equivalently strong bound here would be $k_{\varepsilon}= 2\delta$.
[^12]: For example, one might make progress by direct computation of $S(r\cup
b)$ and $S(b)$ as was performed in [@Mathur:2009hf], but this would involve solving an eigenvalue problem for a four-dimensional density matrix with arbitrary coefficients.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
**Background**
Dielectric spectra of human blood reveal a rich variety of dynamic processes. Achieving a better characterization and understanding of these processes not only is of academic interest but also of high relevance for medical applications as, e.g., the determination of absorption rates of electromagnetic radiation by the human body.
**Methods**
The dielectric properties of human blood are studied using broadband dielectric spectroscopy, systematically investigating the dependence on temperature and hematocrit value. By covering a frequency range from 1 Hz to 40 GHz, information on all the typical dispersion regions of biological matter is obtained.
**Results and conclusions**
We find no evidence for a low-frequency relaxation (“$\alpha$-relaxation”) caused, e.g., by counterion diffusion effects as reported for some types of biological matter. The analysis of a strong Maxwell-Wagner relaxation arising from the polarization of the cell membranes in the 1-100 MHz region (“$\beta$-relaxation”) allows for the test of model predictions and the determination of various intrinsic cell properties. In the microwave region beyond 1 GHz, the reorientational motion of water molecules in the blood plasma leads to another relaxation feature (“$\gamma$-relaxation”). Between $\beta$- and $\gamma$-relaxation, significant dispersion is observed, which, however, can be explained by a superposition of these relaxation processes and is not due to an additional “$\delta$-relaxation” often found in biological matter.
**General significance**
Our measurements provide dielectric data on human blood of so far unsurpassed precision for a broad parameter range. All data are provided in electronic form to serve as basis for the calculation of the absorption rate of electromagnetic radiation and other medical purposes. Moreover, by investigating an exceptionally broad frequency range, valuable new information on the dynamic processes in blood is obtained.
address: 'Experimental Physics V, Center for Electronic Correlations and Magnetism, University of Augsburg, 86135 Augsburg, Germany'
author:
- 'M. Wolf'
- 'R. Gulich'
- 'P. Lunkenheimer'
- 'A. Loidl'
title: Broadband Dielectric Spectroscopy on Human Blood
---
dielectric spectroscopy ,blood ,relaxation ,specific absorption rate ,dielectric loss ,dielectric constant
Introduction {#intro}
============
Blood is a highly functional body fluid, it delivers oxygen to the vital parts, it transports nutrients, vitamins, and metabolites and it also is a fundamental part of the immune system. Therefore the precise knowledge of its constituents, its physical, biological, and chemical properties and its dynamics is of great importance. Especially its dielectric parameters are of relevance for various medical applications [@Grant1978], like cell separation (e.g., cancer cells from normal blood cells [@Becker1995]), checking the deterioration of preserved blood [@Hayashi2010a], and dielectric coagulometry [@Hayashi2010]. In addition, the precise knowledge of the dielectric properties of blood is prerequisite for fixing limiting values for electromagnetic pollution (via the conductivity in the specific absorption rate (SAR))[@Ahlbom1998; @IEEE2002; @IEEE2005; @Loidl2008].
Early measurements of the electrical properties of blood contributed significantly to unravel the constitution of red blood cells (RBC). For example, the results by Höber [@Hoeber1913] provided the first indications of a dispersion (i.e. frequency dependence), caused by the membrane of RBCs, in the radio frequency (RF) spectrum of the dielectric properties of blood. This relaxation process is nowadays identified as being of Maxwell-Wagner type [@Maxwell1873; @Wagner1914] and termed $\beta$-relaxation in biophysical literature [@Schwan1957a; @Schwan1983].
Various early works [@Fricke1923; @Fricke1924; @Fricke1925a; @Fricke1925b; @Cole1928; @Daenzer1934b; @Bruggemann1935; @Rajewsky1948] were followed by measurements at very high frequencies [@Schwan1953; @England1950; @Cook1951; @Cook1952]. Some of them revealed an additional dispersion with a relaxation rate near 18 GHz, which can be assigned to the reorientation of water molecules and which is named $\gamma$-dispersion. Furthermore, a third relaxation, termed $\alpha$-relaxation and located in the low-frequency regime, $\nu<100$ kHz was detected in some biological materials [@Schwan1954; @Schwan1957]. However, interestingly, an $\alpha$-relaxation seems to be absent in whole blood [@Bothwell1956] and only is found in hemolyzed blood cells [@Schwan1957]. This was speculated to be due to a higher ion permeability of the membranes in the latter case, shifting the relaxation spectrum into the experimental frequency window [@Schwan1957]. The origin of the $\alpha$-relaxation is a matter of controversy; most commonly it is assumed to arise from counterion diffusion effects [@Schwan1983; @Schwan1962]. Finally, a dispersion with low dipolar strength located in the frequency regime between the $\beta$- and $\gamma$-dispersion was identified by Schwan [@Schwan1957a]. The origin of the $\delta$-dispersion and the possible role of bound water in its generation is controversially discussed [@Schwan1965; @Grant1965; @Grant1968; @Pennock1969; @Nandi2000; @Feldman2003; @Knocks2001; @Thomas2008; @Sasisanker2008].
Taking together all these results, it is clear that there are three main dispersion regions in the dielectric frequency spectrum of blood between some Hz and 50 GHz, termed $\beta$, $\gamma$, and $\delta$ [@Schwan1957a; @Schwan1983]. This nomenclature should not be confused with that used in the investigation of glassy matter like supercooled liquids or polymers. Within the glass-physics community the terms $\alpha$-, $\beta$-relaxation, etc. are commonly applied to completely different phenomena than those considered above (see, e.g., refs and ). In the present work we follow the biophysical nomenclature.
A lot of additional research has been done until the early 1980s [@Grant1978; @Pauly1966; @Krupa1972; @Schanne1978; @Pethig1979] and a detailed review was given in 1983 by Schwan [@Schwan1983]. Later on, in the course of the upcoming debates about electromagnetic pollution, dielectric properties of body tissues and fluids received renewed interest as they determine the SAR, a measure for the absorption of electromagnetic fields by biological tissue [@Loidl2008; @Gabriel1996a; @Gabriel1996b; @Foster1989; @Foster1995]. But also various other important medical questions can be addressed by dielectric spectroscopy [@Grant1978; @Becker1995; @Hayashi2010a; @Hayashi2010; @Markx1999]. In the last two decades, a number of papers on dielectric spectroscopy on blood and erythrocyte solutions were published [@Feldman2003; @Lisin1996; @Zhao1993; @Bordi1997; @Jaspard2003; @Beving1994a; @Bao1994; @Chelidze2002; @Hayashi2008], most of them treating special aspects only.
On the theoretical side, a number of models for the description of cell suspensions have been proposed. Most models focus on the $\beta$-relaxation [@Schwan1957a; @Fricke1923; @Fricke1924; @Fricke1925a; @Fricke1925b; @Bruggemann1935; @Pauly1959; @Hanai1960a; @Looyenga1965; @Asami2002; @Hanai1961; @Hanai1968; @Katsumoto2008], including the often employed Pauly-Schwan model [@Schwan1957a; @Pauly1959], discussed below. Some of them also account for the non-spherical shape of cells [@Fricke1923; @Fricke1924; @Fricke1925a; @Fricke1925b; @Asami2002]. It seems clear that diluted solutions and whole blood with a hematocrit value of 86% have to be treated differently. The Bruggeman-Hanai model [@Bruggemann1935; @Hanai1960a] was specially developed for highly concentrated suspensions. A recent summary of various models can be found in ref. [@Asami2002].
Concluding this introduction, it has to be stressed that, after more than one century of research, many aspects of the dielectric properties of blood (e.g., the presence and origins of $\alpha$- and $\delta$-dispersion) are still unclear. It should be noted that, in addition to the three main dispersion effects, from a theoretical viewpoint a number of further relaxation features may show up in blood. For example, it is well known that RBC’s are far from being of spherical shape and in principle for shelled ellipsoidal particles up to six relaxations can be expected [@Asami2002]. Furthermore, the hemoglobin molecules within the RBC’s should show all the typical complex dynamics as found in other proteins. Based on the available literature data (e.g., [@Schwan1983]), it seems that most, if not all, of these additional processes do not or only weakly contribute to the experimentally observed spectra. However, one should carefully check for possible deviations from the simple three-relaxation scenario mentioned above, which may well be ascribed to these additional processes.
Maybe the best and most cited broadband spectra of blood covering several dispersion regions are those by Gabriel *et al* [@Gabriel1996b], taken at 37 C, which are commonly used for SAR calculations and for medical purposes. However, even these data are hampered by considerable scatter and they are composed from data collected by different groups on different samples. Clearly, high-quality spectra covering a broad frequency range measured on identical samples are missing. A systematic investigation of the hematocrit and temperature dependence is essential to achieve a better understanding of the different dispersion contributions of blood. The present work provides the dielectric constant $\varepsilon'$, the loss $\varepsilon''$, and the conductivity $\sigma'$ of human blood in a broad frequency range (1 Hz to 40 GHz), by using a combination of different techniques of dielectric spectroscopy applied to identical samples. In addition, the temperature (280 K - 330 K) and hematocrit value (0 - 86%) dependence is thoroughly investigated.
Models and Data Analysis {#Models}
========================
Dielectric spectroscopy is sensitive to dynamical processes that involve the reorientation of dipolar entities or displacement of charged entities, which can cause a dispersive behavior of the dielectric constant and loss. However, also non-intrinsic Maxwell-Wagner effects caused by interfacial polarization in heterogeneous samples can lead to considerable dispersion [@Maxwell1873; @Wagner1914; @Lunkenheimer2002]. As mentioned above, biological matter shows various dispersions in the frequency regime 1 Hz to 40 GHz, which have different microscopic and mesoscopic origins and therefore have to be described differently.
Intrinsic Relaxations {#Intrinsic}
---------------------
Intrinsic processes like, e.g., the cooperative reorientation of dipolar molecules are often described by the Debye formula [@Debye1929]:
$$\label{Db}
\varepsilon^{*}(\nu)=\varepsilon_{\infty}+\frac{\Delta\varepsilon}{1+\mathrm{i}2\mathrm{\pi}\nu\tau}\:$$
The relaxation strength is given by $\Delta\varepsilon=\varepsilon_{\mathrm{s}}-\varepsilon_{\mathrm{\infty}}$. $\varepsilon_{\mathrm{s}}$ and $\varepsilon_{\infty}$ are the limiting values of the real part of the dielectric constant for frequencies well below and above the relaxation frequency $\nu_{\mathrm{relax}}=1/(2\pi\tau)$, respectively. This frequency is characterized by an inflection point in the frequency dependence of the dielectric constant and a peak in the dielectric loss. If taking into account an additional dc-conductivity contribution, the conductivity shows a steplike increase close to $\nu_{\mathrm{relax}}$. The Debye theory assumes that all entities do relax with the same relaxation time $\tau$. In reality, a distribution of relaxation times often leads to a considerable smearing out of the spectral features [@Sillescu1999; @Ediger2000]. Those can be described, e.g., by the Havriliak-Negami formula, which is an empirical extension of the Debye formula by the additional parameters $\alpha$ and $\beta$ [@Havriliak1966; @Havriliak1967]:
$$\label{hn}
\varepsilon^{*}(\nu)=\varepsilon_{\mathrm{\infty}}+\frac{\Delta\varepsilon}{\left[1+(\mathrm{i}2\pi\nu\tau)^{1-\alpha}\right]^{\beta}}\:$$
Special cases of this formula are the Cole-Cole formula [@Cole1941] with $0\leq\alpha<1$ and $\beta=1$ and the Cole-Davidson formula [@Davidson1950; @Cole1952] with $\alpha=0$ and $0<\beta\leq1$.
In most materials with intrinsic relaxations, the inevitable dc conductivity $\sigma_{\mathrm{dc}}$ arising from ionic or electronic charge transport cannot be neglected. Usually it leads to a $1/\nu$ divergence in the loss at frequencies below the loss peak and can be taken into account by including a further additive term $\varepsilon''_{\mathrm{dc}}=\sigma_{\mathrm{dc}}/(\varepsilon_{0}\omega)$ in Eq. \[hn\] ($\varepsilon_{0}$ denotes the permittivity of free space, $\omega$ is the circular frequency).
Maxwell-Wagner Relaxations {#Mw}
--------------------------
The $\beta$-dispersion in biological matter is commonly accepted to be of Maxwell-Wagner type [@Schwan1957a; @Schwan1983; @Pethig1979; @Foster1989; @Foster1995]. As shown by Maxwell and Wagner [@Maxwell1873; @Wagner1914], strong dispersive effects mimicking those of intrinsic dipolar relaxations can arise in samples composed of two or more regions with different electrical properties (e.g., plasma, cytoplasma, and cell membranes in the case of RBCs). It should be noted that this dispersion can be completely understood from the heterogeneity of the investigated samples without invoking any frequency-dependent microscopic processes within the involved dielectric materials. If one of the regions in the sample is of interfacial type and relatively insulating, e.g., an insulating surface layer [@Lunkenheimer2002] or the membranes of biological cells [@Schwan1957a; @Schwan1983], very high apparent values of the dielectric constant are detected at low frequencies. A straightforward approach for understanding the dielectric behavior of heterogeneous systems is an equivalent-circuit analysis. Here any interfacial layer can be modeled by a parallel RC element with the resistance $R$ and capacitance $C$ of the interfacial element much higher than the corresponding bulk values [@Lunkenheimer2002]. This leads to a relaxation spectrum where the low-frequency capacitance and conductance are dominated by the interface. For the calculation of $\varepsilon'(\nu)$ from the measured capacitance, usually the overall geometry of the sample is used instead of that of the thin interfacial layer (i.e., the assumed $C_{0}$ is much smaller than that of the layer). Thus, an artificially high dielectric constant is detected at low frequencies. At high frequencies, the interface capacitor becomes shorted and the bulk properties are detected. This leads to the steplike decrease of $\varepsilon'(\nu)$ and increase of $\sigma'(\nu)$ with increasing frequency, typical for relaxational behavior. The increase of $\sigma'(\nu)$ arises from the fact that the bulk conductance usually is much higher than the interface conductance (the cell membrane in the case of RBCs), the latter being shorted by the interface capacitor at high frequencies.
Instead of an equivalent-circuit analysis [@Lunkenheimer2002], for biological matter it is common practice to treat the $\beta$-relaxation analogous to an intrinsic relaxation process, i.e., to fit it with the Debye equation or its extensions (Eqs. \[Db\] and \[hn\]). A variety of models have been developed to connect the obtained fitting parameters with the intrinsic dielectric properties of the different regions of the samples (see section \[cellmod\]) and a lot of modeling work of experimental data was performed [@Schwan1957a; @Fricke1923; @Fricke1924; @Fricke1925a; @Fricke1925b; @Bruggemann1935; @Hayashi2008; @Pauly1959; @Hanai1960a; @Looyenga1965; @Asami2002; @Hanai1961; @Hanai1968; @Katsumoto2008; @Asami1999; @Gheorghiu1999; @DiBiasio2005; @Merla2006; @Sudsiri2007; @Hayashi2009].
Electrode Polarization {#BE}
----------------------
Blood exhibits strong ionic conductivity. At low frequencies the ions arrive at the metallic electrodes and accumulate in thin layers immediately below the sample surface [@Oncley1942; @Schwan1968; @MacDonald1987; @Bordi2001]. The frequency, below which this effect sets in, critically depends on electrode distance and ionic mobility and in biological matter typically is located in the kHz - MHz region. These insulating layers represent large capacitors leading to an apparent increase of $\varepsilon'(\nu)$ and decrease of $\sigma'(\nu)$ at low frequencies, quite similar to the Maxwell-Wagner effects discussed above. These non-intrinsic contributions can hamper the unequivocal detection of the parameters of the $\beta$-relaxation. Various experimental techniques have been applied to avoid the influence of electrode polarization (see, e.g., refs. ). An alternative way is the exact modeling of these non-intrinsic contributions. The most common models are a parallel RC circuit or a so-called constant-phase element, both connected in series to the bulk sample [@Schwan1968; @MacDonald1987; @Bordi2001].
A parallel RC circuit corresponds to an additional impedance
$$\label{RC}
Z_{\mathrm{RC}}^{*}(\nu)=\frac{R_{\mathrm{RC}}}{1+\mathrm{i}2\pi\nu
R_{\mathrm{RC}}C_{\mathrm{RC}}},$$
which has to be added to the bulk impedance. $R_{\mathrm{RC}}$ and $C_{\mathrm{RC}}$ are the resistance and capacitance of the insulating layers, respectively. From the resulting total impedance, the total capacitance and conductance (and thus $\varepsilon'(\nu)$ and $\sigma'(\nu)$) can be calculated resulting in a behavior equivalent to a Debye-relaxation (this scenario corresponds to a conventional Maxwell-Wagner relaxation). Alternatively a “constant phase element”, which is an empirical impedance function, given by $Z_{\mathrm{CPE}}=A(\mathrm{i\omega})^{-\alpha}$ (refs ), can be used. When defining $\tau_{\mathrm{RC}}=R_{\mathrm{RC}}C_{\mathrm{RC}}$, Eq. \[RC\] formally has the same mathematical structure as Eq. \[Db\]. Thus, in analogy to Eq. \[hn\] a distributed RC-circuit can be introduced by writing
$$\label{RCverteilt}
Z_{\mathrm{RC}}^{*}(\nu)=\frac{R_{\mathrm{RC}}}{\left[1+(\mathrm{i}2\pi\nu\tau_{\mathrm{RC}})^{1-\alpha}\right]^{\beta}}\:.$$
We want to emphasize, that in contrast to Eq. \[RC\], which leads to a frequency dependence identical to that of a Debye relaxation, an equivalent-circuit evaluation using Eq. \[RCverteilt\] does not lead to fit curves identical to those of a Havriliak-Negami relaxation: For the latter case, the relaxation time $\tau$ in Eq. \[Db\] is assumed to be distributed. In the equivalent-circuit case, the corresponding quantity, determining, e.g., the loss peak position, is $\tau=R_{b}C_{RC}$ with $R_{b}$ the bulk resistance [@Lunkenheimer2002]. However, the distributed quantity in the equivalent-circuit case is $\tau_{RC}=R_{RC}C_{RC}$, thus leading to different curve shapes.
Temperature Dependence
----------------------
The fitting of relaxation spectra directly provides the relaxation times ($\tau$), the width parameters ($\alpha$ and $\beta$), the relaxation strengths ($\Delta\varepsilon$), the dielectric constant for $\nu\longrightarrow\infty$ ($\varepsilon_{\infty}$), and the dc conductivity ($\sigma_{\mathrm{dc}}$) (see section \[Intrinsic\]). For the temperature dependence of $\tau$ and $\sigma_{\mathrm{dc}}$, thermally activated behavior
$$\label{tautemp}
\tau=\tau_{0}\exp\left(\frac{E_{\tau}}{k_{\mathrm{B}}T}\right)$$
and
$$\label{sdc}
\sigma_{\mathrm{dc}}=\frac{\sigma_{0}}{T}\exp\left(-\frac{E_{\sigma}}{k_{\mathrm{B}}T}\right),$$
can be assumed. $\sigma_{0}$ is a prefactor. $E_{\tau}$ and $E_{\sigma}$ denote the hindering barriers for the relaxational process and the diffusion of the charge carriers (i.e., dissolved ions of the plasma in the present case), respectively. $\tau_{0}$ is an inverse attempt frequency, often assumed to be of the order of a typical phonon frequency. Equation \[sdc\], with the extra $1/T$ term, is derived by considering the difference of the forward and backward hopping probabilities of an ion between two sites in a potential that becomes asymmetric due to the external field [@Zarzycki1991]. Here the field drives the ionic motion. In contrast, for dielectric relaxations, within the framework of the fluctuation-dissipation theorem it is assumed that the dielectric measurement is sensitive to reorientational fluctuations, which are present even without field. Thus, $\tau$ in Eq. \[tautemp\] is proportional to the inverse of the reorientation probability of a molecule experiencing a hindering barrier, which is just given by the exponential term.
The temperature dependence of the dielectric strength of dipolar relaxation mechanisms often can be characterized by the Curie law [@Debye1912; @Onsager1936]:
$$\label{Curie} \Delta\varepsilon=\frac{C}{T}\$$
Deviations from the Curie law are usually thought to signify dipole-dipole interactions.
For most dielectric materials, the broadening of the loss peak diminishes with increasing temperature, i.e., $\alpha\rightarrow0$ and $\beta\rightarrow1$ for high temperatures. This finding can be ascribed to the fast thermal fluctuations, which cause every relaxing entity “seeing” the same environment [@Lunkenheimer2000], thus leading to an identical relaxation time for each entity, which implies Debye-like behavior.
Cell Models {#cellmod}
-----------
Two commonly employed models for the description of the $\beta$-dispersion of cell suspensions and tissue will be introduced now and applied to the experimental data in section \[app\]. Both models are claimed to be applicable to high cell concentrations and thus are especially suited for the samples of the present work. Based on the Maxwell-Wagner model [@Maxwell1873; @Wagner1914], the Pauly and Schwan takes into account the membranes of cells [@Pauly1959; @Foster1995]. Using appropriate approximations (e.g., a negligible membrane conductance) some simple relations are derived:
$$\label{deltaeps}
\Delta\varepsilon_{\beta}=\frac{9prC_{\mathrm{m}}}{4\varepsilon_{0}\cdot
(1+p/2)^2}$$
and
$$\label{au}
\sigma_{\mathrm{dc\beta}}=\frac{1-p}{1+p/2}\cdot\sigma_{\mathrm{a}}$$
Equation \[deltaeps\] allows the calculation of the membrane capacitance per area unit, $C_{\mathrm{m}}$, from the relaxation strength of the $\beta$-dispersion, $\Delta\varepsilon_{\beta}$, and the volume fraction $p$ and radius $r$ of the suspended particles. Via Eq. \[au\], the conductivity of the suspending medium (plasma in the case of blood), $\sigma_{\mathrm{a}}$, can be determined from the volume fraction and the measured dc conductivity of the suspension, i.e. the limiting low frequency conductivity of the $\beta$-dispersion, $\sigma_{\mathrm{dc\beta}}$. Moreover, resolving the expression for the $\beta$-relaxation time $\tau_{\mathrm{\beta}}$ [@Pauly1959],
$$\label{taubet}
\tau_{\beta}=rC_{\mathrm{m}}\frac{2\sigma_{\mathrm{a}}+\sigma_{\mathrm{i}}-p(\sigma_{\mathrm{i}}-\sigma_{\mathrm{a}})}{\sigma_{\mathrm{i}}\sigma_{\mathrm{a}}(2+p)},$$
one can approximate the conductivity of the cell interior (cytoplasma), $\sigma_{\mathrm{i}}$ by
$$\label{sp}
\sigma_{\mathrm{i}}=\frac{\sigma_{\mathrm{a}}rC_{\mathrm{m}}(2+p)}{\sigma_{\mathrm{a}}\tau_{\beta}(2+p)-(1-p)rC_{\mathrm{m}}}\:.$$
An alternative access to $\sigma_{\mathrm{i}}$ is provided by the following relation [@Pauly1959]:
$$\label{spsu}
\sigma_{\mathrm{i}}=\frac{2\sigma_{\mathrm{a}}^{2}(1-p)-\sigma_{\mathrm{a}}\sigma_{\infty\beta}(2+p)}{\sigma_{\infty\beta}(1-p)-\sigma_{\mathrm{a}}(1+2p)}$$
Here $\sigma_{\infty\beta}$ denotes the high-frequency plateau of the step in $\sigma'(\nu)$. Also the dielectric constant of the cytoplasma, $\varepsilon_{\mathrm{i}}$, can be determined [@Pauly1959]:
$$\label{sp2}
\varepsilon_{\mathrm{i}}=\frac{2\varepsilon_{\mathrm{a}}^{2}(1-p)-\varepsilon_{\mathrm{a}}\varepsilon_{\infty\beta}(2+p)}{\varepsilon_{\infty\beta}(1-p)-\varepsilon_{\mathrm{a}}(1+2p)}$$
$\varepsilon_{\mathrm{\infty\beta}}$ is the limiting high-frequency dielectric constant of the $\beta$-relaxation (cf. $\varepsilon_{\mathrm{\infty}}$ in Eq. \[Db\]) and $\varepsilon_{\mathrm{a}}$ is the dielectric constant of the suspending medium (plasma).
The Pauly-Schwan model includes correction factors for high concentrations (e.g., the $1+p/2$ factor in eq. \[deltaeps\]) and is claimed to be valid for all values of $p$ (see, e.g., ref. [@Foster1995]). Another model, especially developed for highly concentrated suspensions, is the one by Bruggemann [@Bruggemann1935] and Hanai [@Hanai1960a; @Hanai1961; @Hanai1968] taking into account the polarization of particles in the presence of neighboring ones. The model leads to the equations
$$\label{hb1}
\varepsilon_{\mathrm{p}}=\frac{\varepsilon_{\mathrm{a}}(1-p)-\varepsilon_{\mathrm{\infty\beta}}k}{1-p-k},\hspace{0.5cm}
k=\left(\frac{\varepsilon_{\mathrm{a}}}{\varepsilon_{\mathrm{\infty\beta}}}\right)^{1/3}\hspace{0.5cm}$$
and
$$\label{hb2}
\sigma_{\mathrm{p}}=\frac{\sigma_{\mathrm{a}}(1-p)-\sigma_{\mathrm{dc\beta}}k}{1-p-k},\hspace{0.5cm}
k=\left(\frac{\sigma_{\mathrm{a}}}{\sigma_{\mathrm{dc\beta}}}\right)^{1/3}$$
for the dielectric properties $\varepsilon_{\mathrm{p}}$ and $\sigma_{\mathrm{p}}$ of the particles. It should be mentioned that this model assumes homogeneous particles (i.e. without shell) and Eqs. \[hb1\] and \[hb2\] can be considered providing average values of the whole cell only.
One should note that the exact solution of the dielectric theory of suspensions of ellipsoidal particles leads to the prediction of six separate relaxation processes, namely two per ellipsoid axis, arising from the Maxwell-Wagner relaxation of the shell and of the particle interior [@Asami2002]. For spheroids, i.e. ellipsoids with two equal semi-diameters, which may be a good approximation of RBCs, still four relaxations are expected. Usually in the application of the Maxwell-Wagner model to cell suspensions, including the above treated models, various reasonable approximations are made (e.g., that the membrane thickness is much smaller than the cell radius) that lead to the prediction of a single relaxation only.
Materials and methods {#Materials}
=====================
To determine the complex dielectric permittivity and conductivity in a broad frequency range (from 1 Hz to 40 GHz), different measurement techniques were combined [@Schneider2001]. In the frequency range 1 Hz - 10 MHz, high precision measurements were performed by means of a Novocontrol Alpha-A Analyzer. This frequency response analyzer directly measures the sample voltage and the sample current by the use of lock-in technique. The ac voltage is applied to a parallel-plate capacitor made of platinum containing the sample material (diameter 5 mm, plate distance 0.6 mm). In our earlier measurements of various materials, platinum was found to minimize contributions from dissolved ionic impurities arising from the electrode material. The capacitor is mounted into a N$_{2}$-gas cryostat (Novocontrol Quatro) for temperature-dependent measurements. For the measurements in the frequency range 1 MHz - 3 GHz a coaxial reflection method was used employing the Agilent Impedance/Material Analyzer E4991A. Here the sample, again placed in the same parallel-plate capacitor, is connected to the end of a specially designed coaxial line, thereby bridging inner and outer conductor [@Boehmer1989]. For additional measurements between 40 Hz-110 MHz, the autobalance bridge Agilent 4294A was used. Its measurement range overlaps with that of the other devices. In all the measurements described above, the applied ac voltage was 0.1 V. The Agilent “Dielectric Probe Kit” 85070E using the so called “performance probe” with an E8363B PNA Series Network Analyzer covered the high frequency range from 100 MHz to 40 GHz. It uses a so-called open-end coaxial reflection technique, where the end of a coaxial line is immersed into the sample liquid. The applied ac voltage was 32 mV. Calibration was performed with the standards Open, Short, and Water. As any contributions from parasitic elements are excluded by this technique, the obtained absolute values were used to correct the results obtained with the low-frequency techniques, discussed above, for contributions from stray capacitance.
Blood samples from a healthy person were taken at the hospital “Klinikum Augsburg”. All blood samples were taken from the same person and various vital parameters of the blood samples were checked. All samples were taken before a meal, at the same time of the day. We did not find any significant difference in the measurement results obtained on samples taken at different days. To avoid clotting, the samples were prepared with EDTA (ethylenediaminetetraacetic acid). The influence of different coagulation inhibitors on the dielectric properties were tested and found to be insignificant. Besides the whole blood, which was measured as taken from the body, four other samples with different hematocrit values ($Hct$) were prepared. $Hct$ is given by the ratio of volume fraction of the corpuscles (erythrocytes, leukocytes, and thrombocytes) of the blood and the total volume. The whole blood used in the present work was found to have $Hct=0.39$. After centrifugation of the whole blood, the corpuscular parts could be separated from the plasma by pipetting. By remixing with the plasma obtained in this way, four additional samples were prepared: plasma ($Hct<5$) and blood with $Hct=0.23$, 0.57, and 0.86. The exact $Hct$ values were determined by taking hemograms with a Beckman-Coulter Hematology Analyzer at the hospital. For each measurement run with the different devices fresh blood samples were used.
One may suspect that sedimentation of RBCs and other cells during the temperature-dependent measurement runs could influence the measurements. Covering the whole investigated temperature range with the different measurement devices did take about 2-3 hour only. In the high-frequency measurements with the “open-end” coaxial technique, where relatively large amounts of material (about 25 ml) held in transparent test tubes were used, visible inspection revealed no indications of sedimentation even after much longer time. Nevertheless the sample material was thoroughly stirred before each frequency sweep at the different temperatures. The results did well match those at lower frequencies, where small sample amounts of about 0.01 ml contained in a platinum capacitor were used. With this capacitor, up to three separate temperature-dependent measurement runs were performed, using different devices and partly using different thermal histories. All the results did match very well. Finally, the measurements usually were done by first cooling the sample from room temperature and subsequently heating it up to the highest temperatures. The results from the cooling and heating runs, which were done at differently aged samples, did always agree within experimental resolution.
The integrity of the erythrocytes was retained during most of the dielectric measurements, which was checked by a comparison of the room temperature results before and after cooling or heating. However, as expected this was no longer the case when the samples were subjected to the highest temperatures investigated, extending up to 330 K. Therefore these measurements were performed at the end of each temperature-dependent measurement run.
If not otherwise indicated, the experimental data points shown in the plots of the present work have errors that are not exceeding the size of the symbols.
Results and Discussion {#results}
======================
Broadband Spectra
-----------------
Figure \[fig:all\] shows the broadband spectra of the different samples at body temperature ($\approx310$ K), extending from 1 Hz to 40 GHz.
![\[fig:all\] (a) Dielectric constant, (b) dielectric loss, and (c) real part of the conductivity of whole blood, blood plasma ($Hct=0.39$), and blood samples with different hematocrit values as function of frequency, measured at body temperature (310 K). The lines are fits assuming a distributed RC equivalent circuit to account for the electrodes and, for samples with $Hct\geq0.23$, two Cole-Cole functions for the $\beta$- and $\gamma$-relaxation. For the plasma data, a single Cole-Cole function was used instead.](fig1rev.eps){width="8cm"}
The dielectric constant of the blood plasma (Fig. \[fig:all\](a), orange circles) reveals a low-frequency plateau between 1 and 100 Hz, followed by a steplike decrease towards higher frequencies that passes into another plateau between about 1 MHz and 10 GHz. The behavior below about 1 MHz can be ascribed to electrode polarization (see section \[alpha\] for a detailed discussion). At frequencies beyond about 1 GHz a further decrease of $\varepsilon'(\nu)$ indicates the beginning $\gamma$-dispersion arising from the reorientational motion of the water molecules (see section \[gamma\]). $\varepsilon''(\nu)$ shows a plateau at low frequencies, followed by a strong decrease above about 300 Hz and the $\gamma$-relaxation peak at ca. 20 GHz. Accordingly, the conductivity $\sigma'(\nu)$ exhibits a strong increase at low frequencies, followed by a plateau between 1 kHz and 1 GHz. At $\nu>1$ GHz another strong increase at the end of the measured spectrum shows up, again corresponding to the $\gamma$-relaxation.
Just as the plasma, the dielectric spectra of the other samples also show a $\gamma$-relaxation and a electrode-polarization contribution, the latter leading to a strong increase of $\varepsilon'(\nu)$ below about 10 - 100 kHz and decrease of $\sigma'(\nu)$ below about 1 - 10 kHz. However, between these two features, a further process shows up, the well-known $\beta$-dispersion, located at about 1 - 100 MHz in $\varepsilon'(\nu)$. It is caused by the Maxwell-Wagner relaxation arising from the heterogeneity of the solute/cell system (see section \[Mw\]). It is evidenced by a steplike decrease in the dielectric constant at about 1 - 100 MHz, an s-shaped bend in the decrease of the dielectric loss around 1 MHz, and a steplike increase in the conductivity around 1 MHz. Comparing the different samples, it becomes evident that the absolute values of $\varepsilon'(\nu)$, $\varepsilon''(\nu)$, and $\sigma'(\nu)$ decrease with increasing hematocrit value over almost the whole frequency range. A detailed analysis of the $\beta$-dispersion will be provided in section \[beta\]. The $\delta$-dispersion, which is supposed to be located in the frequency range between the $\beta$- and $\gamma$-relaxation cannot be detected on this scale and will be treated in section \[further\] below.
In the present work the complete broadband spectra are fitted by combining several relaxational dispersions and the electrode-polarization contribution. In addition, the dc conductivity has to be included in the fitting routine, since the blood plasma contains free ions that contribute to the conductivity. The lines shown in Fig. \[fig:all\] are fits with the sum of two relaxational dispersions described by Eq. \[hn\] (with $\beta=1$) and the dc-conductivity, which are assumed to be connected in series to the electrode impedance given by Eq. \[RCverteilt\] (with $\beta=1$). The fits were simultaneously performed for real and imaginary part of the permittivity. A qualitative inspection of Fig. \[fig:all\] reveals that an excellent match of the experimental spectra could be achieved in this way, which also is the case for the other temperatures investigated. In the following sections, we discuss the different contributions to the spectra and the resulting relevant fit parameters in detail. To serve for SAR calculations and medical purposes, the fit curves for all temperatures and hematocrit values investigated in the present work are provided in electronic form.
Electrode polarization and $\alpha$-Dispersion {#alpha}
----------------------------------------------
To examine the contributions from electrode polarization to the spectra in more detail, as an example in Fig. \[fig:alphaRCCC\] the spectra of whole blood at 310 K are shown in the low frequency range, 1 Hz - 200 kHz. As demonstrated by the dash-dotted magenta line, the observed relaxationlike feature cannot be satisfactorily fitted by assuming a simple RC equivalent circuit (Eq. \[RC\]) in series to the bulk sample, which leads to a symmetric loss peak just like an intrinsic Debye relaxation (see section \[BE\]). The deviations are strongest in $\varepsilon''$ below the peak frequency, where the measured $\varepsilon''(\nu)$ obviously varies far too weakly with frequency to be describable by the strongly increasing fit curve. Using a Debye relaxation function (Eq. \[Db\]) with an additional conductivity contribution $\sigma'_{\mathrm{dc}}$, which leads to a minimum in $\varepsilon''(\nu)$ below the peak frequency, also cannot account for the experimental data (dotted blue line). Replacing the Debye by a Cole-Cole function (Eq. \[hn\] with $\beta=1$; green line) only leads to a marginal improvement of the fits. Instead, a distributed RC equivalent circuit as described by Eq. \[RCverteilt\] with $\beta=1$ but $\alpha\neq0$ (Cole-Cole case) was found to provide very accurate fits of the measured spectra (solid red line). The same can be said for the other blood samples with different hematocrit values investigated in the present work. A constant phase element in series with the bulk also very well accounts for the experimental data (black dashed line). However, in our further analysis we decided to use the distributed RC circuit instead because its parameters seem to have more physical signification than those of the constant phase element.
![\[fig:alphaRCCC\]$\varepsilon'(\nu)$ (a), $\varepsilon''(\nu)$ (b), and $\sigma'(\nu)$ (c) of whole blood ($Hct=0.39$) at 310 K and low frequencies (circles). The lines are fits using different functions: RC equivalent circuit (Eq. \[RC\]; dash dotted magenta line), distributed RC equivalent circuit (Eq. \[RCverteilt\]; red line), Debye function with additional dc-conductivity contribution (dotted blue line), Cole-Cole function, also with dc contribution (green line), and constant phase element (black dashed line).](fig2.eps){width="8cm"}
Good fits with this approach can also be achieved for the results obtained at different temperatures. As an example, Fig. \[fig:alphaVoll\] shows the dielectric quantities of whole blood in the frequency range, dominated by electrode polarization, for selected temperatures. The lines represent the fits of the complete broadband spectra (cf. Fig. \[fig:all\]) where a distributed RC circuit, Eq. \[RCverteilt\], was used for the description of the low-frequency data. In all cases the agreement of fits and experimental curves are excellent. The onset of the electrode effects, i.e. the increase of $\varepsilon'$ and the decrease of $\sigma'$ when lowering the frequency, is found to shift to lower frequencies with decreasing temperature. This can be ascribed to the reduced mobility of the ionic charge carriers at low temperatures, which thus arrive at the electrodes for smaller frequencies only. As revealed by Fig. \[fig:all\], for increasing hematocrit value the onset of the electrode effects shifts to lower frequencies, too. This is in accord with the well-known increase of the viscosity of blood with increasing hematocrit value, corresponding to a reduction of ion mobility.
![\[fig:alphaVoll\]$\varepsilon'(\nu)$ (a), $\varepsilon''(\nu)$ (b), and $\sigma'(\nu)$ (c) of whole blood ($Hct=0.39$) in the low-frequency regime dominated by electrode effects shown for selected temperatures (symbols). The lines represent fit curves as in Fig. \[fig:all\] using a distributed RC equivalent circuit for the description of the electrode polarization.](fig3rev.eps){width="8cm"}
The fits reveal a width parameter $\alpha$ close to 0.15 and nearly independent of temperature and hematocrit value (not shown), except for $Hct=0.86$, where $\alpha\approx0.20$ is found, slightly increasing with temperature. $\alpha$ characterizes the distribution of relaxation times of the RC equivalent circuit that describes the electrode polarization (see section \[BE\]). The deviations from Debye behavior may be explained, e.g., by the surface roughness of the electrodes [@Nyikos1985; @Liu1985a; @Pajkossy1986]. The fits reveal electrode capacitances $C_{\mathrm{RC}}$ of the order of 10 $\mu$F. $C_{\mathrm{RC}}$ is found to be only weakly temperature dependent and it shows a tendency to decrease with increasing hematocrit value. This can be ascribed to the mentioned reduction of the ionic mobility leading to a less effective formation of the insulating electrode layers. The fits do not reveal reliable information on the electrode resistance as no clear low-frequency plateau in $\sigma'(\nu)$ is seen (cf. Fig. \[fig:all\] and Fig. \[fig:alphaVoll\]).
From the presented results, it is clear that the electrode polarization is the dominant effect in the low frequency spectrum of blood. The equivalent-circuit description in terms of a distributed RC circuit provides nearly perfect fits of the experimental data. The typical deviations between fit and measured data are around 10$\%$ or less, which is negligible compared to the many decades the dielectric quantities vary with frequency. Thus the presence of an additional contribution from a possible $\alpha$-relaxation seems unlikely but due to the mentioned deviations, a weak $\alpha$-relaxation cannot be fully excluded. However, it should be noted that in earlier investigations also no indications for an $\alpha$-relaxation in blood were found [@Bothwell1956].
$\beta$-Dispersion {#beta}
------------------
### Phenomenological Evaluation {#sub:beta}
Figure \[betaVollblut\] shows the spectra of whole blood in the frequency range of the $\beta$-dispersion (10 kHz to 200 MHz) at different temperatures. Except for the plasma, not containing any RBC’s that would cause a $\beta$-process, all samples show a similar relaxational behavior and temperature dependence in this frequency regime (see Fig. \[fig:all\]). Just as for intrinsic relaxations, the permittivity curves in the $\beta$-dispersion regime shift to higher frequencies with increasing temperature. The lines in Fig. \[betaVollblut\] again represent the fits of the complete broadband spectra (cf. Fig. \[fig:all\]) using a Cole-Cole function (Eq. \[hn\] with $\beta=1$) for the $\beta$-relaxation. As discussed in detail in section \[Mw\], the $\beta$-relaxation is generated by the heterogeneity of the sample material, which is composed of plasma and RBC’s. $\varepsilon'(\nu)$ exhibits the typical steplike decrease with increasing frequency (Fig. \[betaVollblut\](a)). Its additional increase towards the lowest frequencies observed in Fig. \[betaVollblut\](a) corresponds to the onset of the electrode-polarization effects (cf. Fig. \[fig:alphaVoll\]), well taken into account in the fits by the distributed RC equivalent circuit (see section \[alpha\]). In $\varepsilon''(\nu)$ a loss peak is expected but only its high-frequency flank can be seen (Fig. \[betaVollblut\](b)). Its low-frequency part is superimposed by the strong ionic dc conductivity, which leads to a contribution $\varepsilon''_{\mathrm{dc}}=\sigma_{\mathrm{dc}}/(\varepsilon_{0}\omega)$. Thus, instead of a loss peak, only a slight shoulder at about 3 MHz is revealed.
![\[betaVollblut\]$\varepsilon'(\nu)$ (a), $\varepsilon''(\nu)$ (b), and $\sigma'(\nu)$ (c) of whole blood ($Hct=0.39$) in the $\beta$-dispersion region for selected temperatures. The lines represent fit curves as in Fig. \[fig:all\] using the Cole-Cole function for the description of the $\beta$-relaxation.](fig4.eps){width="8cm"}
The dc conductivity also leads to the low-frequency plateau in $\sigma'(\nu)$ (Fig. \[betaVollblut\](c)) while the shoulders observed around 3 MHz arise from the relaxation. Via the relation $\sigma'=\varepsilon_{0}\varepsilon''\omega$, the nearly Debye-like behavior of the $\beta$-relaxation (implying $\varepsilon''(\nu)\sim\nu^{-1}$ on the high frequency side of the peaks) leads to the nearly frequency independent $\sigma'(\nu)$ at $\nu>10$ MHz. The low- and high-frequency plateaus of $\sigma'$ are labeled as $\sigma_{\mathrm{dc\beta}}$ and $\sigma_{\mathrm{\infty\beta}}$, respectively. The steplike increase of $\sigma'(\nu)$ from $\sigma_{\mathrm{dc\beta}}$ to $\sigma_{\mathrm{\infty\beta}}$ can be qualitatively understood assuming a shorting of the cell membrane capacitances at high frequencies. Thus, at high frequencies the RBC’s no longer obstruct the current path and an enhanced conductivity is detected. Therefore $\sigma_{\mathrm{\infty\beta}}$ can be regarded as good approximation of the intrinsic conductivity of the plasma, denoted as $\sigma_{\mathrm{a}}$ in section \[cellmod\] (in fact it is a mixture of plasma and cytoplasma conductivity, which we here assume to be of not too different magnitude). This is nicely corroborated by the approximate agreement of this plateau value with the conductivity of the pure plasma as seen in Fig. \[fig:all\](c) for all $Hct$ values (the small deviations for higher $Hct$ values are due to the larger volume fraction of cytoplasma having somewhat lower conductivity; see section \[app\]). The absolute values of both conductivity plateaus revealed in Fig. \[betaVollblut\](c) increase with increasing temperature, mirroring the thermally activated ionic charge transport in the plasma.
As mentioned above, the best fitting results of the $\beta$-relaxation were achieved by using a Cole-Cole function. Only for the highest hematocrit values, significant deviations of fits and experimental data were observed, which will be treated in section \[further\]. The temperature dependence of the width parameter $\alpha_{\beta}$, the relaxation strength $\Delta\varepsilon_{\beta}$, and the relaxation time $\tau_{\beta}$ obtained from the fits are shown in Fig. \[betapar\]. The present fitting of the complete broadband spectra, including the contributions from electrode polarization and the $\gamma$-relaxation, minimizes the influence of any additional processes on the obtained fit parameters.
![\[betapar\] Temperature dependence of width parameter (a), relaxation strength (b), and relaxation time (c) as obtained from fits assuming a Cole-Cole-function for the description of the $\beta$-relaxation (cf Fig. \[betaVollblut\]). The lines in the Arrhenius plot of $\tau_{\beta}$ (c) are linear fits corresponding to thermally activated behavior, Eq. \[tautemp\].](fig5rev.eps){width="8cm"}
The width parameter $\alpha$ (Fig. \[betapar\](a)), which for intrinsic relaxations usually is assumed to arise from a distribution of relaxation times [@Sillescu1999; @Ediger2000], is almost temperature independent. As $\alpha$ assumes rather small values between 0.07 and 0.11, the $\beta$ process shows nearly Debye-like behavior. The width parameter increases with increasing RBC content, i.e., the deviations from the Debye case become stronger. According to the Pauly-Schwan model (see section \[cellmod\], Eq. \[taubet\]) the relaxation time $\tau_{\beta}$ depends on the conductivity outside ($\sigma_{\mathrm{a}}$) and inside of the cell ($\sigma_{\mathrm{i}}$) and on the membrane capacitance ($C_{\mathrm{m}}$). It is unlikely that the cell parameters $\sigma_{\mathrm{i}}$ or $C_{\mathrm{m}}$ should be influenced by the hematocrit value and thus a distribution of the outer plasma conductivity seems the most likely cause of the non-Debye behavior. But also an alternative explanation seems possible: $\alpha\neq0$ implies a shallower high-frequency flank of the $\beta$-peak. This flank is essentially determined by the intrinsic plasma conductivity and corresponds to the high-frequency plateaus seen in Fig. \[betaVollblut\](c). Thus $\alpha\neq0$ implies an increase of $\sigma'(\nu)$ with frequency, which is typical for hopping conductivity as commonly found for ionic charge transport [@Jonscher1983; @Elliott1987; @Elliott1989; @Funke1993]. Finally, it has to be mentioned that additional relaxations arising from the other cell types in blood could also influence the observed $\beta$-relaxation. As RBCs are by far the dominating cell species (e.g., volume fraction about 45% vs. $\sim1\%$ of white blood cells), these contributions can be expected to be small. Nevertheless, it cannot be excluded that they may contribute to the observed deviations from Debye behavior.
As revealed by Fig. \[betapar\](b), the relaxation strength of the $\beta$-relaxation is nearly temperature independent. Thus, according to Eq. \[deltaeps\], the membrane capacity also can be assumed to be temperature independent. The strong drop of $\Delta\varepsilon_{\beta}$ at $T>320$ K, observed for all samples except for $Hct=0.86$, is most likely due to the hemolysis of the RBC’s at high temperatures. This assumption is supported by the fact that no such deviations can be found for the $\gamma$-relaxation (see section \[gamma\]), which is independent of the RBC’s. However, it is not clear why the sample with the highest RBC content (black triangles in Fig. \[betapar\](b)) remains unaffected. Possibly, cell-cell interactions prevent hemolysis at high Hct.
Figure \[betapar\](b) also reveals a decrease of the relaxation strength with increasing content of erythrocytes. This is a rather surprising result and cannot be explained within the proposed theories. Especially, it contradicts the increase of $\Delta\varepsilon_{\beta}$ with $p$ predicted by Eq. \[deltaeps\]. In literature, suspensions of erythrocytes and other cells using non-plasma solvents, like phosphate buffered saline, quite generally show a continuous increase of $\Delta\varepsilon_{\beta}$ with $Hct$ [@Lisin1996; @Hayashi2008; @Fricke1953; @Davey1992; @Bordi2002]. However, blood samples (i.e. with the suspending medium being plasma) can show more complex behavior, especially for higher $Hct$ values [@Chelidze2002; @Pfutzner1984] and as shown in ref , substitution of plasma by some other solute can strongly influence $\Delta\varepsilon_{\beta}$. In the framework of a simple equivalent-circuit picture, $\Delta\varepsilon_{\beta}$ is determined by a complex superposition of the membrane capacitances of all RBCs. Increasing the number of RBCs could lead to an increase or decrease of $\Delta\varepsilon_{\beta}$ depending on whether parallel or series connections of the cell capacitances (relative to the field direction) are prevailing. The latter seems to be the case in our blood samples. The reason is unclear but cell aggregation as, e.g., rouleaux formation may play a role here [@Beving1994a; @Pfutzner1984; @Pribush1999; @Pribush2000].
According to Eq. \[taubet\], the $\beta$-relaxation time should depend on $\sigma_{\mathrm{i}}$, $\sigma_{\mathrm{a}}$, and $C_{\mathrm{m}}$. It was shown above that the membrane capacitance is nearly temperature independent. Thus, the conductivities should dominate the temperature dependence of $\tau_{\beta}$. Indeed the Arrhenius representation of Fig. \[betapar\](c) reveals that $\tau_{\beta}(T)$ is in accord with the expected thermally activated behavior, typical for ionic conductivity. The hindering barriers $E_{\tau}$, calculated from the slopes in Fig. \[betapar\](c) (cf. Eq. \[tautemp\]), seem to slightly increase with growing $Hct$ and barriers varying between 0.11 and 0.15 eV were obtained. However, due to the rather small temperature region that could be covered in these biological samples (compared, e.g., to supercooled liquids [@Lunkenheimer2000; @Kremer2002]), the significance of these values should not be overemphasized. In addition, Fig. \[betapar\](c) reveals a decrease of the relaxation times with increasing hematocrit value. In principle, such a behavior seems to be consistent with Eq. \[taubet\] but the observed variation by about a factor of three is stronger than expected. However, one should be aware that the relaxation time is directly proportional to $C_{\mathrm{m}}$ (Eq. \[taubet\]) while $C_{\mathrm{m}}$ itself is proportional to $\Delta\varepsilon_{\beta}$ (Eq. \[deltaeps\]). Thus it is clear that the observed rather strong $Hct$-dependent variation of $\tau_{\beta}$ is directly connected to that of $\Delta\varepsilon_{\beta}$ revealed by Fig. \[betapar\](b).
To compare the results on the $\beta$-relaxation parameters presented above with earlier publications only partly is possible, because (to the best knowledge of the authors) no such systematic (temperature and hematocrit dependent) and broadband research on human blood has been done before. Moreover, the available literature values deviate quite strongly from each other. For various erythrocyte suspensions, the reported values of the relaxation time $\tau_{\beta}$ are, for example, 254 ns ($Hct=0.07$, room temperature) [@Lisin1996], 29 ns ($Hct=0.30$, $T=298$ K) [@Bordi1990], or 230 ns ($Hct=0.47$, $T=310$ K) [@Bordi1997; @Davey1989]. In blood the following values were found: $\tau_{\beta}=133$ ns (sheep blood, $T=310$ K) [@Gabriel1996b], $\tau_{\beta}=89-65$ ns (human blood$, T=288-308$ K, $Hct=0.43$) [@Cook1952], and $\tau_{\beta}=53.1$ ns (bovine blood, room temperature, $Hct=0.50$) [@Schwan1983]. The values in the present work vary between 35.9 ns ($Hct=0.86$, $T=330$ K) and 274.1 ns ($Hct=0.23$, $T=280$ K). The literature results for the relaxation strength also show rather strong variation. In ref , literature values between 1100 and 5000 were reported. Fricke found, dependent on $Hct$, $\varepsilon_{\mathrm{s}}\approx\Delta\varepsilon=900-4000$ for dog, rabbit, and sheep blood [@Fricke1953]. For human blood, Pfützner published values between approximately 2000 and 6000 ($Hct=0.10-0.90$) [@Pfutzner1984]. In the present work we have obtained $\Delta\varepsilon\approx3300-13800$ (for $Hct=0.86$, $T=330$ K and $Hct=0.23$, $T=320$ K, respectively). Even less data are available for the width parameter $\alpha_{\beta}$, because often other fitting functions were used. But mostly they are around 0.1 [@Gabriel1996b; @Bordi1997; @Gabrielinet], similar to the values in the present work.
### Application of Cell Models {#app}
As mentioned in section \[cellmod\], by using appropriate models it should be possible to determine intrinsic cell parameters as the membrane capacitance or the conductivity of the cytoplasma from the parameters of the $\beta$-relaxation. Using the fitting parameter $\Delta\varepsilon_{\mathrm{\beta}}$ and Eq. \[deltaeps\], the membrane capacitance $C_{\mathrm{m}}$ can be calculated. As discussed in the previous section, the experimentally determined dielectric strength decreases with increasing $Hct$, in contrast to the increase predicted by Eq. \[deltaeps\]. The use of Eq. \[deltaeps\] therefore would imply a strongly $Hct$-dependent membrane capacitance (see Table \[tab:Aus310\] for the results at 310 K). This can hardly be interpreted in a physical way. Literature values vary between 0.17 $\mu$F/$\mathrm{cm}^2$ (ref ) and 3 $\mu$F/$\mathrm{cm}^2$ (ref ). However, most authors assume a membrane capacity of about 1 $\mu$F/$\mathrm{cm}^2$ [@Bordi1997; @Fricke1953; @Davey1992; @Ballario1984a; @Asami1989], whereas some report $Hct$-dependent membrane capacities [@Zhao1993; @Chelidze2002]. Possible reasons for the unexpected behavior of $\Delta\varepsilon_{\mathrm{\beta}}(Hct)$ and thus of $C_{\mathrm{m}}(Hct)$ were discussed in the previous section. It seems that Eq. \[deltaeps\] is not able to account for the observations in “real” blood samples, in contrast to suspensions of erythrocytes in common solvents.
[ccccc]{} & $C_{\mathrm{m}}\,(\frac{\mu\mathrm{F}}{\mathrm{cm^2}}$) & $\sigma_{\mathrm{a}}\,(\frac{10^{-2}}{\Omega \mathrm{cm}})$ & $\sigma_{\mathrm{i}}\,(\frac{10^{-2}}{\Omega \mathrm{cm}})$ &$\varepsilon_{\mathrm{i}}$\
0.23 & 11 ($\pm 3$) & 0.79 ($\pm 0.15$)& 0.75 ($\pm 0.08$)& 36.9 ($\pm 3.0$)\
0.39 & 4.9 ($\pm 1.4$)& 0.63 ($\pm 0.15$)& 0.50 ($\pm 0.08$) & 41.5 ($\pm 2.5$)\
0.57 & 3.5 ($\pm 1.1$)& 0.74 ($\pm 0.15$)& 0.78 ($\pm 0.09$)& 42.0 ($\pm 2.5$)\
0.86 & 1.2 ($\pm 0.4$) & 0.78 ($\pm 0.15$) & 0.83 ($\pm 0.11$)& 44.7 ($\pm 2.0$)\
Using Eq. \[au\] should allow for the calculation of the conductivity of the suspending medium $\sigma_{\mathrm{a}}$ from the dc conductivity $\sigma_{\mathrm{dc}\beta}$ of the blood samples. $\sigma_{\mathrm{a}}$ also can be directly determined from the dc conductivity of the plasma, measured in the present work ($1.7\times10^{-2}~\Omega^{-1}\mathrm{cm}^{-1}$ at 310 K). The calculated values are provided in Table \[tab:Aus310\]. While being nearly $Hct$-independent as expected, they differ from the directly measured value by a factor of about two. Again cell aggregation causing a lowering of the experimentally observed $\sigma_{\mathrm{dc}\beta}$ may explain this finding.
Via Eq. \[sp\], the conductivity of the cell interior $\sigma_{\mathrm{i}}$ can be calculated from $\sigma_{\mathrm{a}}$, $C_{\mathrm{m}}$, and the measured $\beta$-relaxation times. However, as $C_{\mathrm{m}}$ determined from Eq. \[deltaeps\] shows an unphysical $Hct$ dependence (Table \[tab:Aus310\]), also an unreasonable $Hct$ dependence of $\sigma_{\mathrm{a}}$ would result. An alternative determination of $\sigma_{\mathrm{i}}$ is provided by Eq. \[spsu\]. Using the experimentally determined $\sigma_{\mathrm{a}}$ and $\sigma_{\infty\beta}$ at 310 K we arrive at the values for $\sigma_{\mathrm{i}}$ listed in Table \[tab:Aus310\]. Literature results are distributed around $0.6\times10^{-2}(\pm0.1)~\Omega^{-1}\mathrm{cm}^{-1}$ [@Pauly1966; @Krupa1972; @Beving1994a; @Chelidze2002; @Hayashi2008; @Bordi2002; @Ballario1984a; @Asami1989; @Bianco1979]. Deviating results were reported by Cook [@Cook1951; @Cook1952] (ca. $1.0\times10^{-2}~\Omega^{-1}\mathrm{cm}^{-1}$) and Asami [@Asami1980] (ca. $0.32\times10^{-2}~\Omega^{-1}\mathrm{cm}^{-1}$). Our values for $\sigma_{\mathrm{i}}$ are by about a factor 2-3 smaller than the measured conductivity of the plasma, $\sigma_{\mathrm{a}}\approx1.7\times10^{-2}~\Omega^{-1}\mathrm{cm}^{-1}$. It seems reasonable that the conductivity of the cytoplasma should be lowered by the presence of the large hemoglobin molecules and their bound water shells within the RBCs (about 37% volume fraction [@Pauly1966]), if compared to the conductivity of the outer plasma (see following discussion of Fig. \[fig:innen\] for a quantitative treatment). Indeed such behavior was found previously [@Krupa1972; @Beving1994a; @Chelidze2002]. The reported ratios between about 1.5 and 2.7 are consistent with our findings. Obviously, Eq. \[spsu\] is able to provide reasonable estimates for $\sigma_{\mathrm{i}}$. It is based on the determination of $\sigma_{\infty\beta}$, which is read off at high frequencies, where the cell membranes are shorted and thus cell aggregation has no effect on the results.
![\[fig:innen\] Temperature dependent conductivity (Arrhenius plot) (a) and dielectric constant (b) of the cell interior calculated from Eqs. \[spsu\] and \[sp2\], respectively. For comparison, in (a) also data for pure plasma are provided. The solid line in (a) indicates approximately linear behavior implying thermally activated charge transport (Eq. \[sdc\]). The dashed line shows an approximate description of the blood data (except for $Hct=0.39$) using the same energy barrier as for the plasma.](fig6rev.eps){width="8cm"}
In Fig. \[fig:innen\](a) the temperature dependence of $\sigma_{\mathrm{i}}$ is shown in the Arrhenius type of presentation ($\log(\sigma_{\mathrm{i}}T)$ vs $1000/T$) commonly used for ionic conductivity (cf Eq. \[sdc\]). For comparison, also the dc conductivity determined from fits of the spectra of pure blood plasma (cf. Fig. \[fig:all\]) is included. In the determination of $\sigma_{\mathrm{i}}(T)$ via Eq. \[spsu\], for $\sigma_{\mathrm{a}}$ the plasma dc conductivity was used and $\sigma_{\infty\beta}$ was calculated from the fit parameters of the $\beta$-relaxation shown in Fig. \[betapar\]. As the $\beta$-relaxation shows slight deviations from Debye behavior, $\sigma'(\nu)$ exhibits a slight increase in the region of its high-frequency plateau (see, e.g., Fig. \[betaVollblut\](c)). As an estimate of $\sigma_{\infty\beta}$, we used the value of $\sigma'(\nu)$ at a frequency 1.5 decades above the peak frequency. Except for whole blood, the temperature dependence of $\sigma_{\mathrm{i}}$ is nearly independent of $Hct$ and seems to follow thermally activated behavior (dashed line in Fig. \[fig:innen\](a)). The deviations at the two highest temperatures are directly related to similar problems in the $\beta$-relaxation parameters (Fig. \[betapar\]). As discussed in section \[beta\], this may arise from an onset of hemolysis of the RBCs. Interestingly, the energy barrier of 0.17 eV, deduced from the slope of the linear fit curve of the plasma data (solid line) is in good accord with the results on the blood samples (dashed line). Nevertheless, the absolute values of the conductivity of the cytoplasma are about a factor of 2-3 lower than those of the plasma. As mentioned in the previous paragraph, this can be explained by the presence of the hemoglobin molecules within the cell. For a quantitative estimate one can use Maxwell’s mixture equation for the effective conductivity $\sigma_{\mathrm{eff}}$ of a suspension of particles with volume concentration $p$ [@Maxwell1873]:
$$\label{mixeq}
\frac{\sigma_{\mathrm{eff}}-\sigma_{\mathrm{s}}}{\sigma_{\mathrm{eff}}+2\sigma_{\mathrm{s}}}=p\frac{\sigma_{\mathrm{p}}-\sigma_{\mathrm{s}}}{\sigma_{\mathrm{p}}+2\sigma_{\mathrm{s}}}$$
Here $\sigma_{\mathrm{s}}$ and $\sigma_{\mathrm{p}}$ are the conductivity of the solute and the particle, respectively. If we regard the hemoglobin molecules as insulating particles (i.e., $\sigma_{\mathrm{p}}=0$) suspended in plasma with the same conductivity as the extracellular plasma, we arrive at the following ratio of solute conductivity ($\sigma_{\mathrm{s}}=\sigma_{\mathrm{a}}$) and effective conductivity ($\sigma_{\mathrm{eff}}=\sigma_{\mathrm{i}}$) [@Pauly1966]:
$$\label{mixeq2}
\frac{\sigma_{\mathrm{a}}}{\sigma_{\mathrm{i}}}=\frac{1+p/2}{1-p}$$
Using $p=0.37$ (ref ), a ratio of about two is obtained, which is in quite reasonable agreement with the findings of Fig. \[fig:innen\](a). The energy barrier of 0.17 eV for charge transport within the intra- and extracellular plasma is of the same order of magnitude as the one deduced from $\tau_{\beta}(T)$ (0.11 - 0.15 eV, see previous section). This seems reasonable as the temperature dependence of $\tau_{\beta}$ should be mainly governed by the conductivity of inner and outer plasma (Eq. \[taubet\]). In any case one should bear in mind that the absolute values of the energy barriers have rather high uncertainty due to the restricted temperature range.
The dielectric constant of the cell interior $\varepsilon_{\mathrm{i}}$ was calculated using Eq. \[sp2\]. The results for 310 K are listed in Table \[tab:Aus310\] and the temperature dependence is shown in Fig. \[fig:innen\](b). One should be aware that $\varepsilon_{\mathrm{i}}$ is the dielectric constant at frequencies below the onset of the $\gamma$-relaxation. As expected, the obtained values are smaller than the dielectric constant of the suspending medium ($\varepsilon_{\mathrm{a}}\approx73$ - $67$ for $T=290$ - $310$ K, respectively), deduced from the fits of the spectra of pure plasma (see Fig. \[fig:all\] for 310 K). This difference is reasonable because the main contribution to the $\varepsilon'$ of the plasma arises from the highly dipolar water molecules ($\varepsilon_{\mathrm{s}}$ of water $\approx74$ at 310 K (ref )) and the additional constituents of the cytoplasma (mainly hemoglobin) should lower its permittivity. In literature, $\varepsilon_{\mathrm{i}}$ values ranging around 40-70 were reported [@Pauly1966; @Asami1989; @Bianco1979; @Asami1980]. The calculation of $\varepsilon_{\mathrm{i}}$ by Eq. \[sp2\] should provide the same results for each hematocrit value. However, as revealed by Table \[tab:Aus310\] and Fig. \[fig:innen\](b), the calculated $\varepsilon_{\mathrm{i}}$ increases by about 20% with increasing $Hct$, pointing out the limits of the model. Similar behavior was also reported in ref . The decrease of $\varepsilon_{\mathrm{i}}$ with increasing temperature revealed by Fig. \[fig:innen\](b) is consistent with Curie behavior (Eq. \[Curie\]) expected for the dielectric strength (and thus approximately also for the static dielectric constant) of dipolar materials. Here one should be aware that $\varepsilon_{\mathrm{i}}$ represents the static dielectric constant of a $\gamma$-like relaxation of the cell interior that will take place at higher frequencies and in fact this relaxation contributes to the actually observed $\gamma$-relaxation of blood (see section \[gamma\]). The uncertainty of the data in Fig. \[fig:innen\](b) is too large to allow for a quantitative evaluation in terms of Eq. \[Curie\]. Overall, Eq. \[sp2\] seems to lead to reasonable values of the dielectric constant of the cytoplasma. It is based on the determination of the plasma dielectric constant (experimentally determined) and $\varepsilon_{\infty\beta}$, which is not influenced by possible cell aggregation effects.
An alternative determination of the dielectric properties of the RBCs is provided by the Hanai-Bruggemann model, Eqs. \[hb1\] and \[hb2\], which was especially proposed for highly concentrated suspensions. From Eq. \[hb2\] the cell conductivity was calculated, leading, however, to negative values. Using Eq. \[hb1\], the dielectric constant $\varepsilon_{\mathrm{p}}$ of the cell is found to vary between 34 ($Hct=0.57$) and 42 ($Hct=0.39$) at room temperature. Those values are slightly smaller than the ones calculated from the Pauly-Schwan theory. This is a reasonable result since the Hanai-Bruggemann model does not account for the shelled structure of the cells and thus the obtained values represent the average of cell membrane and interior. The dielectric constant of the membrane can be expected to be much lower than that of the cytoplasma (in contrast to its capacitance, which is high due to its small thickness), which leads to the reduced values of the total dielectric constant.
$\gamma$-Dispersion {#gamma}
-------------------
![\[gammaVollblut\] $\varepsilon'(\nu)$ (a), $\varepsilon''(\nu)$ (b), and $\sigma'(\nu)$ (c) of whole blood ($Hct=0.39$) in the $\gamma$-dispersion region for selected temperatures. The lines represent fit curves as in Fig. \[fig:all\] using the Cole-Cole function for the description of the $\gamma$-relaxation](fig7rev.eps){width="8cm"}
.
The dielectric spectra of bulk water exhibit a strong relaxation feature near 18 GHz (at room temperature) [@Kaatze1989], which is also observed in electrolytic solutions [@Gulich2009]. It is commonly ascribed to the reorientational dynamics of the dipolar water molecules (but also alternative scenarios are discussed; see, e.g., ref ) and denoted as $\alpha$-relaxation within the nomenclature of dipolar liquids and glass formers. The same relaxational process also arises from the free water molecules in blood samples. Figure \[gammaVollblut\] shows real and imaginary part of the permittivity (a, b) and the real part of the conductivity (c) of whole blood in the frequency range 1 to 40 GHz at different temperatures. The lines represent fits of the broadband spectra as shown in Fig. \[fig:all\], using a Cole-Cole function for the $\gamma$-relaxation. In $\varepsilon'(\nu)$ (Fig. \[gammaVollblut\](a)), the onset of the typical relaxation steps is seen but their high frequency plateaus, $\varepsilon_{\infty\gamma}$, are located beyond the investigated frequency range. Thus, exact values for $\varepsilon_{\mathrm{\infty}\gamma}$ could not be determined and in the fitting procedure a lower limit of 2.5 was used leading to values between 2.5 and 6. The low-frequency plateau $\varepsilon_{\mathrm{s}\gamma}$ of the $\gamma$-dispersion decreases with increasing temperature. The relaxation steps and loss peaks (Fig. \[gammaVollblut\](b)) show a strong temperature-dependent frequency shift due to the slowing down of the molecular dynamics with decreasing temperature. We find the Cole-Cole formula (Eq. \[hn\] with $\beta=1$) to provide the best fits of the $\gamma$-relaxation. Figure \[gammaVollblut\](c) shows the conductivity spectra with the corresponding rise and the onset of the high frequency plateau. The low-frequency plateau of $\sigma'(\nu)$ corresponds to the combined conductivity of plasma and cytoplasma as discussed in section \[app\]. The $\gamma$-dispersion shows similar behavior for the other investigated samples.
![\[gammapar\]Temperature dependence of width parameter (a), relaxation strength (b), and relaxation time (c) as obtained from fits assuming a Cole-Cole function for the description of the $\gamma$-relaxation (cf Fig. \[gammaVollblut\]). The dashed line in (b) shows literature data for pure water (using the I.U.P.A.C. standard data for $\varepsilon_{\mathrm{s}}(T)$) [@Kienitz1981]. The solid lines in (b) are fits with a Curie-law, Eq. \[Curie\]. The line in the Arrhenius plot of $\tau_{\gamma}$ (c) is a linear fit of the curve for $Hct=0.39$ (whole blood) corresponding to thermally activated behavior, Eq. \[tautemp\]. The dashed line shows the curve for pure water [@Kaatze1989].](fig8rev.eps){width="7.5cm"}
The fitting parameters of the $\gamma$-relaxation, $\alpha_{\gamma}$, $\Delta\varepsilon_{\gamma}$, and $\tau_{\gamma}$, are provided in Fig. \[gammapar\]. The width parameter (a) shows a tendency to increase with increasing $Hct$. This seems reasonable as a higher number of RBC’s should lead to stronger disorder in the system and therefore the distribution of relaxation times should broaden. The observed decline of $\alpha_{\gamma}$ with increasing temperature, corresponding to an approach of Debye behavior, is a common phenomenon in dipolar liquids [@Lunkenheimer2000; @Schonhals1993]. It can be explained by the growing thermal fluctuations of the environment of the water dipoles. At very high temperatures, each dipole “sees” the time average of the quickly fluctuating environment, which is the same for every dipole, leading to Debye behavior [@Lunkenheimer2000].
The hematocrit dependence of the relaxation strength (Fig. \[gammapar\](b)) shows the expected tendency: Increasing $Hct$ values cause a decrease of the volume fraction of plasma and thus of water in the sample, causing the reduction of the $\gamma$-relaxation strength. The temperature dependence of $\Delta\varepsilon_{\gamma}$ can be well parameterized by a Curie-law, Eq. \[Curie\], (solid lines) with some deviations for $Hct=0.23$ only. The obtained Curie constant, $C$, increases smoothly from 13400 to 19800 with decreasing $Hct$. The dashed line in Fig. \[gammapar\](b) corresponds to $\Delta\varepsilon(T)$ of pure water, calculated from the I.U.P.A.C. (International Union of Pure and Applied Chemistry) standard values for the static permittivity $\varepsilon_{\mathrm{s}}$ of bulk water [@Kienitz1981] (see also [@Ellison2007]) via $\Delta\varepsilon=\varepsilon_{\mathrm{s}}-\varepsilon_{\infty}$ assuming $\varepsilon_{\infty}=4$ [@Hasted1973; @Buchner1999]. The relaxation strength of water matches the general trend revealed by the other curves in Fig. \[gammapar\](b). However, obviously it shows a somewhat stronger temperature dependence. This may indicate weaker interactions between the water molecules in blood than in pure water, which can be rationalized by the presence of the other constituents of blood (e.g., proteins or salt ions).
Figure \[gammapar\](c) shows the temperature dependence of the relaxation times $\tau_{\gamma}$ in an Arrhenius plot. The observed linear increase is in accord with thermally activated behavior, Eq. \[tautemp\]. As an example, a linear fit of the data at $Hct=0.39$ is shown (solid line). From its slope an energy barrier of 0.19 eV is obtained. There seems to be a slight increase of energy barriers with growing $Hct$ value (from 0.18 to 0.20 eV). However, this variation is too small to be considered significant, especially when taking into account the rather small temperature range that could be investigated in the present experiments due to the restricted robustness of blood to stronger temperature variations. The present results agree reasonably with those reported by Cook [@Cook1951] who found values for $\tau_{\gamma}$ of whole blood varying between 11.9 and 7.0 ps at three temperatures between 298 and 308 K. Gabriel et al.[@Gabriel1996b] reported 8.4 ps at 310 K, about $30\%$ higher than our result of 6.5 ps. The dashed line in Fig. \[gammapar\](c) represents $\tau(T)$ of pure water as measured by Kaatze [@Kaatze1989]. Obviously the $\gamma$-relaxations in the investigated blood samples exhibit nearly identical dynamics as the main relaxation of pure water. Water shows some small deviations from Arrhenius behavior, which seem to be absent in blood but these differences are of limited significance. However, one may speculate that the non-Arrhenius behavior of water arises from increasing cooperativity of the molecular motions at low temperatures as often invoked to explain corresponding findings in glass forming liquids [@Ediger1996; @Ngai2000]. In blood, its other constituents can be expected to lead to a reduction of the direct interactions between the water molecules and thus to less cooperativity. In addition, there are speculations of a first-order phase transition in supercooled water [@Angell2008], which may lead to critical power-law behavior even in the normal liquid state, thus also explaining the deviation from Arrhenius behavior in Fig. \[gammapar\](c). Both scenarios are consistent with the different temperature dependence of $\Delta\varepsilon(T)$ of water and blood discussed in the previous paragraph.
Further Dispersions {#further}
-------------------
 Comparison of $\varepsilon'(\nu)$ of the blood samples with the lowest ($Hct=0.23$) and highest hematocrit value ($Hct=0.86$) in the frequency range of the $\beta$- and $\delta$-dispersion. The lines represent fit curves as in Fig. \[fig:all\] and Fig. \[betaVollblut\] using the Cole-Cole function for the description of the $\beta$-relaxation. (b) $\varepsilon'(\nu)$ for $Hct=0.86$ (triangles: same data as in (a), crosses: measurement with different experimental setup). The solid line in (b) shows an alternative fit with two Cole-Cole functions for the $\beta$-relaxation. The two separate relaxation steps are indicated by the dashed lines. The inset shows a magnified view of the high-frequency region.](fig9.eps){width="8cm"}
In section \[beta\] it was shown that fits using the Cole-Cole function provide a reasonable description of the $\beta$-relaxation region. However, there are some minor deviations of fits and experimental data especially for the higher hematocrit values. This is demonstrated in Fig. \[deltaneu\](a) where dielectric-constant data for the blood samples with the highest and lowest hematocrit values are shown. In contrast to the 23% sample, the fit of the spectrum of the highly concentrated sample clearly is of inferior quality. Similar deviations were previously also observed in Cole-Cole fits of data on disc-shaped rabbit-erythrocyte suspensions [@Hayashi2008]. A close inspection of the spectrum at $Hct=0.86$ (Fig. \[deltaneu\](b)) seems to indicate that it may be composed of two separate relaxation steps. However, one could suspect an experimental artifact because in the $\beta$-dispersion region the spectrum is composed of results from two different experimental methods with the transition close to 10 MHz (see section \[Materials\]). To exclude this, in Fig. \[deltaneu\](b) additional data extending from 100 kHz to 50 MHz obtained with a different apparatus (autobalance bridge Agilent 4294A) are shown, which exactly reproduce the two other data sets. Thus, a fit using the sum of two separate relaxation contributions was performed (solid line in Fig. \[deltaneu\](b)). It provides an excellent description of the spectrum revealing relaxation times of $16$ ns and $146$ ns. A very similar fit using two Cole-Cole functions was shown by Asami and Yamaguchi [@Asami1999] to provide a good description of data on human erythrocyte suspensions.
In blood there are various possibilities for additional relaxational processes, in addition to those considered for the explanation of the $\alpha$-, $\beta$-, and $\gamma$-relaxation: (i) the reorientation of protein-bound water molecules, (ii) the hemoglobin $\beta$-relaxation (i.e., the tumbling motion of the protein molecules), (iii) the motion of polar protein subgroups, (iv) the Maxwell-Wagner relaxation of the cell interior, or (v) the additional Maxwell-Wagner relaxations due to the non-spherical cell shape, to name just the most likely ones. Most of them can be simply excluded based on the very large amplitude of $\Delta\varepsilon_{\mathrm{s}}\approx1000$ of the additional relaxation suggested by the fit shown in Fig. \[deltaneu\](b): (i) Bound water cannot have a larger $\Delta\varepsilon$ than free water. (ii) The hemoglobin $\beta$-relaxation in aqueous solution was found to have a $\Delta\varepsilon$ of the order of 100 [@Schwan1957a; @Schwan1983; @Pennock1969]. It seems unreasonable that it should be higher in the hemoglobin/cytoplasma solution of the cell interior. (iii) Polar protein subgroups can be expected to have smaller relaxation strength that the main tumbling relaxation. (iv) In principle, the capacitance of the cell interior should also be shorted at high frequencies and, indeed, the Maxwell-Wagner model of shelled particles predicts a corresponding relaxation [@Asami2002]. However, for any reasonable choice of parameters this capacitance is too small to lead to any considerable contribution to $\varepsilon'$ and this additional relaxation usually is considered negligible [@Schwan1957a; @Schwan1983; @Pennock1969]. Thus, the non-spherical shape of the RBCs seems to be the most likely cause of the additional relaxation observed in Fig. \[deltaneu\](b). Already in ref deviations from simple relaxation behavior of erythrocyte suspensions were ascribed to the non-spherical form of RBCs and also Asami and Yamaguchi [@Asami1999] explained their results in this way. As mentioned in section \[cellmod\], the Maxwell-Wagner model for suspensions of spheroid particles predicts up to four relaxations [@Asami2002] (two of them arise from the cell interior and can be neglected). However, the found relaxation-time ratio of the order of 10 is too high to be explainable by this model, at least if assuming a reasonable ratio of the two semi-diameters of the spheroids [@Asami2002; @Asami2002a]. The spectra on rabbit erythrocytes, mentioned above, also could not be described by the Maxwell-Wagner model for spheroid particles [@Hayashi2008]. However, one should be aware that RBCs only roughly can be approximated by spheroids and most likely their biconcave shape plays a role in the found discrepancies.
An additional $\delta$-dispersion between $\beta$- and $\gamma$-relaxation is often invoked to explain a slow continuous decrease of $\varepsilon'(T)$, observed in the region from several 10 MHz to about 3 GHz in various biological materials, including protein solutions [@Schwan1957a; @Schwan1965; @Pennock1969; @Grant1966; @Grant1974] and blood [@Schwan1957a]. It has been ascribed to various mechanisms as the dynamics of protein-bound water molecules or polar subgroups of proteins. Indeed such a dispersion is also found in our present results on blood and becomes most pronounced for the high hematocrit values (see inset of Fig. \[deltaneu\] for an example). However, it can be completely described by the broadband fits promoted in the previous sections (line in inset), especially if including a second relaxation in the $\beta$-relaxation region as shown in Fig. \[deltaneu\](b). Thus in our blood samples the apparent dispersion in this region arises from the superposition of $\beta$- and $\gamma$-relaxation and we find no evidence for a $\delta$-relaxation. However, the presence of a small $\delta$-relaxation is not completely excluded by this finding.
Summary and Conclusions
=======================
In the present work, we have provided dielectric spectra of human blood for an exceptionally broad frequency range and at different temperatures and hematocrit values. A combination of models for the different dispersion regions enabled nearly perfect fits of the broadband spectra. The obtained fit curves represent an excellent estimate of the dielectric properties of blood for a wide range of parameters. They are provided for electronic download in the supporting information and can be employed for SAR calculations and other application purposes. The different dispersion regions have been analyzed in detail. The observed electrode-polarization effects are accounted for by an equivalent circuit model assuming a distribution of relaxation times. While our analysis of the low-frequency region cannot completely rule out the presence of an $\alpha$-dispersion in blood, we can satisfactorily describe our low-frequency data without invoking such a relaxation. This finding agrees with earlier results stating the absence of an $\alpha$-relaxation in blood [@Bothwell1956]. The analysis of the $\beta$-relaxation using standard cell models partly leads to unreasonable results for the intrinsic dielectric properties. This most likely can be ascribed to cell aggregation playing an important role in “real” blood samples, in contrast to suspensions of erythrocytes prepared by standard solutes, often reported in literature. Cell aggregation seems to be important especially for the dielectric behavior at the low-frequency side of the $\beta$-relaxation. In contrast, using only parameters determined at frequencies beyond the $\beta$-peak frequency, leads to reasonable estimates of the conductivity and dielectric constant of the cell interior. In addition, we find strong hints that the $\beta$-relaxation is in fact composed of two separate relaxation processes, which we ascribe to the marked deviations of the RBCs from spherical geometry.
We observe clear dispersion effects in the region between the $\beta$- and $\gamma$-relaxation, which often is ascribed to a so-called $\delta$-relaxation. However, our analysis of the broadband spectra including electrode polarization, $\beta$-dispersion, and $\gamma$-relaxation leads to excellent fits in this region, which thus is revealed to be a superposition of different contributions and not due to a separate relaxation process. Thus, while there clearly is dispersion in blood between the $\beta$- and $\gamma$-relaxation, there is no evidence for a $\delta$-relaxation. Finally, detailed information on the $\gamma$-relaxation in blood is provided. Its properties closely resemble those of the relaxation caused by reorientational molecular motions in pure water. However, some minor differences arise, which seem to indicate less cooperative motions of the water molecules in blood samples.
Overall, the dielectric spectra of blood are of astonishing simplicity if considering the complexity of blood, being composed of a variety of different constituents. In fact we have described our broadband spectra without assuming any intrinsic frequency dependence in the complete range from 1 Hz up to about 1 GHz and only the $\gamma$-dispersion arising from the tumbling motion of the water molecules is of intrinsic nature. Of course, there is the strong $\beta$-relaxation, which may be regarded as quasi-intrinsic but as it is of Maxwell-Wagner type, in a narrower sense it should be considered as artificial. However, of course for many applications (e.g., the calculation of SAR values) the overall dielectric properties and not only the intrinsic ones are of essential importance and we hope our work will serve for these purposes in the future.
Acknowledgements
================
We gratefully acknowledge the help of Dr. W. Behr, Dr. K. Doukas, and Prof. Dr. W. Ehret at the “Klinikum Augsburg” in the taking and preparing of the blood samples.
Supplementary data
==================
Fit curves of the broadband spectra and fit parameters for all samples and temperatures investigated in the present work are available for electronic download.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'We present the aperture synthesis imaging of C$_3$H$_2$ 2$_{12}$–$1_{01}$ and HCO$^+$ 1–0 lines and of continuum emission at $\lambda=3.4$ mm toward the class 0 young stellar object, IRAS 18148-0440 in the L483 molecular cloud. The continuum emission is detected at the IRAS position at a level of 16 mJy, indicating a compact source with a mass of $\sim 0.13 M_{\odot}$. The C$_3$H$_2$ delineates an envelope near the IRAS position, with a size of $\sim$ 3000 AU $\times$ 2000 AU in R.A. and Dec. directions, respectively. A velocity gradient detected only along the axis of outflow can be explained in terms of free-fall motion of the envelope. The HCO$^+$ line wing extends up to the velocity of $\pm 6$ km s$^{-1}$ relative to the systemic one, and the high velocity material shows a symmetric bipolarity and extends over $2''$ or 0.1 pc along the east-west direction. It is found that the outflow material is clumpy and the opening angle is widest for the slowest moving component. The core component of the HCO$^+$ 1–0 line exhibits an anti-infall asymmetry not only in interferometric but also in single dish observations. It is attributable to the slow isotropic outward motion of gas between the flattened envelope and the collimated outflow.'
author:
- 'Y.-S. Park, J.-F. Panis, N. Ohashi, M. Choi, and Y.C. Minh'
title: Interferometric observation of the L483 molecular core
---
Introduction
============
Young stellar objects (YSOs) in the early phase of star formation continuously aggregates material from parent dense molecular cores. This process seems to be followed by the formation of disk and outflows, and protostars come into existence by dispersing debris around them ([@kit97]). This scenario is generally accepted at least for low mass stars. The classification or evolutionary sequence of YSOs from class 0 to III, mainly based on their spectral energy distribution, has been established and became a framework to which subsequent studies should refer ([@and93]).
However, one needs to find direct observational evidence of inward motion leading to a drastic density enhancement of central source which is prerequisite for the star formation. The method of diagnosing the collapse motion had been developed earlier ([@leu77]; [@ang87]), but it was first detected only in the 1990’s ([@zho93]). The method is based on the idea that, in the presence of the infall motion of which magnitude is not so large as that of intrinsic (turbulent + thermal) motion, an opaque line has a self-absorption with the blue peak stronger than the red one, while an optically thin line with a single peak is located between the two peaks of the opaque line. Since class 0 and I objects are believed to be in main accretion phase, prompt attention is paid to them in an attempt to understand the earliest phase of star forming process. Many class 0 and I sources have been surveyed initially with single dish telescopes mainly for the detection of infall signatures, resulting in a statistical preference for inward motion rather than outward one for these objects ([@gre97]; [@mar97]). These surveys also served as a basis for detailed studies of higher angular resolution. Imaging with high spatial resolution is particularly important since the star formation process takes place at a smaller scale. Only interferometric observations enable us to separate envelopes, disks, and outflow lobes and to disentangle motions associated with them. Choi et al. (1999) carried out a survey of 9 class 0 sources using the Berkeley-Illinois-Maryland-Association (BIMA) interferometer, and Hogerheijde et al. (1999) on the YSOs in the Serpens molecular cloud. Detailed investigation has been undertaken for a limited number of YSOs as well ([@oha97]; [@yan97]). Eventually all YSOs will have to be observed with interferometers in a variety of transitions.
As a step toward these high resolution studies, we conducted an interferometric observation of L483. The L483 is one of the so-called Myers’ cores containing an embedded infrared source, IRAS 18148-0440, at a distance of 200 pc. Its size is $\sim 1'$ from the observations of HC$_3$N ([@ful93]) and C$^{18}$O ([@ful97]). A CO outflow with a good collimation was found in the east-west direction, inclined by $\sim 45^{\circ}$ to the sky plane, and an infrared lobe also lies in the axis of molecular outflow ([@par91]; [@ful95]). Recently, Hatchell, Fuller, & Ladd (1999) and Buckle, Hatchell, & Fuller (1999) examined the temperature variation in the outflow lobe and the shock structure of jet embedded in the lobe, respectively. However, the kinematics of the central condensation remains poorly understood, since the line asymmetry of optically thick transitions, known as a probe of internal motions of cores, differs from molecule to molecule. Mardones et al. (1997) and Mardones (1998) obtained CS 2–1, H$_2$CO $2_{12}$–$1_{11}$, and N$_2$H$^+$ 1–0 lines with $20''\sim 30''$ beams and mapped their spatial distributions. They found that the CS and H$_2$CO lines are self-reversed and the blue peak is slightly brighter than the red. On the other hand, Gregersen et al. (1997) show that the red peak is twice as strong as the blue one in the HCO$^+$ 3–2 transition in a $\sim 20''$ beam. The three hyperfine components of HCN 1–0 again exhibit the usual blue asymmetry with a $\sim 60''$ beam: a single peak in the blue side with a shoulder in the red of the thinnest component ($F$=0–1) changes into a deep self-absorption with a stronger blue peak of the thickest one ($F$=2–1) ([@par99]). A recent VLA observation of NH$_3$ indicates an inward motion down to $\sim 10''$ scale ([@ful00]). All these features suggest diverse velocity fields in this source, and this could be understood much better by aperture synthesis imaging.
In section 2, we briefly describe the observation and the data reduction. Results of our interferometric observation are presented in section 3. In section 4, we discuss kinematics of the envelope and outflow as well as molecular abundance. In the final section, we summarize our results.
Observation and data reduction
==============================
Since the HCO$^+$ molecule is most peculiar in line shape when compared with other molecules in single dish observations, the HCO$^+$ molecule is the one to be explored with a prime importance in this observational study. The HCO$^+$ 1–0 transition has a critical density of $\sim 2.5 \times 10^5$ cm$^{-3}$. The HCO$^+$ molecule may not be depleted in an envelope since there is no chemical reaction route to consume it ([@raw96]; [@van98]). Moreover, it often delineates well outflow lobes ([@hog98]; [@hog99]). Thus it may be a good tracer of the density and motion of the envelope and the outflow. However, since the transition may be opaque, we selected C$_3$H$_2$ 2$_{12}$–$1_{01}$ as a complementary probe to cold gas in the envelope. The C$_3$H$_2$ molecule has been extensively surveyed toward numerous Galactic objects ([@mad98]; [@ben98]). The single-peaked line profile of C$_3$H$_2$ toward L483 obtained from a single dish observation suggests that the transition is not so optically thick ([@mye95]). Furthermore, in L1157 it is found to trace an envelope mainly and is insensitive to an outflow or shocked region ([@bac97]). Thus, if a disk-like envelope is embedded in the core of L483, we may be able to discern, by using C$_3$H$_2$, rotation and/or radial (inward/outward) motions which are not contaminated by the outflow.
The observation of the L483 was carried out with the 10 elements BIMA interferometer in its C configuration in October and November 1998 in the transitions of HCO$^+$ 1–0 (89.188523 GHz) and C$_3$H$_2$ 2$_{12}$–1$_{01}$ (85.338905 GHz). We configured correlator for HCN 1–0 line (88.631847 GHz) as well, but the HCN data were discarded because of interference in the IF band. System temperatures were typically $150\sim 500$ K during the observation. Projected baselines range from 1.7 k$\lambda$ to 23 k$\lambda$, and phase center was the position of IRAS 18148-0440, $\alpha{\rm(2000)}= 18^h 17^m 29.\!\!^s83$ and $\delta{\rm(2000)}=-4^{\circ} 39' 38.\!\!''33$. The phase and pass band were calibrated using 1743-038 and 3C273, respectively. We also made observations of Uranus for flux calibration and the flux of 1743-038 was $1.8 \sim 2.0$ Jy at the time of observation. The IF window with the band width of 12.5 MHz was fed into the 256 channels of the correlator for each transition, resulting in the velocity resolutions of 0.164 km s$^{-1}$ and 0.172 km s$^{-1}$ in the HCO$^+$ and C$_3$H$_2$ transitions, respectively. Continua in both upper and lower side bands were simultaneously observed with a 800 MHz bandwidth in total.
We reduced the data using the MIRIAD package. The data were flagged and Fourier transformed with natural weighting. The resulting dirty map was CLEANed and restored with a Gaussian beam of a FWHM size of $14'' \times 10''$ at position angle $5.\!\!^{\circ}5$ measured from North to East.
In addition, in January 1999, one point observation of HCO$^+$ 1–0 was undertaken toward the IRAS source using the radome-enclosed 14 meter telescope (beam size = $60''$) of Taeduk Radio Astronomy Observatory, Korea. The SIS receiver was used for the frontend and the system temperature was 400 K at the time of observation. The backend was an autocorrelation spectrometer with a 20 KHz or 0.067 km s$^{-1}$ resolution.
Results
=======
Continuum emission at $\lambda=3.4$ mm
--------------------------------------
Fig. 1 represents the continuum emission detected around the position of IRAS 18148-0440. There is diffuse emission in the NE–SW direction. The continuum source is barely resolved in the direction of major beam axis, but extended in the E-W direction. By deconvolving the beam, we roughly estimate the size of condensation, 2000 AU and 3000 AU in the N-S and E-W directions, respectively. The estimated total flux is 16 mJy, which is consistent with spectral energy distribution measured from centimeter to near infrared ([@ful95]). Using a mass absorption coefficient, $\kappa = 0.01 ({1.3 {\rm mm} \over \lambda})^{1.5}$ cm$^2$ g$^{-1}$ ([@mot98]), we obtain the mass of 0.13 M$_{\odot}$ for the condensation.
C$_3$H$_2$ line emission
------------------------
We illustrate the distribution of C$_3$H$_2$ in Fig. 2. The line is single-peaked with the FWHM width of 0.32 km s$^{-1}$ and centered at the systemic velocity of 5.4 km s$^{-1}$, which is in accordance with the single dish observations of optically thin N$_2$H$^+$ and C$_3$H$_2$ lines ([@mye95]). The emission peak at $\approx$ ($-9'', -3''$) is displaced by $6''$ to the west of the continuum emission. The C$_3$H$_2$ emission is extended from North to South and its half-power contour larger than the synthesized beam is almost circular. From the figure, we note that the C$_3$H$_2$ emission delineates the envelope around the protostar fairly well. The single dish observation at high spatial resolution in the C$^{18}$O 3–2 transition revealed the structure that is similar, but slightly elongated in the N-S direction ([@ful97]).
In order to examine the kinematics of the envelope, we plot channel maps in Fig. 3, where there are mainly three components. The main component at $\approx$($-10'', 0''$) with $v\approx 5.4$ km s$^{-1}$ contains most of the mass of the envelope. The remaining ones are at $\approx$($5'', 0''$) with $v\approx 5.0$ km s$^{-1}$ and at $\approx$($15'', 20''$) with $v\approx 6.0$ km s$^{-1}$. Extended features in the N–S direction in the figure frame of 5.41 km s$^{-1}$ may result from side lobe effect. The 5.0 and 6.0 km s$^{-1}$ components are minor fragments of the envelope. The systematic displacement of peak positions of the main component along with the LSR velocity suggests a velocity gradient which is seen more clearly in the position-velocity diagrams of Fig. 4. Along the outflow axis of ${\rm P.A.}=95^{\circ}$ (see section 3.3) through the C$_3$H$_2$ emission peak at ($-9'', -3''$), is a uniform velocity gradient of the main component, amounting to $\sim 0.5$ km s$^{-1}$ over $40''$ or 13 km s$^{-1}$ pc$^{-1}$. It is found that line wing is of little importance as expected in section 2 and the line [*itself*]{} is gradually shifted. The figure also shows that the 5.0 km s$^{-1}$ feature is isolated indeed. On the other hand, the position-velocity diagram across the outflow axis shows little sign of systematic motion. We are not confident with the current resolution that the component at $\Delta \delta \approx -20 ''$ with 5.6 km s$^{-1}$ is a discrete one. A plausible explanation of those velocity shifts would be that the envelope has a flattened structure and the velocity gradient represents a gas infall motion toward the central source, whereas the existence of rotation is uncertain ([@oha97]). However, one has to be careful since the sense of the velocity shift of infalling flat disk or envelope is always the same as that of outflow motion. For instance, the shift of C$_3$H$_2$ line shown in L1228 results from the fragments of core entrained to an outflow ([@taf97]). It is difficult to figure out which one is dominant for L483, given the C$_3$H$_2$ data only. However, the velocity shift by outflow is less likely for L483, since the HCO$^+$ line suggests that there is expanding motion as a part of outflow around the envelope and the velocity field is almost isotropic (see section 4.2). If the C$_3$H$_2$ line is affected by this motion, there will be little velocity gradient. Fuller and Wooten (2000) recently found the inward motion by carrying out the VLA observation using the NH$_3$ inversion lines.
The infall motion of L483 inferred from the position-velocity map of Fig. 4 is different from a free-fall motion of $v(r)=\sqrt{2 G M_* / r}$ toward a central star with a mass of $M_*$ which has usually been invoked for, e.g., L1527, HH111, and other YSOs ([@oha97]; [@yan97]; [@mom98]). The velocity gradient is rather linear, suggestive of a collapse of a (pressure-free) gas sphere with a uniform density. We would argue that it is not an accretion toward the central source, but a global contraction of the envelope. This is consistent with the current understanding of the class 0 objects ([@bac96]). Our interpretation, however, may be subject to the spatial and spectral resolutions of observation. Further observations with higher resolutions are necessary to elucidate the kinematics and structure of the envelope.
HCO$^+$ emission
----------------
In Fig. 5, we plot the distribution of the red and blue HCO$^+$ 1–0 wing emission. The red wing is integrated from 6.5 to 11.0 km s$^{-1}$, while the blue wing from $-1.5$ to 4.5 km s$^{-1}$. It is found that the bipolar outflow is well collimated and the peaks of the two outflow lobes are separated each other by $\sim 50''$. The shape of the outflow is similar to those found from CO 2–1 ([@par91]), from CO 3–2 ([@ful95]), and from CO 4–3 ([@hat99]). The outflow lies close to the R.A. axis (P.A.$\approx 95^{\circ}$). The different amounts of extinction towards the lobes in the IR band suggest an inclination of the outflow of $\sim 45^{\circ}$ to the sky plane ([@ful95]).
In order to see details of the outflow, we display channel maps of HCO$^+$ in Fig. 6. It is found that there are at least three discrete components in the red shifted outflows: the 5.9 km s$^{-1}$ component around $\approx$($35'', 10''$), the 6.1 km s$^{-1}$ component centered at $\approx$($10'', -5''$), and the 7.2 km s$^{-1}$ component near $\approx$($35'', 0''$). The 7.2 km s$^{-1}$ component spans a wide range of velocity from 6.7 to 10 km s$^{-1}$. The 5.9 km s$^{-1}$ component is much more widely distributed than the others which are relatively compact. The strong emission around ($-11'', 0''$) in the velocity range of $5.9 \sim 6.4$ km s$^{-1}$ represents the red side of line core component coming from the envelope. There is little emission in the vicinity of line center ($v=5.2 \sim 5.7$ km s$^{-1}$) over the whole field of view, due to heavy self-absorption. The emission on the blue side ($v=4.5 \sim 5.1$ km s$^{-1}$) of the systemic component is absent, contrary to the one in the red side (which will be discussed in the next section). Interestingly, the blue outflow lobe also seems to have three discrete components and to share similar kinematics as the red lobe. One can find the 3.3 km s$^{-1}$ component farthest from the systemic velocity, located around $\approx$($-10'', 5''$) whose velocity range is wide from 1.4 to 4.3 km s$^{-1}$. The other two components have velocities of $\approx$ 4.4 and 5.0 km s$^{-1}$ which reside at $\approx$($5'', 5''$) and ($15'', -10''$), respectively. The 5.0 km s$^{-1}$ component closest to the rest velocity covers the largest area as the 5.9 km s$^{-1}$ one does in the red lobe. The fact that the individual velocity components can be identified and grouped into three pairs supports the idea of symmetric and episodic mass ejections. It may be claimed from the number of pairs that the mass loss phenomena have taken place at least three times in the past. There must be an ambient cloud or a cavity wall with velocity close to the systemic one which embeds these different velocity components, but this is probably resolved out, since its distribution would be extended. The shortest baseline mentioned in section 2 suggests that we are blind to structures larger than $\sim 100''$. The clumpiness and intermittency of the outflow is rather common in the case of extreme high velocity outflows ([@bac96]). It seems that standard or low velocity outflows exhibit such features as well (cf. [@hog98]; [@gom99]).
Furthermore, we note that the component whose velocity is closer to the systemic one occupies larger area. This can be evidence that an opening angle of the outflow becomes wider as the YSO evolves ([@bac96]), provided that the ejected material is accelerated after the ejection from around zero velocity. If this assumption holds, then the high velocity component will be located far away from the driving source. Actually the velocity roughly proportional to the distance away from the source is a reasonable approximation to the velocity field of the well-collimated flow, as can be seen in HH211, NGC1333-IRAS2, NGC2071, and NGC2264G ([@mas93]; [@gue99]; [@san94]; [@che92]; [@fic98]). It is approximately true for the L483 as well, which is shown in the position-velocity diagram of Fig. 7. The velocity increase together with the distance from the source could also be made by other mechanisms like the simultaneous outburst of material with a wide range of velocities. However, this may not be applicable to the well-collimated outflows with narrow throats.
On the other hand, one can imagine that the opening angle has been kept wide from the beginning, but the fastest mass ejection has taken place preferentially normal to the flat disk or envelope. Although this is not the case for the general class 0 objects, this possibility can not be ruled out. The slowly moving material with wider opening angle could also be regarded as a remnant of a pre-existing shell which was slowly expanding and is swept up by the fast jet. All these invoke detailed understanding of the outflow and hence should be tested by further observation.
Discussion
==========
Abundance estimate of HCO$^+$ in the outflow
--------------------------------------------
The line strength of HCO$^+$ in the wings indicates that the molecule may be enhanced in the outflow region. We estimate the column density of HCO$^+$ and CO in the outflow lobe in order to derive the relative abundance of HCO$^+$ to CO. Since the line wings are as wide as 5 km s$^{-1}$ along any line of sights, the condition of the [*large velocity gradient*]{} is applicable. Then we can use an expression by Goldreich & Kwan (1974): $$\tau={8 \pi^3 \mu^2 \over 3 h }{R \over V}
n_0 (1-e^{-T_0 / T_{\rm ex}}),$$ where $\mu(=3.4 {\rm debye})$ is the dipole moment of HCO$^+$, $R/V$ is the velocity gradient, $n_0$ is the level population in the level $J=0$, $T_{\rm ex}$ is the excitation temperature, and $T_0(=h \nu_{10} / k$) is 4.2 K. If we use an approximation, $(n_0+n_1+n_2+\cdots)/n_0 \equiv n_t/n_0 \simeq 2 k T_{\rm ex} / h \nu_{10}$, $$\tau={2 \pi^3 \mu^2 \nu_{10} \over 3 k T_{\rm ex} V } N({\rm HCO}^+)
(1-e^{-T_0 / T_{\rm ex}}),$$ where the column density $N$(HCO$^+$) is equal to $2 R n_t$.
The $\tau$ is derived from $$T_{\rm b}=(J(T_{\rm ex})-J(2.7 K)) (1-e^{-\tau}),$$ where $J(T)=T_0/(e^{T_0/T}-1)$. Since $T_{\rm b}\approx 1$ K in the wings (see Fig. 8), $\tau \approx 0.15$ for an assumed $T_{\rm ex}$ of 10 K. $N$(HCO$^+$) is then $1.4\times 10^{13}$ cm$^{-2}$ for $V=5$ km s$^{-1}$. In the case that $T_{\rm ex}= 20$ K, $\tau \approx 0.051$ and $N({\rm HCO}^+) = 1.7\times 10^{13}$ cm$^{-2}$.
We derive the column density of CO in the outflow lobe in a similar way by using the $J$=2–1 transitions of both $^{12}$CO and $^{13}$CO in Hatchell, Fuller, & Ladd (1999). The resulting CO column density is $N({\rm CO})=4.2\times 10^{17}$ cm$^{-2}$, where we adopted $\tau = 4.0$ and $T_{\rm ex}=25$ K. Finally it is found that the abundance of HCO$^+$ relative to CO is $(3.3 \sim 4.0) \times 10^{-5}$. This is comparable to the value of $\sim 10^{-4}$ inferred either for the Orion extended ridge or for the TMC-1 region ([@irv87]). Thus we find no significant abundance enhancement of HCO$^+$ in the outflow. The line wings look rather strong simply due to the depression of the line core.
Anti-infall signature of the HCO$^+$ lines
------------------------------------------
Line profiles of HCO$^+$ 1–0 toward the envelope are displayed in Fig. 8, where a spectrum obtained toward the IRAS source with the single dish telescope is shown together. Emission from line core is mildly self-absorbed in the single dish observation, but suffers from severe self-absorption in the array observation, falling off close to zero. The self-absorption of the line core is usually amplified in the interferometric observation, since the rest velocity component is most extended and the lack of visibility data at short spacing filters it out. Here it should be pointed out that the blue peak is weaker than the red one in the single dish observation and even almost missing in the interferometric one. This asymmetry is consistent with those of higher transitions of HCO$^+$ ([@gre97]). All these observations strongly suggest that the stronger red peak is not an artifact caused by the limited UV coverage of interferometric observation, but an intrinsic property of the envelope of L483. The stronger red peak of optically thick line is generally known as an [*anti-infall*]{} signature, indicative of an outward motion. However, it contradicts the observations of HCN, H$_2$CO, and NH$_3$, all implying the inward motion ([@par99]; [@mye95]; [@ful00]).
What gives rise to the anti-infall asymmetry of HCO$^+$? In answering this question, however, we should not violate observational facts of HCN, H$_2$CO, and NH$_3$ supporting the collapse motion. First, we can consider the influence of the outflow. In addition to the self-absorption by the surface layer of the envelope, the blue lobe of the outflow in front of the envelope may further obscure the blue part of line core emission. However, since the outflow will have a higher excitation temperature than the envelope ([@hat99]), it makes the emission from the envelope brighter irrespective of its optical depth: if the outflow is optically thin, the emission from the outflow should be added to that of envelope, while if the outflow is opaque, then the emission from outflow will replace it. Therefore the outflow can not explain the anti-infall asymmetry of the line core.
The asymmetry seems to arise from the envelope itself or very close to it. The envelope or material in its vicinity may be really in a state of almost isotropic expansion. Here we recall that the material (traced by HCO$^+$) whose velocity is close to the systemic one moves outward with a wide opening angle, whereas the envelope (traced by C$_3$H$_2$) is collapsing. Thus one viable option may be a nearly isotropic outward motion of gas between the disk-like envelope and the collimated outflow, which is actually a part of the outflow. If this motion takes place over a large amount of volume around the flattened envelope, then we will have line profiles with the anti-infall signature. The detail of the line shape depends on the velocity field in this expanding volume, but in any case the line will have the red peak stronger than the blue ([@leu78]). To summarize, there is a highly collimated jet-like flow far from the central source, while there is a wide angle wind-like mass ejection at the base of the outflow. Therefore the geometry and kinematics of the outflow of L483 seems to be in favor of the X-wind model as a low velocity version (cf. [@bac96]; [@li96]). The Cep A-HW2 is one of the recent examples showing similar line shapes as L483 ([@gom99]). Then one may have to look for reason why only the HCO$^+$ reflects the expansion. This could be explained in terms of its abundance and excitation condition. The main route to form HCO$^+$ is the reaction between H$_3^+$ and CO. The HCO$^+$ is unlikely to be depleted, since both species are abundant in the quiescent and relatively diffuse clouds ([@van98]; [@lan96]). As a consequence, although HCO$^+$ has a large critical density, it is collisionally excited easily at lower densities due to the line trapping. In this case, the critical density, $n_c$, should be modified as $A_{ji}/(C_{ji} \tau_{ji})$, where $\tau_{ji}$ is an optical depth, as already pointed out by Genzel (1991). It is also shown with a simple LVG analysis that HCO$^+$ 1–0 transition requires the lowest effective density, $n_{eff}$, among CS, H$_2$CO, HCN, and HCO$^+$ to produce the line intensity of 1 K for a given column density per unit velocity interval of $\log (N / \Delta V) = 13.5$, where the column density, $N$, is given in cm$^{-2}$ and the line width, $\Delta V$, in km s$^{-1}$ ([@eva99]). (At T$_k=10$ K, $n_{eff}=2.4 \times 10^3$ cm$^{-3}$ for HCO$^+$ 1–0, while $n_{eff}=1.8 \times 10^4$, $6.0 \times 10^4$, and $2.9 \times 10^4$ cm$^{-3}$, for CS 2–1, H$_2$CO 2$_{12}$–1$_{11}$, and HCN 1–0, respectively.) Thus the expanding part of the envelope could be more effectively traced by the HCO$^+$ 1–0 transition than by other transitions. Park, Kim, and Minh (1999) compared the line profiles of HCO$^+$ 3–2 & 4–3, H$_2$CO 2$_{12}$–1$_{11}$, CS 2–1, and HCN 1–0 of 9 class 0 and I objects, and found that the HCO$^+$ molecule seems to prefer the anti-infall asymmetry than the HCN and possibly the CS does. The preference is very marginal due to a small number of samples, but this could be indirect evidence supporting the idea mentioned above.
Position of the young stellar object in L483
--------------------------------------------
As shown in Fig. 1, the position of continuum peak coincides with that of IRAS 18148-0440 within $2'' \sim 3''$, while the peak of C$_3$H$_2$ emission is displaced to the west by $6''$ from it. Although HCO$^+$ is severely saturated, the emission peak of the red part of the systemic component at $v\approx 5.9$ km s$^{-1}$ is roughly coincident with that of C$_3$H$_2$, as represented in Fig. 6.
Are the two sources really different condensations? The error ellipse of the IRAS source position is as large as $35'' \times 8''$ with a position angle of $86^{\circ}$ (IRAS Point Source Catalogue). However, it is found from the VLA observation that the 3.6 cm and 6 cm continuum source is detected exactly at the IRAS position and there is no emission at the position of the C$_3$H$_2$ emission peak. Although the VLA beam is $19''\times 9''$ and $10''\times 5''$ at 3.6 cm and 6 cm, respectively, the accuracy in determining the peak position should be better than $1'' \sim 2''$ ([@ang97]; [@ang00]; [@bel00]). The far-infrared ($100 \sim 190\mu$m) and submillimeter ($450 \sim 1100\mu$m) continuum emission are also detected at the IRAS source position, but the beams are as large as $\approx 50''$ and $\approx 20''$, respectively ([@ful95]; [@lad91]). Recently Fuller and Wooten (2000) mapped the region at $\lambda = 450$ and $850 \mu m$ and found an elongated condensation at the IRAS position with ${\rm P.A.}\approx 70^{\circ}$ whose (beam-convolved) FWHM size is $\approx 30'' \times 15''$. Thus there may be an object at $\approx$ ($-9'', -3''$) which is unveiled only in millimeter line emission and possibly in the continuum from sub-millimeter to far-infrared. However, it is more likely that the line emission of C$_3$H$_2$ and HCO$^+$ and the continuum emission emanate from the same condensation but that the molecular line emission is depressed around the dust condensation by some reasons. One of the reasons may be that the gas is frozen out onto the grain in the cold and dense environment. One can find such examples in VLA 1623 and NGC 2024 ([@and93]).
Summary
=======
The BIMA array observation has been undertaken toward the IRAS 18148-0440, a YSO embedded in the L483 molecular core. An envelope with a beam-deconvolved size of $\sim 3000$ AU (R.A.) $\times$ 2000 AU (Dec.) and with a mass of $\sim 0.13 M_{\odot}$ is found to collapse towards its center. The collapse motion is different from the Shu type ($v\propto r^{-1/2}$), but more likely linear with a velocity gradient of $\sim 13$ km s$^{-1}$ pc$^{-1}$. The YSO seems to undergo main accretion phase without any conclusive signature of rotation. The molecular outflow traced by HCO$^+$ 1–0 suggests that there have been successive (at least three times) mass loss events to date and that the recent ejection might take place with a wide opening angle compared with previous ones. The rest velocity component of the HCO$^+$ line is found to exhibit the anti-infall signature, while the other transitions of density tracing molecules like CS, H$_2$CO, and HCN are not. If the outflow is poorly collimated at its base, the outward motion directed radially would be pervasive in both sides of the flattened envelope and responsible for the HCO$^+$ 1–0 line biased to the red. Due to the ease of collisional excitation and little chance of depletion, the HCO$^+$ 1–0 is the most likely transition with which one can sense the outward motion.
We thank the anonymous referee for his careful reading of the manuscript. YSP was partially supported by the BK21 program of Ministry of Education, Korea.
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|
{
"pile_set_name": "ArXiv"
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---
abstract: 'In relational approach to general rough sets, ideas of directed relations are supplemented with additional conditions for multiple algebraic approaches in this research paper. The relations are also specialized to representations of general parthood that are upper-directed, reflexive and antisymmetric for a better behaved groupoidal semantics over the set of roughly equivalent objects by the first author. Another distinct algebraic semantics over the set of approximations, and a new knowledge interpretation are also invented in this research by her. Because of minimal conditions imposed on the relations, neighborhood granulations are used in the construction of all approximations (granular and pointwise). Necessary and sufficient conditions for the lattice of local upper approximations to be completely distributive are proved by the second author. These results are related to formal concept analysis. Applications to student centered learning and decision making are also outlined.'
address:
- |
HBCSE, Tata Institute of Fundamental Research\
9/1B, Jatin Bagchi Road\
Kolkata (Calcutta)-700029, India\
Homepage: <http://www.logicamani.in>\
Orchid: <https://orcid.org/0000-0002-0880-1035>
- |
Institute of Mathematics, University of Miskolc\
3515 Miskolc-Ergyetemv'aros\
Miskolc, Hungary\
Homepage: <https://www.uni-miskolc.hu/~matradi>
author:
- A Mani
- 'S''andor Radeleczki'
bibliography:
- 'algroughf690.bib'
title: Algebraic Approach to Directed Rough Sets
---
General Approximation Spaces,Up-Directed Relations,Non transitive Parthoods,Granular Rough Semantics,Groupoidal Algebraic Semantics,Malcev Varieties,Directed Rough Sets.
Introduction
============
In relational approach to general rough sets various granular, pointwise or abstract approximations are defined, and rough objects of various kinds are studied [@am501; @am240; @ppm2; @gc2018; @pp2018; @gcd2018]. These approximations may be derived from information tables or may be abstracted from data relating to human (or machine) reasoning. A *general approximation space* is a pair of the form $S = \left\langle \underline{S}, R \right\rangle$ with $\underline{S}$ being a set and $R$ being a binary relation ($S$ and $\underline{S}$ *will be used interchangeably throughout this paper*). Often, approximations of subsets of $\underline{S}$ are generated from these and studied at different levels of abstraction in theoretical approaches to rough sets. It is also of interest to understand ideas of closeness of other relations to the relation $R$ – this includes the problem of computing reducts of a type.
Parthood (part of) relations [@ham2017; @rgac15; @katk06; @ur; @lp2011] of different kinds play a major role in human reasoning over multiple perspectives. They may be between objects and properties, or collections of objects or properties, or between concepts. For example, one can assert that *red is part of maroon* or that *red is a substantial part of pink* or that *redness is part of pinkness* – a key feature of such relations is the connection with ontology [@ham2017; @am3930].
Rough Y-systems and granular operator spaces, introduced and studied extensively by the first author [@am501; @am9969; @am9411; @am240], are essentially higher order abstract approaches in general rough sets in which the primitives are ideas of approximations, parthood, and granularity. In the literature on mereology [@av; @vie; @ur; @rgac15; @am3930; @seibtj2015], it is argued that most ideas of binary *part of* relations in human reasoning are at least antisymmetric and reflexive. *A major reason for not requiring transitivity of the parthood relation is because of the functional reasons that lead to its failure* (see [@seibtj2015]), and to accommodate *apparent parthood* [@am9969]. In the context of approximate reasoning interjected with subjective or pseudo-quantitative degrees, transitivity is again not common. The role of such parthoods in higher order approaches are distinctly different from theirs in lower order approaches – specifically, general approximation spaces of the form $S$ mentioned above with $R$ being a parthood relation are also of interest. Given two concepts ($A$ and $B$ say), it often happens that there are concepts like $E$ of which $A$ and $B$ are part of. This is, loosely speaking, the idea of the parthood relation being up-directed. In approximate reasoning with vague objects or concepts, this property is more common than the existence of supremums (in a general sense).
From a purely mathematical perspective, the property of up-directedness (also referred to as directedness) of partial orders and semilattice orders is widely used in literature, it has also been used in studying concepts of *ideals of binary relations* (see [@jc1977; @am9204]. But the groupoidal approach of [@ichlps2015; @ichlweak2013] is not known in earlier work.
In this research general approximation spaces, in which the relation $R$ is an up-directed parthood relation, are studied in detail by the authors. It is also shown that the algebraic semantics of such spaces is very distinct from those in which $R$ is a directed or partial or quasi-order. More specifically, two of the algebraic models are groupoids with additional operations (these correspond to granular approximations), while the third is based on completely distributive lattices (this corresponds to mixed local approximations).
In the following section, some of the essential background is mentioned. In the third section, directed rough sets are introduced and basic results are proved. Illustrative examples are invented in the following section. Algebraic semantics on the power set and subsets thereof are explored in depth by the first author in the fifth section. In the sixth section, groupoidal semantics over quotients are investigated by the first author. Algebraic semantics of local approximations, connections with formal concept analysis and induced groupoids on subsets of the power set are explored in the following section by the second author. Subsequently knowledge interpretation over the three semantic approaches is discussed and an application to student-centred learning is invented in the next section. Further directions are provided in the ninth section.
Some Background
===============
Information Tables
------------------
The concept of *information* can also be defined in many different and non-equivalent ways. In the first author’s view *anything that alters or has the potential to alter a given context in a significant positive way is information*. In the contexts of general rough sets, the concept of information must have the following properties:
- [information must have the potential to alter supervenience relations in the contexts (A set of properties $Q$ supervene on another set of properties $T$ if there exists no two objects that differ on $Q$ without differing on $T$),]{}
- [information must be formalizable and ]{}
- [information must generate concepts of roughly similar collections of properties or objects.]{}
The above can be read as a minimal set of desirable properties. In practice, additional assumptions are common in all approaches and the above is about a minimalism. This has been indicated to suggest that comparisons may work well when ontologies are justified.
The concept of an information system or table is not essential for obtaining a granular operator space or higher order variants thereof. As explained in [@am501; @am9222; @am9969], in human reasoning contexts it often happens that they arise from such tables.
Information tables (also referred to as descriptive systems or knowledge representation system in the literature) are basically representations of structured data tables. Often these are referred to as *information systems* in the rough set literature, while it refers to an integrated heterogeneous system that has components for collecting, storing and processing data in AI, computer science and ML. From a mathematical point of view, the latter can be described using heterogeneous partial algebraic systems. In rough set contexts, this generality has not been exploited as of this writing. It is therefore suggested in [@cd2017] to avoid plural meanings for the same term.
An *information table* $\mathcal{I}$, is a relational system of the form $$\mathcal{I}\,=\, \left\langle \mathfrak{O},\, \mathbb{A},\, \{V_{a} :\, a\in \mathbb{A}\},\, \{f_{a} :\, a\in \mathbb{A}\} \right\rangle$$ with $\mathfrak{O}$, $\mathbb{A}$ and $V_{a}$ being respectively sets of *Objects*, *Attributes* and *Values* respectively. $f_a \,:\, \mathfrak{O} \longmapsto \wp (V_{a})$ being the valuation map associated with attribute $a\in \mathbb{A}$. Values may also be denoted by the binary function $\nu : \mathbb{A} \times \mathfrak{O} \longmapsto \wp{(V)} $ defined by for any $a\in \mathbb{A}$ and $x\in \mathfrak{O}$, $\nu(a, x) = f_a (x)$.
An information table is *deterministic* (or complete) if $$(\forall a\in At)(\forall x\in \mathfrak{O}) f_a (x) \text{ is a singleton}.$$ It is said to be *indeterministic* (or incomplete) if it is not deterministic that is $$(\exists a\in At)(\exists x\in \mathfrak{O}) f_a (x) \text{ is not a singleton}.$$
Relations may be derived from information tables by way of conditions of the following form: For $x,\, w\,\in\, \mathfrak{O} $ and $B\,\subseteq\, \mathbb{A} $, $ \sigma xw $ if and only if $(\mathbf{Q} a, b\in B)\, \Phi(\nu(a,\,x),\, \nu (b,\, w),) $ for some quantifier $\mathbf{Q}$ and formula $\Phi$. The relational system $S = \left\langle \underline{S}, \sigma \right\rangle$ (with $\underline{S} = \mathbb{A}$) is said to be a *general approximation space*.
This universal feature of the definition of relations in general approximation spaces do not hold always in human reasoning contexts.
In particular if $\sigma$ is defined by the condition Equation \[pawl\], then $\sigma$ is an equivalence relation and $S$ is referred to as an *approximation space*. $$\label{pawl}
\sigma xw \text{ if and only if } (\forall a\in B)\, \nu(a,\,x)\,=\, \nu (a,\, w)$$
*In this research, prefix or Polish notation is uniformly preferred for relations and functions defined on a set. So instances of a relation $\sigma$ are denoted by $\sigma a b$ instead of $a \sigma b$ or $(a, b) \in \sigma$. If-then relations (or logical implications) in a model are written in infix form with $\longrightarrow$.* In Equation \[pawl\], *if and only if* is used because the definition is not done in an obvious model.
Context of this Research
------------------------
This research is relevant to all theoretical approaches to rough sets including the contamination avoidance based axiomatic granular approach due to the first author [@am240; @am501; @am6900; @am9969; @am9204; @am9114; @am9006; @am9222], modal approaches (for the pointwise approximations) [@pp2018; @ppm2], and other abstract approaches [@jpr; @cc5; @ie2011]. For additional clarifications on the context, readers may refer to the references suggested.
In fact, the specific approximation spaces studied in this paper can be used to generate a number of High granular operator spaces and variants thereof studied by the first author [@am9114; @am9969; @am9006; @am501; @am9222]. These will be taken up in a separate paper.
Algebraic Concepts
------------------
For basics of partial algebras, the reader is referred to [@bu; @lj].
A *partial algebra* $P$ is a tuple of the form $$\left\langle\underline{P},\,f_{1},\,f_{2},\,\ldots ,\, f_{n}, (r_{1},\,\ldots ,\,r_{n} )\right\rangle$$ with $\underline{P}$ being a set, $f_{i}$’s being partial function symbols of arity $r_{i}$. The interpretation of $f_{i}$ on the set $\underline{P}$ should be denoted by $f_{i}^{\underline{P}}$, but the superscript will be dropped in this paper as the application contexts are simple enough. If predicate symbols enter into the signature, then $P$ is termed a *partial algebraic system*.
In this paragraph the terms are not interpreted. For two terms $s,\,t$, $s\,\stackrel{\omega}{=}\,t$ shall mean, if both sides are defined then the two terms are equal (the quantification is implicit). $\stackrel{\omega}{=}$ is the same as the existence equality (also written as $\stackrel{e}{=}$) in the present paper. $s\,\stackrel{\omega ^*}{=}\,t$ shall mean if either side is defined, then the other is and the two sides are equal (the quantification is implicit). Note that the latter equality can be defined in terms of the former as $$(s\,\stackrel{\omega}{=}\,s \, \longrightarrow \, s\,\stackrel{\omega}{=} t)\&\,(t\,\stackrel{\omega}{=}\,t \, \longrightarrow \, s\,\stackrel{\omega}{=} t)$$
Various kinds of morphisms can be defined between two partial algebras or partial algebraic systems of the same or even different types. If $$X\, =\, \left\langle\underline{X},\,f_{1},\,f_{2},\,\ldots ,\, f_{n} \right\rangle \text{ and } W\, =\, \left\langle\underline{W},\,g_{1},\,g_{2},\,\ldots ,\, g_{n} \right\rangle$$ are two partial algebras of the same type, then a map $\varphi \, :\, X\, \longmapsto\, W$ is said to be a
- [*morphism* if for each $i$, $$(\forall (x_1,\, \ldots \, x_k)\,\in \, dom (f_i)) \varphi (f_{i}(x_1 , \ldots , \, x_k))\,=\, g_i (\varphi(x_1),\, \ldots , \, \varphi (x_k))$$]{}
- [*closed morphism*, if it is a morphism and the existence of\
$g_{i} (\varphi(x_1),\, \ldots , \, \varphi (x_k))$ implies the existence of $f_{i}(x_1 , \ldots , \, x_k)$.]{}
Usually it is more convenient to work with closed morphisms.
### Lattice Concepts
The reader may refer to [@gra1998; @gra2014; @jj; @dp2002] for lattice theoretical concepts. Some are stated below for convenience.
In a complete lattice $L$, an element $x\neq 0$ is said to be *completely join-irreducible* if and only if $$(\forall K\subseteq L)(\bigvee K = x \longrightarrow (\exists z\in K) z = x)$$ The set of join-irreducible elements of $L$ will be denoted by $CJ(L)$. The lattice $L$ is said to be *CJ-generated* or *spatial* if and only if every element of $L$ is represented as a join of some elements of $CJ(L)$.
A lattice in which every descending chain is finite is said to satisfy the descending chain condition (DCC). In particular, if a complete lattice satisfies DCC, then it is necessarily spatial.
Groupoids and Binary Relations
------------------------------
Under certain conditions, groupoidal operations can correspond to binary relations on a set. More generally, all binary relations can be read as partial groupoidal operations in a perspective ([@ichlps2015]) and therefore all general approximation spaces can be transformed into partial groupoids. The connections will be explored by the first author in a forthcoming paper. In this subsection known results for groupoids are stated for convenience.
Let $S=\left\langle \underline{S}, R \right\rangle $ be a relational system, define $$U_R (a, b) = \{x :\, Rax \,\&\, Rbx\}$$ $S$ is said to be *up-directed* if and only if $U_R (a, b)$ is never empty. That is, $$(\forall a, b) \, \neg U_R (a, b) = \emptyset \tag{up-directed}$$
\[updg\] If a relational system is up-directed, then it corresponds to a number of groupoids defined by $$(\forall a, b )\, ab = \left\lbrace \begin{array}{ll}
b & \text{if } Rab\\
c & c\in U_R(a, b) \,\&\, \neg Rab\\
\end{array} \right. \tag{updg}$$
These are studied in [@icl2013]. The collection of groupoids satisfying the above condition will be denoted by $\mathfrak{B}(S)$ and an arbitrary element of it will be denoted by $\mathsf{B}(S)$. It may be noted that *up-directed sets* (partially ordered sets that are up-directed) and related constructions are well-known in topology and algebra, but the specific association of up-directedness mentioned is new.
*Join directoids* [@jjq90] are groupoids of the form $S$ that admit of a partial order relation $\leq$ that satisfies $(\forall a, b)\, a, b \leq ab$ and if $\max\{a, b\}$ exists then $ab = \max \{a, b\}$. Clearly the results of [@icl2013] may also be read as a severe generalization of known results for join directoids. It may also be noted that lambda lattices (that are commutative join and meet directoids) are related special cases (see [@sva; @am105]).
For a groupoid $A$, the following are equivalent
- [A up-directed reflexive relational system $S$ corresponds to $A$]{}
- [$A$ satisfies the equations $$aa = a \, \&\, a(ab) = b(ab) = ab$$]{}
If $A$ is a groupoid, then two relational systems corresponding to it are $\Re (A) = \left\langle \underline{A}, R_A \right\rangle$ and $\Re^* (A) = \left\langle \underline{A}, R_A^* \right\rangle$ with $$\begin{aligned}
R_A = \{(a, b):\, ab = b\}\\
R_A^* = \bigcup \{(a, ab),\,(b, ab)\}\end{aligned}$$
- [If $A$ is a groupoid then $\Re^* (A)$ is up-directed.]{}
- [If a groupoid $A\models a(ab) = b(ab) = ab$ then $\Re (A) = \Re^* (A)$.]{}
- [If $S$ is an up-directed relational system then $\Re (\mathsf(B) (S)) = S$.]{}
If $S = \left\langle \underline{S}, R \right\rangle$ is a up-directed relational system, then all of the following hold:
- [$R$ is reflexive if and only if $\mathsf{B}(S) \models aa = a$.]{}
- [$R$ is symmetric if and only if $\mathsf{B}(S) \models (ab)a = a$.]{}
- [$R$ is transitive if and only if $\mathsf{B}(S) \models a((ab)c) = (ab)c$.]{}
- [If $\mathsf{B}(S) \models ab = ba$ then $R$ is antisymmetric.]{}
- [If $\mathsf{B}(S) \models (ab)a = ab$ then $R$ is antisymmetric.]{}
- [If $\mathsf{B}(S) \models (ab)c = a(bc)$ then $R$ is transitive.]{}
Morphisms between up-directed relational systems are preserved by corresponding groupoids. A *relational morphism* (as in [@mal]) from a relational system $S = \left\langle \underline{S}, R \right\rangle$ to another $K = \left\langle \underline{K},Q \right\rangle$ is a map $f: S \longmapsto K$ that satisfies $$(\forall a, b)\, (Rab \, \longrightarrow Qf(a)f(b)).$$ $f$ is said to be *strong* if it satisfies $$(\forall c, e\in Q)(\exists a, b\in S )\, Qf(a)f(b)\, \&\, f(a) = c, \&\, f(b) = e$$
Approximation Spaces and Groupoids {#apprsp}
----------------------------------
It should be noted that up-directedness is not essential for a relation to be represented by groupoidal operations. The following construction that differs in part from the above strategy can be used for partially ordered sets as well, and has been used by the first author in [@amdsc2016; @am909] in the context of knowledge generated by approximation spaces. The method relates to earlier algebraic results including [@jjm; @jj1978; @kt1981; @fjjm]. The groupoidal perspective can be extended for quasi ordered sets.
If $S = \left\langle \underline{S}, R \right\rangle$ is an approximation space, then define (for any $a, b\in S$) $$a\cdot b \,=\, \left\{
\begin{array}{ll}
a, & \text{if } Rab \\
b, & \text{if } \neg Rab
\end{array}
\right.$$
Relative to this operation, the following theorem (see [@jjm]) holds:
\[ab\] $\left\langle S,\, \cdot \right\rangle$ is a groupoid that satisfies the following axioms (braces are omitted under the assumption that the binding is to the left, e.g. ’$abc$’ is the same as ’$(ab)c$’): $$\begin{aligned}
{x x = x} \tag{E1}\\
{x (a z) = (x a) (x z) } \tag{E2}\\
{x a x = x} \tag{E3}\\
{azxauz = auz } \tag{E4}\\
{u(azxa)z = uaz } \tag{E5}\end{aligned}$$
The following are consequences of the defining equations of $\mathbb{E}_{0}$ (from **E1,E2,E3**): $$\begin{aligned}
{x(ax) = x ;\; x(xa) = xa ;\; (xa)a = xa } \\
{x(xaz) = x(az) ;\; (xz)(az) = xz ;\; (xa)(zx) = xazx } \\
{xazxa = xa;\; xazaz = xaz;\; xcazaxa = xaza} \\
{(xazx)(za) = x(za) ;\; x(az)a = xaza ;\; (xaz)(ax) = (xza)(zx) ;\; xazxz = xzaz.} \\
{(\forall x)(ex=ea \longrightarrow x=a) \;\equiv \; (\forall x) xe = e } \end{aligned}$$
Meta Explanation of Terms
-------------------------
This purpose of this list is to help with the terminology relating to general rough sets (and also high granular operator spaces [@am501; @am9222]).
- [: That which has been designated as *crisp* or is an approximation of some other object.]{}
- [: That whose approximations do not coincide with the object or that which has been designated as a *vague* object.]{}
- [: That which is available for computations in a rough semantic domain (in a contamination avoidance perspective). ]{}
- [: Many definitions and representations are possible relative to the context. From the representation point of view these are usually functions of definite or crisp objects.]{}
- [: An object that is invariant relative to an approximation process. In actual semantics a number of concepts of definiteness is possible. In some approaches, as in [@yzm2012; @hmy2019], these are taken as granules. Related theory has a direct connection with closure algebras and operators as indicated in [@am501].]{}
Up-Directed Rough sets: Basic Results
=====================================
In a general approximation space $S = \left\langle \underline{S}, R \right\rangle$ consider the following conditions: $$\begin{aligned}
(\forall a, b )(\exists c) R ac \, \&\, Rbc \tag{up-dir}\\
(\forall a) Raa \tag{reflexivity}\\
(\forall a, b)(Rab \, \&\, Rba \longrightarrow a=b) \tag{anti-sym}\end{aligned}$$ If $S$ satisfies up-dir, then it will be said to be a *upper directed approximation space*. If it satisfies all three conditions then it will be said to be a *up-directed parthood space*.
In general, partial/quasi orders, and equivalences need not satisfy . When they do satisfy the condition, then the corresponding general approximation spaces will be referred to as *up-directed general approximation spaces*.
The neighborhood granulations used for defining approximations are specified next.
For any element $a\in S$, the following neighborhoods are associated with it $$\begin{aligned}
[a] = \{x :\, Rxa\} \tag{neighborhood}\\
[a]_i = \{x: \, Rax\} \tag{inverse-neighborhood}\\
[a]_o = \{x: \, Rax \, \&\, Rxa \} \tag{symmetric neighborhood}\\\end{aligned}$$ A subset $A\subseteq S$ will be said to be *nbd-closed* if and only if $$(\forall x\in A)\, [x] \subseteq A$$ Let the set of all nbd-closed subsets of $S$ be $\mathcal{E}(S)$
$[a]$ is the set of things that relate to $a$ and $[a]_i$ is the set of things that $a$ relates to. $[a]$, $[a]_i$ and $[a]_o$ are respectively denoted by $R^{-1}(a)$, $R(a)$ and $(R\cap R^{-1})(a)$
For any subset $A\subseteq S$, the following approximations can be defined: $$\begin{aligned}
A^{l}\,=\, \bigcup \{[a]:\, [a]\subseteq A\} \tag{lower}\\
A^{l_i}\,=\, \bigcup \{[a]_i:\, [a]_i\subseteq A\} \tag{i-lower}\\
A^{u}\,=\, \bigcup \{[a]:\, \exists z \in [a]\cap A\} \tag{upper}\\
A^{u_i}\,=\, \bigcup \{[a]_i:\, \exists z \in [a]_i\cap A\} \tag{i-upper}\\
A^{l_s}\,=\, \bigcup \{[a]_o:\, [a]_o\subseteq A\} \tag{s-lower}\\
A^{u_s}\,=\, \bigcup \{[a]_o:\, \exists z \in [a]_o\cap A\} \tag{s-upper}\end{aligned}$$
In the context of the previous definition, the pointwise and local approximations are defined as follows:
$$\begin{aligned}
\tag{Point-wise Upper} A^{u+} \, =\, \{ x \, :\, [x]\cap A \neq \emptyset \}.\\
\tag{Point-wise Lower} A^{l+} \, =\, \{x \,:\,[x]\subseteq A \}\\
\tag{Point-wise i-Upper} A^{ui+} \, =\, \{ x \, :\, [x]_i\cap A \neq \emptyset \}.\\
\tag{Point-wise i-Lower} A^{li+} \, =\, \{x \,:\,[x]_i\subseteq A \}\\
\tag{u-Triangle} A^{\triangle} \, =\, \{ x \, :\, Rax \, \& a\in A \}\\ \tag{l-Triangle} A^{\triangledown} \, =\, \{x \,:\,[x]_i\subseteq A\,\&\, x\in A \}\\ \tag{ub-Triangle} A^{\blacktriangle} \, =\, \bigcup \{[x] :\, x\in A\}.\\
\tag{lb-Triangle} A^{\blacktriangledown} \, =\, \{x \,:\,[x]\subseteq A\,\&\, x\in A \}\end{aligned}$$
The \*-triangle approximations are *local* in the sense that they are defined relative to points (as opposed to subsets) in the set being approximated. It is shown below that while $\triangledown$ and $\blacktriangledown$ are pointwise approximation operators, $\triangle$ and $\blacktriangle$ are granular approximations because they can be represented as terms involving neighborhood granules and set operations alone. Because, neighborhoods and inverse-neighborhoods are used, the granular and pointwise approximations are inter related in a complex way.
In the above context, for any subset $A$, $$\begin{aligned}
A^{\triangle} = \bigcup \{[x]_i :\, x\in A\} \subseteq A^{u+}\subseteq A^{u_i}\\
A^{\blacktriangle} = \bigcup \{[x]:\, x\in A\}\subseteq A^{u}\end{aligned}$$
In the above context, when $R$ is reflexive and $c$ is the complementation operation $$\begin{aligned}
A^{l} = A^{\triangledown\triangle} \text{ and } A^{l_i} = A^{\blacktriangledown\blacktriangle}\\
A^{u} = A^{\triangle\blacktriangle} \text{ and } A^{u_i} = A^{\blacktriangle\triangle}\\
A^{\triangle} = \bigcup \{[x]_i :\, x\in A\} \subseteq A^{u+}\subseteq A^{u_i}\\
A\subseteq A^{\blacktriangle} = \bigcup \{[x]:\, x\in A\}\subseteq A^{u}\\
A^{\triangledown} \subseteq A\subseteq A^{\triangle}\,\&\, A^{\blacktriangledown} \subseteq A\subseteq A^{\blacktriangle}\\
A^{\triangle c} = A^{c\triangledown}\,\&\, A^{\blacktriangle c} = A^{c\blacktriangledown}\end{aligned}$$
\[luprop\] In a reflexive up-directed approximation space $S$, the following properties hold for elements of $\wp (S)$: $$\begin{aligned}
(\forall a) a^{ll} = a^{l} \subseteq a \tag{l-id}\\
(\forall a) a \subseteq a^{u} \subseteq a^{uu} \tag{u-wid}\\
(\forall a) a^{l}\subseteq a^{lu} \subseteq a^u \tag{lu-inc}\\
(\forall a, b)(a\subseteq b \longrightarrow a^l \subseteq b^l) \tag{l-mo}\\
(\forall a, b)(a\subseteq b \longrightarrow a^u \subseteq b^u) \tag{u-mo}\\
S^u = S = S^l \, \&\, \emptyset^l = \emptyset = \emptyset^u \tag{bnd} \\
(\forall a, b)(a\cup b)^u = a^u \cup b^u \tag{u-union}\\
(\forall a, b) a^l \cup b^l \subseteq (a\cup b)^l \tag{l-union}\\
(\forall a, b) (a\cap b = \emptyset \longrightarrow a^l \cup b^l = (a\cup b)^l ) \tag{l-union0}\\
(\forall a, b) (a\cap b)^l \subseteq a^l \cap b^l \tag{l-cap}\\
(\forall a, b) (a\cap b)^u \subseteq a^u \cap b^u \tag{u-cap}\end{aligned}$$
l-id
: [If $x\in a^{ll}$ then there exists a $b\in S$ such that $x\in [b] \subseteq a^l$. So $a^{ll} \subseteq a^{l}$. This proves .]{}
u-wid
: [If $x\in a$ then because of reflexivity of $R$, $x\in [x]\subseteq a^u$. So holds.]{}
lu-inc
: [The proof of u-wid carries over to that of lu-inc because $a^l\subseteq a^u$ implies $a^{lu} \subseteq a^u$.]{}
l-mo
: [If $a\subseteq b$ then $(\forall x\in S)([x]\subseteq a \longrightarrow [x] \subseteq b)$. This ensures that $a^l \subseteq a \subseteq b $ and $b^l \subseteq b$ and $a^l \subseteq b^l$. ]{}
u-mo
: [The proof is similar to that of l-mo.]{}
bnd
: [Follows from $(\forall x\in S) \, x\in [x]$.]{}
u-union
: - [If $x\in (a\cup b )^u $, then $(\exists z\in S)\, x\in [z]\,\&\, [z]\cap (a\cup b) \neq \emptyset$]{}
- [the latter condition is $([z]\cap a)\cup ([z]\cap b) \neq \emptyset$]{}
- [or $([z]\cap a)\neq \emptyset$ or $([z]\cap b) \neq \emptyset$. So $x\in a^u \cup b^u$.]{}
- [Conversely, if $h\in a^u \cup b^u$ then $(\exists z\in S)\, h\in [z]\,\&\, ([z]\cap a) \neq \emptyset \,\vee \,([z]\cap b) \neq \emptyset $ ]{}
- [the latter condition is $([z]\cap a)\cup ([z]\cap b) \neq \emptyset$]{}
- [So $h\in (a\cup b)^u$]{}
l-union
: - [$x\in a^l \cup b^l$ $\Leftrightarrow$ $x\in a^l$ or $x\in b^l$]{}
- [$\Leftrightarrow (\exists z, h \in S)\, x\in [z]\subseteq a\, \vee \, x\in [h]\subseteq b $]{}
- [$\Leftrightarrow (\exists z, h \in S)\, x\in [z] \subseteq a \cup b $ and so $x\in (a\cup b )^l$.]{}
Examples for the failure of the converse inclusion are easy to construct.
l-union0
: - [$x\in (a \cup b)^l$ then $(\exists z \in S)\, x\in [z] \subseteq a \cup b $]{}
- [So $(\exists z \in S)\, x\in [z]\subseteq a\, \text{ Xor } \, x\in [z]\subseteq b $]{}
- [So $x\in a^l$ xor $x\in b^l$, which implies $x\in a^l \cup b^l$.]{}
l-cap
: $x\in (a\cap b)^l$
- [if and only if $(\exists z\in S) x\in [z] \subseteq a\cap b$]{}
- [if and only if $(\exists z\in S) x\in [z] \subseteq a \,\&\,[z] \subseteq a $]{}
- [implies $x\in a^l$ and $x\in b^l$]{}
- [To see possible reasons for the failure of the converse, let $x\in a^l$ and $x\in b^l$]{}
- [then $(\exists z_1 \in S)\, x\in [z_1] \subseteq a $ and $(\exists z_2 \in S)\, x\in [z_2] \subseteq b $]{}
- [so $x\in [z_1]\cap [z_2] \subseteq a\cap b$, but it can happen that $[z_1]\cap [z_2]$ is not of the form $z$ for some $z\in a\cap b$.]{}
The nature of failure of $a^l \cap b^l \subseteq (a\cap b)^l$ shown in the proof suggests that it can be fixed at a semantic level in many ways.
\[lup\] In a up-directed approximation space $S$, the following properties hold for elements of $\wp (S)$: $$\begin{aligned}
(\forall a) a^{ll} = a^{l} \subseteq a \tag{l-id0}\\
(\forall a) a^{u} \subseteq a^{uu} \tag{u-wid0}\\
(\forall a) a^{l}\subseteq a^{lu} \subseteq a^u \tag{lu-inc}\\
(\forall a, b)(a\subseteq b \longrightarrow a^l \subseteq b^l) \tag{l-mo}\\
(\forall a, b)(a\subseteq b \longrightarrow a^u \subseteq b^u) \tag{u-mo}\\
S^l =S^u \subseteq S \, \&\, \emptyset^l = \emptyset = \emptyset^u \tag{bnd0}\\
(\forall a, b)(a\cup b)^u = a^u \cup b^u \tag{u-union}\\
(\forall a, b) a^l \cup b^l \subseteq (a\cup b)^l \tag{l-union}\\
(\forall a, b) (a\cap b)^l \subseteq a^l \cap b^l \tag{l-cap}\\
(\forall a, b) (a\cap b)^u \subseteq a^u \cap b^u \tag{u-cap}\end{aligned}$$
Most of the proof of Theorem \[luprop\] carries over. Because of the absence of reflexivity, the weaker properties u-wid0, and bnd0 hold.
It may be noted that the upper cone of a subset $A$ (that is the set $\{b: \, (\exists a, c \in A)\, Rab \,\&\, Rcb\}$) is contained in $A^u$.
Illustrative Examples
=====================
Abstract and practical examples are constructed in this section for illustrating various aspects of up-directed approximation spaces.
Abstract Example {#absexample}
----------------
Let $\underline{S}$ be the set $$\underline{S} \,=\, \{a, b, c, e, f\} \text{ and let }$$ $$R \, =\, \{ac, ae, af, bc, bf, ca, cb, cf, ea, ef, fa, fb \}$$ be a binary relation on it ($ac$ $(a, c)$ and so on for other elements). In Figure \[udr\], the general approximation space $S = \left\langle \underline{S}, R \right\rangle$ is depicted. An arrow from $e$ to $f$ is drawn because $Ref$ holds.
\(F) [$f$]{}; (O) \[below of=F\] ; (E) \[below of=O\] [$e$]{}; (C) \[below of=E\] [$c$]{}; (A) \[left of=C\] [$a$]{}; (B) \[right of=C\] [$b$]{}; (E) to node (F); (F) to node (A); (E) to node (A); (B) to node (F); (E) to node (B); (A) to node (C); (B) to node (C);
The up-directed approximation space $S = \left\langle \underline{S}, R \right\rangle$ is irreflexive and $R$ is not antisymmetric. The antisymmetric completion $R^+$ of $R$ coincides with its reflexive completion and is defined by $$R^+ = R \cup \{aa, bb, cc, ee, ff \}$$
The groupoid corresponding to $S$ is given by Table \[grps\]
a b c e f
--- --- --- --- --- ---
a e c c e f
b e c c e f
c a b f f f
e a b f f f
f a b a a a
: A Groupoid of $S$[]{data-label="grps"}
The neighborhood granules determined by the elements of $S$ are as in Table \[nbdg\]
$x $ $ [x] $ $ [x]_i $ $ [x]_o $
--------- ------------------------- ---------------------- -------------------
$a \, $ $ \, \{c, e, f\}\, $ $ \,\{c, e, f\}\, $ $ \, \{c, e, f\}$
$b\, $ $ \, \{c, e, f\}\, $ $ \,\{c, f\}\,$ $ \, \{c, f\}$
$c \,$ $ \,\{a, b \}\, $ $ \,\{a, b, f \} \,$ $ \, \{a, b\}$
$e \,$ $ \,\{a\}\, $ $\, \{a, b, f\} \,$ $\, \{a\}$
$f \, $ $ \,\{a, b, c, e \}\, $ $ \,\{a, b\}\, $ $\, \{a, b\}$
: Neighborhood Granules[]{data-label="nbdg"}
Since $\wp(S)$ has $32$ elements, approximations of specific subsets are alone considered next.
Let $A =\{e, c\} $, then its approximations are as below:
- [$A^l = \emptyset$ and $A^u = S$]{}
- [$A^{\triangledown} =\emptyset $ and $A^{\triangle} = \{a, b, f\}$]{}
- [$A^{\blacktriangledown} =\emptyset $ and $A^{\blacktriangle} = \{a, b\}$]{}
- [$A^{l_i} =\emptyset = A^{l_o} $ and $A^{u_i} = \{c, e, f\} = A^{u_o}$]{}
- [$A^{l+} =\emptyset $ and $A^{u+} = {a, b, f}$]{}
- [$A^{l_i+} =\emptyset $ and $A^{u_i+} = \{a, b\}$]{}
Reasoning about Vague Concepts
------------------------------
Suppose a set $\underline{S}$ of concepts relating to a classroom lesson are given, and that some of these are vague. For any two concepts $a$ and $b$, assume that a concept $c$ that apparently contains the two exists – this type of search for a $c$ amounts to taking decisions. Let this concept of apparent parthood be denoted by $R$. Depending on the context, the relation $R$ may be a up-directed, reflexive and antisymmetric relation. Thus $S= \left\langle \underline{S}, R \right\rangle$ may be a up-directed parthood space or definitely an up-directed space.
*Apparent parthood* relation has been considered by the first author in [@am9969] – in general it is not antisymmetric.
For two concepts $a$ and $b$, $ab = b$ may mean that $b$ fulfils the functions of $a$ in some sense (for example). If, on the other hand, $ab\in U_R(a, b)$ then there is a implicit reference to a choice function in the search for a concept that fulfils the role of both $a$ and $b$.
For a concept $a$, the neighborhood $[a]$ is the set of concepts that are *apparently part of* it, while $[a]_i$ is the set of concepts that it is apparently part of, and $[a]_o$ is the set of concepts that it is apparently part of and conversely. Obviously, when antisymmetry holds, the set $[a]_o$ will be a singleton. Note that these concepts have a directional character – because of up-directedness of $R$. Each granule of the form $[a]$ may be associated with at least one element of $S$. Is $[a]$ *determined by $a$*? The actual interpretation depends on the application context. In this case, it can be said the *investigation of $a$ leads to the set* $[a]$.
For a subset of concepts $A$, the lower approximation is an aggregation of directed granules that are included in $A$. It may also be read as the collection of *relatively definite concepts* that are attainable from $A$ (using common sense methods or through common knowledge).
Algebraic Semantics-1 {#algs1}
=====================
In this section, possible semantics of the approximations $l$ and $u$ on their image set are investigated. From Theorem \[luprop\] and Theorem \[lup\], it follows that a semantics over $\wp(S)$ without additional constructions is not justified because they do not distinguish between closely related general approximation spaces.
On the set $(\wp(S))^u = \{x^u :\, x\in \wp(S)\} = S_u$, the following operations can be defined (apart from the induced $\cup$ operation): $$\begin{aligned}
a\wedge b \,=\, (a\cap b)^u \tag{iu1}\\
a\vee b \,=\, (a\cup b) \tag{iu2}\\
\bot = \emptyset \tag{iu3}\\
\top = S^u \tag{iu4}\\\end{aligned}$$ and the resulting algebra $S_u \,=\, \left\langle \underline{S_u}, \vee, \wedge, \cup, l, u, \bot , \top \right\rangle$ will be called the *algebra of upper approximations in a up-directed space* (UUA algebra). If $R$ is a up-directed parthood relation or a reflexive up-directed relation respectively, then it will be said to be a up-directed parthood algebra of upper approximations (UAP algebra) or a reflexive algebra of upper approximations (UAR algebra) respectively.
The UUA, UAP and UAR algebras are well-defined, and an algebra of upper approximations satisfies all of the following: $$\begin{aligned}
(\forall a)\, a\vee a = a = a\vee \bot \tag{idemp1}\\
(\forall a, b)\, a\wedge b = b\wedge a \tag{comm2}\\
(\forall a, b)\, a\vee b = b\vee a \tag{comm1}\\
(\forall a, b, c)\, a\vee (b\vee c) = (a\vee b)\vee c \tag{assoc1}\\
(\forall a)\, (a\wedge a)\vee a = a\wedge a = a^u \tag{absfail}\\
(\forall a, b, c)\,(a\vee b = b \longrightarrow (a\wedge c)\vee (b\wedge c) = b\wedge c) \tag{mo1}\\\end{aligned}$$
The lower approximation operation is redundant and so the algebras are well-defined.
idemp1
: [$a\vee a = a\cup a = a $.]{}
comm2
: [$a\wedge b = (a\cap b)^u = (b\cap a)^u = b\wedge a$.]{}
comm1
: [Follows from definition.]{}
assoc1
: [Follows from associativity of set union.]{}
absfail
: [$a\wedge a = a^u$. So absorptivity fails in general.]{}
Absorptivity can be improved by defining the operations differently.
Let $S_{lu} = \{ x: \, x = a^l \text{ or } x= a^u \, \&\, a\in S\}$
On $S_{lu}$, the following operations can be defined (apart from $l$ and $u$ by restriction): $$\begin{aligned}
a\Cap b = (a\cap b )^l \tag{Cap}\\
a \Cup b = (a\cup b)^u \tag{Cup}\\
\bot = \emptyset \tag{iu3}\\
\top = S^u \tag{iu4}\\\end{aligned}$$ The resulting algebra $S_{lu} = \left\langle \underline{S_{lu}},\Cap , \Cup ,\cup, l, u, \bot , \top \right\rangle$ will be called the *algebra of approximations in a up-directed space* (UA algebra). If $R$ is a up-directed parthood relation or a reflexive up-directed relation respectively, then it will be said to be a up-directed parthood algebra of approximations (AP algebra) or a reflexive up-directed algebra of upper approximations (AR algebra) respectively.
A AP algebra $S_{lu}$ satisfies all of the following: $$\begin{aligned}
(\forall a) a\Cap a = a \,\&\, (a\Cup a)\Cap a = a \tag{idemp3}\\
(\forall a) a\Cup a = a^u \tag{quasi-idemp4}\\
(\forall a, b) a\Cap b = b\Cap a \,\&\, a\Cup b = b\Cup a \tag{comm12}\\
(\forall a , b) a\Cap ( b \Cup a) = a \tag{half-absorption}\\
(\forall a, b, c) a\Cup (b\Cup c) = (a \Cup b^u)\Cup c^u \tag{quasi-assoc1}\\
(\forall a, b, c) (a\Cup (b\Cup c))\Cup ((a\Cup b)\Cup c) = ((a\Cup a)\Cup (b\Cup b))\Cup (c\Cup c \Cup c) \tag{quasi-assoc0}\end{aligned}$$
idemp3
: - [$a\Cap a = (a\cap a)^l = a^l = a$]{}
- [$a\Cup a = a^u $ and $a^u \cap a = a$]{}
quasi-idemp4
: [$a\Cup a = (a\cup a)^u = a^u$.]{}
comm12
: [This follows from definition.]{}
half-absorption
: - [$a\Cap (b\Cup a) = (a\cap (b\cup a)^u)^l = ((a\cap a^u)\cup (a\cap b^u))^l$]{}
- [$= (a \cup (a\cap b^u))^l = a^l = a$]{}
quasi-assoc1
: - [$a\Cup (b\Cup c) = (a\cup (b\cup c)^u)^u = (a^u \cup b^{uu} \cup c^{uu})$]{}
- [$= (a \cup b^u))^u \cup c^{uu} = (a \Cup b^u)\Cup c^u$]{}
quasi-assoc0
: [This can be proved by writing all terms in terms of $\cup$. In fact $(a\Cup (b\Cup c))\Cup ((a\Cup b)\Cup c) = a^{uuu}\cup b^{uuu} \cup c^{uuu}$. The expression on the right can be rewritten in terms of $\Cup$ by .]{}
The above two theorems in conjunction with the properties of the approximations on the power set, suggest that it would be useful to enhance UA-, AP-, and AR-algebras with partial operations for defining an abstract semantics.
A partial algebra of the form $$S_{lu}^* = \left\langle \underline{S_{lu}},\Cap , \Cup ,\cup,\sqcap, ^{\kappa}
, l, u, \bot , \top \right\rangle$$ will be called the *algebra of approximations in a up-directed space* (UA partial algebra) whenever $S_{lu} = \left\langle \underline{S_{lu}},\Cap , \Cup ,\cup, l, u, \bot , \top \right\rangle$ is a UA algebra and $\sqcap$ and $^{kappa}$ are defined as follows ($\cap$ and $^c$ being the intersection and complementation operations on $\wp (S)$): $$(\forall a, b\in S_{lu} )\, a\sqcap b = \left\lbrace \begin{array}{ll}
a\cap b & \text{if } a\cap b \in S_{lu}\\
\text{undefined} & \text{ otherwise}
\end{array} \right.$$
$$(\forall a \in S_{lu} )\, a^{\kappa} = \left\lbrace \begin{array}{ll}
a^c & \text{if } a^c \in S_{lu}\\
\text{undefined} & \text{ otherwise}
\end{array} \right.$$
If $R$ is an up-directed parthood relation or a reflexive up-directed relation respectively, then it will be said to be a up-directed parthood partial algebra of approximations (AP partial algebra) or a reflexive algebra of upper approximations (AR partial algebra) respectively.
If $S$ is a up-directed approximation space, then its associated enhanced up-directed parthood partial algebra $S_{lu}^*= \left\langle \underline{S_{lu}},\Cap , \Cup ,\cup,\sqcap, ^{\kappa}
, l, u, \bot , \top \right\rangle$ satisfies all of the following: $$\begin{aligned}
\left\langle \underline{S_{lu}},\Cap , \Cup ,\cup, l, u, \bot , \top \right\rangle \text{ is a AP algebra} \tag{app1}\\
(\forall a) \, a\sqcap a = a \,\&\, a\sqcap \bot = \bot \, \&\, a\sqcap \top = a \tag{app2}\\
(\forall a, b, c) \, a\sqcap b \stackrel{\omega}{=} b \sqcap a\, \&\, a\sqcap (b\sqcap c) \stackrel{\omega}{=} (a\sqcap b)\sqcap c \tag{app3}\\
a\sqcap a^u = a = a\sqcap a^l \,\&\, a^{\kappa \kappa} \stackrel{\omega}{=} a \tag{app4}\\
a\sqcap (b\cup c) \stackrel{\omega}{=} (a\sqcap b)\cup (a\sqcap c) \,\&\, a\cup (b\sqcap c) \stackrel{\omega}{=} (a\cup b)\sqcap (a\cup c) \tag{app5}\\
(\forall a, b)\, (a\sqcap b)^{\kappa} \stackrel{\omega}{=} a^{\kappa}\cup b^{\kappa} \,\&\, (a\cup b)^{\kappa} \stackrel{\omega}{=} a^{\kappa}\sqcap b^{\kappa} \tag{app6}\end{aligned}$$
The theorem follows from the previous theorems in this section.
Groupoidal Semantics
====================
In the powerset $\wp (S)$ generated by a upper directed approximation space $S$, the following operation can be defined (apart from the rough approximations and induced Boolean operations) $$(\forall A, B\in \wp(S))\, A\cdot B = \{ab : \,a\in A \, \&\, b\in B \} \tag{g0}$$ The resulting algebra, $S^b = \left\langle \underline{\wp(S)}, \cdot, \cup, \cap, l, u, ^c, \bot, \top \right\rangle$ of type $(2, 2, 2, 1, 1, 1, 0, 0 )$ will be called a *basic power up-directed algebra* (-algebra). If $l$ and $u$ are replaced by $l_s$ and $u_s$, then the resulting algebra will be called a *basic symmetric power up-directed algebra* (-algebra)
$a\subseteq b$ will be used as an abbreviation for $a\cup b = b$ in what follows.
The algebra $\left\langle \underline{\wp(S)}, \cup, \cap, ^c, \emptyset, \wp(S) \right\rangle$ is a Boolean algebra. Further, the following properties are satisfied by a BP-algebra $S^b$: $$\begin{aligned}
(\forall a, b, c)(a\cup b = b \longrightarrow ac \cup bc = bc) \tag{order-comp}\\
(\forall a) \emptyset a = a\emptyset = \emptyset \,\&\, a S \subseteq S\, \&\, S a \subseteq S \tag{bnd2}\\
(\forall a, b, h)(a\cup b) h = (ah)\cup (bh) \,\&\, (a\cap b) h = (ah)\cap (bh) \tag{comp2}\\
\text{ Conditions mentioned in eqn.\ref{luprop}. } \tag{lu-properties} \end{aligned}$$
order-comp
: [ If $x\in ac $, then it is of the form $ef$ with $e\in a$ and $f\in c$. By the premise, $e\in b$, so the conclusion follows.]{}
comp2
: [$x\in (a\cup b)h$ if and only if $x\in ah$ or $x\in bh$. Similarly for the second part.]{}
Note that $x\in a^c h$ then $x$ is of the form $ef$ with $e\in a^c$ and $f\in h$, but $ef$ may be in $ah$ or $(ah)^c$. So, in general, $a^c h\neq (ah)^c$.
Meaning of the Groupoidal Operation
-----------------------------------
In the first author’s opinion, the groupoid operation can be read in at least two ways. The operation obviously adds information to the general approximation space – *this addition can be read as a decision because it involves choice among alternatives*. In fact, the collection of all possible groupoidal operations can be used to generate a decision space. As such this aspect can be investigated in the given form or by taking the exact region to which the result of the operation belongs relatively. For the latter perspective, the groupoidal operation over $\wp (S)$ can be read as a combination of operations that are relatively better behaved relative to the approximations, aggregation and commonality operations. This permits easier interpretation, and semantics.
For any $A, B\in \wp(S)$, the following operations can be defined: $$n(A, B) \,=\, \{b:\, (\exists a\in A \exists b\in B)\, ab =b\} \tag{normal}$$ $$o_1(A, B) \,=\, \{c:\, (\exists a\in A \exists b\in B)\, ab =c\in U_R(a, b)\setminus A \} \tag{outer-1}$$ $$o_2(A, B) \,=\, \{c:\, (\exists a\in A \exists b\in B)\, ab =c\in U_R(a, b)\setminus B \} \tag{outer-2}$$ $$i_1(A, B) \,=\, \{c:\, (\exists a\in A \exists b\in B)\, ab =c\in U_R(a, b)\cap A \} \tag{inner-1}$$ $$i_2(A, B) \,=\, \{c:\, (\exists a\in A \exists b\in B)\, ab =c\in U_R(a, b)\cap B \} \tag{inner-2}$$ $$o(A, B) = o_1(A, B)\cap o_2(A, B) \tag{outer}$$
In the above definition, the global groupoid operation has been split into multiple operations based on the relative values assumed. For any two sets $A, B\in \wp(S)$,
- [$n(A, B)$ is the set of things in $B$ that have some part or approximate part in $A$,]{}
- [$o_1(A, B$ is the set of things in the outer core determined by elements of $A\times B$ that are not in $A$,]{}
- [$o_2(A, B$ is the set of things in the outer core determined by elements of $A\times B$ that are not in $B$,]{}
- [$i_1(A, B$ is the set of things determined by elements of $A\times B$ that are in $A$,]{}
- [$i_2(A, B$ is the set of things determined by elements of $A\times B$ that are in $B$, and]{}
- [$o(A, B$ is the set of things determined by elements of $A\times B$ that are not in $A$ or $B$.]{}
These can also be read as generalizations of natural concepts of g-ideals in the context that can be defined as follows:
A subset $A$ of the groupoid ${S =\left\langle\underline{S}, \cdot \right\rangle}$ is an *g-ideal* if and only if $A$ is a subgroupoid and $$a b =b \,\&\, b\in A \longrightarrow a\in A$$ A subset $B$ of the groupoid $S$ is a *g-filter* if and only if $B$ is a subgroupoid and $$a b = b \,\&\, a\in A \longrightarrow b\in B$$
The *g-ideal generated by a subset* $A$ will be the smallest g-ideal $\mathbf{I}(A)$ containing the subgroupoid $Sg(A)$ generated by $A$. If $A$ is a singleton, then the g-ideal will be said to be *principal*. The set of all g-ideals (resp principal, finitely generated) on $S$ will be denoted by $\mathcal{I}(S)$ (resp. $\mathcal{I}_p(S)$, $\mathcal{I}_f (S)$).
If $S$ is an up-directed parthood space, then the algebra $$S^{\sharp} = \left\langle \underline{\wp(S)}, n, i_1, i_2, o_1, o_2, o, \cup, \cap, l, u, ^c, \emptyset, \wp(S) \right\rangle$$ defined above will be referred to as the *expanded up-directed parthood\
groupoidal Boolean algebra* (EUPGB)
In the context of a EUPGB $S^{\sharp}$, all of the following hold (for any $A, B\in S^{\sharp}$): $$\begin{aligned}
n(A, B) \subseteq B \tag{n}\\
o_1(A, B) \subseteq A^c \tag{o1}\\
o_2(A, B) \subseteq B^c \tag{o2}\\
i_1(A, B) \subseteq A \tag{i1}\\
i_2(A, B) \subseteq B \tag{i2}\\
o(A, B) \subseteq (A\cup B)^c \tag{o} \end{aligned}$$
- [For any $a\in A$ and $b\in B$, $ab=b$ yields $ab\in B$.]{}
- [For any $a\in A$ and $b\in B$, $ab=c\in U_R(a, b)\setminus A$ yields $ab\in A^c$. So o1 follows.]{}
- [Note that if $a\in A$, $b\in B$, and $ab\in U_R(a, b)$ then it is possible that $ab\in B$. $o_2$ ensures that this does not happen. ]{}
- [Other parts can be verified from definition.]{}
\[subsetg\] If $B\subseteq A$ in the context of the previous theorem then $$\begin{aligned}
n(A, B) = B \tag{1}\\
i_2(A, B) \subseteq i_1(A, B) \subseteq A \tag{2}\\
o_1(A, B)\subseteq o_2(A, B) \tag{3}\\
AB = B\cup i_1(A, B)\cup o_2(A, B) \tag{summary}\end{aligned}$$
Clearly the operations $n, \, i_1, \, i_2, \, o_1, \, 0_2$ and $o$ are better behaved than the groupoid operation $\cdot$.
From the above considerations, it can also be deduced that
In a EUPGB algebra $S^{\sharp}$, for any $a, b\in S$
- [$n([a],[b]) \subseteq [b]$]{}
- [$i_1([a],[b])\subseteq [a]$]{}
- [$i_2([a],[b])\subseteq [b]$]{}
- [$o_1([a],[b]) \subseteq [a]^c$ and]{}
- [$o_2([a],[b]) \subseteq [b]^c$]{}
Rough Equalities and Inequalities
---------------------------------
Particular rough equalities of natural interest are defined next.
For any $a, b\in S^*$, let $$\begin{aligned}
a\approx b \text{ if and only if } a^l = b^l \, \&\, a^u = b^u \tag{standard req}\\
a\approx_l b \text{ if and only if } a^l = b^l \tag{l-standard req}\\
a\approx_u b \text{ if and only if } a^u = b^u \tag{u-standard req}\end{aligned}$$ $\approx$, $\approx_s$, $\approx_l$, $\approx_u$ and $\tau$ will respectively be referred to as the standard, l-standard, and u-standard rough equalities respectively.
Obviously, the relations $\approx$, $\approx_s$, $\approx_l$, and $\approx_u$ are equivalences on $\wp (S)$. The meaning of the above relations is closely connected with the following rough inequalities on $\wp(S)$:
In a EUGB $H$, the following relations are definable: $$\begin{aligned}
a\sqsubseteq_l b \text{ if and only if } a^l \subseteq b^l \tag{l-rough inequality}\\
a\sqsubseteq_u b \text{ if and only if } a^u \subseteq b^u \tag{u-rough inequality}\\
a\sqsubseteq b \text{ if and only if } a\sqsubseteq_l b \, \&\, a\sqsubseteq_u b \tag{rough inequality}\end{aligned}$$
The relations $\sqsubseteq_l$, $\sqsubseteq_u$ and $\sqsubseteq$ are quasi orders on the EUGB $H$. Moreover, they are partly compatible with the operations $\cup$ and $\cap$ in the following sense: $$\begin{aligned}
(\forall a, b, c)(a\sqsubseteq_l b \longrightarrow a\cap c \sqsubseteq_l b\cap c) \tag{cs1}\\
(\forall a, b, c)(a\sqsubseteq_u b \longrightarrow a\cup c \sqsubseteq_u b\cup c) \tag{cs2}\\\end{aligned}$$
It is obvious that $\sqsubseteq_l$ is reflexive. It is transitive because for any $a, b$ and $c$, $a^l \subseteq c^l$ follows from $a^l \subseteq b^l$ and $b^l \subseteq c^l$.
In general, antisymmetry does not hold because $a^l \subseteq b^l$ and $b^l \subseteq a^l$need not imply $a=b$.
The property can be verified by considering the neighborhoods that may be included in $a$, $b$ and $c$, and observing that the neighborhoods included in $a\cap c$ must also be included in $b\cap c$.
The proof for $\sqsubseteq_u$ is analogous. For , note that neighborhoods having nonempty intersection with $a\cup c$ must also have nonempty intersection with $b\cup c$.
In the context of a EUPGB, on the quotient $\wp (S)|\approx$, the following operations can be defined.
\[eugb\] In the quotient $\wp(S)|\approx$ generated on a EUPGB $$\wp(S) = \left\langle \underline{\wp(S)}, n, i_1, i_2, o_1, o_2, o, \cup, \cap, l, u, ^c, \emptyset, \wp(S) \right\rangle,$$ the following operations can be defined $$(\forall A, B\in \wp(S))\, \breve{\alpha}([A]_{\approx}, [B]_{\approx}) \,=\, [\bigcup \{\alpha(F, H):\, F\in [A]_{\approx}\, \&\,H\in[B]_{\approx} \}]_{\approx}$$ where $\alpha$ is any of $\cdot, n, i_1, i_2, o_1, o_2, o, \cup$ and $\cap $. Further, $$(\forall A, B\in \wp(S))\, [A]_{\approx} \circledast [A]_{\approx}\,=\, \left[\bigcap \{F\cap H:\, F\in [A]_{\approx} \,\&\, H\in [B]_{\approx} \}\right]_{\approx}$$ $$(\forall A\in \wp(S))\, \neg([A]_{\approx}) \,=\, \left[\bigcup \{F^c\in [A]_{\approx} \}\right]_{\approx}$$ $$(\forall A \in \wp(S))\, L([A]_{\approx}) \,=\, \left[\bigcup \{F^l:\, F\in [A]_{\approx} \} \right]_{\approx}$$ $$(\forall A \in \wp(S))\, U([A]_{\approx}) \,=\, \left[\bigcup \{F^u :\, F\in [A]_{\approx}\}\right]_{\approx}$$
In addition, the $0$-ary operations $\bot$ and $\top$ can be defined as $[\emptyset]_\approx$ and $[\wp(S)]_\approx$ respectively.
The algebra $$Z = \left\langle \underline{\wp(S)}|\approx, \breve{n}, \breve{i}_1, \breve{i}_2, \breve{o}_1, \breve{o}_2, \breve{o}, \breve{\cup}, \breve{\cap}, L, U, \neg, \bot, \top \right\rangle$$ will be referred to as a *up-directed rough parthood algebra* (RPA).
In the above definition, the $L$ and $U$ operations are not likely to behave as modal operators, and this is consistent with the semantic intent.
If $a$ is an element of $\wp(S)|\approx$ then it can also be interpreted as a subset of $\wp(S)$, and its *representative approximations* $a_l$ and $a_u$ are $$a_l \,=\, x^l \text{ for any } x\in a \text{ and }$$ $$a_u \,=\, x^u \text{ for any } x\in a$$
All operations in Def.\[eugb\] are well defined.
The definition of each operation over $\wp(S)|\approx$ is based on forming a set of *equivalent* sets from a union and so is well defined.
In the next theorem, key relations between representatives and operations on a RPA are established.
\[repop\] If $x\in a\in \wp(S)|\approx$, then $x$ can be represented in the form $a_l \cup K$ subject to the condition $a_l \cup K^u = a_u$ and $K^l =\emptyset$.
Let $a = \{A_1, \ldots, A_n\}$ for some integer $n\leq \infty$ with $A_i \in \wp(S)$, then $$A_i = a_l \cup K_i \text{ for some set } K_i$$ Because $a_l^l \cup K_i^l \subseteq A_i^l = a_l$, it can be assumed that $K_i^l = \emptyset$ and that $K_i\cap a_l = \emptyset$.
In this situation, $A_i^u = (a_l \cup K_i)^u = a_l^u \cup K_i^u = a_u$.
In the context of Prop. \[repop\], all of the following hold: $$\begin{aligned}
(\forall a) a_u \subseteq (Ua)_u \tag{Uu}\\
(\forall a)a_l = (La)_l \subseteq (Ua)_l \tag{Ll}\\
(\forall a, b) (a\breve{\cup} b)_u = a_u \cup b_u \tag{ujoins}\\
(\forall a) (\neg a)_u \subseteq a_l^{cu} \tag{uc}\\\end{aligned}$$
Uu
: [Using the representation of $a$ in Prop. \[repop\], it follows that $a_u = (a_l \cup K_i )^u = a_l ^u \cup K_i^u $ for any $i$, while $$Ua = [\bigcup \{a_l \cup b\, : \, b\cap K_i\neq \emptyset \text{ or } b\cap a\neq \emptyset \,\&\, b\in \mathcal{G} \}]$$ So $Ua\, =\, (a_l)^u \cup \bigcup K_i^u $. So $a_u \subseteq (Ua)_u$. ]{}
Ll
: [In the same representation, $La = [(\bigcup A_i)^l] = [a_l \cup (\bigcup K_i)^l]$. So $a_l = (La)_l$. ]{}
ujoins
: [Suppose $$a\breve{\cup}b = \left[\bigcup \{X_i ; X_i \in a \text{ or } X_i \in b \} \right]$$ The $X_i$ can be written as $A_i \cup B_i = a_l \cup K_i \cup b_l \cup J_i$. Further for each $i$ $(a_l \cup K_i)^u = a_u = a_l^u \cup K_i^u$ ans similarly $b_u = b_l^u \cup J_i^u$. Using these for substituting $X_i$ results in $(a\breve{\cup} b)_u = a_u \cup b_u$.]{}
uc
: Using the same strategy as in the proof of the previous properties,
- [$\neg a = \left[ \{\bigcup A_i^c \, :\, A_i \in a^ \}\right] =$]{}
- [$= \left[ \{\bigcup (a_l \cup K_i)^c \}\right] = \left[ a_l^c \cap (\bigcup K_i^c) \right].$]{}
- [So $(\neg a)_u \subseteq a_l^{cu}$]{}
This result suggests that a better (but relatively difficult) operation on the quotient $\wp(S)|\approx$ can be $$\begin{aligned}
\mathfrak{L} a = \left[\{\bigcup \{ A : A\in a \}\}^l\right]_{\approx} \tag{bL}\\
\mathfrak{U} a = \left[\{\bigcup \{ A : A\in a \}\}^u\right]_{\approx} \tag{bU}\end{aligned}$$
It can be checked that while $\mathfrak{L} a $ is the same as $La$, but $(Ua)_l \subseteq (\mathfrak{U} a)_l$ and $(Ua)_u \subseteq (\mathfrak{U} a)_u$ in general.
If $Z$ is an RPA, then $$\begin{aligned}
(\forall a \in Z)\, a_l \sqsubseteq (La)_l \sqsubseteq (La)_u \tag{rep1}\\
(\forall a \in Z)\, a_u \sqsubseteq (Ua)_u \tag{rep2}\\
(\forall a, b \in Z)\, a_l \subseteq (a \breve{\cup} b)_l \tag{rep3}\\
(\forall a, b \in Z)\, a_u \subseteq (a \breve{\cup} b)_u \tag{rep4}\end{aligned}$$ The converse of holds if $Z$ is reflexive.
The proof depends on Thm \[luprop\], and Thm. \[lup\].
- [Suppose $a = \{b_1, b_2, \ldots , b_n \}$ with $b_i ^l = b_j^l = a_l$ for all $i,\, j$. By definition, $La = [\bigcup b_i^l]_{\approx}$. So $a_l \sqsubseteq (La)_l $. The converse holds if $Z$ is reflexive. ]{}
- [Suppose $a = \{b_1, b_2, \ldots , b_n \}$ with $b_i ^u = b_j^u = a_u$ for all $i,\, j$. By definition, $Ua = [\bigcup b_i^u]_{\approx}$. So $a_u \sqsubseteq (Ua)_u $.]{}
- [Suppose $a = \{a_1, \ldots a_v \}$ and $b = \{b_1, b_2, \ldots , b_n \}$, then $a\breve{\cup} b$ is by definition equal to $[\bigcup \{a_i\cup b_j\}]_\approx$. Since $a_i \subseteq a_i\cap b_j $ for all $i$, follows. ]{}
- [The proof of is similar to that of .]{}
Note that the following can fail to hold in general: $$\begin{aligned}
(\forall a, b \in Z)\, (a \breve{\cap} b)_l \subseteq a_l \tag{rep5}\\
(\forall a, b \in Z)\, (a \breve{\cap} b)_u \subseteq a_u \tag{rep6}\end{aligned}$$
All of the following properties hold in a RPA $Z$: $$\begin{aligned}
\breve{n}(a, a) = a \tag{n-idemp}\\
a\breve{\cup} b \,=\, b\breve{\cup} a \tag{join-comm}\\
a\breve{\cap} b \,=\, b\breve{\cap} a \tag{meet-comm}\\
a\sqsubseteq a\breve{\cup} a \tag{join-explosion}\\
a \sqsubseteq a \breve{\cap} a \tag{meet-explosion}\\
a\circledast b = b\circledast a \tag{star-comm}\\
a\sqsubseteq (a\breve{\cup} b)\breve{\cap} a \tag{abs-fail}\end{aligned}$$
Let $$Z = \left\langle \underline{\wp(S)}|\approx, \breve{n}, \breve{i}_1, \breve{i}_2, \breve{o}_1, \breve{o}_2, \breve{o}, \breve{\cup}, \breve{\cap}, L, U, \neg, \bot, \top \right\rangle$$
Directoids
----------
As mentioned in the introduction, join directoids were introduced in [@jjq90]. A better equational way of defining these is as follows:
Directoids (join) are groupoids of the form $H = \left\langle \underline{H}, \cdot \right\rangle$ that satisfy the following conditions: $$\begin{aligned}
aa = a \tag{dir1}\\
(ab)a = ab \tag{dir2}\\
b(ab) = ab \tag{dir3}\\
a((ab)c) = (ab)c \tag{dir4}\end{aligned}$$
A groupoid of the form $H = \left\langle \underline{H}, \cdot \right\rangle$ is a join directoid if and only if there exists a partial order $\leq$ on $H$ that satisfies $$\begin{aligned}
(\forall a, b) a, b \leq ab \tag{jd1}\\
(\forall a, b) (a\leq b \longrightarrow ab = ba = b) \tag{jd2}\end{aligned}$$
So it follows that a up-directed partially ordered set can be written as a groupoid and the groupoid in turn determines the partial order uniquely.
From the proposition it follows that
When an up-directed parthood space is also transitive, then a join directoid operation is definable on it (as per Equation \[updg\]).
Algebraic Semantics of Local Approximations
===========================================
If $\wp({S})^\triangle = \{A^\triangle:\, A\in \wp(S)\} $, then let $$\begin{aligned}
(\forall A, B\in \wp(S)^{\triangle})\, A\curlyvee B = A\cup B \tag{T1}\\
(\forall A, B\in \wp(S)^{\triangle})\, A\curlywedge B = (A\cap B)^{\blacktriangledown\triangle} \tag{T2}\\
(\forall \{A_j\}_{j\in J} \in \wp(S)^{\triangle})\, \curlyvee_{j\in J} A_j = \bigcup A_j \tag{ET1}\\
(\forall \{A_j\}_{j\in J}\in \wp(S)^{\triangle})\, \curlywedge_{j\in J} A_j = (\bigcap_j A_j)^{\blacktriangledown\triangle} \tag{T2}
\end{aligned}$$
\[tr1\] The algebra $\left\langle \underline{\wp(S)^{\triangle}}, \curlyvee, \curlywedge \right\rangle $ is a complete bounded lattice. The corresponding lattice order on the algebra is $\subseteq$ (induced from set inclusion on $\wp(S)$).
If $A = X^{\triangle}$ for some $X$, then $A = \bigcup_{x\in X} [x]_i$. For arbitrary collections $\{A_j\}_J$ in $\wp(S)^\triangle$, it is easy to see that $$(\forall B)(\& A_j\subseteq B \longrightarrow \, \bigcup A_j \subseteq B)$$ This ensures that the union is a complete join semilattice operation and $B = Z^{\triangle}$ for some $Z$, then $A = \bigcup_{x\in Z} [x]_i$
$$\begin{aligned}
\curlywedge_{j\in J}A_j = \bigcup \{Z\,:\, Z\in \wp(S)^{\triangle}\, \&\, Z\subseteq \bigcap_{j\in J}A_j\} = \bigcup \{\cup[x]_i:\, \cup [x]_i\subseteq \bigcap_{j\in J}\, A_j\} =\\
\bigcup \{[x]_i:\, [x]_i\subseteq \bigcap_{j\in J}\, A_j\} = (\bigcap_{j\in J} A_j )^{\blacktriangledown\triangle}\end{aligned}$$
\[tr2\] $ \left\langle \wp (S)^{\triangledown
},\subseteq \right\rangle $ is dually isomorphic to $ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle $ as a complete lattice.
Define a map $f: \wp(S)^{\triangle}\longmapsto \wp (S)^{\triangledown}$ according to $$(\forall Z \in \wp(S)^{\triangle})\, f(Z) = Z^c$$
- [Since any $Z\in\wp (S)^{\triangle}$ has the form $Z=X^{\triangle}$ for some $X\subseteq S$, so $f(Z)=X^{\triangle c}=X^{c\triangledown}\in\wp (S)^{\triangledown}$. ]{}
- [This ensures that the map $f$ is well defined.]{}
- [For any $Z_{1},Z_{2}\in\wp(S)^{\triangle}$, $$Z_{1}\subseteq Z_{2}\leftrightarrow Z_{2}^{c}\subseteq Z_{1}^{c}\leftrightarrow f(Z_{1})\supseteq f(Z_{2}).$$ From this it follows that $f$ is a dual order-isomorphism.]{}
Hence in view of Theorem \[tr1\], $ \left\langle \wp (S)^{\triangledown},\subseteq \right\rangle $ is also a complete lattice.
On the image $\wp (S)^{\blacktriangle}=\{X^{\blacktriangle}: X\subseteq
S\}$ of $\blacktriangle$, the induced relation $\subseteq$ can be associated with the following operations: $$\underset{i\in I}{\curlyvee ^*}A_{i}=\underset{i\in I}{\ \bigcup}A_{i} \tag{bt-join}$$ $$\underset{i\in I}{\curlywedge^*}A_{i}= ( \underset{i\in I}{\bigcap}A_{i}) ^{\triangledown\blacktriangle} \tag{bt-meet}$$
Note that relation of $\wp (S)^{\blacktriangle}$ to $R^{-1}$ corresponds to the relation of $\wp (S)^{\triangle}$ with $R$.
\[tr3\] $ \left\langle \wp (S)^{\blacktriangle},\subseteq \right\rangle $ and $ \left\langle \wp (S)^{\blacktriangledown},\subseteq \right\rangle $ are dually isomorphic complete lattices.
The proof is analogous to that of Theorem \[tr2\].
Definable sets in rough sets can be described in different ways. From a lattice-theoretical perspective, it is of interest to see if the set of lower or upper definable or at least the set of lower and upper approximations form distributive lattices. In this section, it is shown that the algebras formed by the set of approximations $\wp (S)^{\triangle}$, $\wp (S)^{\triangledown}$, $\wp (S)^{\blacktriangle}$, and $\wp (S)^{\blacktriangledown}$ are completely distributive lattices. It may be noted that the second author has studied these sets from a similar perspective in the context of approximations generated by tolerance relations in [@jjsr2017].
In view of Theorem \[tr3\], this condition is equivalent to the condition that the concept lattice $\mathcal{L}(S,S,I)$ is (completely) distributive. In [@gw99] several conditions equivalent to the complete distributivity of $\mathcal{L}(S,S,I)$ are formulated. For instance, the following was established:
\[th40\] A concept lattice $\mathcal{L}(G,M,I)$ is completely distributive if and only if for any object attribute pair $(g,m)\notin I$ there exists an object $h\in G$ and an attribute $n\in M$ with $(g,n)\notin I$, $(h,m)\notin I$ and such that $h\in\{k\}^{II}$, for any $k\in G\diagdown\{n\}^{I}$.
As an immediate consequence, in case of the concept lattice $\mathcal{L}(S,S,I)$ and the lattice $\wp (S)^{\triangle}$ we can formulate the following:
\[tr23\] The lattice $ \left\langle \wp(S)^{\triangle},\subseteq \right\rangle $ is completely distributive if and only if for any $a,b\in S$ satisfying $Rab$ there exist some elements $n,h\in S$ satisfying $Ran \, \&\, Rhb $ and such that for any $x\in S$ satisfying $(Rxn$ we have $[h]_i \subseteq [x]_i$. That is $$(\forall a, b) Rab \longrightarrow (\exists n, h)(\forall x) Ran\,\&\, Rhb \, \&\, [h]_i \subseteq [x]_i$$
In view of Theorem \[tr1\], $ \left\langle \wp(S)^{\triangle},\subseteq \right\rangle $ is completely distributive if and only if the concept lattice $\mathcal{L}(S,S,I)$ is completely distributive. This is equivalent to the condition formulated in Theorem \[tr2\].
- [Let $a, b\in S$ and $Rab$.]{}
- [In the context $\mathcal{L}(S,S,I)$, $Rab \leftrightarrow \neg Iab$. So the above theorem applies with $g:=a$ and $m:=b$ and there exists $n,h\in S$ with $Ran\, \&\, Rhb$ and satisfying $h\in\{x\}^{II}$, for any $x\in S\diagdown\{n\}^{I}$.]{}
- [ As $S\diagdown\{n\}^{I}=\{s\in S:\neg Isn \}$, $x\in S\diagdown\{n\}^{I}$ means that $Rxn$. Since $h\in\{x\}^{II}$ is equivalent to $[x]_i^{c}=\{x\}^{I}\subseteq\{h\}^{I}=[h]_i^{c}$, we deduce that $ Rxn$ implies $[h]_i\subseteq [x]_i$, for any $x\in S$.]{}
Therefore the condition in the present theorem is equivalent to the condition formulated in Theorem \[th40\] and the conclusion follows.
By using this theorem, two characterizations of the (complete) distributivity of $ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle $ can be deduced. Also note that it is easy to check that any completely distributive element of $ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle $ has the form $[s]_i$ (for some $s\in S$) – but the converse statement is not true in general.
\[tr24\] If the lattice $ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle $ is spatial, then the following assertions are equivalent:
i
: [The lattice $ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle $ is completely distributive.]{}
ii
: [If $[s]_i$ is an arbitrary completely join-irreducible element of $ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle $, then $$[s]_i\nsubseteqq\bigcup\{[x]_i: [x]_i\nsupseteqq [s]_i\} \tag{ei3}$$]{}
\[i\] $\Rightarrow$ \[ii\]
- [Let $[s]_i$ be a completely join-irreducible element of $ \left\langle \wp (S)^{\triangle},\subseteq
\right\rangle $.]{}
- [Assume the contrary $[s]_i\subseteq\bigcup \{[x]_i: [x]_i\nsupseteqq [s]_i\}$.]{}
- [Since the lattice $ \left\langle \wp(S)^{\triangle},\cup,\wedge \right\rangle $ is completely distributive, we have $[s]_i=[s]_i\wedge \bigcup\{[x]_i: [x]_i\nsupseteqq [s]_i\} =\bigcup\{[x]_i\wedge [s]_i: [x]_i\nsupseteqq [s]_i\}$.]{}
- [Since $[s]_i$ is a completely join-irreducible, we obtain $[s]_i=[s]_i\wedge [x]_i$, i.e. $[s]_i\subseteq [x]_i$, for some $x\in S$ with $[x]_i\nsupseteqq [s]_i$ – a contradiction.]{}
- [This proves the implication.]{}
\[ii\] $\Rightarrow$ \[i\]
- [Assume that (ii) holds, and let $Rab$ for some $a,b\in S$. Then $b\in [a]_i$.]{}
- [If $[a]_i$ is completely join-irreducible, then in view of (ii), there exists an element $n\in [a]_i\setminus \bigcup\{[x]_i:\, [x]_i\nsupseteqq [a]_i\}$. If we set $h:=a$, then $Ran \,\&\, Rhb$.]{}
- [For any $k\in S$ satisfying $Rkn$, $n\in [k]_i$ excludes the case $[k]_i\nsupseteqq [a]_i$, hence we obtain $[k]_i\supseteq [a]_i= [h]_i$.]{}
- [Now suppose that $[a]_i$ is not completely join-irreducible. Then $b\in [a]_i=\bigcup\{[p]_i:\, [p]_i\in\ CJ(\wp (S)^{\triangle})\}$, and this yields $b\in [p]_i$ for some completely join-irreducible element $[p]_i)$ of $\wp (S)^{\triangle}$ with $[p]_i\subseteq [a]_i$. Further $Rpb$ and $$[p]_i\nsubseteqq\bigcup\{[x]_i: [x]_i\nsupseteqq [p]_i\}$$]{}
- [Therefore there exists an element $n\in [p]_i\setminus \bigcup\{[x]_i: [x]_i\nsupseteqq R(p)\} \subseteq [a]_i$ and hence we get $Ran$. Set $h:=p$. This yields $Rhb$.]{}
- [For any $k\in S$ satisfying $Rkn$, $n\in [k]_i$ and $n\notin \bigcup\{[x]_i: [x]_i\nsupseteqq [p]_i\}$ exclude $[k]_i\nsupseteqq [p]_i$. Hence we obtain $[h]_i\subseteq [k]_i$.]{}
From this it follows that the lattice $ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle $ is completely distributive.
Replacing the relation $R$ with $R^{-1}$ in the above theorem we obtain:
\[cor2.5\] If the lattice $ \left\langle \wp(S)^{\blacktriangle},\subseteq \right\rangle$ is spatial, then the following assertions are equivalent:
1. [The lattice $ \left\langle \wp (S)^{\blacktriangle},\subseteq \right\rangle $ is completely distributive.]{}
2. [If $R^{-1}(s)$ is a completely join-irreducible element of $\left\langle \wp (S)^{\blacktriangle},\subseteq \right\rangle$, then $$[s]\nsubseteqq\bigcup\{[x]:\, [x] \nsupseteqq [s] \} \tag{ei4}$$]{}
\[cor2.6\] Let $R$ be a reflexive antisymmetric relation. Then
i
: [$ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle$ is completely distributive if and only if $R$ is transitive.]{}
ii
: [$ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle $ is completely distributive if and only if $ \left\langle \wp(S)^{\blacktriangle},\subseteq \right\rangle$ is completely distributive.]{}
- [If $R$ is a reflexive and antisymmetric relation, then in view of Theorem \[tr23\], $ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle $ and $ \left\langle \wp (S)^{\blacktriangle},\subseteq \right\rangle $ are spatial lattices and for any $s\in S$, $[s]_i$ is a completely join-irreducible element of $\wp (S)^{\triangle}$, and $[s]$ is completely join-irreducible in $\wp (S)^{\blacktriangle}$.]{}
- [Therefore, in view of Theorem \[tr24\] and Theorem \[cor2.5\], the relations (ei3) and (ei4) are satisfied for all $s\in S$.]{}
i
: - [Suppose that $ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle $ is completely distributive, and let $Rus \,\&\, Rsv$.]{}
- [Then $[s]_i\nsubseteqq\bigcup\{[x]_i: [x]_i\nsupseteqq [s]_i\}$, according to (ei3).]{}
- [Let $n\in [s]_i\setminus \bigcup\{[x]_i: [x]_i\nsupseteqq [s]_i\}$, then $Rsn$ and $n\in [n]_i$ implies that $[s]_i\subseteq [n]_i$.]{}
- [Since $s\in [s]_i$, we also get $Rns$. By the antisymmetry of $R$ we obtain $n=s$. Hence $s\in [s]_i\setminus \bigcup\{[x]_i: [x]_i\nsupseteqq [s]_i\}$.]{}
- [Since $s\in [u]_i$, we have $[s]_i\subseteq [u]_i$. As $v\in [s]_i$ (by assumption), we obtain $v\in [u]_i
\,\&\, Ruv$. This proves the transitive property of $R$.]{}
- [Conversely, if it is assumed that $R$ is transitive, then $R$ is a partial order, and by the result in [@jrv09] $\wp (S)^{\triangle}$ and $\wp(S)^{\blacktriangle}$ are completely distributive lattices.]{}
ii
: - [Assume that $ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle $ is completely distributive. Then, in view of (i) $R$ and $R^{-1}$ are partial orders.]{}
- [Then by [@jrv09] $ \left\langle \wp (S)^{\blacktriangle},\subseteq \right\rangle $ is also completely distributive.]{}
- [The proof of the converse implication is completely analogous.]{}
By applying the above definitions and Proposition 5, we obtain:
Let $(S,R)$ be a directed relational system and $ \left\langle B(S),\cdot \right\rangle $ the corresponding groupoid. Then $ \left\langle \wp (S)^{\triangle},\subseteq \right\rangle $ is completely distributive if and only if $$\begin{gathered}
(\forall a, b\in S)(ab= b \longrightarrow (\exists n, h\in S)(\forall k, s\in S)\\ \, an=n \,\&\, hb= b \, \&\, kn =n\, \& (hs = s \rightarrow ks =s) ) \tag{triagrp}\end{gathered}$$
Knowledge Perspective {#seckno}
=====================
In a general approximation space $S$, if $R$ is an equivalence, a partial order or a quasi order, then it is also possible to associate other groupoidal operations (see [@amdsc2016; @am501; @am5019; @am909]) on $S$. This is discussed in brief in Sec.\[apprsp\]. But the associated operation is distinct from the one considered in this paper.
General and classical rough sets have been associated with concepts of knowledge and studied from that perspective in a number of papers by the first author [@am9114; @am9006; @am9501; @am9969; @am99] and others[@zpb; @zp6; @ppm2; @chp3; @bgc12]. The basic idea in the context of classical approximation spaces [@zpb] is to associate definite objects with concepts and consequently the equivalence relation $R$ is associated with knowledge. In more general situations, granularity has a bigger role to play, and knowledge is defined relative to granular axioms used and other desirable properties. Examples of such conditions are
GK1
: [Individual granules are atomic units of knowledge.]{}
GK2
: [If collections of granules combine subject to a concept of mutual independence, then the result would be a concept of knowledge. The ’result’ may be a single entity or a collection of granules depending on how one understands the concept of *fusion* in the underlying mereology.]{}
GK3
: [Maximal collections of granules subject to a concept of mutual independence are admissible concepts of knowledge.]{}
GK4
: [Parts common to subcollections of maximal collections may be interpreted as knowledge.]{}
GK5
: [All stable concepts of knowledge consistency should reduce to correspondences between granular components of knowledges. In particular, two relations $R_1$ and $R_2$ may be said to be *consistent* if and only if the set of granules associated with the two general approximation spaces have bijective correspondence. ]{}
In [@am99] and [@dtl2017] choice operations over granules are involved. But they do not generate groupoid operations on the general approximation space itself. Neither do the granular knowledge axioms of the kind mentioned. All this means that the groupoid operation provides an additional layer of decision making that needs to integrated with existing work. A concrete practical example is considered next to illustrate key aspects of this.
Applications to Student Centred Learning
----------------------------------------
In student-centered learning students are put at the center of the learning process, and are encouraged to learn through active methods. Arguably, students become more responsible for their learning in such environments. In traditional teacher-centered classrooms, teachers have the role of instructors and are intended to function as the only source of knowledge. By contrast, teachers are typically intended to perform the role of facilitators in student-centered learning contexts. A number of best practices for teaching in such contexts [@jrp2016] have evolved over time. Teachers need to constantly improve their methods in such teaching contexts because that is part of the methodology.
Because of the open-ended aspect of the learning process, it is not expected that teachers have absolute control over the concepts learned. Students may themselves arrive at new methods of solution or define new concepts as part of the learning process. In this scenario it is of interest to suggest potential higher concepts that relate to the progress of the work in question. Teachers can possibly provide some initial suggestions and subsequently these can be worked upon by algorithms relying upon datasets of concepts for improved suggestions. From the perspective of this research this becomes the problem of construction of the best groupoid operations.
In more precise terms,
L1
: [Let $A$ and $B$ be two concepts arrived at by the learner. The open-ended nature of the learning process means that a general rough set model of concepts must be adaptive or permit supervision. ]{}
T1
: [Teacher observes that concept $C$ among others contains $A$ and $B$ in some sense, and offers suggestions relating to the scenario.]{}
S1
: [Software aid for the learning context provides better suggestions based on and using a groupoidal decision model instead of the former alone. In general available strategies that can be used to arrive at suggestions based on alone are likely to be unintelligent.]{}
It may be noted that the impact of AI on enhancing classroom learning and learning in general has been very limited (see [@chos2018] and related references). In fact digital technology in the context of mathematics teaching has been stagnating because most of the effort has been on non-intelligent software that merely aid communication. There is no dearth of motivation for such work – Often teachers do not have sufficient knowledge about the working of their students mind, have an excess of work load at hand and may be suffering from cognitive dissonances of specific types.
In a forthcoming paper by the first author, the rough methodology suggested in this subsection is applied to specific practices such as opening of exercises in the context of mathematics teaching [@sko2011; @milani2019], use of explicit mathematical language [@usz2012], and software for student expression [@alp2019; @chos2018].
Further Directions and Remarks
==============================
In this research
- [the concepts of up-directed and up-directed parthood approximation\
spaces are invented,]{}
- [their potential role in weak decision making is illustrated, ]{}
- [algebraic semantics of sets of granular, nongranular and local approximations are invented and investigated in depth and shown to be nonequivalent, ]{}
- [algebraic semantics of roughly equivalent objects that involve additional groupoidal operations of decision making are invented and investigated,]{}
- [their connection with knowledge and formal concept analysis are explored, and]{}
- [possible applications to student centred learning is proposed.]{}
The results on connection with FCA supplement the work in [@jjs2014].
Parthood and apparent parthood relations have been the focus in higher order granular approaches to rough sets in a number of papers by the first author [@am9114; @am9969; @am9006; @am501; @am9222]. The results of this paper motivate connections between those and the lower order approach of this paper. Specifically it is of interest to identify the cases that are representable in terms of lower order semantics. The groupoidal approach of this paper is also extended to the higher order approaches in a separate paper.
A groupoid $S$ is *tolerance trivial* if every definable compatible tolerance on it is a congruence. Key results can be found in [@sandor1991; @chtol1991]. This concept extends to all algebras including the AR, AP, EUPGB and algebras of local approximations. In relation to knowledge interpretation, tolerance triviality amounts to a *self organizing* aspect of knowledge. In other words, much less computational effort would be required to impose an interpretation on the semantics. This aspect is also explored in concrete terms by the first author in a forthcoming paper in the frameworks proposed in this research.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a study of thermoelectric coefficients in CeCoIn$_{5}$ down to 0.1 K and up to 16 T in order to probe the thermoelectric signatures of quantum criticality. In the vicinity of the field-induced quantum critical point, the Nernst coefficient $\nu$ exhibits a dramatic enhancement without saturation down to lowest measured temperature. The dimensionless ratio of Seebeck coefficient to electronic specific heat shows a minimum at a temperature close to threshold of the quasiparticle formation. Close to $T_{c}(H)$, in the vortex-liquid state, the Nernst coefficient behaves anomalously in puzzling contrast with other superconductors and standard vortex dynamics.'
author:
- 'K. Izawa$^{1,2,3}$, K. Behnia$^{4}$, Y. Matsuda$^{3,5}$, H. Shishido$^{5,6}$, R.Settai$^{6}$, Y. Onuki$^{6}$ and J. Flouquet$^{2}$'
title: 'Thermoelectric response near a quantum critical point: the case of CeCoIn$_{5}$ '
---
CeCoIn$_{5}$ is an unconventional superconductor with an intriguing normal state[@petrovic]. Its behavior is peculiar near the upper critical field, where the energy scale governing various electronic properties is vanishingly small and increases with increasing magnetic field[@paglione; @bianchi], a behavior expected in presence of a Quantum Critical Point(QCP)[@stewart]. The proximity of this QCP to the upper critical field in CeCoIn$_{5}$ is puzzling[@bauer; @ronning; @paglione2]. The possible existence of a FFLO state[@fflo] and/or an elusive magnetic order are a subject of recent intense research. On the other hand, even in the absence of magnetic field, the normal state presents strong deviation from the standard Fermi-liquid behavior[@petrovic; @nakajima]. The application of pressure leads to the destruction of superconductivity and the restoration of the Fermi liquid[@sidorov; @ronning; @nakajima2]. The link between the field-induced and the pressure-induced routes to the Fermi liquid is yet to be clarified.
During the last three years, the anomalous properties of CeCoIn$_{5}$ near the field-induced QCP have been reported thanks to measurements of specific heat[@bianchi], electric resistivity[@paglione], thermal transport[@paglione2] and Hall effect[@singh]. In this paper, new insight on the quantum criticality is given via thermoelectric response down to 0.1K. As far as we know, this is the first experimental investigation of the thermoelectric tensor in the vicinity of a QCP, a subject of several theoretical studies[@paul; @miyake; @podolsky]. Single crystals were grown by self-flux method. Thermoelectric coefficients were measured with one heater and two RuO$_{2}$ thermometers in magnetic field along $c$-axis. The heat current was applied along the basal plane. Previous studies of thermoelectricity in CeCoIn$_{5}$ detected a large Nernst coefficient and a field-dependent Seebeck coefficient in the non-Fermi liquid regime above $T_{c}$[@bel] and an additional field scale at 23 T[@sheikin]. Here, we find that the most spectacular thermoelectric signature of quantum criticality is a drastic enhancement of the Nernst coefficient, $\nu$. The vanishingly small Fermi energy, which was previously detected by a nearly diverging enhancement of the $A$ coefficient of resistivity ($\rho=\rho_{0}+AT^{2}$)[@paglione] and the Sommerfeld coefficient of specific heat ($\gamma=C_{el}/T$)[@bianchi], leads also to an apparently diverging $\nu/T$. These results show two distinct anomalies close to $H_{c2}(0)$ and $T_{c}(0)$ which are different in the origin. This conclusion cannot be derived from other probes mentioned above. We also find a milder enhancement of the Seebeck coefficient near the QCP. Moreover, the ratio of thermopower to electronic specific heat, expressed in appropriate units[@behnia], remains close to unity even in the vicinity of the QCP. The temperature dependence of this ratio presents a minimum at a temperature roughly marking the formation of well-defined quasi-particles[@paglione2].
Figure 1 presents the data obtained by measuring the Nernst and the Seebeck coefficients at various magnetic fields. Since the thermoelectric response of Fermions is expected to be $T$-linear well below their Fermi temperature, what is plotted in the figure is the temperature dependence of the two coefficients divided by temperature. As seen in fig. 1(a), the Seebeck coefficient, $S$ vanishes in the superconducting state. In the normal state, $S/T$ increases with decreasing temperature for all fields. For fields exceeding 5.4 T, the normal state extends down to zero temperature and a finite $S/T$ in the zero-temperature limit can be extracted. For a field of 16 T (which is well above the quantum critical region) $S/T$ saturates to a value of about 13 $\mu$VK$^{-2}$. For fields between 5.4 T and 16 T, $S/T$ presents a non-monotonous temperature dependence. An upturn below 0.15 K is visible for $\mu_{0}H \simeq$ 5.5 T (i.e. in the vicinity of the QCP) curves. Note that this upturn leads to a moderate enhancement of $S/T$. The overall change in the magnitude of $S/T$ is about 70 %. On the other hand, the temperature dependence of the Nernst coefficient divided by temperature $\nu/T$ reveals a more dramatic signature of Quantum criticality. As seen in figure 1(b), for $\mu_{0}H=$ 5.5 T and $\mu_{0}H=$ 6 T, below 1 K, $\nu/T$ is steadily increasing with decreasing temperature. No such enhancement occurs for $\mu_{0}H=$16 T, far above QCP. At the lowest measured temperature ($\sim$ 0.1 K), $\nu/T$ is five-fold enhanced near the QCP ($\sim$6 T) compared to its 16 T value. Since the thermal Hall conductivity $\kappa_{xy}$ in CeCoIn$_{5}$ becomes large at low temperatures due to enhancement of the mean-free-path of the electrons [@kasahara], the transverse thermal gradient $\nabla_{y} T$ could generate a finite transverse electric field $E_{y}$. Therefore, the (measured) adiabatic and the (theoretical) isothermal Nernst coefficients are not identical in CeCoIn$_{5}$. However, using the value of $|\nabla_{y} T|/|\nabla_{x} T| \sim$ 0.1 at 5.2 T reported in Ref. [@kasahara], the difference between these two is estimated to be about 10 %, indicating that the observed enhancement is not due to a finite $\nabla_{y} T$. We will argue below that this enhancement reflects a concomitant decrease in the magnitude of the normalized Fermi energy as previously documented by specific heat and resistivity measurements.
The thermoelectric response of CeCoIn$_{5}$ in the vicinity of QCP can be better understood by complementing our data with the information extracted by other experimental probes[@bianchi; @paglione], which originally detected a quantum critical behavior near $H_{c2}$. In particular, an interesting issue to address is the fate of the correlation observed between thermopower and specific heat of many Fermi liquids in the zero-temperature limit In a wide range of systems, the dimensionless ratio linking these two is of the order of unity ($q=\frac{S N_{A} e}{T\gamma}\simeq\pm 1$, with $\gamma=C_{el}/T$, $N_{A}$ the Avogadro number and $e$ the charge of electron)[@behnia]. What happens to such a correlation at a quantum critical point? Combining the specific heat data reported by Bianchi *et al.*[@bianchi] with our thermopower results allows us to address these questions. Fig. 2(a) presents $q$ computed in this way as a function of temperature. The first feature to remark is that $q$ remains of the order of unity even in the quantum critical region. Note that, theoretically, this correlation arises because $S/T$ and $\gamma$ are both inversely proportional to the normalized Fermi energy and thus $q$ is expected to be of the order of (and $not$ rigorously equal to) unity[@miyake]. According to our result ($q\simeq 0.9$ at 6 T and 0.1 K), this correlation holds even when the normalized Fermi Energy becomes vanishingly small. The second feature of interest in figure 2(a) is the temperature dependence of $q$, which presents a minimum. For both fields, the temperature at which this minimum occurs is close to the one where the Lorenz number($L=\frac{\kappa}{\sigma T}$) linking thermal, $\kappa$, and electric,$\sigma$, conductivities present also a minimum. Paglione and co-workers, who report this latter feature, argue that this temperature marks the formation of well-defined quasi-particles[@paglione2]. This is a temperature below which both thermal and electric resistivities display a $T^{3/2}$ temperature dependence. Remarkably, Miyake and Kohno, who provided a theoretical framework in a periodic Anderson model for the correlation between thermopower and specific heat, predicted that $q$ should deviate downward from unity in presence of an antiferromagnetic (AF) QCP leading to hot lines on the Fermi surface[@miyake].
We now turn to the Nernst coefficient. In a simple picture, it is proportional to the energy derivative of the relaxation time at the Fermi energy[@sondheimer]. In a first approximation, it tracks a magnitude set by the cyclotron frequency, the scattering time and the Fermi energy[@behnia2]. Since it scales inversely with the Fermi Energy, there is no surprise that it becomes large in heavy-fermion metals[@bel; @sheikin] and in particular in heavy-Fermion semi-metals[@bel2; @pourret], where both the heavy mass of electrons and the smallness of the Fermi wave-vector contribute to its enhancement($\nu/T \propto
1/(k_{F}\epsilon_{F}$)). Now, since the Fermi energy (broadly defined as the characteristic energy scale of the system) becomes very small near a QCP, one would expect a large Nernst coefficient in agreement with the experimental observation reported here.
With these phenomenological considerations in mind let us compare the behavior of the Nernst coefficient with specific heat and resistivity. Both $\gamma$ and $A$, the $T^2$ term of the resistivity ($\rho=\rho_{0}+AT^{2}$) inversely scale with the Fermi Energy, $\epsilon_{F}$. Therefore, both are enhanced when the Fermi energy is small. Since these two quantities are linked by the Kadowaki-Woods relation ($\gamma^{2}\propto A $), the enhancement is more pronounced in $A$ than in $\gamma$. Figure 2(b) compares the field-dependence of $A^{1/2}$, $\gamma$ and $\nu/T$. In a naive picture, the enhancement of the three quantities are comparable in magnitude. This quantitative correlation suggests that the main reason for the enhancement of $\nu/T$ near QCP is due to a small $\epsilon_{F}$. It is instructive to trace a contour plot of this quantity in the temperature-field plane. This is done in Fig. 3 with a logarithmic color scale in order to enhance the contrast. Note that contrary to the other probes, there is no need to subtract an offset from the Nernst data. In the case of specific heat, one should subtract the Schottky contribution at low temperature[@bianchi] and high-field, and the phonon contribution at high temperature. In the case of resistivity the $T^2$ behavior is interrupted at low temperature and high-fields by an upturn due to the temperature-dependent magnitude of $\omega_{c}\tau$[@paglione]. As seen in Fig. 3, $\nu/T$ becomes very large near the QCP, which constitutes the main hearth of the figure. However, there is a second one at zero field just above $T_c$, which was identified by previous measurements[@bel]. This zero-field hot region corresponds to a purely linear resistivity and anomalously enhanced Hall coefficient[@nakajima] due to strong anisotropic scattering by AF fluctuations[@nakajima2], which can also enhance the Nernst coefficient[@kontani]. On the other hand, close to the QCP, the magnitude of Hall coefficient[@singh] is comparable to its value at room-temperature or in LaCoIn$_{5}$[@nakajima2]. Therefore, there appears to be two distinct sources for the enhancement of the Nernst coefficient. In the zero-field regime just above $T_c$, it is enhanced mostly because of strong inelastic scattering associated with AF fluctuations, but in the zero-temperature regime just above $H_{c2}$, it becomes large because of the smallness of the Fermi energy. The occurrence of superconductivity impedes to explore the route linking together these two hot regions of the (B,T) plane. The inset of the figure compares the evolution of energy scales detected by different experimental probes near the QCP.
We now turn to the puzzling behavior of the Nernst coefficient in the vicinity of the superconducting transition. Deep into the superconducting state, there is no measurable Nernst signal, as illustrated by the existence of the black area in Fig. 3. On the other hand, close to $H_{c2}$(T) (or alternatively, near $T_{c}(H)
$), vortices can move and an additional contribution to the Nernst signal is expected. In the entire range of our study, the Nernst coefficient keeps the same sign which is presented in the inset of Fig. 1. Such a Nernst coefficient is negative according to a textbook convention on the sign of the thermoelectric coefficients[@nolas]. However, the literature on the vortex Nernst effect[@wang] usually takes for positive the Nernst signal generated by vortices moving from hot to cold, which leads to an opposite convention. The sign of the Nernst effect in CeCoIn$_{5}$ is negative according to the textbook convention[@nolas], but positive according to the vortex one[@wang; @huebener]. Indeed, contrary to quasi-particles, the Nernst signal produced by vortices should have a fixed sign. A thermal gradient $\nabla_{x} T$ generates a force on a vortex because its core has an excess of entropy. The direction of this force is thus thermodynamically determined; vortices move along the thermal gradient from hot to cold region. The orientation of electric field is also unambiguously set by the direction of the vortex movement and the vortex Nernst signal is not expected to have an arbitrary sign. In order to separate the vortex and the quasi-particle contributions to the Nernst signal, we put under careful scrutiny the effect of superconducting transition on three coefficients : $\rho(T)$, $S(T)$ and $N(T)$ . As illustrated in fig. 4(a) and 4(b), with the onset of superconductivity, the Nernst signal, $N$, collapses faster than both resistivity and the Seebeck coefficient. This robust feature was observed for *all* magnetic fields. On the other hand, the collapse in $\rho(T)$ and $S(T)$ closely track each other. This latter feature, which was also observed in cuprates[@huebener], suggests that the Seebeck response is essentially generated by quasi-particles. Therefore, the most natural assumption regarding their contribution to the Nernst signal in the vortex liquid regime is that $N_{qp}(T)$ also follows $\rho(T)$ and $S(T)$ and the vortex contribution to the Nernst signal can be obtained by subtracting the normalized Seebeck coefficient off the normalized Nernst one. Fig. 4(c) and 4(d) show that this procedure clearly resolves a signal of opposite sign. Thus, the most straightforward interpretation of the faster collapse of $N(T)$ implies an additional source of Nernst signal in the vortex liquid regime with a sign *opposite* to the predominant one and also to the one expected for vortices moving along the heat flow.
This result appears incompatible with the standard picture of vortex dynamics driven by a thermal gradient. However, one shall not forget that additional forces on vortices besides thermal force may be present. CeCoIn$_{5}$ is distinguished from other superconductors by the possible occurrence of an anti-ferromagnetic state in the normal core of its vortices. This feature could decrease the entropy excess of the vortices and reduce the intensity of the thermal force, which can therefore be vanquished by another source of vortex movement. As first noted by Ginzburg[@ginzburg], in a superconductor subject to a thermal gradient, a quasi-particle current (which carry heat) and a supercurrent (which does not) counterflow in order to keep the charge current zero[@huebener]. In ordinary conditions, this counterflow generates a transverse Magnus force on vortices[@ri]. Its role in the context of superclean CeCoIn$_{5}$[@kasahara] needs an adequate theoretical treatment.
Another remarkable feature of Fig. 4 is the presence of a small shoulder in the temperature dependence of the Nernst effect at the end of the transition. The shoulder is present in an extended range of magnetic fields and only disappears in the proximity of $H_{c2}$. There seems to be a narrow temperature window, where a thermal gradient can create a transverse electric field, but a current does not produce any electric field. The simplest explanation for such a discrepancy would imply a threshold force to depin vortices, $f_{dp}$ attained by the applied temperature gradient, but not by the applied current. However, this feature was found to be robust and no change was detected by modifying the magnitude of the applied thermal gradient. Clearly, the sign and the fine structure of the Nernst effect in the vortex liquid regime of CeCoIn$_{5}$ need further investigation.
We thank J-P. Brison, H. Kontani, N. Kopnin, K. Maki and K. Miyake for helpful discussions and specially N. P. Ong for his illuminating input on the sign of the Nernst effect. K.I. acknowledges a European Union Marie Curie fellowship. This work was supported by the Agence Nationale de la Recherche through the ICENET project.
[99]{} C. Petrovic *et al.*, J. Phys. Condens. Matter **13**, L337 (2001). J. Paglione *et al.*,Phys. Rev. Lett. **91**, 246405 (2003). A. Bianchi *et al.*, Phys. Rev. Lett. **91**, 257001 (2003). G. R. Stewart, Rev. Mod. Phys. **73**, 797 (2001). E. D. Bauer *et al.*, Phys. Rev. Lett. **94**, 047001 (2005). F. Ronning *et al.*, Phys. Rev. B **73**, 064519 (2006). J. Paglione *et al.*,Phys. Rev. Lett. **97**, 106606 (2006). Y. Matsuda and H. Shimahara, J. Phys. Soc. Jpn **76**, 051005 (2007). Y. Nakajima *et al.*, J. Phys. Soc. Jpn. **73**, 5 (2004). V. A. Sidorov *et al.*, Phys. Rev. Lett. **89**, 157004 (2002). Y. Nakajima *et al.*, J. Phys. Soc. Jpn. **76**, 024703 (2007). Y.Kasahara [*et al.*]{}, Phys. Rev. B [**72**]{}, 214515 (2005). S. Singh *et al.*, Phys. Rev. Lett. **98**, 057001 (2007). R. Bel *et al.*, Phys. Rev. Lett. **92**, 217002 (2004). I. Sheikin *et al.*, Phys. Rev. Lett. **96**, 077207 (2006). I. Paul and G. Kotliar, Phys. Rev. B **64**, 184414 (2001). K. Miyake and Kohno, J. Phys. Soc. Jpn. **74**, 254 (2005). D. Podolsky *et al.*, Phys. Rev. B **75**, 014520 (2007). K. Behnia *et al.*, J. Phys.: Condens. Matter **16**, 5187 (2004). E. H. Sondheimer, Proc. R. Soc. London, Ser. A 193, **484**(1948). K. Behnia *et al.*, Phys. Rev. Lett. **98**, 076603 (2007). R. Bel *et al.*, Phys. Rev. B **70**, 220501(R) (2004). A. Pourret *et al.* Phys. Rev. Lett. **96**, 176402 (2006). H. Kontani, Phys. Rev. Lett. **89**, 237003 (2002). G. S. Nolas *et al.*, Thermoelectrics, Springer (2001). Y. Wang *et al.*, Phys. Rev. B **73**, 024510 (2006). R. P. Huebener, Supercond. Sci. Technol. **8**, 189 (1995). V. L. Ginzburg, Sov. Phys. Usp. **34**, 101(1991). H. -C. Ri *et al.*, Phys. Rev. B **47**, 12312 (1993).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $f=ax+x^{r(q-1)+1}\in \mathbb{F}_{q^2}^*[x], r\in \{5,7\}.$ We give explicit conditions on the values $(q,a)$ for which $f$ is a permutation polynomials of $\mathbb{F}_{q^2}.$'
address: 'Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620'
author:
- 'Stephen D. Lappano'
title: 'A Note Regarding Permutation Binomials over $ \mathbb{F}_{q^2}$'
---
Introduction
============
A polynomial $f \in \mathbb{F}_q[x]$ is called a *permutation polynomial* (PP) if the mapping $\texttt{x} \mapsto f(\texttt{x})$ induces a permutation on $ \mathbb{F}_q$. Permutation polynomials taking simple algebraic forms are particularly interesting. While permutation monomials are quite obvious, the situation for permutation binomials is not as well understood. As a result much research has been concerned with finding, classifying and understanding permutation binomials. The main results of the present paper are the following theorems.
Let $f(x)=ax+x^{5q-4}\in\mathbb{F}_{q^{2}}[x]$. Then $f$ is a PP of $\mathbb{F}_{q^{2}}$ iff one of the following occurs:
1. $q=2^{4k+2}$ and $a^{\frac{q+1}{5}}\neq1$ is a fifth root of unity
2. $q=3^{2}$ and $a^{2}$ is a root of $(1+x) (1+x^2)(2+x+x^2)(2+2 x+x^2) (1+x+x^2+x^4) (1+x^2+x^3+x^4) (1+2 x+x^2+2 x^3+x^4)$
3. $q=19$ and $a^{4}$ is a root of $(1+x)(2+x)(3+x)(4+x)(5+x)(9+x)(10+x)(13+x)(17+x)(16+3x+x^{2})(1+4x+x^{2})(6+18x+x^{2})$
4. $q=29$ and $a^{6}\in\{15,18,22,23\}$
5. $q=7^{2}$ and $a^{10}$ is a root of $(1+4x+x^{2})$
6. $q=59$ and $a^{12}$ is a root of $(4+x)(55+x)(x^{2}+36)$
7. $q=2^{6}$ and $a^{13}$ is a root of $(1+x+x^2)(1+x+x^3)$
Let $f(x)=ax+x^{7q-6}\in\mathbb{F}_{q^{2}}[x]$. Then $f$ is a PP of $\mathbb{F}_{q^{2}}$ iff one of the following occurs:
1. $q=13$ and $a^2$ is a root of $(1+x) (2+x) (3+x) (4+x) (5+x) (6+x) (7+x) (8+x) (9+x) (10+x) (11+x) (12+x+x^2) (9+2 x+x^2) (10+3 x+x^2) (9+4 x+x^2) (12+4 x+x^2) (10+5 x+x^2) (3+6 x+x^2) (1+7 x+x^2) (4+7 x+x^2) (1+8 x+x^2) (12+9 x+x^2) (1+10 x+x^2) (3+12 x+x^2) (4+12 x+x^2) (12+12 x+x^2)$
2. $q=3^3$ and $a^4$ is a root of $(2+x+x^2+x^3) (1+2 x+x^2+x^3) (1+x+2 x^2+x^3) (2+2 x+2 x^2+x^3) (1+2 x+x^2+2 x^3+x^4+2 x^5+x^6)$
3. $q=41$ and $a^6$ is a root of $(9+x) (10+x) (26+x) (30+x) (32+x) (34+x) (35+x) (37+x) (39+2 x+x^2) (1+14 x+x^2) (20+40 x+x^2)$
It is worth noting the polynomial $f$ in Theorems 1.1 and 1.2 can be written as $f=xh(x^{q-1})$, where $h(x)=a+x^r$. Zieve [@Zieve] considered polynomials of this type and under the assumption $a\in \mathbb{F}_{q^2}$ is a (q+1)st root of unity Theorem 1.1 (1) follows from \[5,Cor 5.3\]. The approach of this paper is quite different from that of [@Zieve], as no restrictions are placed on the nature of $a \in \mathbb{F}_{q^2}$. We employ a method similar to that of [@Hou1; @Hou2] and expand the sum $\sum_{x \in \mathbb{F}_{q^2}}f(x)^s$. Surprisingly, considering only a few values of $s$ give explicit results. In [@Hou; @and; @Lappano] permutation binomials of the form $f=ax+x^{3q-2}$ were completely characterized. The present paper extends this result to polynomials of the form $f=ax+x^{5q-4}$ and $f=ax+x^{7q-6}$. We conclude with a conjecture regarding the behavior of $f=ax+x^{r(q-1)+1}$ for odd primes $r$.
Computations
============
Let $f=ax+x^{5q-4}\in \mathbb{F}_{q^2}^*[x]$ and let $0 \leq \alpha , \beta \leq q-1$. We compute
$$\begin{aligned}
\label{2.1}
\sum_{x\in \mathbb{F}_{q^2}}f(x)^{\alpha +\beta q}
&= \sum_{x\in \mathbb{F}_{q^2}}(ax+x^{5q-4})^\alpha (a^q x^q + x^{5-4q})^\beta \notag \\
&= \sum_{x\in \mathbb{F}_{q^2}} \sum_{i,j} \binom{\alpha}{i} (ax)^{\alpha -i}x^{(5q-4)i} \binom{\beta}{j}(a^q x^q )^{(\beta-j)}x^{(5-4q)j} \notag \\
&=a^{\alpha+\beta q} \sum_{i,j} \binom{\alpha}{i} \binom{\beta}{j} a^{-i-jq} \sum_{x\in \mathbb{F}_{q^2}} x^{\alpha + \beta q +5(q-1)(i-j)}.\end{aligned}$$
It is clear the inner sum is $0$ unless $\alpha + \beta q \equiv (\text{ mod } q-1)$, thus $\alpha +\beta = q-1$.
With $0 \leq \alpha \leq q-1$ and $\beta = q-1-\alpha$ (2.1) becomes $$-a^{(\alpha+1)(1-q)} \sum_{-\alpha -1 +5(i-j) \equiv 0 (\text{mod} q+1)} \binom{\alpha}{i} \binom{q-1-\alpha}{j} a^{-i-jq}$$ Since $0 \leq i \leq q-1$ and $0 \leq \beta \leq q-1-\alpha$ it follows $$4\alpha +4 - 5q \leq -\alpha -1 +5(i-j) \leq 4\alpha -1$$ Define $\Gamma(q,\alpha):= \{n \in (q+1)\mathbb{Z} : 4\alpha +4 - 5q \leq n \leq 4\alpha -1 \}$. Now we have $$\sum_{x\in \mathbb{F}_{q^2}}f(x)^{\alpha +\beta q} = -a^{(\alpha +1)(1-q)}\Lambda(q,\alpha,a)$$ where $$\Lambda(q,\alpha,a)= \sum_{-\alpha -1 +5(i-j)\in \Gamma(q,\alpha)} \binom{\alpha}{i} \binom{q-1-\alpha}{j} a^{-i-jq}$$ Now Hermite’s criterion implies $f$ is a PP of $\mathbb{F}_{q^2}$ if and only if $0$ is the only root of $f$ in $\mathbb{F}_{q^2}$ and $$\Lambda(q,\alpha,a)=0 \text{ for each } 0 \leq \alpha \leq q-1.$$
Notice if $q+1\equiv 0 \pmod{5},$ then $0$ is the only root of $f$ if and only if $a^{\frac{q+1}{5}} \ne 1.$
If $f$ is a PP of $\mathbb{F}_{q^2}$, then $q+1 \equiv 0 \text{ mod } 5.$
Assume $f$ is a PP and $q \geq 5$. First suppose $5 \leq q <8$. Note that $$\Gamma(q,0)=\{-3(q+1),-2(q+1),-(q+1)\}.$$ By (2.6) we have $$0 = \Lambda(5,0,a)= \binom{5-1}{1}a^{-5}=-a^{-5}$$ and $$0 = \Lambda(7,0,a)= \binom{7-1}{1}a^{-21}=-a^{-21}.$$ In either case we have a contradiction. Now suppose $q \geq 8$. In this case notice $$\Gamma(q,0)=\{ -4(q+1), -3(q+1), -2(q+1), -(q+1) \}.$$ So again by (2.6) we have $$0=\Lambda(q,0,a)=\sum_{k=1}^{4} \binom{q-1}{ \frac{k(q+1)-1}{5}}^{*} a^{-(\frac{k(q+1)-1}{5})q}$$ where $$\binom{n}{m}^{*} = \begin{cases} \binom{n}{m} &\mbox{ if } m \in \mathbb{Z} \\ 0 &\mbox{ otherwise }
\end{cases}.$$ Now if $(q+1) \not \equiv 0 \pmod{5}$ then exactly one of $\binom{q-1}{ \frac{k(q+1)-1}{5}}^{*}$ is nonzero which contradicts (2.6). Thus if $f$ is a PP of $\mathbb{F}_{q^2}$ we must have $q+1 \equiv 0 \pmod{5}$.
Assume $q+1 \equiv 0 \pmod{5} \text{ and } \alpha>0.$ The previous lemma together with (2.5) imply the sum $\Lambda(q,\alpha,0)$ is empty unless $\alpha+1 \equiv 0 \pmod{5}.$
Assume $q+1 \equiv 0 \pmod{5}$, $\alpha >0$, $\alpha + 1 \equiv 0 \pmod{5}$ and $q \geq 4 \alpha +8$. Set $v= a^{-\frac{q+1}{5}}$, then $$\Lambda(q,\alpha,a)= (-a)^{\frac{\alpha +1}{5}q} \sum_{i=0}^{\alpha}(-1)^i \binom{\alpha}{i} \sum_{l=0}^{4} \binom{i+\frac{4 \alpha - 1 + l}{5}}{\alpha} v^{5i + l}.$$
Since $q \geq 4\alpha +8$ we have $$\Gamma(q,\alpha)=\{-4(q+1),-3(q+1),-2(q+1),-(q+1),0\}.$$ Using (2.5) we see $$\begin{split}
\Lambda(q,\alpha,a) &= \sum_{-\alpha -1 +5(i-j) \in\Gamma(q,\alpha)} \binom{\alpha}{i} \binom{q-1-\alpha}{j} a^{-i-jq}\\
&= \sum_{i=0}^{\alpha} \binom{\alpha}{i} \sum_{l=0}^{4} \binom{-1-\alpha}{\frac{1}{5}(l(q+1)-\alpha-1)+i}a^{-i-[\frac{1}{5}(l(q+1)-\alpha-1)+i]q}\\
&=(-a)^{\frac{q+1}{5}q}\sum_{i=0}^\alpha \binom{\alpha}{i} \sum_{l=0}^4 (-1)^{i} \binom{\frac{1}{5}(l(q+1)-\alpha-1)+i+\alpha}{\alpha}a^{-\frac{q+1}{5}(l+5i)}\\
&= (-a)^{\frac{\alpha +1}{5}q} \sum_{i=0}^{\alpha}(-1)^i \binom{\alpha}{i} \sum_{l=0}^{4} \binom{i+\frac{4 \alpha - 1 + l}{5}}{\alpha} v^{5i + l}.
\end{split}$$ Between the second and third line we use $\binom{-m}{n}=(-1)^n \binom{n+m-1}{m-1}.$
Assume $q+1 \equiv 0 \pmod{5}$ and $\alpha > 0.$ $\Gamma(q,\alpha)$ contains exactly five consecutive multiples of $q+1$ unless $\alpha=\frac{q-1}{2} \in \mathbb{Z}$ or $\alpha=\frac{q-3}{4}\in \mathbb{Z}.$
We may assume $\alpha +1 \equiv 0 \pmod{5}.$ Since $\Gamma(q,\alpha)$ in contained in the interval $[4\alpha+4-5q, 4\alpha -1]$ which has length $5(q-1)$ we must have $ 4 \leq | \Gamma(q,\alpha)| \leq 5.$
Suppose $ 4 =| \Gamma(q,\alpha)|.$ Choose $k$ so $\{(k-3)(q+1), (k-2)(q+1), (k-1)(q+1), k(q+1) \} = \Gamma(q,\alpha).$ Note $q \geq 6$ and $\alpha >0$ force $k \in \{0,1,2,3 \}.$ We have the following inequalities $$\begin{cases}
4 \alpha -1 \leq k(q+1) +q \\
(k-3)(q+1)-q \leq 4 \alpha +4 -5q.
\end{cases}$$ Since $4 \alpha -1, 4\alpha +4 -5q \equiv 0 \pmod{5}$ it follows $$\begin{cases}
4 \alpha -1 \leq k(q+1) +q - 4 \\
(k-3)(q+1)-q + 4 \leq 4 \alpha +4 -5q.
\end{cases}$$ Since $a \leq q-1$ and $0\leq k \leq 3$, taking the difference of the inequalities in (2.8) reveals $k=0$ which gives $\alpha = \frac{q-3}{4}$ or $k=1$ which gives $\alpha = \frac{q-1}{2}$.
Assume $q+1 \equiv 0 \pmod{5}$ and $y:=a^{\frac{q+1}{5}} \neq 1$ is a 5th root of unity. Then for $1 \leq \alpha \leq q-1$ we have $$\Lambda(q, \alpha, a) = \begin{cases}
-a^{-\frac{1}{5}(\alpha+1)}(y^{-1}+1+y+y^2) &\mbox{ if } \alpha=\frac{q-1}{2}, \hspace{2mm} \alpha \in \mathbb{Z}\\
-a^{-\frac{1}{5}(\alpha+1)}(1+y+y^2+y^3) &\mbox{ if } \alpha=\frac{q-3}{4}, \hspace{2mm} \alpha \in \mathbb{Z}\\
0 &\mbox{ otherwise. }
\end{cases}$$
We may assume $\alpha +1 \equiv 0 \pmod{5}.$ First suppose $\alpha \ne \frac{q-3}{4}, \frac{q-1}{2}.$ Let $K$ be a set of five consecutive integers such that $K(q+1) = \Gamma(q,\alpha).$ Now we have $$\begin{split}
\Lambda(q,\alpha,a) &= \sum_{-\alpha -1 +5(i-j) \in \Gamma(q,\alpha)} \binom{\alpha}{i} \binom{q-1-\alpha}{j} a^{-i-jq}\\
&= \sum_{k\in K} \sum_{-\alpha-1+5(i-j)=k(q+1)}\binom{\alpha}{i} \binom{q-1-\alpha}{j} a^{-i+j}\\
&= \sum_{k\in K} a^{-\frac{1}{5}[\alpha+1+k(q+1)]} \sum_{i-j=\frac{1}{5}[\alpha+1+k(q+1)]} \binom{\alpha}{\alpha - i} \binom{q-1-\alpha}{j}\\
&= a^{-\frac{1}{5}(\alpha+1)} \sum_{k \in K}y^{-k} \sum_{\alpha-i+j=\frac{1}{5}[4 \alpha -1 - k(q+1)]} \binom{\alpha}{\alpha-i} \binom{q-1-\alpha}{j}\\
&= a^{-\frac{1}{5}(\alpha+1)} \sum_{k \in K}y^{-k} \binom{q-1}{\frac{1}{5}[4 \alpha - 1 - k(q+1)]}\\
&= -a^{-\frac{1}{5}(\alpha+1)} \sum_{k \in K}y^{-k}=0.
\end{split}$$
Now suppose $\alpha= \frac{q-1}{2}.$ By the above computation and the previous lemma we have $K =\{-2,-1,0,1\}$, thus $$\Lambda(q,\alpha,a)= -a^{-\frac{1}{5}(\alpha+1)}(y^{-1}+1+y+y^2).$$ Similarly if $\alpha= \frac{q-3}{4},$ we have $$\Lambda(q,\alpha,a) =-a^{-\frac{1}{5}(\alpha+1)}(1+y+y^2+y^3).$$
Proof of the Theorem
====================
(Theorem1.1)\
$(\Leftarrow)$ Cases (ii)-(vii) are easily verified by a computer. Assume (i), that is $q=2^{4k+2}$ and $ a^{ \frac{q+1}{5}} \neq 1$ is a fifth root of unity. Lemma 2.6 gives $\Lambda(q,\alpha,a)=0$ for each $0 \leq \alpha \leq q-1$ and $0$ is the only root of $f$, so $f$ is a PP by (2.6)\
\
$(\Rightarrow)$ Assume $f$ is a PP. By Lemma (2.2) we have $q+1 \equiv 0 \pmod{5}.$ Let $y:=a^{\frac{q+1}{5}}.$ If $y \ne 1$ is a fifth root of unity then Lemma 2.6 implies $q$ must be even. Thus $q=2^{4k+2}$ and we have case (i).
Now suppose $1+y+y^2+y^3+y^4 \neq 0$. The sum in the RHS of Lemma 2.4 is a polynomial in $v(=y^{-1})$ and can be easily computed for small values of $\alpha$ with the help of a computer algebra system. For a few values of $\alpha$ we find $$\Lambda(q,\alpha,a) = (-a)^\frac{\alpha+1}{5}v(1+v+v^2+v^3+v^4)
\begin{cases}
5^{-4}g_4 (v) &\mbox{ if } \alpha = 4, q\geq 24\\
5^{-10}g_9 (v) &\mbox{ if } \alpha = 9, q\geq 44\\
5^{-16}g_{14} (v) &\mbox{ if } \alpha = 14, q\geq 64\\
5^{-28}g_{24} (v) &\mbox{ if } \alpha = 24, q\geq 104.\\
\end{cases}$$ The polynomials $g_\alpha$ are given in the appendix. Write $R(p_1,p_2)$ for the resultant of polynomials $p_1, p_2.$ Then $$GCD(R(g_4,g_9),R(g_4,g_{14}))=2^{15}3^{3}5^{197}.$$ Thus if $q\geq 64$ we must have $p \hspace{1mm} (=char \mathbb{F}_{q^2}) \in \{2,3\}.$ Since $q+1 \equiv 0 \pmod{5}$ there are only a few prime powers $q<64$ with $p\hspace{1mm} (=char \mathbb{F}_{q^2})\neq 2,3.$
- When $q=19$, a computer search results in case (iii)
- When $q=29$, a computer search results in case (iv)
- When $q=49$, a computer search results in case (v)
- When $q=59$, a computer search results in case (vi)
When $p=2$ we have $GCD(g_4,g_{24})=x$ thus $q<104.$ Since $q+1 \equiv 0 \pmod{5}$ and $q>4$ we only need to consider $q=64.$ A computer search results in case (vii).
When $p=3$ we have $GCD(g_4,g_9)=1$ thus $q<44.$ Again since $q+1 \equiv 0 \pmod{5}$ we only need to consider $q=9.$ A computer search results in case (ii).
Theorem 1.2 is proved using a similar method which leads us to the following conjecture.
Let $ r>2$ be a fixed prime. If both $ (q+1) \equiv 0 \pmod{r}$ and $a^{\frac{q+1}{r}}$ is not an $r-th$ root of unity; we conjecture there are only finitely many values $(q,a)$ for which $f=ax+x^{r(q-1)+1}\in \mathbb{F}_{q^2}^*[x]$ is a permutation polynomial of $\mathbb{F}_{q^2}.$
Appendix
========
$$\begin{split}
g_4(x) &=g_4(x)=44+75 x+115 x^2+165 x^3-2899 x^4-1152 x^5-1465 x^6-1825 x^7\\
&-2235 x^8+25427 x^9+4122 x^{10}+4805 x^{11}+5555 x^{12}+6375 x^{13}-58357 x^{14}\\
&-3514 x^{15}-3915 x^{16}-4345 x^{17}-4805 x^{18}+38454 x^{19}.
\end{split}$$
$$\begin{split}
g_9(x) &=35464+47564 x+41954 x^2-3211 x^3-121771 x^4+3307018 x^5+7237538 x^6\\
&+13764443 x^7+24034088 x^8-399905587 x^9-247096418 x^{10}-374180158 x^{11}\\
&-550629013 x^{12}-791033308 x^{13}+10166063897 x^{14}+3488769646 x^{15}\\
&+4732145066 x^{16}+6330108851 x^{17}+8362735316 x^{18}-90589540129 x^{19}\\
&-20005926320 x^{20}-25545256270 x^{21}-32342528845 x^{22}-40628485270 x^{23}\\
&+389225321705 x^{24}+56878393066 x^{25}+69794856416 x^{26}+85151720951 x^{27}\\
&+103326084566 x^{28}-901674492499 x^{29}-84876624338 x^{30}-101203850008 x^{31}\\
&-120170621113 x^{32}-142129134058 x^{33}+1152208354517 x^{34}+63621134698 x^{35}\\
&+74215593188 x^{36}+86304925343 x^{37}+100064515838 x^{38}-764098747192 x^{39}\\
&-18879570941 x^{40}-21644171461 x^{41}-24754360696 x^{42}-28246292086 x^{43}\\
&+205243145184 x^{44}.
\end{split}$$
$$\begin{split}
g_{14}(x) &=35781192+45103176 x+34642700 x^2-11075651 x^3-104451417 x^4+3431018744 x^5\\&+5425819552 x^6+5815099340 x^7+1030592053 x^8-15702529689 x^9+367570246196 x^{10}\\&+915262570648 x^{11}+1948805856320 x^{12}+3779801961517 x^{13}-62553432822181 x^{14}\\&-47505978162672 x^{15}-79121737743896 x^{16}-127716748824160 x^{17}-200786722216499 x^{18}\\&+2746238364681602 x^{19}+1263694841543105 x^{20}+1868009627371040 x^{21}\\&+2718322826664950 x^{22}+3900065855839735 x^{23}-46407807995168830 x^{24}\\&-15130762640643679 x^{25}-20922156001611712 x^{26}-28643147196518350 x^{27}\\&-38852104359262613 x^{28}+415138746369911354 x^{29}+101011800565346642 x^{30}\\&+133546181274464111 x^{31}+175316634655627970 x^{32}+228624694439538979 x^{33}\\&-2240965131247477702 x^{34}-414819831279200662 x^{35}-530625761486025481 x^{36}\\&-675157922777947390 x^{37}-854715990669328349 x^{38}+7803518358751564382 x^{39}\\&+1100711546227581974 x^{40}+1372388078602423607 x^{41}+1703988861163039670 x^{42}\\&+2107260628357829383 x^{43}-18124791008882124634 x^{44}-1923722972889132730 x^{45}\\&-2349566656784986915 x^{46}-2860023585150150200 x^{47}-3470096113600539985 x^{48}\\&+28364072170221684830 x^{49}+2200015572073495862 x^{50}+2641585104385038536 x^{51}\\&+3162990481363018525 x^{52}+3777155092039529639 x^{53}-29542409091657957562 x^{54}\\&-1584053771764262386 x^{55}-1874825156457447288 x^{56}-2213821365876774435 x^{57}\\&-2608223671181041657 x^{58}+19628014114205307016 x^{59}+651404631123094716 x^{60}\\&+761528771717834208 x^{61}+888518897148324570 x^{62}+1034700084660340082 x^{63}\\&-7525847208868343576 x^{64}-116633198562800052 x^{65}-134898641118903336 x^{66}\\&-155761841308153260 x^{67}-179556106888260384 x^{68}+1266995051549992032 x^{69}
\end{split}$$
$$\begin{split}
g_{24}(x) &=44199864566676+53277891868890 x+36951353492365 x^2-18447292166275 x^3\\&-115981817761656 x^4+5683172059809972 x^5+7671280571862570 x^6\\&+6327550580716885 x^7-1472857823809435 x^8-18209145388579992 x^9\\&+496676536774208388 x^{10}+771501576163548690 x^{11}+774677809879397185 x^{12}\\&+37151453118769745 x^{13}-2080007375935924008 x^{14}+44904298634276465124 x^{15}\\&+85192441936752637650 x^{16}+111451370786904498265 x^{17}+47809116140932255865 x^{18}\\&-289357227498865856904 x^{19}+6805964762978995053180 x^{20}+19701780140886237080850 x^{21}\\&+47920556078520020855875 x^{22}+105337499516914088443775 x^{23}\\&-1763163188957551294558680 x^{24}-1682149911521612848440924 x^{25}\\&-3144091106960055505926510 x^{26}-5682560762926174819291685 x^{27}\\&-9983113995337864503302425 x^{28}+145844209029690653966024044 x^{29}\\&+90577048940932553337666992 x^{30}+149016337368570701236013970 x^{31}\\&+240987977179133628538814635 x^{32}+383707733869425939573259415 x^{33}\\&-5054637186099365312920130012 x^{34}-2383073542636350764433826072 x^{35}\\&-3646247878744859791470683310 x^{36}-5516927650609039918496642165 x^{37}\\&-8260762058311010597210260105 x^{38}+99947418327478671197343833527 x^{39}\\&+37844995423440695266939743959 x^{40}+55100153059398963201126006300 x^{41}\\&+79573062038413273868028870415 x^{42}+114034864385838660589245973115 x^{43}\\&-1283855920027521585601121843789 x^{44}-401631430484182503011941004665 x^{45}\\&-563418559621522038915547815300 x^{46}-785422361095522275273631000625 x^{47}\\&-1088343656606667747839725560325 x^{48}+11515350760355635501320142255915 x^{49}\\&+3023936138319755361442261262591 x^{50}+4119412652650088403766862240940 x^{51}\\&+5583398973821401437427782781615 x^{52}+7530907539377792043231905759675 x^{53}\\&-75471967365836479567646155794821 x^{54}-16787688185166611656225020168433 x^{55}\\&-22327203688191441773802872732580 x^{56}-29570337359172614836200572834465 x^{57}\\&-39004838361833906826391115026885 x^{58}+372587302842602423374479932324863 x^{59}\\&+70520166819123145016529674257763 x^{60}+91921490099273043258578272890040 x^{61}\\&+119393417476195244286408801766635 x^{62}+154543277170360802146843655221595 x^{63}\\&-1414460450943061212979844779136033 x^{64}-228058586275344486040909532074621 x^{65}\\&-292198163240869437036528542283800 x^{66}-373236999859930139181490117490085 x^{67}\\&-475340219041052709214076298517285 x^{68}+4186641921387463113636090427865791 x^{69}\\&+573982399597440241104412076252915 x^{70}+724508808520889474869872176906800 x^{71}\\&+912087128194582180606507235839875 x^{72}+1145264181936242472433040520458075 x^{73}\\&-9742873878315091411250160134457665 x^{74}-1130413280103442753936876665890989 x^{75}\\&-1408264364840845831044139437784160 x^{76}-1750319535256268238132497892063885 x^{77}\end{split}$$
$$\begin{split}
&-2170502922925974506143566661090725 x^{78}+17890834639608180012805054289642259 x^{79}\\&+1742739271574108271273793805711627 x^{80}+2145966682431740273469435944163595 x^{81}\\&+2637022907200101153997087624591860 x^{82}+3233886164548639605146540321869515 x^{83}\\&-25897969665493387445366349883836597 x^{84}-2092770412365513238435254665605237 x^{85}\\&-2550276778698834519482774414234285x^{86}-3102050153955665578994040465961140 x^{87}\\&-3766364889683287116775343845510605 x^{88}+29374000062448380447889073547561267 x^{89}\\&+1935240033819795981039415092100619 x^{90}+2336265403685016505132971195469975 x^{91}\\&+2815697619541879892767440305985240 x^{92}+3387974339410094717815325956323015 x^{93}\\&-25785924105952569948928004112378849 x^{94}-1350431582795782203758136667410805 x^{95}\\&-1616445075520877395405741712969225 x^{96}-1931945507456689143942634660617000 x^{97}\\&-2305611169359156003531542005529025 x^{98}+17156418696304763600550408562151055 x^{99}\\&+687379082077441716165690906155771 x^{100}+816415958977468475853193747835215 x^{101}\\&+968348850584524348447504741362840 x^{102}+1147010610713807759039043095096625 x^{103}\\&-8358195150530508420416077021700451 x^{104}-240714899339982433449408005104533 x^{105}\\&-283875024040741105674985093276305 x^{106}-334355514938579365555961289992040 x^{107}\\&-393328645903551948398787580245735 x^{108}+2810835580815912962362589234243613 x^{109}\\&+51827832972624672481156065143283 x^{110}+60721533591007894432573966947615 x^{111}\\&+71059972468668725537659588986360 x^{112}+83064891280671737260332843826245 x^{113}\\&-582904099186926848986624808653503 x^{114}-5170193326947891817908520747581 x^{115}\\&-6020861165457984420843719642625 x^{116}-7004114740388167029285613941960 x^{117}\\&-8139491600243587212407368112835 x^{118}+56154582678607837122596101351251 x^{119}\end{split}$$
[99]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We use a hydrodynamic model to describe the relaxation of optically injected currents in quantum wells on a picosecond time scale, numerically solving the continuity and velocity evolution equations with the Hermite-Gaussian functions employed as a basis. The interplay of the long-range Coulomb forces and nonlinearity in the equations of motion leads to rather complex patterns of the calculated charge and current densities. We find that the time dependence of even the first moment of the electron density is sensitive to this complex evolution.'
author:
- 'R.M. Abrarov, E. Ya. Sherman, and J. E. Sipe'
title: 'Hydrodynamic model for relaxation of optically injected currents in quantum wells [^1]'
---
The dynamics of hot electron currents, which determines the distance that injected carriers can propagate and the charge density patterns that can form, is important for the design of various semiconductor devices, including transistors, charge-coupled light detectors, cascade lasers and light emitting diodes [@Chicago]. The full dynamics of systems strongly out of equilibrium requires a complicated analysis based either on extended numerical modeling employing Monte Carlo simulations [@MonteCarlo], or on quantum versions of the kinetic equations[@Wu06; @Duc06; @Rumyantsev04; @Kuhn04]. For two-dimensional (2D), systems quantum kinetic approaches are numerically demanding and often unfeasible.
The direct injection of strong electrical and spin currents [@Bhat00] is possible using the interference of one-photon (frequency $2\omega $) and two-photon (frequency $\omega$) absorption. The high speed of the injected electrons, $v\approx 1000$ km/s, is determined by the excess photon energy $2\hbar\omega-E_{g},$ where $E_{g}$ is the band gap for the quantum well (QW). In an experiment injecting electrical current, coherent control is achieved by adjusting the relative phase parameter $\Delta\phi =2\phi
_{\omega }-\phi _{2\omega}$ of the two beams, where $\phi _{\omega }$ $\left( \phi _{2\omega }\right) $ is the phase of the beam at $\omega $ $\left( 2\omega \right).$ The resulting current density is approximately given by $Nv\sin(\Delta\phi),$ where $N$ is the concentration of optically injected carriers.
In experiments done on multiple quantum well (MQW) samples, with the beams incident along the growth direction, these injected lateral currents (see Fig.1) have been detected using various techniques [@Stevens02; @Hubner03]. The QWs are well-separated and unbiased, so electrons do not have sufficient energy either to tunnel through the barriers or overcome them by thermal activation. For this reason no dynamics along the “vertical” growth direction is possible, and in these systems we do not explore the rich nonlinear behavior seen in vertical transport in superlattices.[@Bonilla05]. Instead, our subject is only the lateral dynamics of the carriers subsequent to their injection.
Nonetheless, the analysis of this lateral dynamics is more complicated than in many transport problems because of the largely inhomogeneous long-range Coulomb field, leading to complicated space-charge effects. As the electrons and holes move away from each other, the characteristic range of inhomogeneity is on the order of the size of the electron and hole puddles themselves. A qualitative approach to this problem is to adopt a “rigid spot” approximation, which reduces the problem to the motion of two coupled and damped linear oscillators, representing the centers of the electron and hole puddles, each moving with a uniform velocity and thus exactly preserving its shape.[@Sherman06] Here the entire dynamics is described by four parameters: displacements of electron and hole spots and their velocities.
An important question is to what extent this “rigid spot” model can describe the dynamics, as Coulomb and other concentration-dependent forces come into play when the puddles separate. Typically, the change in carrier densities at any point is small, since the relaxation and Coulomb interaction prevent any large separation of the centers of charge. However, can the spatial dependence of this small change depart significantly from what is predicted in the “rigid spot” model, leading to a picture involving more generally “distorted puddles” as more appropriate ?
Answering this question is important for the ultimate design of numerical schemes to efficiently explore the system dynamics. What is needed initially is a practical model and calculation procedure that captures the essential physics, is transparent enough to allow extraction of the important features of the dynamics, and reasonable enough to allow comparison with experimental results, at least at a semiquantitative level. Here we propose a phenomenological hydrodynamic model for the dynamics of optically injected charge currents subject to space-charge effects. The advantage of the hydrodynamic approach is that it is insensitive to the details of the carriers’ distribution functions, with the dynamics described by a coupled system of partial differential equations for the concentrations and velocities. The collisions between carriers, and interaction between the carriers and the environment, are modeled by a set of characteristic times determined by the experimental conditions.
As an example calculation we consider a MQW structure (Fig. 1), assuming the excitation and subsequent dynamics in each quantum well is the same; within our model we perform a full 2D calculation of the carriers’ motion. The $\omega $- and $2\omega $-beams produce initial distribution of electron and hole densities and velocities which then evolve in time and space. The initial distribution of carriers in each single QW is $N_{e,h}^{(s)}\left(\mathbf{r},t=0\right)\equiv N_{s}\exp\left(-r^{2}/2\Lambda^{2}\right)$, where $\Lambda$ is governed by the beam sizes (see caption of Fig.1). The total concentration $N_{e,h}\left( \mathbf{r},t\right)=qN_{e,h}^{(s)}\left(\mathbf{r},t\right)$, where $\mathbf{r}=(x,y)$ is the 2D coordinate, and $q$ is the total number of single QWs, neglecting the distance from the first to the last single QW in the structure compared to $\Lambda$. The spin indices are omitted since here the electrons are assumed to be unpolarized. The $\left(\mathbf{r},t\right)$ arguments will be omitted below for brevity.
For simplicity, in this letter we present results for the dynamics of electrons assuming that the holes are infinitely heavy; we will include hole dynamics in a later publication. The equations describing the motion of carriers consist of the continuity, and momentum and energy evolution sets. In the effective mass approximation, the analysis of the dynamics shows that a first description can be provided even without taking the energy relaxation into account. For this reason we restrict ourselves to the first two sets of variables. The continuity and the evolution of the local mean velocity $\mathbf{u}\left(\mathbf{r},t\right)$ of electrons, are described by: $$\begin{aligned}
&&\frac{\partial N_{e}}{\partial t}+\nabla \left( N_{e}\mathbf{u}\right) =0, \\
&&\frac{\partial \mathbf{u}}{\partial t}+\left( \mathbf{u}\mathbf{\cdot \nabla
}\right) \mathbf{u}=-\frac{e\mathbf{E}}{m_{e}}-\frac{\mathbf{u}}{\tau _{eh}}\frac{N_{h}}{N_{0}}
-\frac{\mathbf{u}}{\tau _{e}},\end{aligned}$$where $\mathbf{E}=\mathbf{E}^{e}+\mathbf{E}^{h}$ is the macroscopic Coulomb field produced by electrons and holes, and $e$ is the elementary charge. Here $m_{e}$ is the electron effective mass, $\tau _{eh}$ describes momentum-conserving drag due to the Coulomb forces at electron-hole collisions [@Hopfel] and can be weakly concentration-dependent itself [Zhao07]{}, $\tau _{e}$ is the relaxation time for electrons due to external factors, such as phonons and impurities leading to the relaxation of the total momentum, and $N_{0}=qN_{s}$. In addition to the explicit time scales $\tau _{e}$ and $\tau _{eh}$ a third important time scale in the problem is $\Omega_{\rm pl}^{-1}$, where $\Omega_{\rm pl}=$ $\left(\pi /2\right)^{3/4}\sqrt{N_{0}e^{2}/\varepsilon m_{e}\sqrt{2}\Lambda}$ is the characteristic two-dimensional plasma frequency[@Sherman06] ($\varepsilon$ is the dielectric constant); it characterizes the strength of the Coulomb interaction. In our sample calculations we take $\tau_{e}=80$ fs and $\tau_{eh}=150$ fs; a typical value $N_{s}=10^{11}$ cm$^{-2}$ with $q=1$ and our choice of $\Lambda=1$ $\mu$m leads to $\Omega_{\rm pl}^{-1}\approx 1.4$ ps. These times agree with the set of parameters of Duc [*et al.*]{} \[\] and are shorter than those for vertical transport [@Weber06] due to the higher dimensionality of carrier motion here.
{width="0.4\columnwidth"}
To reduce the system of partial differential equations, we introduce a full basis set in the form of the harmonic oscillator wave functions $$\Psi _{n_{1},n_{2}}\left(\mathbf{r}\right) =\psi _{n_{1}}(x)\psi
_{n_{2}}(y),\qquad \psi _{n}(x)=\frac{H_{n}\left( x/\Lambda \right) }{\pi
^{1/4}\sqrt{n!2^{n}}}e^{-x^{2}/2\Lambda ^{2}},$$where $H_{n}\left(x/\Lambda\right)$ is the Hermite polynomial of the $n-$th order. The key point of our approach is the expansion of the possible solutions in a finite basis in the form: $$N_{e} =\sum_{n_{1},n_{2}}^{n_{\max }}N_{n_{1},n_{2}}^{e}(t)\Psi
_{n_{1},n_{2}}\left(\mathbf{r}\right); \qquad
u_{i} =\sum_{n_{1},n_{2}}^{n_{\max }}u_{n_{1},n_{2}}^{(i)}(t)\Psi
_{n_{1},n_{2}}\left(\mathbf{r}\right) +U_{i}\left( t\right) .$$ Here $i=x,y$ is the Cartesian index. To improve the convergence, we include a known function of time $U_{i}\left( t\right) $ in the right-hand side of Eq.(4). This function is obtained by solving the linear equations of motion in the rigid spot approximation [Sherman06]{}. In the geometry considered here, $U_{y}\left( t\right) =0$, and, therefore, we drop the Cartesian index of $U_{x}\left( t\right) $. The electric fields at the point $\mathbf{r}$ are $$\mathbf{E}^{e,h}\left(\mathbf{r},t\right) =\mp \frac{e}{\varepsilon }\sum_{n_{1},n_{2}}^{n_{\mathrm{max}}}N_{n_{1},n_{2}}^{e,h}(t)
\int
\Psi_{n_{1},n_{2}}\left(\widetilde{\mathbf{r}}\right)\frac{\mathbf{r}-\widetilde{\mathbf{r}}}{\left\vert \mathbf{r}-\widetilde{\mathbf{r}}\right\vert ^{3}}d^{2}\widetilde{r},$$where the upper (lower) sign corresponds to electrons (holes). If a QW is located close to the semiconductor-air interface, the role of the image charges which can be taken into account with the Green function technique [@Sipe81], increases the field by a factor of two. Since $\Lambda $ is typically much larger than the distance between the QWs, within our assumption that the excitation of all the wells is the same, the electric field at a point $\mathbf{r}$ in each well and the subsequent dynamics is the same.
The state and the motion of the electron spot is then fully described by a $3\times \left( n_{\max }+1\right) ^{2}-$component vector $S_{\alpha },$ where $\alpha $ is a two-component index corresponding to indices $n_{1}$ and $n_{2}.$ By projecting the equations (1),(2) and (5) on the set of $\Psi
_{n_{1},n_{2}}\left( \mathbf{r}\right) $, we obtain the system of ordinary nonlinear differential equations in the form $$\frac{dS_{\alpha }}{dt}=\sum_{\eta,\mu}\mathcal{C}\left( \alpha ;\eta ,\mu
\right) S_{\eta }S_{\mu }-\sum_{\zeta}{\Gamma }\left( \alpha ;\zeta
\right) S_{\zeta }.$$Here the matrix $\mathcal{C}\left( \alpha ;\eta ,\mu \right) $ is determined by the spatial dependence of concentration and velocity, and the electron-hole drag, while the matrix ${\Gamma }\left( \alpha ;\zeta \right) $ depends on the Coulomb forces and momentum relaxation times. The full matrices will be presented elsewhere.
The initial conditions are given by: (i) $N_{0,0}^{e}(0)=\sqrt{\pi }N_{0},\;N_{n_{1},n_{2}}^{e}(0)=0$ (if $n_{1}>0$ or $n_{2}>0$), and (ii) $u_{n_{1},n_{2}}^{(i)e}(0)=0.$ Conditions (i) correspond to the initial injection of density in the Gaussian mode $\Psi_{0,0}\left( \mathbf{r}\right) $ only. Due to the symmetry of the problem $N_{n_{1},2m+1}^{e}(t)$, $u_{n_{1},2m+1}^{(x)}(t)$ and $u_{n_{1},2m}^{(y)}(t)$ will remain zero. The initial speed of the electron spot $U(0)$ is determined [@Bhat00] by the photon excess energy and $\Delta\phi$.
At very short times after the injection the carriers propagate ballistically, and then the motion becomes diffusive and influenced by the Coulomb forces. A simple characterization of the motion of the charge is given by the first moment of the concentration,$$\left\langle x(t)\right\rangle =\frac{1}{N_{\rm t}}{\int xN_{e}\left( \mathbf{r},t\right)
d^{2}r},\quad
N_{\mathrm{t}}=\pi N_{0}\Lambda ^{2},
\label{firstmoment}$$where $N_{\mathrm{t}}$ is the total number of injected electrons. The relaxation and Coulomb interaction guarantee that $\xi\equiv\left\langle x(t)\right\rangle/\Lambda $ is always small in our systems. Were the rigid spot approximation valid, the amplitude of the nonzero higher terms $N_{n,0}^{e}$ would drop off as $\xi^{n}$. The main qualitative result of our investigations is that *this is often not so* for reasonable excitation parameters. Instead, the concentration-dependent Coulomb forces and drag lead to a deformation of the spot and, therefore, to an increase in the number of $\Psi _{n_{1},n_{2}}$ terms with non-negligible amplitude in the sums, and to a complicated spot structure. In the course of time, a few of the $N_{n_{1},n_{2}}^{e}(t)$ components can become of the same order of magnitude, although all small compared to $N_{0,0}(0)$ due to a relatively small spot displacement $\xi\ll 1$. The number of higher terms that are important depends mainly on the structure of the $\mathcal{C}\left( \alpha ;\eta ,\mu \right) $-matrix, with their relative contribution dependent on the parameters $\tau _{e}\Omega _{\mathrm{pl}}$ and $\tau _{e}/\tau _{eh}$. In contrast to the scattering by phonons and disorder, which stabilizes the spot motion and does not lead to its distortion, the role of electron-hole collisions is two-fold. On one hand, it stops the motion of the spot as a whole, on the other hand, it generates higher components in the velocity pattern due to nonuniform velocity evolution. The nonuniform velocity distribution produced by the Coulomb forces and the drag then leads to a nonuniform density distribution due to continuity. So the joint effects of the Coulomb forces and the drag lead to a “buckling” of the electron spot along the direction of its velocity. The buckling becomes more pronounced as the space-charge effects, characterized by the parameter $\tau _{e}\Omega _{\mathrm{pl}}$, increase.
{width="0.55\columnwidth"}
This is illustrated in Fig.2, where we show the pattern of the change in the electron density at two times, for samples with $q=1$ and $q=16$ buried deep within bulk Al$_{x}$Ga$_{1-x}$As where Eq.(5) is applicable. On the time scales of the ballistic regime, $t\leq\tau _{e},$ in both samples the spot moves as a rigid distribution with $$N_{e}=N_{0}\exp \left[ -\left( x-\left\langle x(t)\right\rangle \right)
^{2}/2\Lambda ^{2}\right] \exp \left[ -y^{2}/2\Lambda ^{2}\right] ,$$and the corresponding density component develops as $N_{1,0}^{e}(t)=N_{0,0}^{e}(0)\xi/\sqrt{2}$. This is clearly seen in the left column of the Fig. 2, where the density profile is almost the same for systems with weak ($q=1$) and much stronger ($q=16$) space-charge effects at $t=80$ fs, and is well described by Eq.(8). At 1.5 ps, the profile for the $q=1$ system buckles only slightly; a “rigid spot” picture is applicable. But for $q=16$ the behavior at these times is much more complicated, with several $\Psi_{n_{1},n_{2}}$ states contributing, and the profile changing from oval to bow-shaped. Here the “rigid spot” approximation is no longer reasonable, and a more general “distorted puddle” picture emerges. More insight can be obtained from the current density $N_{e}(\mathbf{r})u_{x}(\mathbf{r})$ distribution in Fig.3. At $t=80$ fs the patterns for $q=1$ and $q=16$ look similar. At $t=160$ fs, the distributions already differ very considerably, and the pattern for $q=16$ is much more nonuniform than that for $q=1$.
{width="0.55\columnwidth"}
It is interesting to see the effect of the real spot shape on gross features of the density distribution, such as its first moment (\[firstmoment\]). We evaluate this from our calculations as $$\left\langle x(t)\right\rangle =\sum_{n_{1}=1}^{n_{\lim }}\sum_{n_{2}=0}^{n_{\max }}\frac{N_{n_{1},n_{2}}^{e}(t)}{N_{\mathrm{t}}}\int x\psi
_{n_{1}}\left( x\right) dx\int \psi _{n_{2}}\left( y\right) dy,$$and present in Fig.4 the results for $q=4$ and $q=16$ samples. The upper limit $n_{\lim }$ here is less than the number of Hermite-Gaussian states in the basis $n_{\max }$, since the contributions of the upper states cannot be calculated with a high precision; this problem is typical for calculations with finite basis. [Sherman06a]{} For this set of parameters we find that $n_{\max }=14$ and $n_{\lim }=7$ give converged results. In addition, Fig.4 shows the displacement in the rigid spot approximation [@Sherman06], where the dynamics yields $\left\langle x_{\mathrm{rsa}}(t)\right\rangle =
U(0)e^{-\gamma t}\mathrm{sinh}\left({\gamma}_{\rm pl}t\right)/{\gamma}_{\rm pl}$ with $2\gamma =1/\tau_{e}+1/2\tau_{eh}$, ${\gamma}_{\rm pl}^{2}=\gamma^{2}-\Omega_{\mathrm{pl}}^2$. With the stronger space-charge effects that arise for a larger number of wells, the deviation from the rigid spot approximation becomes more severe. However, both $\left\langle x_{\mathrm{rsa}}(t)\right\rangle $ and $\left\langle x(t)\right\rangle $ reach their maxima at almost the same time $t_{0}=2\tau _{e}\left\vert\ln(\Omega _{\mathrm{pl}}\tau _{e})\right\vert$, and the maximum values are very close to each other. The difference begins developing at $t>t_{0}$, when the buckling effects become more pronounced. The curves for small value of $\tau_{eh}=50$ fs show that a decrease in $\tau_{eh}$ does not qualitatively modify the dynamics in our range of parameters. However, in a regime with $\tau_{eh}\ll\tau_{e}$ a decrease in $\tau_{eh}$ can lead to an increase in the puddle distortion. We turn to the influence of this distortion on experimental results in a separate publication.
{width="0.30\columnwidth"}
To conclude, we have shown that within a hydrodynamic model a Hermite-Gaussian expansion can be used to study the dynamics of currents injected optically by coherent control in MQW samples. As samples with increasing numbers of QWs are considered, we find the resulting dynamics shows a complicated buckling behavior in the density distribution that is driven by space-charge effects, with the charge density departing more and more from what a rigid spot approximation would predict. The details of our calculated results reflect both the simple dynamics (1),(2) we adopt, and the assumption of equal excitation of all the quantum wells. But it is unlikely that a relaxation of these approximations, and the richer dynamics that would result, would lead to a [*more rigid*]{} behavior of the charge puddle.
For $\tau_{eh}=150$ fs, we find that the complex behavior, associated with a number of comparable Hermite-Gaussian amplitudes, emerges at $\Omega_{\rm pl}\tau _{e}\approx 0.1$, well within the overdamped regime, and is determined by this parameter rather than by the electron-hole separation. How sensitive various experimental probes will be to this deviation from rigid spot dynamics has yet to be examined; at least it has significant consequences on the evolution of the first moment of density on the timescale of a picosecond.
Our numerical consideration relies on the finite basis analysis. In the overdamped regime we have considered here a fast decay of the off-diagonal Coulomb matrix elements with the distance between the states ensures the applicability of relatively small basis sets. As we consider systems with stronger Coulomb interaction, the basis should be extended. We can naturally expect the distortion of the carrier puddles to get more pronounced, and an interesting problem is whether it can lead to a chaotic behavior [@Pershin07] when many Hermite-Gaussian terms form the density and velocity patterns.
We are grateful to H. van Driel, J. McLeod, A. Najmaie, I. Rumyantsev, and A.L. Smirl for numerous valuable discussions. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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[^1]: A slightly shorter version published in Appl. Phys. Lett. **91**, 232113 (2007)
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'A. Belloche'
- 'P. André'
- 'D. Despois'
- 'S. Blinder'
date: 'Received April 16, 2002; accepted July 12, 2002'
title: |
Molecular line study of the very young protostar\
IRAM 04191 in Taurus: Infall, rotation, and outflow
---
Introduction {#intro}
============
The enigmatic onset of protostellar collapse {#intro_background}
--------------------------------------------
Despite recent progress, the initial conditions of star formation and the first phases of protostellar collapse remain poorly known [e.g. @Myers99; @Andre00 for reviews]. In the standard theory of isolated, low-mass star formation [e.g. @Shu87], the initial conditions correspond to essentially static singular isothermal spheres (SISs, which have $\rho = (\as^2/2\pi G )r^{-2}$), assumed to be in slow, solid-body rotation [@Terebey84 – TSC84]. This leads to a strictly constant mass accretion rate, $\maccm \sim \as^3/G$ (where $\as$ is the isothermal sound speed), and to a growth of the centrifugal disk as $R_{disk} \propto t^3$ (cf. @Terebey84) during the protostellar accretion phase ($t > 0$). Other theoretical models exist, however, that predict a time-dependent accretion history if the collapse initial conditions are either not singular or not scale-free [e.g. @Foster93; @Henriksen97; @Basu97; @Ciolek98; @Hennebelle02]. Starting from realistic, finite-sized prestellar cores with $\rho \approx cte$ in their central region [c.f. @Ward99; @Bacmann00; @Alves01], these models yield supersonic inward velocities close to the center prior to point mass formation (i.e. at $t<0$) and result in denser, nonequilibrium density distributions with strong differential rotation at the onset of the main accretion phase, i.e., at $t=0$. In these models, the accretion rate is initially significantly larger than in the Shu model, then quickly converges toward the standard $\sim \as^3/G$ value, and finally declines much below $\as^3/G$ because of the finite reservoir of mass [see, e.g., @Foster93; @Henriksen97]. Conservation of angular momentum during dynamical collapse at $t<0$ produces a differential rotation profile at $t=0$ [e.g. $\Omega \propto r^{-1}$ in the magnetically-controlled model of @Basu98]. This rotation profile in turn implies a more rapid growth of $R_{disk}$ initially (i.e., $R_{disk} \propto t$ at small $t>0$ in the Basu model) than in the @Terebey84 model.
Getting at a better, more quantitative knowledge of protostellar collapse is crucial, e.g., to gain insight into the origin of stellar masses and disk formation. Observationally, there are two complementary approaches to estimating the initial conditions of protostar formation. The first approach consists in studying the structure and kinematics of “prestellar cores” such as L1544 [e.g. @Ward99; @Tafalla98] , representative of times $t \simlt 0$. The second approach, adopted here, is the detailed study of Class 0 accreting protostars observed at $t \simgt 0$, such as IRAM 04191 (see § \[intro\_iram04191\]), which should still retain detailed memory of their initial conditions.
IRAM 04191: A very young Class 0 protostar in Taurus {#intro_iram04191}
----------------------------------------------------
The massive ($M_{tot} \sim 1.5\, M_\odot $) dense core/envelope of the Class 0 object, IRAM 04191 + 1522 (hereafter IRAM 04191), was originally discovered in the millimeter dust continuum with the IRAM 30m telescope in the southern part of the Taurus molecular cloud [@Andre99 – hereafter AMB99]. Follow-up observations revealed a highly collimated CO bipolar outflow (see Fig. \[n2h+flow\]), a weak 3.6 cm VLA radio continuum source located at its center of symmetry, and spectroscopic evidence of spatially extended infall motions in the bulk of the envelope. These are typical attributes of a Class 0 protostar [@Andre93; @Andre00].
\[!ht\]
The very high envelope mass to luminosity ratio of IRAM 04191 ($M_\mathrm{env}^{< 4200 \mathrm{ AU}}/L_\mathrm{bol}\simgt 3$ M$_\odot$/L$_\odot$) and its position in the $L_\mathrm{bol}-T_\mathrm{bol}$ evolutionary diagram ($L_\mathrm{bol} \sim 0.15$ L$_\odot$ and $T_\mathrm{bol} \sim 18$ K) suggest an age $t \sim 1-3 \times 10^4$ yr since the beginning of the accretion phase (see @Andre99). This is significantly younger than all of the IRAS candidate protostars of Taurus [e.g. @Kenyon93], including L1527 which has $t \simlt 10^5$ yr [e.g. @Ohashi97a]. IRAM 04191 thus appears to be the youngest accreting protostar known so far in Taurus, although the collapsing protostellar condensation MC 27 discovered by @Onishi99 may be in a comparable evolutionary state.
As IRAM 04191 is particularly young, nearby (d $= 140$ pc), and relatively isolated, the study of its velocity structure based on molecular line observations provides a unique opportunity to set constraints on collapse models. This is especially true since the viewing angle is favorable. The CO(2–1) outflow map of @Andre99 (see Fig. \[n2h+flow\]) shows well separated outflow lobes with almost no overlap between blue-shifted and red-shifted emission, indicating that the flow lies out of the plane of the sky at an intermediate inclination angle [e.g. @Cabrit90]. In addition, @Andre99 estimate an aspect ratio of $\sim 0.65$ for the circumstellar dust/N$_2$H$^+$ envelope, whose major axis is perpendicular to the outflow axis (cf. Fig. \[n2h+flow\]). Both characteristics are consistent with an inclination angle of the outflow axis to the line of sight of $i \sim 50 \degr$.
Here, we present and discuss the results of a comprehensive set of molecular line observations toward IRAM 04191. The layout of the paper is as follows. Sect. \[obs\_set\] summarizes observational details. Sect. \[obs\_ana\] interprets the observations in terms of infall, rotation, and outflow motions in the envelope. We then model the observed spectra using radiative transfer simulations computed in 1D spherical geometry with radial infall motions (Sect. \[simul\_1d\]) and in 2D axial geometry with both infall and rotational motions (Sect. \[simul\_2d\]). Sect. \[discuss\] compares the derived constraints on the velocity structure of the IRAM 04191 envelope with the predictions of collapse models. Our conclusions are summarized in Sect. \[concl\].
Observations {#obs_set}
============
We carried out millimeter line observations with the IRAM 30m telescope at Pico Veleta, Spain, during 7 nights in July and August 1999, in the following molecular transitions: N$_2$H$^+$(1-0), CS(2-1), C$^{34}$S(2-1), HCO$^+$(1-0), H$^{13}$CO$^+$(1-0), C$_3$H$_2$(2$_{12}$-1$_{01}$) at 3 mm, CS(3-2), C$^{34}$S(3-2), H$_2$CO(2$_{12}$-1$_{11}$) at 2 mm, and CS(5-4), H$_2$CO(3$_{12}$-2$_{11}$), C$^{18}$O(2-1), DCO$^+$(3-2) at 1.2 mm. Our adopted set of line frequencies is given in Table \[tab\_freq\]. The half-power beamwidths were $\sim 26 ~\arcsec$, $\sim 17 ~\arcsec$ and $\sim 10 ~\arcsec$ at 3 mm, 2 mm and 1.2 mm, respectively. We used four SIS heterodyne receivers simultaneously and an autocorrelation spectrometer as backend, with a spectral resolution of 20 kHz at 3 mm and 40 kHz at 2 mm and 1.2 mm. The corresponding velocity resolutions ranged from 0.05 to 0.08 km s$^{-1}$ per channel. The observations were done in single sideband mode, with sideband rejections of 0.01, 0.1 and 0.05 at 3 mm, 2 mm and 1.2 mm, respectively. The forward and beam efficiencies of the telescope used to convert antenna temperatures $T^*_\mathrm{A}$ into main beam temperatures $T_\mathrm{mb}$ are listed in Table \[tab\_freq\]. The system temperatures ($T^*_\mathrm{A}$ scale) ranged from $\sim 110$ K to $\sim 150$ K at 3 mm, $\sim 280$ K to $\sim 410$ K at 2 mm, and $\sim 300$ K to $\sim 550$ K at 1.2 mm. The telescope pointing was checked every $\sim 2$ hours on Saturn, 0528+134, and/or 0420-014 and found to be accurate to $\sim 3 ~\arcsec$ (rms). The telescope focus was optimized on Saturn every $\sim 3$ hours. Position switching observations were done with a reference position located at either ($\Delta\alpha$,$\Delta\delta$) = ($1200 ~\arcsec$,$-1200 ~\arcsec$) or ($80 ~\arcsec$,$-80 ~\arcsec$) relative to the (0,0) position (envelope center as measured in the 1.3mm continuum). Extensive mapping was performed in the “on-the-fly” mode [@Ungerechts00]. A few additional C$^{18}$O(1-0), C$^{18}$O(2-1), CS(3-2), C$^{34}$S(3-2), CS(5-4) and H$^{13}$CO$^+$(3-2) position-switch observations, performed in September 1997, November 2000 and October 2001, will also be used here. All of these single-dish data were reduced with the CLASS software package [@Buisson02].
In addition, we also observed IRAM 04191 with the 5-antenna IRAM Plateau de Bure interferometer (PdBI) in its B1, C2, D configurations between January and April 1999. The two receivers of each antenna were tuned to the CS(2-1) and H$_2$CO(3$_{12}$-2$_{11}$) lines, with spectral resolutions of 40 kHz and 80 kHz, corresponding to velocity resolutions of 0.12 km s$^{-1}$ and 0.10 km s$^{-1}$, respectively. The four remaining windows of the PdBI correlator were used to record the continuum emission with a total bandwidth of 300 MHz at both 98 GHz ($\lambda \sim 3$ mm) and 227 GHz ($\lambda \sim 1.3$ mm). The (naturally-weighted) synthesized half-power beamwidths were $4.5~\arcsec \times 4.4~\arcsec$ (630 AU $\times$ 620 AU) at 98 GHz and $1.9~\arcsec \times 1.8~\arcsec$ (270 AU $\times$ 250 AU) at 227 GHz, and the (FWHP) primary beams $\sim 50 ~\arcsec$ and $\sim 25 ~\arcsec$, respectively. The correlator bandpass was calibrated on the strong source 3C273. Several nearby phase calibrators were observed to determine the time-dependent complex antenna gains. The absolute calibration uncertainty was estimated to be $\sim 15\% $. The data were calibrated and imaged using the CLIC [@Lucas99] and Mapping [@Guilloteau02] packages in the IRAM software.
------------------------------- ------------------- ----------------------------- ---------- -------------------------- --------------------------
$\sigma_\mathrm{v}$$^{(2)}$ Ref. F$_\mathrm{eff}$$^{(4)}$ B$_\mathrm{eff}$$^{(4)}$
$^{(3)}$ ($\%$) ($\%$)
C$_3$H$_2$(2$_{12}$-1$_{01}$) $85338.905(6)$ 0.02 (1) 92 73
N$_2$H$^+$[(101-012)]{} $93176.258(7)$ 0.02 (2) 92 73
C$^{34}$S(2-1) $96412.952(1)$ 0.003 (3) 92 73
CS(2-1) $97980.953(1)$ 0.003 (3) 90 73
C$^{18}$O(1-0) $109782.175(2)$ 0.005 (4) 92 73
C$^{34}$S(3-2) $144617.101(1)$ 0.002 (3) 87 65
90 54
CS(3-2) $146969.026(1)$ 0.002 (3) 90 54
C$^{18}$O(2-1) $219560.3541(15)$ 0.002 (5) 86 42
CS(5-4) $244935.555(1)$ 0.001 (3) 84 42
------------------------------- ------------------- ----------------------------- ---------- -------------------------- --------------------------
: Adopted line rest frequencies and telescope efficiencies.[]{data-label="tab_freq"}
[The frequency uncertainty in units of the last significant digit is given in parentheses.]{}
[Frequency uncertainty converted in units of velocity.]{}
[(1) Laboratory measurement from @Vrtilek87; (2) Observational result from @Lee01; (3) Laboratory measurement from @Gottlieb02. (4) Laboratory measurement from Klapper (2001, private communication). (5) Laboratory measurement from @Klapper01.]{}
[Forward and beam efficiencies of the IRAM 30m telescope (@Wild99 and http://www.iram.es/).]{}
Results and qualitative interpretation : Evidence for rotation and infall {#obs_ana}
=========================================================================
Weak 1.3 mm continuum detection at PdBI {#obs_pdbi}
---------------------------------------
The PdBI continuum image at 227 GHz (see Fig. \[n2h+flow\]) reveals weak, point-like emission centered at $\alpha_\mathrm{(2000)}= 04^\mathrm{h}21^\mathrm{m}56^\mathrm{s}_{^.}91$, $\delta_\mathrm{(2000)}= 15\degr 29\arcmin 46.1\arcsec$, a position which should be accurate to better than $0.5\arcsec$. (The PdBI position is offset by $\sim 5 ~\arcsec$ from the N$_2$H$^+$(1-0) centroid observed at the 30m telescope, which is only marginally significant given the $\sim 3 ~\arcsec$ single-dish pointing accuracy.) We measure a peak 227 GHz flux density of $S_\mathrm{peak}^{1.9\arcsec} = 6.1 \pm 0.4$ mJy/$1.9\arcsec$-beam at the central position. The 227 GHz emission is slightly resolved and a Gaussian fit performed in the uv-plane yields a deconvolved FWHM size of $(1.1 \arcsec \pm 0.4 \arcsec) \times (0.6 \arcsec \pm 0.3 \arcsec)$ with a position angle P.A. $= 84~\degr$. At 98 GHz, the rms noise level is 0.14 mJy/$4.5\arcsec$-beam and we do not detect any emission above 0.6 mJy/$4.5\arcsec$-beam ($\sim 4\, \sigma $)[^1].
For comparison, using the IRAM 30m telescope equipped with the MPIfR bolometer array (MAMBO), @Motte01 measured a peak flux density of $S_\mathrm{peak}^{11\arcsec} = 110 \pm 7$ mJy/$11\arcsec$-beam at 1.3 mm [$\nu_{eff} \sim 240$ GHz – e.g. @Broguiere02] and a radial intensity profile of the form $I(\theta) \propto \theta^{-0.6 \pm 0.1}$ in the range of angular radii $\theta = 11~\arcsec$ to $\theta = 100~\arcsec$. This extended 1.3 mm continuum source is clearly the dust counterpart of the circumstellar gas envelope/core observed in N$_2$H$^+$ (cf. Fig. \[n2h+flow\]). Assuming the same radial intensity profile holds at smaller angular radii, one expects the peak flux density of the envelope to scale as $S_\mathrm{peak}(\theta_b) \propto \theta_b^{1.4 \pm 0.1}$ with beamsize $\theta_b $. If we adopt a dust opacity index $\beta = 1.5$ to account for the slight difference in observing frequency between the 30m and PdBI measurements, this flux-density scaling predicts $S_\mathrm{peak}^{1.9\arcsec} \approx 7.7$ mJy/$1.9\arcsec$-beam, which is only 25% larger than the PdBI peak flux density quoted above. Given the relative calibration uncertainties, this comparison suggests that the weak 227 GHz emission detected at PdBI arises from the inner part of the envelope seen at the 30m telescope rather than from an accretion disk surrounding the central protostellar object. Furthermore, a single power law $I(\theta) \propto \theta^{-0.5 \pm 0.2}$ appears to characterize the radial intensity profile of the envelope over the whole range of angular radii from $\sim 1 \arcsec$ to $\sim 100 \arcsec$. Such an intensity profile implies either a relatively flat $\rho \propto r^{-1}$ density profile if central heating with $T_\mathrm{dust} \propto r^{-0.4}$ applies in the inner part of the envelope (see Fig. \[model\_1d\] below), or a $\rho \propto r^{-1.5}$ density profile if the dust temperature is uniform.\
The weak emission detected at PdBI sets strong constraints on the mass and size of any central accretion disk. Assuming optically thin emission, a dust mass opacity of 0.02 cm$^2$ g$^{-1}$ typical of circumstellar disks [cf. @Beckwith90] and a mean dust temperature of 20 K, the PdBI peak flux density yields $M_\mathrm{disk} < 1 \times 10^{-3}$ M$_\odot$. Alternatively, the assumption of optically thick dust emission at 20 K implies $R_\mathrm{disk} < 10$ AU.
Fast, differential rotation {#obs_rotation}
---------------------------
All of the centroid velocity (first-order moment) maps taken at the 30m telescope in small optical depth lines, such as C$_3$H$_2$(2$_{12}$-1$_{01}$), N$_2$H$^+$(101-012), H$^{13}$CO$^+$(1-0), and C$^{34}$S(2-1), show a clear velocity gradient across the envelope (see, e.g., Fig. \[map\_vcent\]). The (south)-east part is redshifted with respect to the source systemic velocity, while the (north)-west part is blueshifted. We have applied the sector method described by @Arquilla86 to measure the direction of this velocity gradient in the C$_3$H$_2$(2$_{12}$-1$_{01}$) map using Gaussian fits to the spectra. We obtain a position angle P.A. $\sim 114\degr \pm 8\degr$ [^2] (cf. Fig. \[map\_vcent\]). The velocity gradient thus lies along the major axis of the elongated dust/N$_2$H$^+$ core (P.A. $\sim 120\degr$), i.e., perpendicular to the outflow axis. Coupled to the high degree of symmetry of the position-velocity diagrams shown in Fig. \[pvdiag\] with respect to envelope center, this strongly suggests that the envelope is rotating about an axis coinciding with the outflow axis (cf. Fig. \[n2h+flow\]). Turbulent motions would produce a more random velocity field [cf. @Burkert00] and are weak here anyway (see § \[obs\_width\] and Fig. \[model\_vel\]b below).
From now on, we adopt a position angle P.A. $= 28\degr$ for the projection of the rotation/outflow axis onto the plane of the sky. To minimize contamination by the outflow, we analyze the velocity structure of the envelope along the axis perpendicular to the outflow (P.A. $= 28\degr$) and going through the center. Along this axis, the magnitude of the velocity gradient estimated from linear fits to the C$_3$H$_2$(2$_{12}$-1$_{01}$), H$^{13}$CO$^+$(1-0), C$^{34}$S(2-1), and N$_2$H$^+$(101-012) position-velocity diagrams is $\sim 6$ km s$^{-1}$ pc$^{-1}$, $\sim
7$ km s$^{-1}$ pc$^{-1}$, $\sim 6$ km s$^{-1}$ pc$^{-1}$, and 10 km s$^{-1}$ pc$^{-1}$, respectively, increasing from north-west to south-east over $40 ~\arcsec$ (see Fig. \[pvdiag\]). The mean velocity gradient is $7 \pm 2$ km s$^{-1}$ pc$^{-1}$. Given the estimated viewing angle $i = 50\degr$ of the flattened envelope (cf. § \[intro\_iram04191\]), this implies a mean rotational angular velocity of $\Omega = 9 \pm 3$ km s$^{-1}$ pc$^{-1}$ in the inner $r \sim 2800$ AU radius region[^3].
The C$_3$H$_2$(2$_{12}$-1$_{01}$), H$^{13}$CO$^+$(1-0), and N$_2$H$^+$(101-012) position-velocity diagrams shown in Fig. \[pvdiag\] may be slightly contaminated by a secondary, redshifted component toward the east. We have used the GAUSSCLUMPS algorithm of @Stutzki90 [see also @Kramer98] to try and subtract this secondary component. The algorithm finds a Gaussian component at approximately ($+50~\arcsec$, $+30~\arcsec$) in C$_3$H$_2$(2$_{12}$-1$_{01}$) and ($+30\arcsec$, $+20\arcsec$) in H$^{13}$CO$^+$(1-0). The removal of this secondary component reduces the magnitude of the velocity gradient measured in C$_3$H$_2$(2$_{12}$-1$_{01}$) and H$^{13}$CO$^+$(1-0) by $\sim 30\%$. GAUSSCLUMPS fails to identify any secondary component near the same location in the N$_2$H$^+$(101-012) data. The velocity gradient thus seems to be significantly larger in N$_2$H$^+$(101-012) than in the other three lines (see below).
Remarkably, all of the position-velocity diagrams shown here present a S shape, clearly seen on the peak velocity curves derived from Gaussian fits (see Fig. \[pvdiag\]). On either side of the source, the velocity shift with respect to the systemic velocity increases in absolute value up to an angular radius $\theta_\mathrm{m} \sim 25 \pm 10\arcsec$ (i.e., one full beamwidth of the 30m telescope at 3mm, corresponding to $r_\mathrm{m} \sim 3500 \pm 1400$ AU), and then decreases for $\theta > \theta_\mathrm{m}$. This is further illustrated in Fig. \[spec\_rot\] which shows the C$_3$H$_2$(2$_{12}$-1$_{01}$) and C$^{34}$S(2-1) spectra observed every half beamwidth (Nyquist sampling) at symmetric positions with respect to source center along the direction perpendicular to the outflow axis. It can be seen that the velocity shift between symmetric positions decreases from $0.17$ km s$^{-1}$ (i.e., 2.5 channels) at $\pm 26 \arcsec$ to $0.09$ km s$^{-1}$ (1.3 ch.) at $\pm 52 \arcsec$ in C$_3$H$_2$(2$_{12}$-1$_{01}$). In C$^{34}$S(2-1), the velocity shift varies from $0.16$ km s$^{-1}$ (2.6 ch.) at $\pm 26 \arcsec$ to $0.14$ km s$^{-1}$ (2.3 ch.) at $\pm 39 \arcsec$. These values translate into a decrease of the C$_3$H$_2$(2$_{12}$-1$_{01}$) velocity gradient from 4.8 km s$^{-1}$ pc$^{-1}$ to 1.3 km s$^{-1}$ pc$^{-1}$ between $26 \arcsec$ and $52 \arcsec$, and to a decrease of the C$^{34}$S(2-1) velocity gradient from 4.5 km s$^{-1}$ pc$^{-1}$ to 2.6 km s$^{-1}$ pc$^{-1}$ between $26 \arcsec$ and $39 \arcsec$. The decrease of the velocity shift with radius beyond $\pm 26 \arcsec$ is [*in marked contrast with the linear increase expected in the case of solid-body rotation*]{}. We thus conclude that there is strong evidence for differential rotation in the envelope beyond $\sim 3500$ AU.\
Correcting for inclination and taking all four lines of Fig. \[pvdiag\] into account, we estimate that the rotational angular velocity decreases by a factor of $\sim 5$ from $\Omega = 9 \pm 3$ km s$^{-1}$ pc$^{-1}$ ($\sim 3 \times 10^{-13}$ rad s$^{-1}$) at $r = 2800$ AU ($20 \arcsec$) to $\Omega = 1.9 \pm 0.2$ km s$^{-1}$ pc$^{-1}$ ($\sim 6 \times 10^{-14}$ rad s$^{-1}$) at $r = 7000$ AU ($50 \arcsec$). At a radius $r \sim 11000$ AU, our C$^{18}$O(1-0) and C$^{18}$O(2-1) observations suggest an even smaller angular velocity, $\Omega \simlt 0.5-1$ km s$^{-1}$ pc$^{-1}$ ($\sim 1.5-3 \times 10^{-14}$ rad s$^{-1}$). On scales smaller than the beam, the intrinsic rotation velocity pattern is uncertain due to insufficient spatial resolution (the beam HWHM angular radius is $\sim 13~\arcsec$ at 3 mm, corresponding to a physical radius of $\sim 1800$ AU). However, two indirect arguments suggest that the differential rotation pattern observed here between $\sim 3500$ AU and $\sim 11000$ AU continues down to smaller ($\sim 1000-2000$ AU) scales. First, such a differential rotation pattern, combined with a lower level of molecular depletion near envelope center in N$_2$H$^+$ (cf. § \[obs\_width\] and § \[molec\_depl\] below), would explain the higher velocity gradient (10 km s$^{-1}$ pc$^{-1}$) measured in N$_2$H$^+$(101-012) over $\pm 20 \arcsec$ compared to the other three lines shown in Fig. \[pvdiag\]. Second, using NH$_3$ interferometric observations sensitive to $\sim 5 \arcsec-15 \arcsec $ scales, @Wootten01 have recently reported an even larger gradient ($\sim 15$ km s$^{-1}$ pc$^{-1}$) than our present N$_2$H$^+$ value.
Spectroscopic signature of collapse {#obs_infall}
-----------------------------------
As already pointed out by @Andre99, the classical spectroscopic signature of infall motions [cf. @Evans99; @Myers00] is seen toward IRAM 04191. Optically thick lines such as CS(2-1), CS(3-2), CS(5-4), H$_2$CO(2$_{12}$-1$_{11}$) and H$_2$CO(3$_{12}$-2$_{11}$) are double-peaked and skewed to the blue, while low optical depth lines such as C$^{34}$S(2-1) and C$^{34}$S(3-2) peak in the dip of the self-absorbed lines (see spectra observed at the central position in Fig. \[spec\_center\]). Blue-skewed CS(2-1) and CS(3-2) spectra are observed in an extended region, up to an angular radius of at least $40 ''$ (5600 AU) from source center (see cut taken perpendicular to the outflow axis in Fig. \[compare\_1d\] below). Such asymmetric line profiles with a blue peak stronger than the red peak are expected in a collapsing envelope when the line excitation temperature increases toward the center. There is therefore strong evidence for the presence of extended inward motions in the IRAM 04191 envelope.
The blue-to-red peak intensity ratio of the CS lines is weaker toward the south-east (i.e. the envelope hemisphere red-shifted by rotation), while the asymmetry is stronger toward the north-west (i.e. the hemisphere blue-shifted by rotation) where the red peak is even barely visible. This behavior is in qualitative agreement with the expected distortion of the infall asymmetry due to rotation when the rotation velocity does not dominate over the infall velocity[^4] [@Zhou95].
The dips of the optically thick CS(2-1), CS(3-2), and CS(5-4) lines have velocities of $6.75 \pm 0.03$ km s$^{-1}$, $6.81 \pm 0.04$ km s$^{-1}$, and $6.86 \pm 0.05$ km s$^{-1}$, respectively, and are redshifted relative to the source systemic velocity of $6.66 \pm 0.03$ km s$^{-1}$ (see Fig. \[spec\_center\]). The latter value results from a 7-component Gaussian fit to the N$_2$H$^+$(1-0) multiplet, using the hyperfine structure (hfs) method of the CLASS reduction software, and assuming the relative frequencies and intensities of the 7 hyperfine components determined by @Caselli95, and the N$_2$H$^+$(101-012) rest frequency of @Lee01 (see Table \[tab\_freq\]). The uncertainty on the @Lee01 frequency is estimated to be $\sim 0.02$ km s$^{-1}$, to which we conservatively add (in quadrature) an uncertainty of 0.015 km s$^{-1}$ arising from a maximum pointing error of $\sim 5\arcsec $, given the velocity gradient discussed in § \[obs\_rotation\] above. This gives a final uncertainty of $\sim 0.03$ km s$^{-1}$ on the source systemic velocity. The dips of the self-absorbed CS(2-1), CS(3-2), and CS(5-4) spectra are thus redshifted relative to the systemic velocity by $0.09 \pm 0.04$ km s$^{-1}$, $0.15 \pm 0.05$ km s$^{-1}$, and $0.20 \pm 0.06$ km s$^{-1}$, respectively. These absorption dips are presumably produced by the outer layers of the front hemisphere of the envelope. The fact that they are redshifted provides a second indication that inward motions are present in the outer envelope where the opacity of the CS lines is of order unity. More quantitatively, the main-beam brightness temperature of the dip in the central CS(2-1) spectrum is $\sim 0.7$ K, which matches the peak temperature of the spectra taken at $\sim 70 ~\arcsec$ from source center. These spectra are still optically thick since we measure a CS(2-1) to C$^{34}$S(2-1) integrated intensity ratio of only $\sim 5$, i.e., $\sim 4$ times less than the standard CS to C$^{34}$S isotopic ratio of $\sim 22$ [@Wilson94]. Assuming a spherically symmetric envelope, we conclude that the absorbing shell producing the dip in the central CS(2-1) spectrum has an angular radius larger than $\sim 70\arcsec$. The observed $\sim 0.1$ km s$^{-1}$ redshift of the dip is thus suggestive of inward motions $\sim 0.1$ km s$^{-1}$ extending up to a radius of at least $10000$ AU. Radiative transfer simulations confirm this conclusion (see Sect. \[simul\_1d\] below).
On the other hand, the CS(5-4) emission is much more concentrated spatially than the CS(2-1) emission, and confined to the inner $20 ~\arcsec$ (FWHM) region (see the non-detection at $10\arcsec$ in Fig. \[compare\_1d\] below). The radius of the shell producing the absorption dip in the central CS(5-4) spectrum must therefore be smaller than $10 ~\arcsec$. The observed $\sim 0.2$ km s$^{-1}$ redshift of the CS(5-4) dip then suggests faster inward motions in the central ($r < 1400$ AU) region.
Linewidths: Evidence for turbulent infall ? {#obs_width}
-------------------------------------------
The C$^{34}$S(2-1) and C$^{34}$S(3-2) spectra displayed in Fig. \[spec\_center\] are slightly asymmetric and skewed to the blue, which suggests they are marginally optically thick and showing some infall asymmetry [see, e.g., Fig. 1 of @Myers95]. We measure (FWHM) linewidths of $0.38
\pm 0.01$ km s$^{-1}$ for C$^{34}$S(2-1) and $0.32 \pm 0.03$ km s$^{-1}$ for C$^{34}$S(3-2), which are 3.8 and 3.2 times larger than the thermal broadening of C$^{34}$S at a kinetic temperature T$_\mathrm{K} = 10$ K, respectively (see § \[mass\_tk\] for constraints on T$_\mathrm{K}$). Radiative transfer simulations (cf. Sect. \[simul\_1d\]) indicate opacities of $\sim 1.5$ and $\sim 1$ for C$^{34}$S(2-1) and C$^{34}$S(3-2), respectively. Therefore, line saturation effects cannot broaden the C$^{34}$S(2-1) and C$^{34}$S(3-2) spectra by more than $\sim 30 \%$ and $\sim 20 \%$, respectively, and the linewidths are primarily nonthermal. Motions such as infall, rotation, outflow, or “turbulence” along the line of sight are required to explain such nonthermal linewidths. The nonthermal motions do not dominate over thermal motions, however, since the C$^{34}$S(2-1) and C$^{34}$S(3-2) linewidths represent only $\sim 80 \%$ of the thermal velocity dispersion for a mean particle of molecular weight $\mu =2.33$. The N$_2$H$^+$(101-012) line is slightly broader, with a FWHM $\sim 0.55$ km s$^{-1}$, i.e., 1.2 times broader than the (mean particle) thermal velocity dispersion and $\sim 1.5$ times larger than the C$^{34}$S linewidths. The hyperfine structure fit to the N$_2$H$^+$ multiplet (see § \[obs\_infall\]) yields an optical depth of $\sim 0.85$ for the isolated N$_2$H$^+$(101-012) component, suggesting negligible ($\simlt 15\% $) optical depth broadening. The level of optical depth broadening should thus be more pronounced in C$^{34}$S and cannot explain the difference in linewidth between N$_2$H$^+$ and C$^{34}$S.
We propose that this difference in linewidth between N$_2$H$^+$ and C$^{34}$S results from a combination of higher infall/rotation velocities and lower N$_2$H$^+$ depletion toward the center. @Bergin97 have shown that sulphur-bearing molecules such as CS are strongly depleted when the density increases, whereas N$_2$H$^+$ remains in the gas phase, at least up to densities n$_{\mbox{\tiny H}_2}
\simlt 10^6$ cm$^{-3}$. Indeed, we measure a decrease of the C$^{34}$S(2-1)/N$_2$H$^+$(101-012) integrated intensity ratio by a factor of 2 from $\sim 5000$ AU to $\sim 2000$ AU. As both lines have nearly the same critical density and are approximately optically thin[^5], the decrease of the integrated intensity ratio may be interpreted as a decrease of the C$^{34}$S/N$_2$H$^+$ abundance ratio toward the center.
Finally, we note that the N$_2$H$^+$(101-012) linewidth peaks at the central position (as shown by a linewidth-position plot along the direction perpendicular to the ouflow axis). N$_2$H$^+$ may thus be more sensitive to higher velocity material produced by, e.g., infall, rotation, or outflow near the central protostellar object. As N$_2$H$^+$ is generally underabundant in molecular outflows [e.g. @Bachiller97], the central broadening of N$_2$H$^+$(1-0) is most likely due to infall and/or rotation motions.
The CS line wing emission {#obs_wing}
-------------------------
The morphology of the single-dish CS(2-1) maps integrated over the 4.6–6.1 km s$^{-1}$ (blue) and 7.1–8.6 km s$^{-1}$ (red) velocity ranges strongly suggests that the CS line wing emission is dominated by material associated with the outflow (see Fig. \[wing\_map\]a & Fig. \[wing\_map\]b). Some CS(2-1) emission is detected at the edges of both the red and the blue lobe of the CO outflow: the CS(2-1) red wing is relatively weak and concentrated at the south-eastern edge and the tip of the red CO lobe (Fig. \[wing\_map\]a), while the CS(2-1) blue wing is much stronger and distributed in two spots on either side of the blue CO lobe (Fig. \[wing\_map\]b). This “high-velocity” CS(2-1) emission is likely to arise from dense, shocked material entrained by the outflow.
The redshifted CS(2-1) emission detected on smaller scales by the PdBI interferometer in the 7–8 km s$^{-1}$ velocity range also appears to be associated with the outflow. This emission arises from the base of the red CO lobe and its shape closely follows the edges of the outflow lobe (see Fig. \[wing\_map\]c). No emission was detected by PdBI in the blue wing range (see Fig. \[wing\_map\]d) or at the source systemic velocity. The blueshifted CS(2-1) emission seen in the single-dish map (Fig. \[wing\_map\]b) thus appears to be more extended than the redshifted emission and is likely resolved out by the interferometer. Such a difference in spatial extent between blueshifted and redshifted emission, opposite to what infall motions would produce, is expected if the CS line wing emission arises from outflowing material. Likewise, the emission detected by the 30m telescope near the systemic velocity, i.e., close to the dip of the CS(2-1) line (Fig. \[spec\_center\]), arises from extended foreground material on scales $\sim 70\arcsec $ (see § \[obs\_infall\]) and is also resolved out by PdBI.
Low degree of ionization {#obs_ionization}
------------------------
Following, e.g., @Caselli98, we can estimate the degree of ionization in the IRAM 04191 envelope from the observed value of the abundance ratio R$_\mathrm{D} = [\mathrm{DCO}^+]/[\mathrm{HCO}^+]$. We measure an integrated intensity ratio $I_{\mathrm{DCO}^+}/I_{\mathrm{H}^{13}\mathrm{CO}^+} = 3.3 \pm 0.6$ at the central position. Assuming optically thin H$^{13}$CO$^+$(3-2) and DCO$^+$(3-2) emission, this implies $[\mathrm{DCO}^+]/[$H$^{13}$CO$^+$$] = 3.3 \pm 0.6$. Adopting a $[^{12}\mathrm{C}]/[^{13}\mathrm{C}]$ abundance ratio of 77 in the local ISM [@Wilson94], we thus derive R$_\mathrm{D} = 0.04 \pm 0.01$, which is identical to the ratio measured by @Caselli01b toward the central position of L1544. As the same density ($\sim 10^6$ cm$^{-3}$) is probed by the observations in both cases, we conclude that the ionization degree $x_i \sim 2 \times
10^{-9}$ derived in L1544 by @Caselli01b based on their chemical models should be representative of the ionization degree in the IRAM 04191 envelope at this density (assuming similar depletion factors for CO, which seems likely – see § \[molec\_depl\] below).
Radiative transfer modeling: 1D spherical simulations {#simul_1d}
=====================================================
Here, we use the radiative transfer code MAPYSO [@Blinder97] to model the observed line spectra and set quantitative constraints on the kinematics of the IRAM 04191 envelope (see Appendix for details about the code). For simplicity, we use a piecewise powerlaw description for the spatial variations of the kinetic temperature, density, molecular abundance, infall velocity, and rotational velocity (in Sect. \[simul\_2d\] below) across the envelope (see Fig. \[model\_1d\]).
Model inputs: mass distribution and kinetic temperature profile {#mass_tk}
---------------------------------------------------------------
We use the envelope mass distribution derived by @Motte01 (hereafter @Motte01) from 1.3 mm dust continuum observations with the 30m telescope. @Motte01 estimate that the envelope mass contained within a radius of 4200 AU is $M_\mathrm{env}(r < 4200~ \mathrm{AU}) = 0.45$ M$_\odot$ with an uncertainty of a factor of 2 on either side, assuming a mean dust temperature $T_\mathrm{dust} = 12.5 \pm 2.5$ K obtained by @Andre99 from a graybody fit to the $\lambda > 90$ $\mu$m portion of the SED (see Fig. 3 of @Andre99). In addition, the radial structure analysis of @Motte01 indicates an average radial intensity profile $I \propto \theta^{-m}$ with $m = 0.6 \pm 0.1$ in the range of angular radii $\theta \sim 11''-100''$ (i.e., $r \sim $ 1500-14000 AU). Assuming a dust temperature profile $T_\mathrm{dust} \propto r^{-q}$ with $q = -0.2 \pm 0.2$, @Motte01 obtain a density profile $\rho \propto r^{-p}$ with $p = m+1-q = 1.8 \pm 0.3$. Given the low bolometric luminosity of IRAM 04191, the dust temperature is indeed expected to rise outward ($q < 0$) due to external heating by the interstellar radiation field [e.g. @Masunaga00; @Evans01; @Zucconi01]. The regime of central heating by the accreting protostar ($q \sim 0.4$) is likely confined to the inner $\sim 2000$ AU radius region (cf. @Motte01). Since a good thermal coupling between gas and dust grains is expected for densities $n_{\mbox{\tiny H$_2$}} > 10^5$ cm$^{-3}$ [e.g. @Ceccarelli96; @Doty97], the gas kinetic temperature profile should track the dust temperature profile, at least up to a radius of $\sim 4000$ AU.
We also have some constraints on the gas kinetic temperature from our C$^{18}$O, N$_2$H$^+$, and CS observations. We measure a C$^{18}$O(2-1) to C$^{18}$O(1-0) integrated intensity ratio $I_{\mbox {\tiny C$^{18}$O(2-1)}}/I_{\mbox {\tiny C$^{18}$O(1-0)}} = 1.5 \pm
0.3$ in the range of angular radii $0''-60''$, using main beam temperatures and after degrading the resolution of the $J= 2-1$ data to that of the $J= 1-0$ data. As the critical densities of C$^{18}$O(1-0) and C$^{18}$O(2-1) are $3 \times 10^3$ cm$^{-3}$ and $2 \times 10^4$ cm$^{-3}$ at 10 K, respectively, both lines should be thermalized in most of the envelope. Assuming local thermodynamic equilibrium (LTE) and optically thin emission, we derive an excitation temperature $T_{\mbox{\tiny ex}}(\mbox{C$^{18}$O}) = 10 \pm 2$ K, which should be a good estimate of the gas kinetic temperature in the low-density outer ($r \sim 6000$ AU) part of the envelope probed by C$^{18}$O. Likewise, the excitation temperature of the dense-gas tracer N$_2$H$^+$(101-012) may be estimated from the relative intensities of the seven components of the N$_2$H$^+$(1-0) multiplet, assuming the same excitation temperature for all components (cf. § \[obs\_infall\]). This method yields $T_{\mbox{\tiny ex}}(\mbox{N$_2$H$^+$}) = 5.5 \pm 0.5$ K. With a critical density $\sim 2 \times 10^5$ cm$^{-3}$, the N$_2$H$^+$(1-0) multiplet is probably thermalized only in the inner ($r < 2000$ AU) envelope. We thus obtain a lower limit of $\sim 6$ K for the gas kinetic temperature in the dense ($n_{\mbox{\tiny H$_2$}} > 10^5$ cm$^{-3}$), inner part of the envelope. Finally, the weak intensities of the optically thick CS(2-1) and CS(3-2) lines require a low gas kinetic temperature $\sim$ 6-7 K in the range of radii $2000 - 6000$ AU. Given the density profile shown in Fig. \[model\_1d\]a, a uniform gas temperature of 10 K would produce CS(2-1) and CS(3-2) spectra with main beam temperatures about 1-2 K stronger than the observed temperatures. Beyond $r \sim 6000$ AU, the gas temperature is likely to increase to the typical $\sim 10$ K temperature of the Taurus cloud [see, e.g., @Benson89]. However, the gas temperature profile in the outer parts of the envelope has little influence on the CS and C$^{34}$S spectra since the observed lines are far from LTE there.
In summary, the gas kinetic temperature profile is likely to present a minimum of $\sim 6$ K at $\sim 10\arcsec-20\arcsec$ (i.e., $r \sim $ 1400-2800 AU) and to reach a value of $\sim 10$ K in the outer parts of the envelope.
Molecular depletion {#molec_depl}
-------------------
Assuming a standard isotopic ratio $\chi_{\mbox{\tiny CS}}/\chi_{\mbox{\tiny C$^{34}$S}} = 22$, a $[\mathrm{CS}]/[\mathrm{H}_2]$ abundance ratio of $8 \times 10^{-9}$ is required to match the C$^{34}$S(2-1) integrated intensity at an angular radius $40~\arcsec-50~\arcsec$. But a uniform abundance with such a value produces too strong C$^{34}$S(2-1), C$^{34}$S(3-2), and CS(5-4) spectra toward the center. A good fit to the C$^{34}$S(2-1) integrated line intensities is obtained by assuming that the relative CS abundance drops by a factor of $\sim 20$ toward the center (see also § \[obs\_width\]). Such a depletion factor for CS is comparable to those observed in starless cores such as L1544 [e.g. @Tafalla02].
Two regimes of infall {#analyze_1d}
---------------------
\[!ht\] {width="15cm"}
\[!ht\]
In Fig. \[compare\_1d\], we present a series of synthetic spectra emitted by a spherically symmetric model envelope with the input structure shown in Fig. \[model\_1d\] and our best estimate of the infall velocity field (shown by the solid line in Fig. \[model\_vel\]a below). The model spectra are overlaid on the multitransition CS and C$^{34}$S spectra observed along the direction perpendicular to the outflow axis, so as to minimize the effects of the outflow. The blue infall asymmetry of the model optically thick lines and the position of the CS(2-1) dip match the observations well. The widths of the optically thin lines are also well reproduced. The main shortcoming of the model is that it does not reproduce the fairly strong emission present in the wings of the observed CS spectra. As discussed in § \[obs\_wing\], we attribute these wings to the fraction of envelope material entrained by the outflow.
In order to determine the range of input model parameters that yield reasonably good fits to the observed CS and C$^{34}$S spectra, we have performed a comprehensive exploration of the parameter space, as illustrated in Figs. \[zoom\_vinfext\], \[zoom\_vinfint\] and \[zoom\_vturb\]. Strong constraints on the infall velocity arise from the small optical depth lines, i.e., C$^{34}$S(2-1) and C$^{34}$S(3-2). The widths of these lines set firm upper limits to the absolute value of the infall velocity on the size scale of the beam. We obtain $v_\mathrm{inf} \leq 0.15$ km s$^{-1}$ at $\sim 1750$ AU and $v_\mathrm{inf} \leq 0.2$ km s$^{-1}$ at $\sim 1150$ AU from C$^{34}$S(2-1) and C$^{34}$S(3-2), respectively. On the other hand, the amplitude of the blue asymmetry seen in the self-absorbed CS(2-1) and CS(3-2) lines, as well as the redshifted position of the corresponding absorption dips (see § \[obs\_infall\]), both require a relatively flat, extended infall velocity field with $v_\mathrm{inf} \sim 0.10 \pm 0.05$ km s$^{-1}$ up to $r \sim 10000-12000$ AU. The latter value approximately corresponds to the radius where the bulk of the absorption occurs in CS(2-1) and CS(3-2) (see § \[obs\_infall\]). These constraints are illustrated in Fig. \[zoom\_vinfext\] which shows the effect of varying the infall velocity field on the central CS(2-1) and C$^{34}$S(2-1) spectra. Three models are compared: the preferred model displayed in Fig. \[compare\_1d\] is shown in the central panel (Fig. \[zoom\_vinfext\]b), while models with lower and higher infall velocities in the outer part of the envelope are shown in the left (a) and right (c) panels, respectively. It can be seen that the position of the CS(2-1) absorption dip is not redshifted enough in model (a) and too redshifted in model (c) to match the observations. Furthermore, the CS(2-1) blue-to-red asymmetry is too weak in model (a) and the C$^{34}$S(2-1) line becomes too broad in model (c) compared to the observations. Only model (b) approximately reproduces the observed position of the CS(2-1) dip and the width of the C$^{34}$S(2-1) line.
In the context of a pure infall model, larger velocities ($v_\mathrm{inf} \sim 0.2-0.4 $ km s$^{-1}$) in the inner ($r \sim 700$ AU) part of the envelope are suggested by the broad linewidth of the central CS(5-4) spectrum. This is shown in Fig. \[zoom\_vinfint\] which compares three models differing in the magnitude of their infall velocity at 700 AU. A similar trend is indicated by the broadening of the N$_2$H$^+$(101-012) line toward the center (cf. § \[obs\_width\]). However, the central CS(5-4) spectrum (which does show infall asymmetry – cf. Fig. \[spec\_center\]) may be partly contaminated by small-scale structure in the outflow. In particular, the redshifted portion of the CS(5-4) spectrum, not reproduced by the model of Fig. \[compare\_1d\], may be related to the redshifted CS(2-1) emission detected by the interferometer on small scales (see § \[obs\_wing\]). Such a switch between infall-dominated CS emission on large scales and outflow-dominated CS emission on small scales is also observed in the Class 0 object B335 [cf. @Wilner00].
In addition to infall, a “turbulent” velocity field is needed to match the width of the dip, and consequently the velocity difference between the blue and the red peak, in the optically thick CS(2-1) and CS(3-2) spectra. Assuming a uniform turbulent velocity dispersion for simplicity, a good compromise between the upper limit set by the linewidth of the optically thin C$^{34}$S(2-1) and C$^{34}$S(3-2) spectra and the lower limit set by the width of the CS(2-1) and CS(3-2) dips is obtained for $\sigma_\mathrm{turb} = 0.085 \pm 0.02 $ km s$^{-1}$ (cf. Fig. \[zoom\_vturb\]). This is equivalent to $\Delta v_\mathrm{turb}^\mathrm{FWHM} = \sigma_\mathrm{turb} \times
\sqrt{8\,\rm{ln}2} = 0.20 \pm 0.05 $ km s$^{-1}$ and corresponds to only half the thermal broadening of the mean molecular particle at 10 K, showing that the IRAM 04191 envelope is “thermally-dominated” (see also § \[obs\_width\]) as are Taurus dense cores in general [e.g. @Myers99].
The main conclusions of our 1D exploration of the parameter space are summarized in Fig. \[model\_vel\]a and Fig. \[model\_vel\]b, where the shaded areas represent the ranges of infall velocities (a) and turbulent velocity dispersion (b) for which acceptable fits are found. Two infall regimes seem to stand out in Fig. \[model\_vel\]a: the infall velocity is relatively large ($v_\mathrm{inf} \simgt
0.2$ km s$^{-1}$, supersonic) and increases toward the center for $r \simlt 2000-3000$ AU, while it is smaller and roughly uniform at $v_\mathrm{inf} \sim 0.10 \pm 0.05$ km s$^{-1}$ between $\sim 2000-3000$ AU and $\sim 10000-12000$ AU. Given the density profile of Fig. \[model\_1d\]a, such an infall velocity field implies a mass infall rate of $\dot{M}_\mathrm{inf} \sim 3 \times 10^{-6}$ M$_\odot$ yr$^{-1}$ at $r = 1750$ AU. (The density and velocity profiles shown in Fig. \[model\_1d\]a and Fig. \[model\_vel\]a are such that $\dot{M}_\mathrm{inf} $ is roughly independent of radius.) Inside the $r \sim 11000 $ AU region (where non-zero inward motions are inferred), the fraction of envelope mass with supersonic ($\simgt 0.16-0.2$ km s$^{-1}$) infall velocities is estimated to be only $\sim 1-10\%$, depending on the exact value of the sound speed and exact form of the infall profile (see Fig. \[model\_vel\]a).
Radiative transfer modeling: Simulations with infall and rotation {#simul_2d}
=================================================================
\[!ht\]
Quasi 2D simulations {#mapyso_2d}
--------------------
To account for the effects of rotation in the envelope (see § \[obs\_rotation\] and Fig. \[pvdiag\]), we have performed “quasi”-2D simulations with the following approximation. The non-LTE level populations are still calculated with a 1D Monte Carlo method (see Appendix) assuming a spherical envelope with the same characteristics as the model described in § \[analyze\_1d\]. We then add a cylindrical rotation velocity field to the 1D model of § \[analyze\_1d\] and use the 2D version of the MAPYSO code to compute a proper radiative transfer integration along each line of sight. If we ignore departures from a spherical density distribution, this approach would remain strictly exact in the case of solid-body rotation, since the velocity difference between any couple of points projected on the axis joining these points is insensitive to the addition of a solid-body rotation component [cf. @Ward01]. In practice, however, the rotation observed here departs from solid body and the density distribution is not spherical. We therefore assume that, to first order, the line excitation is much more sensitive to the density distribution (averaged over angles) than to the velocity field. (In particular, we have checked that the profiles of excitation temperature are essentially insensitive to the infall velocity field as long as the turbulent velocity dispersion is of the same order as the infall velocity.) The rotation velocity field has nevertheless important effects on the shape of the line profiles, which we properly take into account here.
Two regimes of rotation {#analyze_2d}
-----------------------
{width="16.cm"}
The cylindrical rotation velocity field that we have added to the 1D spherical model of § \[analyze\_1d\] is shown in Fig. \[model\_vel\]c. The rotation axis is taken to coincide with the outflow axis, at an inclination angle $i = 50^{\circ}$ to the line of sight (see § \[intro\_iram04191\]) and a position angle P.A. $= 28~\degr$ in projection onto the plane of the sky (cf. Fig. \[n2h+flow\]).
Adding rotation yields two major improvements in the fits to the CS and C$^{34}$S spectra (see Fig. \[pvdiagmod\] and Fig. \[compare\_2d\]). First, the velocity gradient of the small optical depth C$^{34}$S(2-1) line along the direction perpendicular to the outflow is well reproduced, as shown in Fig. \[pvdiagmod\]. Both the model C$^{34}$S(2-1) position-velocity diagram (Fig. \[pvdiagmod\]a) and the centroid velocity curve agree well with the observations (Fig. \[pvdiagmod\]b).
Second, the blue asymmetry of the model CS(2-1) and CS(3-2) spectra is enhanced toward the north-west and attenuated toward the south-east, as seen on the observed spectra (see § \[obs\_infall\] and Fig. \[compare\_2d\]).
In agreement with the discussion of the position-velocity diagrams (§ \[obs\_rotation\] above), the present 2D modeling indicates that the envelope can be divided into two regions with distinct rotational characteristics (see the radial profiles derived for the rotational velocity and angular velocity in Fig. \[model\_vel\]c and Fig. \[model\_vel\]d, respectively). First, solid-body rotation is ruled out in the outer $3500 < r < 7000$ AU radius envelope, where a good fit to the centroid velocity curve is obtained with $v_\mathrm{rot} \propto r^{- 1.5 \pm 0.5}$, corresponding to an angular velocity $\Omega \propto r^{- 2.5 \pm 0.5}$. Second, in the inner $r < 3500$ AU radius region, our simulations suggest a velocity profile $v_\mathrm{rot} \propto r^{0.1 \pm 0.4}$, i.e., $\Omega \propto r^{-0.9 \pm 0.4}$, although the form of the position-velocity diagram is significantly influenced by the finite resolution of the observations. A rotation velocity $v_\mathrm{rot} = 0.20 \pm 0.04$ km s$^{-1}$, corresponding to $\Omega = 12 \pm 3$ km s$^{-1}$ pc$^{-1}$, is derived at a radius $r = 3500$ AU after correcting for inclination[^6].
Discussion: Comparison with collapse models {#discuss}
===========================================
In this section, we first summarize the predictions of collapse models and the main constraints derived from our observations (§ \[discuss\_models\] and § \[discuss\_summ\]), and then discuss the applicability of various models to IRAM 04191 (§ \[discuss\_shu\] to § \[discuss\_magnet\]). Some implications for the distribution and evolution of angular momentum during protostellar collapse are discussed in § \[discuss\_ang\].
Overview of model predictions {#discuss_models}
-----------------------------
@Whitworth85 have shown that there is a two-dimensional continuum of similarity solutions to the problem of isothermal spherical collapse. In this continuum, the well-known solutions proposed by @Shu77 and @Larson69-@Penston69 represent two extreme limits. All isothermal similarity solutions share a universal evolutionary pattern. At early times ($t < 0$), a compression wave (initiated by, e.g., an external disturbance) propagates inward at the sound speed, $\as$, leaving behind it a $\rho(r) \propto r^{-2}$ density profile and a uniform infall velocity field. This compression wave has zero amplitude in the limiting case of the Shu ‘inside-out’ collapse solution. At $t = 0$, the compression wave reaches the center and a point mass forms which subsequently grows by accretion. At later times ($t > 0$), this wave is reflected into a rarefaction or expansion wave, propagating outward (also at the sound speed) through the infalling gas, and leaving behind it free-fall density and velocity distributions (i.e., $\rho(r) \propto r^{-1.5}$ and $v(r) \propto r^{-0.5}$). The various solutions can be distinguished by the [*absolute*]{} values of the density and velocity at $t \sim 0$. The @Shu77 solution has the $\rho(r) = (\as^2/2\pi\,G)\ r^{-2}$ density distribution of a static ($v = 0$) singular isothermal sphere (SIS) at $t =0$, while the Larson-Penston (1969) solution is $\sim 4.4$ times denser and far from equilibrium ($v \approx -3.3\ \as$). The recent finding of inward motions of subsonic amplitude $\sim 0.02-0.10$ km s$^{-1}$, extended over $\sim 0.1$ pc ($\sim 20000$ AU) in the prestellar core L1544 [see @Tafalla98; @Williams99; @Caselli01a] suggests that true protostellar collapse in Taurus proceeds in a manner which is neither the Shu nor the Larson-Penston flow, and is perhaps more reminiscent of an intermediate similarity solution.
In practice, the initial conditions for fast protostellar collapse are not strictly self-similar and involve a density profile that is flat at small radii [e.g. @Ward94; @Andre96] and bounded or sharp-edged at some outer radius $\rout $ [e.g. @Motte98; @Bacmann00] like a finite-sized Bonnor-Ebert isothermal sphere [e.g. @Bonnor56; @Alves01]. A number of recent numerical (magneto)hydrodynamic simulations or simplified analytical calculations attempt to describe the collapse in such a situation, either in the absence [e.g. @Foster93; @Henriksen97; @Masunaga98; @Hennebelle02] or in the presence [e.g. @Tomisaka96; @Basu97; @Safier97; @Li98; @Ciolek98] of magnetic fields. The Larson-Penston similarity solution is found to describe the collapse quite satisfactorily near $t= 0$ (at least for small radii), but the Shu solution is more adequate at intermediate $t \geq 0$ times, before the expansion wave reaches the edge of the initial, pre-collapse dense core.
When rotation is included, a rotationally-supported disk develops at the center of the infalling envelope during the accretion phase (i.e. at $t > 0$). The size scale of this disk is determined by the centrifugal radius, $R_C $, which defines the position where the centrifugal force balances gravity in the equatorial plane. Strong departures from a spherical density distribution and a purely radial inflow in the envelope are expected to occur on size scales of order (or smaller than) $R_C $ [cf. @Chevalier83; @Hartmann98]. Most collapse models predict that $R_C \equiv j^2/Gm$ should increase with time as material of higher and higher specific angular momentum $j = \Omega \, R^2$ falls in, but the exact dependence on time $t$, or alternatively accumulated central mass $m$, varies from model to model, according to the distributions of mass and angular momentum at $t = 0$. For instance, $R_C $ scales as $m^3$ or $t^3$ in the Shu model (@Terebey84), which assumes solid-body rotation at point mass formation. By contrast, the dependence of $R_C $ on $m$ is only linear in the magnetically-controlled collapse model of @Basu98 [see also @Krasnopolsky02].
Summary of observational constraints {#discuss_summ}
------------------------------------
The analysis of our line observations (§ \[simul\_1d\] and § \[simul\_2d\]) indicates that both the infall and rotation velocity fields of the IRAM 04191 envelope are characterized by an inner and an outer regime (see Fig. \[model\_vel\]).\
Our 1D radiative transfer simulations (§ \[simul\_1d\]) indicate that the infall velocity profile is flat with $v_\mathrm{inf} \sim
0.1$ km s$^{-1}$ between $r_\mathrm{i} \sim 2000$ AU and $r_\mathrm{i,o} \sim
10000-12000$ AU. Higher infall velocities at radii $r < r_\mathrm{i}$ are suggested the CS observations, which are consistent with a free-fall velocity field ($v_\mathrm{inf} \propto r^{-0.5}$) at $r < r_\mathrm{i}$. The width of the optically thin C$^{34}$S lines strongly constrains $v_\mathrm{inf}$ to be $\simlt 0.15$ km s$^{-1}$ at $r \sim r_\mathrm{i}$ (cf. Fig. \[model\_vel\]a).
The position-velocity diagrams observed in optically thin lines (§ \[obs\_rotation\]) show that the envelope is differentially rotating with an angular velocity profile $\Omega \propto r^{-2.5 \pm 0.5}$ between $r_\mathrm{m} \sim 3500$ AU and $r_\mathrm{m,o} \sim 7000$ AU, and $\Omega \sim 12$ km s$^{-1}$ pc$^{-1}$ at $r \sim r_\mathrm{m}$. Although the limited spatial resolution of our observations prevents us from deriving accurate values in the inner region, the rotation profile is definitely shallower for $r < r_\mathrm{m}$ (cf. Fig. \[model\_vel\]d). A follow-up interferometric study is underway to provide more accurate constraints on the angular velocity in this inner region.
Although the two critical radii $r_\mathrm{i}$ and $r_\mathrm{m}$ differ by less than a factor of $\sim 2$, it is unclear whether they are physically related or not. The inner $r < r_\mathrm{i}$ envelope may correspond to the free-fall region developing inside the expansion wavefront, while the outer $r > r_\mathrm{i}$ envelope may be dominated by the flat, extended inward velocity field set up by the compression wave at $t < 0$ (see § \[discuss\_models\] above). If the current radius of the expansion wave is indeed $\sim r_\mathrm{i}$, then the age of the central protostellar object should be $t \sim r_\mathrm{i}/a_s \simlt 5 \times 10^4$ yr (assuming a propagation speed $a_s \sim 0.2$ km s$^{-1}$), in rough agreement with the estimated age of $\simlt 3 \times 10^4$ yr (§ \[intro\_iram04191\]). On the other hand, the fastly rotating $r < r_\mathrm{m}$ region may be a dynamically collapsing ‘supercritical’ core in the process of decoupling from the ambient medium, and the outer $r_\mathrm{m} < r <
r_\mathrm{m,o}$ envelope may be a transition region between the protostar and the background cloud (see § \[discuss\_magnet\] below).
### Centrifugal support {#discuss_cent}
Using our derived model for the structure and kinematics of the IRAM 04191 envelope (e.g. Fig. \[model\_1d\] and Fig. \[model\_vel\]), we can estimate the dynamical importance of rotation in the envelope. The ratio of the centrifugal acceleration $a_\mathrm{centr} = v_{\mathrm{rot}}^2 / r$ to the local gravitational field $g_\mathrm{r} = G \times [M_{\mathrm{env}}(r)+M_\star] / r^2$ is shown in Fig. \[ratioacc\]. Here, $M_{\mathrm{env}}(r)$ is the envelope mass within radius $r$ corresponding to the density profile of Fig. \[model\_1d\]a, and $M_\star$ is the uncertain mass of the central protostellar object $\sim 0.03-0.1\, M_\odot$ (cf. @Andre99). We estimate a ratio $a_\mathrm{centr}/g_\mathrm{r} \sim 0.4 \pm 0.2$ at the radius $r_\mathrm{m} = 3500$ AU of the rapidly rotating inner envelope, showing that the centrifugal acceleration is a sizeable fraction of the gravitational acceleration in this region. Comparable values of $a_\mathrm{centr}/g_\mathrm{r} $ are nevertheless obtained in some magnetic models of cloud collapse [see Fig. 4 of @Basu94 and § \[discuss\_magnet\] below].
We estimate the centrifugal radius to be $R_\mathrm{c} < 400$ AU, assuming the maximum rotation velocity profile consistent with the observations (cf. Fig. \[model\_vel\]c) and a stellar mass $M_\star = 0.03\, M_\odot$. In principle, a large centrifugal radius should imply the presence of a large centrifugal disk around the central object. @Stahler94 have investigated the initial growth of protostellar disks in the context of the rotating collapse picture of TSC84 and have shown that a 3-component structure should develop inside $R_\mathrm{c} $: a dense inner accretion disk of radius $R_\mathrm{disk }\sim 0.34 R_\mathrm{c} $, a ring where material and angular momentum pile up at $R_\mathrm{disk }$, and a low-density outer disk where material travels at high velocity between $ R_\mathrm{c} $ and $R_\mathrm{disk }$. If $R_\mathrm{c} \sim 200$ AU, we may thus expect $R_\mathrm{disk } \sim 70$ AU. Our 1.3 mm continuum interferometric observations set a firm upper limit to the radius of any dense inner circumstellar structure around IRAM 04191: $R_\mathrm{disk} < 10$ AU (cf. § \[obs\_pdbi\]). This suggests that the actual centrifugal radius is significantly smaller than 200 AU or else that the presence of a protostellar companion at $r \sim 30$ AU may have cleared a substantial gap in the inner accretion disk [cf. @Jensen96]. @Artymowicz94 show that, due to tidal disk truncation in binary systems, any individual disk must have an outer radius less than half the binary separation, and any circumbinary disk must have an inner radius more than roughly twice the binary separation. The non-detection of a dense $\sim 70$ AU radius disk by the PdBI interferometer could thus be accounted for by this effect if IRAM 04191 were a protobinary of separation $\sim 30$ AU (i.e. $\sim 0.2\arcsec $).
Problems with the inside-out collapse model {#discuss_shu}
-------------------------------------------
The inside-out, isothermal collapse model described by @Shu77, and its 2D extension including rotation (@Terebey84), has been widely used to explain the infall spectral signature seen in the envelopes of low-mass protostars [e.g. @Zhou93; @Myers95; @Choi95; @Zhou96; @Hoger00]. It accounts relatively well for the densities and accretion rates measured in Taurus protostellar envelopes [e.g. @Ohashi99; @Motte01; this paper]. This model is, however, inconsistent with the combined density and infall velocity profiles measured for IRAM 04191 (cf. Fig. \[model\_1d\]a and Fig. \[model\_vel\]a). An expansion wave radius $\simgt 10000$ AU is indeed required to reproduce the blue asymmetry of the central CS(2-1) spectrum in the context of the inside-out collapse picture. But such a large infall radius then implies high infall velocities at small radii $r \sim 1500$ AU which in turn yield central C$^{34}$S(2-1) and C$^{34}$S(3-2) spectra that are 2–3 times too wide compared to the observed linewidths. Conversely, a model with an expansion wave radius of only $\sim 2000$ AU yields correct C$^{34}$S(2-1) and C$^{34}$S(3-2) linewidths but fails to reproduce the strong asymmetry observed in CS(2-1) and CS(3-2). It is clearly because of the absence of significant inward motions at $t < 0$ that the inside-out collapse model cannot account for the infall velocity profile observed here. A similar inconsistency with the inside-out collapse model has been noted previously in prestellar cores such as L1544 by @Tafalla98 and @Lee01.
Comparison with other thermal models of collapse {#discuss_extend}
------------------------------------------------
With an age $\simlt 3-5 \times 10^4$ yr for the central IRAM 04191 protostellar object (see § \[intro\_iram04191\] and § \[discuss\_summ\]) and a sound speed $\as \sim 0.19$ km s$^{-1}$ ($T_\mathrm{K} = 10$ K), the radius of the expansion wave must be smaller than the inner radius $r_\mathrm{i} \sim 2000$ AU of the extended region where a flat infall velocity field is inferred (see Fig. \[model\_vel\]a). The kinematics and density structure of the outer $r > r_\mathrm{i}$ envelope should thus still reflect the physical conditions at $t=0$. Accordingly, the observation of substantial infall velocities between $r_\mathrm{i}$ and $r_\mathrm{i,o} \sim 11000$ AU points to collapse models that are more dynamical than the Shu solution and involve the propagation of a finite-amplitude compression wave prior to the formation of the central object [cf. § \[discuss\_models\] and @Whitworth85]. On the other hand, the infall velocities derived in the outer ($r > r_\mathrm{i}$) envelope are only subsonic (approximately half the sound speed), and clearly inconsistent with the supersonic infall velocities $\sim 3.3\as$ characterizing the Larson-Penston isothermal similarity solution at $t = 0$ (cf. § \[discuss\_models\]). The infall velocity field derived here (cf. Fig. \[model\_vel\]a) is suggestive of a moderately dynamical collapse model, intermediate between the Shu similarity solution (zero-amplitude compression wave at $t < 0$) and the Larson-Penston solution (strong compression wave at $t < 0$).
Qualitatively at least, such a moderately dynamical infall velocity field resembles that achieved during the collapse of a finite-sized, Bonnor-Ebert isothermal sphere. For instance, in their numerical simulations of the collapse of critically stable Bonnor-Ebert spheres (without magnetic field or rotation), @Foster93 found infall velocities at $t=0$ ranging from $3.3\as$ near the origin to 0 at the outer boundary radius. Assuming that the IRAM 04191 dense core was initially a marginally stable Bonnor-Ebert sphere with a center-to-edge density contrast of $\sim 14$, the central density at the onset of collapse must have been $\sim 5 \times 10^4$ cm$^{-3}$, given the typical outer density $\sim 3 \times 10^3$ cm$^{-3}$ observed in prestellar cores [e.g. @Bacmann00]. Adopting a temperature of 10 K, Fig. 1 of @Foster93 then predicts an infall velocity at $t=0$ varying from $\as \sim 0.2$ km s$^{-1}$ at $\sim 6000$ AU to $0.5\as \sim 0.1$ km s$^{-1}$ at $\sim 15000$ AU. Even if this represents a somewhat stronger variation of infall velocity with radius than derived in the case of IRAM 04191 (Fig. \[model\_vel\]a), the @Foster93 model clearly provides a much better fit to the observations than either the Shu or the Larson-Penston similarity solution. A definite problem with the critical Bonnor-Ebert model, however, is that as much as 44% of the envelope mass is predicted to flow in supersonically at $t = 0$ [@Foster93], and progressively more mass at $t > 0$. This is much larger than the $\sim 1-10\% $ mass fraction derived in § \[analyze\_1d\] for IRAM 04191. We conclude that spherically symmetric collapse models in which thermal pressure provides the only force opposing gravity tend to be too dynamical and are only marginally consistent with our observational constraints. It is likely that the inclusion of rotation in thermal models would improve the comparison with observations and may even account for the flattened shape of the protostellar envelope seen in Fig. \[n2h+flow\] (P. Hennebelle, private communication). However, a generic feature of rotating thermal collapse models is that, due to simultaneous conservation of energy and angular momentum, they tend to predict similar forms for the rotation and infall velocity profiles at radii (much) larger than the centrifugal radius [e.g. @Saigo98]. This is at variance with the steeply declining rotation velocity profile and flat infall velocity profile we observe beyond 3500 AU (a radius much larger than the estimated centrifugal radius – see § \[discuss\_cent\] above). In fact, [*the strong decline of the rotation velocity profile beyond 3500 AU suggests that angular momentum is not conserved in the outer envelope.*]{} In the next subsection, we propose that this results from magnetic braking.
Comparison with magnetically-controlled collapse models {#discuss_magnet}
-------------------------------------------------------
Magnetic ambipolar diffusion models [e.g. @Ciolek94; @Basu94; @Basu95a; @Basu95b – hereafter @Basu94, @Basu95a, @Basu95b] are another class of models which yield non-zero inward velocities in an extended region prior to point mass formation. Ambipolar diffusion has been invoked by @Ciolek00 to explain the extended inward motions observed in the Taurus prestellar core L1544 [see @Tafalla98; @Williams99; @Caselli01a]. The models start from a magnetically subcritical cloud, initially supported against gravitational collapse by a static magnetic field, and predict an evolution in two phases. During the first, quasistatic phase, the subcritical cloud contracts along directions perpendicular to the field lines through ambipolar diffusion. The gravitationally-induced inward drift of the neutral species is slowed down by collisions with the ions which are well coupled to the magnetic field. The central mass to magnetic flux ratio increases with time, until it reaches the critical value for collapse, $(1/2\pi)G^{-0.5}$. The inner region of the cloud then becomes magnetically supercritical and collapses dynamically, while the outer envelope remains subcritical. This effect introduces a spatial scale in the collapse process, which corresponds to the boundary between the magnetically supercritical inner core and the subcritical outer envelope. These two regions are characterized by distinct rotational properties. The supercritical inner core evolves with conservation of angular momentum and rapidly spins up. It achieves a power-law angular velocity profile $\Omega (r) \propto \Sigma (r) \propto 1/r $ at $t=0$, where $\Sigma $ is the (mass) column density [e.g. @Basu97]. By contrast, due to magnetic braking, the subcritical envelope loses (part of) its angular momentum on the timescale $\tau_J \approx t_{ff}^{b} $ (where $t_{ff}^{b} $ is the free-fall time at the density of the background – cf. @Spitzer78 and @Tomisaka00) and is progressively brought to near corotation with the background medium, assumed to rotate at the uniform rate $\Omega_b $ (e.g. @Basu94). This separation generates a break in the angular velocity profile at the radius $R_{crit}$ of the magnetically supercritical core.
Qualitatively at least, the rotation and infall profiles of the IRAM 04191 envelope (see Fig. \[model\_vel\] and § \[discuss\_summ\]) can be accounted for in the framework of such magnetic models[^7] if we identify $R_{crit}$ with the radius $r_\mathrm{m} \sim 3500$ AU beyond which the observed rotation profile exhibits a marked steepening (cf. Fig. \[model\_vel\]c). Indeed, the angular velocity profile of the inner $r < r_\mathrm{m}$ region is consistent with the $\Omega \propto 1/r $ power law expected in the supercritical core at $t =0$. Furthermore, the models predict a steepening of the rotation profile beyond $R_{crit}$ when magnetic braking does not have enough time to bring the system to corotation with the background before the formation of the supercritical core. In the parameter study presented by @Basu95a and @Basu95b, this happens in models 6 and 8, for two different reasons. First, if the cloud is already critical (or close to critical) initially near the center (but still subcritical in its outer parts) as in model 6, there is no quasistatic phase and the dynamical contraction of the supercritical core starts right away on a timescale $\sim t_{ff}^{c} $ (free-fall time at cloud center) shorter than $\tau_J $. This produces a transition region beyond $R_{crit}$ with a steep angular velocity profile between the supercritical inner core and the background outer region (see Fig. 6b of @Basu95b). At the same time, the collapse of the supercritical core is retarded by the magnetic forces so that supersonic infall velocities develop only very close to the center (at $r \simlt 0.01 \times R_{crit}$).\
Second, if the ionization fraction is low enough that the magnetic braking timescale is only slightly shorter than the ambipolar diffusion timescale as in model 8 of @Basu95a, then the supercritical core can begin its dynamical evolution with a rotational angular velocity much larger than that of the background. This also results in the apparition of a transition region with a steep rotation profile between the supercritical core and the external background (cf. Fig. 5 of @Basu95a). In model 8 of @Basu95a, the outer radius of the transition region is $\sim 3 \times R_{crit} $, which is consistent with the extent of the $\Omega \propto r^{-2.5}$ zone in Fig. \[model\_vel\]d. In this model, the infall velocity becomes supersonic for $r \simlt 0.3 \times R_{crit}$ and exhibits a relatively flat profile beyond $ R_{crit} $, also in agreement with the observational constraints of Fig. \[model\_vel\]a. The ionization degree $x_i = n_i/n(\mathrm{H}_2)$ assumed in the model is low ($2 \times 10^{-9}$ at a density of $10^{6}$ cm$^{-3}$), but comparable to the value derived at the center of the IRAM 04191 envelope in § \[obs\_ionization\].
Quantitatively, however, it is more difficult to obtain a good match of the observations of IRAM 04191 with published ambipolar diffusion models. These models rotate a factor of $\sim 3-10$ more slowly[^8] than does the IRAM 04191 envelope and have supercritical core radii that are a factor $\sim 3-30$ bigger than the observed break radius $r_\mathrm{m} \sim 3500$ AU (cf. Table 2 of @Basu95b). Physically, the radius of the magnetically supercritical core corresponds to the Jeans length for the density $n_{crit}$ (or equivalently surface density $\Sigma_{crit}$) at which “decoupling” occurs, i.e., the density (or surface density) at which gravity overcomes magnetic support. In the disk-like geometry of ambipolar diffusion core models, the critical Jeans radius is $R_{crit} = \as^2/(2G\Sigma_{crit})$ and one has $n_{crit} = \frac{\pi G}{2\mu m_H} \times (\Sigma_{crit}/\as)^2
= \frac{\pi}{8G\mu m_H} \times (\as/R_{crit})^2 $ (see @Basu95a). For $T_{core} = 7$ K and $ R_{crit} = 3500$ AU, this gives $N_{crit} \equiv \frac{\Sigma_{crit}}{\mu m_H} \sim
9.3 \times 10^{21}$ cm$^{-2}$ and $n_{crit} \approx 1.4 \times 10^5 $ cm$^{-3}$, the latter being remarkably similar to the volume density estimated at $r = 3500$ AU in the envelope (as expected in the models – see, e.g., Fig. 6a of @Basu95b). If the mass–to–flux ratio $M/\Phi = \Sigma/B$ is just critical in the supercritical core, then the magnetic field strength of the core should be $B_{crit} = 2\pi G^{1/2} \Sigma_{crit} \sim 60\, \mu$G at $r \sim 3500$ AU. The reason why published magnetic models have larger values of $R_{crit}$ is that their critical “decoupling” densities and field strengths are typically lower than these estimates by factors $\sim 10$ and $\sim 3$, respectively. Direct Zeeman measurements suggest that the magnetic field strength is only $\sim 10\, \mu $G in the low-density ($\sim 10^4$ cm$^{-3}$) outer parts of prestellar cores such as L1544 [@Crutcher99; @Crutcher00]. Only relatively poor upper limits ($< 100\, \mu $G) exist for the field strength in the central parts of these cores [@Levin01]. Albeit quite large the above value $B_{crit} \sim 60 \, \mu $G, which refers to the $\simgt 10^5$ cm$^{-3}$ region, thus remains realistic. Furthermore, it should be noted that published ambipolar diffusion models include only a static magnetic field and do not take turbulent support into account. If a turbulent magnetic field is present, then a weaker static field may be sufficient to yield a decoupling density as high as $n_{crit} \sim 10^5 $ cm$^{-3}$. According to this interpretation, the observed radius of “decoupling” $\sim 3500$ AU would correspond to the cutoff wavelength for MHD waves, i.e., $\lambda_A \sim 6200\, \rm{AU} \times (\frac{B}{10\mu G}) \times
(\frac{n_{H_2}}{3 \times 10^3 \rm{cm}^{-3}})^{-1}$ [cf. @Mouschovias91], and the IRAM 04191 dense core would have initially formed through the dissipation of MHD turbulence [e.g. @Nakano98; @Myers98] rather than ambipolar diffusion.
Evolution of angular momentum during protostar formation {#discuss_ang}
--------------------------------------------------------
It is instructive to compare the rotational properties of the IRAM 04191 envelope with the characteristics of other prestellar and protostellar objects in Taurus. Our 2D radiative transfer modeling indicates a rotational velocity $v_\mathrm{rot} = 0.20 \pm 0.04$ km s$^{-1}$ at $r = 3500$ AU, using an inclination angle of $i = 50\degr$ (cf. § \[analyze\_2d\]). This corresponds to a local specific angular momentum $j = 3.4 \times 10^{-3}$ km s$^{-1}$ pc at $r_\mathrm{m} = 3500$ AU (roughly equal to the mean half-power radius of the N$_2$H$^+$ dense core: $FWHM \sim 8800$ AU $\times \, 5200$ AU). Quite remarkably, the mean specific angular momentum $J/M \sim 1.5 \times 10^{-3}$ km s$^{-1}$ pc we measure inside the rapidly rotating inner $r < r_\mathrm{m}$ envelope of IRAM 04191 is very similar to the nearly constant $\sim 10^{-3}$ km s$^{-1}$ pc value found by @Ohashi99 for a number of small-scale “envelopes” and “disks” around Taurus Class I sources. More precisely, in the diagram of specific angular momentum versus radius presented by @Ohashi97b (see their Fig. 6), IRAM 04191 lies close to the intersection between the “dense core” regime, where the angular velocity is approximately locked to a constant background value $\Omega_b \sim 1-2$ km s$^{-1}$ pc$^{-1}$ (presumably as a result of magnetic braking – see @Basu94 and § \[discuss\_magnet\] above), and the “protostellar” regime, where the specific angular momentum is roughly constant in time[^9] (cf. Fig. \[joverm\]). According to @Ohashi97b, the transition between these two regimes at a radius $\sim 5000$ AU (i.e., $\sim 0.03$ pc) characterizes the size scale for dynamical collapse, inside which evolution proceeds with near conservation of angular momentum. Interestingly, this size scale is comparable to the radius $r_\mathrm{m} \sim 3500$ AU found here for the rapidly rotating inner envelope of IRAM 04191. Our suggestion that the inner envelope is a magnetically supercritical core decoupling from a subcritical environment (§ \[discuss\_magnet\] above) is thus fully consistent with the finding and interpretation of @Ohashi97b.
Summary and conclusions {#concl}
=======================
We have carried out a detailed study of the structure and kinematics of the envelope surrounding the Class 0 protostar IRAM 04191 in Taurus. Our main results and conclusions are as follows:
1. Extended, subsonic infall motions with $v_\mathrm{inf} \sim 0.5\, a_s \sim 0.1$ km s$^{-1}$, responsible for a marked ‘blue infall asymmetry’ in self-absorbed CS and H$_2$CO lines, are present in the bulk of the envelope, up to at least $r_\mathrm{i,o} \sim 10000-12000$ AU. The observations are also consistent with larger infall velocities scaling as $v_\mathrm{inf} \propto r^{-0.5}$ in an inner region of radius $r_\mathrm{i} \approx 2000$ AU. The corresponding mass infall rate is estimated to be $\dot{M}_\mathrm{inf} \sim 2-3 \times a_s^3/G
\sim 3 \times 10^{-6}$ M$_\odot$ yr$^{-1}$.
2. The protostellar envelope is differentially rotating with an angular velocity profile $\Omega \propto r^{-2.5 \pm 0.5}$ between $r_\mathrm{m} \approx 3500$ AU and $r_\mathrm{m,o} \sim 7000$ AU. The rotation profile is shallower, albeit more poorly constrained, in the inner $r < r_\mathrm{m}$ region, i.e., $\Omega \propto r^{-0.9 \pm 0.4}$. The angular velocity is estimated to be $\Omega \sim 12$ km s$^{-1}$ pc$^{-1}$ at $r \sim 3500$ AU and only $\Omega \simlt 0.5- 1$ km s$^{-1}$ pc$^{-1}$ at $r \sim 11000$ AU. The present value of the centrifugal radius is estimated to be less than 400 AU.
3. The extended infall velocity profile is inconsistent with the inside-out collapse picture of @Shu87 and only marginally consistent with isothermal collapse models starting from marginally stable equilibrium Bonnor-Ebert spheres. The latter tend to produce somewhat faster infall velocities than are observed.
4. The contrast observed between the (steeply declining) rotation velocity profile and the (flat) infall velocity profile beyond $r_\mathrm{m} \approx 3500$ AU suggests that angular momentum is [*not*]{} conserved in the outer envelope. This is difficult to account for in the context of non-magnetic collapse models.
5. Based on a qualitative comparison with magnetic ambipolar diffusion models of cloud collapse (e.g. @Basu94), we propose that the rapidly rotating inner envelope of IRAM 04191 corresponds to a magnetically supercritical core decoupling from an environment still supported by magnetic fields and strongly affected by magnetic braking. In this view, the outer ($r_\mathrm{m} < r < r_\mathrm{m,o}$) envelope represents a transition region between the forming protostar and the slowly rotating ambient cloud. Although published ambipolar diffusion models have difficulty explaining supercritical cores as small as $R_{crit} \sim 3500$ AU, we speculate that more elaborate versions of these models, including the effects of MHD turbulence in the outer envelope, would be more satisfactory.
6. Interestingly, the steepening of $\Omega(r)$ in IRAM 04191 occurs at a radius comparable to the $\sim 5000$ AU scale inside which the specific angular momentum of Taurus dense cores appears to be conserved [cf. @Ohashi97b and Fig. \[joverm\]]. Our results therefore support @Ohashi97b’s proposal that $r \sim 5000$ AU represents the typical size scale for dynamical collapse in Taurus. More generally, we suggest that the rotation/infall properties observed here for IRAM 04191 are representative of the physical conditions prevailing in isolated protostellar envelopes shortly ($\sim 10^4$ yr) after point mass formation.
We would like to thank Shantanu Basu for enlightening discussions on ambipolar diffusion models and Carl A. Gottlieb for providing his laboratory measurements of the CS and C$^{34}$S frequencies prior to publication. We acknowledge the contribution of Aurore Bacmann during the 1999 observing run at the 30m telescope. We are also grateful to the IRAM astronomers in Grenoble for their help with the Plateau de Bure interferometric observations.
The numerical code we have used first calculates the non-LTE level populations with a 1D (spherical) Monte-Carlo method [@Bernes78; @Bernes79]. Radiative transfer along each line of sight and convolution with the antenna beam, approximated by a Gaussian, are then computed with the MAPYSO package [@Blinder97]. The latter works in both 1D and 2D geometry.
We have tested the Monte Carlo code for two test problems (1 and 2) available on the web page of the workshop on Radiative Transfer in Molecular Lines held in Leiden in May 1999 (). These tests correspond to a low-abundance and high-abundance HCO$^+$ 12-level problem, respectively, in the context of the @Shu77 spherical collapse model. The level populations and the excitation temperatures computed by our Monte-Carlo code without any reference field [see @Bernes79; @Pagani98] agree quite well with those calculated by the workshop participants. The only significant difference occurs for the high-abundance case in the central region (inside $\sim$ 1300 AU) where our HCO$^+$(2-1), (3-2), and (4-3) excitation temperatures are lower by $\sim 20 \%$ compared to the main group results.
The CS and C$^{34}$S Monte-Carlo calculations reported in § \[simul\_1d\] and § \[simul\_2d\] used 9 levels and 27 concentric shells. This number of levels should be sufficient as the 9$^{\mbox{\tiny th}}$ level is 85 K above the ground level while the kinetic temperature in the envelope does not exceed 20 K (see § \[mass\_tk\]). We used the CS collision rates computed in the 20-300 K range by @Turner92 and extrapolated these to 5-300 K with polynomials [@Choi95 and N. Evans, private communication]. Each simulation was performed without any reference field and resulted from two successive Monte-Carlo runs. The first run started from LTE, used packets of 1000 model photons, and computed 100 iterations, reinitializing the counters after each iteration [see @Bernes79]. It converged rapidly but still suffered from a high level of statistical noise. The second run improved the convergence and reduced the noise level by starting from the output of the first run, computing 40 iterations with packets of 40000 model photons, and reinitializing the counters after each group of five iterations. We checked that this number of iterations was large enough to reach convergence on the populations of the first five levels with an accuracy better than a few percents.
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[^1]: We measure a peak 98 GHz flux density of 0.6 mJy/$4.5\arcsec$-beam close to ($\sim 4\arcsec$ north-east of) the central 227 GHz position but the cleaned image contains negative contours down to $-0.8$ mJy/$4.5\arcsec$-beam.
[^2]: Before fitting the direction of the velocity gradient, we have masked the redshifted emission seen in the north-east part of the C$_3$H$_2$(2$_{12}$-1$_{01}$) map shown in Fig. \[map\_vcent\], probably due to the redshifted lobe of the outflow or to the nearby Class I source IRAS 04191+1523 [@Tamura91].
[^3]: With such an angular velocity, the gas at $r \sim 2800$ AU would make a complete turn in $\sim 0.7$ Myr, which is comparable to the typical lifetime of prestellar cores with central densities $\simgt 10^5$ cm$^{-3}$ [e.g. @Jessop00]. Thus, although the observed velocity gradient is probably not indicative of a well-developed circular motion, it may have already induced significant rotational distortion in the envelope.
[^4]: When the rotation velocity becomes comparable to the infall velocity, the asymmetry may even be reversed [e.g. @Walker94; @Ward01].
[^5]: Based on the opacities derived above, optical depth effects cannot account for more than a factor of 1.3 decrease in the C$^{34}$S(2-1)/N$_2$H$^+$(101-012) ratio.
[^6]: The reason why the value of $12$ km s$^{-1}$ pc$^{-1}$ found here for $\Omega$ is larger than the apparent velocity gradient of $9$ km s$^{-1}$ pc$^{-1}$ given in § \[obs\_rotation\] is that the 2D model shown in Fig. \[pvdiagmod\] properly accounts for the finite ($\sim 25\arcsec $) resolution of the observations.
[^7]: Strictly speaking, the ambipolar diffusion models of Basu & Mouschovias describe the evolution of core properties only during the prestellar phase ($t<0$), while the properties of the IRAM 04191 envelope beyond $r_\mathrm{i} \sim 2000$ AU are representative of the physical conditions at $t \approx 0$ (cf § \[discuss\_extend\]). However, the models are easily extrapolated to $t=0$ by considering the limiting profiles achieved for an infinite central density [cf. @Basu97; @Basu98].
[^8]: Although rotating more slowly than the IRAM 04191 envelope, some of the models are characterized by high ratios of centrifugal to gravitational acceleration at the supercritical radius, comparable to the $0.4\pm 0.2$ ratio observed here (cf. model 1 of @Basu94 and model 6 of @Basu95b). Qualitatively, a faster initial rotation rate is not expected to change the evolution significantly compared to model 6 of @Basu95b (S. Basu, private communication).
[^9]: Assuming that all Taurus sources follow a similar time evolution, Fig. \[joverm\] may be viewed as an evolutionary diagram where the radius plotted on the x-axis represents the “contraction state” of a core as a function of time. This is conceptually different from a plot showing the spatial distribution of angular momentum/velocity in a given core at a given time [such as Fig. 1c of @Basu97 or Fig. \[model\_vel\]d of the present paper].
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'V. Caloi and F. D’Antona'
date: 'Received ; accepted '
title: 'Helium self–enrichment in globular clusters and the second parameter problem in M3 and M13'
---
Introduction
============
The processes that gave origin to globular clusters (GCs) begin to appear more and more complex, in the light of chemical inhomogeneities and peculiarities, difference in radial distribution of the various cluster stellar components, the permanence of the second parameter mistery, etc. (f.e., Kraft 1994, Gratton et al. 2001, Briley et al. 2004; Catelan et al. 2001). In preceding papers (D’Antona et al. 2002, D’Antona & Caloi 2004) we examined the possibility of self–enrichment in helium as a consequence of star formation episodes out of the material ejected by massive asymptotic giant branch (AGB) stars of the first generation. Such material will be enriched in N, possibly in Na and Al, but surely enriched in helium, owing to the second and third phases of dredge–up. The presence of a spread in helium content may help in explaining features such as blue tails and bimodality in horizontal branches (HBs) (see the above quoted papers), the main point being the reduction in the evolving mass, at a given age, with increasing helium content.
The most obvious candidates to the helium enrichment phenomenon appear the clusters with an extended HB, populated in regions difficult to reach with a standard mass loss of about 0.2 and a dispersion of 0.02 (f.e., Lee et al. 1994, Catelan et al. 1998). Among these are many of the so called second parameter clusters, which have an intermediate heavy element content and a HB mainly populated on the blue side of the RR Lyrae region. We shall look for features which may be caused by an enhanced helium content, from a morphological and evolutionary point of view, leaving aside other considerations (formation, dynamics, ...), since our purpose is to explore the possibilities opened by the original helium enhancement hypothesis without the presumption of offering a complete solution to the many problems involved.
The models
==========
For the purpose of this investigation, we computed isochrones and HB models, which are shown in Figs. 1 and 2. The code used is ATON2.0 and its adjournments (Ventura et al. 1998). The heavy element content Z = 0.001 has been chosen as close to the value observed in the intermediate metal poor GCs that we shall consider, and in particular, close to the metallicity of M3 and M13 (f.e., Sneden et al. 2004). A helium content of 0.24 has been assumed, close to the cosmological value (Izotov & Thuan 1998, Spergel et al. 2003).
As we see from Fig. 1, the turnoff in isochrones with the same metal content and increasing helium content becomes bluer and fainter. The turnoff at 13 Gyr and Y = 0.28 is 0.06 mag fainter than with Y = 0.24, and the shift in colour is of $\sim$ 0.007 mag, both in (B–V) and (V–I). The HB evolutionary tracks with varying Y have been computed with the evolutionary helium core masses (M$_{\rm c}$ in Fig. 2) and surface helium content deriving from the first dredge–up (Y$_{\rm sur}$ in Fig. 2), and are terminated when central Y is reduced to 0.10. The most important feature of the tracks is the increase in luminosity with increasing Y (Sweigart & Gross 1976): the HB with Y = 0.28 is on the average $\sim$ 0.20 mag more luminous than the one with Y = 0.24.
The bump on the red giant branch
================================
Selection of the sample
-----------------------
The slowing down of red giant (RG) evolution which takes place when the hydrogen shell reaches the point of the maximum inner expansion of the convective envelope gives origin to the so called red giant bump (Thomas 1967, Iben 1968); this feature has been observed by now in many clusters (Ferraro et al. 1999, Zoccali et al. 1999, Riello et. 2003). We consider intermediate metallicity clusters with very similar heavy element content, and report in Table 1 the magnitude difference between turnoff and RG bump. We considered only clusters for which the data of the turnoff and bump come from the same photometry.[^1] Unfortunately, it is not possible to use recent large data sets: i) Rosenberg et al. (1999) give a homogeneous data set for turnoffs and HBs, but not for RG bumps; ii) RG bumps magnitudes are given by Ferraro et al. (1999) (the ones we used), by Zoccali et al. (1999), based on HST observations transformed into the standard B and V system, and by Riello et al. (2003), who leave the data in the HST system. They choose to do so because of possible “deceptive errors in the estimate of the visual magnitudes” introduced by the transformation to the standard Johnson system, which requires knowledge of the cluster reddening. Both Zoccali et al. (1999) and Riello et al. (2003) do not give turnoff estimates.
In any case, we used Rosenberg’s turnoff magnitudes together with the ones quoted in present paper to reduce the errors, except for NGC 7089 and NGC 6934, not present in Rosenberg’s compilation. A meaningful improvement is found only for M3 and M13, for which the errors in Rosenberg’s data are substantially lower than those in Table 1. As for the bump magnitudes, the Riello ones are useless for our purpose, and the ones by Zoccali et al. may be used for clusters with low reddening, to reduce the calibration errors mentioned before. We stress that there are no other intermediate metallicity clusters in the literature for which turnoff and bump magnitude are derived from the same photometry.
A hint for differences in $\Delta V_{\rm TO}^{\rm bump}$
--------------------------------------------------------
In Table 1 we compare the data for clusters with mostly blue HBs with the data for clusters with HBs well populated in the B, V and R regions. We found only two clusters of this latter type, one of which is the well studied M3.[^2]
[18cm]{}[ccccccccc]{}\
Cluster & Messier & \[Fe/H\]$_{\rm CG}$& V$_{\rm TO}$ & V$_{\rm bump}$ & $\Delta V_{\rm TO}^{\rm bump}$ & V$_{\rm RR}$ & N$_{\rm RR}$\
Globular Clusters with predominantly blue HB &&\
NGC 1904& M79 &-1.37 & 19.70 $\pm$ 0.10 & 15.95 $\pm$ 0.10 & 3.75 $\pm$ 0.11 & & &\
NGC 6093& M80 &-1.41 & 19.75 $\pm$ 0.10 & 15.95 $\pm$ 0.10 & 3.80 $\pm$ 0.14 & 15.98 & 3\
NGC 6205& M13 &-1.39 & 18.55 $\pm$ 0.10 & 14.75 $\pm$ 0.05 & 3.80 $\pm$ 0.11 &14.83 $\pm$ 0.02& 7\
NGC 6752& &-1.42 & 17.40 $\pm$ 0.10 & 13.65 $\pm$ 0.05 & 3.75 $\pm$ 0.11 & &\
NGC 7089& M2 &-1.38 & 19.60 $\pm$ 0.10 & 15.85 $\pm$ 0.05 & 3.75 $\pm$ 0.11 & 15.926 $\pm$ 0.021& 11\
Globular Clusters with HB populated in the B, V and R regions &&\
NGC 5272 & M3& -1.34& 19.10 $\pm$ 0.10 & 15.45 $\pm$ 0.05 & 3.65 $\pm$ 0.11 & 15.665 $\pm$0.013 &35\
NGC 6934 & & -1.30 & 20.40 $\pm$ 0.15 & 16.78 $\pm$ 0.10 & 3.62 $\pm$ 0.18 & 16.873 $\pm$0.017 &24\
\
Bibl. CM diagram: M79: Kravtsov et al. 1997; M80: Brocato et al. 1998; M13: Paltrinieri et al. 1998; NGC 6752: Buonanno et al. 1986 as in Ferraro et al. 1999; M2, M3: Lee & Carney 1999b; NGC 6934: Piotto et al. 1999.
Bibl. RR Lyr: M80: Wehlau et al. 1990, the 3 RR Lyr with the best photometry; M13: Kopacki et al. 2003; M2: Lee & Carney 1999a, 11 RRab with stable light curves; M3: Carretta et al. 1998, 35 RRab with stable light curves chosen by Lee & Carney 1999b; NGC 6934: Kaluzny et al. 2001, 24 RRab with stable light curves. \[bump\]
We notice that the the average $\Delta V_{\rm TO}^{\rm bump}$ for the blue HB clusters is always larger than for the M3–type clusters: for the former ones we have an average of 3.77 $\pm$ 0.07 mag, while for the latter clusters we have 3.64 $\pm$ 0.11 mag. To be precise, we should correct this latter value for the difference in average metallicity between the two groups (–1.39 vs. –1.32), since both turnoff and bump magnitudes increase with metallicity: this small correction would move the M3–like average to 3.67 mag. However, we do not enter in the details of the procedure, because an inspection of the data in Table \[bump\] shows that the error on the difference in $\Delta V_{\rm TO}^{\rm bump}$ between the two groups is of the same order of magnitude of the difference itself. So we limit the discussion to the clusters M3 and M13, for which a substantial reduction of the errors is possible, as mentioned before. In dealing with these two clusters, we shall follow the universal opinion of considering them of the same metallicity (see, f.e., Rosenberg et al. 1999).
The case of M3 and M13
----------------------
In Table 2 we give the available data for M3 and M13 from Rosenberg et al. (1999) and Zoccali et al. (1999), the weighted means between these values and the data in Table 1, and the resulting estimate for $\Delta V_{\rm TO}^{\rm bump}$. The difference in $\Delta V_{\rm
TO}^{\rm bump}$ between the two clusters is 0.14 $\pm$ 0.09 mag. Even if this result is meaningful only at one $\sigma$ level, the fact that in [*all*]{} blue clusters for which a consistent measure was possible $\Delta V_{\rm TO}^{\rm bump}$ turned out larger than in M3–type clusters, appears intriguing enough to be worth of further investigation.
[18cm]{}[ccccccccc]{}\
Cluster & Messier & V$_{\rm TO}$(R)& V$_{\rm bump}$(Z) &V$_{\rm
TO}$(mean) &V$_{\rm bump}$(mean) & $\Delta V_{\rm TO}^{\rm bump}$ & $\Delta V_{\rm TO}^{\rm RR}$\
NGC 6205& M13 & 18.50 $\pm$ 0.06 & 14.70 $\pm$ 0.04 & 18.51 $\pm$ 0.05&14.72$\pm$ 0.03 & 3.79 $\pm$ 0.06 & 3.68 $\pm$ 0.05\
NGC 5272& M3 & 19.10 $\pm$ 0.04 & — & 19.10 $\pm$ 0.04 &15.45$\pm$ 0.05 & 3.65 $\pm$ 0.06 & 3.44 $\pm$ 0.04\
\
\[m3m13\]
Possible causes of the difference in $\Delta V_{\rm TO}^{\rm
bump}$: age or helium difference?
============================================================
The main candidates to an influence on CM diagram features are age and helium content, since in present case differences in heavy element content are excluded. Age influences $\Delta V_{\rm TO}^{\rm bump}$ because the turnoff and the bump fade at a different rate. For the metallicity of these clusters (Z $\sim$ 0.001) and for ages of 12–15 Gyr, the fading of the turnoff with age is of about 0.075 mag/Gyr (present models, D’Antona et al. 1997, Cassisi et al. 1999). For the bump magnitude we have a fading of 0.035 - 0.04 mag/Gyr (Ferraro et al. 1999, Cho & Lee 2002, and our own models). So for each Gyr of difference, $\Delta V_{\rm TO}^{\rm bump}$ would increase by about 0.04 mag.
The difference of 0.14 $\pm$ 0.09 mag would correspond to an age difference of 3.5 $\pm$ 2.2 Gyr. A similar estimate of 3 Gyr is often obtained from the difference in HB morphology for the age interval 12–15 Gyr (see, f.e., the discussion in Johnson & Bolte 1998). For lower ages – M3 of about 10 Gyr – the age difference becomes of the order of 2 Gyr (Rey et al. 2001).
Let us now consider the effects of an increase in the helium content in M13. In clusters with Y $\sim$ 0.28 the turnoff luminosity would be fainter by about 0.06 mag (our models, see Fig. 1 and D’Antona et al. 2002), while the luminosity of the RG bump would increase by $\sim$ 0.08 – 0.09 mag, according to our models, and of 0.09 mag according to Riello et al. (2003).[^3] So $\Delta V_{\rm
TO}^{\rm bump}$ = 0.14 $\pm$ 0.09 would be given by a helium variation $\Delta$Y = 0.04 + 0.02 / - 0.03.
The role of RR Lyrae variables
------------------------------
We consider now the characteristics of the RR Lyrae variables in the two clusters. M3 has a population of almost 200 RR Lyr variables, with a large majority of RRab–type; the average periods are of 0.56 d and of 0.32 d, for RRab and RRc, respectively (see, e.g., Jurcsik et al. 2003, Clement et al. 2001). In M13 only 9 RR Lyr stars have been identified, out of which only one is an RRab; the average period of 7 well observed RRc stars is 0.36 d (Kopacki et al. 2003). Both the average periods and the percentage of RRc stars indicate that variables in M3 belong to the Oosterhoff type I (a well established fact), while those in M13 more likely belong to the Oosterhoff type II.
Carretta et al. (1998) and Lee & Carney (1999a,b) tie their RR Lyr photometries for M3 to the one of the non variable stars (Ferraro et al. 1997a), obtaining the same average magnitude of 15.66 mag. The same value is found independently by Jurcsik et al. (2003). So for this cluster we have a unique photometry for all the data in Table 1.
For M13 the photometry of RR Lyr stars is different from the one of the CM diagram. The RR Lyr magnitudes are given by Kopacki et al. (2003), who give also mean values for some bright red giant variables. For these latter stars, the average of their mean magnitudes differs by $\sim$ 0.04 mag from the average obtained from the photometry given by Pilachowski et al. (1996) for the same objects.[^4] This indicates that the two photometries should be consistent within 0.04 magnitudes.
The difference in Oosterhoff type between M3 and M13 is part of the well known second parameter problem: GCs with the same heavy element content and very different HB populations. From Table 2 we see that the luminosity difference between turnoff and RR Lyraes $\Delta V_{\rm
TO}^{\rm RR}$ is much larger in M13 than in M3: 3.68 $\pm$ 0.05 vs. 3.44 $\pm$ 0.04 mag (Table 2), so that the difference in $\Delta V_{\rm TO}^{\rm RR}$ between the two clusters is 0.24 $\pm$ 0.06 mag. In the literature, such a difference is generally interpreted as due to a higher luminosity of the blue and RR Lyr regions in M13; these stars are supposed to be in a more advanced, and more luminous, evolutionary phase than the ZAHB and its vicinity, where most of RR Lyraes in M3 are found. This behaviour would follow from the higher age of M13: the smaller HB masses would populate the ZAHB only on the blue side of the RR Lyr variables.
The case of a difference in age
-------------------------------
If the difference of 0.14 $\pm$ 0.09 mag between $\Delta V_{\rm TO}^{\rm
bump}$ in M3 and M13 (Table 2) is all due to an age difference, M13 should be older by 3.5 $\pm 2.2$ Gyr. In this case, the turnoff in M13 should be fainter than in M3 by $\sim 0.075$ mag/Gyr $\times 3.5$ Gyr = 0.26 mag. However, available observations are in strong disagreement with such a large difference in turnoff luminosity (e.g., Ferraro et al. 1997b, Johnson & Bolte 1998, Rey et al. 2001). Besides, the RR Lyr luminosity level in M13, following the observed $\Delta V_{\rm TO}^{\rm
RR}$ (Table 2) would turn out 0.02 mag fainter than in M3, and the apparent Oo II type of the RR Lyr variables in M13 would remain unexplained.
Only considering the [*minimum*]{} difference in age required by the observed difference in $\Delta V_{\rm TO}^{\rm bump}$ taking into account the estimated errors (see above), that is 1.3 Gyr, we obtain a consistent scenario: the turnoff in M13 would be fainter by 0.10 mag than the turnoff in M3, and the RR Lyrs in M13 would be 0.14 mag brighter than in M3. Still, we must remember that such a small difference in age could lead to the observed difference in HB morphology only for very low ages, $\sim 10$ Gyr for M3 (Rey et al. 2001).
The case of a difference in helium
----------------------------------
If the age is the same and helium in M13 is enhanced up to Y = 0.28, we have that the turnoff in M13 would be fainter by 0.06 mag and the HB would be brighter by 0.20 mag (see Fig. 1 and Fig. 2). This explanation is consistent with the observed difference in $\Delta V_{\rm TO}^{\rm RR}$ between M13 and M3 (0.24 $\pm 0.06$ mag). In addition, it predicts for the RR Lyraes in M13 a larger luminosity (by $\sim$ 0.20 mag) than for those in M3.[^5]
As a conclusion, the hypothesis of a difference in the helium content of the stars in these two clusters is the more appealing one, if the $\Delta V_{\rm TO}^{\rm bump}$ will be confirmed to the level used in this work. Therefore, we stress that it would be extremely important to have homogenous sets of data for the turnoff [*and*]{} bump. In fact, we could carry on this kind of analysis only for the famous couple M3 – M13.
The detailed fitting of M3 and M13 with differing helium contents
=================================================================
Beside the relative luminosity levels, we have to check whether the (subtle) differences in the CM diagram introduced by a helium increase find some support in the observations. In Fig. 3a it is shown the fit of the fiducial lines for M3 given by Johnson & Bolte (1998) with the isochrone of 13 Gyr, Z=0.001, Y=0.24, and the relative HB models. Similarly, in Fig. 3b the fit is shown for M13, but with an isochrone corresponding to Y = 0.28; the average position of six RR Lyr variables from Kopacki et al. (2003) is also indicated. Different values for Y not only provide a better explanation for the relative luminosities of the turnoff, HB and bump, but also a better morphological fit. To further stress this point, we show in Fig. 3c a fit of M13 fiducial sequence with isochrones with Y=0.24, ages of 13 and 14 Gyr: there is an evident difficulty in obtaining a satisfactory fit, in contrast with the increased helium case.
Finally, the HB stellar distribution (second parameter problem) finds a natural explanation. In D’Antona et al. (2002) we examined the consequences on HB population of an increase in Y from 0.24 to 0.28. To complete the discussion, we summarize the main effects. In M3, assuming Y = 0.24 on the main sequence, we have that the bulk of the HB population covers the mass interval 0.60–0.70 , with the peak at the RR Lyr region; a sparse tail toward the blue would require lower masses. Since the evolving giant (in absence of mass loss) at the helium flash is of about 0.82 , the implied mass loss is 0.17 $\pm$ 0.05 .
With a helium content of 0.28 and 13 Gyr of age, the evolving giant at the helium flash is of 0.766 (Z = 0.001, as before). With the same mass loss as in M3, the maximum HB mass turns out 0.646 , in the middle of the RR Lyr region (see Fig. 2), while the minimum mass is of about 0.546, almost down at the level of the turnoff. If mass loss along the giant branch increases with decreasing mass (e.g., Lee et al. 1994), lower masses can be reached, but see also Lee & Carney (1999b). So M13 would naturally shift its HB population according to observations. Let us note that we have been mentioning a helium content of 0.28 for simplicity, while, most likely, a spread in Y is to be presumed, starting from $\sim$ 0.28 and reaching well beyond 0.30 (D’Antona & Caloi 2004). We elaborate briefly on the subject of self–enrichment in the following section.
The self–enrichment scenario
============================
In a preceding paper (D’Antona et al. 2002) we considered helium enrichment as mostly related to the origin of very blue HB tails in second parameter clusters such as M13 and NGC 6752. We are now proposing that the overall difference between M3 and M13 CM diagram morphologies is due to the different helium content. Thus [*all*]{} stars in M13 should have an enriched helium content with respect to the Big Bang abundance. How is this possible? We suggest that M13 represents an extreme case of self–enrichment. Among GCs which show abundance spreads, we have identified NGC 2808 as a cluster in which at present half of the stellar population has a normal Y, and the other half is helium enriched at various degrees (D’Antona & Caloi 2004). The normal Y population is responsible for the red part of the HB, and the enriched one is responsible for the blue side of the HB and for the blue tails. We have shown that this may happen only if the number of low mass stars with normal Y content present in the cluster is much smaller than expected on the basis of an initial mass function which is, on the one side not too implausible, and on the other, able to explain the blue HB stars as born from the helium enriched ejecta of the massive AGBs. Then many low mass stars of the first generation must have been lost by the cluster, perhaps because the intermediate mass stars were much more concentrated into the central regions (D’Antona & Caloi 2004). In M13 [*there is no red HB clump*]{}: in this case, the first generation stars must have been completely lost by the cluster. This hypothesis is certainly difficult to be digested without a detailed modeling: in fact M3 and M13 are very similar on any other respect: present day mass, central density, relaxation time. However, this kind of scenario is still at the beginning. Notice in addition, that Salaris et al. (2004), in a detailed analysis of the R parameter in GCs, detect a significant spread of helium content towards higher abundances, in the clusters having very blue HBs. This is a further indication in favour of our hypothesis.
Beside the difficulties with the cluster dynamics, there is the other one that, in this scheme, [*all*]{} M13 stars should be helium enriched, while the observations show that not all of them are oxygen poor (f.e., Sneden et al. 2004), a feature which is considered to accompany processed matter. However, it has been shown that, while most of the AGB matter from which the second generation stars are born is helium enriched due to the action of the second dredge–up, not necessarily all this same matter is oxygen depleted (Ventura et al. 2001, 2002).
Is then M3 devoid of self–enrichment? Actually this may be not the case, as suggested by the presence of luminous RR Lyrae variables.
The case of the luminous RR Lyrae variables in M3
-------------------------------------------------
Recently it has been noticed that a few RR Lyr variables in M3, the prototype of the Oo I type clusters, show Oo II characteristics (Clement & Sheldon 1999, Corwin & Carney 2001, who quote similar observations by Belserene 1954). Jurcsik et al. (2003) made a detailed study of about 150 RR Lyr stars (of both types ab and c), and found that they could be classified in four groups, according to their mean magnitudes and periods. The most luminous sample has statistically Oo II properties regarding the mean periods and RR$_{\rm ab}$/RR$_{\rm c}$ number ratio. They found also that the various samples can be identified with different stages of HB evolution, with some difficulty for the most luminous one. In fact, this has a luminosity at which evolution is rather fast (last phases of central helium burning), too fast to support the existence of more than 20% of the observed RR Lyr stars. As Jurcsik et al. (2003) remark, the discrepancy could be removed by the presence of the (infamous)“breathing pulses”, that is, of a final phase of helium mixing in convective cores, when helium abundance is approaching zero (f.e., Castellani et al. 1985, Dorman & Rood 1993, Caloi & Mazzitelli 1993), but at present the phenomenon is not considered real and HB evolution is computed ignoring it.
Keeping in mind such a possibility, we can anyway consider other solutions to the problem of Oo II RR Lyrs in M3. On the line of multiple star generations with helium enrichment, we can hypothise that also this cluster presents a certain amount of second generation members formed from the helium rich ejecta of AGB stars. Beside the luminous RR Lyrs, other features could be interpreted in terms of variable helium content: i) the large range in period covered by RR$_{\rm d}$ variables (Clementini et al. 2004); ii) the great length of the HB, since the lower mass of helium rich red giants, at a given age, helps to originate the very blue HB stars hotter than the main body of HB population (Ferraro et al. 1997a, Fig. 18); the spread in luminosity and colour in the subgiant region (as noted by Clementini et al., see Fig. 4 in Corwin & Carney 2001, and Fig. 15 in Ferraro et al. 1997a). In this respect, we notice that, according to recent estimates (Sneden et al. 2004), about one–third of the observed giants in M3 are oxygen poor. In the hypothesis mentioned before (conversion of oxygen to nitrogen in the envelopes of AGB stars of a preceding generation), these stars could give origin to the luminous RR Lyr variables, being also helium enriched.
Discussion
==========
The recently acquired impressive amount of detailed information on the chemical composition of GC members down to the main sequence has profoundly changed our perception of these most ancient stellar systems. Complex processes, both chemical and dynamical, must have taken place in their formation phase, leaving characteristic marks on stellar chemistry and population.
Here we have tried to relate some photometric features in CM diagrams to chemical peculiarities, such as helium enrichment. Johnson & Bolte (1998) suggested a higher helium content in M13 as the best explanation for the CM diagram morphology differences between this cluster and M3, and discussed widely the subject. Beside the turnoff shape (see Fig. 3), we consider the luminosity of the RG bump with respect to the turnoff, and find that it correlates with the HB type: the metal content being the same, clusters with a blue HB have a larger $\Delta V_{\rm TO}^{\rm bump}$ than clusters with a uniformly populated HB.
Considering in detail M3 and M13, we examine both the difference in age and in helium content as possible causes for this feature, and find that the latter is the more plausible, if one considers the luminosity level of the RR Lyr variables. The presence of Oo II type RR Lyrs in M3 can be also interpreted in terms of a helium enriched stellar component.
The difference in helium content between M3 and M13 appears as a possible “second parameter”, at least with regard to this pair of clusters. The difficulties are various, mainly deriving from our ignorance of GC formation phases and of the initial mass function(s). Anyway, with respect to similar suggestions in the past, the hypothesis presented here has the support of detailed investigations on the AGB ejecta composition and of the recognized necessity of a primordial contamination of main sequence stars in many clusters, beside the interpretation of CM features otherwise not easily understood (HB and bump luminosities). Partial pollution does not help much to give origin to very blue HB structures (Caloi 2001, D’Antona et al. 2002), while an increased structural helium content does (D’Antona et al. 2002). An important contribution to the problem would be to check the presence of chemical peculiarities in the luminous, OoII type RR Lyr variables in M3, that we proposed as candidates for second generation, helium increased structures.
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[^1]: For this reason for NGC 6752 we used the photometry by Buonanno et al. (1986), the only set of data in which turnoff and RG bump are both identified.
[^2]: We did not consider NGC 6229 (\[Fe/H\] = -1.30) for which the bump magnitude is uncertain (Borissova et al. 1997), and NGC 3201 which is 0.1 dex more metal rich than M3 (\[Fe/H\] = -1.23, Carretta & Gratton 1997).
[^3]: We assumed $\Delta M_{\rm
bol}\sim 1.21\Delta M_{\rm V}$ as in Cho & Lee 2002.
[^4]: The variables V11, V15, V17, V18, V19 and V24 have been considered.
[^5]: To be precise, the RR Lyrs become brighter by 0.18 mag.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Julien Sorci
title: 'Minimal Codes From Characteristic Functions Not Satisfying The Ashikhmin-Barg Condition'
---
Introduction
============
Linear codes have found applications in areas far beyond error-correction. For example, an $(S,T)$ *secret-sharing scheme* is a collection of $S$ “shares" of a $q$-ary secret such that knowledge of any $T$ shares determines the secret, but knowledge of $T-1$ or fewer shares gives no information. Massey showed that certain codewords in the dual of a linear code, called *minimal codewords*, can be used to construct a secret sharing scheme [@M]. However, determining the set of minimal codewords in a linear code is a hard problem in general, which galvanized the search for codes where every codeword is minimal, referred to as *minimal codes*.
In 1998, Ashikhmin and Barg gave a sufficient condition for a code to be minimal based on the ratio of the maximum and minimum weight of the code [@AB].
A $q$-ary linear $[n,k]$ code $\mathcal C$ is minimal if $$\frac{w_{\max}}{w_{\min}} < \frac{q}{q-1},$$ where $w_{\min}$ and $w_{\max}$ are the minimum and maximum weights of $\mathcal C$, respectively.
For many years following this result all known examples of minimal codes satisfied Ashikhmin and Barg’s condition, until 2017 when Chang and Hyun [@CH] constructed a minimal binary code from a simplicial complex that did not. As for finding a family of $q$-ary minimal codes not satisfying Ashikhmin and Barg’s condition for $q$ an odd prime power, a particular code that has received much attention is the code $\mathcal C_f$, which we will now define. Let $f: {\mathbb{F}}_q^n \rightarrow {\mathbb{F}}_q$ be an arbitrary but fixed function. We define $\mathcal C_f$ to be the ${\mathbb{F}}_q$-subspace of ${\mathbb{F}}_q^{q^n-1}$ spanned by vectors of the form $$c(u,v):= (uf(x)+v \cdot x)_{x \in {\mathbb{F}}_q^n \setminus \{0\}}$$ where $u \in {\mathbb{F}}_q$, $v \in {\mathbb{F}}_q^n$, and $v\cdot x$ is the usual dot product. We record below the basic parameters of the code, which are well known.
If $f:{\mathbb{F}}_q^n \rightarrow {\mathbb{F}}_q$ is not linear and $f(a) \neq 0$ for some $a \in {\mathbb{F}}_q^n \setminus \{0 \}$, then $\mathcal C_f$ is a $[q^n-1, n+1]$ linear code.
In [@DHZ], Ding et. al. gave necessary and sufficient conditions for the code $\mathcal C_f$ to be minimal when $q=2$ based on the Walsh-Hadamard transform of $f$, which is defined to be the function $\hat{f}(x):= \sum_{v \in {\mathbb{F}}_2^n}(-1)^{f(x)+v\cdot x}$.
\[dingminimal\] If $f:{\mathbb{F}}_2^n \rightarrow {\mathbb{F}}_2$ is not linear and $f(a) \neq 0$ for some $a \in {\mathbb{F}}_2^n \setminus \{0 \}$ then the binary code $\mathcal C_f$ is minimal if and only if $\hat{f}(x) + \hat{f}(y) \neq 2^n$ and $\hat{f}(x)-\hat{f}(y) \neq 2^n$ for every pair of distinct vectors $x, y \in {\mathbb{F}}_2^n$.
A boolean function $f: {\mathbb{F}}_2^n \rightarrow {\mathbb{F}}_2$ with $n$ a positive even integer is said to be *bent* if $|\hat{f}(x)|=2^{n/2}$ for all $x \in {\mathbb{F}}_2^n$. Using Lemma \[dingminimal\], it is easy to see that the binary code $\mathcal C_f$ is minimal when $f$ is a bent function.
Bonini et. al. studied the code $\mathcal C_f$ for arbitrary prime powers $q$ and functions $f$, and showed that if the zero set of $f$ satisfies certain geometric properties then $\mathcal C_f$ is a minimal code [@BB]. We continue this line of work by considering the code $\mathcal C_f$ when $f$ is the indicator function of a set, and give sufficient conditions for $\mathcal C_f$ to be minimal and not satisfying the condition of Ashikhmin and Barg in terms of the geometric properties of the support of $f$. We give a tight lower bound on the size of sets satisfying our geometric conditions, and give an explicit example of a set meeting the lower bound. In section 2 we lay out the notation used, and in section 3 we present the main results.
Notation
========
A *linear* $[n,k]$ *code* is a $k$-dimensional subspace $\mathcal C$ of ${\mathbb{F}}_q^n$.
A codeword $c \in \mathcal C$ is said to be *minimal* if whenever ${\textnormal{supp}}(c) \subseteq {\textnormal{supp}}(c^\prime)$ for some codeword $c^\prime \in \mathcal C$ it implies that $c = \lambda c^\prime$ for some $\lambda \in {\mathbb{F}}_q^\times$. A linear code $\mathcal C$ is *minimal* if all codewords of $\mathcal C$ are minimal.
We summarize some of the notation used in the paper:
- $e_i$ will denote the $i^{th}$ standard basis vector.
- For $v \in {\mathbb{F}}_q^n$, we will let $H(v)$ denote the set $\{u \in {\mathbb{F}}_q^n : v \cdot u =0 \}$.
- For a function $f : {\mathbb{F}}_q^n \rightarrow {\mathbb{F}}_q$, we will let $V(f)$ denote the set of zeros of $f$, $\{ u \in {\mathbb{F}}_q^n : f(u)=0 \}$.
- If $U$ is any set, we will let $U^*:=U \setminus \{0 \}$, and $\overline{U}$ will denote the complement of $U$.
- By a *hyperplane*, we will mean an $(n-1)$-dimensional subspace of ${\mathbb{F}}_q^n$.
- By an *affine hyperplane*, we will mean a coset of an $(n-1)$-dimensional subspace of ${\mathbb{F}}_q^n$. Note that by our convention, a hyperplane is also an affine hyperplane.
The Main Results
================
The next theorem is our main result.
\[main\]
Let $q$ be an arbitrary prime power, and let $S \subseteq {\mathbb{F}}_q^n \setminus \{0\}$ be a set of points such that
1. $S$ is not contained in any affine hyperplane,
2. $S$ meets every affine hyperplane,
3. $|S| < q^{n-2}(q-1)$
Then $\mathcal C_f$ with $f$ the indicator function of $S$ is a minimal code that does not satisfy the Ashikhmin-Barg condition.
Suppose that ${\textnormal{supp}}( c(u^\prime,v^\prime)) \subseteq {\textnormal{supp}}(c(u,v))$ for some codewords $c(u,v), c(u^\prime, v^\prime)$ of $\mathcal C_f$. Equivalently, $$\label{assump}
V(uf(x)+v \cdot x)^* \subseteq V(u^\prime f(x) + v^\prime \cdot x)^*$$ We proceed by cases to show that $c(u,v)=\lambda c(u^\prime,v^\prime)$ for some $\lambda \in {\mathbb{F}}_q^\times$.
**Case 1:** If $v=0$, then it implies $V(f)^* \subseteq H(v^\prime)^*$, so that $\overline{H(v^\prime)^*} \subseteq S$, contradicting that $|S| < q^{n-2}(q-1)$.
**Case 2:** If $v^\prime =0$ then $V(uf(x)+v\cdot x)^* \subseteq V(f)^*$. From the partition $$\label{partition}
V(uf(x) + v\cdot x)^* = (V(f)^* \cap H(v)^*) \cup ( S \cap \{v \cdot x = -u \})$$ it implies that $S$ does not meet the affine hyperplane $\{v \cdot x =-u \}$, a contradiction.
**Case 3:** If $v, v^\prime \neq 0$, then from equation \[assump\] and the partition of equation \[partition\] we have $$V(f)^* \cap H(v)^* \subseteq V(f)^* \cap H(v^\prime)^* \subseteq H(v^\prime)^*$$ Here $|H(v)^* \cap V(f)^*| = |H(v)^* \setminus S| \geq q^{n-1} - |S| \geq q^{n-2}+1$, so that $H(v)^* \cap V(f)^*$ is not contained in a hyperplane other than $H(v)$, i.e. $H(v)=H(v^\prime)$. Thus we have $v^\prime = \lambda v$ for some $\lambda \in {\mathbb{F}}_q^\times$. Since $S$ meets every affine hyperplane, we can choose some $y$ in $S \cap \{v\cdot x = -u \}$. Equation \[assump\] then implies $u=-v\cdot y$ and $u^\prime=-\lambda v \cdot y$, so that $u^\prime = \lambda u$. We therefore conclude $c(u^\prime, v^\prime) = \lambda c(u,v)$ in this case, as was required.
**Case 4:** If $u=0$ and $H(v) \neq H(v^\prime)$, then equation \[assump\] reads as $H(v)^* \subseteq V(u^\prime f(x)+v^\prime \cdot x)^*$. Using the partition of equation \[partition\] and the assumption that $S$ is not contained in any affine hyperplane, we have $$\begin{split}
q^{n-1}-1 &= |H(v)^*| \\
&= |V(f)^* \cap H(v^\prime)^* \cap H(v)^*| + |S \cap \{ v^\prime \cdot x =-u^\prime \} \cap H(v)^* | \\
& \leq |H(v)^* \cap H(v^\prime)^*| + |S \cap \{ v^\prime \cdot x =-u^\prime \} | \\
& \leq q^{n-2}-1 + q^{n-2}(q-1) - 1 \\
&= q^{n-1} - 2\\
\end{split}$$ This clear contradiction means we therefore must have $H(v)=H(v^\prime)$, so that $v^\prime = \lambda v$ for some $\lambda \in {\mathbb{F}}_q^\times$. But the containment $H(v)^* \subseteq V(u^\prime f(x) + \lambda v \cdot x)^*$ implies that $H(v)^* \subseteq V(f)^*$, or equivalently $S \subseteq \overline{H(v)^*}$. This contradicts that $S$ meets the hyperplane $H(v)$.
**Case 5:** If $u^\prime =0$, then we have $V(uf(x)+v\cdot x)^* \subseteq H(v^\prime)^*$, which follows by case 3.
Lastly we check that the code $\mathcal C_f$ does not satisfy the Ashikhmin-Barg condition. The maximum weight is at least the weight of $c(0,1)$, which is the number of points in $H(v)$, and the minimum weight is at most the weight of $c(1,0)$, which is $|S|$. Thus: $$\frac{w_{\max}}{w_{\min}} \geq \frac{q^{n-1}}{|S|} > \frac{q}{q-1}$$
When $q=2$ the conditions of Theorem \[main\] simplify considerably, which we record in the following corollary.
\[mainbinary\]
Let $S \subseteq {\mathbb{F}}_2^n \setminus \{0 \}$ be a set of points such that
1. $S$ is not contained in any hyperplane,
2. $S$ meets every hyperplane,
3. $|S| < 2^{n-2}$
Then the binary code $C_f$ with $f$ the indicator function of $S$ is a minimal code not satisfying the Ashikhmin-Barg condition.
It suffices to check that when $q=2$, and $S$ is a set of points that is not contained in a hyperplane and meets every hyperplane, then $S$ is not contained in an affine hyperplane, and meets every affine hyperplane.
To see that $S$ is not contained in an affine hyperplane, suppose that $H$ is an affine hyperplane not containing the origin, and that $S \subseteq H$. Since $q=2$, then $\overline{H}$ is a hyperplane, so that $S$ does not meet the hyperplane $\overline{H}$, a contradiction.
Similarly, if $S$ does not meet the affine hyperplane $H$ not containing the origin, then $S$ is contained in $\overline{H}$, which is a hyperplane. Therefore $S$ meets every affine hyperplane, so that $S$ satisfies the conditions of Theorem \[main\].
Assume that $n \geq 6$ is an even positive integer, and let $q=2$. A *partial spread* of order $s$ is a set of $n/2$-dimensional subspaces $\{U_1,...,U_s\}$ of ${\mathbb{F}}_2^n$ such that $U_i \cap U_j = \{ 0 \}$ for all $1 \leq i, j \leq s$. It is easy to see that a partial spread of order $s$ has at most $2^{n/2}+1$ elements.
In [@DHZ], Ding et. al. showed that when $1 \leq s \leq 2^{n/2}+1$, $s \notin \{1, 2^{n/2}, 2^{n/2}+1 \}$, then $\mathcal C_f$ with $f$ the indicator function of the set $S = \cup_{i=1}^s U_i^*$ is a minimal code. Moreover, they showed that if, in addition, we have $s \leq 2^{\frac{n}{2}-2}$ then $\mathcal C_f$ does not satisfy Ashikhmin and Barg’s condition. They proved this by computing the Walsh-Hadamard transform of $f$ and then applying Lemma \[dingminimal\], but we can alternatively check that the set $S$ satisfies the conditions of Corollary \[mainbinary\].
Since $s \geq 2$, then $S$ clearly spans ${\mathbb{F}}_2^n$, and the assumption that $n \geq 6$ means that $\dim(U_i) \geq 3$, so $S$ meets every hyperplane. Finally, we have in general that $|S|=s(2^{n/2}-1)$, so if we assume that $s \leq 2^{\frac{n}{2}-2}$ then an easy computation shows that $|S| \leq 2^{n-2}-2^{\frac{n}{2}-2}< 2^{n-2}$. Therefore $S$ indeed satisfies the conditions of Corollary \[mainbinary\].
Let $n \geq 7$ and $2 \leq k \leq \lfloor \frac{n-3}{2} \rfloor$. Let $S$ be the set of vectors of ${\mathbb{F}}_2^n$ with weight at most $k$. In [@DHZ], Ding et. al. showed that $\mathcal C_f$ with $f$ the indicator function of $S$ is a minimal $[2^n-1, n+1, \sum_{i=1}^k {n \choose i}]$ binary code, and moreover that $\mathcal C_f$ does not satisfy the Ashikhmin-Barg condition if and only if $$\label{dingbd}
1 + 2 \sum_{i=1}^k {n \choose i} \leq 2^{n-1} + {n-1 \choose k}$$
We alternatively check that the set $S$ satisfies the conditions of Corollary \[mainbinary\]. Since $S$ contains the standard basis vectors, $S$ is clearly not contained in any hyperplane. Given any hyperplane $H(v)$, at least one of the vectors $e_1$, $e_2$, or $e_1+e_2$ is an element of $H(v)$, and each of these vectors is also an element of $S$. Therefore $S$ also meets every hyperplane. In general the size of $S$ is $\sum_{i=1}^k {n \choose k}$, so to apply Corollary \[mainbinary\] we lastly need to impose the restriction that $\sum_{i=1}^k {n \choose k} < 2^{n-2}$. We note that this is equivalent to the inequality $$1+ 2 \sum_{i=1}^k {n \choose k} \leq 2^{n-1}$$ which is a more restrictive condition than the inequality given in Equation \[dingbd\].
We lastly give a tight lower bound on the size of a set of points satisfying the conditions of Theorem \[main\]. The following lemma was first proved by Jameson [@J]. There are many known proofs of the result; for a survey on them we refer the reader to [@B].
\[bound\]
If $S$ is a set of points in ${\mathbb{F}}_q^n$ meeting every affine hyperplane then $|S| \geq n(q-1)+1$.
The lower bound of Lemma \[bound\] clearly gives a lower bound on the size of a set of points satisfying the conditions of Theorem \[main\]. However, it is not obvious that this bound should be tight since the set of points in Theorem \[main\] does not contain the origin.
\[boundmain\]
Let $q$ be an arbitrary prime power. If $S \subseteq {\mathbb{F}}_q^n \setminus \{0\}$ is a set of points such that
1. $S$ is not contained in any affine hyperplane,
2. $S$ meets every affine hyperplane,
3. $|S| < q^{n-2}(q-1)$,
then $|S| \geq n(q-1)+1$. Moreover, this lower bound is tight.
To show that this lower bound is tight, consider the set of points $$S: =\{ a+ \lambda e_i : \lambda \in {\mathbb{F}}_q, 1 \leq i \leq n \}$$ where $a \in {\mathbb{F}}_q^n \setminus \{0 \}$ is any point not equal to $\lambda e_i$ for any $\lambda \in {\mathbb{F}}_q$, $1 \leq i \leq n$. By our choice of $a$, the origin is not an element of $S$. Clearly $S$ is not contained in an affine hyperplane, and $|S|=n(q-1)+1 <q^{n-2}(q-1)$. Lastly, if $u_1X_1+u_2X_2+...+u_nX_n=\alpha$ is the equation of an affine hyperplane, then it is easily checked that $a+\lambda e_i$ is a point on the affine hyperplane, where $i$ is chosen such that $u_i \neq 0$, and $\lambda$ is chosen to be the element $\lambda = \frac{1}{u_i}(\alpha- u \cdot a)$.
The sufficient conditions given in Theorem \[main\] and Corollary \[mainbinary\] give geometric conditions for the code $\mathcal C_f$ to be minimal and not satisfy the Ashikhmin-Barg condition when $q$ is any prime power and $f$ is an indicator function. Moreover, since the minimum weight of $\mathcal C_f$ is at most $|S|$, then the tight lower bound on the size of a set of points satisfying these conditions given in Theorem \[boundmain\] shows that it is possible for $\mathcal C_f$ to additionally have small minimum weight.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank Prof. Peter Sin for his sage advice and helpful comments in preparing this paper.
[99]{}
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Julien Sorci, <span style="font-variant:small-caps;">Department of Mathematics, University of Florida, P. O. Box 118105, Gainesville FL 32611, USA</span>
*E-mail address*: `[email protected]`
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We study a large sample of $625$ low-redshift brightest cluster galaxies (BCGs) and link their morphologies to their structural properties. We derive visual morphologies and find that $\sim57$% of the BCGs are cD galaxies, $\sim13$% are ellipticals, and $\sim21$% belong to the intermediate classes mostly between E and cD. There is a continuous distribution in the properties of the BCG’s envelopes, ranging from undetected (E class) to clearly detected (cD class), with intermediate classes (E/cD and cD/E) showing the increasing degrees of the envelope presence. A minority ($\sim7$%) of BCGs have disk morphologies, with spirals and S0s in similar proportions, and the rest ($\sim2$%) are mergers. After carefully fitting the galaxies light distributions by using one-component () and two-component (+Exponential) models, we find a clear link between the BCGs morphologies and their structures and conclude that a combination of the best-fit parameters derived from the fits can be used to separate cD galaxies from non-cD BCGs. In particular, cDs and non-cDs show very different distributions in the – plane, where is the effective radius and (the residual flux fraction) measures the proportion of the galaxy flux present in the residual images after subtracting the models. In general, cDs have larger and values than ellipticals. Therefore we find, in a statistically robust way, a boundary separating cD and non-cD BCGs in this parameter space. BCGs with cD morphology can be selected with reasonably high completeness ($\sim 75\%$) and low contamination ($\sim 20\%$). This automatic and objective technique can be applied to any current or future BCG sample with good quality images.'
author:
- |
Dongyao Zhao$^{1}$[^1], Alfonso Aragón-Salamanca$^{1}$[^2], Christopher J. Conselice$^{1}$[^3]\
\
$^{1}$School of Physics and Astronomy, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK
bibliography:
- 'BCG\_paper1.bib'
date: 'Accepted 2015 January 24. Received 2015 January 20; in original form 2014 December 12'
title: 'The Link Between Morphology and Structure of Brightest Cluster Galaxies: Automatic Identification of cDs'
---
\[firstpage\]
galaxies: clusters: general — galaxies: elliptical and lenticular, cD — galaxies: structure
Introduction
============
The brightest cluster galaxies (BCGs) are the most luminous and massive galaxies in today’s universe. Their stellar masses reach beyond $\sim 10^{11} M_{\odot}$, and they reside at the bottom of the gravitational potential well of galaxy clusters and groups. Their formation and evolution relate closely to the evolution of the host clusters [@whiley08] and further tie to the history of large-scale structures in universe (@Conroy07). BCGs are typically classified as elliptical galaxies [@LP92], but a fraction of them possess an extended, low surface brightness envelope around the central region. These are referred to as cD galaxies (e.g. @Dressler84; @OH01).
The surface brightness profile of elliptical galaxies was originally modelled using the empirical $R^{1/4}$ de Vaucouleurs law [@deVaucl48]. However, [@Lugger84] and [@Schombert86] showed that the $r^{1/4}$ model cannot properly describe the flux excess at large radii for most elliptical galaxies, and an additional parameter $n$ was introduced in the so-called ($r^{1/n}$) law [@Sersic63]. For the most massive early-type galaxies, however, a single profile still does not reproduce their luminosity distribution accurately. @Gonzalez05 found that a sample of $30$ BCGs were best fitted using a double $r^{1/4}$ de Vaucouleurs profile rather than a single law. Furthermore, @Donzelli11 suggested that a two-component model with an inner and an outer exponential profile is required to properly decompose the light distribution of $\sim48\%$ of the BCGs in their 430 galaxy sample. A similar conclusion was obtained by @Seigar07.
The light profiles of BCGs need to be explained by any successful model of galaxy formation and evolution. In hierarchical models of structure formation, a two-phase scenario is currently favoured. [@Hopkins09] proposed that a early central starburst could give rise to the bulge (elliptical) component of these galaxies, while the outer envelope was subsequently formed by the violent relaxation of stars originating in galaxies which merged with the central galaxy. Alternatively, [@Oser10] and [@Johansson12] suggested that intense dissipational processes such as cold accretion or gas-rich mergers could rapidly build up an initially compact progenitor and, after the star formation is quenched, a second phase of slower, more protracted evolution is dominated by non-dissipational processes such as dry minor mergers to form the low-surface-brightness outskirts.
To shed light on the mechanism(s) leading to the formation of BCGs, especially of cD galaxies, we need to answer questions such as: are elliptical and cD BCGs two clearly distinct and separated classes of galaxies? if so, are elliptical and cD BCGs formed by different processes or in different environments? are the extended envelopes of cD galaxies intrinsically different structures which formed separately from the central bulge? To help answer these questions, in this paper we explore statistically how the visual classification of BCGs into different morphological classes (e.g., elliptical, cD; here referred to as “morphology”), relates to the quantitative structural properties of their light profiles (e.g., effective radius $R_{\rm e}$, -index $n$; generically called “structure” in this paper). Moreover, finding an automatic and objective way to select cD BCGs is nontrivial for the future databases and study. Recent study such as @Liu08 identified cD BCGs by Petrosian parameter profiles (@Petrosian76), but their method does not give an unambiguous criterion to separate cD galaxies from non-cD BCGs.
In this paper, we visually-classify $625$ BCGs from the sample of @Linden07 [hereafter L07] and fit accurate models to their light profiles. We find clear links between the visual morphologies and the structural parameters of BCGs, and these allow us to develop a quantitative and objective method to separate cDs galaxies from ellipticals BCGs. In a later paper (Zhao et al., in preparation) we will study how the visual morphology and structural properties of BCGs correlate with their intrinsic properties (stellar masses) and their environment (cluster mass and galaxy density), and explore the implications that such correlations have for the formation mechanisms and histories of cDs/BCGs.
The paper is organized as follows. In §\[sec:data\] we introduce the BCG samples and the visual morphological classification of the BCGs. In §\[sec:BCGstruct\] we describe the light distribution models and the fitting methods we use, and discuss how the results are affected by sky-subtraction uncertainties. This section also presents a quantitative evaluation of the quality of the fits. In §\[sec:structureresult\] we present the structural properties of the BCGs in the sample. In §\[sec:border\] we introduce an objective diagnostic to separate cDs from non-cD BCGs using quantitative information from their light profiles. We summarise our main conclusions in §\[sec:conclude\].
Data {#sec:data}
====
BCG Catalogue and Images
------------------------
To study the structural properties of BCGs in galaxy groups and clusters, we use the BCG catalogue published by L07. The groups and clusters that host these BCGs come from the C4 cluster catalogue [@Miller05] extracted from the Sloan Digital Sky Survey [SDSS; @York00] third data release spectroscopic sample. The cluster-finding algorithm used to build the C4 catalogue identifies clusters as over-densities in a seven-dimensional parameter space of position, redshift and colour, minimising projection effects. The C4 catalogue gives a very clean widely-used cluster sample which is well supported by simulations. Based on the C4 catalogue, L07 restricted their cluster sample to the $0.02\leqslant z \leqslant 0.10$ redshift range to avoid problems related to the 55 arcsec “fiber collision” region of SDSS. Within each cluster, L07 applied an improved algorithm to identify the BCG as the galaxy being closest to the deepest point of the potential well of the cluster (see @Linden07 for a detailed discussion of this identification), and developed an iterative algorithm to measure the cluster velocity dispersion within the virial radius $R_{200}$[^4]. The catalogue created by L07 contains $625$ BCGs in galaxy groups and clusters with redshifts $0.02\leqslant z \leqslant 0.10$ spanning a wide range of cluster velocity dispersions, from galaxy groups ($\,\leqslant 200\,$) to very massive clusters ($\sim 1000\,$). $75\%$ of the BCGs in L07 are in dark matter halos with $\sigma_{r200} \geqslant 309\,$, where the completeness of the halos identified by the C4 algorithm is expected to be above $50\%$. Obviously, for larger halo masses the completeness is higher.
The images we use to classify the BCGs and analyse their structural properties come from the SDSS Seventh Data Release (DR7) $r$-band images. We also use SDSS-DR7 $g$-band images of the BCGs in Section \[sec:uncertainty\]. The BCG catalogue used in this paper together with their main properties are presented in appendix A.
Visual Classification {#sec:visualclassification}
---------------------
The 625 BCGs in L07 sample were visually classified by careful inspection of the SDSS images. BCGs were displayed using a logarithmic scale between the sky level and the peak of the surface brightness distribution. The contrast was adjusted manually to ensure that the low-surface-brightness envelopes were revealed if present. cD galaxies are identified by a visually extended envelope, while the envelope is not visible in our elliptical BCGs. Finally the BCGs were classified into three main types: 414 cDs, including pure cD (356), cD/E (53) and cD/S0 (5); 155 ellipticals, including pure E (80), E/cD (72), and E/S0 (3); 46 disk galaxies, containing spirals (24) and S0s (22). The main morphological classes of BCGs are illustrated in Figure \[fig:atlas\]. There are also 10 BCGs undergoing major mergers, but we will not discuss them in this paper in any detail.
Over half of the BCGs in the sample are classified as cDs. Separating cD BCGs and non-cD elliptical BCGs is a very hard problem since there is no sharp distinction between these two classes [e.g., @Patel06; @Liu08]. Detecting the extended stellar envelope that characterises cD galaxies depends not only on its dominance, but also on the quality and depth of the images, and on the details of the method(s) employed. We used intermediate classes such as cD/E (probably a cD, but could be E) and E/cD (probably E, but could be cD) to account for the uncertainty inherent in the visual classification.
Our careful inspection of the images clearly reveals that there is a wide range in the brightness and extent of the envelopes. There seems to be a continuous distribution in the envelope properties, ranging from undetected (pure E class) to clearly detected (pure cD class), with the intermediate classes (E/cD and cD/E) showing increasing degrees of envelope presence. This continuous distribution in envelope detectability will also be made evident in the structural analysis carried out later in this paper. The classification we present here does not intend to be a definitive one since such a thing is probably unachievable. Our aim is to obtain a homogeneous and systematic visual classification of the BCGs and then study how such classification correlates with quantitative and objective structural properties of the BCGs. The visual morphological types of all the galaxies in the sample are presented in Appendix A.
We checked the effect that the redshift of BCGs may have on the visual classification. cDs might be mistakenly identified as elliptical if they are more distant since the extended low-surface-brightness envelope may be harder to resolve at higher redshifts. Figure \[fig:redshift\] illustrates the redshift distribution of the three main types. cD galaxies generally share the same redshift distribution with elliptical BCGs, especially at $z\geqslant0.05$. At $z<0.05$ we identify a slightly higher proportion (by $\sim10$%) of cD galaxies. However, if we compare the structural properties of cD and elliptical BCGs which are at $z\geqslant0.05$, the results we obtain do not significantly differ from those using the full-redshift sample. As an additional check, we artificially redshifted some of the lowest redshift galaxies ($z\sim0.02$–$0.03$) to $z=0.1$, the highest redshift of the sample, taking into account cosmological effects such as surface-brightness dimming. Because the redshift range of the BCGs we study is very narrow, the effect on the images is minimal and does not have any significant impact on the visual classification. We are therefore confident that our visual classification is robust and that in the relatively narrow redshift range explored here any putative redshift-related biases will not affect our results.
Quantitative Characterisation of BCG Structure {#sec:BCGstruct}
==============================================
The surface brightness profiles of galaxies provide valuable information on their structure and clues to their formation. It has become customary to fit the radial surface brightness distribution using theoretical functions which have parameters that include a measurement of size (e.g., half-light radius or scale length), a characteristic surface brightness, and other parameter(s) describing the shape and properties of the surface brightness profiles. In this paper we use GALFIT (@Peng02) to fit the 2-D luminosity profile of each BCG using two parametric models, and thus determine the best-fitted parameters of each model. GALFIT can simultaneously fit model profiles to several galaxies in one image, which is particularly important for BCGs since they usually inhabit very dense environments. In this way, the light contamination from nearby galaxies can be accounted for appropriately.
We explore two models to represent the luminosity profile of the BCGs. A model commonly used to fit a variety of galaxy light profiles is the generalization of the $r^{1/4}$ [@deVaucl48] law introduced by [@Sersic63]. The model has the form $$\label{eq:sersic}
I(r)=I_{\rm e} \exp \{-b[(r/r_{\rm e})^{1/n}-1]\},$$ where $I(r)$ is the intensity at distance $r$ from the centre, , the effective radius, is the radius that encloses half of the total luminosity, $I_e$ is the intensity at , $n$ is the index representing concentration, and $b \simeq 2n-0.33$ (@Caon93). The function provides a good model for galaxy bulges and massive elliptical galaxies. Since BCGs are mostly early-type galaxies, it is reasonable to fit their structure with single models first. Subsequently, in order to explore the complexity introduced by the extended envelopes of cD galaxies, we will also fit the light profile of BCGs adding an additional exponential component to the profile. Adding this exponential component is the simplest way to describe the “extra-light” from the extended envelope. Note that the exponential profile $I(r)=I_0 \exp (-r/r_{\rm s})$ is just a model with $n=1$. The models assume that the isophotes have elliptical shapes, and the ellipticity and orientation of each model component are parameters determined in the fitting process.
In order to run GALFIT, we require a postage stamp image for each BCG with appropriate size to measure its structure over the full extent of the object, a mask image with the same size as the stamp image, an initial guess for the fitting parameters, an estimate of the background sky level, and a point spread function (PSF). Details on how these ingredients are produced and the fitting procedures are given below.
Pipeline for One-Component Fits: Modified GALAPAGOS {#sec:galapagos}
---------------------------------------------------
We run GALFIT using the GALAPAGOS pipeline [@Barden12]. GALAPAGOS has been successfully applied to a wide variety of ground- and space-based images [@Haubler13; @Vika13; @Vulcani14]. @Guo09 specifically tested a modified version of GALAPAGOS to fit the central galaxies of local clusters using SDSS $r$-band images. We applied the same modified version of GALAPAGOS to fit the SDSS $r$-band images of the BCGs in our sample. The starting point are SDSS images with a size of $2047 \times 1488$ pixels. For each BCG, the pipeline carries out four main tasks before running GALFIT itself: (i) detection of all the sources present in the image; (ii) cutting out the appropriate postage stamp and preparing the mask image; (iii) estimation of the sky background; (iv) preparation of the input file for GALFIT. After completing these tasks, GALAPAGOS will run GALFIT using the appropriate images and input parameters. We describe now these tasks in detail.
*(i) Source Detection:* SExtractor (@BA96) is used to detect galaxies in the SDSS images. A set of configuration parameters defines how SExtractor detects sources. The values of the SExtractor input parameters follows @Guo09: DETECT\_MINAREA$\,=25$, DETECT\_THRESH$\,=3.0$, and DEBLEND\_MINCONT$\,=0.003$. This set of parameters were tested to perform well on SDSS $r$-band images so that the bright and extended BCGs were isolated from other sources without artificially deblending them into multiple components. SExtractor also provides estimates of several properties for the target BCGs and nearby objects such as their magnitude, size, axis ratio and position angle. These values are used to calculate the initial guesses of the model parameters that are needed as inputs by GALFIT.
*(ii) Postage Stamp creation:* GALAPAGOS cuts out a rectangular postage stamp centred on the target BCG which will be used by GALFIT as input image. We define the “Kron ellipse” for a galaxy image as an ellipse whose semi-major axis is the Kron radius[^5] (), with the ellipticity and orientation determined by SExtractor. The postage stamp size is determined in such a way that it will fully contain an ellipse 3.5 times larger than the Kron ellipse, i.e., its semi-major axis is $3.5$, and has the same ellipticity and orientation. The 3.5 factor represents a compromise between computational speed and ensuring that virtually all the BCG’s light is included in the postage stamp. At this stage, a mask image is also created, identifying and masking out all pixels belonging to objects in the postage stamp which will not be simultaneously fitted by GALFIT. The aim is to reduce the computational time by excluding objects too far from the BCG or too faint to have any significant effect on the fit. Following [@Barden12], an “exclusion ellipse” is defined for each galaxy with a semi-major axis $1.5$$\, + 20\,{\rm pixels}$, and the same ellipticity and orientation as the Kron ellipse. GALAPAGOS masks out all objects whose exclusion ellipse does not overlap with the exclusion ellipse of the target BCG. These objects are deemed to be too far away from the BCG to require simultaneous fitting. Furthermore, all objects more than $2.5\,$magnitudes fainter than the BCG are also masked out since they are too faint to affect the BCG fit. The pixels that belong to these objects according to the SExtractor segmentation maps are masked out and excluded from the fits. All the remaining objects will be simultaneously fitted by GALFIT at the same time as the BCG. For a detailed description of this process and a justification of the parameter choice see [@Barden12].
*(iii) Sky Estimation:* Accurate estimates of the sky background level is crucial when fitting galaxy profiles, particularly when interested in the low-surface-brightness outer regions. Overestimating the sky level will result in the underestimation of the galaxy flux, size, and index $n$, and vice-versa. GALAPAGOS uses a flux growth curve method to robustly estimate the local sky background around the target galaxy. SDSS DR7 also provides a global sky value for the whole $2047 \times 1488$ image frame and local sky values for each galaxy. The SDSS *PHOTO* pipeline estimates the sky background using the median flux of all the pixels in the image after $2.33\sigma$-clipping. However, according to the SDSS-III website, the version of *PHOTO* used in DR7 and earlier data releases tended to overestimate both the global and local sky values. The sky measurement is improved by SDSS-III in later data releases, but since we use the images from DR7 we cannot use the SDSS sky value with enough confidence. @Haubler07 demonstrated that the sky measurement that GALAPAGOS produces is highly reliable for single-band fits because it takes into account the effect of all the objects in the image. Therefore, in this study we use the local sky background estimated by GALAPAGOS. The accurate sky measurement provided by GALAPAGOS indicates that we can reach a surface brightness limit in the $r$-band of $\sim 27$ mag/arcsec$^{2}$. This is deep enough to study the faint extended structures of BCGs. For each galaxy, its local sky background is included in the GALFIT input file and is fixed during the fitting procedure. Given the importance of accurate sky subtraction, in Section \[sec:skyuncer\] we will carry out an explicit comparison of our results using SDSS and GALAPAGOS sky estimates.
*(iv) GALFIT Input:* GALAPAGOS produces an input file which includes initial guesses for the fitting parameters based on the SExtractor output. As mentioned above, all objects which are not masked out are fitted simultaneously using a model. The initial-guess model parameters for these nearby companions are also determined from SExtractor. In order to obtain reasonable results, we impose some constraints on the acceptable model parameter range. Our constrains on position, magnitude, axis ratio and position angle follow @Haubler07. Additionally, the half-light radius is constrained within $0.3 \leqslant \rex \leqslant 800\,$pixels. This prevents the code from yielding unreasonably small or large sizes. Since the pixel size of the SDSS images is $0.396\,$arcsec, is constrained to be larger than $0.12\,$arcsec, which is much smaller than the PSF, and smaller than half the size of the original input images, reasonable for the range of redshifts explored. In the original GALAPAGOS pipeline, the constraint on the index is $0.2 \leqslant n \leqslant 8$. These are reasonably conservative limits, since normal galaxies with $n>8$ are rarely seen and are often associated with poor model fits. However, some studies have shown that very luminous elliptical galaxies with $n>8$ do exist [e.g., @Graham05], therefore for the target BCGs we allow $n$ to be as large as 14 to keep the fits as free as possible. For the companion galaxies, which are fitted simultaneously, we still keep the constraint $0.2 \leqslant n \leqslant 8$. The final ingredient needed by GALFIT is a PSF image appropriate for each BCG. These are extracted from the SDSS DR7 data products[^6] according to the photometric band used and the position of the BCG on the SDSS image.
Effect of the Sky Background Subtraction: Comparing SDSS and GALAPAGOS Sky Estimates {#sec:skyuncer}
------------------------------------------------------------------------------------
As described in Section \[sec:galapagos\], in this study we rely on the sky measurements provided by GALAPAGOS. However, it is important to test the effect that the choice of sky background has on our results. We do this by comparing the fitted model parameters $n$ and using the GALAPAGOS and SDSS sky estimates. As mentioned before, SDSS DR7 provides a global sky value for the whole $2047 \times 1488$ image and local sky values for each galaxy. @Guo09 found that the local background estimates are generally larger than the global ones due to contamination from the outskirts of extended and bright sources, making them unreliable. We therefore restrict our comparison to the global SDSS sky values. We fit the BCG light profiles twice using exactly the same procedure and input parameters (see §\[sec:galapagos\]) but changing only the sky background estimates. The first set of fits use the GALAPAGOS-determined sky values, while the second set use the SDSS DR7 global ones.
Figure \[fig:skydiff\] shows the distribution of the difference between the SDSS DR7 global sky and the sky measured by GALAPAGOS. It is clear that the SDSS global sky is generally larger than the local sky from GALAPAGOS. The effect from different sky values on the best-fitted structural parameters ( index $n$ and effective radius ) is shown in Figure \[fig:skyparam\]. It is clear that the SDSS larger sky values result in the values of $n_{\rm sdss}$ and $r_{\rm e,sdss}$ being smaller than the corresponding GALAPAGOS ones. The effect becomes more severe for those BCGs with large $n$ and , most of which are cD galaxies. This means the overestimated sky values would particularly affect the measurements on the low-surface-brightness envelopes of cD galaxies. Although it is difficult to know *a priori* which the *true* value of the sky background is, based on the fact that the SDSS-III provides evidence that DR7 sky values are overestimated while @Haubler07 showed reasonable proof of the reliability of the GALAPAGOS sky measurements, in what follows we will therefore trust and use the GALAPAGOS-determined sky values.
Two-Component Fits {#sec:BDfit}
------------------
Although the light profiles of many early-type galaxies can be reproduces reasonably well with single models, the extended envelopes of cD galaxies may require an additional component. We therefore fitted all the BCGs using a two-component model consisting of a profile plus an exponential. The input postage stamp, mask image, PSF, and sky values required by GALFIT remain the same as for the single- fits. To ensure that we are fitting exactly the same light distribution, the location of the centre of the BCG is fixed to the X and Y coordinates determined in the single fit, and we also force the initial guesses of the model parameters to be the single-component fit results. The BCG companions are simultaneously fitted still with single- profiles but with initial-guess parameters determined by the single profile fits.
Residual Flux Fraction and Reduced $\chi^2$ {#sec:rff}
-------------------------------------------
Although the models we are fitting are generally reasonably good descriptions of the BCG light profiles, real galaxies can be more complicated, with additional features and structures such as star-forming regions, spiral arms, and extended halos. It is therefore desirable to quantify how good the fits are and what residuals remain after subtracting the best-fit models. A visual inspection of the residual images can generally give a good feel for how good a fit is, and sometimes tell us whether an additional component or components are required. However, more quantitative, repeatable and objective diagnostics are also needed. The residual flux fraction [; @Hoyos11] provides one such diagnostic. It is defined as $$\label{eq:rff}
RFF=\dfrac{\sum_{i,j\in A} |I_{i,j}-I_{i,j}^{\rm model}|-0.8\times \Sigma_{i,j\in A}\sigma_{i,j}^{\rm bkg}}{\Sigma_{i,j\in A} I_{i,j}},$$ where $A$ is the particular aperture used to calculate . Within A, $I_{i,j}$ is the original flux of pixel $(i.j)$, $I_{i,j}^{\rm model}$ is the model flux created by GALFIT, and $\sigma_{i,j}^{\rm bkg}$ is the *rms* of the background. measures the fraction of the signal contained in the residual image that cannot be explained by background noise. The $0.8$ factor ensures that the expectation value of the for a purely Gaussian noise error image of constant variance is $0.0$. See [@Hoyos11] for details. Obviously, this diagnostic can be applied to both single- and two-component profiles, or any other model. The aperture A we use to calculate is the “Kron ellipse” defined in Section \[sec:galapagos\] (an ellipse with semi-major axis and the ellipticity and orientation determined by SExtractor for the BCG). $\Sigma_{i,j\in A} I_{i,j}$, the denominator of Equation (\[eq:rff\]), is computed as the total BCG flux contained inside the Kron ellipse, which is one of the SExtractor outputs, and therefore independent of the model fit.
Since BCGs usually reside in dense environments, sometimes there are some faint nearby objects contained within the Kron ellipse that have not been fitted by GALFIT (those more than $2.5\,$mag fainter than the BCG, see §\[sec:galapagos\]). These objects will be present in the residual image. Moreover, brighter companions that have been simultaneously fitted may also leave some residuals due to inaccuracies in their fits. Therefore, even if the BCG light distribution has been accurately fitted, can be affected by the residuals from the companion galaxies, failing to provide an accurate measure of the quality of the fit. To minimise the effect from companion galaxies on , we mask out the pixels belonging to all companions within the Kron ellipse using SExtractor segmentation maps. The will therefore measure the residuals from the BCG fit alone, excluding, as far as possible, those belonging to nearby galaxies.
An additional measurement of the fit accuracy is the reduced $\chi^2$, which is minimised by GALFIT when finding the best-fit models. It is defined as $$\label{eq:chi2}
\chi^2_\nu=\frac{1}{N_{\rm dof}} \sum_{i,j\in A} \frac{(I_{i.j}-I^{\rm model}_{i,j})^2}{\sigma^2_{i,j}},$$ where $A$ is the aperture used to calculate $\chi^2_\nu$, $N_{\rm dof}$ is the number of degrees of freedom in the fit, $I_{i,j}$ is the original image flux of pixel $(i,j)$. $I^{\rm model}_{i,j}$ represents, for each pixel, the sum of the flux of the models fitted to all the galaxies in the aperture, and $\sigma_{i,j}$ is the noise corresponding to pixel $(i,j)$. This noise is calculated by GALFIT taking into account the contribution of the Poisson errors and the read-out-noise of the image [@Peng02].
Similarly to , also measures the deviation of the fitted model from the original light distribution. The value of that GALFIT minimises to find the best-fit model is calculated over the whole postage stamp, and includes contributions from all the objects fitted. To make sure that we only take into account the contribution to from the BCG fit, we calculate it within the Kron ellipse of the BCG, masking out the nearby objects as we did when calculating .
The choice of aperture (Kron ellipse with semi-major axis of ) over which we evaluate and represents a good compromise between covering a large fraction of the galaxy light while minimising the impact of close companions. We carried out several tests to evaluate the sensitivity of our results to the changes in aperture size. If we reduce the semimajor axis of the aperture by $20$% or more we lose significant information on the extended halo of BCGs, which we must avoid. If we increase the semimajor axis of the aperture by $20$% or more, we potentially increase the sensitivity to the galaxy halos but in the crowded central cluster regions contamination from companion galaxies becomes a serious problem, generally increasing and . Changes in the aperture semimajor axis within $\pm20$% would have no effect on the conclusions of this paper.
Evaluating One-Component and Two-Component Fits {#sec:1cvs2c}
-----------------------------------------------
Since and can quantify the residual images after subtracting the model fits, we attempt to use them to assess whether a one-component () fit or a two-component (+Exponential) fit is more appropriate to describe the light profile of individual BCGs. In order to do this, we first evaluate the effectiveness of and at quantifying the goodness-of-fit. We visually examine the fits and residuals obtained from both one- and two-component models for all the BCGs in our sample. In some cases, two of which are illustrated in Figure \[fig:exp12BCG\], it is obvious which model is clearly favoured.
For those BCGs where such a clear distinction can confidently be made, we classify them into what we call 1C (one-component) BCGs and 2C (two-component) BCGs. Explicitly, 1C BCGs (e.g., galaxy 1 in the top panel of Figure \[fig:exp12BCG\]) are those for which a one-component model represents their light distribution very well, and therefore the residuals left are small and show no significant visible structure. For these galaxies, adding a second component does not visibly improve the residuals. Conversely, 2C BCGs (e.g., galaxy 2 in the bottom panel of Figure \[fig:exp12BCG\]) are not well fitted by a one-component model, and the residuals are significant. These residuals often show excess light at large radii which can be identified as an exponential component or halo. Additionally, the fit to these galaxies visibly improves when using a two-component model. With these criteria we confidently identify $53$ 1C BCGs and $25$ 2C BCGs. Since we want to test the sensitivity of and , we concentrate for now on this small but robust subsample. The rest of the BCGs (537) cannot be confidently classified into 1C or 2C BCGs because it is too hard to tell visually due to the residuals containing significant structures which cannot be accurately fitted by such simple models.
Figure \[fig:comp12BCG\] presents a comparison of the and values for the one- and two-component fits of the $53$ 1C BCGs and $25$ 2C BCGs. For 1C BCGs, the and distributions of one- and two-component fits are virtually indistinguishable. Neither nor improve significantly when the second component is added. However, and are significantly smaller for the two-component fits of 2C BCGs. It is clear therefore that the quantitative information that and provide agrees very well with the visual assessments of the fits. Both and are sensitive to changes in the residuals, but appears to be more sensitive. As shown in the bottom panels of Figure \[fig:comp12BCG\], the improvement in the two-component fit for 2C BCGs is around $40\%$–$60\%$ when measured by , while it is only $\sim 20\%$ when measured by . A further useful piece of information obtained from this test is that the typical values of $\log$ and $\log$ for fits deemed to be good by visual inspection are $\log$$\,\simeq -1.7^{+0.11}_{-0.06}$, and of $\log$$\,\simeq 0.042^{+0.033}_{-0.025}$ (median $+$/$-$ 1st and 3rd quartiles of the parameter distributions).
As mentioned before, the majority of the BCGs cannot be visually classified into 1C or 2C BCGs with high certainty because their light distributions are too complex to be accurately represented by such simple models. Nevertheless, we can use the quantitative information provided by and to gauge to what extent the BCGs are better fit by a two-component model than by a one-component model. This will be discussed later.
We would like to point out that this is the first time that the residual flux is calculated considering *only* the contribution of the target galaxies when estimating both and , explicitly excluding the contribution due to the companion galaxies. For instance, @Hoyos11 also used to evaluate the goodness-of-fit, but they measured the residuals over all pixels within a specific area around the target galaxies, without excluding nearby companions. Similarly, the values from GALFIT have also been applied to evaluate which fitting model is better (e.g., @Bruce12), but the effect of nearby objects on the values was also overlooked. Using the 2C BCG sample, we assessed the importance of this improvement. If the and are calculated considering the residuals in all the pixels inside the relevant aperture, the and distributions for the two-component fits of 2C BCGs cannot be distinguished from the one-component results. The effect of the contribution to the residuals from companion galaxies is so severe that it renders such a comparison useless. Our method therefore represents a significant step forward. It is extremely important to exclude the contibution of the companion galaxies when calculating and in this kind of analysis.
Structural Properties of BCGs {#sec:structureresult}
=============================
Our morphologically-classified BCGs provide a large sample to statistically study their structural properties and link them to their morphological properties. In what follows we consider the three main morphological classes of BCGs: cDs (including all BCGs classified as pure cD, cD/E and cD/S0); ellipticals (including pure E, E/cD and E/S0) and disk (spiral and S0) BCGs. The $10$ BCGs classified as mergers are excluded (see Section \[sec:visualclassification\] for details). We decided to include the galaxies with “uncertain” morphologies (such as cD/E and E/cD) in our analysis to reflect the difficulties involved in visual classification. However, to ensure the robustness of our analysis, at every stage we have checked that considering only “pure” cD and elliptical BCGs (i.e., excluding the cD/E, cD/S0, E/cD and E/S0 classes) would not change our conclusions.
Since most BCGs are early-type galaxies, we will first consider and discuss single models when fitting their SDSS $r$-band images. We will subsequently use +Exponential models to see whether the fits are improved. But before embarking in the analysis of the parameters derived from these model fits, we first evaluate their uncertainties.
Structural Parameter Uncertainties {#sec:uncertainty}
----------------------------------
The parameter uncertainties that GALFIT reports are calculated using the covariance matrix derived from the Hessian matrix computed by the Levenberg-Marquardt algorithm that the program uses [@Peng10]. These formal uncertainties are only meaningful when the model provides a good fit to the image, in which case the fluctuations in the residual image are only due to Poisson noise. However, for real galaxy images the residual images contain not only Poissonian noise, but also systematics from non-stochastic and stochastic factors due to additional components not included in the fitting function (e.g., spiral arms, star-forming regions), asymmetries, shape mismatch, flat-fielding errors and so on. These non-random factors usually dominate the uncertainty of the parameters, and the uncertainties inferred from the covariance matrices are only lower-limit estimates [@Peng10]. Therefore, if we rely on the errors reported by GALFIT the uncertainties in the structural parameters of the BCGs could be severely underestimated. Indeed, these formal errors seem unrealistically small: typical GALFIT uncertainties for and $n$ are only $\sim1$–$2\%$. A more robust and realistic way of determining these uncertainties is clearly needed.
We have measured the structural parameters of the BCGs in our sample using the SDSS $r$-band images. Independent measurements can also be obtained using the SDSS $g$-band images. In principle, the structural parameters could be wavelength-dependent. However, the $g-r$ colours of massive early-type galaxies with old stellar populations are quite spatially uniform and do not change much from galaxy-to-galaxy [e.g., @fukugita95]. Furthermore, morphological $k$-corrections are negligible for early-type galaxies between these two bands (e.g., @TaylorMager07), so it is reasonable to expect that the intrinsic structural parameters will not change much between $g$ and $r$ band. Therefore, any differences in the measured parameters between these two bands should be largely dominated by measurement errors. Moreover, if there are significant wavelength-dependent differences in the measured parameters that are driven by real physical differences, it is reasonable to expect that these may correlate with other galaxy properties such as their colour, morphology, redshift, cluster velocity dispersion, etc. No such correlations were found, so we are confident that the intrinsic differences are not significant in these two bands.
We use GALAPAGOS to fit the SDSS $g$-band images of the BCGs in our sample in exactly the same way as we did for the $r$-band images. Figure \[fig:error\] shows a comparison of the and values obtained in both bands. Similar comparisons were carried out for the rest of the structural parameters. The scatter around the 1-to-1 relations is due, in principle, to both intrinsic wavelength-dependent differences and measurement errors. Since, as we have argued, the intrinsic differences are not expected to be significant between these two bands, the measurement errors should dominate the scatter. We can thus use this scatter as an estimate of realistic, albeit perhaps marginally pessimistic, parameter uncertainties. The average errors are $\delta (n) \simeq 0.9$, $\delta(\log r_{\rm e}) \simeq 0.16$, and $\delta(\log RFF_{1c}) \simeq 0.13$.
The right-hand panel of Figure \[fig:error\] shows that the errors in and are not correlated. This is an important point since these two are the main parameters that we will use as diagnostics in our analysis in Section \[sec:border\].
Single Models {#sec:1cresult}
--------------
We analyse now the behaviour of four parameters derived from the best-fitting single- models along with the morphological classifications. Two of them, the index $n$ and the effective radius , provide information on the intrinsic properties of the BCGs. The other two, and , show how well the models fit the real light distribution of the BCGs and also provide information about their detailed structure. The values of these parameters are listed in Appendix A. Figure \[fig:paradistr\] shows the distribution of these parameters for the three main BCG morphologies. The $\sigma$ value in each panel indicates the significance (confidence level) of the observed differences between the cD and elliptical BCG parameter distributions. These are derived from two-sample Kolmogorov-Smirnov tests.
### Index $n$ {#sec:1cn}
The index $n$ measures the concentration of the light profile, with larger $n$ corresponding to higher concentration. The upper left panel of Figure \[fig:paradistr\] presents the $n$ distributions for the three main BCG morphologies. It is clear that disk (spiral and S0) BCGs tend to have smaller values of $n$, as expected. However, the $n$ distribution for disk BCGs is skewed towards larger values ($n\gtrsim3$) than those of the normal disk galaxy population [e.g., $n=2.5$ in @Shen03]. This is because most disk BCGs are early-type bulge-dominated spirals and S0s. Elliptical and cD BCGs tend to have larger $n$ values ($n\ge4$). The $n$ distributions of cD and elliptical BCGs are quite similar. A K–S test indicates that the distributions are not significantly different: the significance of any possible difference is just $2.04\sigma$.
### Effective Radius
The effective radius is a measurement of the extent (or size) of the light distribution. The upper right panel of Figure \[fig:paradistr\] shows the distributions of $\log R_{\rm e}$. Disk BCGs tend to have relatively small sizes, and the vast majority of them ($\sim 85\%$) have smaller than $\sim 15\,$. About $75\%$ of the elliptical BCGs also have $R_{\rm e} \lesssim 15\,$, while cD galaxies tend to be significantly larger. More than $60\%$ of cDs have $R_{\rm e} \gtrsim 15\,$. A K–S test demonstrates that the difference in distributions between cD and elliptical BCGs is very significant. This suggests that could be a good discriminator to separate cD and elliptical BCGs.
### Residual Flux Fraction and Reduced $\chi^2$ {#sec:rff1c}
The lower left panel of Figure \[fig:paradistr\] presents the distributions in a $\log_{10}$ scale, where denotes for one-component models. The of disk BCGs has a much broader distribution and reaches significantly larger values than those of cDs and ellipticals. This reflects the fact that a single- model is not a good representation of the light distribution of galaxies with clear disks, spiral arms and star-forming regions. Early-type BCGs have smoother light distributions that can be reasonably well reproduced with a profile, and their tend to be smaller. However, there are statistically significant differences between the distributions of cD and elliptical BCGs. About $60\%$ of elliptical BCGs have values in the range corresponding to good fits (see Section \[sec:1cvs2c\] and Figure \[fig:comp12BCG\]), while just $\sim 25\%$ of cD galaxies do. This suggests that most elliptical BCGs can be well represented by single models, while most cD galaxies are harder to model with such a simple profile. Since an extended envelope is a general property of cD galaxies, their deviation from a single profile may be due, at least partially, to this extended envelope. This suggests that an additional model component may be required for them. We will re-visit two-component models in Section \[sec:2cresult\]. The clear difference in suggests that could be another good discriminator to separate cD and elliptical BCGs.
Similar conclusions can be reached from the the distributions of shown in the lower right panel of Figure \[fig:paradistr\], albeit less clearly. This is not surprising since, as shown in Section \[sec:1cvs2c\], both and measure the strength of the residuals, but is significantly less sensitive. Therefore, is expected to be more efficient for separating cD and elliptical BCGs than .\
These results show a clear link between the visual morphologies of BCGs and their structural properties. Although cD galaxies tend to have similar shapes to elliptical BCGs, they usually have larger sizes and their structures generally deviate more from single profiles. In contrast, elliptical BCGs tend to be smaller, and their light profiles are statistically more consistent with single models. These structural differences, especially in and , could therefore provide quantitative ways to separate elliptical and cD BCGs without relying on visual inspection. We will explore these issues in Section \[sec:border\].
+Exponential Models {#sec:2cresult}
-------------------
The distributions shown in Section \[sec:1cresult\] indicate that elliptical BCGs are statistically better fitted by a single model than cDs. Since a distinctive feature of cD galaxies is their extended luminous halo, two-component models may be more appropriate to describe accurately the light distributions of cD BCGs. Following [@Seigar07] and [@Donzelli11], we explore here how a model consisting of an inner profile and an outer exponential envelope performs when fitting BCG images. The fitting process was described in detail in Section \[sec:BDfit\].
As shown in Section \[sec:1cvs2c\], both and can provide quantitative information to assess whether BCGs are better fitted by a two-component model than by a one-component model, at least in very clear cases. Figure \[fig:RFF12c\] shows a comparison of these parameters obtained for single and +Exponential models. In the left panel we show a histogram of the fractional differences in the values $(\rffsx-\rffbdx)/\rffsx$ for all three BCG types. The right panel shows the corresponding fractional differences $(\chisx-\chibdx)/\chisx$. It is clear that for disk BCGs, the +Exponential model does a better job. This is not surprising since spiral and lenticular galaxies contain clearly distinct bulges and disks. For elliptical BCGs the improvement in and for two-component models is generally quite small, as expected: elliptical galaxies are known to be reasonably well fitted by models, so the extra component does not improve the residuals significantly. Perhaps surprisingly, the improvement is also only marginally better for cDs: the typical fractional differences for cD galaxies are $(\rffsx-\rffbdx)/\rffsx=0.11^{+0.14}_{-0.08}$ and $(\chisx-\chibdx)/\chisx=0.035^{+0.053}_{-0.029}$ (median $+$/$-$ 1st and 3rd quartiles of the parameter distributions).
Since the distributions shown in Figure \[fig:RFF12c\] for ellipticals and cDs are statistically indistinguishable, there is no clear separation that could be used to distinguish elliptical and cD BCGs by comparing one-component and two-component fits. Moreover, on average, +Exponential model does not fit the profile of cD BCGs clearly better than single model. The reason is that for cD BCGs the values of and are generally not dominated by the presence or absence of a second exponential model component but by other structures present in the residual images, such as double cores. Since there is no clear improvement in the +Exponential model, the model with the smallest number of parameters (i.e., single model) will be preferred for simplicity. The following discussions are based on the results from the single fits.
Summary of Section \[sec:structureresult\] {#sec:1cdiscuss}
------------------------------------------
In this section we have analysed the differences in the structural properties of BCGs as a function of morphology. These structural parameters have been derived from one-component () and two-component (+exponential) model fits. Disk BCGs (a small minority) have smaller indices ($n$) than elliptical and cD BCGs, as expected. They also have different, generally broader, distributions of and . Elliptical and cD BCGs have similar $n$ values, but cDs tend to have larger values of , and . These differences do not depend strongly on whether we use one- or two-component models.
The observed structural differences could provide quantitative ways to separate elliptical and cD BCGs without relying on visual inspection. We explore these in section \[sec:border\]. Furthermore, the differences we have found in the structural parameters suggest that the formation histories of elliptical and cD BCGs may be different. For instance, gas-rich major mergers and other dissipative processes may be responsible for building the inner (-like) component, while dissipationless minor mergers may contribute to the build-up of the outer extended envelope and to the growth of galaxy sizes (e.g., @Oser10; @Johansson12; @Huang13). We will explore in a subsequent paper (Zhao et al., in preparation) whether the morphological and structural properties of BCGs are linked to other intrinsic BCG properties such as their stellar mass, and/or to the properties of their environment. These links will provide more clues to the formation history of cDs/BCGs.
Separating elliptical and cD BCGs {#sec:border}
=================================
The results of Section \[sec:1cresult\] suggest that we may be able to use the different distributions of cD and non-cD BCGs on the $\log$–$\log$ plane to separate them in an objective, quantitative and automatic way. Figure \[fig:Abeta125\] shows that cDs are clearly segregated from other BCGs in this two-dimensional parameter space. We attempt to find a robust, well-defined way to separate, statistically, cD and non-cD BCGs using the information provided by this diagram. In other words, we suppose to find an “optimal border” that can separate them.
Method Description and the Optimal Border {#sec:bestborder}
-----------------------------------------
Ideally, any process that selects cD galaxies from a sample of BCGs needs to have high completeness (i.e., select as many of the cDs present in the sample as possible), while avoiding contamination from non-cDs (i.e., maximising the purity of the sample). These two requirements compete with each other, and increasing completeness often results in a decrease in sample purity, and vice-versa. We need therefore to find the best compromise between these competing requirements. In general, the optimal solution will depend on the specific intent for the selected sample, and therefore on the decision of how much weight to give to completeness and to purity. It is useful to define a measurement on the quality of the selection method that combines both requirements in a well-defined way. The optimal solution will then be obtained by maximising this quality parameter.
Following [@Hoyos11] the *sensitivity*, which is often known as *completeness* in astronomy, is defined as: $$r=\dfrac{\rm \# True Positives}{\rm \# True Positives+ \# False Negatives}.$$ Similarly, we define *specificity* as: $$p=\dfrac{\rm \# True Negatives}{\rm \# True Negatives+ \# False Positives}.$$ A “True Positive” is an object retrieved by the selection process with the required properties (i.e., a cD galaxy that is correctly selected as such). A “False Negative” is an item that is not retrieved by the selection process but does present the needed properties (a cD galaxy that is not selected). A “True Negative” is an item that is rightfully rejected by the selection process since it does not have the required properties (for instance, an elliptical galaxy that is not selected as a cD). A “False Positive” is an item that is incorrectly picked up by the selection process, but does not have the properties of interest (for example, an elliptical galaxy that is wrongly selected as a cD).
*Sensitivity* and *specificity* can be combined into a single number, known as the (@vanRijsbergen79), which provides a single measure on the quality of the selection process. The is just a weighted harmonic average of $r$ and $p$, $$F_\beta=\dfrac{(1+\beta ^2)\times p \times t}{\beta ^2 \times p+r},$$ where $\beta$ is a control parameter that regulates the relative importance of completeness with respect to specificity. This is a user-supplied value that depends on the particular goals of the study. We will explore later how the choice of $\beta$ affects our selecting results. At this stage, a value of $\beta= 1.25$ is used, which can be thought of as weighing completeness more than the lack of contamination. For our BCG samples, the is used to grade the performance of the diagnostics we use when separating cD galaxies from the parent population.
The selection process that we will apply to the parent population of BCGs in order to select cD galaxies will be defined by a “border” in the $\log$–$\log$ plane (see Figure \[fig:Abeta125\]). This border will be represented by a second-order polynomial in the horizontal coordinate. Higher-order polynomials (or more conplex functions) could be used, but the additional complexity is not required here. In our specific problem, the cD galaxies play the role of the “items presenting the required properties” discussed above, and the parent population is the complete sample of BCGs.
According to the definition of *sensitivity* and *specificity*, the BCGs in the parent sample are classified into four categories by their position relative to the border. In the $\log$–$\log$ plane, cD galaxies dominate the region of large and . We therefore define this region as the “cD side”. Thus
- cD galaxies that fall on the cD side of the border are True Positives.
- cD galaxies that do not fall on the cD side of the border are called False Negatives.
- elliptical and disk (spiral and S0) BCGs that fall on the cD side are regarded as False Positives.
- elliptical and disk (spiral and S0) BCGs that do not fall on the cD side of the border are True Negatives.
The optimal border is found by maximising the value. Following the method described in [@Hoyos11], we use the Amoeba algorithm [@Press88] to carry out this maximization and find the polynomial defining the border.
It is clear from Figure \[fig:Abeta125\] that the selected galaxy sample on the cD side of the optimal border will not contain only cD galaxies, and a degree of contamination will be present. We define contamination [@Hoyos11] as: $$C=\dfrac{\textrm{\#non-cDs tested as positive}}{\textrm{\#all positives}}.$$ The numerator are the non-cD BCGs which are on the cD side of the optimal border. The denominator of this fraction includes both cD galaxies and non-cD BCGs on the cD side.
Figure \[fig:Abeta125\] shows the $\log\rex$–$\log\rffsx$ plane for the BCGs in our sample. The Amoeba algorithm requires a first guess for the border, shown by the black horizontal dotted line. The optimal border determined by the algorithm does not depend on the exact initial guess. The blue solid curve is the optimal border determined when we consider all cD galaxies (cD, cD/E and cD/S0) as cD galaxies. This border, computed using $\beta=1.25$, has $\,=0.69$. $75\%$ of all the cD galaxies are above the border ($r=0.75$), and thus selected from the parent sample. The remaining $25\%$ are mixed with the elliptical and disk BCGs in the region below the border. This selection therefore yields 75% completeness. The galaxy sample above the border contains $311$ cD galaxies and $79$ non-cD BCGs resulting in a $\sim 20\%$ contamination in the selected cD samples. In the region below the border there are $103$ cD galaxies and $122$ and non-cD BCGs. Thus, the non-cD BGC sample has a contamination of $46\%$ from cD galaxies. This indicates that this technique is more effective (cleaner) at selecting cD galaxies than at selecting non-cD BCGs. Note that if we consider a “cleaner” sample that contains only pure cD and pure elliptical BCGs (excluding all cD/E, cD/S0, E/cD, E/S0, spiral and S0 BCGs), the optimal border (blue dashed curve in Figure \[fig:Abeta125\]) does not change significantly, but the quality of the selection as determined by the value, the completeness $r$ and the specificity $p$ improves. This is not surprising: the identification of BCGs as pure cDs/Es (as opposed to the “dubious” ones) depends on more secure morphological characteristics which should be linked more clearly to the structural parameters. However, considering only this cleaner sample is not a realistic scenario since in practical cases we would like to start from a full sample of BCGs and find which ones are cDs. Nevertheless, it is reassuring that the border we determine does not depend very strongly on the exact training set used.
On the selected cD side, spiral BCGs are an important source of contamination. However, since most of them appear in the large region, it would be possible to go a step further to implement a simple further refinement in our method to separate spirals from the selected cDs: very few cD galaxies have $\log \rffsx$ larger than $\sim -1.1$. This would significantly improve the purity of the cD sample at very little cost in terms of its completeness.
Moreover, it is clear from Figure \[fig:Abeta125\] that all disk BCGs (spirals and S0s) contribute significantly to the contamination of either the cD or the elliptical samples separated by the best border. However, we can use the fact that disk BCGs distribute over a distinct area on the $\log\rex$–$\log\rffsx$ plane to apply a two-step process to exclude them from our cD selection. First, the disk BCGs can be separated from the elliptical and cD BCGs, and then the cD BCGs can be selected out of the rest BCG sample. Figure \[fig:twoborder\] illustrates the results of this two-step selection. The blue dashed curve is the determined in the the first step. By excluding disk BCGs using this border, a very complete ($r=0.93$) and pure ($p=0.87$) non-disk BCG sample is built. The cDs can then be separated from the ellipticals using the shown by the blue solid curve with a completeness of $77\%$ ($305$ cDs are selected), and a contamination of only $14\%$. Compared to the single-step cD selection ($311$ cDs were selected with $20\%$ contamination), the two-step process clearly selects a very similar number of cDs but with better purity. The decision on whether the increase in purity is worth the additional complexity is left to the reader. In the reminder of this paper we will use the single-step selection process for simplicity.
The automatic techniques we have developed can be applied to any BCG sample, but the needs to be adapted and calibrated using the imaging data from which the parent sample was derived. The calibration can be performed using a sub-sample of visually-classified BCGs, and then automatically applied to the complete sample using the structural parameters determined from standard single- fits.
A $\beta$ value needs to be chosen depending on whether we are more interested in the completeness of the cD sample or in its purity, but we suggest that $\beta=1.25$ represents a reasonable compromise (see section \[sec:betaparameter\]). Furthermore, it is important to remember that this method works better at selecting a sample of cD galaxies rather than a sample of non-cDs.
Distance to the Optimal Border {#sec:distancetoborder}
------------------------------
It is informative to explore the distribution of the points in the $\log \rex$–$\log \rffsx$ plane (Figure \[fig:Abeta125\]) in terms of their minimum (perpendicular) distance to the . We define the distance from each point to the as $$D=\sqrt{\left( \frac{\Delta \log \rffsx}{\sigma_{\log \rffsx}}\right) ^2 + \left( \frac{\Delta \log \rex}{\sigma_{\log \rex}}\right) ^2},$$ where $\Delta \log \rffsx$ is the difference in $\log \rffsx$ between the data point and the , and $\sigma_{\log \rffsx}$ is the dispersion in $\log \rffsx$ computed for all the points. $\Delta \log \rex$ and $\sigma_{\log \rex}$ have a similar meaning but for $\log \rex$. Note that, because the units of the $x$ and $y$ axes are different, the distance is measured in units of the scatter of each parameter. For each point, the minimum distance $D_{\rm min}$ can be then determined. Figure \[fig:borderdistanceall\] shows the distribution of these minimum distances for the different morphologies. As expected, the vast majority ($>80\%$) of the cDs show positive distances (they are above the line) while most of the ellipticals have negative ones. Under $20\%$ of the cDs spill over to the negative region, severely contaminating the non-cD sample, while a few ellipticals weakly contaminate the cD region. The measurement errors in $\log \rex$ ($\sim0.16$) and $\log \rffsx$ ($\sim0.13$) result in distance errors on the order of $0.7$ in this metric. This contributes to the cDs’ “spillover”, but does not completely explain it. Reducing the measurement errors would certainly improve the performance of our method, but it would never make it perfect.
Interestingly, the spiral and S0 BCGs are quite well separated: the former show mostly positive distances while the later have mostly negative ones. This is mainly due to spirals having generally larger values because the spiral arms and star-forming regions are not included in the models, while the S0s are smoother. This clear separation provides a possible way to separate spiral and S0 galaxies, but this needs to be further tested with large disk samples.
Another interesting result is that BCGs classified as pure and uncertain cDs (e.g., cD/E) have very different minimum distance distributions (Figure \[fig:borderdistamcecDs\], top panel). About half of the cD/E BCGs have negative distances (i.e., are on the wrong side of the border), but only $\simeq20$% of the pure cDs do. Most of the spillover of the pure cDs into the negative region, however, can be explained by the measurement errors. It should be noticed that the difficulties inherit in the visual morphological classification are directly reflected in the structural parameters: when the visual classifier is certain that a BCG is a cD, its structural parameters almost always confirm it, while in uncertain cases (e.g., cD/E) the structural parameters reflect this uncertainty. Similar conclusions can also be obtained from the pure elliptical BCGs and uncertain ones (e.g., E/cD), as shown in the bottom panel of Figure \[fig:borderdistamcecDs\].
This analysis confirms the visual impression in terms of the BCG structure that there is a continuous distribution in the properties of the BCG extended envelopes, ranging from undetected (pure E class) to clearly detected (pure cD class), with the intermediate classes (E/cD and cD/E) showing increasing degrees of envelope presence. This continuous distribution in envelope detectability is reflected quantitatively in the structural parameters of the BCGs, by the minimum distance to the providing some indication of the relative importance of the envelope.
Effect of the $\beta$ Parameter {#sec:betaparameter}
-------------------------------
In the definition, the $\beta$ parameter is used to apportion weight to the completeness and the specificity. For larger values of $\beta$ the completeness is given a larger weight than the lack of contamination. Conversely, smaller values of $\beta$ prioritise lack of contamination above completeness. To test how changing $\beta$ affects the results of the selection process, we repeat the exercise carried out in Section \[sec:bestborder\] but using $\beta=2.0$ and $\beta=0.5$ in the determination of the optimal border.
Figure \[fig:Abeta0520\] shows the for $\beta=2.0$ (upper panel) and $\beta=0.5$ (lower panel). It is clear that the $\beta$ parameter has a decisive impact on the selection of potential cD galaxies. As shown in the upper panel, when compared to the $\beta=1.25$ results, $11\%$ more galaxies are correctly identified as cDs, significantly increasing the completeness. The price paid is that the specificity goes down from $61\%$ to $46\%$ since more non-cD BCGs are included. Conversely, in the lower panel ($\beta=0.5$) the selected cD sample is purer ($p=0.85$), but at the expense of completeness, with $20\%$ fewer cD galaxies selected when compared with the $\beta=1.25$ result.
With $\beta=2.0$, the contamination of the cD sample by non-cDs is $23\%$, while the contamination of the non-cD sample by cDs is $39\%$. With $\beta=0.5$, the corresponding values are $12\%$ and $52\%$ respectively. Therefore, for any value of $\beta$ this selecting technique is cleaner and more effective at selecting cD galaxies than at selecting non-cD BCGs.
As before, if we consider a cleaner sample that contains only pure cD and pure elliptical BCGs, the optimal border (blue dashed curve) does not change significantly, but the value, the completeness $r$ and the specificity $p$ improve. However, we have argued that this does not represent a realistic scenario.
We conclude that $\beta=1.25$ represents a good compromise, as its picks up a cD galaxy sample reasonably complete, and with relatively small contamination. However, no single value of $\beta$ can be considered to be “correct” and needs to be set according to the scientific goals of the study.
Conclusions {#sec:conclude}
===========
In this paper we have analysed a well-defined sample of $625$ low-redshift Brightest Cluster Galaxies published in @Linden07 with the aim of linking their morphologies to their structural properties. We morphologically classified the BCGs using SDSS $r$-band images and found that over half of them ($\sim57$%) are pure cD galaxies and pure elliptical BCGs constitute $\sim13$% of the sample. The intermediate classes (mostly cD/E or E/cD) account for $\sim21$%. It suggests a continuous distribution in the properties of the BCG extended envelopes, ranging from undetected (pure E class) to clearly detected (pure cD class), with the intermediate classes (E/cD and cD/E) showing increasing degrees of envelope presence. We found this continuous distribution in envelope detectability is reflected quantitatively in the structural parameters of the BCGs. There is also a minority of BCGs that are neither cD nor elliptical. About $7$% are disk galaxies (spirals and S0s, in similar proportions) and the rest ($\sim2$%) are in merging (see appendix A).
In order to link the morphologies of the BCGs to their structural properties, we have fitted the BCGs light distributions with the SDSS $r$-band images using one-component () and two-component (+Exponential) models. We first characterised how well the models fit the target BCG by using two quantitative diagnostics. One diagnostic is the residual flux fraction (), which measures the fraction of the galaxy flux presenting in the residual images after subtracting the models. The other diagnostic is the reduced . We concluded that generally it is very difficult to find a robust diagnostic to decide, in a statistic way, whether a one-component or a two-component model is preferred for BCGs, especially for cD galaxies. Since there is no evident improvement by using two-component model fits, our other conclusions rely on the one-component fits.
From simple one-component profile fits, we have found a clear link between the BCGs morphologies and their structures, and claimed that a combination of the best-fit parameters can be used to separate cD galaxies from non-cD BCGs. In particular, cDs and non-cDs show very different distributions in the – plane, where is the effective radius and is the residual flux fraction, both determined from fits. cDs have, generally, larger and values than ellipticals. Therefore we found, in a statistically robust way, a boundary to separate cD and non-cD BCGs in this parameter space. BCGs with cD morphology can be selected with reasonably high completeness ($\sim 75\%$) and low contamination ($\sim 20\%$).
This automatic and objective technique can be applied to any current or future BCG samples which have good quality images. The method needs to be adapted and calibrated using the imaging data from which the parent sample was derived. Once calibrated with a representative sub-sample of visually-classified BCGs, this technique can be applied to the complete sample using the structural parameters determined from standard single- fits.
In a subsequent paper (Zhao et al., in preparation) we will explore how the morphological and structural properties of BCGs are linked to other intrinsic BCG properties such as their stellar mass, and/or to the properties of their environments. These links will provide more clues to the formation history of cDs/BCGs.
Acknowledgments {#acknowledgments .unnumbered}
===============
DZ’s work is supported by a Research Excellence Scholarship from the University of Nottingham and the China Scholarship Council. AAS and CJC acknowledge financial support from the UK Science and Technology Facilities Council. This paper is partially based on SDSS data. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
Data Table
==========
Table A1 contains the main properties of the BCGs discussed in this paper. The full table is published electronically.
\[tab:L07data\]
[rrrrrrrrrrll]{} ID2 &ID3 &RA &DEC &$z$ &$\sigma_{\rm cl}$ & $\log R_{\rm e,1c}$ &$n_{\rm 1c}$ &$RFF_{\rm 1c}$ &$\chi^2_{\rm 1c}$ & Type &Comments\
(1) & (2) ° (3) ° (4) & (5) &km$\,$s$^{-1}$ (6) &kpc (7) & (8) & (9) & (10) & (11) & (12)\
\[0.5ex\] 1011 & 1013 & 227.107346 & $-$0.266291 & 0.091 & 748 & 1.527 & 5.38 & 0.08190 & 1.752 & cD & Clear halo; perhaps interacting\
1023 & 1025 & 153.409478 & $-$0.925413 & 0.045 & 790 & 1.908 & 6.25 & 0.05052 & 1.374 & cD & Clear halo; interacting with fainter galaxies\
1064 & 1075 & 153.437067 & $-$0.120224 & 0.094 & 875 & 1.312 & 4.49 & 0.02648 & 1.086 & E/cD &\
– & 1027 & 191.926938 & $-$0.137254 & 0.088 & 1020 & 1.063 & 4.42 & 0.06594 & 1.903 & E & Interacting/merging with bright early-type\
– & 1389 & 202.337884 & 0.749685 & 0.080 & 853 & 1.044 & 6.02 & 0.01990 & 1.087 & E/cD & Faint/small halo\
2040 & 2050 & 17.513187 & 13.978117 & 0.059 & 759 & 2.408 & 9.77 & 0.04122 & 1.224 & cD & Several bright-ish companions\
1052 & 1058 & 195.719058 & $-$2.516350 & 0.083 & 749 & 1.627 & 4.89 & 0.04694 & 1.455 & cD & Multiple merger\
1034 & 1036 & 192.308670 & $-$1.687394 & 0.085 & 771 & 0.977 & 4.86 & 0.02102 & 1.115 & E &\
1041 & 1044 & 194.672887 & $-$1.761463 & 0.084 & 771 & 2.318 & 5.64 & 0.05023 & 1.280 & cD & Very large, elongated halo; some faint companions\
– & 1126 & 192.516071 & $-$1.540383 & 0.084 & 878 & 2.039 & 9.12 & 0.04520 & 1.348 & cD & Interacting with faint companions\
3002 & 3004 & 258.120056 & 64.060761 & 0.080 & 1156 & 1.667 & 4.81 & 0.02561 & 0.991 & cD &\
3096 & 3283 & 135.322540 & 58.279747 & 0.098 & 756 & 1.866 & 6.96 & 0.05535 & 1.144 & cD & Merging with bright companion\
1045 & 1048 & 205.540176 & 2.227213 & 0.077 & 828 & 0.883 & 2.52 & 0.10689 & 11.280 & E/cD & Multiple merger\
1003 & 1004 & 184.421356 & 3.655806 & 0.077 & 966 & 1.753 & 4.75 & 0.05233 & 1.225 & cD/E & Interacting/merging with early-type\
– & 1456 & 173.336242 & 2.199054 & 0.099 & 746 & 1.696 & 8.09 & 0.02573 & 1.128 & cD &\
1053 & 1061 & 228.220703 & 4.514004 & 0.038 & 789 & 0.875 & 7.54 & 0.01749 & 1.074 & cD &\
2163 & 2074 & 314.975446 & $-$7.260758 & 0.079 & 765 & 1.231 & 8.03 & 0.04481 & 1.315 & E/cD &\
2002 & 2002 & 358.557007 & $-$10.419200 & 0.076 & 812 & 2.660 & 11.12 & 0.03832 & 1.201 & cD & Many faint and bright-ish companions\
2006 & 2013 & 10.460272 & $-$9.303146 & 0.056 & 903 & 1.433 & 1.62 & 0.04140 & 1.477 & cD & Several faint companions\
1355 & 1460 & 175.554108 & 5.251709 & 0.097 & 1074 & 0.952 & 5.30 & 0.01557 & 1.052 & cD & Interacting with faint galaxy; faint but clear halo\
1058 & 1069 & 184.718166 & 5.245665 & 0.076 & 721 & 1.988 & 7.98 & 0.04144 & 1.251 & cD & Interacting with faint galaxies\
1002 & 1002 & 159.777581 & 5.209775 & 0.069 & 800 & 1.740 & 8.40 & 0.03838 & 1.321 & cD/E & Clear halo\
– & 1276 & 183.271286 & 5.689677 & 0.081 & 729 & 0.995 & 5.30 & 0.02142 & 1.151 & E &\
1039 & 1042 & 228.808792 & 4.386210 & 0.098 & 857 & 1.800 & 8.77 & 0.04365 & 1.205 & E/cD & Some halo? faint companions\
– & 3332 & 124.471428 & 40.726395 & 0.063 & 802 & 1.463 & 6.40 & 0.08125 & 2.639 & SB0 &\
3011 & 3028 & 204.034694 & 59.206401 & 0.070 & 872 & 2.120 & 7.86 & 0.08172 & 1.556 & cD & Several faint companions\
1001 & 1001 & 208.276672 & 5.149740 & 0.079 & 746 & 1.820 & 7.85 & 0.02720 & 1.128 & E/cD &\
3004 & 3012 & 255.677078 & 34.060024 & 0.099 & 1127 & 1.717 & 3.54 & 0.08433 & 1.949 & cD & Late merger?\
– & 3094 & 254.933115 & 32.615319 & 0.098 & 875 & 1.291 & 3.50 & 0.02878 & 1.069 & cD & Very faint companions\
– & 1066 & 202.795126 & $-$1.730259 & 0.085 & 814 & 1.942 & 9.09 & 0.03653 & 1.161 & E/cD & Interacting/merging with bright galaxy and fainter one\
– & 2214 & 321.599487 & 10.777511 & 0.095 & 741 & 0.818 & 3.98 & 0.02260 & 1.199 & E &\
2096 & 2109 & 359.836166 & 14.670211 & 0.093 & 786 & 1.161 & 6.56 & 0.03572 & 1.242 & cD/E &\
2085 & 2085 & 334.197449 & $-$9.724778 & 0.094 & 806 & 0.779 & 3.43 & 0.02861 & 1.348 & cD &\
2027 & 2035 & 4.177309 & $-$0.445436 & 0.065 & 1084 & 1.436 & 8.89 & 0.02417 & 1.168 & cD & Several companions\
– & 3084 & 118.360820 & 29.359459 & 0.061 & 781 & 1.584 & 3.95 & 0.06632 & 1.382 & cD & Several faint and bright companions\
– & 3347 & 119.679733 & 30.773809 & 0.076 & 902 & 1.354 & 6.04 & 0.01470 & 1.019 & E/cD &\
– & 1283 & 125.745443 & 4.299105 & 0.095 & 754 & 2.747 & 10.47 & 0.04483 & 1.094 & cD & Several faint-ish companions\
– & 1039 & 186.878093 & 8.824560 & 0.090 & 846 & 1.962 & 6.94 & 0.06100 & 1.965 & cD & Clear halo, bright companion (dumbbell galaxy)\
\[lastpage\]
[^1]: E-mail: : `[email protected]`
[^2]: E-mail: : `[email protected]`
[^3]: E-mail: : `[email protected]`
[^4]: $R_{200}$ is the radius within which the average mass density is $200\rho_{c}$, where $\rho_{c}$ is the critical density of the universe.
[^5]: In this paper we use the following definition of “Kron radius”: $\,=2.5r_1$, where $r_1$ is the first moment of the light distribution [@Kron80; @BA96]. For an elliptical light distribution, this is, strictly speaking, the semi-major axis.
[^6]: http://www.sdss.org/DR7/products/images/read\_psf.html
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The relativistic chiral $SU(3)$ Lagrangian is used to describe kaon-nucleon scattering imposing constraints from the pion-nucleon sector and the axial-vector coupling constants of the baryon octet states. We solve the covariant coupled-channel Bethe-Salpeter equation with the interaction kernel truncated at chiral order $Q^3$ where we include only those terms which are leading in the large $N_c$ limit of QCD. The baryon decuplet states are an important explicit ingredient in our scheme, because together with the baryon octet states they form the large $N_c$ baryon ground states of QCD. Part of our technical developments is a minimal chiral subtraction scheme within dimensional regularization, which leads to a manifest realization of the covariant chiral counting rules. All SU(3) symmetry-breaking effects are well controlled by the combined chiral and large $N_c$ expansion, but still found to play a crucial role in understanding the empirical data. We achieve an excellent description of the data set typically up to laboratory momenta of $p_{\rm lab} \simeq $ 500 MeV.'
address:
- |
$^a$ Gesellschaft für Schwerionenforschung (GSI),\
Planck Str. 1, D-64291 Darmstadt, Germany
- |
$^b$ ECT$^*$, Villa Tambosi, I-38050 Villazzano (Trento)\
and INFN, G.C. Trento, Italy
author:
- 'M.F.M. Lutz$^a$ and E.E. Kolomeitsev$^{a,b}$'
title: |
Relativistic chiral SU(3) symmetry, large $N_c$ sum rules\
and\
meson-baryon scattering
---
GSI-Preprint-2001-12 and ECT\*-Preprint-2001-10
Introduction
============
The meson-baryon scattering processes are an important test for effective field theories which aim at reproducing QCD at small energies, where the effective degrees of freedom are hadrons rather than quarks and gluons. In this work we focus on the strangeness sector, because there the acceptable effective field theories are much less developed and also the empirical data set still leaves much room for different theoretical interpretations. In the near future the new DA$\Phi$NE facility at Frascati could deliver new data on kaon-nucleon scattering [@DAPHNE] and therewith help to establish a more profound understanding of the role played by the $SU(3)$ flavor symmetry in hadron interactions. At present the low-energy elastic $K^+$-proton scattering data set leads to a rather well established $K^+p$ scattering length with $a_{K^+p}\simeq 0.28-0.34$ fm [@Dover]. Uncertainties exist, however, in the $K^+$-neutron channel where the elastic cross section is extracted from the scattering data on the $K^+d\to K^+ p\,n $ and $K^+d \to K^0 p \,p$ reactions. Since data are available only for $p_{\rm lab} > 350$ MeV, the model dependence of the deuteron wave function, the final state interactions and the necessary extrapolations down to threshold lead to conflicting values for the $K^+$-neutron scattering length [@Dumbrajs]. A recent analysis [@Barnes] favors a repulsive and small value $a_{K^+n}\simeq 0.1$ fm. Since low-energy polarization data are not available for $K^+$-nucleon scattering, the separate strength of the various p-wave channels can only be inferred by theory at present. This leads to large uncertainties in the p-wave scattering volumes [@Dumbrajs].
The $K^-$-proton scattering length was only recently determined convincingly by a kaonic-hydrogen atom measurement [@Iwasaki]. In contrast the $K^-$-neutron scattering length remains model dependent [@A.D.Martin; @martsakit]. This reflects the fact that at low energies there are no $K^-$ deuteron scattering data available except for some $K^-d$ branching ratios [@Veirs] commonly not included in theoretical models of kaon-nucleon scattering. The rather complex multi-channel dynamics of the strangeness minus one channel is constrained by not too good quality low-energy elastic and inelastic $K^-p$ scattering data [@Landoldt] but rather precise $K^-p$ threshold branching ratios [@branch-rat]. Therefore the isospin one scattering amplitude is constrained only indirectly for instance by the $\Lambda \pi^0$ production data [@mast-pio]. That leaves much room for different theoretical extrapolations [@A.D.Martin; @martsakit; @kim; @sakit; @gopal; @oades; @Juelich:2; @dalitz; @keil]. As a consequence the subthreshold $\bar K N$ scattering amplitudes, which determine the $\bar K$-spectral function in nuclear matter to leading order in the density expansion, are poorly controlled. In the region of the $\Lambda(1405)$ resonance the isospin zero amplitudes of different analyses may differ by a factor of two [@Kaiser; @Ramos].
Therefore it is desirable to make use of the chiral symmetry constraints of QCD. First intriguing works in this direction can be found in [@Kaiser; @Ramos; @Hirschegg; @Oller-Meissner]. The reliability of the extrapolated subthreshold scattering amplitudes can be substantially improved by including s- [*and*]{} p-waves in the analysis of the empirical cross sections, because the available data, in particular in the strangeness minus one channel, are much more precise for $p_{\rm lab}>200$ MeV than for $p_{\rm lab}<200$ MeV, where one expects s-wave dominance.
In this work we use the relativistic chiral $SU(3)$ Lagrangian including an explicit baryon decuplet resonance field with $J^P\!= \!\frac{3}{2}^+$. The baryon decuplet field is an important ingredient, because it is part of the baryon ground state multiplet which arises in the large $N_c$ limit of QCD [@Hooft; @Witten]. We also consider the effects of a phenomenological baryon nonet d-wave resonance field with $J^P\!=\!\frac{3}{2}^-$, because some of the p-wave strengths in the $\bar K N$ system can be extracted from the available data set reliably only via their interference effects with the d-wave resonance $\Lambda (1520)$. As to our knowledge this is the first application of the chiral $SU(3)$ Lagrangian density to the kaon-nucleon and antikaon-nucleon systems including systematically constraints from the pion-nucleon sector. We propose a convenient minimal chiral subtraction scheme for relativistic Feynman diagrams which complies manifestly with the standard chiral counting rule [@Becher; @nn-lutz; @Gegelia]. Furthermore it is argued that the relatively large kaon mass necessarily leads to non-perturbative phenomena in the kaon-nucleon channels in contrast to the pion-nucleon system where standard chiral perturbation theory ($\chi$PT) can be applied successfully [@Gasser; @Bernard; @Meissner]. In the strangeness sectors a partial resummation scheme is required [@Siegel; @Kaiser; @Ramos]. We solve the Bethe-Salpeter equation for the scattering amplitude with the interaction kernel truncated at chiral order $Q^3$ where we include only those terms which are leading in the large $N_c$ limit of QCD [@Hooft; @Witten; @DJM; @Carone; @Luty]. The s-, p- and d-wave contributions with $J=\frac{1}{2},\frac{3}{2}$ in the scattering amplitude are considered. As a novel technical ingredient we construct a covariant projector formalism. It is supplemented by a subtraction scheme rather than a cutoff scheme as employed previously in [@Kaiser; @Ramos]. The renormalization scheme is an essential input of our chiral $SU(3)$ dynamics, because it leads to consistency with chiral counting rules and an approximate crossing symmetry of the subthreshold kaon-nucleon scattering amplitudes. Our scheme avoids, in particular, breaking the $SU(3)$ symmetry by channel-dependent cutoff parameters as suggested in [@Kaiser] and also a sensitivity of the $\Lambda(1405)$ resonance structure to the cutoff parameter implicit in [@Ramos].
We successfully adjust the set of parameters to describe the existing low-energy cross section data on kaon-nucleon and antikaon-nucleon scattering including angular distributions to good accuracy. At the same time we achieve a satisfactory description of the low-energy s- and p-wave pion-nucleon phase shifts as well as the empirical axial-vector coupling constants of the baryon octet states. We make detailed predictions for the poorly known meson-baryon coupling constants of the baryon octet and decuplet states. Furthermore the many $SU(3)$ reaction amplitudes and cross sections like $\pi \Lambda \to \pi \Lambda, \pi \Sigma , \bar K \,N $, relevant for transport model simulations of heavy-ion reactions, will be presented. As a result of our analysis, particularly important for any microscopic description of antikaon propagation in dense nuclear matter, we predict sizable contributions from p-waves in the subthreshold $\bar K$-nucleon forward scattering amplitude.
In section 2 we construct the parts of the relativistic chiral Lagrangian relevant for this work. All interaction terms are analyzed systematically in the $1/N_c$ expansion of QCD. In section 3 we develop the formalism required for the proper treatment of the Bethe-Salpeter equation including details on the renormalization scheme. In section 4 we derive the coupled channel effective interaction kernel in accordance with the scheme presented in section 3. The reader interested primarily in the numerical results can go directly to section 5, which can be read rather independently.
Relativistic chiral $SU(3)$ interaction terms in large $N_c$ QCD
================================================================
In this section we present all chiral interaction terms to be used in the following sections to describe the low-energy meson-baryon scattering data set. Particular emphasis is put on the constraints implied by the large $N_c$ analysis of QCD. The reader should not be discouraged by the many fundamental parameters introduced in this section. The empirical data set includes many hundreds of data points and will be reproduced rather accurately. Our scheme makes many predictions for poorly known or not known observable quantities like for example the p-wave scattering volumes of the kaon-nucleon scattering processes or $SU(3)$ reactions like $\pi \Lambda \to \pi \Sigma $. In a more conventional meson-exchange approach, which lacks a systematic approximation scheme, many parameters are implicit in the so-called form factors. In a certain sense the parameters used in the form factors reflect the more systematically constructed and controlled quasi-local counter terms of the chiral Lagrangian.
We recall the interaction terms of the relativistic chiral $SU(3)$ Lagrangian density relevant for the meson-baryon scattering process. For details on the systematic construction principle, see for example [@Krause]. The basic building blocks of the chiral Lagrangian are $$\begin{aligned}
&& U_\mu = \frac{1}{2}\,e^{-i\,\frac{\Phi}{2\,f}} \left(
\partial_\mu \,e^{i\,\frac{\Phi}{f}}
+ i\,\Big[A_\mu , e^{i\,\frac{\Phi}{f}} \Big]_+
\right) e^{-i\,\frac{\Phi}{2\,f}} \;,\qquad \!\!
B \;, \qquad \! \!\Delta_\mu \,, \qquad \! \! B^{*}_\mu \;,
\label{def-fields}\end{aligned}$$ where we include the pseudo-scalar meson octet field $\Phi(J^P\!\!=\!0^-)$, the baryon octet field $B(J^P\!\!=\!{\textstyle{1\over2}}^+)$, the baryon decuplet field $\Delta_\mu(J^P\!\!=\!{\textstyle{3\over2}}^+)$ and the baryon nonet resonance field $B^*_\mu(J^P\!\!=\!{\textstyle{3\over2}}^-)$ (see [@Tripp:1; @Tripp:2; @Plane]). In (\[def-fields\]) we introduce an external axial-vector source function $A_\mu $ which is required for the systematic evaluation of matrix elements of the axial-vector current. A corresponding term for the vector current is not shown in (\[def-fields\]) because it will not be needed in this work. The axial-vector source function $A^\mu =\sum A_a^\mu\,\lambda^{(a)} $, the meson octet field $\Phi=\sum \Phi_a\,\lambda^{(a)}$ and the baryon octet fields $B= \sum B_a\,\lambda^{(a)}/\sqrt{2}$, $B_\mu^*= B_{\mu,0}^*/\sqrt{3}+\sum B_{\mu,a}^*\,\lambda^{(a)}/\sqrt{2}$ are decomposed using the Gell-Mann matrices $\lambda_a$ normalized by $\tr \lambda_a \,\lambda_b =
2\,\delta_{ab}$. The baryon decuplet field $\Delta^{abc} $ is completely symmetric and related to the physical states by $$\begin{aligned}
\begin{array}{llll}
\Delta^{111} = \Delta^{++}\,, & \Delta^{113} =\Sigma^{+}/\sqrt{3}\,, &
\Delta^{133}=\Xi^0/\sqrt{3}\,, &\Delta^{333}= \Omega^-\,, \\
\Delta^{112} =\Delta^{+}/\sqrt{3}\,, & \Delta^{123} =\Sigma^{0}/\sqrt{6}\,, &
\Delta^{233}=\Xi^-/\sqrt{3}\,, & \\
\Delta^{122} =\Delta^{0}/\sqrt{3}\,, & \Delta^{223} =\Sigma^{-}/\sqrt{3}\,, &
& \\
\Delta^{222} =\Delta^{-}\,. & & &
\end{array}
\label{dec-field}\end{aligned}$$ The parameter $f $ in (\[def-fields\]) is determined by the weak decay widths of the charged pions and kaons properly corrected for chiral SU(3) effects. Taking the average of the empirical decay parameters $f_\pi = 92.42 \pm 0.33 $ MeV and $f_K \simeq 113.0 \pm 1.3$ MeV [@fpi:exp] one obtains the naive estimate $f \simeq 104$ MeV. This value is still within reach of the more detailed analysis [@GL85] which lead to $f_\pi/f = 1.07 \pm 0.12$. As emphasized in [@MO01], the precise value of $f$ is subject to large uncertainties.
Explicit chiral symmetry-breaking effects are included in terms of scalar and pseudo-scalar source fields $\chi_\pm $ proportional to the quark-mass matrix of QCD $$\begin{aligned}
\chi_\pm = \frac{1}{2} \left(
e^{+i\,\frac{\Phi}{2\,f}} \,\chi_0 \,e^{+i\,\frac{\Phi}{2\,f}}
\pm e^{-i\,\frac{\Phi}{2\,f}} \,\chi_0 \,e^{-i\,\frac{\Phi}{2\,f}}
\right) \,,
\label{def-chi}\end{aligned}$$ where $\chi_0 \sim {\rm diag} (m_u,m_d,m_s)$. All fields in (\[def-fields\]) and (\[def-chi\]) have identical properties under chiral $SU(3)$ transformations. The chiral Lagrangian consists of all possible interaction terms, formed with the fields $U_\mu, B, \Delta_\mu, B^*_\mu$ and $\chi_\pm$ and their respective covariant derivatives. Derivatives of the fields must be included in compliance with the chiral $SU(3)$ symmetry. This leads to the notion of a covariant derivative ${\mathcal D}_\mu$ which is identical for all fields in (\[def-fields\]) and (\[def-chi\]). For example, it acts on the baryon octet field as $$\begin{aligned}
\Big[{\mathcal D}_\mu , B\Big]_- &=& \partial_\mu \,B +
\frac{1}{2}\,\Big[ e^{-i\,\frac{\Phi}{2\,f}} \left(
\partial_\mu \,e^{+i\,\frac{\Phi}{2\,f}}\right)
+e^{+i\,\frac{\Phi}{2\,f}} \left(
\partial_\mu \,e^{-i\,\frac{\Phi}{2\,f}}\right), B\Big]_-
\nonumber\\
&+& \frac{i}{2}\,\Big[ e^{-i\,\frac{\Phi}{2\,f}} \,
A_\mu \,e^{+i\,\frac{\Phi}{2\,f}}- e^{+i\,\frac{\Phi}{2\,f}} \, A_\mu
\,e^{-i\,\frac{\Phi}{2\,f}}, B\Big]_- \,.
\label{}\end{aligned}$$
The chiral Lagrangian is a powerful tool once it is combined with appropriate power counting rules leading to a systematic approximation strategy. One aims at describing hadronic interactions at low-energy by constructing an expansion in small momenta and the small pseudo-scalar meson masses. The infinite set of Feynman diagrams are sorted according to their chiral powers. The minimal chiral power $Q^{\nu }$ of a given relativistic Feynman diagram, $$\begin{aligned}
\nu = 2-{\textstyle {1\over2}}\, E_B + 2\, L
+\sum_i V_i \left( d_i +{\textstyle {1\over2}}\, n_i-2 \right) \;,
\label{q-rule}\end{aligned}$$ is given in terms of the number of loops, $L$, the number, $V_i$, of vertices of type $i$ with $d_i$ ’small’ derivatives and $n_i$ baryon fields involved, and the number of external baryon lines $E_B$ [@Weinberg]. Here one calls a derivative small if it acts on the pseudo-scalar meson field or if it probes the virtuality of a baryon field. Explicit chiral symmetry-breaking effects are perturbative and included in the counting scheme with $\chi_0 \sim Q^2$. For a discussion of the relativistic chiral Lagrangian and its required systematic regrouping of interaction terms we refer to [@nn-lutz]. We will encounter explicit examples of this regrouping later. The relativistic chiral Lagrangian requires a non-standard renormalization scheme. The $MS$ or $\overline{MS}$ minimal subtraction schemes of dimensional regularization do not comply with the chiral counting rule [@Gasser]. However, an appropriately modified subtraction scheme for relativistic Feynman diagrams leads to manifest chiral counting rules [@Becher; @nn-lutz; @Gegelia]. Alternatively one may work with the chiral Lagrangian in its heavy-fermion representation where an appropriate frame-dependent redefinition of the baryon fields leads to a more immediate manifestation of the chiral power counting rule (\[q-rule\]). We will return to this issue in section 3.1 where we propose a simple modification of the $\overline {MS}$-scheme which leads to consistency with (\[q-rule\]). Further subtleties of the chiral power counting rule (\[q-rule\]) caused by the inclusion of an explicit baryon resonance field $B^*_\mu$ are addressed in section 4.1 when discussing the u-channel resonance exchange contributions.
In the $\pi N$ sector, the $SU(2)$ chiral Lagrangian was successfully applied [@Gasser; @Bernard] demonstrating good convergence properties of the perturbative chiral expansion. In the $SU(3)$ sector, the situation is more involved due in part to the relatively large kaon mass $m_K \simeq m_N/2$. The perturbative evaluation of the chiral Lagrangian cannot be justified and one must change the expansion strategy. Rather than expanding directly the scattering amplitude one may expand the interaction kernel according to chiral power counting rules [@Weinberg; @LePage]. The scattering amplitude then follows from the solution of a scattering equation like the Lipmann-Schwinger or the Bethe-Salpeter equation. This is analogous to the treatment of the $e^+\,e^-$ bound-state problem of QED where a perturbative evaluation of the interaction kernel can be justified. The rational behind this change of scheme lies in the observation that reducible diagrams are typically enhanced close to their unitarity threshold. The enhancement factor $(2\pi)^n$, measured relative to a reducible diagram with the same number of independent loop integrations, is given by the number, $n$, of reducible meson-baryon pairs in the diagram, i.e. the number of unitary iterations implicit in the diagram. In the $\pi N$ sector this enhancement factor does not prohibit a perturbative treatment, because the typical expansion parameter $m^2_\pi/(8 \pi \,f^2) \sim 0.1 $ remains sufficiently small. In the $\bar K N$ sector, on the other hand, the factor $(2\pi)^n$ invalidates a perturbative treatment, because the typical expansion parameter would be $m^2_K/(8 \pi\,f^2) \sim 1$. This is in contrast to irreducible diagrams. They yield the typical expansion parameters $m_\pi/(4 \pi \,f)$ and $m_K/(4\pi \,f)$ which justifies the perturbative evaluation of the scattering kernels. We will return to this issue later and discuss this phenomena in terms of the Weinberg-Tomozawa interaction in more detail.
In the next section we will develop the formalism to construct the leading orders interaction kernel from the relativistic chiral Lagrangian and then to solve the Bethe-Salpeter scattering equation. In the remainder of this section, we collect all interaction terms needed for the construction of the Bethe-Salpeter interaction kernel. We consider all terms of chiral order $Q^2$ but only the subset of chiral $Q^3$-terms which are leading in the large $N_c$ limit. Loop corrections to the Bethe-Salpeter kernel are neglected, because they carry minimal chiral order $Q^3$ and are $1/N_c$ suppressed. The chiral Lagrangian $$\begin{aligned}
{\mathcal L} = \sum_n \,{\mathcal L}^{(n)}+\sum_n\,{\mathcal L}^{(n)}_\chi
\label{}\end{aligned}$$ can be decomposed into terms of different classes ${\mathcal L}^{(n)}$ and ${\mathcal L}^{(n)}_\chi$. With an upper index $n$ in ${\mathcal L}^{(n)}$ we indicate the number of fields in the interaction vertex. The lower index $\chi $ signals terms with explicit chiral symmetry breaking. We assume charge conjugation symmetry and parity invariance in this work. To leading chiral order the following interaction terms are required: $$\begin{aligned}
{\mathcal L}^{(2)} &=&
\tr \bar B \left(i\,\partialslash-\m0_{[8]}\right) \, B
+\frac{1}{4}\,\tr (\partial^\mu \,\Phi )\,
(\partial_\mu \,\Phi )
\nonumber\\
&+&\tr \bar \Delta_\mu \cdot \Big(
\left( i\,\partialslash -\m0_{[10]} \right)g^{\mu \nu }
-i\,\left( \gamma^\mu \partial^\nu + \gamma^\nu \partial^\mu\right)
+i\,\gamma^\mu\,\partialslash\,\gamma^\nu
+\,\m0_{[10]} \,\gamma^\mu\,\gamma^\nu
\Big) \, \Delta_\nu
\nonumber\\
&+&\tr \bar B^*_\mu \cdot \Big(
\left( i\,\partialslash -\m0_{[9]} \right)g^{\mu \nu }
-i\,\left( \gamma^\mu \partial^\nu + \gamma^\nu \partial^\mu\right)
+i\,\gamma^\mu\,\partialslash\,\gamma^\nu
+\,\m0_{[9]} \,\gamma^\mu\,\gamma^\nu
\Big) \, B^*_\nu
\nonumber\\
{\mathcal L}^{(3)} &=&\frac{F_{[8]}}{2\,f} \,\tr \bar B
\,\gamma_5\,\gamma^\mu \,\Big[\left(\partial_\mu\,\Phi\right),B\Big]_-
+\frac{D_{[8]}}{2\,f} \,\tr \bar B
\,\gamma_5\,\gamma^\mu \,\Big[\left(\partial_\mu\,\Phi\right),B\Big]_+
\nonumber\\
&-&\frac{C_{[10]}}{2\,f}\,
\tr \left\{
\Big( \bar \Delta_\mu \cdot
(\partial_\nu \,\Phi ) \Big)
\Big( g^{\mu \nu}-\half\,Z_{[10]}\, \gamma^\mu\,\gamma^\nu \Big) \,
B +\mathrm{h.c.}
\right\}
\nonumber\\
&+&\frac{D_{[9]}}{2\,f}\,
\tr \left\{ \bar B^*_\mu \cdot \Big[
(\partial_\nu \,\Phi ) \,
\Big( g^{\mu \nu}-\half\,Z_{[9]}\, \gamma^\mu\,\gamma^\nu \Big)\,\gamma_5 ,
B \Big]_++\mathrm{h.c.} \right\}
\nonumber\\
&+&\frac{F_{[9]}}{2\,f}\,
\tr \left\{
\bar B^*_\mu \cdot \Big[
(\partial_\nu \,\Phi ) \,
\Big( g^{\mu \nu}-\half\,Z_{[9]}\, \gamma^\mu\,\gamma^\nu \Big)\,\gamma_5 ,
B \Big]_-+\mathrm{h.c.}
\right\}
\nonumber\\
&+&\frac{C_{[9]}}{8\,f}\,
\tr \left\{ \bar B^*_\mu \,\tr \Big[
(\partial_\nu \,\Phi ) \,
\Big( g^{\mu \nu}-\half\,Z_{[9]}\, \gamma^\mu\,\gamma^\nu \Big)\,\gamma_5 ,
B \Big]_++\mathrm{h.c.}
\right\} \;,
\nonumber\\
{\mathcal L}^{(4)}&=& \frac{i}{8\,f^2}\,\tr\bar B\,\gamma^\mu \Big[\Big[ \Phi ,
(\partial_\mu \,\Phi) \Big]_-,B \Big]_- \,,
\label{lag-Q}\end{aligned}$$ where we use the notations $[A,B]_\pm = A\,B\pm B\,A$ for $SU(3)$ matrices $A$ and $B$. Note that the complete chiral interaction terms which lead to the terms in (\[lag-Q\]) are easily recovered by replacing $i\,\partial_\mu \,\Phi /f \to U_\mu $. A derivative acting on a baryon field in (\[lag-Q\]) must be understood as the covariant derivative $\partial_\mu \,B \to [{\mathcal D}_\mu ,B ]_- $ and $\partial_\mu \,\Delta_\nu \to [{\mathcal D}_\mu ,\Delta_\nu ]_- $ .
The $SU(3)$ meson and baryon fields are written in terms of their isospin symmetric components $$\begin{aligned}
\Phi &=& \tau \cdot \pi
+ \alpha^\dagger \!\cdot \! K + K^\dagger \cdot \alpha +
\eta \,\lambda_8 \;,
\nonumber\\
\sqrt{2}\,B &=& \alpha^\dagger \!\cdot \! N(939)+\lambda_8 \,\Lambda(1115)+ \tau \cdot \Sigma(1195)
+\Xi^t(1315)\,i\,\sigma_2 \!\cdot \!\alpha \, ,
\nonumber\\ \nonumber\\
\alpha^\dagger &=&
{\textstyle{1\over\sqrt{2}}}\left( \lambda_4+i\,\lambda_5 ,
\lambda_6+i\,\lambda_7 \right)
\;,\;\;\;\tau = (\lambda_1,\lambda_2,\lambda_3)\;,
\label{field-decomp}\end{aligned}$$ with the isospin doublet fields $K =(K^+,K^0)^t $, $N=(p,n)^t$ and $\Xi = (\Xi^0,\Xi^-)^t$. The isospin Pauli matrix $\sigma_2$ acts exclusively in the space of isospin doublet fields $(K,N,\Xi)$ and the matrix valued isospin doublet $\alpha$ (see Appendix A). For our work we chose the isospin basis, because isospin breaking effects are important only in the $\bar K N$ channel. Note that in (\[field-decomp\]) the numbers in the parentheses indicate the approximate mass of the baryon octet fields and $(...)^t$ means matrix transposition. Analogously we write the baryon resonance field $B_\mu^*$ as $$\begin{aligned}
\sqrt{2}\,B^*_\mu &=&
\Big( \sqrt{{\textstyle{2\over 3}}}\,\cos \vartheta- \lambda_8\,\sin \vartheta\Big) \,\Lambda_\mu(1520)
+\alpha^\dagger \!\cdot\! N_\mu(1520)
\nonumber\\
&+& \Big( \sqrt{{\textstyle{2\over 3}}}\,\sin \vartheta+ \lambda_8\,\cos \vartheta\Big) \,\Lambda_\mu(1690)
+\tau \!\cdot \!\Sigma_\mu(1670) +\Xi_\mu^t(1820)\,i\,\sigma_2 \!\cdot\! \alpha \, ,
\label{def-b-stern}\end{aligned}$$ where we allow for singlet-octet mixing by means of the mixing angle $\vartheta $ (see [@Plane]). The parameters $\m0_{[8]}$, $\m0_{[9]}$ and $\m0_{[10]}$ in (\[lag-Q\]) denote the baryon masses in the chiral $SU(3)$ limit. Furthermore the products of an anti-decuplet field $\bar
\Delta$ with a decuplet field $\Delta$ and an octet field $\Phi$ transform as $SU(3)$ octets $$\begin{aligned}
\Big(\bar \Delta \cdot \Delta \Big)^a_b &=&
\bar \Delta_{bcd}\,\Delta^{acd}\,, \qquad
\Big( \bar \Delta \cdot \Phi \Big)^a_b
=\epsilon^{kla}\,\bar \Delta_{knb}\,\Phi_l^n \,,
\nonumber\\
\Big( \Phi \cdot \Delta \Big)^a_b
&=&\epsilon_{klb}\,\Phi^l_n\,\Delta^{kna} \; ,
\label{dec-prod}\end{aligned}$$ where $\epsilon_{abc}$ is the completely anti-symmetric pseudo-tensor. For the isospin decomposition of $\bar \Delta \cdot \Delta$, $\bar \Delta \cdot \Phi$ and $\Phi \cdot \Delta$ we refer to Appendix A.
The parameters $F_{[8]}\simeq 0.45$ and $D_{[8]}\simeq 0.80$ are constrained by the weak decay widths of the baryon octet states [@Okun] (see also Tab. \[weak-decay:tab\]) and $C_{[10]}\simeq 1.6$ can be estimated from the hadronic decay width of the baryon decuplet states. The parameter $Z_{[10]}$ in (\[lag-Q\]) may be best determined in an $SU(3)$ analysis of meson-baryon scattering. While in the pion-nucleon sector it can be absorbed into the quasi-local 4-point interaction terms to chiral accuracy $Q^2$ [@Tang] (see also Appendix H), this is no longer possible if the scheme is extended to $SU(3)$. Our detailed analysis reveals that the parameter $Z_{[10]}$ is relevant already at order $Q^2$ if a simultaneous chiral analysis of the pion-nucleon and kaon-nucleon scattering processes is performed. The resonance parameters may be estimated by an update of the analysis [@Plane]. That leads to the values $F_{[9]} \simeq 1.8$, $D_{[9]} \simeq 0.84 $ and $C_{[9]} \simeq 2.5 $. The singlet-octet mixing angle $\vartheta \simeq $ 28$^\circ$ confirms the finding of [@Tripp:2] that the $\Lambda(1520)$ resonance is predominantly a flavor singlet state. The value for the background parameter $Z_{[9]}$ of the $J^P\!\!=\!{\textstyle{3\over2}}^-$ resonance is expected to be rather model dependent, because it is unclear so far how to incorporate the $J^P\!\!=\!{\textstyle{3\over2}}^-$ resonance in a controlled approximation scheme. As will be explained in detail $Z_{[9]}$ will drop out completely in our scheme (see sections 4.1-4.2).
Large $N_c$ counting
--------------------
In this section we briefly recall a powerful expansion strategy which follows from QCD if the numbers of colors ($N_c$) is considered as a large number. We present a formulation best suited for an application in the chiral Lagrangian leading to a significant parameter reduction. The large $N_c$ scaling of a chiral interaction term is easily worked out by using the operator analysis proposed in [@Dashen]. Interaction terms involving baryon fields are represented by matrix elements of many-body operators in the large $N_c$ ground-state baryon multiplet $| {\mathcal B} \rangle $. A n-body operator is the product of n factors formed exclusively in terms of the bilinear quark-field operators $J_i, G_i^{(a)}$ and $T^{(a)}$. These operators are characterized fully by their commutation relations, $$\begin{aligned}
&&[ G^{(a)}_i\,,G^{(b)}_j]_- ={\textstyle{1\over 4}}\,i\,\delta_{ij}\,f^{ab}_{\;\;\;\,c}\,T^{(c)}
+{\textstyle{1\over 2}}\,i\,\epsilon_{ij}^{\;\;\;k}
\left({\textstyle{1\over 3}}\,\delta^{ab}\,J_{k}+d^{ab}_{\;\;\;\,c}\,G^{(c)}_k\right), \;\;
\nonumber\\
&&[ J_i\,,J_j]_- =i\,\epsilon_{ij}^{\;\;\;k}\,J_{k}\, , \quad
[ T^{(a)}\,,T^{(b)}]_- =i\,f^{ab}_{\;\;\;\,c}\,T^{(c)}\,,\quad
\nonumber\\
&& [ T^{(a)}\,,G^{(b)}_i]_- =i\,f^{ab}_{\;\;\;c}\,G^{(c)}_i \;,\quad \!
[ J_i\,,G^{(a)}_j]_- =i\,\epsilon_{ij}^{\;\;\;k}\,G^{(a)}_k \;, \quad \!
[ J_i\,,T^{(a)}]_- = 0\;.
\label{comm}\end{aligned}$$ The algebra (\[comm\]), which refelcts the so-called contracted spin-flavor symmetry of QCD, leads to a transparent derivation of the many sum rules implied by the various infinite subclasses of QCD quark-gluon diagrams as collected to a given order in the $1/N_c$ expansion. A convenient realization of the algebra (\[comm\]) is obtained in terms of non-relativistic, flavor-triplet and color $N_c$-multiplet field operators $q$ and $q^\dagger$ $$\begin{aligned}
&& J_i = q^\dagger \Bigg( \,\frac{\sigma_i^{(q)}}{2} \otimes 1\Bigg) \,q \,, \qquad
T^{(a)} = q^\dagger \Bigg( 1 \otimes \frac{\lambda^{(a)}}{2} \Bigg) \,q \,, \;\,
\nonumber\\
&& G^{(a)}_i = q^\dagger \Bigg(\, \frac{\sigma_i^{(q)}}{2} \otimes \frac{\lambda^{(a)}}{2} \Bigg) \,q \,.
\label{}\end{aligned}$$ If the fermionic field operators $q$ and $q^\dagger $ are assigned anti-commutation rules, the algebra (\[comm\]) follows. The Pauli spin matrices $\sigma^{(q)}_i$ act on the two-component spinors of the fermion fields $q, q^\dagger $ and the Gell-Mann matrices $\lambda_a$ on their flavor components. Here one needs to emphasize that the non-relativistic quark-field operators $q$ and $q^\dagger $ should not be identified with the quark-field operators of the QCD Lagrangian [@DJM; @Carone; @Luty]. Rather, they constitute an effective tool to represent the operator algebra (\[comm\]) which allows for an efficient derivation of the large $N_c$ sum rules of QCD. A systematic truncation scheme results in the operator analysis, because a $n$-body operator is assigned the suppression factor $N_c^{1-n}$. The analysis is complicated by the fact that matrix elements of $ T^{(a)}$ and $G_i^{(a)}$ may be of order $N_c$ in the baryon ground state $|{\mathcal B} \rangle$. That implies for instance that matrix elements of the (2$n$+1)-body operator $(T_a\,T^{(a)})^n\,T^{(c)}$ are not suppressed relative to the matrix elements of the one-body operator $T^{(c)}$. The systematic large $N_c$ operator analysis relies on the observation that matrix elements of the spin operator $J_i$, on the other hand, are always of order $N_c^0$. Then a set of identities shows how to systematically represent the infinite set of many-body operators, which one may write down to a given order in the $1/N_c$ expansion, in terms of a finite number of operators. This leads to a convenient scheme with only a finite number of operators to a given order [@Dashen]. We recall typical examples of the required operator identities $$\begin{aligned}
&&[T_a,\, T^{(a)}]_+ - [J_i,\,J^{(i)}]_+ ={\textstyle {1\over 6}}\,N_c\,(N_c+6)\;,
\quad [ T_{a}\,,G^{(a)}_i]_+ ={\textstyle {2\over 3}}\,(3+N_c)\,J_i \;,
\nonumber\\
&& 27\,[T_a,\, T^{(a)}]_+-12\,[G_i^{(a)},\, G_a^{(i)}]_+
= 32\,[J_i,\,J^{(i)}]_+ \;,
\nonumber\\
&& d_{abc}\,[T^{(a)},\,T^{(b)}]_+ -2\,[J_{i},\,G_c^{(i)}]_+
= -{\textstyle {1\over 3}}\,(N_c+3)\,T_c \;,
\nonumber\\
&& d^a_{\;\;bc}\,[G_a^{(i)},\,G_i^{(b)}]_+
+{\textstyle{9\over 4}}\, d_{abc}\,[T^{(a)},\,T^{(b)}]_+
= {\textstyle {10\over 3}}\,[J_{i},\,G_c^{(i)}]_+ \;,
\nonumber\\
&& d_{ab}^{\;\;\; c}\,[T^{(a)},\,G^{(b)}_i]_+ = {\textstyle{1\over 3}}\,[J_{i},\,T^{(c)}]_+
-{\textstyle{1\over 3}}\,\epsilon_{ijk}\,f_{ab}^{\;\;\;c}\,[G^{(j)}_a\,,G^{(k)}_b]_+ \;.
\label{operator-ex}\end{aligned}$$ For instance the first identity in (\[operator-ex\]) shows how to avoid the infinite tower $(T_a\,T^{(a)})^n\,T^{(c)}$ discussed above. Note that the ’parameter’ $N_c$ enters in (\[operator-ex\]) as a mean to classify the possible realizations of the algebra (\[comm\]).
As a first and simple example we recall the large $N_c$ structure of the 3-point vertices. One readily establishes two operators with parameters $g$ and $h$ to leading order in the $1/N_c$ expansion [@Dashen]: $$\begin{aligned}
\langle {\mathcal B}' |\, {\mathcal L}^{(3)}\,| {\mathcal B} \rangle =\frac{1}{f}\,
\langle {\mathcal B}' |\, g\,G_i^{(c)}+h\,J_i\,T^{(c)}| {\mathcal B} \rangle \,
\tr \,\lambda_c\,\nabla^{(i)}\,\Phi + {\mathcal O}\left( \frac{1}{N_c}\right) \;.
\label{3-point-vertex}\end{aligned}$$ Further possible terms in (\[3-point-vertex\]) are either redundant or suppressed in the $1/N_c$ expansion. For example, the two-body operator $ i\,f_{abc}\,G_i^{(a)} \,T^{(b)} \sim N_c^0$ is reduced by applying the relation $$\begin{aligned}
&& i\,f_{ab}^{\;\;\;c}\,\Big[G_i^{(a)} \,,T^{(b)} \Big]_- = i\,f_{ab}^{\;\;\;c}\,i\,f^{ab}_{\;\;\;\,d}\,G_i^{(d)} =
-3\,G_i^{(c)} \;. \nonumber
\label{}\end{aligned}$$ In order to make use of the large $N_c$ result, it is necessary to evaluate the matrix elements in (\[3-point-vertex\]) at $N_c=3$ where one has a $\bf 56$-plet with $| {\mathcal B} \rangle= |B(a) ,\Delta(ijk) \rangle $. Most economically this is achieved with the completeness identity $1=|B\rangle \langle B|+ |\Delta\rangle \langle \Delta | $ in conjunction with $$\begin{aligned}
&&T_c\,| B_a(\chi)\rangle = i\,f_{abc}\,| B^{(b)}(\chi)\rangle \;,\qquad
J^{(i)} \,| B_a(\chi )\rangle
= \frac{1}{2}\,\sigma^{(i)}_{\chi' \chi}| B_a(\chi')\rangle \;,
\nonumber\\
&& G^{(i)}_c\,| B_a(\chi)\rangle =
\left(\frac{1}{2}\,d_{abc}
+ \frac{1}{3}\,i\,f_{abc}\right)\,\sigma^{(i)}_{\chi'\chi}\,| B^{(b)}(\chi')\rangle
\nonumber\\
&& \qquad \qquad \quad \;
+ \frac{1}{\sqrt{2}\,2}\,\Big(\epsilon_{l}^{\;jk}\,\lambda^{(c)}_{mj}
\,\lambda_{nk}^{(a)}\Big)\,S^{(i)}_{\chi' \chi}
| \Delta^{\!(lmn)}(\chi') \rangle \;,
\label{matrix-el}\end{aligned}$$ where $S_i\,S^\dagger_j=\delta_{ij}-\sigma_i\,\sigma_j/3$ and $\lambda_a\,\lambda_b = {\textstyle{2 \over 3}}\,\delta_{ab}
+(i\,f_{abc}+d_{abc})\,\lambda^{(c)}$. In (\[matrix-el\]) the baryon octet states $| B_b(\chi)\rangle $ are labelled according to their $SU(3)$ octet index $a=1,...,8$ with the two spin states represented by $\chi=1,2$. Similarly the decuplet states $| \Delta_{lmn}(\chi') \rangle$ are listed with $l,m,n=1,2,3$ as defined in (\[dec-field\]). Note that the expressions (\[matrix-el\]) may be verified using the quark-model wave functions for the baryon octet and decuplet states. The result (\[matrix-el\]) is however much more general than the quark-model, because it follows from the structure of the ground-state baryons in the large $N_c$ limit of QCD only. Matching the interaction vertices of the relativistic chiral Lagrangian onto the static matrix elements arising in the large $N_c$ operator analysis requires a non-relativistic reduction. It is standard to decompose the 4-component Dirac fields $B$ and $\Delta_\mu $ into baryon octet and decuplet spinor fields $B(\chi)$ and $\Delta (\chi)$: $$\begin{aligned}
\Big(B, \Delta_\mu \Big) \to \left(
\begin{array}{c}
\left(\frac{1}{2}+\frac{1}{2}\,\sqrt{1+\frac{\nabla^2}{M^2}}
\right)^{\frac{1}{2}} \Big(B(\chi ),S_\mu\,\Delta (\chi )\Big) \\
\frac{(\sigma \cdot \nabla )}{\sqrt{2}\,M}
\left(1+\sqrt{1+\frac{\nabla^2}{M^2} }\,\right)^{-\frac{1}{2}}
\Big(B(\chi ),S_\mu\,\Delta (\chi )\Big)
\end{array}
\right) \,,
\label{}\end{aligned}$$ where $M$ denotes the baryon octet and decuplet mass in the large $N_c$ limit. To leading order one finds $S_\mu =(0, S_i) $ with the transition matrices $S_i$ introduced in (\[matrix-el\]). It is then straightforward to expand in powers of $\nabla/M$ and achieve the desired matching. This leads for example to the identification $D_{[8]}=g $, $F_{[8]}= 2\,g/3+h$ and $C_{[10]}=2\,g$. The empirical values of $F_{[8]},D_{[8]}$ and $C_{[10]}$ are quite consistent with those large $N_c$ sum rules [@Jenkins]. Note that operators at subleading order in (\[3-point-vertex\]) then parameterize the deviation from $C_{[10]}\simeq 2 \,D_{[8]}$.
Quasi-local interaction terms
-----------------------------
We turn to the two-body interaction terms at chiral order $Q^2$. From phase space consideration it is evident that to this order there are only terms which contribute to the meson-baryon s-wave scattering lengths, the s-wave effective range parameters and the p-wave scattering volumes. Higher partial waves are not affected to this order. The various contributions are regrouped according to their scalar, vector or tensor nature as $$\begin{aligned}
{\mathcal L}^{(4)}_2= {\mathcal L}^{(S)}+{\mathcal L}^{(V)}+{\mathcal L}^{(T)}
%+{\mathcal L}^{(R)}
\,,
\label{l42}\end{aligned}$$ where the lower index k in ${\mathcal L}^{(n)}_k$ denotes the minimal chiral order of the interaction vertex. In the relativistic framework one observes mixing of the partial waves in the sense that for instance ${\mathcal L}^{(S)}, {\mathcal L}^{(V)}$ contribute to the s-wave channels and ${\mathcal L}^{(S)}, {\mathcal L}^{(T)}$ to the p-wave channels. We write $$\begin{aligned}
{\mathcal L}^{(S)}&=&\frac{g^{(S)}_0}{8\!\,f^2}\,\tr\bar B\,B
\,\tr (\partial_\mu\Phi) \, (\partial^\mu\Phi)
+\frac{g^{(S)}_1}{8\!\,f^2}\,\tr \bar B \,(\partial_\mu\Phi)
\,\tr(\partial^\mu\Phi) \, B
\nonumber\\
&+&\frac{g^{(S)}_F}{16\!\,f^2} \,\tr\bar B \Big[
\Big[(\partial_\mu\Phi),(\partial^\mu\Phi)
\Big]_+ ,B\Big]_-
\!+\frac{g^{(S)}_D}{16\!\,f^2}\,\tr \bar B \Big[
\Big[(\partial_\mu\Phi),(\partial^\mu\Phi)
\Big]_+, B \Big]_+ \;,
\nonumber\\
{\mathcal L}^{(V)}&=&
\frac{g^{(V)}_0}{16\!\, f^2}\,
\Big(\tr \bar B \,i\,\gamma^\mu\,( \partial^\nu B) \,
\tr(\partial_\nu\Phi) \, ( \partial_\mu\Phi)
+\mathrm{h.c.}\Big)
\nonumber\\
&+&\frac{g^{(V)}_1}{32\!\,f^2}\,
\tr \bar B \,i\,\gamma^\mu\,\Big( ( \partial_\mu\Phi)
\,\tr(\partial_\nu\Phi) \, ( \partial^\nu B)
+ ( \partial_\nu\Phi) \,
\tr ( \partial_\mu \Phi) \, ( \partial^\nu B)
+\mathrm{h.c.}\Big)
\nonumber\\
&+&\frac{g_F^{(V)}}{32\!\,f^2}\,\Big(
\tr \bar B \,i\,\gamma^\mu\,\Big[
\Big[(\partial_\mu \Phi) , (\partial_\nu\Phi)\Big]_+,
( \partial^\nu B) \Big]_-
+\mathrm{h.c.} \Big)
\nonumber\\
&+& \frac{g^{(V)}_D}{32\!\,f^2}\,\Big(\tr
\bar B \,i\,\gamma^\mu\,\Big[
\Big[( \partial_\mu\Phi) , (\partial_\nu\Phi)\Big]_+,
( \partial^\nu B) \Big]_+
+\mathrm{h.c.} \Big)\,,
\nonumber\\
{\mathcal L}^{(T)}&=&
\frac{g^{(T)}_1}{8\!\,f^2}\,\tr\bar B \,( \partial_\mu\Phi)
\,i\,\sigma^{\mu \nu}\,\tr( \partial_\nu \Phi) \, B
\nonumber\\
&+& \frac{g^{(T)}_D}{16\!\,f^2}\,
\tr \bar B \,i\,\sigma^{\mu \nu}\,\Big[
\Big[(\partial_\mu \Phi) ,( \partial_\nu \Phi) \Big]_-, B \Big]_+
\nonumber\\
&+&\frac{g^{(T)}_F}{16\!\,f^2}\,
\tr \bar B \,i\,\sigma^{\mu \nu}\,\Big[
\Big[( \partial_\mu \Phi) ,( \partial_\nu\Phi) \Big]_- ,B\Big]_-
\,.
\label{two-body}\end{aligned}$$ It is clear that if the heavy-baryon expansion is applied to (\[two-body\]) the quasi-local 4-point interactions can be mapped onto corresponding terms of the heavy-baryon formalism presented for example in [@CH-Lee]. Inherent in the relativistic scheme is the presence of redundant interaction terms which requires that a systematic regrouping of the interaction terms is performed. This will be discussed below in more detail when introducing the quasi-local counter terms at chiral order $Q^3$.
We apply the large $N_c$ counting rules in order to estimate the relative importance of the quasi-local $Q^2$-terms in (\[two-body\]). Terms which involve a single-flavor trace are enhanced as compared to the double-flavor trace terms. This is because a flavor trace in an interaction term is necessarily accompanied by a corresponding color trace if visualized in terms of quark and gluon lines. A color trace signals a quark loop and therefore provides the announced $1/N_c$ suppression factor [@Hooft; @Witten]. The counting rules are nevertheless subtle, because a certain combination of double trace expressions can be rewritten in terms of a single-flavor trace term [@Fearing] $$\begin{aligned}
&&\tr \left( \bar B \, B \right)
\,\tr \Big(\Phi \, \Phi \Big)
+2\,\tr \left( \bar B \, \Phi \right)
\,\tr \Big(\Phi \, B \Big)
\nonumber\\
=&&\tr \Big[\bar B, \Phi \Big]_-\,\Big[B, \Phi
\Big]_-
+\frac{3}{2}\,\tr \bar B \Big[
\Big[\Phi , \Phi \Big]_+, B \Big]_+ \;.
\label{trace-id}\end{aligned}$$ Thus one expects for example that both parameters $g_{0}^{(S)}$ and $g_{1}^{(S)}$ may be large separately but the combination $2\,g_0^{(S)}-g_1^{(S)}$ should be small. A more detailed operator analysis leads to $$\begin{aligned}
&&\langle {\mathcal B}' | {\mathcal L}^{(4)}_2 | {\mathcal B} \rangle = \frac{1}{16\,f^2}\,
\langle {\mathcal B}' | \,O_{ab}(g_1,g_2)\, | {\mathcal B} \rangle
\,\tr [(\partial_\mu \Phi),\lambda^{(a)}]_-\,[(\partial^\mu \Phi),\lambda^{(b)}]_-
\nonumber\\
&& \qquad \qquad +\frac{1}{16\,f^2}\,
\langle {\mathcal B}' | \,O_{ab}(g_3,g_4)\, | {\mathcal B} \rangle
\,\tr [(\partial_0 \Phi),\lambda^{(a)}]_-\,[(\partial_0 \Phi),\lambda^{(b)}]_-
\nonumber\\
&& \qquad \qquad + \frac{1}{16\,f^2}\,
\langle {\mathcal B}' | \,O^{(ij)}_{ab}(g_5,g_6)\, | {\mathcal B} \rangle
\,\tr [(\nabla_i \Phi),\lambda^{(a)}]_-\,[(\nabla_j \Phi),\lambda^{(b)}]_- \;,
\nonumber\\ \nonumber\\
&& O_{ab}(g,h) = g\,d_{abc}\,T^{(c)} + h\,[T_a,T_b]_+
+{\mathcal O}\left(\frac{1}{N_c} \right) \;,
\nonumber\\
&& O_{ab}^{(ij)}(g,h) = i\,\epsilon^{ijk }\,i\,f_{abc}\left(
g\,G_k^{(c)} + h\,J_{k}\,T^{(c)} \right)
+{\mathcal O}\left(\frac{1}{N_c} \right) \;.
\label{Q^2-large-Nc}\end{aligned}$$ We checked that other forms for the coupling of the operators $O_{ab}$ to the meson fields do not lead to new structures. It is straight forward to match the coupling constants $g_{1,..,6}$ onto those of (\[two-body\]). Identifying the leading terms in the non-relativistic expansion, we obtain: $$\begin{aligned}
&& g_0^{(S)}= \frac{1}{2}\,g_1^{(S)} = \frac{2}{3}\,g_D^{(S)}
= -2\,g_2\,, \qquad \; g_F^{(S)}= -3\,g_1 \,,
\nonumber\\
&& g_0^{(V)}= \frac{1}{2}\,g_1^{(V)}=\frac{2}{3}\,g_D^{(V)}
=-2\,\frac{g_4}{M}\,, \qquad
g_F^{(V)}= -3\,\frac{g_3}{M} \,,
\nonumber\\
&& g_1^{(T)}= 0 \,, \qquad
g_F^{(T)}= -g_5-\frac{3}{2}\,g_6 \,, \qquad g_D^{(T)}= -\frac{3}{2}\,g_5 \;,
\label{Q^2-large-Nc-result}\end{aligned}$$ where $M$ is the large $N_c$ value of the baryon octet mass. We conclude that to chiral order $Q^2$ there are only six leading large $N_c$ coupling constants.
We turn to the quasi-local counter terms to chiral order $Q^3$. It is instructive to discuss first a set of redundant interaction terms: $$\begin{aligned}
{\mathcal L}^{(R)}&=&\frac{h^{(1)}_0}{8 f^2}\,
\tr(\partial^\mu\bar B)\,
(\partial_\nu B)\,
\tr( \partial_\mu \Phi) \, ( \partial^\nu\Phi)
\nonumber\\
&+&\frac{h^{(1)}_1}{16 f^2}\,
\tr(\partial^\mu\bar B)\, ( \partial_\nu\Phi)\,
\tr(\partial_\mu\Phi) \, (\partial^\nu B)
\nonumber\\
&+&\frac{h^{(1)}_1}{16 f^2}\,
\tr(\partial^\mu\bar B)\,( \partial_\mu \,\Phi)\,
\tr(\partial^\nu\Phi) \, (\partial_\nu B) \,,
\nonumber\\
&+&\frac{h^{(1)}_F}{16 f^2}\,
\tr (\partial^\mu\bar B)\, \Big[
\Big[( \partial_\mu\Phi) , (\partial^\nu \Phi) \Big]_+ ,
(\partial_\nu B) \Big]_-
\nonumber\\
&+& \frac{h^{(1)}_D}{16 f^2}\,
\tr (\partial^\mu\bar B)\, \Big[
\Big[( \partial_\mu\Phi) , (\partial^\nu \Phi)\Big]_+,
(\partial_\nu B )\Big]_+ \,.
\label{redundant}\end{aligned}$$ Performing the non-relativistic expansion of (\[redundant\]) one finds that the leading moment is of chiral order $Q^2$. Formally the terms in (\[redundant\]) are transformed into terms of subleading order $Q^3$ by subtracting ${\mathcal L}^{(V)}$ of (\[two-body\]) with $g^{(V)} = \m0_{[8]} \,h^{(1)}$. Bearing this in mind the terms (\[redundant\]) define particular interaction vertices of chiral order $Q^3$. Note that in analogy to (\[Q\^2-large-Nc\]) and (\[Q\^2-large-Nc-result\]) we expect the coupling constants $h_F^{(1)}$ and $h_D^{(1)}$ with $h_1^{(1)}=2\,h_0^{(1)}= 4\,h_D^{(1)}/3$ to be leading in the large $N_c$ limit. A complete collection of counter terms of chiral order $Q^3$ is presented in Appendix B. Including the four terms in (\[redundant\]) we find ten independent interaction terms which all contribute exclusively to the s- and p-wave channels. Here we present the two additional terms with $h^{(2)}_{F}$ and $h^{(3)}_{F}$ which are leading in the large $N_c$ expansion: $$\begin{aligned}
{\mathcal L}^{(4)}_3 &=& {\mathcal L}^{(R)}- {\mathcal L}^{(V)} [g^{(V)}\! = \m0_{[8]}\,h^{(1)}]
\nonumber\\
&+& \frac{h^{(2)}_{F}}{32\!\,f^2}\,
\tr \bar B \,i\,\gamma^\mu\,\Big[ \big[ ( \partial_\mu\Phi)
, (\partial_\nu \Phi)\big]_- , ( \partial^\nu B) \Big]_- +{\rm h.c}
\nonumber\\
&+&\frac{h^{(3)}_{F}}{16\!\,f^2}\,
\tr \bar B \,i\,\gamma^\alpha\,\Big[ \big[ ( \partial_\alpha \,\partial_\mu\Phi)
,( \partial^\mu \Phi) \big]_- , B \Big]_- \;.
\label{local-q-3}\end{aligned}$$ The interaction vertices in (\[local-q-3\]) can be mapped onto corresponding static matrix elements of the large $N_c$ operator analysis: $$\begin{aligned}
&&\langle {\mathcal B}' | {\mathcal L}^{(4)}_3 | {\mathcal B} \rangle = \frac{h_2}{16\,f^2}\,
\langle {\mathcal B}' | \,i\,f_{abc}\,T^{(c)}\, | {\mathcal B} \rangle
\,\partial_\mu \,\Big(
\tr [(\partial_0 \Phi),\lambda^{(a)}]_-\,[(\partial^\mu \Phi),\lambda^{(b)}]_- \Big)
\nonumber\\
&& \qquad \qquad \;\; +\frac{h_3}{16\,f^2}\,
\langle {\mathcal B}' | \,i\,f_{abc}\,T^{(c)}\, | {\mathcal B} \rangle
\, \tr [(\partial_0 \,\partial_\mu \Phi),\lambda^{(a)}]_-\,[(\partial^\mu \Phi),\lambda^{(b)}]_- \,,
\label{}\end{aligned}$$ where $h^{(2)}_{F} \sim h_2$ and $h_{F}^{(3)} \sim h_3$. We summarize our result for the quasi-local chiral interaction vertices of order $Q^3$: at leading order the $1/N_c$ expansion leads to four relevant parameters only. Also one should stress that the $SU(3)$ structure of the $Q^3$ terms as they contribute to the s- and p-wave channels differ from the $SU(3)$ structure of the $Q^2$ terms. For instance the $g^{(S)}$ coupling constants contribute to the p-wave channels with four independent $SU(3)$ tensors. In contrast, at order $Q^3$ the parameters $h^{(2)}_{F}$ and $h^{(3)}_{F}$, which are in fact the only parameters contribution to the p-wave channels to this order, contribute with a different and independent $SU(3)$ tensor. This is to be compared with the static $SU(3)$ prediction that leads to six independent tensors: $$8\otimes 8= 1
\oplus 8_S\oplus 8_A \oplus 10\oplus \overline{10}\oplus 27 \;.
\label{}$$ Part of the predictive power of the chiral Lagrangian results, because chiral $SU(3)$ symmetry selects certain subsets of all $SU(3)$ symmetric tensors at a given chiral order.
Explicit chiral symmetry breaking
---------------------------------
There remain the interaction terms proportional to $\chi_\pm $ which break the chiral $SU(3)$ symmetry explicitly. We collect here all relevant terms of chiral order $Q^2$ [@Gasser; @Kaiser] and $Q^3$ [@Mueller]. It is convenient to visualize the symmetry-breaking fields $\chi_\pm$ of (\[def-chi\]) in their expanded forms: $$\begin{aligned}
\! \!\! \chi_+ = \chi_0 -\frac{1}{8\,f^2}
\Big[ \Phi, \Big[ \Phi ,\chi_0 \Big]_+\Big]_+ \!+{\mathcal O} \left(\Phi^4 \right) \,,
\;\,
\chi_- = \frac{i}{2\,f}\, \Big[ \Phi ,\chi_0 \Big]_+
\!+{\mathcal O} \left(\Phi^3 \right) \,.
\label{chi-exp}\end{aligned}$$ We begin with the 2-point interaction vertices which all follow exclusively from chiral interaction terms linear in $\chi_+$. They read $$\begin{aligned}
{\mathcal L }_{\chi}^{(2)}&=& -\frac{1}{4}\,\tr \Phi\,\Big[ \chi_0, \Phi \Big]_+
+2\,\tr \bar B \left( b_D\,\Big[ \chi_0 , B \Big]_+ +b_F\,\Big[ \chi_0 , B \Big]_-
+b_0 \, B \,\tr \chi_0 \right)
\nonumber\\
&+&2\,d_D\,\tr \Big(\bar \Delta_\mu\cdot \Delta^\mu \Big) \,\chi_0
+2\,d_0 \,\tr \left(\bar \Delta_\mu\cdot \Delta^\mu\right) \,\tr \chi_0
\nonumber\\
&+&\tr \bar B \left(i\,\partialslash-\m0_{[8]}\right) \,
\Big( \zeta_0\,B \, \tr \chi_0
+ \zeta_D\,[B, \chi_0 ]_+ + \zeta_F\, [B, \chi_0 ]_- \Big) \;,
\nonumber\\ \nonumber\\
\chi_0 &=&\frac{1}{3} \left( m_\pi^2+2\,m_K^2 \right)\,1
+\frac{2}{\sqrt{3}}\,\left(m_\pi^2-m_K^2\right) \lambda_8 \;,
\label{chi-sb}\end{aligned}$$ where we normalized $\chi_0$ to give the pseudo-scalar mesons their isospin averaged masses. The first term in (\[chi-sb\]) leads to the finite masses of the pseudo-scalar mesons. Note that to chiral order $Q^2$ one has $m_\eta^2 = 4\,(m_K^2-m_\pi^2)/3$. The parameters $b_D$, $b_F$, and $d_D$ are determined to leading order by the baryon octet and decuplet mass splitting $$\begin{aligned}
&&m_{[8]}^{(\Sigma )}-m_{[8]}^{(\Lambda )}=
{\textstyle{16\over 3}}\,b_D\,(m_K^2-m_\pi^2)\,, \quad
m_{[8]}^{(\Xi )}-m_{[10]}^{(N )} =-8\,b_F\,(m_K^2-m_\pi^2)\,,
\nonumber \\
&&m_{[10]}^{(\Sigma )}-m_{[10]}^{(\Delta )}=m_{[8]}^{(\Xi )}-m_{[10]}^{(\Sigma )}
=m_{[10]}^{(\Omega )}-m_{[10]}^{(\Xi )}
=-\textstyle{4\over 3}\, d_D\, (m_K^2-m_\pi^2)\,.
\label{mass-splitting}\end{aligned}$$ The empirical baryon masses lead to the estimates $b_D \simeq 0.06$ GeV$^{-1}$, $b_F \simeq -0.21$ GeV$^{-1}$, and $d_D\simeq -0.49$ GeV$^{-1}$. For completeness we recall the leading large $N_c$ operators for the baryon mass splitting (see e.g. [@Jenkins]): $$\begin{aligned}
&&\langle {\mathcal B}' |{\mathcal L }_{\chi}^{(2)} | {\mathcal B} \rangle
=\langle {\mathcal B}' |\,
b_1\,T^{(8)} + b_2\,[J^{(i)}, G^{(8)}_i]_+ \,| {\mathcal B} \rangle
+{\mathcal O}\left(\frac{1}{N_c^{2}}\right)\,,
\nonumber\\
&&b_D = -\frac{\sqrt{3}}{16}\,\frac{3\,b_2}{m_K^2-m_\pi^2} \,, \qquad
b_F = -\frac{\sqrt{3}}{16}\,\frac{2\,b_1+b_2}{m_K^2-m_\pi^2}\,,
\nonumber\\
&& d_D = -\frac{3\,\sqrt{3}}{8}\,\frac{b_1+2\,b_2}{m_K^2-m_\pi^2} \,,
\label{}\end{aligned}$$ where we matched the symmetry-breaking parts with $\lambda_8$. One observes that the empirical values for $b_D+b_F$ and $d_D$ are remarkably consistent with the large $N_c$ sum rule $b_D+b_F\simeq {\textstyle{1\over 3}}\,d_D$. The parameters $b_0$ and $d_0$ are more difficult to access. They determine the deviation of the octet and decuplet baryon masses from their chiral $SU(3)$ limit values $\m0_{[8]}$ and $\m0_{[10]}$: $$\begin{aligned}
&& m_{[8]}^{(N)} = \m0_{[8]}-2\,m_\pi^2\,(b_0+2\,b_F)-4\,m_K^2\,(b_0+b_D-b_F) \;,
\nonumber\\
&& m_{[10]}^{(\Delta )} = \m0_{[10]}-2\,m_\pi^2 (d_0 +d_D)-4\,m_K^2 \,d_0 \;,
\label{piN-sig-term}\end{aligned}$$ where terms of chiral order $Q^3$ are neglected. The size of the parameter $b_0$ is commonly encoded into the pion-nucleon sigma term $$\sigma_{\pi N}= -2\,m_\pi^2\,(b_D+b_F+2\,b_0) +{\mathcal O}\left(Q^3\right)\,.
\label{spin:naive}$$ Note that the former standard value $\sigma_{\pi N}=(45\pm 8)$ MeV of [@piN-sigterm] is currently under debate [@pin-news].
The parameters $\zeta_0,\zeta_D$ and $\zeta_F$ are required to cancel a divergent term in the baryon wave-function renormalization as it follows from the one loop self-energy correction or equivalently the unitarization of the s-channel baryon exchange term. It will be demonstrated explicitly that within our approximation they will not have any observable effect. They lead to a renormalization of the three-point vertices only, which can be accounted for by a redefinition of the parameters in (\[chi-sb-3\]). Thus one may simply drop these interaction terms.
The predictive power of the chiral Lagrangian lies in part in the strong correlation of vertices of different degrees as implied by the non-linear fields $U_\mu $ and $\chi_\pm $. A powerful example is given by the two-point vertices introduced in (\[chi-sb\]). Since they result from chiral interaction terms linear in the $\chi_+$-field (see (\[chi-exp\])), they induce particular meson-octet baryon-octet interaction vertices: $$\begin{aligned}
{\mathcal L }_{\chi}^{(4)}&=&
\frac{i}{16\,f^2}\,\tr\bar B\,\gamma^\mu \Big[\Big[ \Phi ,
(\partial_\mu \,\Phi) \Big]_-,\zeta_0\,B \, \tr \chi_0
+ \zeta_D\,[B, \chi_0 ]_+ + \zeta_F\, [B, \chi_0 ]_- \Big]_-
\nonumber\\
&+&\frac{i}{16\,f^2}\,\tr \Big[\zeta_0\,\bar B \, \tr \chi_0
+ \zeta_D\,[\bar B, \chi_0 ]_+ + \zeta_F\, [\bar B, \chi_0 ]_- ,
\Big[ \Phi ,(\partial_\mu \,\Phi) \Big]_- \Big]_- \,\gamma^\mu\,B
\nonumber\\
&-&\frac{1}{4\,f^2}\,\tr \bar B \left(b_D\,
\Big[ \Big[ \Phi, \Big[ \Phi ,\chi_0 \Big]_+\Big]_+ , B \Big]_+
+b_F\,\Big[ \Big[ \Phi, \Big[ \Phi ,\chi_0 \Big]_+\Big]_+ , B \Big]_-
\right)
\nonumber\\
&-&\frac{b_0}{4\,f^2}\,\tr\bar{B}\, B \,\tr \Big[ \Phi, \Big[ \Phi ,\chi_0 \Big]_+\Big]_+
\;.
\label{chi-sb-4}\end{aligned}$$ To chiral order $Q^3$ there are no further four-point interaction terms with explicit chiral symmetry breaking.
We turn to the three-point vertices with explicit chiral symmetry breaking. Here the chiral Lagrangian permits two types of interaction terms written as ${\mathcal L }_{\chi }^{(3)}={\mathcal L }_{\chi,\, +}^{(3)}+{\mathcal L }_{\chi, -}^{(3)}$. In ${\mathcal L }_{\chi, +}^{(3)}$ we collect 16 axial-vector terms, which result form chiral interaction terms linear in the $\chi_+$ field (see (\[chi-exp\])), with a priori unknown coupling constants $F_{0,..,9}$ and $C_{0,...,5}$, $$\begin{aligned}
{\mathcal L }_{\chi, +}^{(3)}&=&
\frac{1}{4\,f} \,\tr \bar B
\,\gamma_5\,\gamma^\mu \,\Big(
\Big[\left(\partial_\mu\,\Phi\right),F^{}_0\,\Big[\chi_0, B \Big]_+
+F^{}_1\,\Big[\chi_0, B \Big]_-\Big]_+ \Big)+{\rm h.c.}
\nonumber\\
&+&\frac{1}{4\,f} \,\tr \bar B
\,\gamma_5\,\gamma^\mu \,\Big(
\Big[\left(\partial_\mu\,\Phi\right),F^{}_2\,\Big[\chi_0, B \Big]_+
+F^{}_3\,\Big[\chi_0, B \Big]_-\Big]_- \Big)+{\rm h.c.}
\nonumber\\
&+&\frac{1}{2\,f} \,\tr \bar B
\,\gamma_5\,\gamma^\mu \,\Big( F_4\,\Big[
\Big[\chi_0, \left(\partial_\mu\,\Phi\right)\Big]_+ , B \Big]_+
+F^{}_5\,\Big[\Big[\chi_0, \left(\partial_\mu\,\Phi\right)\Big]_+ , B \Big]_- \Big)
\nonumber\\
&+&\frac{1}{4\,f} \,\tr \bar B
\,\gamma_5\,\gamma^\mu \,\Big(
F^{}_{6}\,B \,\tr \Big( \chi_0 \left(\partial_\mu\,\Phi\right) \Big)
+F^{}_{7} \left(\partial_\mu\,\Phi\right)
\tr \Big(\chi_0 \, B \Big) \Big) +{\rm h.c.}
\nonumber\\
&+& \frac{1}{2\,f} \,\tr \bar B
\,\gamma_5\,\gamma^\mu \,\Big(
F_8\,\Big[\left(\partial_\mu\,\Phi\right),B\Big]_+ \,\tr \chi_0
+F_9 \,\Big[\left(\partial_\mu\,\Phi\right),B\Big]_-\,\tr \chi_0 \Big)
\nonumber\\
&-& \frac{1}{2\,f}\,
\tr \left\{C_0 \,\Big( \bar \Delta_\mu \cdot
\Big[ \chi_0 ,(\partial_\mu \,\Phi )\Big]_+
+C_1 \,\Big( \bar \Delta_\mu \cdot
\Big[ \chi_0 ,(\partial_\mu \,\Phi )\Big]_- \Big)\,B +\rm{h.c.}\right\}
\nonumber\\
&-&\frac{1}{2\,f}\,
\tr \left\{
\Big( \bar \Delta_\mu \cdot (\partial_\mu \,\Phi ) \Big)\,
\Big( C_2\,\Big[\chi_0,B\Big]_++C_3\,\Big[\chi_0,B\Big]_- +\rm{h.c.}\right\}
\nonumber\\
&-&\frac{C_4}{2\,f}\,
\tr \left\{ \Big( \bar \Delta_\mu \cdot \chi_0
\Big)\,\big[(\partial_\mu \,\Phi ),B \big]_- +\rm{h.c.}\right\}
\nonumber\\
&-&\frac{C_5}{2\,f}\,
\tr \left\{ \Big( \bar \Delta_\mu \cdot
(\partial_\mu \,\Phi )\Big)\,
B +\mathrm{h.c.}\right\} \tr \, \chi_0 \;.
\label{chi-sb-3}\end{aligned}$$ Similarly in ${\mathcal L }_{\chi, -}^{(3)}$ we collect the remaining terms which result from chiral interaction terms linear in $\chi_-$. There are three pseudo-scalar interaction terms with $\bar F_{4,5,6}$ and four additional terms parameterized by $\delta F_{4,5,6}$ and $\delta C_0$ $$\begin{aligned}
{\mathcal L }_{\chi, -}^{(3)}&=& \frac{1}{2\,f} \,\tr \bar B
\Big[
2\,i\,\gamma_5\,\m0_{[8]}\,
\bar F_4\,\Big[\chi_0,\Phi\Big]_+
+\gamma_5\,\gamma^\mu \, \delta F_4\,
\Big[\chi_0, \left(\partial_\mu\,\Phi\right)\Big]_+
, B \Big]_+
\nonumber\\
&+&\frac{1}{2\,f} \,\tr \bar B
\Big[
2\,i\,\gamma_5\,\m0_{[8]}\,
\bar F_5\,\Big[\chi_0,\Phi\Big]_+
+\gamma_5\,\gamma^\mu \, \delta F_5\,
\Big[\chi_0, \left(\partial_\mu\,\Phi\right)\Big]_+
, B \Big]_-
\nonumber\\
&+&\frac{1}{2\,f} \,\tr \bar B\,
\Big(
2\,i\,\gamma_5\,\m0_{[8]}\,
\bar F_6\,\tr \big(\chi_0\,\Phi\big)
+\gamma_5\,\gamma^\mu \, \delta F_6\,\tr
\big( \chi_0\, \left(\partial_\mu \,\Phi\right)\big) \Big)\, B
\nonumber\\
&-& \frac{\delta C_0}{2\,f}\,
\tr \,\Big( \bar \Delta_\mu \cdot
\Big[ \chi_0 ,(\partial_\mu \,\Phi )\Big]_+ \,B +\rm{h.c.} \Big) \,.
\label{chi-sb-3:p}\end{aligned}$$ We point out that, while the parameters $F_i$ and $C_i$ contribute to matrix elements of the $SU(3)$ axial-vector current $A^\mu_a$, none of the terms of (\[chi-sb-3:p\]) contribute. This follows once the external axial-vector current is restored. In (\[chi-sb-3\]) this is achieved with the replacement $\partial^\mu\,\Phi_a \to \partial^\mu\,\Phi_a +2\,f\, A^\mu_a$ (see (\[def-fields\])). Though it is obvious that the pseudo-scalar terms in (\[chi-sb-3:p\]) proportional to $\bar F_i$ do not contribute to the axial-vector current, it is less immediate that the terms proportional to $\delta F_i$ and $\delta C_i$ also do not contribute. Moreover, the latter terms appear redundant, because terms with identical structure at the 3-point level are already listed in (\[chi-sb-3:p\]). Here one needs to realize that the terms proportional to $\delta F_i$ and $\delta C_i$ result from chiral interaction terms linear in $[{\mathcal D}_\mu ,\chi_-]_- $ while the terms proportional to $F_i, C_i$ result from chiral interaction terms linear in $U_\mu $.
The pseudo-scalar parameters $\bar F_i$ and also $\delta F_{i}$ lead to a tree-level Goldberger-Treiman discrepancy. For instance, we have $$\begin{aligned}
f \,g_{\pi N N} -g_A \,m_N = 2\,m_N\,m_\pi^2 \,
(\bar F_4+\delta F_4+\bar F_5+\delta F_5 ) + {\mathcal O} \left( Q^3\right) \,,
\label{GTD}\end{aligned}$$ where we introduced the pion-nucleon coupling constant $g_{\pi N N}$ and the axial-vector coupling constant of the nucleon $g_A$. The corresponding generalized Goldberger-Treiman discrepancies for the remaining axial-vector coupling constants of the baryon octet states follow easily from the replacement rule $F_i \to F_i+\bar F_i+\delta F_i $ for $i=4,5,6$ (see also [@Goity:Lewis]). We emphasize that (\[GTD\]) must not be confronted directly with the Goldberger-Treiman discrepancy as discussed in [@GTD; @GTD:pi; @Goity:Lewis], because it necessarily involves the $SU(3)$ parameter $f $ rather than $f_\pi \simeq 92$ MeV or $f_K \simeq 113$ MeV.
The effect of the axial-vector interaction terms in (\[chi-sb-3\]) is twofold. First, they lead to renormalized values of the $F_{[8]}, D_{[8]}$ and $C_{[10]}$ parameters in (\[lag-Q\]). Secondly they induce interesting $SU(3)$ symmetry-breaking effects which are proportional to $(m_K^2-m_\pi^2)\,\lambda_8$. Note that the renormalization of the $F_{[8]},D_{[8]}$ and $C_{[10]}$ parameters requires care, because it is necessary to discriminate between the renormalization of the axial-vector current and the one of the meson-baryon coupling constants. We introduce $F_R, D_R$ and $C_R$ as they enter matrix elements of the axial-vector current : $$\begin{aligned}
&& F_R =F_{[8]}+ (m_\pi^2+2\,m_K^2)\left(F_9
+{\textstyle{2\over 3}}\,\big( F_2+F_5\big) \right) \;, \qquad
\nonumber\\
&& D_R= D_{[8]}+ (m_\pi^2+2\,m_K^2)
\left(F_8 +{\textstyle{2\over 3}}\,\big( F_0+F_4\big) \right)\;,
\nonumber\\
&& C_R=C_{[10]} +(m_\pi^2+2\,m_K^2)\,\left(C_5 +{\textstyle{2\over 3}}\,\big( C_0+C_2\big)\right)\,,
\label{ren-FDC}\end{aligned}$$ and the renormalized parameters $F_{A,P}, D_{A,P}$ and $ C_A$ as they are relevant for the meson-baryon 3-point vertices: $$\begin{aligned}
&& F_A= F_R + {\textstyle{2\over 3}}\,(m_\pi^2 +2\,m_K^2)\,\delta F_5\;, \quad
D_A= D_R + {\textstyle{2\over 3}}\,(m_\pi^2 +2\,m_K^2)\,\delta F_4\;, \quad
\nonumber\\
&& F_P= {\textstyle{2\over 3}}\,(m_\pi^2 +2\,m_K^2)\,\bar F_5\;, \quad
D_P= {\textstyle{2\over 3}}\,(m_\pi^2 +2\,m_K^2)\,\bar F_4\;, \quad
\nonumber\\
&& C_A= C_R + {\textstyle{2\over 3}}\,(m_\pi^2 +2\,m_K^2)\,\delta C_0\;.
\label{ren-FDC-had}\end{aligned}$$ The index $A$ or $P$ indicates whether the meson couples to the baryon via an axial-vector vertex (A) or a pseudo-scalar vertex (P). It is clear that the effects of (\[ren-FDC\]) and (\[ren-FDC-had\]) break the chiral symmetry but do not break the $SU(3)$ symmetry. In this work we will use the renormalized parameters $F_R, D_R$ and $C_R$. One can always choose the parameters $F_8,F_9$ and $C_5$ as to obtain $F_R=F_{[8]}$, $D_R=D_{[8]}$ and $C_R=C_{[10]}$. In order to distinguish the renormalized values from their bare values one needs to determine the parameters $F_8, F_9$ and $C_5$ by investigating higher-point Green functions. This is beyond the scope of this work.
The number of parameters inducing $SU(3)$ symmetry-breaking effects can be reduced significantly by the large $N_c$ analysis. We recall the five leading operators presented in [@Dashen] $$\begin{aligned}
&&\langle {\mathcal B}' | \,{\mathcal L}_\chi^{(3)}\,| {\mathcal B} \rangle
=\frac{1}{f}\,\langle {\mathcal B}'| \,O_i^{(a)}(\tilde c) |{\mathcal B}\rangle \,
\tr\,\lambda_a\,\nabla^{(i)}\, \Phi
+ 4\,\langle {\mathcal B}'| \,O_i^{(a)}(c) |{\mathcal B}\rangle \,
A^{(i)}_a \;,
\nonumber\\ \nonumber\\
&& O_i^{(a)}(c) = c_1\,\big( d^{8a}_{\;\;\;b}\,G^{(b)}_{i}
+{\textstyle {1\over 3}}\,\delta^{a8} \,J_{i}\big)
+c_2\,\big( d^{8a}_{\;\;\;b}\,J_{i}\,T^{(b)}+\delta^{a8} \,J_{i}\big)
\nonumber\\
&& \qquad \qquad \quad +\;c_3\,[G^{(a)}_i, T^{(8)}]_+ +c_4 \, [T^{(a)},G^{(8)}_{i}]_+
+c_5 \, [J^2,[T^{(8)},G^{(a)}_i]_-]_- \;.
\label{ansatz-3}\end{aligned}$$ It is important to observe that the parameters $c_i$ and $\tilde c_i$ are a priori independent. They are correlated by the chiral $SU(3)$ symmetry only. With (\[matrix-el\]) it is straightforward to map the interaction terms (\[ansatz-3\]), which all break the $SU(3)$ symmetry linearly, onto the chiral vertices of (\[chi-sb-3\]) and (\[chi-sb-3:p\]). This procedure relates the parameters $c_i $ and $\tilde c_i$. One finds that the matching is possible for all operators leaving only ten independent parameters $c_{i}$, $\tilde c_{1,2}$, $\bar c_{1,2}$ and $a$, rather than the twenty-three $F_i$, $C_i$ and $\delta F_i, \delta C_i$ and $\bar F_i$ parameters in (\[chi-sb-3\],\[chi-sb-3:p\]). We derive $c_i=\tilde c_i$ for $i=3,4,5$ and $$\begin{aligned}
&& \!\!\!\!F_1 = -\frac{\sqrt{3}}{2}\,\frac{c_3}{m_K^2-m_\pi^2}\;,\quad
F_2 = -\frac{\sqrt{3}}{2}\,\frac{c_4}{m_K^2-m_\pi^2} \,,\quad
F_3 =-\frac{1}{\sqrt{3}}\,\frac{c_3+c_4}{m_K^2-m_\pi^2} \;,
\nonumber\\
&& \!\!\!\!F_4 = -\frac{\sqrt{3}}{4}\,\frac{c_1}{m_K^2-m_\pi^2} \;,\quad
F_5 = -\frac{\sqrt{3}}{4}\,\frac{{\textstyle{2\over 3}}\,c_1+c_2}{m_K^2-m_\pi^2} \;,\quad
F_6=-\frac{\sqrt{3}}{2}\,\frac{{\textstyle{2\over 3}}\,c_1+c_2}{m_K^2-m_\pi^2} \;,
\nonumber\\
&& \!\!\!\! C_0 = -\frac{\sqrt{3}}{2}\,\frac{c_1}{m_K^2-m_\pi^2} \,, \quad
C_1 = -\frac{\sqrt{3}}{2}\,\frac{c_3-c_4+3\,c_5}{m_K^2-m_\pi^2} \;,
\nonumber\\
&& \!\!\!\! C_3 =-\sqrt{3}\,\frac{c_3}{m_K^2-m_\pi^2}\,, \quad
C_4 =-\sqrt{3}\,\frac{c_4}{m_K^2-m_\pi^2}\,, \quad
F_{0,7}=C_{2,5}=0 \,,
\label{large-Nc-FDC}\end{aligned}$$ and $$\begin{aligned}
&& \bar F_4= -\frac{\sqrt{3}}{4}\,\frac{\bar c_1}{m_K^2-m_\pi^2} \;,\quad
\delta F_4 = -\frac{\sqrt{3}}{4}\,\frac{\delta c_1}{m_K^2-m_\pi^2} \;,
\nonumber\\
&& \bar F_5 = -\frac{\sqrt{3}}{4}\,\frac{{\textstyle{2\over 3}}\,\bar c_1+\bar c_2+a}{m_K^2-m_\pi^2}
\,, \quad \delta F_5 = -\frac{\sqrt{3}}{4}\,\frac{{\textstyle{2\over 3}}\,\delta c_1+\delta c_2-a}{m_K^2-m_\pi^2}
\,,
\nonumber\\
&& \bar F_6 =-\frac{\sqrt{3}}{2}\,\frac{{\textstyle{2\over 3}}\,\bar c_1+\bar c_2+a}{m_K^2-m_\pi^2} \,,
\quad \delta F_6 =-\frac{\sqrt{3}}{2}\,\frac{{\textstyle{2\over 3}}\,\delta c_1+\delta c_2-a}{m_K^2-m_\pi^2} \,,
\nonumber\\
&& \delta C_0=-\frac{\sqrt{3}}{2}\,\frac{ \tilde c_1- c_1}{m_K^2-m_\pi^2} \;, \qquad
\delta c_i = \tilde c_i -c_i -\bar c_i \;.
\label{large-Nc-FDC:delta}\end{aligned}$$ In (\[large-Nc-FDC:delta\]) the pseudo-scalar parameters $\bar F_{4,5,6}$ are expressed in terms of the more convenient dimension less parameters $\bar c_{1,2}$ and $a$. Here we insist that an expansion analogous to (\[ansatz-3\]) holds also for the pseudo-scalar vertices in (\[chi-sb-3:p\]). =1.4mm
$g_A $ (Exp.) $ F_R$ $ D_R$ $c_1$ $c_2 $ $c_3$ $c_4 $ $c_5 $
--------------------------------------------- -------------------- ----------------------- ----------------------- -------------------------- -------------------------- -------------------------- -------------------------- -------------
$ n \to p\,e^-\,\bar \nu_e $ $1.267 \pm 0.004$ 1 1 $\frac{5}{3\,\sqrt{3}}$ $\frac{1}{\sqrt{3}}$ $\frac{5}{\sqrt{3}}$ $ \frac{1}{\sqrt{3}}$ $\;\;0\;\;$
$ \Sigma^- \to \Lambda \,e^-\,\bar \nu_e $ $0.601 \pm 0.015$ 0 $\sqrt{\frac{2}{3}}$ $\frac{\sqrt{2}}{3}$ 0 0 0 0
$ \Lambda \to p \,e^-\,\bar \nu_e $ $-0.889 \pm 0.015$ $-\sqrt{\frac{3}{2}}$ $-\frac{1}{\sqrt{6}}$ $\frac{1}{2\,\sqrt{2}}$ $\frac{1}{2\,\sqrt{2}}$ $-\frac{3}{2\,\sqrt{2}}$ $\frac{1}{2\,\sqrt{2}}$ 0
$ \Sigma^- \to n \,e^-\,\bar \nu_e $ $0.342 \pm 0.015$ $-1$ 1 $-\frac{1}{6\,\sqrt{3}}$ $\frac{1}{2\,\sqrt{3}}$ $\frac{1}{2\,\sqrt{3}}$ $-\frac{\sqrt{3}}{2}$ 0
$ \Xi^- \to \Lambda \,e^-\,\bar \nu_e $ $0.306 \pm 0.061 $ $\sqrt{\frac{3}{2}}$ $-\frac{1}{\sqrt{6}}$ $-\frac{1}{6\,\sqrt{2}}$ $-\frac{1}{2\,\sqrt{2}}$ $-\frac{1}{2\,\sqrt{2}}$ $-\frac{5}{2\,\sqrt{2}}$ 0
$ \Xi^- \to \Sigma^0 \,e^-\,\bar \nu_e $ $0.929 \pm 0.112 $ $\frac{1}{\sqrt{2}}$ $\frac{1}{\sqrt{2}}$ $-\frac{5}{6\,\sqrt{6}}$ $-\frac{1}{2\,\sqrt{6}}$ $-\frac{5}{2\,\sqrt{6}}$ $-\frac{1}{2\,\sqrt{6}}$ 0
: Axial-vector coupling constants for the weak decay processes of the baryon octet states. The empirical values for $g_A$ are taken from [@Dai]. Here we do not consider $SU(3)$ symmetry-breaking effects of the vector current. The last seven columns give the coefficients of the axial-vector coupling constants $g_A$ decomposed into the $F_R$, $D_R$ and $c_i$ parameters.[]{data-label="weak-decay:tab"}
The analysis of [@Dai], which considers constraints from the weak decay processes of the baryon octet states and the strong decay widths of the decuplet states, obtains $c_2 \simeq -0.15$ and $c_3\simeq 0.09$ but finds values of $c_1$ and $c_4$ which are compatible with zero[^1]. In Tab. (\[weak-decay:tab\]) we reproduce the axial-vector coupling constants as given in [@Dai] relevant for the various baryon octet weak-decay processes. Besides $C_A+2\,\tilde c_1/\sqrt{3}$, the empirical strong-decay widths of the decuplet states constrain the parameters $c_{3}$ and $c_4$ only, as is evident from the expressions for the decuplet widths $$\begin{aligned}
&& \Gamma_{[10]}^{(\Delta )} = \frac{m_N+E_N}{2\,\pi \,f^2}\,\frac{p_{\pi N}^3}{12\,m_{[10]}^{(\Delta )}}
\,\Big( C_A +{\textstyle{2\over \sqrt{3}}}\,(\tilde c_1+3\,c_3)\Big)^2 \,,
\nonumber\\
&& \Gamma_{[10]}^{(\Sigma )} = \frac{m_\Lambda+E_\Lambda}{2\,\pi \,f^2}\,
\frac{p_{\pi \Lambda}^3}{24\,m_{[10]}^{(\Sigma )}}
\,\Big( C_A +{\textstyle{2\over \sqrt{3}}}\,\tilde c_1 \Big)^2
\nonumber\\
&& \qquad +\frac{m_\Sigma+E_\Sigma}{3\,\pi \,f^2}\,
\frac{p_{\pi \Sigma}^3}{24\,m_{[10]}^{(\Sigma )}}
\,\Big( C_A +{\textstyle{2\over \sqrt{3}}}\,(\tilde c_1+6\,c_4)\Big)^2 \,,
\nonumber\\
&& \Gamma_{[10]}^{(\Xi )} = \frac{m_\Xi+E_\Xi}{2\,\pi \,f^2}\,\frac{p_{\pi \Xi}^3}{24\,m_{[10]}^{(\Xi )}}
\,\Big( C_A +{\textstyle{2\over \sqrt{3}}}\,(\tilde c_1-3\,c_3+3\,c_4)\Big)^2 \,,
\label{decuplet-decay}\end{aligned}$$ where for example $m_\Delta =\sqrt{m_N^2+p_{\pi N}^2}+\sqrt{m_\pi^2+p_{\pi N}^2}$ and $E_N= \sqrt{m_N^2+p_{\pi N}^2}$. For instance, the values $ C_A+2\,\tilde c_1/\sqrt{3} \simeq 1.7$, $c_3 \simeq 0.09$ and $c_4 \simeq 0.0$ together with $f\simeq f_\pi \simeq 93$ MeV lead to isospin averaged partial decay widths of the decuplet states which are compatible with the present day empirical estimates. It is clear that the six data points for the baryon octet decays can be reproduced by a suitable adjustment of the six parameters $F_R$, $D_R$ and $c_{1,2,3,4}$. The non-trivial issue is to what extent the explicit $SU(3)$ symmetry-breaking pattern in the axial-vector coupling constants is consistent with the symmetry-breaking pattern in the meson-baryon coupling constants. Here a possible strong energy dependence of the decuplet self-energies may invalidate the use of the simple expressions (\[decuplet-decay\]). A more direct comparison with the meson-baryon scattering data may be required. Finally we wish to mention an implicit assumption one relies on if Tab. \[weak-decay:tab\] is applied. In a strict chiral expansion the $Q^2$ effects included in that table are incomplete, because there are various one-loop diagrams which are not considered but carry chiral order $Q^2$ also. However, in a combined chiral and $1/N_c$ expansion it is natural to neglect such loop effects, because they are suppressed by $1/N_c$. This is immediate with the large $N_c$ scaling rules $m_\pi \sim N_c^0$ and $f \sim \sqrt{N_c}$ [@Hooft; @Witten] together with the fact that the one-loop effects are proportional to $m_{K,\pi}^2/(4\pi\,f^2) $ [@Bijnens:ga]. On the other hand, it is evident from (\[3-point-vertex\]) and (\[ansatz-3\]) that the $SU(3)$ symmetry-breaking contributions are not necessarily suppressed by $1/N_c$. Our approach differs from previous calculations [@Bijnens:ga; @Jenkins:ga1; @Jenkins:ga2; @Luty-2] where emphasis was put on the one-loop corrections of the axial-vector current rather than the quasi-local counter terms which were not considered. It is clear that part of the one-loop effects, in particular their renormalization scale dependence, can be absorbed into the counter terms considered in this work.
We will return to the large $N_c$ counting issue when discussing the approximate scattering kernel and also when presenting our final set of parameters, obtained from a fit to the data set.
Relativistic meson–baryon scattering
====================================
In this section we prepare the ground for our relativistic coupled-channel effective field theory of meson–baryon scattering. We first develop the formalism for the case of elastic $\pi N$ scattering for simplicity. The next section is devoted to the inclusion of inelastic channels which leads to the coupled-channel approach. Our approach is based on an ’old’ idea present in the literature for many decades. One aims at reducing the complexity of the relativistic Bethe-Salpeter equation by a suitable reduction scheme constrained to preserve the relativistic unitarity cuts [@Sugar; @Gross; @Tjon]. A famous example is the Blankenbecler-Sugar scheme [@Sugar] which reduces the Bethe-Salpeter equation to a 3-dimensional integral equation. For a beautiful variant developed for pion-nucleon scattering see the work by Gross and Surya [@Gross]. In our work we derive a scheme which is more suitable for the relativistic chiral Lagrangian. We do not attempt to establish a numerical solution of the four dimensional Bethe-Salpeter equation based on phenomenological form factors and interaction kernels [@Afnan]. The merit of the chiral Lagrangian is that a major part of the complexity is already eliminated by having reduced non-local interactions to ’quasi’ local interactions involving a finite number of derivatives only. Our scheme is therefore constructed to be particularly transparent and efficient for the typical case of ’quasi’ local interaction terms. In the course of developing our approach we suggest a modified subtraction scheme within dimensional regularization, which complies manifestly with the chiral counting rule (\[q-rule\]).
Consider the on-shell pion-nucleon scattering amplitude $$\begin{aligned}
\langle \pi^{i}(\bar q)\,N(\bar p)|\,T\,| \pi^{j}(q)\,N(p) \rangle
=(2\pi)^4\,\delta^4(0)\,
\bar u(\bar p)\,
T^{ij}_{\pi N \rightarrow \pi N}(\bar q,\bar p ; q,p)\,u(p) \,,
\label{on-shell-scattering}\end{aligned}$$ where $\delta^4(0)$ guarantees energy-momentum conservation and $u(p)$ is the nucleon isospin-doublet spinor. In quantum field theory the scattering amplitude $T_{\pi N \rightarrow \pi N}$ follows as the solution of the Bethe-Salpeter matrix equation $$\begin{aligned}
T(\bar k ,k ;w ) &=& K(\bar k ,k ;w )
+\int \frac{d^4l}{(2\pi)^4}\,K(\bar k , l;w )\, G(l;w)\,T(l,k;w )\;,
\nonumber\\
G(l;w)&=&-i\,S_N({\textstyle
{1\over 2}}\,w+l)\,D_\pi({\textstyle {1\over 2}}\,w-l)\,,
\label{BS-eq}\end{aligned}$$ in terms of the Bethe-Salpeter kernel $K(\bar k,k;w )$ to be specified later, the nucleon propagator $S_N(p)=1/(\pslash-m_N+i\,\epsilon)$ and the pion propagator $D_\pi(q)=1/(q^2-m_\pi^2+i\,\epsilon)$. Self energy corrections in the propagators are suppressed and therefore not considered in this work. We introduced convenient kinematics: $$\begin{aligned}
w = p+q = \bar p+\bar q\,,
\quad k= \half\,(p-q)\,,\quad
\bar k =\half\,(\bar p-\bar q)\,,
\label{def-moment}\end{aligned}$$ where $q,\,p,\, \bar q, \,\bar p$ are the initial and final pion and nucleon 4-momenta. The Bethe-Salpeter equation (\[BS-eq\]) implements Lorentz invariance and unitarity for the two-body scattering process. It involves the off-shell continuation of the on-shell scattering amplitude introduced in (\[on-shell-scattering\]). We recall that only the on-shell limit with $ \bar p^2, p^2\rightarrow m_N^2 $ and $\bar
q^2,q^2\rightarrow m_\pi^2 $ carries direct physical information. In quantum field theory the off-shell form of the scattering amplitude reflects the particular choice of the pion and nucleon interpolating fields chosen in the Lagrangian density and therefore can be altered basically at will by a redefinition of the fields [@off-shell; @Fearing].
It is convenient to decompose the interaction kernel and the resulting scattering amplitude in isospin invariant components $$\begin{aligned}
K_{ij}(\bar k ,k ;w ) =\sum_I\,K_I(\bar k ,k ;w )\,P^{(I)}_{ij}
\,,\,\,
T_{ij}(\bar k ,k ;w ) &=& \sum_I\,T_I(\bar k ,k ;w )\,P^{(I)}_{ij} \,,
\label{isospin-decom}\end{aligned}$$ with the isospin projection matrices $P^{(I)}_{ij}$. For pion-nucleon scattering one has: $$\begin{aligned}
P^{\left(\frac{1}{2}\right)}_{ij} &=& \frac{1}{3}\,\sigma_i\,\sigma_j \;,\;\;\;
P^{\left(\frac{3}{2}\right)}_{ij} = \delta_{ij}-\frac{1}{3}\,\sigma_i\,\sigma_j
\;,\;\;\;
\sum_k\,P^{\left(I\right)}_{ik}\,P^{\left(I'\right)}_{kj}=
\delta_{I\,I'}\,P^{\left(I\right)}_{ij}\;.
\label{iso:proj}\end{aligned}$$ The Ansatz (\[isospin-decom\]) decouples the Bethe-Salpeter equation into the two isospin channels $I=1/2$ and $I=3/2$.
On-shell reduction of the Bethe-Salpeter equation
-------------------------------------------------
For our application it is useful to exploit the ambiguity in the off-shell structure and choose a particularly convenient representation. We decompose the interaction kernel into an ’on-shell irreducible’ part $\bar K $ and ’on-shell reducible’ terms $K_L$ and $K_R$ which vanish if evaluated with on-shell kinematics either in the incoming or outgoing channel respectively $$\begin{aligned}
&&K=\bar K+K_L+K_R+K_{LR} \;,
\nonumber\\
&&\bar u_N(\bar p)\,K_L \Big|_{\mathrm{on-shell}} = 0 = K_R
\,u_N(p)\Big|_{\mathrm{on-shell}} \;.
\label{k-decomp}\end{aligned}$$ The term $K_{LR}$ disappears if evaluated with either incoming or outgoing on-shell kinematics. Note that the notion of an on-shell irreducible kernel $\bar K$ is not unique per se and needs further specifications. The precise definition of our particular choice of on-shell irreducibility will be provided when constructing our relativistic partial-wave projectors. In this subsection we study the generic consequences of decomposing the interaction kernel according to (\[k-decomp\]). With this decomposition of the interaction kernel the scattering amplitude can be written as follows $$\begin{aligned}
T &=& \bar T
-\Big(K_L+K_{LR}\Big)\!\cdot\! \Big( 1-G\!\cdot\! K\Big)^{-1}\!\cdot
\!G\! \cdot\!
\Big(K_R+K_{LR}\Big)-K_{LR}
\nonumber\\
&+&\Big(K_L+K_{LR}\Big)\! \cdot\! \Big( 1-G\!\cdot\! K\Big)^{-1}
+\Big( 1-K\!\cdot\! G\Big)^{-1}\!\cdot\!\Big(K_R+K_{LR} \Big)\,,
\nonumber\\
\bar T &=& \Big(1-V\!\cdot\! G \Big)^{-1} \!\cdot\! V \;,
\label{t-eff}\end{aligned}$$ where we use operator notation with, e.g., $T=K+ K\!\cdot \!G\!\cdot \!T $ representing the Bethe-Salpeter equation (\[BS-eq\]). The effective interaction $V$ in (\[t-eff\]) is given by $$\begin{aligned}
V&=&\Big( \bar K+K_R\!\cdot\! G\!\cdot\! X \Big)\!\cdot\!
\Big(1-G\!\cdot\! K_L-G\!\cdot\! K_{LR}\!\cdot\! G\!\cdot\! X \Big)^{-1} \;,
\nonumber\\
X &=& \Big( 1-(K_R+K_{LR})\!\cdot\! G\Big)^{-1}\!
\cdot\! \Big( \bar K+K_L\Big)
\label{v-eff}\end{aligned}$$ without any approximations. We point out that the interaction kernels $V$ and $K$ are equivalent on-shell by construction. This follows from (\[t-eff\]) and (\[k-decomp\]), which predict the equivalence of $T$ and $\bar T$ for on-shell kinematics $$\begin{aligned}
\bar{u}_N(\bar p)\, T \,u_N(p)\Big|_{\mathrm{on-shell}}
\equiv \bar{u}_N(\bar
p)\,\bar T \,u_N(p)\Big|_{\mathrm{on-shell}} .
\nonumber\end{aligned}$$ As an explicit simple example for the application of the formalism (\[t-eff\],\[v-eff\]) we consider the s-channel nucleon pole diagram as a particular contribution to the interaction kernel $K(\bar k,k;w)$ in (\[BS-eq\]). In the isospin $1/2$ channel its contribution evaluated with the pseudo-vector pion-nucleon vertex reads $$\begin{aligned}
K(\bar k, k;w ) &=& -\frac{3\,g_A^2}{4\,f^2}\,
\gamma_5\,\left( {\textstyle{1\over 2}}\,\wslash-\barkslash \right)\,
\frac{1}{\wslash-m_N-\Delta m_N(w)}\,\gamma_5\,
\left( {\textstyle{1\over 2}}\,\wslash- \kslash \right)\;,
\label{nucleon-pole}\end{aligned}$$ where we included a wave-function and mass renormalization $\Delta m_N(w)$ for later convenience[^2]. We construct the various components of the kernel according to (\[k-decomp\]) $$\begin{aligned}
\bar K&=&-\frac{3\,g_A^2}{4\,f^2}\,
\frac{\big( \wslash-m_N \big)^2}{\wslash+\bar m_N}\,,\quad
K_{LR}=\frac{3\,g_A^2}{4\,f^2}\,
\Big( {\barpslash}-m_N \Big)\,
\frac{1}{\wslash+\bar m_N}\,\Big( \pslash- m_N\Big)\;,
\nonumber\\
K_L&=&\frac{3\,g_A^2}{4\,f^2}\,
\Big(\barpslash-m_N \Big)\,
\frac{\wslash- m_N}{\wslash+\bar m_N}\,,\quad
K_R=\frac{3\,g_A^2}{4\,f^2}\,
\frac{\wslash-m_N}{\wslash+\bar m_N}\,\Big( \pslash- m_N\Big)
\,,
\label{nucleon-pole-off}\end{aligned}$$ where $\bar m_N = m_N+\gamma_5\,\Delta m_N(w)\,\gamma_5$. The solution of the Bethe-Salpeter equation is derived in two steps. First solve for the auxiliary object $X$ in (\[v-eff\]) $$\begin{aligned}
X &=& \frac{3\,g_A^2}{4\,f^2}\,
\Big( \barpslash-\wslash\Big)\,
\frac{1}{\wslash+\bar m_N}\,\Big( \wslash- m_N\Big)
\nonumber\\
&+&\frac{3\,g_A^2}{4\,f^2}\,
\Big( \barpslash+\wslash-2\,m_N\Big)\,
\frac{3\,g_A^2\,I_\pi^{(l)}}
{4\,f^2\,(\wslash+\bar m_N)+3\,g_A^2\,I_\pi}\,
\frac{ \wslash- m_N}{\wslash+\bar m_N} \;,
\label{x-example}\end{aligned}$$ where one encounters the pionic tadpole integrals: $$\begin{aligned}
I_\pi &=&i\,
\int \frac{d^d\,l}{(2\,\pi)^d }
\frac{\mu^{4-d}}{l^2-m_\pi^2+i\,\epsilon }
\,,\quad
I^{(l)}_\pi =i\,
\int \frac{d^d\,l}{(2\,\pi)^d }
\frac{\mu^{4-d}\,\lslash }{l^2-m_\pi^2+i\,\epsilon }\,,
\label{def-tadpole}\end{aligned}$$ properly regularized for space-time dimension $d$ in terms of the renormalization scale $\mu $. Since $I_\pi^{(l)}=0$ and $K_R\cdot G\cdot
\!(\bar K+K_L)\equiv 0$ for our example the expression (\[x-example\]) reduces to $X=\bar K+K_L $. The effective potential $V(w)$ and the on-shell equivalent scattering amplitude $\bar T $ follow $$\begin{aligned}
V(w)&=& -\frac{3\,g_A^2}{4\,f^2}\,\frac{ (\wslash- m_N)^2}{\wslash+\bar m_N}
\left(1+
\frac{3\,g_A^2\,I_\pi}{4\,f^2}\,\frac{\wslash- m_N}{\wslash+\bar m_N}
\right)^{-1} \;,
\nonumber\\
\bar T(w) &=& \frac{1}{1-V(w)\,J_{\pi N}(w)}\,V(w) \;.
\label{pin-example}\end{aligned}$$ The divergent loop function $J_{\pi N}$ in (\[pin-example\]) defined via $\bar{K}\!\cdot\!G\!\cdot\!\bar{K}= \bar K\, J_{\pi N}\,\bar K$ may be decomposed into scalar master-loop functions $I_{\pi N}(\sqrt{s}\,)$ and $I_{N},I_\pi$ with $$\begin{aligned}
J_{\pi N}(w)&=& \left(m_N +
\frac{w^2+m_N^2-m_\pi^2}{2\,w^2}\,
\wslash\right)\,I_{\pi N}(\sqrt{s}\,)
+\frac{I_N-I_{\pi}}{2\,w^2}\,\wslash \;,
\nonumber\\
I_{\pi N}(\sqrt{s}\,)&=&-i\,\int
\frac{d^dl}{(2\pi)^d}\,\frac{\mu^{4-d}}{l^2-m_\pi^2}\,\frac{1}{(w-l)^2-m_N^2}
\;,
\nonumber\\
I_N &=&i\,
\int \frac{d^d\,l}{(2\,\pi)^d }
\frac{\mu^{4-d}}{l^2-m_N^2+i\,\epsilon }
\label{jpin-def}\end{aligned}$$ where $s=w^2$. The result (\[pin-example\]) gives an explicit example of the powerful formula (\[t-eff\]). The Bethe-Salpeter equation may be solved in two steps. Once the effective potential $V$ is evaluated the scattering amplitude $\bar T$ is given in terms of the loop function $J_{\pi N}$ which is independent on the form of the interaction. In section 3.3 we will generalize the result (\[pin-example\]) by constructing a complete set of covariant projectors which will define our notion of on-shell irreducibility explicitly. Before discussing the result (\[pin-example\]) in more detail we wish to consider the regularization and renormalization scheme required for the relativistic loop functions in (\[jpin-def\]).
Renormalization program
-----------------------
An important requisite of the chiral Lagrangian is a consistent regularization and renormalization scheme for its loop diagrams. The regularization scheme should respect all symmetries built into the theory but should also comply with the power counting rule (\[q-rule\]). The standard $MS$ or $\overline{MS}$ subtraction scheme of dimensional regularization appears inconvenient, because it contradicts standard chiral power counting rules if applied to relativistic Feynman diagrams [@gss; @kambor]. We will suggest a modified subtraction scheme for relativistic diagrams, properly regularized in space-time dimensions $d$, which complies with the chiral counting rule (\[q-rule\]) manifestly.
We begin with a discussion of the regularization scheme for the one-loop functions $I_\pi$, $I_N$ and $I_{\pi N}(\sqrt{s}\,)$ introduced in (\[jpin-def\]). One encounters some freedom in regularizing and renormalizing those master-loop functions, which are typical representatives for all one-loop diagrams. We first recall their well-known properties at $d=4$. The loop function $I_{\pi N}(\sqrt{s}\,)$ is made finite by one subtraction, for example at $\sqrt{s}=0$, $$\begin{aligned}
\!\!\!\!I_{\pi N}(\sqrt{s}\,)&=&\frac{1}{16\,\pi^2}
\left( \frac{p_{\pi N}}{\sqrt{s}}\,
\left( \ln \left(1-\frac{s-2\,p_{\pi N}\,\sqrt{s}}{m_\pi^2+m_N^2} \right)
-\ln \left(1-\frac{s+2\,p_{\pi N}\sqrt{s}}{m_\pi^2+m_N^2} \right)\right)
\right.
\nonumber\\
&+&\left.
\left(\frac{1}{2}\,\frac{m_\pi^2+m_N^2}{m_\pi^2-m_N^2}
-\frac{m_\pi^2-m_N^2}{2\,s}
\right)
\,\ln \left( \frac{m_\pi^2}{m_N^2}\right) +1 \right)+I_{\pi N}(0)\;,
\nonumber\\
p_{\pi N}^2 &=&
\frac{s}{4}-\frac{m_N^2+m_\pi^2}{2}+\frac{(m_N^2-m_\pi^2)^2}{4\,s} \;.
\label{ipin-analytic}\end{aligned}$$ One finds $I_{\pi N}(m_N)-I_{\pi N}(0)= (4\pi)^{-2}+{\mathcal O}(m_\pi/m_N)\sim Q^0$ in conflict with the expected minimal chiral power $Q$. On the other hand the leading chiral power of the subtracted loop function complies with the prediction of the standard chiral power counting rule (\[q-rule\]) with $I_{\pi N}(\sqrt{s}\,)-I_{\pi N}(\mu_S ) \sim Q$ rather than the anomalous power $Q^0$ provided that $\mu_S \sim m_N$ holds. The anomalous contribution is eaten up by the subtraction constant $I_{\pi N}(\mu_S)$. This can be seen by expanding the loop function $$\begin{aligned}
I_{\pi N}(\sqrt{s}\,)&=&i\,\frac{\sqrt{\phi_{\pi N}}}{8\,\pi\,m_N}
+\frac{\sqrt{\phi_{\pi N} }}{16\,\pi^2\,m_N}\,
\ln \left( \frac{\sqrt{s}-m_N+\sqrt{\phi_{\pi N} }}
{\sqrt{s}-m_N-\sqrt{\phi_{\pi N} }} \right)
\nonumber\\
&+&\frac{\sqrt{m^2_\pi-\mu_N^2}}{8\,\pi\,m_N}
-\frac{\sqrt{\mu_N^2-m_\pi^2}}{16\,\pi^2\,m_N}\,
\ln \left( \frac{\mu_N+\sqrt{\mu_N^2-m_\pi^2}}
{\mu_N-\sqrt{\mu_N^2-m_\pi^2}} \right)
\nonumber\\
&-&\frac{\sqrt{s}-\mu_S}{16\,\pi^2\,m_N}\,\ln \left( \frac{m_\pi^2}{m_N^2}\right)
+{\mathcal O}\left( \frac{(\mu_S-m_N)^2}{m_N^2},Q^2 \right)+I_{\pi N}(\mu_S)\;,
\label{ipin-expand}\end{aligned}$$ in powers of $\sqrt{s}-m_N\sim Q$ and $\mu_N=\mu_S -m_N $. Here we introduced the approximate phase-space factor $\phi_{\pi N}= (\sqrt{s}-m_N)^2-m_\pi^2$.
It should be clear from the simple example of $I_{\pi N}(\sqrt{s}\,)$ in (\[ipin-analytic\]) that a manifest realization of the chiral counting rule (\[q-rule\]) is closely linked to the subtraction scheme implicit in any regularization scheme. A priori it is unclear in which way the subtraction constants of various loop functions are related by the pertinent symmetries[^3]. Dimensional regularization has proven to be an extremely powerful tool how to regularize and how to subtract loop functions in accordance with all symmetries. Therefore we recall the expressions for the master-loop function $I_N$, and $I_\pi$ and $I_{\pi N}(m_N)$ as they follow in dimensional regularization: $$\begin{aligned}
&& I_N = m_N^2\,\frac{\Gamma (1-d/2)}{(4\pi)^2}
\left(\frac{m_N^2}{4\,\pi \,\mu^2} \right)^{(d-4)/2}
\nonumber\\
&& \quad \;\,= \frac{m_N^2}{(4\,\pi)^2}
\left( -\frac{2}{4-d}+\gamma-1 -\ln (4 \pi)+\ln \left( \frac{m_N^2}{\mu^2}\right)
+ {\mathcal O}\left(4-d \right)\right) \;,
\label{n-tadpole}\end{aligned}$$ where $d$ is the dimension of space-time and $\gamma $ the Euler constant. The expression for the pionic tadpole follows by replacing the nucleon mass $m_N$ in (\[n-tadpole\]) by the pion mass $m_\pi$. The merit of dimensional regularization is that one is free to subtract all poles at $d=4$ including any specified finite term without violating any of the pertinent symmetries. In the $\overline{MS }$-scheme the pole $1/(4-d)$ is subtracted including the finite part $\gamma- \ln (4 \pi)$. That leads to $$\begin{aligned}
&&I_{N,\overline{MS}}= \frac{m_N^2}{(4 \pi)^2}
\left( -1+\ln \left(\frac{m_N^2}{\mu^2}\right) \right)\,,\quad \;
I_{\pi,\overline{MS}} =\frac{m_\pi^2}{(4\pi)^2}
\left(-1+\ln \left(\frac{m_\pi^2}{\mu^2}\right) \right)\,,
\nonumber\\
&& I_{\pi N,\overline{MS}}\,(m_N)= -\frac{1}{(4\pi)^2} \,
\ln \left(\frac{m_N^2}{\mu^2} \right)
-\frac{m_\pi}{16\,\pi \,m_N}+
{\mathcal O}\left( \frac{m_\pi^2}{m_N^2}\right) \,.
\label{master-dim}\end{aligned}$$ The result (\[master-dim\]) confirms the expected chiral power for the pionic tadpole $I_\pi \sim Q^2$. However, a striking disagreement with the chiral counting rule (\[q-rule\]) is found for the $\overline {MS}$-subtracted loop functions $I_{\pi N}\sim Q^0$ and $I_N \sim Q^0$. Recall that for the subtracted loop function $I_{\pi N}(\sqrt{s}\,)-I_{\pi N}(m_N)\sim Q $ the expected minimal chiral power is manifest (see (\[ipin-expand\])). It is instructive to trace the source of the anomalous chiral powers. By means of the identities $$\begin{aligned}
&& I_{\pi N}(m_N) = \frac{I_\pi-I_N}{m_N^2-m_\pi^2} + I_{\pi N}(m_N)-I_{\pi N}(0) \;,
\nonumber\\
&& I_{\pi N}(m_N)-I_{\pi N}(0)=
\frac{1}{(4\pi)^2}-\frac{m_\pi}{16\,\pi\,m_N} \left(1-\frac{m_\pi^2}{8\,m_N^2} \right)
\nonumber\\
&& \qquad \qquad \qquad +\frac{1}{(4 \pi)^2}\left( 1
-\frac{3}{2}\,\ln \left( \frac{m_\pi^2}{m_N^2}\right) \right)
\frac{m_\pi^2}{m_N^2}
+{\mathcal O}\left( \frac{m_\pi^4}{m_N^4}, d-4 \right) \,,
\label{ipin-dim}\end{aligned}$$ it appears that once the subtraction scheme is specified for the tadpole terms $I_\pi$ and $I_N$ the required subtractions for the remaining master-loop functions are unique. In (\[ipin-dim\]) we used the algebraic consistency identity[^4] $$\begin{aligned}
I_\pi-I_N= \Big(m_N^2-m_\pi^2\Big)\,I_{\pi N}(0) \,,
\label{algebra-1}\end{aligned}$$ which holds for any value of the space-time dimension d, and expanded the finite expression $I_{\pi N}(m_N)-I_{\pi N}(0)$ in powers of $m_\pi/m_N$ at $d=4$. The result (\[ipin-dim\]) seems to show that one either violates the desired chiral power for the nucleonic tadpole, $I_N$, or for the loop function $I_{\pi N}(m_N)$. One may for example subtract the pole at $d=4$ including the finite constant $$\gamma-1 -\ln (4 \pi)+\ln \left( \frac{m_N^2}{\mu^2}\right) \,.$$ That leads to a vanishing nucleonic tadpole $I_N \to 0$, which would be consistent with the expectation $I_N\sim Q^3$, but we find $I_{\pi N}(m_N) \to 1/(4\pi)^2 +{\mathcal O}\left(m_\pi \right)$, in disagreement with the expectation $I_{\pi N}(m_N)\sim Q$. This problem can be solved if one succeeds in defining a subtraction scheme which acts differently on $I_{\pi N}(0)$ and $I_{\pi N}(m_N)$. We stress that this is legitimate, because $I_{\pi N}(0)$ probes our effective theory outside its applicability domain. Mathematically this can be achieved most economically and consistently by subtracting a pole in $I_{\pi N}(\sqrt{s}\,)$ at $d=3$ which arises in the limit $m_\pi /m_N \to 0$. To be explicit we recall the expression for $I_{\pi N}(\sqrt{s}\,)$ at arbitrary space-time dimension $d$ (see eg. [@Becher]): $$\begin{aligned}
&& I_{\pi N}(\sqrt{s}\,) = \left(\frac{m_N}{\mu}\right)^{d-4}
\frac{\Gamma(2-d/2)}{(4\pi)^{d/2}}\,\int_0^1 \,d z \; C^{d/2-2} \;,
\nonumber\\
&& \qquad C= z^2-
\frac{s-m_N^2-m_\pi^2}{m^2_N}\,z \,(1-z)+\frac{m_\pi^2}{m_N^2}\,(1-z)^2-i\,\epsilon \,.
\label{}\end{aligned}$$ We observe that at $d=3$ the loop function $I_{\pi N}(\sqrt{s}\,)$ is finite at $\sqrt{s}=0$ but infinite at $\sqrt{s}=m_N$ if one applies the limit $m_\pi /m_N \to 0$. One finds $$\begin{aligned}
&& I_{\pi N}(m_N\,) = \frac{1}{8\,\pi}\,\frac{\mu}{m_N}\,\frac{1}{d-3}
-\frac{1}{16\,\pi}\,\frac{\mu}{m_N}
\left(\ln (4\,\pi)+ \ln \left(\frac{\mu^2}{m_N^2}\right)+\frac{\Gamma'(1/2)}{\sqrt{\pi}}\right)
\nonumber\\
&&\qquad \qquad \,+\,{\mathcal O}\left( \frac{m_\pi}{m_N},d-3\right) \;.
\label{}\end{aligned}$$ We are now prepared to introduce a minimal chiral subtraction scheme which may be viewed as a simplified variant of the scheme of Becher and Leutwyler in [@Becher]. As usual one first needs to evaluate the contributions to an observable quantity at arbitrary space-time dimension $d$. The result shows poles at $d=4$ and $d=3$ if considered in the non-relativistic limit with $m_\pi/m_N \to 0$. Our modified subtraction scheme is defined by the replacement rules: $$\begin{aligned}
&& \frac{1}{d-3} \to -\frac{m_N}{2\pi\,\mu} \;, \qquad
\frac{2}{d-4} \to \gamma-1 -\ln (4 \pi)+\ln \left( \frac{m_N^2}{\mu^2}\right) \,,
\label{def-sub}\end{aligned}$$ where it is understood that poles at $d=3$ are isolated in the non-relativistic limit with $m_\pi/m_N \to 0$. The $d=4$ poles are isolated with the ratio $m_\pi/m_N$ at its physical value. The limit $d\to 4$ is applied after the pole terms are replaced according to (\[def-sub\]). We emphasize that there are no infrared singularities in the residuum of the $1/(d-3)$-pole terms. In particular we observe that the anomalous subtraction implied in (\[def-sub\]) does not lead to a potentially troublesome pion-mass dependence of the counter terms[^5]. To make contact with a non-relativistic scheme one needs to expand the loop function in powers of $p/m_N $ where $p$ represents any external three momentum.
We collect our results for the loop functions $I_N$, $I_\pi$ and $I_{\pi N}(m_N)$ as implied by the subtraction prescription (\[def-sub\]): $$\begin{aligned}
&& \bar I_N = 0 \,, \qquad
\bar I_\pi = \frac{m_\pi^2}{(4\pi)^2}\,\ln \left( \frac{m_\pi^2}{m_N^2}\right)\;,
\nonumber\\
&& \bar I_{\pi N}(m_N) = -\frac{m_\pi}{16\,\pi\,m_N} \left(1-\frac{m_\pi^2}{8\,m_N^2} \right)
\nonumber\\
&& \qquad \qquad \; +\frac{1}{(4 \pi)^2}\left( 1
-\frac{1}{2}\,\ln \left( \frac{m_\pi^2}{m_N^2}\right) \right)
\frac{m_\pi^2}{m_N^2}
+{\mathcal O}\left( \frac{m_\pi^4}{m_N^4} \right) \,,
\label{result-bar}\end{aligned}$$ where the ’$\bar{\phantom{X}}$’ signals a subtracted loop functions[^6]. The result shows that now the loop functions behave according to their expected minimal chiral power (\[q-rule\]). Note that the one-loop expressions (\[result-bar\]) do no longer depend on the renormalization scale $\mu$ introduced in dimensional regularization[^7]. This should not be too surprising, because the renormalization prescription (\[def-sub\]) has a non-trivial effect on the counter terms of the chiral Lagrangian. The prescription (\[def-sub\]) defines also a unique subtraction for higher loop functions in the same way the $\overline {MS}$-scheme does. The renormalization scale dependence will be explicit at the two-loop level, reflecting the presence of so-called overall divergences.
We checked that all scalar one-loop functions, subtracted according to (\[def-sub\]), comply with their expected minimal chiral power (\[q-rule\]). Typically, loop functions which are finite at $d=4$ are not affected by the subtraction prescription (\[def-sub\]). Also, loop functions involving pion propagators only do not show singularities at $d=3$ and therefore can be related easily to the corresponding loop functions of the $\overline{MS}$-scheme. Though it may be tedious to relate the standard $\overline{MS}$-scheme to our scheme in the nucleon sector, in particular when multi-loop diagrams are considered, we assert that we propose a well defined prescription for regularizing all divergent loop integrals. A prescription which is far more convenient than the $\overline {MS}$-scheme, because it complies manifestly with the chiral counting rule (\[q-rule\])[^8].
If we were to perform calculations within standard chiral perturbation theory in terms of the relativistic chiral Lagrangian we would be all set for any computation. However, as advocated before we are heading towards a non-perturbative chiral theory. That requires a more elaborate renormalization program, because we wish to discriminate carefully reducible and irreducible diagrams and sum the reducible diagrams to infinite order. The idea is to take over the renormalization program of standard chiral perturbatioon theory to the interaction kernel. In order to apply the standard perturbative renormalization program for the interaction kernel one has to move all divergent parts lying in reducible diagrams into the interaction kernel. That problem is solved in part by constructing an on-shell equivalent interaction kernel according to (\[k-decomp\],\[t-eff\],\[v-eff\]). It is evident that the ’moving’ of divergences needs to be controlled by an additional renormalization condition. Any such condition imposed should be constructed so as to respect crossing symmetry approximatively. While standard chiral perturbation theory leads directly to cross symmetric amplitudes at least approximatively, it is not automatically so in a resummation scheme.
Before introducing our general scheme we examine the above issues explicitly with the example worked out in detail in the previous section. Taking the s-channel nucleon pole term as the driving term in the Bethe-Salpeter equation we derived the explicit result (\[pin-example\]). We first discuss its implicit nucleon mass renormalization. The result (\[pin-example\]) shows a pole at the physical nucleon mass with $\wslash =-m_N$, only if the mass-counter term $\Delta m_N$ in (\[nucleon-pole\]) is identified as follows $$\begin{aligned}
\Delta m_N = \frac{3\,g_A^2}{4\,f^2}\,2\,m_N\,\Big (m_\pi^2\,I_{\pi N}(m_N)-I_N \Big)\,.
\label{mass-ren}\end{aligned}$$ In the $\overline{MS}$-scheme the divergent parts of $\Delta m_N$, or more precisely the renormalization scale dependent parts, may be absorbed into the nucleon bare mass [@gss]. In our scheme we renormalize by simply replacing $I_{\pi N}\to \bar I_{\pi N} $, $I_\pi \to \bar I_\pi$ and $I_N \to \bar I_N= 0$. The result (\[mass-ren\]) together with (\[result-bar\]) then reproduces the well known result [@gss; @kambor] $$\Delta m_N = -\frac{3\,g_A^2\,m_\pi^3}{32\,\pi\,f^2}+\cdots \;,$$ commonly derived in terms of the one-loop nucleon self energy $\Sigma_N(p)$. In order to offer a more direct comparison of (\[pin-example\]) with the nucleon self-energy $\Sigma_N(p)$ we recall the one-loop result $$\begin{aligned}
&&\!\!\!\!\!\! \Sigma_N (p) = \frac{3\,g_A^2}{4\,f^2}\, \Bigg(
m_N \,\Big( m_\pi^2\,I_{\pi N}(\sqrt{p^2}\,) -I_N\Big)
-\pslash \,\Bigg(\frac{p^2-m_N^2}{2\,p^2} \, I_\pi
+\frac{p^2+m_N^2}{2\,p^2} \,I_N
\nonumber\\
&& \qquad \qquad \qquad
+\left( \frac{(p^2-m_N^2)^2}{2\,p^2}- m_\pi^2\,\frac{p^2+m_N^2}{2\,p^2}
\right) I_{\pi N}(\sqrt{p^2}\,)
\Bigg) \Bigg)\;,
\label{n-self}\end{aligned}$$ in terms of the convenient master-loop function $I_{\pi N}(\sqrt{p^2})$ and the tadpole terms $I_N$ and $I_\pi$. We emphasize that the expression (\[n-self\]) is valid for arbitrary space-time dimension $d$. The wave-function renormalization, ${\mathcal Z}_N$, of the nucleon can be read off (\[n-self\]) $$\begin{aligned}
{\mathcal Z}^{-1}_N &=& 1-\frac{\partial \,\Sigma_N }{\partial \,\pslash}\Bigg|_{\pslash \,=m_N}
\!\!= 1 -\frac{3\,g_A^2}{4\,f^2}
\left( 2\,m_\pi^2\,m_N \,\frac{\partial \,I_{\pi N}(\sqrt{p^2})}{\partial \,\sqrt{p^2}}
\Bigg|_{p^2=m_N^2}
\!\!-I_\pi
\right)
\nonumber\\
&=& 1- \frac{3\,g_A^2}{4\,f^2}\,\frac{m_\pi^2 }{(4\pi)^2}
\left( -4 -3\,\ln \left(\frac{m_\pi^2}{m_N^2}\right)
+3\,\pi\,\frac{m_\pi}{m_N}
+{\mathcal O}\left( \frac{m^2_\pi}{m^2_N} \right)
\right) \,,
\label{def-zn}\end{aligned}$$ where in the last line of (\[def-zn\]) we applied our minimal chiral subtraction scheme (\[def-sub\]). The result (\[def-zn\]) agrees with the expressions obtained previously in [@gss; @kambor; @Becher]. Note that in (\[def-zn\]) we suppressed the contribution of the counter terms $\zeta_0, \zeta_D$ and $\zeta_F$.
It is illuminating to discuss the role played by the pionic tadpole contribution, $I_\pi$, from $V(w)$ in (\[pin-example\]) and from $J_{\pi N}(w)$ in (\[jpin-def\]). In the mass renormalization (\[mass-ren\]) both tadpole contributions cancel identically. If one dropped the pionic tadpole term $I_\pi$ in the effective interaction $V(w)$ one would find a mass renormalization $\Delta m_N \sim m_N\,I_\pi/f^2 \sim Q^2 $, in conflict with the expected minimal chiral power $\Delta m_N \sim Q^3$. Since we would like to evaluate the effective interaction $V$ in chiral perturbation theory we take this cancellation as the motivation to ’move’ all tadpole contributions from the loop function $J_{\pi N}(w)$ to the effective interaction kernel $V(w)$. We split the meson-baryon propagator $G={\mathcal Z}_N\,(G_R+\Delta G)$ into two terms $G_R$ and $\Delta G$ which leads to a renormalized tadpole-free loop function $G_R$. The renormalized effective potential $V_R(w)$ follows $$\begin{aligned}
\bar T_R= \frac{{\mathcal Z}_N}{1-V\cdot G}\cdot V=\frac{1}{1-V_R\cdot G_R}\cdot V_R \;,
\quad
V_R = \frac{{\mathcal Z}_N}{1-{\mathcal Z}_N\,V\cdot\Delta G}\cdot V \;,
\label{ren-v}\end{aligned}$$ where we introduced the renormalized scattering amplitude $\bar T_R = {\mathcal Z}^{\frac{1}{2}}_N \,\bar T\, {\mathcal Z}^{\frac{1}{2}}_N$ as implied by the LSZ reduction scheme. Now the cancellation of the pionic tadpole contributions is easily implemented by applying the chiral expansion to $V_R(w)$. The renormalized interaction kernel $V_R(w)$ is real by construction.
Our renormalization scheme is still incomplete[^9]. We need to specify how to absorb the remaining logarithmic divergence in $I_{\pi N}(\sqrt{s}\,)$. The strategy is to move all divergences from the unitarity loop function $J_{\pi N}(w)$ into the renormalized potential $V_R(w)$ via (\[ren-v\]). For the effective potential one may then apply the standard perturbative renormalization program. We impose the renormalization condition that the effective potential $V_R(w)$ matches the scattering amplitudes at a subthreshold energy $\sqrt{s}=\mu_S$: $$\begin{aligned}
\bar T_R (\mu_S) = V_R(\mu_S) \;.
\label{ren-cond}\end{aligned}$$ We argue that the choice for the subtraction point $\mu_S$ is rather well determined by the crossing symmetry constraint. In fact one is lead almost uniquely to the convenient point $\mu_S =m_N$. For the case of pion-nucleon scattering one observes that the renormalized effective interaction $V_R$ is real only if $m_N-m_\pi<\mu_S < m_N+m_\pi$ holds. The first condition reflects the s-channel unitarity cut with $\Im I_{\pi N}(\mu_S) =0 $ only if $\mu_S < m_N+m_\pi$. The second condition signals the u-channel unitarity cut. A particular convenient choice for the subtraction point is $\mu_S=m_N$, because it protects the nucleon pole term contribution. It leads to $$\begin{aligned}
\bar T_R (w)&=& V_R(w)+{\mathcal O}\Big(\left( \wslash+m_N\right)^0\Big)
\nonumber\\
&=&-\frac{3\,({\mathcal Z}_N\,g_A)^2}{4\,f^2}\,
\frac{4\,m^2_N }{\wslash+m_N}
+{\mathcal O}\Big(\left( \wslash+m_N\right)^0\Big) \,,
\label{pole-protection}\end{aligned}$$ if the renormalized loop function is subtracted in such a way that $I_{\pi N,R}(m_N) =0 $ and $I_{\pi N,R}'(m_N)=0$ hold.
Note that we insist on a minimal subtraction for the scalar loop function $I_{\pi N}(\sqrt{s}\,)$ ’inside’ the full loop function $J_{\pi N}(w)$. According to (\[ipin-analytic\]) one subtraction suffices to render $I_{\pi N}(\sqrt{s}\,)$ finite. A direct subtraction for $J_{\pi N}(w)$ would require an infinite subtraction polynomial which would not be specified by the simple renormalization condition (\[ren-cond\]). We stress that the double subtraction in $I_{\pi N}(\sqrt{s}\,)$ is necessary in order to meet the condition (\[ren-cond\]). Only with $I_{\pi N,R}'(m_N)=0$ the renormalized effective potential $V_R(w)$ in (\[pole-protection\]) represents the s-channel nucleon pole term in terms of the physical coupling constant ${\mathcal Z}_N\,g_A$ properly. For example, it is evident that if $V_R$ is truncated at chiral order $Q^2$ one finds ${\mathcal Z}_N=1$. The one-loop wave-function renormalization (\[def-zn\]) is needed if one considers the $Q^3$-terms in $V_R$.
We observe that the subtracted loop function $I_{\pi N}(\sqrt{s}\,)-I_{\pi N}(\mu_S)$ is in fact independent of the subtraction point to order $Q^2$ if one counted $\mu_S-m_N\sim Q^2$. In this case one derives from (\[ipin-expand\]) the expression $$\begin{aligned}
I_{\pi N}(\sqrt{s}\,)-I_{\pi N}(\mu_S)
&=&i\,\frac{\sqrt{\phi_{\pi N}}}{8\,\pi\,m_N}
+\frac{m_\pi}{16\,\pi\,m_N}
-\frac{\sqrt{s}-m_N}{16\,\pi^2\,m_N}\,\ln \left( \frac{m_\pi^2}{m_N^2}\right)
\nonumber\\
&+&\frac{\sqrt{\phi_{\pi N}}}{16\,\pi^2\,m_N}\,
\ln \left( \frac{\sqrt{s}-m_N+\sqrt{\phi_{\pi N}}}
{\sqrt{s}-m_N-\sqrt{\phi_{\pi N}}} \right)
+{\mathcal O}\left( Q^2 \right) \;,
\label{ipin-expand:b}\end{aligned}$$ where $\phi_{\pi N}= (\sqrt{s}-m_N)^2-m_\pi^2$. The result (\[ipin-expand:b\]) suggests that one may set up a systematic expansion scheme in powers of $\mu_S-m_N \sim Q^2$. The subtraction-scale independence of physical results then implies that all powers $(\mu_S-m_N)^n$ cancel except for the leading term with $n=0$. In this sense one may say that the scattering amplitude is independent of the subtraction point $\mu_S$. Note that such a scheme does not necessarily require a perturbative expansion of the scattering amplitude. It may be advantageous instead to restore that minimal $\mu_S$-dependence in the effective potential $V$, leading to a subtraction scale independent scattering amplitude at given order in $(\mu_S-m_N)$. Equivalently, it is legitimate to directly insist on the ’physical’ subtraction with $\mu_S=m_N$. There is a further important point to be made: we would reject a conceivable scheme in which the inverse effective potential $V^{-1}$ is expanded in chiral powers, even though it would obviously facilitate the construction of that minimal $\mu_S$-dependence. As examined in detail in [@nn-lutz] expanding the inverse effective potential requires a careful analysis to determine if the effective potential has a zero within its applicability domain. If this is the case one must reorganize the expansion scheme. Note that this is particularly cumbersome in a coupled-channel scenario where one must ensure that $\det V \neq 0$ holds. Since in the $SU(3)$ limit the Weinberg-Tomozawa interaction term, which is the first term in the chiral expansion of $V $, leads to $\det V_{WT}=0$ (see (\[WT-k\])) one should not pursue this path[^10].
We address an important aspect related closely to our renormalization scheme, the approximate crossing symmetry. At first, one may insist on either a strict perturbative scheme or an approach which performs a simultaneous iteration of the s- and u-channel in order to meet the crossing symmetry constraint. We point out, however, that a simultaneous iteration of the s- and u-channel is not required, if the s-channel iterated and u-channel iterated amplitudes match at subthreshold energies $\sqrt{s}\simeq \mu_S \simeq m_N $ to high accuracy. This is a sufficient condition in the chiral framework, because the overlap of the applicability domains of the s- and u-channel iterated amplitudes are restricted to a small matching window at subthreshold energies in any case. This will be discussed in more detail in section 4.3. Note that crossing symmetry is not necessarily observed in a cutoff regularized approach. In our scheme the meson-baryon scattering process is [*perturbative*]{} below the s and u-channel unitarity thresholds by construction and therefore meets the crossing symmetry constraints approximatively. [*Non-perturbative*]{} effects as implied by the unitarization are then expected at energies outside the matching window.
Before turning to the coupled channel formalism we return to the question of the convergence of standard $\chi$PT. The poor convergence of the standard $\chi$PT scheme in the $\bar KN$ sector is illustarted by comparing the effect of the iteration of the Weinberg-Tomozawa term in the $\bar KN$ and $\pi N$ channels.
Weinberg-Tomozawa interaction and convergence of $\chi$PT
---------------------------------------------------------
We consider the Weinberg-Tomozawa interaction term $K^{}_{WT}(\bar k,k;w)$ as the driving term in the Bethe-Salpeter equation. This is an instructive example because it exemplifies the non-perturbative nature of the strangeness channels and it also serves as a transparent first application of our renormalization scheme. The on-shell equivalent scattering amplitude $\bar T_{WT}(w)$ is $$\begin{aligned}
&& \!\! V_{WT}(w) =\left( 1-\frac{c}{4 f^2}\,I_L\right)^{-1}
\!\! \frac{c}{4 f^2} \left(2 \;\wslash -2\!\,M
-I_{LR}\,\frac{c}{4 f^2} \right)
\left(1-I_R\,\frac{c}{4 f^2} \right)^{-1},
\nonumber\\
&&\!\! \bar T_{WT}(w) = \frac{1}{1-V_{WT}(w)\,J(w)}\,V_{WT}(w) \;,\quad
K_{WT}(\bar k,k;w) = c^{}_{}\,\frac{\qslash+\barqslash}{4\,f^2}\;,
\label{wt-tadpole}\end{aligned}$$ where the Bethe-Salpeter scattering equation (\[BS-eq\]) was solved algebraically following the construction (\[k-decomp\],\[t-eff\]). Here we identify $K_R\cdot G \to I_R$, $G\cdot K_L \to I_L$ and $K_R\cdot G\cdot K_L \to I_{LR}$ with the mesonic tadpole loop $I_M$ (see (\[def-tadpole\])) with $I_L=I_R=I_M$ and $I_{LR}=(\wslash-M)\,I_{M}$. The meson-baryon loop function $J(w)$ is defined for the $\pi N$ system in (\[jpin-def\]). The coupling constants $c^{(I)}_{MB\to MB}$ specifies the strength of the Weinberg-Tomozawa interaction in a given meson-baryon channel with isospin $I$. Note that (\[wt-tadpole\]) is written in a way such that coupled channel effects are easily included by identifying the proper matrix structure of its building blocks. A detailed account of these effects will be presented in subsequent sections. We emphasize that the mesonic tadpole $I_M$ cannot be absorbed systematically in $f$, in particular when the coupled channel generalization of (\[wt-tadpole\]) with its non-diagonal matrix $c$ is considered. At leading chiral order $Q$ the tadpole contribution should be dropped in the effective interaction $V$ in any case.
It is instructive to consider the s-wave $\bar K N $ scattering lengths up to second order in the Weinberg-Tomozawa interaction vertex. The coefficients $c^{(0)}_{\bar K N\to \bar KN}=3$ and $c^{(0)}_{\bar K N\to \bar KN}=3$ and $c^{(1)}_{\bar K N\to \bar KN}=1$ lead to $$\begin{aligned}
4\,\pi \left(1+\frac{m_K}{m_N} \right) a_{\bar K N}^{(I=0)} &=&
\frac{3}{2}\,\frac{m_K}{f^2} \left(
1+\frac{3\,m^2_K}{16\,\pi^2\,f^2}
\left(\pi-\ln \left(\frac{m_K^2}{m_N^2}\right) \right)\right)+\cdots \;,
\nonumber\\
4\,\pi \left(1+\frac{m_K}{m_N} \right) a_{\bar K N}^{(I=1)} &=&
\frac{1}{2}\,\frac{m_K}{f^2} \left(
1+\frac{m^2_K}{16\,\pi^2\,f^2}
\left(\pi-\ln \left(\frac{m_K^2}{m_N^2}\right) \right)\right)+\cdots \;,
\label{wt-square-KN}\end{aligned}$$ where we included only the s-channel iteration effects following from (\[wt-tadpole\]). The loop function $I_{KN}(\sqrt{s})$ was subtracted at the nucleon mass and all tadpole contributions are dropped. Though the expressions (\[wt-square-KN\]) are incomplete in the $\chi $PT framework (terms of chiral order $Q^2$ and $Q^3$ terms are neglected) it is highly instructive to investigate the convergence property of a reduced chiral Lagrangian with the Weinberg-Tomozawa interaction only. According to (\[wt-square-KN\]) the relevant expansion parameter $m_K^2/(8\pi f^2) \simeq 1$ is about one in the $\bar KN$ sector. One observes the enhancement factor $2 \pi$ as compared to irreducible diagrams which would lead to the typical factor $ m^2_K/(4\pi \,f)^2$. The perturbative treatment of the Weinberg-Tomozawa interaction term is therefore unjustified and a change in approximation scheme is required. In the isospin zero $\bar KN$ system the Weinberg-Tomozawa interaction if iterated to all orders in the s-channel (\[wt-tadpole\]) leads to a pole in the scattering amplitude at subthreshold energies $\sqrt{s}< m_N+m_K$. This pole is a precursor of the $\Lambda(1405)$ resonance [@dalitz-1; @Siegel; @Kaiser; @Ramos; @Hirschegg].
![Real (l.h.s.) and imaginary (r.h.s.) part of the isospin zero s-wave $K^-$-nucleon scattering amplitude as it follows from the $SU(3)$ Weinberg-Tomozawa interaction term in a coupled channel calculation. We use $f = 93 $ MeV and identify the subtraction point with the $\Lambda(1116)$ mass.[]{data-label="fig:wt"}](fig1.eps){width="12cm"}
In Fig. \[fig:wt\] we anticipate our final result for the leading interaction term of the chiral $SU(3)$ Lagrangian density suggested by Tomozawa and Weinberg. If taken as input for the multi-channel Bethe-Salpeter equation, properly furnished with a renormalization scheme leading to a subtraction point close to the baryon octet mass, a rich structure of the scattering amplitude arises. Details for the coupled channel generalization of (\[wt-tadpole\]) are presented in the subsequent sections. Fig. \[fig:wt\] shows the s-wave solution of the multi-channel Bethe-Salpeter as a function of the kaon mass. For physical kaon masses the isospin zero scattering amplitude exhibits a resonance structure at energies where one would expect the $\Lambda(1405)$ resonance. We point out that the resonance structure disappears as the kaon mass is decreased. Already at a hypothetical kaon mass of $300$ MeV the $\Lambda(1405)$ resonance is no longer formed. Fig. \[fig:wt\] demonstrates that the chiral $SU(3)$ Lagrangian is necessarily non-perturbative in the strangeness sector. This confirms the findings of [@Kaiser; @Ramos]. In previous works however the $\Lambda(1405)$ resonance is the result of a fine tuned cutoff parameter which gives rise to a different kaon mass dependence of the scattering amplitude [@Ramos]. In our scheme the choice of subtraction point close to the baryon octet mass follows necessarily from the compliance of the expansion scheme with approximate crossing symmetry. Moreover, the identification of the subtraction point with the $\Lambda$-mass in the isospin zero channel protects the hyperon exchange s-channel pole contribution and therefore avoids possible pathologies at subthreshold energies.
We turn to the pion-nucleon sector. The chiral $SU(2)$ Lagrangian has been successfully applied to pion-nucleon scattering in standard chiral perturbation theory [@Bernard; @Meissner; @pin-q4]. Here the typical expansion parameter $m^2_\pi/(8\,\pi\,f^2) \ll 1$ characterizing the unitarization is sufficiently small and one would expect good convergence properties. The application of the chiral $SU(3)$ Lagrangian to pion-nucleon scattering on the other hand is not completely worked out so far. In the SU(3) scheme the $\pi N$ channel couples for example to the $K \Sigma $ channel. Thus the slow convergence of the unitarization in the $K \Sigma$ channel suggests to expand the interaction kernel rather than the scattering amplitude also in the strangeness zero channel. This may improve the convergence properties of the chiral expansion and extend its applicability domain to larger energies. Also, if the same set of parameters are to be used in the pion-nucleon and kaon-nucleon sectors the analogous partial resummation of higher order counter terms included by solving the Bethe-Salpeter equation should be applied. We illustrate such effects for the case of the Weinberg-Tomozawa interaction. With $c^{(\frac{3}{2})}_{\pi N\to \pi N}=-1$ and $c^{(\frac{3}{2})}_{\pi N\to K \Sigma}=-1$ the isospin three half s-wave pion-nucleon scattering lengths $a_{\pi N}^{(\frac{3}{2})}$ receive the typical correction terms $$\begin{aligned}
&&4\,\pi \left(1+\frac{m_\pi}{m_N} \right) a_{\pi N}^{(\frac{3}{2})} =
-\frac{m_\pi}{2\,f^2}\, \Bigg(
1-\frac{m^2_\pi}{16\,\pi^2\,f^2}
\left(\pi-\ln \left(\frac{m_\pi^2}{m_N^2}\right) \right)
\nonumber\\
&&\qquad \qquad - \frac{(m_\pi+m_K)^2}{32\,\pi^2\,f^2}
\Bigg( -1-\frac{1}{2}\,\ln \left(\frac{m_K^2}{m_\Sigma^2}\right)
\nonumber\\
&& \qquad \qquad
+\pi \left( \frac{3\,m_K}{4\,m_N}-\frac{m_\Sigma-m_N}{2\,m_K} \right)
+{\mathcal O}\left( Q^2\right)\Bigg)\Bigg)
+ {\mathcal O}\left( m_\pi^2\right)\;,
\label{wt-square-piN}\end{aligned}$$ where we again considered exclusively the unitary correction terms. Note that the ratio $(m_\Sigma -m_N)/m_K$ arises in (\[wt-square-piN\]), because we first expand in powers of $m_\pi$ and only then expand further with $m_\Sigma-m_N \sim Q^2$ and $m_K^2 \sim Q^2 $. The correction terms in (\[wt-square-piN\]) induced by the kaon-hyperon loop, which is subtracted at the nucleon mass, exemplify the fact that the parameter $f$ is renormalized by the strangeness sector and therefore must not be identified with the chiral limit value of $f$ as derived for the $SU(2)$ chiral Lagrangian. This is evident if one confronts the Weinberg-Tomozawa theorem of the chiral $SU(2)$ symmetry with (\[wt-square-piN\]). The expression (\[wt-square-piN\]) demonstrates further that this renormalization of $f$ appears poorly convergent in the kaon mass. Note in particular the anomalously large term $\pi \,m_K/m_N $. Hence it is advantageous to consider the partial resummation induced by a unitary coupled channel treatment of pion-nucleon scattering.
Partial-wave decomposition of the Bethe-Salpeter equation
---------------------------------------------------------
The Bethe-Salpeter equation (\[BS-eq\]) can be solved analytically for quasi-local interaction terms which typically arise in the chiral Lagrangian. The scattering equation is decoupled by introducing relativistic projection operators ${Y}^{(\pm)}_n(\bar q,q;w)$ with good total angular momentum: $$\begin{aligned}
&&{Y}^{(\pm )}_n(\bar q,q;w)=\frac{1}{2}\,\Bigg(\frac{\wslash}{\sqrt{w^2}}\pm 1\Bigg)\,
\bar Y_{n+1}(\bar q,q;w)
\nonumber\\
&&\;\;\;\;\;\;\;\;\;\;-\frac{1}{2}\,\Bigg( \barqslash -\frac{w\cdot \bar q}{w^2}\,\wslash \Bigg)
\Bigg(\frac{\wslash}{\sqrt{w^2}} \mp 1\Bigg)\,
\Bigg( \qslash -\frac{w\cdot q}{w^2}\,\wslash \Bigg)
\bar Y_{n}(\bar q,q;w)\;,
\nonumber\\
&&\bar Y_{n}(\bar q,q;w)= \sum_{k=0}^{[(n-1)/2]}\,\frac{(-)^k\,(2\,n-2\,k) !}{2^n\,k !\,(n-k)
!\,(n-2\,k -1) !}\,Y_{\bar q \bar q}^{k}\,Y_{\bar q q}^{n-2\,k-1}\,Y_{q q}^{k}\;,
\nonumber\\
&&Y_{\bar q \bar q}=\frac{(w\cdot \bar q )\,(\bar q\cdot w)}{w^2}
-\bar q \cdot \bar q
\;,\;\;\;
Y_{q q}=\frac{(w\cdot q )\,( q\cdot w)}{w^2} -q \cdot q \;,
\nonumber\\
&&Y_{\bar q q}=\frac{(w\cdot \bar q )\,(q\cdot w)}{w^2} -\bar q \cdot q \;.
\label{cov-proj}\end{aligned}$$ For the readers convenience we provide the leading order projectors ${Y}_n^{(\pm)}$ relevant for the $J={\textstyle{1\over 2}}$ and $J={\textstyle{3\over 2}}$ channels explicitly: $$\begin{aligned}
{Y}_0^{(\pm )}(\bar q,q;w) &=& \frac{1}{2}\,\left( \frac{\wslash}{\sqrt{w^2}}\pm 1 \right)\;,
\nonumber\\
{Y}_1^{(\pm)}(\bar q,q;w) &=& \frac{3}{2}\,\left( \frac{\wslash}{\sqrt{w^2}}\pm 1 \right)
\left(\frac{(\bar q \cdot w )\,(w \cdot q)}{w^2} -\big( \bar q\cdot q\big)\right)
\nonumber\\
&-&\frac{1}{2}\,\Bigg( \barqslash -\frac{w\cdot \bar q}{w^2}\,\wslash \Bigg)
\Bigg(\frac{\wslash}{\sqrt{w^2}}\mp 1\Bigg)\,
\Bigg( \qslash -\frac{w\cdot q}{w^2}\,\wslash \Bigg)\;.
\label{}\end{aligned}$$
The objects ${Y}^{(\pm)}_n(\bar q,q;w)$ are constructed to have the following convenient property: Suppose that the interaction kernel $K$ in (\[BS-eq\]) can be expressed as linear combinations of the ${Y}^{(\pm)}_n(\bar q,q;w)$ with a set of coupling functions $V^{(\pm)}(\sqrt{s}\,, n)$, which may depend on the variable $s$, $$\begin{aligned}
K(\bar k ,k ;w ) &=& \sum_{n=0}^\infty \left(
V^{(+)}(\sqrt{s};n)\,{Y}^{(+)}_n(\bar q,q;w)
+V^{(-)}(\sqrt{s};n)\,{Y}^{(-)}_n(\bar q,q;w) \right) \, ,
\label{k-sum}\end{aligned}$$ with $ w = p+q$, $ k= (p-q)/2 $ and $\bar k =(\bar p-\bar q)/2 $. Then in a given isospin channel the unique solution reads $$\begin{aligned}
&& T(\bar k ,k ;w ) = \sum_{n=0}^\infty \left(
M^{(+)}(\sqrt{s};n )\,{Y}^{(+)}_n(\bar q,q;w)
+ M^{(-)}(\sqrt{s};n)\,{Y}^{(-)}_n(\bar q,q;w)\right) \;,
\nonumber\\
&& M^{(\pm)}(\sqrt{s};n) = \frac{V^{(\pm )}(\sqrt{s};n)}{1-V^{(\pm)}(\sqrt{s};n)
\,J^{(\pm)}_{\pi N }(\sqrt{s};n)} \;,
\label{t-sum}\end{aligned}$$ with a set of divergent loop functions $J^{(\pm)}_{\pi N }(\sqrt{s};n)$ defined by $$\begin{aligned}
J^{(\pm)}_{\pi N}(\sqrt{s};n)\,{Y}^{(\pm)}_n(\bar q,q;w)
&&= -i\int \frac{d^4l}{(2\pi)^4}\,{Y}^{(\pm)}_n(\bar q,l;w)\,
S_N(w-l)\,
\nonumber\\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\times D_\pi(l)\,{Y}^{(\pm)}_n(l,q;w)\;.
\label{jpin-n-def}\end{aligned}$$ We underline that the definition of the loop functions in (\[jpin-n-def\]) is non trivial, because it assumes that $Y_n^{(\pm )}\cdot G \cdot Y_n^{(\pm )} $ is indeed proportional to $Y_n^{(\pm)}$. An explicit derivation of this property, which is in fact closely linked to our renormalization scheme, is given in Appendix C. We recall that the loop functions $J^{(\pm)}_{\pi N}(\sqrt{s};n)$, which are badly divergent, have a finite and well-defined imaginary part $$\begin{aligned}
\Im\,J^{(\pm)}_{\pi N }(\sqrt{s};n) &=& \frac{p^{2\,n+1}_{\pi N}}{8\,\pi\,\sqrt{s}}
\left( \frac{\sqrt{s}}{2}+ \frac{m_N^2-m_\pi^2}{2\,\sqrt{s}}\pm m_N \right)\;.
\label{}\end{aligned}$$ We specify how to renormalize the loop functions. In dimensional regularization the loop functions can be written as linear combinations of scalar one loop functions $I_{\pi N}(\sqrt{s}\,)$, $I_\pi$, $I_N $ and $I^{(n)}$, $$\begin{aligned}
I^{(n)}=i\,\int \frac{d^4l}{(2\pi)^4}\,\Big(l^2 \Big)^n\;,
\label{tadpole:a}\end{aligned}$$ According to our renormalization procedure we drop $I^{(n)}$, the tadpole contributions $I_\pi$, $I_N $ and replace $I_{\pi N}(s) $ by $I_{\pi N}(\sqrt{s}\,)-I_{\pi N}(\mu_S )$. This leads to $$\begin{aligned}
J^{(\pm)}_{\pi N }(\sqrt{s}; n) &=&
p_{\pi N}^{2\,n}(\sqrt{s}\,)
\left( \frac{\sqrt{s}}{2}+ \frac{m_N^2-m_\pi^2}{2\,\sqrt{s}}\pm m_N \right)
\Delta I_{\pi N}(\sqrt{s}\,)\;,
\nonumber\\
\Delta I_{\pi N}(\sqrt{s}\,) &=& I_{\pi N}(\sqrt{s}\,)-I_{\pi N}(\mu_S ) \;,
\label{result-loop}\end{aligned}$$ with the master loop function $I_{\pi N}(\sqrt{s}\,)$ and $p_{\pi N}(\sqrt{s}\,)$ given in (\[ipin-analytic\]). In the center of mass frame $p_{\pi N}$ represents the relative momentum. We emphasize that the loop functions $J^{(\pm)}_{\pi N }(\sqrt{s}; n)$ are renormalized in accordance with (\[ren-cond\]) and (\[ren-v\]) where $\mu_S=m_N$[^11]. This leads to tadpole-free loop functions and also to $M^{(\pm )}(m_N,n)=V^{(\pm )}(m_N,n)$. The behavior of the loop functions $J^{(\pm )}_{\pi N}(\sqrt{s},n)$ close to threshold $$\begin{aligned}
\Im \,J^{(+)}_{\pi N}(\sqrt{s},n) \sim p_{\pi N}^{2\,n+1} \;, \qquad
\Im \,J^{(-)}_{\pi N}(\sqrt{s},n) \sim p_{\pi N}^{2\,n+3} \,,
\label{}\end{aligned}$$ already tells the angular momentum, $l$, of a given channel with $l=n$ for the $'+'$ and $l=n+1$ for the $'-'$ channel.
The Bethe-Salpeter equation (\[t-sum\]) decouples into reduced scattering amplitudes $M^{(\pm)}(\sqrt{s},n)$ with well-defined angular momentum. In order to unambiguously identify the total angular momentum $J$ we recall the partial-wave decomposition of the on-shell scattering amplitude $T$ [@Landoldt]. The amplitude $T$ is decomposed into invariant amplitudes $F^{(I)}_\pm(s,t)$ carrying good isospin $I$ $$\begin{aligned}
T &=& \sum_{I}\,\left(
\frac{1}{2}\,\Bigg(\frac{\wslash}{\sqrt{w^2}}+1
\Bigg)\, F^{(I)}_+(s,t)+ \frac{1}{2}\,\Bigg(\frac{\wslash}{\sqrt{w^2}}-1
\Bigg)\,F^{(I)}_-(s,t)\right) P_{I}
\label{}\end{aligned}$$ where $s= (p+q)^2=w^2$ and $t=(\bar q-q)^2$ and $P_I$ are the isospin projectors introduced in (\[iso:proj\]). Note that the choice of invariant amplitudes is not unique. Our choice is particularly convenient to make contact with the covariant projection operators (\[cov-proj\]). For different choices, see [@Landoldt]. The amplitudes $F_{\pm}(s,t)$ are decomposed into partial-wave amplitudes $f^{(l)}_{J=l\pm \frac{1}{2}}(\sqrt{s}\,)$ [@Landoldt] $$\begin{aligned}
F_+(s,t) &=& \frac{8\,\pi\,
\sqrt{s}}{E+m_N}\,\sum_{n=1}^\infty
\,\Big( f_{J=n+\frac{1}{2}}^{(n-1)}(\sqrt{s}\,)-f_{J=n- \frac{1}{2}}^{(n+1)}(\sqrt{s}\,)\Big)\,P'_n(\cos \theta)
\;,
\nonumber\\
F_-(s,t) &=&\frac{8\,\pi\,
\sqrt{s}}{E-m_N}\,\sum_{n=1}^\infty
\,\Big( f_{J=n-\frac{1}{2}}^{(n)}(\sqrt{s}\,)-f_{J=n+ \frac{1}{2}}^{(n)}(\sqrt{s}\,)\Big)\,P'_n(\cos \theta)\;,
\nonumber\\
P'_n (\cos \theta)&=& \sum_{k=0}^{[(n-1)/2]}\,\frac{(-)^k\,(2\,n-2\,k) !}{2^n\,k !\,(n-k)
!\,(n-2\,k -1) !}\,\Big(\cos \theta\Big)^{n-2\,k-1} \,,
\label{t-on-decomp}\end{aligned}$$ where $[n/2]= (n-1)/2$ for $n$ odd and $[n/2]= n/2$ for $n$ even. $P'_n(z)$ is the derivative of the Legendre polynomials. In the center of mass frame $E$ represents the nucleon energy and $\theta$ the scattering angle: $$\begin{aligned}
&&E=\frac{1}{2}\,\sqrt{s}+\frac{m_N^2-m_\pi^2}{2\,\sqrt{s}} \;,\;\;\;
t=(\bar q-q)^2=-2\,p_{\pi N}^2\,\Big(1-\cos \theta \Big) \;.
\label{}\end{aligned}$$ The unitarity condition formulated for the partial-wave amplitudes $f_J^{(l)}$ leads to their representation in terms of the scattering phase shifts $\delta^{(l)}_J$ $$\begin{aligned}
&&p_{\pi N}\,f^{(l)}_{J=l\pm \frac{1}{2}}(\sqrt{s}\,) =
\frac{1}{2\,i}\left( e^{2\,i\,\delta^{(l)}_{J=l\pm \frac{1}{2}} (s)}-1 \right)
=\frac{1}{\cot \delta^{(l)}_{J=l\pm \frac{1}{2}} (s)-i}\;.
\label{}\end{aligned}$$ One can now match the reduced amplitudes $M^{(\pm)}_n(s)$ of (\[t-sum\]) and the partial-wave amplitudes $f^{(l)}_{J=l\pm \frac{1}{2}}(s)$ $$\begin{aligned}
f^{(l)}_{J=l\pm\frac{1}{2}}(\sqrt{s}\,) &=& \frac{p^{2\,J-1}_{\pi N}}{8\,\pi\,\sqrt{s}}
\left( \frac{\sqrt{s}}{2}+\frac{m_N^2-m_\pi^2}{2\,\sqrt{s}} \pm m_N\right)
M^{(\pm )}_{}(\sqrt{s},J-{\textstyle{1\over2}})\;.
\label{match}\end{aligned}$$
It is instructive to consider the basic building block $\bar Y_n(\bar q,q;w)$ of the covariant projectors ${Y}_n^{(\pm)}(\bar q,q;w)$ in (\[cov-proj\]) and observe the formal similarity with $P'_n(\cos
\theta )$ in (\[t-on-decomp\]). In fact in the center of mass frame with $w_{cm}=(\sqrt{s},0)$ one finds $p_{\pi N}^{2\,n-2}\,P'_n(\cos \theta )=Y_n(\bar q,q;w_{cm})$. This observation leads to a straightforward proof of (\[t-sum\]) and (\[jpin-n-def\]). It is sufficient to prove the orthogonality of the projectors ${Y}_n^{(\pm)}(\bar q,q;w)$ in the center of mass frame, because the projectors are free from kinematical singularities. One readily finds that the imaginary part of the unitary products ${Y}^{(\pm)}_n\,G\,{Y}^{(\pm)}_m$ vanish unless both projectors are the same. It follows that the unitary product of projectors which are expected to be orthogonal can at most be a real polynomial involving the tadpole functions $I_\pi, I_N$ and $I^{(n)}$. Then our renormalization procedure as described in section 3.2 leads to (\[t-sum\]) and (\[jpin-n-def\]). We emphasize that our argument relies crucially on the fact that the projectors ${Y}_n^{(\pm)}(\bar q,q;w)$ are free from kinematical singularities in $q$ and $\bar q$. This implies in particular that the object $Y_n(\bar q,q;w)$ must not be identified with $p_{\pi N}^{2\,n-2}\,P'_n(\cos \theta )$ as one may expect naively.
We return to the assumption made in (\[k-sum\]) that the interaction kernel $K$ can be decomposed in terms of the projectors $Y_n^{(\pm)}$. Of course this is not possible for a general interaction kernel $K$. We point out, however, that the on-shell equivalent interaction kernel $V$ in (\[t-eff\]) can be decomposed into the $Y_n^{(\pm)}$ if the on-shell irreducible kernel $\bar K$ and the on-shell reducible kernels $K_{L,R}$ and $K_{LR}$ of (\[k-decomp\]) are identified properly. The on-shell irreducible kernel $\bar K$ of (\[k-decomp\]) is defined by decomposing the interaction kernel according to $$\begin{aligned}
&&\bar K ^{(I)}(\bar q, q;w)=
\sum_{n=0}^\infty \left(\bar K^{(I)}_+(\sqrt{s};n)\,{Y}_n^{(+)}(\bar q, q;w)
+\bar K^{(I)}_-(\sqrt{s};n)\,{Y}_n^{(-)}(\bar q, q;w)\right) \;,
\nonumber\\
&&\bar K^{(I )}_\pm(\sqrt{s};n)= \int_{-1}^1 \frac{dz}{2} \,
\frac{K^{(I)}_\pm(s,t)}{\,p^{2\,n}_{\pi N}}\,P_n(z)
\nonumber\\
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \int_{-1}^1 \frac{dz}{2} \,
\left( \frac{1}{2}\,\sqrt{s}+\frac{m_N^2-m_\pi^2}{2\,\sqrt{s}}\mp m_N\right)^2
\frac{K^{(I)}_\mp(s,t)}{p^{2\,n+2}_{\pi N}}\,P_{n+1}(z) \;,
\label{bark-def-pin}\end{aligned}$$ where $t= -2\,p_{\pi N}^2\,(1-x)$ and $K_\pm^{(I)}(s,t)$ follows from the decomposition of the interaction kernel $K$: $$\begin{aligned}
&&K^{(I)}(\bar q, q; w) \Big|_{\rm{on-shell}}=
\frac{1}{2}\,\Bigg(\frac{\wslash}{\sqrt{w^2}}+1
\Bigg)\, K_+^{(I)}(s,t) + \frac{1}{2}\,\Bigg(\frac{\wslash}{\sqrt{w^2}}-1
\Bigg)\,K_-^{(I)}(s,t) \;.
\label{}\end{aligned}$$ Then $K-\bar K $ is on-shell reducible by construction and therefore can be decomposed into $K_L,K_R,K_{LR}$. Note that it does not yet follow that the induced effective interaction $V$ can be decomposed into the $Y_n^{(\pm)}$. This may need an iterative procedure in particular when the interaction kernel shows non-local structures induced for example by a $t$-channel meson-exchange. The starting point of the iteration is given with $ K_0 = K$ and $V_0 = \bar K $ as defined via (\[bark-def-pin\]). Then $K_{n+1} = V[K_n]$, where $V[K_n]$ is defined in (\[v-eff\]) with respect to $\bar K_n$ as given in (\[bark-def-pin\]). The effective interaction $V$ is then identified with $V=^{\rm \;\;\;\,lim}_{\;n \to \infty}\, \bar K_n $. In our work we will not encounter this complication, because the effective interaction kernel is treated to leading orders of chiral perturbation theory.
$SU(3)$ coupled-channel dynamics
================================
The Bethe-Salpeter equation (\[BS-eq\]) is readily generalized for a coupled-channel system. The chiral $SU(3)$ Lagrangian with baryon octet and pseudo-scalar meson octet couples the $\bar K N$ system to five inelastic channels $\pi \Sigma $, $\pi \Lambda $, $\eta \Lambda$, $\eta \Sigma$ and $K \Xi $ and the $\pi N$ system to the three channels $K \Sigma$, $\eta N$ and $K\Lambda$. The strangeness plus one sector with the $K N$ channel is treated separately in the next section when discussing constraints from crossing symmetry. For simplicity we assume in the following discussion good isospin symmetry. Isospin symmetry breaking effects are considered in Appendix D. In order to establish our convention consider for example the two-body meson-baryon interaction terms in (\[two-body\]). They can be rewritten in the following form $$\begin{aligned}
{\mathcal L}(\bar k ,k ;w)&=&
\sum_{I=0,\frac{1}{2},1,\frac{3}{2}}\,R^{(I)\,\dagger }(\bar q,\bar p)\,\gamma_0
\,K^{(I)}(\bar k ,k ;w )\,R^{(I)}(q,p) \,,\;\;\;\;
\nonumber\\
R^{(0)}&=& \left(
\begin{array}{c}
\textstyle{1\over\sqrt{2}}\,K^\dagger\,N \\
\textstyle{1\over\sqrt{3}}\,\vec{\pi}_c \; \vec{\Sigma} \\
\eta_c \,\Lambda\\
\textstyle{1\over\sqrt{2}}\,K^t\,i\,\sigma_2\,\Xi
\end{array}
\right) \;,\;\;\;
\vec{R}^{(1)}= \left(
\begin{array}{c}
\textstyle{1\over\sqrt{2}}\,K^\dagger\,\vec{\sigma}\,N \\
\textstyle{1\over i\sqrt{2}}\,\vec{\pi}_c \, \times \vec{\Sigma} \\
\vec{\pi}_c \,\Lambda\\
\eta_c \,\vec{\Sigma} \\
\textstyle{1\over\sqrt{2}}\,K^t\,i\,\sigma_2\,\vec{\sigma}\,\Xi
\end{array}
\right) \;,
\nonumber\\
R^{(\frac{1}{2})} &=&
\left(
\begin{array}{c}
\textstyle{1\over\sqrt{3}}\,\pi_{c} \cdot \sigma\,N \\
\textstyle{1\over\sqrt{3}}\,\Sigma \cdot \sigma \,K \\
\eta_c \,N \\
K\,\Lambda
\end{array}
\right) \;,\;\;\;\;
R^{(\frac{3}{2})}= \left(
\begin{array}{c}
\pi_c \cdot S\,N \\
\Sigma \cdot S \,K
\end{array}
\right) \;,
\label{r-def}\end{aligned}$$ where $R(q,p)$ in (\[r-def\]) is defined by $R(q,p)= \int d^4x \, d^4y\,e^{-i\,q\,x-i\,p\,y}\,\Phi(x)\,B(y)$, and $\Phi(x)$ and $B(y)$ denoting the meson and baryon fields respectively. In (\[r-def\]) we decomposed the pion field $\vec \pi = \vec \pi_c+\vec \pi_c^\dagger $ and the eta field $\eta = \eta_c+\eta_c^\dagger$[^12]. Also we apply the isospin decomposition of (\[field-decomp\]). The isospin $1/2$ to $3/2$ transition matrices $S_i$ in (\[r-def\]) are normalized by $S^\dagger_i\,S_j=\delta_{ij}-\sigma_i\,\sigma_j/3$. The Lagrangian density ${\mathcal L}(x)$ in coordinate space is related to its momentum space representation through $$\begin{aligned}
\int d^4x\,{\mathcal L}(x) &=& \int \frac{d^4k}{(2\,\pi)^4}\,
\frac{d^4\bar k}{(2\,\pi)^4}\,\frac{d^4w}{(2\,\pi)^4}\,
{\mathcal L}(\bar k ,k ;w)\;.
\label{lag-momentum}\end{aligned}$$ The merit of the notation (\[r-def\]) is threefold. Firstly, the phase convention for the isospin states is specified. Secondly, it defines the convention for the interaction kernel $K$ in the Bethe-Salpeter equation. Last it provides also a convenient scheme to read off the isospin decomposition for the interaction kernel $K$ directly from the interaction Lagrangian (see Appendix A). The coupled channel Bethe-Salpeter matrix equation reads $$\begin{aligned}
T^{(I)}_{ab}(\bar k ,k ;w ) &=& K^{(I)}_{ab}(\bar k ,k ;w )
+\sum_{c,d}\int\!\! \frac{d^4l}{(2\pi)^4}\,K^{(I)}_{ac}(\bar k , l;w )\,
G^{(I)}_{cd}(l;w)\,T^{(I)}_{db}(l,k;w )\;,
\nonumber\\
G^{(I)}_{cd}(l;w)&=&-i\,D_{\Phi(I,d)}(\half\,w-l)\,S_{B(I,d)}(
\half\,w+l)\,\delta_{cd} \,,
\label{BS-coupled}\end{aligned}$$ where $D_{\Phi(I,d)}(q)$ and $S_{B(I,d)}(p)$ denote the meson propagator and baryon propagator respectively for a given channel $d$ with isospin $I$. The matrix structure of the coupled-channel interaction kernel $K_{ab}(\bar k ,k ;w )$ is defined via (\[r-def\]) and $$\begin{aligned}
&&\Phi(0,a)=(\bar K,\pi,\eta , K)_a\;,\;\;\;\;\;\;\ B(0,a)=(N,\Sigma,\Lambda,\Xi)_a\;,
\nonumber\\
&&\Phi(1,a)=(\bar K,\pi,\pi,\eta , K)_a\;,\;\;\ B(1,a)=(N,\Sigma,\Lambda,\Sigma ,\Xi)_a\;,
\nonumber\\
&&\Phi({\textstyle{1\over 2}},a)=(\pi,K, \eta, K)_a\;,\;\;\;\;\;\;\
B({\textstyle{1\over 2}},a)=(N,\Sigma,N, \Lambda )_a\;,
\nonumber\\
&&\Phi({\textstyle{3\over 2}},a)=(\pi,K)_a\;,\;\;\qquad \quad \;\;
B({\textstyle{3\over 2}},a)=(N,\Sigma )_a\;.
\label{def-channel}\end{aligned}$$ We proceed and identify the on-shell equivalent coupled channel interaction kernel $V$ of (\[v-eff\]). At leading chiral orders it is legitimate to identify $V^{(I)}_{ab}$ with $\bar K^{(I)}_{ab}$ of (\[k-decomp\]), because the loop corrections in (\[v-eff\]) are of minimal chiral power $Q^3$ (see (\[q-rule\])). To chiral order $Q^3$ the interaction kernel receives additional terms from one loop diagrams involving the on-shell reducible interaction kernels $K_{L,R}$ as well as from irreducible one-loop diagrams. In the notation of (\[k-decomp\]) one finds $$\begin{aligned}
V = \bar K + K_R \cdot G \cdot \bar K +\bar K\,\cdot G \cdot K_L +
K_R \cdot G \cdot K_L +{\mathcal O}\left(Q^4 \right) \;.
\label{v-idef}\end{aligned}$$ Typically the $Q^3$ terms induced by $K_{L,R}$ in (\[v-idef\]) are tadpoles (see e.g. (\[pin-example\],\[wt-tadpole\])). The only non-trivial contribution arise from the on-shell reducible parts of the u-channel baryon octet terms. However, by construction, those contributions have the same form as the irreducible $Q^3$ loop-correction terms of $\bar K$. In particular they do not show the typical enhancement factor of $2 \pi$ associated with the s-channel unitarity cuts. In the large $N_c$ limit all loop correction terms to $V$ are necessarily suppressed by $1/N_c$. This follows, because any hadronic loop function if visualized in terms of quark-gluon diagrams involves at least one quark-loop, which in turn is $1/N_c$ suppressed [@Hooft; @Witten]. Thus it is legitimate to take $V = \bar K$ in this work.
We do include those correction terms of suppressed order in the $1/N_c$ expansion which are implied by the physical baryon octet and decuplet exchange contributions. Individually the baryon exchange diagrams are of forbidden order $N_c$. Only the complete large $N_c$ baryon ground state multiplet with $J=({\textstyle {1 \over 2}}, ...,{\textstyle {N_c\over 2}} )$ leads to an exact cancellation and a scattering amplitude of order $N_c^0$ [@Manohar]. However this cancellation persists only in the limit of degenerate baryon octet $\m0_{[8]}$ and decuplet mass $\m0_{[10]}$. With $m_\pi < \m0_{[10]}-\m0_{[8]}$ the cancellation is incomplete and thus leads to an enhanced sensitivity of the scattering amplitude to the physical baryon-exchange contributions. Therefore one should sum the $1/N_c$ suppressed contributions of the form $(\m0_{[10]}-\m0_{[8]})^n/m_\pi^n \sim N_c^{-n}$. We take this into account by evaluating the baryon-exchange contributions to subleading chiral orders but avoid the expansion in either of $(\m0_{[10]}-\m0_{[8]})/m_\pi $ or $m_\pi/(\m0_{[10]}-\m0_{[8]})$. We also include the $SU(3)$ symmetry-breaking counter terms of the 3-point meson-baryon vertices and the quasi-local two-body counter terms of chiral order $Q^3$ which are leading in the $1/N_c$ expansion. Note that the quasi-local two-body counter terms of large chiral order are not necessarily suppressed by $1/N_c$ relatively to the terms of low chiral order. This is plausible, because for example a t-channel vector-meson exchange, which has a definite large $N_c$ scaling behavior, leads to contributions in all partial waves. Thus quasi-local counter terms with different partial-wave characteristics may have identical large $N_c$ scaling behavior even though they carry different chiral powers. Finally we argue that it is justified to perform the partial $1/N_c$ resummation of all reducible diagrams implied by solving the Bethe-Salpeter equation (\[t-eff\]). In section 3 we observed that reducible diagrams are typically enhanced close to their unitarity threshold. The typical enhancement factor of $2 \pi$ per unitarity cut, measured relatively to irreducible diagrams (see (\[wt-square-KN\])), is larger than the number of colors $N_c=3$ of our world.
By analogy with (\[t-sum\]) the coupled-channel scattering amplitudes $T^{(I)}_{ab} $ are decomposed into their on-shell equivalent partial-wave amplitudes $M^{(I,\pm)}_{ab} $ $$\begin{aligned}
\bar T^{(I)}_{ab}(\bar k,k; w)&=& \sum_{n=0}^\infty\,M^{(I,+)}_{ab}(\sqrt{s};n)\,{Y}_n^{(+)}(\bar q, q;w)
\nonumber\\
&+&\sum_{n=0}^\infty\,M^{(I,-)}_{ab}(\sqrt{s};n)\,{Y}_n^{(-)}(\bar q, q;w) \;,
\label{tttt}\end{aligned}$$ where $k=\! {\textstyle{1\over 2}}\,w -q$ and $\bar k=\! {\textstyle{1\over 2}}\,w -\bar q$ and $s= w_\mu\, w^\mu$. The covariant projectors ${Y}_n^{(\pm )}(\bar q, q;w)$ were defined in (\[cov-proj\]). Expressions for the differential cross sections given in terms of the partial-wave amplitudes $M_{ab}^{(\pm)}$ can be found in Appendix F. The form of the scattering amplitude (\[tttt\]) follows, because the effective interaction kernel $V^{(I)}_{ab}$ of (\[v-eff\]) is decomposed accordingly $$\begin{aligned}
V^{(I)}_{ab}(\bar k,k;w)&=&
\sum_{n=0}^\infty\,V^{(I,+)}_{ab}(\sqrt{s};n)\,{Y}_n^{(+)}(\bar q, q;w)
\nonumber\\
&+&\sum_{n=0}^\infty\,V^{(I,-)}_{ab}(\sqrt{s};n)\,{Y}_n^{(-)}(\bar q, q;w) \;.
\label{not-def}\end{aligned}$$ The coupled-channel Bethe-Salpeter equation (\[BS-coupled\]) reduces to a convenient matrix equation for the effective interaction kernel $V^{(I)}_{ab}$ and the invariant amplitudes $M^{(I,\pm)}_{ab} $ $$\begin{aligned}
M^{(I,\pm)}_{ab}(\sqrt{s};n) &=& V^{(I,\pm )}_{ab}(\sqrt{s};n)
\nonumber\\
&+& \sum_{c,d}
V^{(I,\pm)}_{ac}(\sqrt{s};n)\,J^{(I,\pm )}_{cd}(\sqrt{s};n)\,M^{(I,\pm)}_{db}(\sqrt{s};n) \;,
\label{}\end{aligned}$$ which is readily solved with: $$\begin{aligned}
M^{(I,\pm)}_{ab}(\sqrt{s};n)&=& \Bigg[\left( 1- V^{(I,\pm )}_{}(\sqrt{s};n)\,J^{(I,\pm )}_{}(\sqrt{s};n)\right)^{-1}
V^{(I,\pm )}_{}(\sqrt{s};n)\Bigg]_{ab} \,.
\label{}\end{aligned}$$ It remains to specify the coupled-channel loop matrix function $J^{(I,\pm )}_{ab} \!\! \sim \delta_{a b }$, which is diagonal in the coupled-channel space. We write $$\begin{aligned}
&& J^{(I,\pm )}_{aa}(\sqrt{s};n)=
\left( \frac{\sqrt{s}}{2}+ \frac{m_{B(I,a)}^2
-m_{\Phi(I,a)}^2}{2\,\sqrt{s}}\pm m_{B(I,a)} \right)
\Delta I^{(k)}_{\Phi(I,a)\,B(I,a)}(\sqrt{s}\, )
\nonumber\\
&& \qquad \qquad \qquad \! \times
\left(\frac{s}{4}-\frac{m_{B(I,a)}^2+m_{\Phi(I,a)}^2}{2}
+\frac{\big(m_{B(I,a)}^2-m_{\Phi(I,a)}^2\big)^2}{4\,s} \right)^n
\;,
\nonumber\\
&& \Delta I^{(k)}_{\Phi(I,a)\,B(I,a)}(\sqrt{s}\,) = I_{\Phi(I,a)\,B(I,a)}
(\sqrt{s}\,)
\nonumber\\
&& \qquad \qquad \qquad \quad -
\sum_{l=0}^k\,\frac{1}{l}\left(\frac{\partial }{\partial \,\sqrt{s}} \right)^l\,
\Bigg|_{\sqrt{s}= \mu_S}\,I_{\Phi(I,a)\,B(I,a)}(\sqrt{s}\,)\;.
\label{result-loop:ab}\end{aligned}$$ The index $a$ labels a specific channel consisting of a meson-baryon pair of given isospin ($\Phi(I,a)$, $B(I,a)$). In particular $m_{\Phi(I,a)}$ and $m_{B(I,a)}$ denote the empirical isospin averaged meson and baryon octet mass respectively. The scalar master-loop integral, $I_{\Phi(I,a)\,B(I,a)}(\sqrt{s}\,)$, was introduced explicitly in (\[ipin-analytic\]) for the pion-nucleon channel. Note that we do not use the expanded form of (\[ipin-expand\]). The subtraction point $\mu_S^{(I)}$ is identified with $\mu_S^{(0)}\!=\!m_\Lambda$ and $\mu_S^{(1)}\!=\!m_\Sigma$ in the isospin zero and isospin one channel respectively so as to protect the hyperon s-channel pole structures. Similarly we use $\mu_S^{(\frac{1}{2})}=\mu_S^{(\frac{3}{2})}=m_N$ in the pion-nucleon sector. We emphasize that in the p-wave loop functions $J^{(-)}(\sqrt{s},0)$ and $J^{(+)}(\sqrt{s},1)$ we perform a double subtraction of the internal master-loop function with $k=1$ in (\[result-loop:ab\]) whenever a large $N_c$ baryon ground state manifests itself with a s-channel pole contribution in the associated partial-wave scattering amplitude. In all remaining channels we use $k=0$ in (\[result-loop:ab\]). This leads to consistency with the renormalization condition (\[ren-cond\]), in particular in the large $N_c$ limit with $m_{[8]}=m_{[10]}$.
We proceed by identifying the on-shell irreducible interaction kernel $\bar K^{(I)}_{ab}$ in (\[v-idef\]). The result (\[bark-def-pin\]) is generalized to the case of inelastic scattering. This leads to the matrix structure of the interaction kernel $K_{ab}$ defined in (\[r-def\]). In a given partial wave the effective interaction kernel $\bar K^{(I,\pm )}_{ab}(\sqrt{s};n)$ reads $$\begin{aligned}
&&\bar K ^{(I)}_{ab}=
\sum_{n=0}^\infty \left(\bar K^{(I,+)}_{ab}(\sqrt{s};n)\,{Y}_n^{(+)}(\bar q, q;w)
+\bar K^{(I,-)}_{ab}(\sqrt{s};n)\,{Y}_n^{(-)}(\bar q, q;w)\right) \;,
\nonumber\\
&&\bar K^{(I,\pm)}_{ab}(\sqrt{s};n)= \int_{-1}^1 \frac{dz}{2} \,
\frac{K^{(I,\pm)}_{ab}(s,t^{(I)}_{ab})}{\big( p^{(I)}_{a}\,
p^{(I)}_{b}\big)^{n}}\,P_n(z)
\nonumber\\
&&\;\;\;\;\;\;\;\;\;\;\;+\int_{-1}^1 \frac{dz}{2} \,
\,\Big(E^{(I)}_a\mp m_{B(I,a)} \Big) \,\Big(E^{(I)}_b \mp m_{B(I,b)} \Big)
\frac{K^{(I,\mp)}_{ab}(s,t^{(I)}_{ab})}{\big( p^{(I)}_{a}\,
p^{(I)}_{b}\big)^{n+1}}\,P_{n+1}(z)\;,
\label{bark-def}\end{aligned}$$ where $P_n(z)$ are the Legendre polynomials and $$\begin{aligned}
&&t^{(I)}_{ab}= m_{\Phi(I,a)}^2+m_{\Phi(I,b)}^2 -2\,\omega^{(I)}_a\,\omega^{(I)}_b
+2\,p^{(I)}_a\,p^{(I)}_b\,z \;,\;\;\;
E^{(I)}_a=\sqrt{s}-\omega^{(I)}_a\;,
\nonumber\\
&&\omega^{(I)}_a=\frac{s+m^2_{\Phi(I,a)}-m^2_{B(I,a)}}{2\,\sqrt{s}}\;,\;\;\;
\Big(p^{(I)}_{a}\Big)^2 =\Big(\omega^{(I)}_a\Big)^2-m_{\Phi(I,a)}^2\;.
\label{}\end{aligned}$$ The construction of the on-shell irreducible interaction kernel requires the identification of the invariant amplitudes $K_{ab}^{(I,\pm )}(s,t) $ in a given channel $ab$: $$\begin{aligned}
&&K^{(I)}_{ab} \Big|_{\rm{on-sh.}}
= \frac{1}{2}\,\Bigg(\frac{\wslash}{\sqrt{w^2}}+1
\Bigg)\, K_{ab}^{(I,+)}(s,t) + \frac{1}{2}\,\Bigg(\frac{\wslash}{\sqrt{w^2}}-1
\Bigg)\,K_{ab}^{(I,-)}(s,t) \;.
\label{}\end{aligned}$$ Isospin breaking effects are easily incorporated by constructing super matrices $V^{(I' I)}$, $J^{(I'I)}$ and $T^{(I'I)}$ which couple different isospin states. We consider isospin breaking effects induced by isospin off-diagonal loop function $J^{(I'I)}$ but impose $V^{(I'I)}\sim \delta_{I'I}$. For technical details we refer to Appendix D.
We underline that our approach deviates here from the common chiral expansion scheme as implied by the heavy-fermion representation of the chiral Lagrangian. A strict chiral expansion of the unitarity loop function $I_{\Phi(I,a)\,B(I,a)}(\sqrt{s}\,)$ does not reproduce the correct s-channel unitarity cut. One must perform an infinite summation of interaction terms in the heavy-fermion chiral Lagrangian to recover the correct threshold behavior. This is achieved more conveniently by working directly with the manifest relativistic scheme, where it is natural to write down the loop functions in terms of the physical masses. In this work we express results systematically in terms of physical parameters avoiding the use of bare parameters like $\m0_{[8]}$ whenever possible.
Construction of effective interaction kernel
--------------------------------------------
We collect all terms of the chiral Lagrangian contributing to order $Q^3$ to the interaction kernel $K^{(I)}_{ab}(\bar k ,k ;w )$ of (\[BS-coupled\]): $$\begin{aligned}
K^{(I)}(\bar k,k;w) &=& K^{(I)}_{WT}(\bar k,k;w)+K^{(I)}_{s-[8]}(\bar k,k;w)
+K^{(I)}_{u-[8]}(\bar k,k;w)
\nonumber\\
&+&K^{(I)}_{s-[10]}(\bar k,k;w)
+K^{(I)}_{u-[10]}(\bar k,k;w)+K^{(I)}_{s-[9]}(\bar k,k;w)
\nonumber\\
&+&K^{(I)}_{u-[9]}(\bar k,k;w)
+K^{(I)}_{[8][8]}(\bar k,k;w)+K^{(I)}_{\chi}(\bar k,k;w) \,.
\label{k-all}\end{aligned}$$ The various contributions to the interaction kernel will be written in a form which facilitates the derivation of $K^{(I,\pm)}_{ab}(s,t)$ in (\[bark-def\]). It is then straightforward to derive the on-shell irreducible interaction kernel $V$.
We begin with a discussion of the Weinberg-Tomozawa interaction term $K^{(I)}_{WT}$ in (\[lag-Q\]). To chiral order $Q^3 $ it is necessary to consider the effects of the baryon wave-function renormalization factors ${\mathcal Z}$ and of the further counter terms introduced in (\[chi-sb-4\]). We have $$\begin{aligned}
\Big[\,K^{(I)}_{W\,T\,}(\bar k,k;w)\Big]_{ab}&=&
\Big[C_{WT}^{(I)}\Big]_{ab}\,
\Big(1+{\textstyle{1\over 2}}\,\Delta \zeta_{B(I,a)}
+{\textstyle{1\over 2}}\,\Delta \zeta_{B(I,b)} \Big)
\,\frac{\barqslash+\qslash}{4\,f^2}\;.
\label{WT-k}\end{aligned}$$ The dimensionless coefficient matrix $C_{WT}^{(I)}$ of the Weinberg-Tomozawa interaction is given in Tab. \[tabpi-1\] for the strangeness zero channels and in the Appendix E for the strangeness minus one channels. The constants $\Delta \zeta_{B(c)}$ in (\[WT-k\]) receive contributions from the baryon octet wave-function renormalization factors ${\mathcal Z}$ and the parameters $\zeta_0,\zeta_D$ and $\zeta_F$ of (\[chi-sb-4\]): $$\begin{aligned}
&&\Delta \zeta = \zeta +{\mathcal Z}-1 +\cdots \,,
\nonumber\\
&&\zeta_N = \big( \zeta_0+2\,\zeta_F \big)\, m_\pi^2
+ \big( 2\,\zeta_0 +2\,\zeta_D-2\,\zeta_F\big) \,m_K^2 \;,
\nonumber\\
&& \zeta_\Lambda =\big( \zeta_0-{\textstyle{2\over 3}}\,\zeta_D \big)\, m_\pi^2
+ \big( 2\,\zeta_0 +{\textstyle{8\over 3}}\,\zeta_D\big) \,m_K^2 \;,
\nonumber\\
&& \zeta_\Sigma =\big( \zeta_0+2\,\zeta_D \big)\, m_\pi^2 + 2\,\zeta_0 \,m_K^2 \;,
\nonumber\\
&& \zeta_\Xi = \big( \zeta_0-2\,\zeta_F \big)\, m_\pi^2
+ \big( 2\,\zeta_0 +2\,\zeta_D+2\,\zeta_F\big) \,m_K^2 \;.
\label{zeta-par}\end{aligned}$$ The dots in (\[zeta-par\]) represent corrections terms of order $Q^3$ and further contributions from irreducible one-loop diagrams not considered here. We point out that the coupling constants $\zeta_0$, $\zeta_D$ and $\zeta_F$, which appear to renormalize the strength of the Weinberg-Tomozawa interaction term, also contribute to the baryon wave-function factor ${\mathcal Z}$ as is evident from (\[chi-sb\]). In fact, it will be demonstrated that the explicit and implicit dependence via the wave-function renormalization factors ${\mathcal Z}$ cancel identically at leading order. Generalizing (\[def-zn\]) we derive the wave-function renormalization constants for the $SU(3)$ baryon octet fields $$\begin{aligned}
&& {\mathcal Z}^{-1}_{B(c)}-1 = \zeta_{B(c)}
-\frac{1}{4\,f^2}\,\sum_{a} \,\xi_{B(I,a)}\,m_{\Phi(I,a)}^2
\left( G^{(B(c))}_{\Phi (I,a)\, B(I,a)} \right)^2 \,,
\label{zeta-result}\\
&& \xi_{B(I,a)}= 2\,m_{B(c)}\,
\frac{\partial \,I_{\Phi (I,a)\, B(I,a)}(\sqrt{s}\,)}{\partial \sqrt{s}}
\Bigg|_{\sqrt{s}=m_{B(c)}} -\frac{I_{\Phi (I,a)}}{m_{\Phi(I,a)}^2 }
+{\mathcal O}\left( Q^4\right)\,,
\nonumber\end{aligned}$$ where the sum in (\[zeta-result\]) includes all $SU(3)$ channels as listed in (\[r-def\]). The result (\[zeta-result\]) is expressed in terms of the dimension less coupling constants $G_{\Phi B}^{(B)}$. For completeness we collect here all required 3-point coupling coefficients: $$\begin{aligned}
&&G_{\pi N}^{(N)} = \sqrt{3}\,\Big( F+D \Big) \;, \quad
G_{K \,\Sigma}^{(N)} = -\sqrt{3}\,\Big( F-D \Big) \;,\quad
G_{\eta \,N}^{(N)} = {\textstyle{1\over\sqrt{3}}}\,\Big( 3\,F-D \Big)\;,
\nonumber\\
&& G_{K \,\Lambda}^{(N)} = -{\textstyle{1\over\sqrt{3}}}\,\Big( 3\,F+D \Big) \;,\quad
G_{\bar K N}^{(\Lambda)} = \sqrt{2}\,G_{K \,\Lambda}^{(N)}
\;, \quad G_{\pi \Sigma}^{(\Lambda)} = 2\,D \;,
\nonumber\\
&& G_{\eta \,\Lambda}^{(\Lambda)} = -{\textstyle{2\over\sqrt{3}}}\,D \;,\quad
G_{K \Xi}^{(\Lambda)} = -\sqrt{{\textstyle{2\over 3}}}\,\Big( 3\,F-D\Big) \;,
\quad G_{\bar K N}^{(\Sigma)} =
\sqrt{{\textstyle{2\over 3}}}\,G_{K \Sigma}^{(N)}\;,
\nonumber\\
&& G_{\pi \Sigma}^{(\Sigma)} = -\sqrt{8}\,F \;, \quad
G_{\pi \Lambda}^{(\Sigma)} =
{\textstyle{1\over\sqrt{3}}}\,G_{\pi \Sigma}^{(\Lambda)} \;,\quad
G_{\eta \Sigma }^{(\Sigma)} = {\textstyle{2\over\sqrt{3}}}\,D\;,
\nonumber\\
&&G_{K\Xi}^{(\Sigma)} = \sqrt{2}\,\Big( F+D\Big) \;,\quad
G_{\bar K \Lambda}^{(\Xi)} = -{\textstyle{1\over\sqrt{2}}}\,
G_{K \Xi}^{(\Lambda)}\;, \quad
G_{\bar K \Sigma}^{(\Xi)} = -\sqrt{{\textstyle{3\over2}}}\,
G_{K \Xi}^{(\Sigma)}\;,
\nonumber\\
&&G_{\eta \,\Xi}^{(\Xi)} = -{\textstyle{1\over\sqrt{3}}}\,\Big( 3\,F+D \Big)\;,\quad
G_{\pi \,\Xi}^{(\Xi)} = \sqrt{3}\,\Big( F-D \Big) \;,
\label{G-explicit}\end{aligned}$$ at leading chiral order $Q^0$. The loop function $I_{\Phi (I,a)\, B(I,a)}(\sqrt{s}\,)$ and the mesonic tadpole term $I_{\Phi(I,a)}$ are the obvious generalizations of $I_{\pi N}(\sqrt{s}\,)$ and $I_\pi$ introduced in (\[ipin-analytic\]). We emphasize that the contribution $\zeta_{B(c)}$ in (\[zeta-result\]) is identical to the corresponding contribution in $ \Delta \zeta_{B(c)}$. Therefore, if all one-loop effects are dropped, one finds the result $$\begin{aligned}
{\mathcal Z}^{-1}=1 +\zeta \;, \qquad
\Delta \zeta =0 +{\mathcal O} \left(Q^3, \frac{1}{N_c}\,Q^2 \right) \;.
\label{delta-zeta}\end{aligned}$$ Given our renormalization condition (\[ren-cond\]) that requires the baryon s-channel pole contribution to be represented by the renormalized effective potential $V_R$, we conclude that (\[delta-zeta\]) holds in our approximation. Note that the loop correction to ${\mathcal Z}$ as given in (\[zeta-result\]) must be considered as part of the renormalized effective potential $V_R$ and consequently should be dropped as they are suppressed by the factor $Q^3/N_c$. It is evident that the one-loop contribution to the baryon ${\mathcal Z}$-factor is naturally moved into $V_R$ by that double subtraction with $k=1$ in (\[result-loop:ab\]), as explained previously.
=0.7mm
$ C_{WT}^{(\frac{1}{2})} $ $ C_{N_{[8]}}^{(\frac{1}{2})}$ $C_{\Lambda_{[8]}}^{(\frac{1}{2})}$ $C_{\Sigma_{[8]}}^{(\frac{1}{2})}$ $C_{\Delta_{[10]}}^{(\frac{1}{2})}$ $C_{\Sigma_{[10]} }^{(\frac{1}{2})}$ $\widetilde{C}_{N_{[8]}}^{(\frac{1}{2})}$ $\widetilde{C}_{\Lambda_{[8]} }^{(\frac{1}{2})}$ $\widetilde{C}_{\Sigma_{[8]}}^{(\frac{1}{2})}$ $\widetilde {C}_{\Xi_{[8]} }^{(\frac{1}{2})}$ $\widetilde {C}_{\Delta_{[10]} }^{(\frac{1}{2})}$ $\widetilde {C}_{\Sigma_{[10]}}^{(\frac{1}{2})}$ $\widetilde C_{\Xi_{[10]}}^{(\frac{1}{2})}$
------ ---------------------------- -------------------------------- ------------------------------------- ------------------------------------ ------------------------------------- -------------------------------------- ------------------------------------------- -------------------------------------------------- ------------------------------------------------ ----------------------------------------------- --------------------------------------------------- -------------------------------------------------- --------------------------------------------- --
$11$ 2 1 0 $0$ 0 0 -$\frac{1}{3}$ 0 0 0 $\frac{4}{3}$ 0 0
$12$ $\frac{1}{2}$ 1 0 $0$ 0 0 0 $\frac{1}{\sqrt{6}}$ -1 0 0 -1 0
$13$ 0 1 0 0 0 0 1 0 0 0 0 0 0
$14$ -$\frac{3}{2}$ 1 0 $0$ 0 0 0 0 $\sqrt{\frac{3}{2}}$ 0 0 $\sqrt{\frac{3}{2}}$ 0
$22$ 2 1 0 $0$ 0 0 0 0 0 -$\frac{1}{3}$ 0 0 -$\frac{1}{3}$
$23$ -$\frac{3}{2}$ 1 0 $0$ 0 0 0 0 $\sqrt{\frac{3}{2}}$ 0 0 $\sqrt{\frac{3}{2}}$ 0
$24$ 0 1 0 $0$ 0 0 0 0 0 -1 0 0 -1
$33$ 0 1 0 0 0 0 1 0 0 0 0 0 0
$34$ -$\frac{3}{2}$ 1 0 $0$ 0 0 0 $\frac{1}{\sqrt{2}}$ 0 0 0 0 0
$44$ 0 1 0 $0$ 0 0 0 0 0 1 0 0 1
$ C_{WT}^{(\frac{3}{2})} $ $ C_{N_{[8]}}^{(\frac{3}{2})}$ $C_{\Lambda_{[8]}}^{(\frac{3}{2})}$ $C_{\Sigma_{[8]}}^{(\frac{3}{2})}$ $C_{\Delta_{[10]}}^{(\frac{3}{2})}$ $C_{\Sigma_{[10]} }^{(\frac{3}{2})}$ $\widetilde{C}_{N_{[8]}}^{(\frac{3}{2})}$ $\widetilde{C}_{\Lambda_{[8]} }^{(\frac{3}{2})}$ $\widetilde{C}_{\Sigma_{[8]}}^{(\frac{3}{2})}$ $\widetilde {C}_{\Xi_{[8]} }^{(\frac{3}{2})}$ $\widetilde {C}_{\Delta_{[10]} }^{(\frac{3}{2})}$ $\widetilde {C}_{\Sigma_{[10]}}^{(\frac{3}{2})}$ $\widetilde C_{\Xi_{[10]}}^{(\frac{3}{2})}$
$11$ -1 0 0 0 1 0 $\frac{2}{3}$ 0 0 0 $\frac{1}{3}$ 0 0
$12$ -1 0 0 0 1 0 0 $\frac{1}{\sqrt{6}}$ $\frac{1}{2}$ 0 0 $\frac{1}{2}$ 0
$22$ -1 0 0 0 1 0 0 0 0 $\frac{2}{3}$ 0 0 $\frac{2}{3}$
: Weinberg-Tomozawa interaction strengths and baryon exchange coefficients in the strangeness zero channels as defined in (\[k-nonlocal\]).[]{data-label="tabpi-1"}
It is instructive to afford a short detour and explore to what extent the parameters $\zeta_0, \zeta_D $ and $\zeta_F $ can be dialed to give ${\mathcal Z}=1$. In the $SU(3)$ limit with degenerate meson and also degenerate baryon masses one finds a degenerate wave-function renormalization factor ${\mathcal Z}$ $$\begin{aligned}
&&{\mathcal Z}^{-1}-1=3\,\zeta_0+2\,\zeta_D
- \frac{m_\pi^2}{(4\pi\,f)^2}
\left( \frac{5}{3}\,D^2+3\,F^2\right)
\Bigg( -4 -3\,\ln \left(\frac{m_\pi^2}{m_N^2}\right)
\nonumber\\
&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+3\,\pi\,\frac{m_\pi}{m_N}
+{\mathcal O}\left( \frac{m^2_\pi}{m^2_N} \right)
\Bigg) \;,
\label{su3-limit:z}\end{aligned}$$ where we used the minimal chiral subtraction prescription (\[def-sub\]). The result (\[su3-limit:z\]) agrees identically with the $SU(2)$-result (\[def-zn\]) if one formally replaces $g^2_A $ by $4\, F^2+ {\textstyle{20\over 9}}\,D^2 $ in (\[def-zn\]). It is clear from (\[su3-limit:z\]) that in the $SU(3)$ limit the counter term $3\,\zeta_0+2\,\zeta_D$ may be dialed such so as to impose ${\mathcal Z}=1$. This is no longer possible once the explicit $SU(3)$ symmetry-breaking effects are included. Note however that consistency of the perturbative renormalization procedure suggests that for hypothetical $SU(3)$-degenerate $\xi$-factors in (\[zeta-result\]) it should be possible to dial $\zeta_0$, $\zeta_D$ and $\zeta_F$ so as to find ${\mathcal Z}_B =1$ for all baryon octet fields. This is expected, because for example in the $\overline {MS}$-scheme one finds a $SU(3)$-symmetric renormalization scale[^13] dependence of $\xi_{B(c)}$ in (\[zeta-result\]) with $\xi_{B(c)} \sim \ln \mu^2$. Indeed in this case the choice $$\begin{aligned}
&&\zeta_0 = \frac{\xi}{4\,f^2}\left(\frac{26}{9}\,D^2+2\,F^2\right)\, , \qquad
\zeta_D = \frac{\xi}{4\,f^2}\left(-D^2+3\,F^2\right)\,, \quad
\nonumber\\
&& \zeta_F = \frac{\xi}{4\,f^2}\,\frac{10}{3}\,D\,F \,,
\label{}\end{aligned}$$ would lead to ${\mathcal Z}=1$ for all baryon octet wave functions. We will return to the $SU(3)$ symmetry-breaking effects in the baryon wave-function renormalization factors when discussing the meson-baryon 3-point vertices at subleading orders.
We proceed with the s-channel and u-channel exchange diagrams of the baryon octet. They contribute as follows $$\begin{aligned}
\Big[K^{(I)}_{s-[8]}(\bar k,k;w)\Big]_{ab} &=& -\sum_{c\,=\,1}^3\,
\Big(\barqslash -R^{(I,c)}_{L,ab}\Big)\,
\frac{\Big[C^{(I,c)}_{[8]}\Big]_{ab}}
{4\,f^2\,\big(\wslash+m^{(c)}_{[8]}\big)}\,\Big( \qslash -R^{(I,c)}_{R,ab}\Big)\;,
\nonumber\\
\Big[K^{(I)}_{u-[8]}(\bar k,k;w)\Big]_{ab} &=&
\sum_{c\,=\,1}^4\,
\frac{\Big[\widetilde C^{(I,c)}_{[8]}\Big]_{ab} }{4\,f^2 }\,\Bigg(
\wslash+m^{(c)}_{\,[8]}+\tilde R^{(I,c)}_{L,ab}+ \tilde R^{(I,c)}_{R,ab}
\nonumber\\
&-& \Big(\barpslash+m^{(c)}_{\,[8]}+ \tilde R^{(I,c)}_{L,ab} \Big)\,
\frac{1}{\barwslash+m^{(c)}_{\,[8]} }
\, \Big(\pslash+m^{(c)}_{\,[8]}+ \tilde R^{(I,c)}_{R,ab}\Big)\Bigg)\;,
\label{wave-ren}\end{aligned}$$ where $\widetilde w_\mu=p_\mu-\bar q_\mu$. The index $c$ in (\[wave-ren\]) labels the baryon octet exchange with $c\to (N,\Lambda, \Sigma ,\Xi)$. In particular $m^{(c)}_{[8]}$ denotes the physical baryon octet masses and $R^{(I,c)}$ and $\tilde R^{(I,c)}$ characterize the ratios of pseudo-vector to pseudo-scalar terms in the meson-baryon vertices. The dimensionless matrices $C^{}_{[8]}$ and $\widetilde C^{}_{[8]}$ give the renormalized strengths of the s-channel and u-channel baryon exchanges respectively. They are expressed most conveniently in terms of s-channel $C^{}_{B(c)}$ and u-channel $ \tilde C^{}_{B(c)}$ coefficient matrices $$\begin{aligned}
&& \Big[C^{(I,c)}_{[8]}\Big]_{ab} = \Big[C^{(I)}_{B(c)}\Big]_{ab}\,
\bar A_{\Phi (I,a) B(I,a)}^{(B(c))}\,\bar A_{\Phi (I,b) B(I,b)}^{(B(c))} \;,
\nonumber\\
&& \Big[\widetilde C^{(I,c)}_{[8]}\Big]_{ab}=
\Big[\widetilde C^{(I)}_{B(c)}\Big]_{ab}\,
\bar A_{\Phi (I,b) B(I,a)}^{(B(c))}\,\bar A_{\Phi (I,a) B(I,b)}^{(B(c))}
\;,
\label{def-not}\end{aligned}$$ and the renormalized three-point coupling constants $\bar A$. According to the LSZ-scheme the bare coupling constants, $A$, are related to the renormalized coupling constants, $\bar A$, by means of the baryon octet ${\mathcal Z}$-factors $$\begin{aligned}
\bar A_{\Phi (I,a) B(I,b)}^{(B(c))}= {\mathcal Z}^{\frac{1}{2}}_{B(c)}\,A_{\Phi (I,a) B(I,b)}^{(B(c))}
\,{\mathcal Z}_{B(I,b)}^{\frac{1}{2}}\,.
\label{g-ren:z}\end{aligned}$$ To leading order $Q^0$ the bare coupling constants $A = G(F,D)+{\mathcal O}\left(Q^2 \right)$ are specified in (\[G-explicit\]). The chiral correction terms to the coupling constants $A$ and $R $ will be discussed in great detail subsequently. In our convention the s-channel coefficients $C_{B(c)}$ are either one or zero and the u-channel coefficients $\tilde C^{}_{B(c)}$ represent appropriate Fierz factors resulting from the interchange of initial and final meson states. To illustrate our notation explicitly the coefficients are listed in Tab. \[tabpi-1\] for the strangeness zero channels but relegated to Appendix E for the strangeness minus one channel. We stress that the expressions for the s- and u-channel exchange contribution (\[wave-ren\]) depend on the result (\[delta-zeta\]), because there would otherwise be a factor ${\mathcal Z}^{-1}/(1+\zeta)$ in front of both contributions. That would necessarily lead to an asymmetry in the treatment of s- versus u-channel exchange contribution, because the s-channel contribution would be further renormalized by the unitarization. In contrast, we observe in our scheme the proper balance of s- and u-channel exchange contributions in the scattering amplitude, as is required for the realization of the large $N_c$ cancellation mechanism [@Manohar].
We turn to the $SU(3)$ symmetry-breaking effects in the meson-baryon coupling constants. The 3-point vertices have pseudo-vector and pseudo-scalar components determined by $F_i+\delta F_i$ and $\bar F_i$ of (\[chi-sb-3\],\[chi-sb-3:p\]) respectively. We first collect the symmetry-breaking terms in the axial-vector coupling constants $A$ introduced in (\[g-ren:z\]). They are of chiral order $Q^2$ but lead to particular $Q^3$-correction terms in the scattering amplitudes. One finds $$\begin{aligned}
&& A = G(F_A,D_A) -\frac{2}{\sqrt{3}}\,(m_K^2-m_\pi^2)\,\Delta A \;,\quad
\tilde F_i = F_i + \delta F_i \;,
\nonumber\\
&&\Delta A_{\pi N}^{(N)} =
3\,(F_1+F_3)-F_0-F_2+2\,(\tilde F_4+\tilde F_5) \;,
\nonumber\\
&&\Delta A_{K \Sigma}^{(N)} =
{\textstyle{3\over 2}}\,(F_1-F_3) +{\textstyle{1\over 2}}\,(F_0-F_2)-\tilde F_4+\tilde F_5 \;,
\nonumber\\
&&\Delta A_{\eta \,N}^{(N)} =
-F_1+3\,F_3+{\textstyle{1\over3}}\,F_0-F_2
+{\textstyle{2\over3}}\,\tilde F_4-2\,\tilde F_5 +2\,(\tilde F_6+{\textstyle{4\over3}}\,\tilde F_4) \;,
\nonumber\\
&&\Delta A_{K \Lambda}^{(N)} =
-{\textstyle{1\over 2}}\,F_1-{\textstyle{3\over 2}}\,F_3+{\textstyle{1\over 2}}\,F_0+
{\textstyle{3\over2}}\,F_2 +{\textstyle{1\over3}}\,\tilde F_4+\tilde F_5
+(F_7+{\textstyle{4\over 3}}\,F_0)\;,
\nonumber\\
&& \Delta A_{\bar K N}^{(\Lambda)} = \sqrt{2}\,\Delta A_{K \,\Lambda}^{(N)} \,,\quad
\nonumber\\
&& \Delta A_{\pi \Sigma}^{(\Lambda)} =
{\textstyle{4\over \sqrt{3}}}\,\tilde F_4 +\sqrt{3}\,(F_7+{\textstyle{4\over3}}\,F_0) \;,
\nonumber\\
&&\Delta A_{\eta \,\Lambda}^{(\Lambda)} =
{\textstyle{4\over 3}}\,(F_0 +\tilde F_4)+2\,(\tilde F_6+{\textstyle{4\over3}}\,\tilde F_4)
+2\,(F_7+{\textstyle{4\over3}}\,F_0) \;,
\nonumber\\
&& \Delta A_{K \Xi}^{(\Lambda)} =
-{\textstyle{1\over \sqrt{2}}}\,(F_0+F_1)
+{\textstyle{3\over \sqrt{2}}}\,(F_2+F_3)
-{\textstyle{\sqrt{2}\over 3}}\,(\tilde F_4-3\,\tilde F_5)
-\sqrt{2}\,(F_7 +{\textstyle{4\over3}}\,F_0) \;,
\nonumber\\
&& \Delta A_{\bar K N}^{(\Sigma)} =
\sqrt{{\textstyle{2\over 3}}}\,\Delta A_{K \Sigma}^{(N)} \;, \quad
\Delta A_{\pi \Sigma}^{(\Sigma)} = -4\,\sqrt{{\textstyle{2\over 3}}}\,(F_2+\tilde F_5) \;, \quad
\Delta A_{\pi \Lambda}^{(\Sigma)} =
{\textstyle{1\over\sqrt{3}}}\,\Delta A_{\pi \Sigma}^{(\Lambda)} \;,
\nonumber\\
&& \Delta A_{\eta \Sigma }^{(\Sigma)} = {\textstyle{4\over 3}}\,(F_0-\tilde F_4)
+2\,(\tilde F_6+{\textstyle{4\over3}}\,\tilde F_4) \;,
\nonumber\\
&& \Delta A_{K \Xi}^{(\Sigma)} = -\sqrt{{\textstyle{3\over 2}}}\,(F_1+F_3)+
\sqrt{{\textstyle{1\over 6}}}\,(F_0+F_2)
-\sqrt{{\textstyle{2\over 3}}}\,(\tilde F_4+\tilde F_5)\;,
\nonumber\\
&& \Delta A_{\bar K \Lambda}^{(\Xi)} = -{\textstyle{1\over\sqrt{2}}}\,
\Delta A_{K \Xi}^{(\Lambda)} \;, \quad
\Delta A_{\bar K \Sigma}^{(\Xi)} = -\sqrt{{\textstyle{3\over2}}}\,
\Delta A_{K \Xi}^{(\Sigma)} \;,
\nonumber\\
&& \Delta A_{\eta \,\Xi}^{(\Xi)} =
F_1+3\,F_3+{\textstyle{1\over3}}\,F_0+F_2
+{\textstyle{2\over3}}\,\tilde F_4+2\,\tilde F_5+2\,(\tilde F_6+{\textstyle{4\over3}}\,\tilde F_4) \;,
\nonumber\\
&& \Delta A_{\pi \Xi}^{(\Xi)} = F_0+3\,F_1-F_2-3\,F_3+2\,(\tilde F_5-\tilde F_4) \;,
\label{G-explicit-new}\end{aligned}$$ where the renormalized coupling constants $D_A$ and $F_A$ are given in (\[ren-FDC-had\]) and $G(F,D)$ in (\[G-explicit\]). We emphasize that to order $Q^2$ the effect of the ${\mathcal Z}$-factors in (\[g-ren:z\]) can be accounted for by the following redefinition of the $F_R,D_R$ and $F_i$ parameters in (\[G-explicit-new\]) $$\begin{aligned}
&& F_R \to F_R +\, \Big( 2\,m_K^2+m_\pi^2\Big)\,
\Big(\zeta_0+{\textstyle{2\over 3}}\,\zeta_D\Big) \,F_R \;, \qquad
\nonumber\\
&& D_R \to D_R + \Big( 2\,m_K^2+m_\pi^2\Big)\,
\Big( \zeta_0+{\textstyle{2\over 3}}\,\zeta_D\Big) \,D_R \;,
\nonumber\\
&& F_0 \to F_0 +\zeta_D\,D_R\,, \qquad F_1 \to F_1+\zeta_F \,F_R
\,, \qquad F_2 \to F_2 +\zeta_D \,F_R\,,
\nonumber\\
&& F_3 \to F_3 + \zeta_F \,F_R\,,\qquad \,F_7 \to F_7 -{\textstyle{4\over 3}}\,\zeta_D\,\,D_R\,.
\label{zeta-redundant}\end{aligned}$$ As a consequence the parameters $\zeta_0$, $\zeta_D$ and $\zeta_F$ are redundant in our approximation and can therefore be dropped. It is legitimate to identify the bare coupling constants $A$ with the renormalized coupling constants $\bar A$. Note that the same renormalization holds for matrix elements of the axial-vector current.
We recall the large $N_c$ result (\[large-Nc-FDC\]). The many parameters $F_i$ and $\delta F_i$ in (\[G-explicit-new\]) are expressed in terms of the seven parameters $c_{i}$ and $\delta c_i= \tilde c_i-c_i-\bar c_i$. For the readers convenience we provide the axial-vector coupling constants, which are most relevant for the pion-nucleon and kaon-nucleon scattering processes, in terms of those large $N_c$ parameters $$\begin{aligned}
&& A_{\pi N}^{(N)} = \sqrt{3}\,\Big( F_A+D_A \Big) +
{\textstyle{5\over 3}}\, (c_1+\delta c_1) + c_2+\delta c_2 -a+ 5\, c_3 + c_4\;,
\nonumber\\
&& A_{\bar K N}^{(\Lambda)} = -\sqrt{{\textstyle{2\over3}}}\,\Big( 3\,F_A+D_A \Big)
+{\textstyle{1\over\sqrt{2}}}\,\Big(c_1+\delta c_1+c_2+\delta c_2-a-3\,c_3+c_4 \Big)\,,
\nonumber\\
&& A_{\bar K N}^{(\Sigma)} = -\sqrt{2 }\,\Big( F_A-D_A \Big)
-{\textstyle{1\over\sqrt{6}}}\,\Big({\textstyle{1\over 3}}\,(c_1+\delta c_1)-c_2-\delta c_2+a-c_3+3\,c_4 \Big)\,,
\nonumber\\
&& A_{\pi \Sigma}^{(\Lambda)} = 2\,D_A +{\textstyle{2\over\sqrt{3}}}\,(c_1+\delta c_1)\;,
\label{c1234}\end{aligned}$$ where $$\begin{aligned}
F_A &=& F_R
-\frac{\beta}{\sqrt{3}}\,\Big({\textstyle{2\over 3}}\,\delta c_1 +\delta c_2 -a\Big) \,,
\qquad \; \;D_A = D_R - \frac{\beta}{\sqrt{3}}\,\delta c_1 \,,
\nonumber\\
\beta &=& \frac{m_K^2+m_\pi^2/2}{m_K^2-m_\pi^2} \,.
\label{c1234-b}\end{aligned}$$ The result (\[c1234\]) shows that the axial-vector coupling constants $A_{\pi N}^{(N)}$, $A_{\bar K N}^{(\Lambda)}$, $A_{\bar K N}^{(\Sigma)}$ and $A_{\pi \Sigma}^{(\Lambda)}$, can be fine tuned with $c_i $ in (\[large-Nc-FDC\]) to be off their $SU(3)$ limit values. Moreover, the SU(3) symmetric contribution proportional to $ F_A$ or $ D_A$ deviate from their corresponding $F_R$ and $D_R$ values relevant for matrix elements of the axial-vector current, once non-zero values for $\delta c_{1,2}$ are established. We recall, however, that the values of $F_R, D_R$ and $c_i$ are strongly constrained by the weak decay widths of the baryon-octet states (see Tab. \[weak-decay:tab\]). Thus the SU(3) symmetry-breaking pattern in the axial-vector current and the one in the meson-baryon axial-vector coupling constants are closely linked.
It is left to specify the pseudo-scalar part, $P$, of the meson-baryon vertices introduced in (\[chi-sb-3:p\]). In (\[wave-ren\]) their effect was encoded into the $R$ and $\tilde R $ parameters with $R, \tilde R \sim \bar F_{}\sim \bar c_{},a_{}$. Applying the large $N_c$ result of (\[large-Nc-FDC:delta\]) we obtain $$\begin{aligned}
&& R_{L,ab}^{(I,c)} \,A_{\Phi (I,a)\, B(I,a)}^{(B(c))} =
\Big(m_{B(I,a)}+m_{B(c)} \Big)\,P_{\Phi (I,a)\, B(I,a)}^{(B(c))}
\;,
\nonumber\\
&& R_{R,ab}^{(I,c)} \,A_{\Phi (I,b)\, B(I,b)}^{(B(c))} =
\Big(m_{B(I,b)}+m_{B(c)} \Big)\,P_{\Phi (I,b)\, B(I,b)}^{(B(c))}
\;,
\nonumber\\
&& \tilde R_{L,ab}^{(I,c)} \,A_{\Phi (I,b)\, B(I,a)}^{(B(c))} =
\Big(m_{B(I,a)}+m_{B(c)} \Big)\,P_{\Phi (I,b)\, B(I,a)}^{(B(c))}
\;,
\nonumber\\
&& \tilde R_{R,ab}^{(I,c)} \,A_{\Phi (I,a)\, B(I,b)}^{(B(c))} =
\Big(m_{B(I,b)}+m_{B(c)} \Big)\,P_{\Phi (I,a)\, B(I,b)}^{(B(c))}
\;,
\nonumber\\ \nonumber\\
&&P_{\pi N}^{(N)} = -\Big({\textstyle{5\over3}}\, \bar c_1 +\bar c_2+a\Big)\, \Big(\beta -1\Big)\;, \quad
P_{K \Sigma}^{(N)} = -\Big( {\textstyle{1\over3}}\,\bar c_1 -\bar c_2-a \Big)\,
\Big( \beta +{\textstyle{1\over2}} \Big) \;,
\nonumber\\
&&P_{\eta \,N}^{(N)} = {\textstyle{7\over3}}\, \bar c_1 +\bar c_2+a
- \beta \,\Big( {\textstyle{1\over3}}\,\bar c_1 +\bar c_2+a \Big)\;, \qquad
\nonumber\\
&& P_{K \Lambda}^{(N)} = \Big( \bar c_1 +\bar c_2+a\Big)\,\Big(\beta +{\textstyle{1\over2}}\Big)\;,
\qquad
P_{\bar K N}^{(\Lambda)} = \sqrt{2}\,P_{K \,\Lambda}^{(N)} \,,\qquad
\nonumber\\
&&P_{\pi \Sigma}^{(\Lambda)} = - {\textstyle{2\over \sqrt{3}}}\,\bar c_1\,\Big( \beta -1 \Big)\;,
\quad P_{\eta \,\Lambda}^{(\Lambda)} = {\textstyle{2\over3}}\,\bar c_1\,\Big( 5+\beta \Big) +2\,\bar c_2+2\,a \;, \qquad
\nonumber\\
&& P_{K \Xi}^{(\Lambda)} = {\textstyle{1\over \sqrt{2}}}\,\Big( {\textstyle{1\over3}}\,\bar c_1+\bar c_2+a \Big)\,
\Big(2\, \beta +1 \Big)\;, \quad
P_{\bar K N}^{(\Sigma)} =
\sqrt{{\textstyle{2\over 3}}}\,P_{K \Sigma}^{(N)} \;, \quad \!\!
\nonumber\\
&& P_{\pi \Sigma}^{(\Sigma)} = \sqrt{{\textstyle{2\over3}}}\,\Big( {\textstyle{4\over 3}}\,\bar c_1 +2\,\bar c_2+2\,a \Big)\,
\Big(\beta -1\Big) \;, \quad
P_{\pi \Lambda}^{(\Sigma)} =
{\textstyle{1\over\sqrt{3}}}\,P_{\pi \Sigma}^{(\Lambda)} \;,
\nonumber\\
&& P_{\eta \Sigma }^{(\Sigma)} = 2\,\bar c_1 \,\Big(1-{\textstyle{1\over 3}}\,\beta \Big)+2\,\bar c_2+2\,a \;,\qquad
\nonumber\\
&& P_{K \Xi}^{(\Sigma)} = -{\textstyle{1\over \sqrt{6}}}\,\Big( {\textstyle{5\over 3}}\,\bar c_1 +\bar c_2+a\Big) \,
\Big(2\,\beta +1\Big) \;,\quad
P_{\bar K \Lambda}^{(\Xi)} = -{\textstyle{1\over\sqrt{2}}}\,
P_{K \Xi}^{(\Lambda)} \;, \quad
\nonumber\\
&& P_{\bar K \Sigma}^{(\Xi)} = -\sqrt{{\textstyle{3\over2}}}\,
P_{K \Xi}^{(\Sigma)} \;,\quad
P_{\eta \,\Xi}^{(\Xi)} = {\textstyle{11\over3}}\,\bar c_1 +3\,\bar c_2 + 3\,a
+\beta \,\Big( \bar c_1 +\bar c_2+a\Big)
\;,\quad \!\!
\nonumber\\
&& P_{\pi \Xi}^{(\Xi)} = \Big({\textstyle{1\over3}}\,\bar c_1 -\bar c_2-a\Big)\,\Big( \beta -1 \Big)\;,
\label{P-result}\end{aligned}$$ where $\beta \simeq 1.12$ was introduced in (\[c1234-b\]). The pseudo-scalar meson-baryon vertices show $SU(3)$ symmetric contribution linear in $\beta \,\bar c_i$ and symmetry-breaking terms proportional to $\bar c_i$. We emphasize that, even though the physical meson-baryon coupling constants, $G=A+P$, are the sum of their axial-vector and pseudo-scalar components, it is important to carefully discriminate both types of vertices, because they give rise to quite different behavior for the partial-wave amplitudes off the baryon-octet pole. We expect such effects to be particularly important in the strangeness sectors since there (\[P-result\]) leads to $P^{(\Lambda, \Sigma )}_{K N} \sim m_K^2$ to be compared to $P_{\pi N}^{(N)} \sim m_\pi^2$.
We turn to the decuplet exchange terms $K^{(I)}_{s-[10] }$ and $K^{(I)}_{u-[10] }$ in (\[k-all\]). Again we write their respective interaction kernels in a form which facilitates the identification of their on-shell irreducible parts $$\begin{aligned}
K^{(I)}_{s-[10] }(\bar k,k;w) &=&\sum_{c\,=\,1}^2\,
\frac{C^{(I,c)}_{[10]}}{4\,f^2 }\,\Bigg(
\frac{\bar q\cdot q}{\wslash-m_{[10]}^{(c)}}\,
-\frac{(\bar q\cdot w)\,(w\cdot q)}
{(m_{[10]}^{(c)})^2\,\big(\wslash-m_{[10]}^{(c)}\big) }
\nonumber\\
&+&\frac{1}{3}\left(\barqslash +\frac{\bar q \cdot w}{m_{[10]}^{(c)}}\right)
\frac{1}{\wslash+m_{[10]}^{(c)}}
\left(\qslash +\frac{w\cdot q}{m_{[10]}^{(c)}}\right) -\frac{Z_{[10]}}{3\,m_{[10]}^{(c)}}\,\barqslash \,\qslash
\nonumber\\
&-&\frac{Z_{[10]}^2}{6\,(m_{[10]}^{(c)})^2}\,\barqslash \,\Big(\wslash-2\,m_{[10]}^{(c)}\Big)\,\qslash
+Z_{[10]}\,\barqslash \,\frac{w\cdot q}{3\,(m_{[10]}^{(c)})^2}
\nonumber\\
&+&Z_{[10]} \,\frac{\bar q \cdot w}{3\,(m_{[10]}^{(c)})^2}\,\qslash \Bigg)\;,
\nonumber\\
K^{(I)}_{u-[10] }(\bar k,k;w) &=&\sum_{c\,=\,1}^3\,
\frac{\widetilde C^{(I,c)}_{[10]}}{4\,f^2 }\,\Bigg(
\frac{\bar q\cdot q}{\barwslash-m_{[10]}^{(c)}}
-\frac{(\bar q\cdot \widetilde w)\,(\widetilde w\cdot q)}
{(m_{[10]}^{(c)})^2\,\big(\barwslash-m_{[10]}^{(c)}\big)}
\nonumber\\
&+&\frac{1}{3}\left(\barpslash +m_{[10]}^{(c)}+\frac{q \cdot \widetilde w}{m_{[10]}^{(c)}}\right)
\frac{1}{\barwslash+m_{[10]}^{(c)}}
\left(\pslash +m_{[10]}^{(c)}+\frac{\widetilde w\cdot \bar q}{m_{[10]}^{(c)}}\right)
\nonumber\\
&-&\frac{\widetilde w\cdot (q+\bar q)}{3\,m_{[10]}^{(c)}}
-\frac{1}{3}\,\Big(\wslash+m_{[10]}^{(c)}\Big)
\nonumber\\
&+&\frac{Z_{[10]}}{3\,(m_{[10]}^{(c)})^2}\,\Big(
\barqslash \,\big(\widetilde w\cdot q\big)
+\big(\bar q \cdot \widetilde w\big) \,\qslash
+m_{[10]}^{(c)}\,\big(
\barqslash\,\qslash-2\,(\bar q\cdot q) \big) \Big)
\nonumber\\
&-&\frac{Z_{[10]}^2}{6\,(m_{[10]}^{(c)})^2}\,
\Big( \barpslash \;\barwslash\,\pslash -\widetilde w^2\,\wslash
+2\,m_{[10]}^{(c)}\,\big(\barqslash\,\qslash -2\,q\cdot q\big) \Big)\Bigg) \;.
\label{k-nonlocal}\end{aligned}$$ The index $c$ in (\[k-nonlocal\]) labels the decuplet exchange with $c\to (\Delta_{[10]} , \Sigma_{[10]}, \Xi_{[10]})$. The dimensionless matrices $C^{(I,c)}_{[10]}$ and $\widetilde C^{(I,c)}_{[10]}$ characterize the strength of the s-channel and u-channel resonance exchanges respectively. Here we apply the notation introduced in (\[def-not\]) also for the resonance-exchange contributions. In particular the decuplet coefficients $C^{(I,c)}_{[10]}$ and $\tilde C^{(I,c)}_{[10]}$ are determined by $$\begin{aligned}
&& A = G( C_A) -\frac{2}{\sqrt{3}}\,(m_K^2-m_\pi^2)\,\Delta A \;,\qquad
\tilde C_0 = C_0 +\delta C_0 \;,
\nonumber\\
&&\Delta A_{\pi N}^{(\Delta_{[10]} )} = \sqrt{2}\,
\big( -{\textstyle{1\over \sqrt{3}}}\,C_2+\sqrt{3}\,C_3+{\textstyle{2\over \sqrt{3}}}\,\tilde C_0
\big) \;, \quad
\nonumber\\
&&\Delta A_{K \,\Sigma}^{(\Delta_{[10]} )} = -\sqrt{2}\,
\big({\textstyle{2\over \sqrt{3}}}\,C_2
- {\textstyle{1\over \sqrt{3}}}\,\tilde C_0+\sqrt{3}\,C_1\big) \;,\quad
\nonumber\\
&& \Delta A_{\bar K N}^{(\Sigma_{[10]})} =\sqrt{{\textstyle{2\over 3}}}\,
\big(-{\textstyle{1\over \sqrt{3}}}\,C_2+\sqrt{3}\,C_3
- {\textstyle{1\over \sqrt{3}}}\,\tilde C_0-\sqrt{3}\,C_1 \big)-\sqrt{2}\,C_4 \;, \quad
\nonumber\\
&&\Delta A_{\pi \Sigma}^{(\Sigma_{[10]})} = -\sqrt{{\textstyle{2\over 3}}}\,
\big({\textstyle{2\over \sqrt{3}}}\,C_2
+ {\textstyle{2\over \sqrt{3}}}\,\tilde C_0\big)-2\,\sqrt{2}\,C_4 \;, \qquad
\nonumber\\
&& \Delta A_{\pi \Lambda}^{(\Sigma_{[10]})} = {\textstyle{2\over \sqrt{3}}}\,(C_2-\tilde C_0) \;, \qquad
\Delta A_{\eta \Sigma}^{(\Sigma_{[10]})} = {\textstyle{2\over \sqrt{3}}}\,C_2
- {\textstyle{2\over \sqrt{3}}}\,\tilde C_0\;,\quad
\nonumber\\
&& \Delta A_{K \Xi}^{(\Sigma_{[10]})} = -\sqrt{{\textstyle{2\over 3}}}\,
\big( -{\textstyle{1\over \sqrt{3}}}\,C_2-\sqrt{3}\,C_3
- {\textstyle{1\over \sqrt{3}}}\,\tilde C_0 +\sqrt{3}\, C_1\big)+\sqrt{2}\,C_4\;,
\nonumber\\
&&\Delta A_{\bar K \Lambda}^{(\Xi_{[10]})} =
-{\textstyle{1\over \sqrt{3}}}\, \tilde C_0 -\sqrt{3}\,C_1
-{\textstyle{1\over \sqrt{3}}}\,(3\,C_4+2\,C_2) \;, \quad
\nonumber\\
&& \Delta A_{\bar K \,\Sigma}^{(\Xi_{[10]})} = {\textstyle{2\over \sqrt{3}}}\,C_2
- {\textstyle{1\over \sqrt{3}}}\,\tilde C_0- \sqrt{3}\, C_1 +\sqrt{3}\,C_4 \;,\quad
\nonumber\\
&&\Delta A_{\eta \,\Xi}^{(\Xi_{[10]})} = {\textstyle{1\over \sqrt{3}}}\,C_2+\sqrt{3}\,C_3
+ {\textstyle{2\over \sqrt{3}}}\,\tilde C_0 +\sqrt{3}\,C_4 \;,\quad
\nonumber\\
&&\Delta A_{\pi \,\Xi}^{(\Xi_{[10]})} = {\textstyle{1\over \sqrt{3}}}\,C_2+\sqrt{3}\,C_3
- {\textstyle{2\over \sqrt{3}}}\,\tilde C_0 -\sqrt{3}\,C_4 \;,
\label{}\end{aligned}$$ where the $SU(3)$ symmetric contributions $G(C)$ are $$\begin{aligned}
&&G_{\pi N}^{(\Delta_{[10]} )} = \sqrt{2}\,C \;, \quad G_{K \,\Sigma}^{(\Delta_{[10]} )} = -\sqrt{2}\,C \;,
\quad
G_{\bar K N}^{(\Sigma_{[10]})} =\sqrt{{\textstyle{2\over 3}}}\,C \;,
\nonumber\\
&& G_{\pi \Sigma}^{(\Sigma_{[10]})} = -\sqrt{{\textstyle{2\over 3}}}\,C \;,\quad
G_{\pi \Lambda}^{(\Sigma_{[10]})} = -C \;, \quad G_{\eta \Sigma}^{(\Sigma_{[10]})} = C\;,
\quad G_{K \Xi}^{(\Sigma_{[10]})} = -\sqrt{{\textstyle{2\over 3}}}\,C\;,
\nonumber\\
&&G_{\bar K \Lambda}^{(\Xi_{[10]})} = C \;, \quad
G_{\bar K \,\Sigma}^{(\Xi_{[10]})} = C \;,\quad
G_{\eta \,\Xi}^{(\Xi_{[10]})} = -C\;,\quad
G_{\pi \,\Xi}^{(\Xi_{[10]})} = -C \;.
\label{}\end{aligned}$$ We recall that at leading order in the $1/N_c$ expansion the five symmetry-breaking parameters $C_i$ introduced in (\[chi-sb-3\]) are all given in terms of the $c_i$ parameters (\[large-Nc-FDC\]). The parameter $\delta C_0 \sim \tilde c_1-c_1$ and also implicitly $\tilde C_R $ (see (\[ren-FDC-had\])) probe the $\tilde c_1$ parameter introduced in (\[ansatz-3\]). We emphasize that all $c_i$ parameters but $c_5$ are determined to a large extent by the decay widths of the baryon octet states and also $\tilde c_1$ is constrained strongly by the $SU(3)$ symmetry-breaking pattern of the meson-baryon-octet coupling constants (\[ansatz-3\]).
The baryon octet resonance contributions $K^{(I)}_{s-[9] }$ and $K^{(I)}_{u-[9] }$ require special attention, because the way how to incorporate systematically these resonances in a chiral SU(3) scheme is not clear. We present first the s-channel and u-channel contributions as they follow from (\[lag-Q\]) at tree-level $$\begin{aligned}
K^{(I)}_{s-[9] }(\bar k,k;w) &=&\sum_{c\,=\,0}^4\,
\frac{C^{(I,c)}_{[9]}}{4\,f^2 }\,\Bigg(
-\frac{\bar q\cdot q}{\wslash+m_{[9]}^{(c)}}\,
+\frac{(\bar q\cdot w)\,(w\cdot q)}
{(m_{[9]}^{(c)})^2\,\big(\wslash+m_{[9]}^{(c)}\big) }
\nonumber\\
&-&\frac{1}{3}\left(\barqslash -\frac{\bar q \cdot w}{m_{[9]}^{(c)}}\right)
\frac{1}{\wslash-m_{[9]}^{(c)}}
\left(\qslash -\frac{w\cdot q}{m_{[10]}^{(c)}}\right) \Bigg)\;,
\nonumber\\
K^{(I)}_{u-[9] }(\bar k,k;w) &=&\sum_{c\,=\,0}^4\,
\frac{\widetilde C^{(I,c)}_{[9]}}{4\,f^2 }\,\Bigg(
-\frac{\bar q\cdot q}{\barwslash+m_{[9]}^{(c)}}
+\frac{(\bar q\cdot \widetilde w)\,(\widetilde w\cdot q)}
{(m_{[9]}^{(c)})^2\,\big(\barwslash+m_{[9]}^{(c)}\big)}
\nonumber\\
&-&\frac{1}{3}\left(\barpslash -m_{[9]}^{(c)}-\frac{q \cdot \widetilde w}{m_{[9]}^{(c)}}\right)
\frac{1}{\barwslash-m_{[9]}^{(c)}}
\left(\pslash -m_{[9]}^{(c)}-\frac{\widetilde w\cdot \bar q}{m_{[9]}^{(c)}}\right)
\nonumber\\
&-&\frac{\widetilde w\cdot (q+\bar q)}{3\,m_{[9]}^{(c)}}
+\frac{1}{3}\,\Big(\wslash-m_{[9]}^{(c)}\Big)\Bigg)
\nonumber\\
&-&\frac{Z_{[9]}}{3\,(m_{[9]}^{(c)})^2}\,\Big(
\barqslash \,\big(\widetilde w\cdot q\big)
+\big(\bar q \cdot \widetilde w\big) \,\qslash
-m_{[9]}^{(c)}\,\big(
\barqslash\,\qslash-2\,(\bar q\cdot q) \big) \Big)
\nonumber\\
&+&\frac{(Z_{[9]})^2}{6\,(m_{[9]}^{(c)})^2}\,
\Big( \barpslash \;\barwslash\,\pslash -\widetilde w^2\,\wslash
-2\,m_{[9]}^{(c)}\,\big(\barqslash\,\qslash -2\,q\cdot q\big) \Big)\Bigg)
\label{8stern}\end{aligned}$$ where the index $c\to (\Lambda(1520),N(1520), \Lambda (1690), \Sigma(1680) ,\Xi(1820))$ extends over the nonet resonance states. The coefficient matrices $C^{(I,c)}_{[9]}$ and $\widetilde C^{(I,c)}_{[9]}$ are constructed by analogy with those of the octet and decuplet contributions (see (\[G-explicit\],\[def-not\])). In particular one has $C_{B^*(c)}^{(I)}=C_{B(c)}^{(I)}$ and $\tilde C_{B^*(c)}^{(I)}=\tilde C_{B(c)}^{(I)}$. The coupling constants $A^{(B^*)}_{\Phi B}=G^{(B)}_{\Phi B}(F_{[9]},D_{[9]})$ are given in terms of the $F_{[9]}$ and $D_{[9]}$ parameters introduced in (\[lag-Q\]) for all contributions except those for the $\Lambda (1520)$ and $\Lambda (1690)$ resonances $$\begin{aligned}
&& G_{\bar K N}^{(\Lambda (1520))} = \sqrt{{\textstyle{2\over 3}}}\,\Big( 3\,F_{[9]}+D_{[9]} \Big)\,\sin \vartheta
+\sqrt{3}\,C_{[9]}\,\cos \vartheta
\;,\quad
\nonumber\\
&& G_{\pi \Sigma}^{(\Lambda(1520))} = -2\,D_{[9]} \,\sin \vartheta
+{\textstyle{3\over \sqrt{2}}}\,C_{[9]}\,\cos \vartheta\;,\quad
\nonumber\\
&& G_{\eta \,\Lambda}^{(\Lambda(1520))} = {\textstyle{2\over\sqrt{3}}}\,D_{[9]}\,\sin \vartheta
+\sqrt{{\textstyle{3\over 2}}}\,C_{[9]}\,\cos \vartheta\;,\quad
\nonumber\\
&& G_{K \Xi}^{(\Lambda(1520))} = \sqrt{{\textstyle{2\over 3}}}\,\Big( 3\,F_{[9]}-D_{[9]}\Big)\,\sin \vartheta
-\sqrt{3}\,C_{[9]}\,\cos \vartheta\;
\label{G-9-explicit-1}\end{aligned}$$ and $$\begin{aligned}
&& G_{\bar K N}^{(\Lambda (1690))} = -\sqrt{{\textstyle{2\over 3}}}\,\Big( 3\,F_{[9]}+D_{[9]} \Big)\,\cos \vartheta
+\sqrt{3}\,C_{[9]}\,\sin \vartheta
\;,\quad
\nonumber\\
&& G_{\pi \Sigma}^{(\Lambda(1690))} = 2\,D_{[9]} \,\cos \vartheta
+{\textstyle{3\over \sqrt{2}}}\,C_{[9]}\,\sin \vartheta\;,\quad
\nonumber\\
&& G_{\eta \,\Lambda}^{(\Lambda(1690))} = -{\textstyle{2\over\sqrt{3}}}\,D_{[9]}\,\cos \vartheta
+\sqrt{{\textstyle{3\over 2}}}\,C_{[9]}\,\sin \vartheta\;,\quad
\nonumber\\
&& G_{K \Xi}^{(\Lambda(1690))} = -\sqrt{{\textstyle{2\over 3}}}\,\Big( 3\,F_{[9]}-D_{[9]}\Big)\,\cos \vartheta
-\sqrt{3}\,C_{[9]}\,\sin \vartheta\;.
\label{G-9-explicit-2}\end{aligned}$$
The baryon octet field $B^*_\mu $ asks for special considerations in a chiral SU(3) scheme as there is no straightforward systematic approximation strategy. Given that the baryon octet and decuplet states are degenerate in the large $N_c$ limit, it is natural to impose $m_{[10]} -m_{[8]} \sim Q$. In contrast with that there is no fundamental reason to insist on $m_{N(1520)}-m_N \sim Q$, for example. But, only with $m_{[9]}-m_{[8]} \sim Q $ is it feasible to establish consistent power counting rules needed for the systematic evaluation of the chiral Lagrangian. Note that the presence of the baryon octet resonance states in the large $N_c$ limit of QCD is far from abvious. Our opinion differs here from the one expressed in [@Carone-1; @Carone-2] where the d-wave baryon octet resonance states are considered as part of an excited large $N_c$ ${\bf 70}$-plet. Recall that reducible diagrams summed by the Bethe-Salpeter equation are typically enhanced by a factor of $2 \pi$ relatively to irreducible diagrams. We conclude that there are a priori two possibilities: the baryon resonances are a consequence of important coupled-channel dynamics or they are already present in the interaction kernel. An expansion in $2 \pi/N_c $, in our world with $N_c=3$, does not appear useful. The fact that baryon resonances exhibit large hadronic decay widths may be taken as an indication that the coupled-channel dynamics is the driving mechanism for the creation of baryon resonances. Related arguments have been put forward in [@Aaron:1; @Aaron:2]. Indeed, for instance the $N(1520)$ resonance, was successfully described in terms of coupled channel dynamics, including the vector-meson nucleon channels as an important ingredient [@QM-lutz], but without assuming a preformed resonance structure in the interaction kernel. The successful description of the $\Lambda (1405)$ resonance in our scheme (see Fig. \[fig:wt\]) supports the above arguments. For a recent discussion of the competing picture in which the $\Lambda (1405)$ resonance is considered as a quark-model state we refer to [@Kimura].
The description of resonances has a subtle consequence for the treatment of the u-channel baryon resonance exchange contribution. If a resonance is formed primarily by the coupled channel dynamics one should not include an explicit bare u-channel resonance contribution in the interaction kernel. The then necessarily strong energy dependence of the resonance self-energy would invalidate the use of (\[8stern\]), because for physical energies $\sqrt{s}> m_N+m_\pi$ the resonance self-energy is probed far off the resonance pole. Our discussion has non-trivial implications for the chiral SU(3) dynamics. Naively one may object that the effect of the u-channel baryon resonance exchange contribution in (\[8stern\]) can be absorbed to good accuracy into chiral two-body interaction terms in any case. However, while this is true in a chiral $SU(2)$ scheme, this is no longer possible in a chiral $SU(3)$ approach. This follows because chiral symmetry selects a specific subset of all possible $SU(3)$-symmetric two-body interaction terms at given chiral order. In particular one finds that the effect of the $Z_{[9]}$ parameter in (\[8stern\]) can not be absorbed into the chiral two-body parameters $g^{(S)}, g^{(V)}$ or $g^{(T)}$ of (\[two-body\]). For that reason we discriminate two possible scenarios. In scenario I we conjecture that the baryon octet resonance states are primarily generated by the coupled channel dynamics of the vector-meson baryon-octet channels. Therefore in this scenario the u-channel resonance exchange contribution of (\[8stern\]) is neglected and only its s-channel contribution is included as a reminiscence of the neglected vector meson channels. Note that this is analogous to the treatment of the $\Lambda (1405)$ resonance, which is generated dynamically in the chiral $SU(3)$ scheme (see Fig. \[fig:wt\]). Here a s-channel pole term is generated by the coupled channel dynamics but the associated u-channel term is effectively left out as a much suppressed contribution. In scenario II we explicitly include the s- and the u-channel resonance exchange contributions as given in (\[8stern\]), thereby assuming that the resonance was preformed already in the large $N_c$ limit of QCD and only slightly affected by the coupled channel dynamics. Our detailed analyses of the data set clearly favor scenario I. The inclusion of the u-channel resonance exchange contributions appears to destroy the subtle balance of chiral s-wave range terms and makes it impossible to obtain a reasonable fit to the data set. Thus all results presented in this work will be based on scenario I. Note that in that scenario the background parameter $Z_{[9]}$ drops out completely (see (\[v-result-1\])).
=1.1mm
[|r||c||c|c||c|c|c||c|c|c|c||c|c|c|]{}\
\
& $C_{\pi,0}^{(\frac{1}{2})}$ & $C_{\pi,D}^{(\frac{1}{2})}$ & $C_{\pi,F}^{(\frac{1}{2})}$ & $C_{K,0}^{(\frac{1}{2})}$ &${C}_{K,D}^{(\frac{1}{2})}$ & ${C}_{K,F}^{(\frac{1}{2})}$ & ${C}_{0}^{(\frac{1}{2})}$ & ${C}_{1}^{(\frac{1}{2})}$ &${C}_{D}^{(\frac{1}{2})}$ & ${C}_{F}^{(\frac{1}{2})}$& $\bar C_{1}^{(\frac{1}{2})}$ & $\bar C_{D}^{(\frac{1}{2})}$& $\bar C_{F}^{(\frac{1}{2})}$\
$11$& -4 & -2 & -2 & 0 & 0& 0 & 2 & 0 & 1 & 1 & 0 & 2 & 2\
$12$& 0 & $\frac{1}{2}$ & -$\frac{1}{2}$ & 0 & $\frac{1}{2}$ & -$\frac{1}{2}$ & 0 & 1 & -$\frac{1}{2}$ & $\frac{1}{2}$ &-1 & -$\frac{1}{2}$ & $\frac{1}{2}$\
$13$& 0 & -2& -2 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0\
$14$& 0 & $\frac{1}{2}$ & $\frac{3}{2}$ & 0 & $\frac{1}{2}$ & $\frac{3}{2}$ & 0 & 0 & -$\frac{1}{2}$ & -$\frac{3}{2}$ & 0 & -$\frac{1}{2}$ & -$\frac{3}{2}$\
$22$& 0 & 0 & 0 & -4 & -2 & 4 & 2 & 0 & 1 & -2 & 0 & -1 & 2\
$23$& 0 & -$\frac{3}{2}$ & $\frac{3}{2}$ & 0 & $\frac{5}{2}$ & -$\frac{5}{2}$ & 0 & 0 & -$\frac{1}{2}$ & $\frac{1}{2}$ & 0 & $\frac{3}{2}$ & -$\frac{3}{2}$\
$24$& 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 1 & 0 & 0 &-1 & 0\
$33$& $\frac{4}{3}$ & 2 & -$\frac{10}{3}$ & -$\frac{16}{3}$& -$\frac{16}{3}$ & $\frac{16}{3}$ & 2 & 0 & $\frac{5}{3}$ & -1 & 0 & 0 & 0\
$34$& 0 & $\frac{1}{2}$ & $\frac{3}{2}$ & 0 & -$\frac{5}{6}$ & -$\frac{5}{2}$ & 0 & 1 & $\frac{1}{6}$ & $\frac{1}{2}$ & -1 & -$\frac{1}{2}$ & -$\frac{3}{2}$\
$44$& 0 & 0 & 0 &-4 & -$\frac{10}{3}$ & 0 &2 & 0 & $\frac{5}{3}$ & 0 & 0 & 1& 0\
& $C_{\pi,0}^{(\frac{3}{2})}$ & $C_{\pi,D}^{(\frac{3}{2})}$ & $C_{\pi,F}^{(\frac{3}{2})}$ & $C_{K,0}^{(\frac{3}{2})}$ & ${C}_{K,D}^{(\frac{3}{2})}$ & ${C}_{K,F}^{(\frac{3}{2})}$ & ${C}_{0}^{(\frac{3}{2})}$ & ${C}_{1}^{(\frac{3}{2})} $ & ${C}_{D}^{(\frac{3}{2})}$ & ${C}_{F}^{(\frac{3}{2})}$ & $\bar C_{1}^{(\frac{3}{2})}$ & $\bar C_{D}^{(\frac{3}{2})}$ & $\bar C_{F}^{(\frac{3}{2})}$\
$11$&-4 & -2 & -2 & 0& 0 & 0 & 2& 0 & 1 & 1 & 0 & -1 & -1\
$12$& 0 & -1 & 1 & 0 & -1 & 1 & 0 & 1 & 1 & -1 & -1 & 1 & -1\
$22$& 0 & 0 & 0 & -4 & -2 & -2 & 2 & 0 & 1 & 1 & 0 & -1 & -1\
We turn to the quasi-local two-body interaction terms $K_{[8][8]}^{(I)}$ and $K_{\chi}^{(I)}$ in (\[k-all\]). It is convenient to represent the strength of an interaction term in a given channel $(I,a)\to (I,b)$ by means of the dimensionless coupling coefficients $\big[C^{(I)}_{..}\big]_{ab}$. The $SU(3)$ structure of the interaction terms ${\mathcal L}^{(S)}$ and ${\mathcal L}^{(V)}$ in (\[two-body\]) are characterized by the coefficients $C^{(I)}_{0}$, $C^{(I)}_{1}$, $C^{(I)}_{D}$, $C^{(I)}_{F}$ and ${\mathcal L}^{(T)}$ by $\bar C^{(I)}_{1}$, $\bar C^{(I)}_{D}$ and $\bar C^{(I)}_{F}$. Similarly the interaction terms (\[chi-sb\]) which break chiral symmetry explicitly are written in terms of the coefficients $C^{(I)}_{\pi,0}$, $C^{(I)}_{\pi,D}$, $C^{(I)}_{\pi,F}$ and $C^{(I)}_{K,0}$, $C^{(I)}_{K,D}$, $C^{(I)}_{K,F}$. We have $$\begin{aligned}
K^{(I)}_{[8][8]}(\bar k, k; w)&=& \frac{\bar q \cdot q }{4 f^2} \,
\Big(g^{(S)}_0\,C^{(I)}_0+g^{(S)}_1\,C^{(I)}_1 + g^{(S)}_D\,C^{(I)}_D
+g^{(S)}_F\,C^{(I)}_F\Big)
\nonumber\\
&+&\frac{1}{16 f^2}\,\Big( \qslash\,\big( p+\bar p\big) \cdot \bar q
+\barqslash \,\big(p+ \bar p\big) \cdot q\Big)\,
\Big( g^{(V)}_0\,C^{(I)}_0+g^{(V)}_1\,C^{(I)}_1 \Big)
\nonumber\\
&+&\frac{1}{16 f^2}\,\Big( \qslash\,\big( p+\bar p\big) \cdot \bar q
+\barqslash \,\big(p+ \bar p\big) \cdot q\Big)
\,\Big( g^{(V)}_D\,C^{(I)}_D +g^{(V)}_F\,C^{(I)}_F \Big)
\nonumber\\
&+&\frac{i}{4 f^2}\,\Big( \bar q^\mu\,\sigma_{\mu \nu}\,q^\nu \Big)
\,\Big( g^{(T)}_1\,\bar C^{(I)}_{1}+g^{(T)}_{D}\,\bar C^{(I)}_{D} +g^{(T)}_F\,\bar C^{(I)}_{F}\Big)\;,
\nonumber\\
K^{(I)}_{\chi}(\bar k, k; w)&=&
\frac{b_0}{f^2}\,\Big(
m_\pi^2\,C^{(I)}_{\pi,0} +m_K^2\,C^{(I)}_{K,0} \Big)
+\frac{b_D}{f^2}\,\Big(
m_\pi^2\,C^{(I)}_{\pi,D} +m_K^2\,C^{(I)}_{K,D} \Big)
\nonumber\\
&+&\frac{b_F}{f^2}\,\Big(
m_\pi^2\,C^{(I)}_{\pi,F} +m_K^2\,C^{(I)}_{K,F} \Big)\;,
\label{def-local-1}\end{aligned}$$ where the coefficients $C^{(I)}_{..,ab}$ are listed in Tab. \[tabpi-2\] for the strangeness zero channel and shown in Appendix E for the strangeness minus one channel. A complete listing of the $Q^3$ quasi-local two-body interaction terms can be found in Appendix B.
Chiral expansion and covariance
-------------------------------
We proceed and detail the chiral expansion strategy for the interaction kernel in which we wish to keep covariance manifestly. Here it is instructive to rewrite first the meson energy $\omega^{(I)}_a$ and the baryon energy $E_a^{(I)}$ in (\[bark-def\]) $$\begin{aligned}
&&\omega^{(I)}_a(\sqrt{s}) =\sqrt{s}-m^{(I)}_{B(I,a)}
-\frac{\phi^{(I)}_a(\sqrt{s})}{2\,\sqrt{s}}\,,
\;\;\;E_a^{(I)}(\sqrt{s}) = m_{B(I,a)}+\frac{\phi^{(I)}_a(\sqrt{s})}{2\,\sqrt{s}}\;,
\nonumber\\
&&\phi^{(I)}_{a}(\sqrt{s})= \Big(\sqrt{s}-m_{B(I,a)}\Big)^2-m_{\Phi(I,a)}^2 \,,
\label{rel-power}\end{aligned}$$ in terms of the approximate phase-space factor $\phi^{(I)}_{a}(\sqrt{s}\,)$. We assign $\sqrt{s}\sim Q^0$ and $\phi^{(I)}_a/ \sqrt{s} \sim Q^2$. This implies a unique decomposition of the meson energy $ \omega^{(I)}_a $ in the leading term $ \sqrt{s}- m_{B(I,a)} \sim Q$ and the subleading term $-\phi_a^{(I)}/(2 \sqrt{s})\sim Q^2$. We stress that our assignment leads to $m_{B(I,a)}$ as the leading chiral moment of the baryon energy $E_a^{(I)}$. We differ here from the conventional heavy-baryon formalism which assigns the chiral power $Q$ to the full meson energy $\omega_a^{(I)}$. Consistency with (\[rel-power\]), in particular with $E_a^{(I)}=\sqrt{s}- \omega_a^{(I)}$, then results in either a leading chiral moment of the baryon energy $E_a^{(I)}$ not equal to the baryon mass or an assignment $\sqrt{s}\; \simslash \;Q^0$.
The implications of our relativistic power counting assignment are first exemplified for the case of the quasi-local two-body interaction terms. We derive the effective interaction kernel relevant for the s- and p-wave channels $$\begin{aligned}
V^{(I,+)}_{[8][8]}(\sqrt{s};0)&=& \frac{1}{4 f^2}\,
\Big(g^{(S)}_0+\sqrt{s}\,g_0^{(V)}\Big)\,
\Big( \sqrt{s}-M^{(I)}\Big)\,C_0^{(I)}\,\Big( \sqrt{s}-M^{(I)}\Big)
\nonumber\\
&+& \frac{1}{4 f^2}\,\Big(g^{(S)}_1+\sqrt{s}\,g_1^{(V)}\Big)\,
\Big( \sqrt{s}-M^{(I)}\Big)\,C_1^{(I)}\,\Big( \sqrt{s}-M^{(I)}\Big)
\nonumber\\
&+& \frac{1}{4 f^2}\, \Big(g^{(S)}_D+\sqrt{s}\,g_D^{(V)}\Big)\,
\Big( \sqrt{s}-M^{(I)}\Big)\,C_D^{(I)}\,\Big( \sqrt{s}-M^{(I)}\Big)
\nonumber\\
&+& \frac{1}{4 f^2}\,\Big(g^{(S)}_F+\sqrt{s}\,g_F^{(V)}\Big)\,
\Big( \sqrt{s}-M^{(I)}\Big)\,C_F^{(I)}\,\Big( \sqrt{s}-M^{(I)}\Big)
\nonumber\\
&+&{\mathcal O} \left(Q^3\right)\;,
\nonumber\\
V^{(I,-)}_{[8][8]}(\sqrt{s};0)&=&
-\frac{1}{3 f^2} \,M^{(I)}\,
\Big(g^{(S)}_0\,C_0^{(I)}+g^{(S)}_1\,C_1^{(I)}\Big)\, M^{(I)}
\nonumber\\
&-&\frac{1}{3 f^2} \,M^{(I)}\,
\Big(g^{(S)}_D\,C_D^{(I)} +g^{(S)}_F\,C_F^{(I)}\Big)\, M^{(I)}
\nonumber\\
&+& \frac{1}{4 f^2} \left(\sqrt{s}+M^{(I)}\right)
\Big(g^{(T)}_1\,\bar C_{1}^{(I)}+g^{(T)}_D\,\bar C_{D}^{(I)}\Big)
\left(\sqrt{s}+M^{(I)}\right)
\nonumber\\
&+& \frac{1}{4 f^2} \left(\sqrt{s}+M^{(I)}\right)
g^{(T)}_F\,\bar C_{F}^{(I)} \left(\sqrt{s}+M^{(I)}\right)
\nonumber\\
&-&\frac{1}{3 f^2}\,M^{(I)}\,
\Big(g^{(T)}_1\,\bar C_{1}^{(I)}+g^{(T)}_D\,\bar C_{D}^{(I)}+g^{(T)}_F\,\bar C_{F}^{(I)} \Big)\,
M^{(I)}
+ {\mathcal O} \left(Q\right)\;,
\nonumber\\
V^{(I,+)}_{[8][8]}(\sqrt{s};1)&=& -\frac{1}{12 f^2}\,
\Big(g^{(S)}_0\,C_0^{(I)}+g^{(S)}_1\,C_1^{(I)}+g^{(S)}_D\,C_D^{(I)}
+g^{(S)}_F\,C_F^{(I)}\Big)
\nonumber\\
&-&\frac{1}{12 f^2}\,
\Big(g^{(T)}_1\,\bar C_{1}^{(I)}+g^{(T)}_D\,\bar C_{D}^{(I)}+g^{(T)}_T\,\bar C_{F}^{(I)}\Big)
+ {\mathcal O} \left(Q \right)\;.
\nonumber\\
V^{(I,+)}_{\chi}(\sqrt{s},0)&=& \frac{b_0}{f^2}\,\Big(
m_\pi^2\,C^{(I)}_{\pi,0} +m_K^2\,C^{(I)}_{K ,0} \Big)
+\frac{b_D}{f^2}\,\Big(
m_\pi^2\,C^{(I)}_{\pi,D} +m_K^2\,C^{(I)}_{K ,D} \Big)
\nonumber\\
&+&\frac{b_F}{f^2}\,\Big(
m_\pi^2\,C^{(I)}_{\pi,F} +m_K^2\,C^{(I)}_{K ,F}\Big)
+{\mathcal O}\left(Q^3\right)\;,
\label{local-v}\end{aligned}$$ where we introduced the diagonal baryon mass matrix $M^{(I)}_{ab}=\delta_{ab}\,m_{B(I,a)}$. Our notation in (\[local-v\]) implies a matrix multiplication of the mass matrix $M^{(I)}$ by the coefficient matrices $C^{(I)}$ in the ’$ab$’ channel-space. Recall that the d-wave interaction kernel $V^{(I,-)}_{[8][8]}(\sqrt{s},1)$ does not receive any contribution from quasi-local counter terms to chiral order $Q^3$. Similarly the chiral symmetry-breaking interaction kernel $V_\chi$ can only contribute to s-wave scattering to this order. We point out that the result (\[local-v\]) is uniquely determined by expanding the meson energy according to (\[rel-power\]). In particular we keep in $V^{(I,+)}_{\Phi B}(\sqrt{s},0)$ the $\sqrt{s}$ factor in front of $g^{(V)}$. We refrain from any further expansion in $\sqrt{s}-\Lambda $ with some scale $\Lambda \simeq m_N$. The relativistic chiral Lagrangian supplied with (\[rel-power\]) leads to well defined kinematical factors included in (\[local-v\]). These kinematical factors, which are natural ingredients of the relativistic chiral Lagrangian, can be generated also in the heavy-baryon formalism by a proper regrouping of interaction terms. The $Q^3$-terms induced by the interaction kernel (\[def-local-1\]) are shown in Appendix B as part of a complete collection of relevant $Q^3$-terms.
Next we consider the Weinberg-Tomozawa term and the baryon octet and decuplet s-channel pole contributions $$\begin{aligned}
V^{(I,\pm)}_{WT}(\sqrt{s};0)&=&\frac{1}{2\,f^2}\,\left( \sqrt{s}\,C^{(I)}_{WT}
\mp \frac{1}{2}\,\Big[M^{(I)}\,,C^{(I)}_{WT}\Big]_+ \right) \;,
\nonumber\\
V^{(I,\pm)}_{s-[8]}(\sqrt{s};0)
&=& -\sum_{c=1}^3\,\Big(\sqrt{s}\mp M^{(I,c)}_5 \Big)\,
\frac{C^{(I,c)}_{[8]}}{4\,f^2\,\big(\sqrt{s}\pm m^{(c)}_{[8]}\big) }\,
\Big(\sqrt{s}\mp M^{(I,c)}_5 \Big)\,,
\nonumber\\
V^{(I,+)}_{s-[10]}(\sqrt{s};0)
&=& -\frac{2}{3}\,\sum_{c=1}^2\,\frac{\sqrt{s}+m^{(c)}_{[10]}}{(m^{(c)}_{[10]})^2}\,
\Big(\sqrt{s}- M^{(I)} \Big)\,\frac{C^{(I,c)}_{[10]}}{4\,f^2 }\,\Big(\sqrt{s}- M^{(I)} \Big)
\nonumber\\
&+&\sum_{c=1}^2\,Z_{[10]}\,\frac{2\,\sqrt{s}-m^{(c)}_{[10]}}{3\,(m^{(c)}_{[10]})^2}\,
\Big(\sqrt{s}- M^{(I)} \Big)\,\frac{C^{(I,c)}_{[10]}}{4\,f^2 }\,\Big(\sqrt{s}- M^{(I)} \Big)
\nonumber\\
&+&\sum_{c=1}^2\,Z_{[10]}^2\,\frac{2\,m^{(c)}_{[10]}-\sqrt{s}}{6\,(m^{(c)}_{[10]})^2}\,
\Big(\sqrt{s}- M^{(I)} \Big)\,\frac{C^{(I,c)}_{[10]}}{4\,f^2 }\,\Big(\sqrt{s}- M^{(I)} \Big)
\nonumber\\
&+&{\mathcal O}\left(Q^3 \right)\;,
\nonumber\\
V^{(I,-)}_{s-[10]}(\sqrt{s};0)
&=&\sum_{c=1}^2\, \frac{Z_{[10]}}{3\,m^{(c)}_{[10]}}\,
\Big(\sqrt{s}+ M^{(I)} \Big)\,\frac{C^{(I,c)}_{[10]}}{4\,f^2 }\,\Big(\sqrt{s}+ M^{(I)} \Big)
\nonumber\\
&-&\sum_{c=1}^2\,Z_{[10]}^2\,\frac{\sqrt{s}+2\,m^{(c)}_{[10]}}{6\,(m^{(c)}_{[10]})^2}\,
\Big(\sqrt{s}+ M^{(I)} \Big)\,\frac{C^{(I,c)}_{[10]}}{4\,f^2 }\,\Big(\sqrt{s}+ M^{(I)} \Big)
\nonumber\\
&+&{\mathcal O}\left(Q \right)\;,
\nonumber\\
V^{(I,+)}_{s-[10]}(\sqrt{s};+1)
&=& -\frac{1}{3}\,\sum_{c=1}^2\,
\frac{C^{(I,c)}_{[10]}}{4\,f^2\,(\sqrt{s}-m^{(c)}_{[10]})}
+{\mathcal O}\left(Q \right)\;,
\nonumber\\
V^{(I,-)}_{s-[9]}(\sqrt{s};-1)
&=& -\frac{1}{3}\,\sum_{c=1}^2\,
\frac{C^{(I,c)}_{[9]}}{4\,f^2\,(\sqrt{s}-m^{(c)}_{[9]})}
+{\mathcal O}\left(Q^0 \right)\;
\label{v-result-1}\end{aligned}$$ where we suppressed terms of order $Q^3$ and introduced: $$\begin{aligned}
\Big[M^{(I,c)}_5 \Big]_{ab} = \delta_{ab}\, \Big( m_{B(I,a)}+R^{(I,c)}_{L,aa}\Big) \;.
\label{}\end{aligned}$$ The Weinberg-Tomozawa interaction term $V_{WT}$ contributes to the s-wave and p-wave interaction kernels with $J=\frac{1}{2}$ to chiral order $Q$ and $Q^2$ respectively but not in the $J=\frac{3}{2}$ channels. Similarly the baryon octet s-channel exchange $V_{s-[8]}$ contributes only in the $J=\frac{1}{2}$ channels and the baryon decuplet s-channel exchange $V_{s-[10]}$ to all considered channels but the d-wave channel. The $Q^3$-terms not shown in (\[v-result-1\]) are given in Appendix B. In (\[v-result-1\]) we expanded also the baryon-nonet resonance contribution applying in particular the questionable formal rule $\sqrt{s}-m_{[9]}\sim Q$. To order $Q^3$ one then finds that only the d-wave interaction kernels are affected. In fact, the vanishing of all contributions except the one in the d-wave channel does not depend on the assumption $\sqrt{s}-m_{[9]}\sim Q$. It merely reflects the phase space properties of the resonance field. We observe that our strategy preserves the correct pole contribution in $V^{(I,-)}_{s-[9]}(\sqrt{s};-1)$ but discards smooth background terms in all partial-wave interaction kernels. That is consistent with our discussion of section 4.1 which implies that those background terms are not controlled in any case. We emphasize that we keep the physical mass matrix $M^{(I)}$ rather than the chiral $SU(3)$ limit value in the interaction kernel. Since the mass matrix follows from the on-shell reduction of the interaction kernel it necessarily involves the physical mass matrix $M^{(I)}$. Similarly we keep $M^{(I)}_5$ unexpanded since only that leads to the proper meson-baryon coupling strengths. This procedure is analogous to keeping the physical masses in the unitarity loop functions $J_{MB}(w)$ in (\[jpin-n-def\]).
We proceed with the baryon octet and baryon decuplet u-channel contributions. After performing their proper angular average as implied by the partial-wave projection in (\[bark-def\]) their contributions to the scattering kernels are written in terms of matrix valued functions ${ h}^{(I)}_{n \pm}(\sqrt{s},m),{q}^{(I)}_{n \pm }(\sqrt{s},m)$ and ${ p}^{(I)}_{n \pm }(\sqrt{s},m)$ as $$\begin{aligned}
\Big[V^{(I,\pm )}_{u-[8]}(\sqrt{s};n)\Big]_{ab} &=& \sum_{c=1}^4\,
\frac{1}{4\,f^2 }\,\Big[\widetilde C^{(I,c)}_{[8]}\Big]_{ab}\,
\Big[{ h}^{(I)}_{n \pm }(\sqrt{s},m^{(c)}_{[8]})\Big]_{ab} \;,
\nonumber\\
\Big[V^{(I,\pm )}_{u-[10]}(\sqrt{s};n)\Big]_{ab} &=&\sum_{c=1}^3\,
\frac{1}{4\,f^2 }\,\Big[\widetilde C^{(I,c)}_{[10]}\Big]_{ab}\,
\Big[{ p}^{(I)}_{n \pm }(\sqrt{s},m_{[10]}^{(c)})\Big]_{ab}\; ,
\nonumber\\
\Big[V^{(I,\pm )}_{u-[9]}(\sqrt{s};n)\Big]_{ab} &=& \sum_{c=1}^4\,
\frac{1}{4\,f^2 }\,\Big[\widetilde C^{(I,c)}_{[9]}\Big]_{ab}\,
\Big[{q}^{(I)}_{n \pm }(\sqrt{s},m^{(c)}_{[9]})\Big]_{ab} \;.
\label{u-result}\end{aligned}$$ The functions ${h}^{(I)}_{n \pm}(\sqrt{s},m), { q}^{(I)}_{n \pm }(\sqrt{s},m)$ and ${p}^{(I)}_{n \pm}(\sqrt{s},m)$ can be expanded in chiral powers once we assign formal powers to the typical building blocks $$\begin{aligned}
\mu_{\pm,ab}^{(I)}({s},m)= m_{B(I,a)}+m_{B(I,b)}-\sqrt{s}\pm m \;.
\label{mupm-scale}\end{aligned}$$ We count $\mu_- \sim Q$ and $\mu_+ \sim Q^0$ but refrain from any further expansion. Then the baryon octet functions ${h}^{(I)}_{n \pm}(\sqrt{s},m)$ to order $Q^3$ read $$\begin{aligned}
&&\Big[{h}_{0+}^{(I)} (\sqrt{s},m)\Big]_{ab}=\sqrt{s}+m +\tilde R^{(I,c)}_{L,ab}
+\tilde R^{(I,c)}_{R,ab}-
\frac{m+M_{ab}^{(L)}}{\mu_{+,ab}^{(I)}({s},m)}\,
\Bigg(\sqrt{s}+
\nonumber\\
&& \qquad\qquad
+\frac{\phi^{(I)}_{a}({s})\,(\sqrt{s}-m_{B(I,b)})}
{\mu_{+,ab}^{(I)}({s},m)\,\mu_{-,ab}^{(I)}({s},m)}
+\frac{(\sqrt s-m_{B(I,a)})\,\phi^{(I)}_{b}({s})}
{\mu_{+,ab}^{(I)}({s},m)\,\mu_{-,ab}^{(I)}({s},m)}
\nonumber\\
&& \qquad\qquad
+\frac{1}{3}\,\frac{\phi^{(I)}_{a}({s})\,(4\,\sqrt s -\mu_{+,ab}^{(I)}({s},m))\,\phi^{(I)}_{b}({s}) }
{\big( \mu_{+,ab}^{(I)}({s},m)\big)^2\,\big(\mu_{-,ab}^{(I)}({s},m)\big)^2}
\Bigg)\,\frac{m+M_{ab}^{(R)}}{\sqrt{s}}+ {\mathcal O}\left( Q^3\right)\,,
\nonumber\\
&&\Big[{h}_{0-}^{(I)} (\sqrt{s},m)\Big]_{ab}=
\frac{m+M_{ab}^{(L)}}{\mu_{-,ab}^{(I)}({s},m)}\,
\Bigg(2\,\sqrt{s}+\mu_{-,ab}^{(I)}({s},m)
\nonumber\\
&&\qquad\qquad
-\frac{8}{3}\,\frac{m_{B(I,a)}\,m_{B(I,b)}}
{ \mu_{+,ab}^{(I)}({s},m)} \Bigg)\,
\frac{m+M_{ab}^{(R)}}{\mu_{+,ab}^{(I)}({s},m)}
+ {\mathcal O}\left( Q\right)\,,
\nonumber\\
&&\Big[{h}_{1+}^{(I)} (\sqrt{s},m)\Big]_{ab}=-\frac23\,
\frac{(m+M_{ab}^{(L)})\,(m+M_{ab}^{(R)}) }{\mu_{-,ab}^{(I)}({s},m)\,
\big(\mu_{+,ab}^{(I)}({s},m)\big)^2}
+{\mathcal O}\left( Q\right)
\nonumber\\
\nonumber\\
&&\Big[{h}_{1-}^{(I)} (\sqrt{s},m)\Big]_{ab}=
-\frac{32}{15}\,
\frac{(m+M_{ab}^{(L)})\,m_{B(I,a)}\,m_{B(I,b)}\,(m+M_{ab}^{(R)}) }{\big(\mu_{-,ab}^{(I)}({s},m)\big)^2\,
\big(\mu_{+,ab}^{(I)}({s},m)\big)^3}
\nonumber\\
&&\qquad\qquad
+\frac43\,
\frac{(m+M_{ab}^{(L)})\,\sqrt{s}\,(m+M_{ab}^{(R)}) }{\big(\mu_{-,ab}^{(I)}({s},m)\big)^2\,
\big(\mu_{+,ab}^{(I)}({s},m)\big)^2}
+{\mathcal O}\left( Q^{-1}\right)
\label{u-approx-1}\;,\end{aligned}$$ where we introduced $M_{ab}^{(L)}=m_{B(I,a)}+\tilde R^{(I,c)}_{L,ab}$ and $M_{ab}^{(R)}=m_{B(I,b)}+\tilde R^{(I,c)}_{R,ab}$. The terms of order $Q^3$ can be found in Appendix G where we present also the analogous expressions for the decuplet and baryon-octet resonance exchanges. We emphasize two important points related to the expansion in (\[u-approx-1\]). First, it leads to a separable interaction kernel. Thus the induced Bethe-Salpeter equation is solved conveniently by the covariant projector method of section 3.4. Secondly, such an expansion is only meaningful in conjunction with the renormalization procedure outlined in section 3.2. The expansion leads necessarily to further divergencies which require careful attention.
We close this section with a more detailed discussion of the u-channel exchange. Its non-local nature necessarily leads to singularities in the partial-wave scattering amplitudes at subthreshold energies. For instance, the expressions (\[u-approx-1\]) as they stand turn useless at energies $\sqrt{s}\simeq m_{B(I,a)}+m_{B(I,b)}-m$ due to unphysical multiple poles at $\mu_-=0$. One needs to address this problem, because the subthreshold amplitudes are an important input for the many-body treatment of the nuclear meson dynamics. We stress that a singular behavior in the vicinity of $\mu_-\sim 0$ is a rather general property of any u-channel exchange contribution. It is not an artifact induced by the chiral expansion. The partial-wave decomposition of a u-channel exchange contribution represents the pole term only for sufficiently large or small $\sqrt{s}$. To be explicit we consider the u-channel nucleon pole term contribution of elastic $\pi N$ scattering $$\begin{aligned}
\sum_{n=0}^\infty \,\int_{-1}^1\frac{d x}{2}\,
\frac{P_n(x)}{\mu_{\pi N}^{(+)}\,\mu_{\pi N}^{(-)}-2\,\phi_{\pi N}\,x
+{\mathcal O}\left( Q^3\right)}\;,
\label{pin-example:b}\end{aligned}$$ where $\mu^{(\pm)}_{\pi N}=2\,m_N -\sqrt{s}\pm m_N$. Upon inspecting the branch points induced by the angular average one concludes that the partial wave decomposition (\[pin-example:b\]) is valid only if $\sqrt{s} >\Lambda_+$ or $\sqrt{s}<\Lambda_-$ with $\Lambda_\pm=m_N\pm m^2_\pi/m_N+{\mathcal O}(Q^3)$. For any value in between, $\Lambda_-<\sqrt{s} <\Lambda_+$, the partial-wave decomposition is not converging. We therefore adopt the following prescription for the $u$-channel contributions. The unphysical pole terms in (\[u-approx-1\]) are systematically replaced by $$\begin{aligned}
&& m\,\Lambda^{(\pm)}_{ab} (m)= m\,(m_{B(I,a)}+m_{B(I,b)})-m^2
\nonumber\\
&& \qquad \qquad \pm \left(\Big((m-m_{B(I,b)})^2 - m_{\Phi(I,a)}^2\Big)\,
\Big((m-m_{B(I,a)})^2-m_{\Phi(I,b)}^2\Big)\right)^{1/2} \;,
\nonumber\\
&& \left(\mu^{-1}_{-,ab}(\sqrt{s},m)\right)^n
\rightarrow
\left(\mu^{-1}_{-,ab}(\Lambda_{ab}^{(-)},m)\right)^n
\,\frac{\sqrt{s}-\Lambda_{ab}^{(+)}}{\Lambda_{ab}^{(-)}-\Lambda_{ab}^{(+)}}
\nonumber\\
&&\qquad \qquad \qquad \quad \;+\,
\left(\mu^{-1}_{-,ab}(\Lambda_{ab}^{(+)},m)\right)^n
\,\frac{\sqrt{s}-\Lambda_{ab}^{(-)}}{\Lambda_{ab}^{(+)}-\Lambda_{ab}^{(-)}} \;,
\label{prescription}\end{aligned}$$ for $\Lambda_{ab}^{(-)} < \sqrt{s}< \Lambda_{ab}^{(+)}$ but kept unchanged for $\sqrt{s}>\Lambda^{(+)}_{ab}$ or $\sqrt{s}<\Lambda^{(-)}_{ab}$ in a given channel (a,b). The prescription (\[prescription\]) properly generalizes the $SU(2)$ sector result $\Lambda_\pm\simeq m_N\pm m^2_\pi/m_N $ to the $SU(3)$ sector. We underline that (\[prescription\]) leads to regular expressions for $h_{\pm n}(\sqrt{s},m)$ but leaves the u-channel pole contributions unchanged above threshold. The reader may ask why we impose such a prescription at all. Outside the convergence radius of the partial-wave expansion the amplitudes do not make much sense in any case. Our prescription is nevertheless required due to a coupled-channel effect. To be specific, consider for example elastic kaon-nucleon scattering in the strangeness minus one channel. Since there are no u-channel exchange contributions in this channel the physical partial-wave amplitudes do not show any induced singularity structures. For instance in the $\pi \Sigma \to \pi \Sigma $ channel, on the other hand, the u-channel hyperon exchange does contribute and therefore leads through the coupling of the $\bar K N$ and $\pi \Sigma $ channels to a singularity also in the $\bar K N$ amplitude. Such induced singularities are an immediate consequence of the approximate treatment of the u-channel exchange contribution and must be absent in an exact treatment. As a measure for the quality of our prescription we propose the accuracy to which the resulting subthreshold forward kaon-nucleon scattering amplitudes satisfies a dispersion integral. We return to this issue in the result section.
$SU(3)$ dynamics in the $S=1$ channels
--------------------------------------
We turn to the $K^+$-nucleon scattering process which is related to the $K^-$-nucleon scattering process by crossing symmetry. The scattering amplitudes $T^{(I)}_{K N \rightarrow K N}$ follow from the $K^-$-nucleon scattering amplitudes $T^{(I)}_{\bar K N \rightarrow \bar K N}$ via the transformation $$\begin{aligned}
&&T^{(0)}_{K N \rightarrow K N}(\bar q, q; w)
=-\frac{1}{2}\, T^{(0)}_{\bar K N \rightarrow \bar K N}(-q, -\bar q; w-\bar q-q)
\nonumber\\
&&\quad \quad \quad \quad \quad
+\frac{3}{2}\,T^{(1)}_{\bar K N \rightarrow \bar K N}(-q, -\bar q; w-\bar q-q) \;,
\nonumber\\
&&T^{(1)}_{K N \rightarrow K N}(\bar q, q; w)
=+\frac{1}{2}\, T^{(0)}_{\bar K N \rightarrow \bar K N}(-q, -\bar q; w-\bar q-q)
\nonumber\\
&&\quad \quad \quad \quad \quad
+\frac{1}{2}\,T^{(1)}_{\bar K N \rightarrow \bar K N}(- q, -\bar q; w-\bar q-q) \;.
\label{cross-symmetry}\end{aligned}$$ It is instructive to work out the implication of crossing symmetry for the kaon-nucleon scattering amplitudes in more detail. We decompose the on-shell scattering amplitudes into their partial-wave amplitudes: $$\begin{aligned}
&&T^{(I)}_{K N \rightarrow K N}(\bar q, q; w) =
\frac{1}{2}\,\Bigg(\frac{\wslash}{\sqrt{w^2}}+1
\Bigg)\, F^{(I,+)}_{KN}(s,t)+ \frac{1}{2}\,\Bigg(\frac{\wslash}{\sqrt{w^2}}-1
\Bigg)\,F^{(I,-)}_{KN}(s,t)
\nonumber\\
&&\quad \quad \quad = \sum_{n=0}^\infty \,
\Big( M^{(I,+)}_{KN}(\sqrt{s};n)\,{Y}^{(+)}_n(\bar q,q;w)+
M^{(I,-)}_{KN}(\sqrt{s};n)\,{Y}^{(-)}_n(\bar q,q;w) \Big)\;,
\nonumber\\
&&T^{(I)}_{\bar K N \rightarrow \bar K N}(\bar q, q; w) =
\frac{1}{2}\,\Bigg(\frac{\wslash}{\sqrt{w^2}}+1
\Bigg)\, F^{(I,+)}_{\bar KN}(s,t)+ \frac{1}{2}\,\Bigg(\frac{\wslash}{\sqrt{w^2}}-1
\Bigg)\,F^{(I,-)}_{\bar KN}(s,t)
\nonumber\\
&&\quad \quad \quad = \sum_{n=0}^\infty \,
\Big( M^{(I,+)}_{\bar KN}(\sqrt{s};n)\,{Y}^{(+)}_n(\bar q,q;w)+
M^{(I,-)}_{\bar KN}(\sqrt{s};n)\,{Y}^{(-)}_n(\bar q,q;w) \Big)
\label{}\end{aligned}$$ where we suppressed the on-shell nucleon spinors. The crossing identity (\[cross-symmetry\]) leads to $$\begin{aligned}
&&F^{(I,\pm )}_{K N}(s,u) = \sum_{n=0}^\infty\,\Bigg(
\pm\,\frac{1}{2}\left(\frac{2\,m_N\mp\sqrt{s}}{\sqrt{u}}+1 \right) \,
\bar Y^{(c)}_{n+1}(\bar q,q; w)
\nonumber\\
&&\quad \quad \quad \mp
\frac{1}{2}\left( \frac{2\,m_N\mp\sqrt{s}}{\sqrt{u}}-1\right)
\left(\bar E+ m_N \right)^2
\,\bar Y^{(c)}_{n}(\bar q,q; w)
\Bigg)\,\bar M^{(I,+)}_{\bar K N}(\sqrt{u};n)
\nonumber\\
&&\quad \quad \quad \quad \quad +\sum_{n=0}^\infty\,\Bigg(
\pm \,\frac{1}{2}\left(\frac{2\,m_N\mp\sqrt{s}}{\sqrt{u}}-1 \right) \,
\bar Y^{(c)}_{n+1}(\bar q,q; w)
\nonumber\\
&& \quad \quad \quad
\mp\,\frac{1}{2}\left( \frac{2\,m_N\mp\sqrt{s}}{\sqrt{u}}+1\right)
\left(\bar E-m_N \right)^2
\,\bar Y^{(c)}_{n}(\bar q,q; w)\Bigg)\,\bar M^{(I,-)}_{\bar K N}(\sqrt{u};n) \,,
\nonumber\\
&& \bar E =\frac{1}{2}\,\sqrt{u}+\frac{m_N^2-m_K^2}{2\,\sqrt{u}}\;,\quad
u = 2\,m_N^2+2\,m_K^2-s+2\,p_{KN}^2\,(1-\cos \theta ) \;,
\label{kplus-amplitudes}\end{aligned}$$ where $$\begin{aligned}
&& \bar M^{(0,\pm )}_{\bar K N}(\sqrt{u};n) =-\frac{1}{2}\, M^{(0,\pm )}_{\bar K N }(\sqrt{u};n)+
\frac{3}{2}\,M^{(1,\pm )}_{\bar K N }(\sqrt{u};n)
\;,\quad \quad
\nonumber\\
&&\bar M^{(1,\pm )}_{\bar K N}(\sqrt{u};n) =+\frac{1}{2}\, M^{(0,\pm)}_{\bar K N}(\sqrt{u};n)+
\frac{1}{2}\,M^{(1,\pm )}_{\bar K N}(\sqrt{u};n)\;.
\label{kplus-amplitudes:b}\end{aligned}$$ Note that $ \bar Y^{(c)}_{n}(\bar q,q; w)=\bar Y_{n}(-q,-\bar q; w-q-\bar q)$ was introduced already in (\[cov-proj\]). The partial-wave amplitudes of the $K^+$-nucleon system can now be deduced from (\[kplus-amplitudes\]) using (\[bark-def\]).
We proceed with two important remarks. First, if the solution of the coupled-channel Bethe-Salpeter equation of the $K^-$-nucleon system of the previous sections is used to construct the $K^+$-nucleon scattering amplitudes via (\[kplus-amplitudes\]) one finds real partial-wave amplitudes in conflict with unitarity. And secondly, in any case our $K^-$-nucleon scattering amplitudes must not be applied far below the $K^-$-nucleon threshold. The first point is obvious because in the $K^+$-nucleon channel only two-particle irreducible diagrams are summed. Note that the reducible diagrams in the $KN$ sector correspond to irreducible contributions in the $\bar K N$ sector and vice versa. Since the interaction kernel of the $\bar K N$ sector is evaluated perturbatively it is clear that the scattering amplitude does not include that infinite sum of reducible diagrams required for unitarity in the crossed channel. The second point follows, because the chiral $SU(3)$ Lagrangian is an effective field theory where the heavy-meson exchange contributions are integrated out. It is important to identify the applicability domain correctly. Inspecting the singularity structure induced by the light t-channel vector meson exchange contributions one observes that they, besides restricting the applicability domain with $\sqrt{s} < \sqrt{m_N^2+m_\rho^2/4}+\sqrt{m_K^2+m_\rho^2/4}\simeq 1640$ MeV from above, induce cut-structures in between the $K^+$ and $K^-$-nucleon thresholds. This will be discussed in more detail below. Though close to the $K^+$ and $K^-$-nucleon thresholds the tree-level contributions of the vector meson exchange are successfully represented by quasi-local interaction vertices, basically the Weinberg-Tomozawa interaction term, it is not justified to extrapolate a loop evaluated with the effective $K^-$-nucleon vertex down to the $K^+$- nucleon threshold. We conclude that one should not identify our $K^-$-nucleon scattering amplitudes with the Bethe-Salpeter kernel of the $K^+$-nucleon system. This would lead to a manifestly crossing symmetric approach, a so-called ’parquet’ approximation, if set up in a self consistent manner. We reiterate the fatal drawback of a parquet type of approach within our present chiral framework: the $K^-$-nucleon amplitudes would be probed far below the $\bar K N$ threshold at $\sqrt{s}\simeq m_N-m_K$ outside their validity domain. A clear signal for the unreliability of the $K^-$-nucleon scattering amplitudes at $\sqrt{s}\simeq m_N-m_K$ is the presence of unphysical pole structures which typically arise at $\sqrt{s}< 700$ MeV. For example, the Fig. \[fig:wt\] of section 4.3 would show unphysical pole structures if extended for $\sqrt{s}< 1$ GeV.
=0.5mm
$ C_{WT}^{(I)} $ $ C_{N_{[8]}}^{(I)}$ $C_{\Lambda_{[8]}}^{(I)}$ $C_{\Sigma_{[8]}}^{(I)}$ $C_{\Delta_{[10]}}^{(I)}$ $C_{\Sigma_{[10]} }^{(I)}$ $\widetilde{C}_{N_{[8]}}^{(I)}$ $\widetilde{C}_{\Lambda_{[8]} }^{(I)}$ $\widetilde{C}_{\Sigma_{[8]}}^{(I)}$ $\widetilde {C}_{\Xi_{[8]} }^{(I)}$ $\widetilde {C}_{\Delta_{[10]} }^{(I)}$ $\widetilde {C}_{\Sigma_{[10]}}^{(I)}$ $\widetilde C_{\Xi_{[10]}}^{(I)}$
------- ------------------ ---------------------- --------------------------- -------------------------- --------------------------- ---------------------------- --------------------------------- ---------------------------------------- -------------------------------------- ------------------------------------- ----------------------------------------- ---------------------------------------- ----------------------------------- --
$I=0$ 0 0 0 0 0 0 0 -$\frac{1}{2}$ $\frac{3}{2}$ 0 0 $\frac{3}{2}$ 0
$I=1$ -2 0 0 0 0 0 0 $\frac{1}{2}$ $\frac{1}{2}$ 0 0 $\frac{1}{2}$ 0
: Weinberg-Tomozawa interaction strengths and baryon exchange coefficients in the strangeness plus one channels as defined in (\[k-nonlocal\]).[]{data-label="tabkp-1"}
For the above reasons we evaluate the $K^+$-nucleon interaction kernel perturbatively from the chiral Lagrangian. Equivalently the interaction kernels, $V_{K N}$, could be derived from the $K^-$-nucleon interaction kernels $V_{\bar K N}$ by applying the crossing identities (\[cross-symmetry\]). The interaction kernels $V_{K N}(\sqrt{s};n)$ in the $K^+$-nucleon channel are given by (\[local-v\],\[v-result-1\],\[u-result\]) with the required coefficients listed in Tab. \[tabkp-1\] and Tab. \[tabkp-2\]. By analogy with the treatment of the $K^-$-nucleon scattering process we take the $K^+$-nucleon interaction evaluated to chiral order $Q^3$ as input for the Bethe-Salpeter equation. Again we consider only those $Q^3$-correction terms which are leading in the $1/N_c$ expansion. The partial-wave scattering amplitudes $M_{KN}^{(I,\pm )}(\sqrt{s};n)$ then follow $$\begin{aligned}
M_{KN}^{(I,\pm )}(\sqrt{s};n) &=& \frac{V_{KN}^{(I,\pm )}(\sqrt{s};n)}
{1-V_{KN}^{(I,\pm)}(\sqrt{s};n)\,J^{(\pm)}_{KN}(\sqrt{s}; n)} \;
\label{M:kplus}\end{aligned}$$ where the loop functions $J^{(\pm)}_{KN}(\sqrt{s};n)$ are given in (\[result-loop:ab\]) with $m_{B(I,a)}=m_N$ and $m_{\Phi(I,a)}=m_K$. The subtraction point $\mu^{(I)}$ is identified with the averaged hyperon mass $\mu^{(I)}=(m_\Lambda+m_\Sigma)/2$.
=1.1mm
[|r||c||c|c||c|c|c||c|c|c|c||c|c|c|]{}\
\
& $C_{\pi,0}^{(I)}$ & $C_{\pi,D}^{(I)}$ & $C_{\pi,F}^{(I)}$ & $C_{K,0}^{(I)}$ & ${C}_{K,D}^{(I)}$ & ${C}_{K,F}^{(I)}$ & ${C}_{0}^{(I)}$ & ${C}_{1}^{(I)}$ & ${C}_{D}^{(I)}$ & ${C}_{F}^{(I)}$ & $\bar C_{1}^{(I)}$ & $\bar C_{D}^{(I)}$ & $\bar C_{F}^{(I)}$\
$I=0$& 0 & 0 & 0 & -4 & 0 & 4 & 2 & -1 & 0& -2 & 1 & 2& 0\
$I=1$&0 & 0 & 0 & -4 & -4& 0 & 2 & 1 & 2& 0 & -1& 0 & -2\
We point out that in our scheme we arrive at a crossing symmetric amplitude by matching the amplitudes $M^{(I,\pm)}_{KN}(\sqrt{s},n)$ and $M^{(I,\pm)}_{\bar KN}(\sqrt{s},n)$ at subthreshold energies. The matching interval must be chosen so that both the $K^+$ and $K^-$ amplitudes are still within their validity domains. A complication arises due to the light vector meson exchange contributions in the t-channel, which we already identified above to restrict the validity domain of the present chiral approach. To be explicit consider the t-channel $\omega $ exchange. It leads to a branch point at $\sqrt{s}= \Lambda_\omega $ with $\Lambda_\omega =(m_N^2-m_\omega^2/4)^{1/2}+(m_K^2-m_\omega^2/4)^{1/2} \simeq $ 1138 MeV. Consequently, one expects the partial-wave $K^+$ and $K^-$-amplitudes to be reliable for $\sqrt{s} > \Lambda_\omega $ only. That implies, however, that in the crossed channel the amplitude is needed for $\sqrt{s}>(2\,m_N^2-\Lambda^2_\omega +2\,m_K^2)^{1/2}\simeq $ 978 MeV $< \Lambda_\omega $. One may naively conclude that crossing symmetry appears outside the scope of our scheme, because the matching window is closed. Note that the minimal critical point $\Lambda_{\rm opt}$ needed to open the matching window is $$\begin{aligned}
\Lambda_{\rm opt.}\simeq \sqrt{m_N^2+m_K^2}\simeq 1061 {\rm MeV}\;.
\label{opt-match}\end{aligned}$$ It is determined by the condition $s=u$ at $\cos \theta =1$. We conclude that matching the $K^+$ and $K^-$ amplitudes requires that both amplitudes are within their applicability domain at $\sqrt{s}= \Lambda_{\rm opt.}$. The point $\sqrt{s}=\Lambda_{\rm opt.}$ is optimal, because it identifies the minimal reliability domain required for the matching of the subthreshold amplitudes. We note that the complication implied by the t-channel vector meson exchanges could be circumvented by reconstructing that troublesome branch point explicitly; after all it is determined by a tree-level diagram. On the other hand, it is clear that one may avoid this complication altogether if one considers the forward scattering amplitudes only. For the latter amplitudes the branch cut at $\sqrt{s}=\Lambda_\omega $ cancels identically and consequently the forward scattering amplitudes should be reliable for energies smaller than $ \Lambda_\omega$ also. For that reason, we demonstrate the matching for our forward scattering amplitudes only. We reconstruct the approximate forward scattering amplitudes $T^{(I)}_{KN}(s)$ in terms the partial-wave amplitudes included in our work $$\begin{aligned}
T^{(I)}_{KN}(s) &=&
\frac{1}{2\,\sqrt{s}}\left(\frac{s+m_N^2-m_K^2}{2\,m_N}+\sqrt{s} \right)
M^{(I,+)}_{KN}(\sqrt{s};0)
\nonumber\\
&+&\frac{1}{2\,\sqrt{s}}\left(\frac{s+m_N^2-m_K^2}{2\,m_N} -\sqrt{s}\right)
M^{(I,-)}_{KN}(\sqrt{s};0)
\nonumber\\
&+& \frac{1}{\sqrt{s}}\left(\frac{s+m_N^2-m_K^2}{2\,m_N}+\sqrt{s} \right)
\,p^2_{KN}\,M^{(I,+)}_{KN}(\sqrt{s};1)
\nonumber\\
&+& \frac{1}{\sqrt{s}}\left(\frac{s+m_N^2-m_K^2}{2\,m_N}-\sqrt{s} \right)
\,p^2_{KN}\,M^{(I,-)}_{KN}(\sqrt{s};1) +\cdots \,,
\label{forward-amplitude}\end{aligned}$$ where $\sqrt{s}=\sqrt{m_N^2+p_{KN}^2}+\sqrt{m_K^2+p_{KN}^2}$. The analogous expression holds for $T^{(I)}_{\bar KN}(s)$. The crossing identities for the forward scattering amplitudes $$\begin{aligned}
T^{(0)}_{KN}(s) &=& -\frac{1}{2}\,T^{(0)}_{\bar K N} (2\,m_N^2+2\,m_K^2-s)
+\frac{3}{2}\,T^{(1)}_{\bar K N} (2\,m_N^2+2\,m_K^2-s) \;,
\nonumber\\
T^{(1)}_{KN}(s) &=& +\frac{1}{2}\,T^{(0)}_{\bar K N} (2\,m_N^2+2\,m_K^2-s)
+\frac{1}{2}\,T^{(1)}_{\bar K N} (2\,m_N^2+2\,m_K^2-s)\;
\label{forward-crossing}\end{aligned}$$ are expected to hold approximatively within some matching window centered around $\sqrt{s}\simeq \Lambda_{\rm opt}$. A further complication arises due to the approximate treatment of the u-channel exchange contribution in the $K^+N$-channel. Since the optimal matching point $\Lambda_{\rm opt.}$ is not too far away from the hyperon poles with $\Lambda_{\rm opt.} \sim m_\Lambda, m_\Sigma$ it is advantageous to investigate approximate crossing symmetry for the hyperon-pole term subtracted scattering amplitudes.
We would like to stress that the expected approximate crossing symmetry is closely linked to our renormalization condition (\[ren-cond\]). Since all loop functions $J^{(\pm )}(\sqrt{s},n)$ vanish close to $\sqrt{s}= (m_\Lambda+m_\Sigma)/2$ by construction, the $K^+$ and $K^-$ amplitudes turn perturbative sufficiently close to the optimal matching point $\Lambda_{\rm opt.} $. Therefore the approximate crossing symmetry of our scheme follows directly from the crossing symmetry of the interaction kernel. In the result section our final $K^\pm N$ amplitudes are confronted with the expected approximate crossing symmetry.
Finally we observe that for pion-nucleon scattering the situation is rather different. Given the rather small pion mass, the t-channel vector meson exchange contributions do not induce singularities in between the $\pi^+$ and $\pi^-$-nucleon thresholds but restrict the applicability domain of the chiral Lagrangian to $\sqrt{s} < \sqrt{m_N^2+m_\rho^2/4}+\sqrt{m_\pi^2+m_\rho^2/4}\simeq 1420$ MeV. The approximate crossing symmetry follows directly from the perturbative character of the pion-nucleon sector.
Results
=======
In this section we present the many results of our detailed chiral $SU(3)$ analysis of the low-energy meson-baryon scattering data. We refer to our theory as the $\chi $-BS(3) approach for chiral Bethe-Salpeter dynamics of the flavor $SU(3)$ symmetry. Before delving into details we briefly summarize the main features and crucial arguments of our approach. We consider the number of colors ($N_c$) in QCD as a large parameter relying on a systematic expansion of the interaction kernel in powers of $1/N_c$. The coupled-channel Bethe-Salpeter kernel is evaluated in a combined chiral and $1/N_c$ expansion including terms of chiral order $Q^3$. We include contributions of s and u-channel baryon octet and decuplet states explicitly but only the s-channel contributions of the d-wave $J^P={\textstyle{3\over 2}}^-$ baryon nonet resonance states. Therewith we consider the s-channel baryon nonet contributions to the interaction kernel as a reminiscence of further inelastic channels not included in the present scheme like for example the $K\,\Delta_\mu $ or $K_\mu \,N$ channel. We expect all baryon resonances, with the important exception of those resonances which belong to the large $N_c$ baryon ground states, to be generated by coupled channel dynamics. Our conjecture is based on the observation that unitary (reducible) loop diagrams are typically enhanced by a factor of $2 \pi $ close to threshold relatively to irreducible diagrams. That factor invalidates the perturbative evaluation of the scattering amplitudes and leads necessarily to a non-perturbative scheme with reducible diagrams summed to all orders.
We are painfully aware of the fact that in our present scheme the explicit inclusion of the baryon resonance nonet states is somewhat leaving the systematic chiral framework, in the absence of a controlled approximation scheme. An explicit baryon resonance contribution is needed because there exist no reliable phase shift analyses of the antikaon-nucleon scattering process so far, in particular at low energies. Part of the empirical information on the p-wave amplitudes stems from the interference effects of the p-wave amplitudes with the $\Lambda(1520)$ resonance amplitude. The $\Lambda(1520)$ resonance is interpreted as a $SU(3)$ singlet state with a small admixture of a resonance octet state. We stress that the s- and p-wave channels are not affected by the baryon nonet states directly, because we discard the nonet u-channel contributions in accordance with our arguments concerning resonance generation. As a consequence all s- and p-channels are treated consistently within the chiral framework. We do not include further resonance fields in the p-wave channels, because those are not required at low energies and also because they would destroy the fundamental chiral properties of our scheme. Ultimately we expect those resonances to be generated by an extended coupled channel theory also.
The scattering amplitudes for the meson-baryon scattering processes are obtained from the solution of the coupled channel Bethe-Salpeter scattering equation. Approximate crossing symmetry of the amplitudes is guaranteed by a renormalization program which leads to the matching of subthreshold amplitudes. We first present a complete collection of the parameters as they are adjusted to the data set. In the subsequent sections we report on details of the fit strategy and confront our results with the empirical data. The result section is closed with a detailed analysis of our scattering amplitudes, demonstrating their good analyticity properties as well as their compliance with crossing symmetry.
Parameters
----------
The set of parameters is well determined by the available scattering data and weak decay widths of the baryon octet states. Typically a data point is included in the analysis if $p_{\rm lab}
< 350$ MeV. There exist low-energy elastic and inelastic $K^-p$ cross section data including in part angular distributions and polarizations. We include also the low-energy differential $K^+p \to K^+ p$ cross sections in our global fit. The empirical constraints set by the $K^+$-deuteron scattering data above $p_{\rm lab} > 350$ MeV are considered by requiring a reasonable matching to the single energy $S_{01}$ and $P_{03}$ phase shifts of [@Hashimoto]. That resolves an ambiguity in the parameter set. Finally, in order to avoid the various multi-energy $\pi N$ phase shifts imperspicuous close to threshold, we fit to the single-energy phase shifts of [@pion-phases]. This will be discussed in more detail below. We aim at a uniform quality of the data description. Therefore we form the $\chi^2$ per data point in each sector and add those up to the total $\chi^2$ which is minimized by the search algorithms of Minuit [@Minuit]. In cases where the empirical error bars are much smaller than the accuracy to which we expect the $\chi$-BS(3) to work to the given order, we artificially increase those error bars in our global fit. In Tab. \[q1param:tab\]-\[q3param:tab\] we present the parameter set of our best fit to the data. Note that part of the parameters are predetermined to a large extent and therefore fine tuned in a small interval only.
=1.4mm
$f$ \[MeV\] $ C_R$ $F_R$ $D_R$
------------- -------- ------- ------- --
90.04 1.734 0.418 0.748
: Leading chiral parameters which contribute to meson-baryon scattering at order $Q$.[]{data-label="q1param:tab"}
A qualitative understanding of the typical strength in the various channels can be obtained already at leading chiral order $Q$. In particular the $\Lambda(1405)$ resonance is formed as a result of the coupled-channel dynamics defined by the Weinberg-Tomozawa interaction vertices (see Fig. \[fig:wt\]). There are four parameters relevant at that order $f$, $C_R$, $F_R$ and $D_R$. Their respective values as given in Tab. \[q1param:tab\] are the result of our global fit to the data set including all parameters of the $\chi$-BS(3) approach. At leading chiral order the parameter $f$ determines the weak pion- and kaon-decay processes and at the same time the strength of the Weinberg-Tomozawa interaction vertices. To subleading order $Q^2$ the Weinberg-Tomozawa terms and the weak-decay constants of the pseudo-scalar meson octet receive independent correction terms. The result $f\simeq 90$ MeV is sufficiently close to the empirical decay parameters $f_\pi \simeq 92.4$ MeV and $f_K \simeq 113.0$ MeV to expect that the $Q^2$ correction terms lead indeed to values rather close to the empirical decay constants. Our value for $f$ is consistent with the estimate of [@GL85] which lead to $f_\pi/f = 1.07 \pm 0.12$. The baryon octet and decuplet s- and u-channel exchange contributions to the interaction kernels are determined by the $F_R, D_R$ and $C_R$ parameters at leading order. Note that $F_R$ and $D_R$ predict the baryon octet weak-decay processes and $C_R$ the strong decay widths of the baryon decuplet states to this order also.
A quantitative description of the data set requires the inclusion of higher order terms. Initially we tried to establish a consistent picture of the existing low-energy meson-baryon scattering data based on a truncation of the interaction kernels to chiral order $Q^2$. This attempt failed due to the insufficient quality of the kaon-nucleon scattering data at low energies. In particular some of the inelastic $K^-$-proton differential cross sections are strongly influenced by the d-wave $\Lambda(1520)$ resonance at energies where the data points start to show smaller error bars. We conclude that, on the one hand, one must include an effective baryon-nonet resonance field and, on the other hand, perform minimally a chiral $Q^3$ analysis to extend the applicability domain to somewhat higher energies. Since the effect of the d-wave resonances is only necessary in the strangeness minus one sector, they are only considered in that channel. The resonance parameters will be presented when discussing the strangeness minus one sector.
$g_F^{(V)}$\[GeV$^{-2}$\] 0.293 $g_F^{(S)}$\[GeV$^{-1}$\] -0.198 $g_F^{(T)}$\[GeV$^{-1}$\] 1.106 $Z_{[10]}$ 0.719
--------------------------- ------- --------------------------- -------- --------------------------- ------- ------------ -------
$g_D^{(V)}$\[GeV$^{-2}$\] 1.240 $g_D^{(S)}$\[GeV$^{-1}$\] -0.853 $g_D^{(T)}$\[GeV$^{-1}$\] 1.607 - -
: Chiral $Q^2$-parameters resulting from a fit to low-energy meson-baryon scattering data. Further parameters at this order are determined by the large $N_c$ sum rules.[]{data-label="q2param:tab"}
At subleading order $Q^2$ the chiral $SU(3)$ Lagrangian predicts the relevance of 12 basically unknown parameters, $g^{(S)}, g^{(V)}$, $g^{(T)}$ and $Z_{[10]} $, which all need to be adjusted to the empirical scattering data. It is important to realize that chiral symmetry is largely predictive in the $SU(3)$ sector in the sense that it reduces the number of parameters beyond the static $SU(3)$ symmetry. For example one should compare the six tensors which result from decomposing $8\otimes 8= 1
\oplus 8_S\oplus 8_A \oplus 10\oplus \overline{10}\oplus 27$ into its irreducible components with the subset of SU(3) structures selected by chiral symmetry in a given partial wave. Thus static $SU(3)$ symmetry alone would predict 18 independent terms for the s-wave and two p-wave channels rather than the 11 chiral $Q^2$ background parameters, $g^{(S)}, g^{(V)}$ and $g^{(T)}$. In our work the number of parameters was further reduced significantly by insisting on the large $N_c$ sum rules $$\begin{aligned}
g_1^{(S)}=2\,g_0^{(S)}= 4\,g_D^{(S)}/3 \,, \qquad
g_1^{(V)}=2\,g_0^{(V)}= 4\,g_D^{(V)}/3 \,, \qquad g_1^{(T)}=0 \,,
\nonumber
\label{}\end{aligned}$$ for the symmetry conserving quasi-local two body interaction terms (see (\[Q\^2-large-Nc-result\])). In Tab. \[q2param:tab\] we collect the values of all free parameters as they result from our best global fit. All parameters are found to have natural size. This is an important result, because only then is the application of the chiral power counting rule (\[q-rule\]) justified. We point out that the large $N_c$ sum rules derived in section 2 implicitly assume that other inelastic channels like $K \,\Delta_\mu $ or $K_\mu\,N $ are not too important. The effect of such channels can be absorbed to some extent into the quasi-local counter terms, however possibly at the prize that their large $N_c$ sum rules are violated. It is therefore a highly non-trivial result that we obtain a successful fit imposing (\[q2param:tab\]). Note that the only previous analysis [@Kaiser], which truncated the interaction kernel to chiral order $Q^2$ but did not include p-waves, found values for the s-wave range parameters largely inconsistent with the large $N_c$ sum rules. This may be due in part to the use of channel dependent cutoff parameters and the fact that that analysis missed octet and decuplet exchange contributions, which are important for the s-wave interaction kernel already to chiral order $Q^2$.
The parameters $b_0, b_D$ and $b_F$ to this order characterize the explicit chiral symmetry-breaking effects of QCD via the finite current quark masses. The parameters $b_D$ and $b_F$ are well estimated from the baryon octet mass splitting (see (\[mass-splitting\])) whereas $b_0$ must be extracted directly from the meson-baryon scattering data. It drives the size of the pion-nucleon sigma term for which conflicting values are still being discussed in the literature [@pin-news]. Our values $$\begin{aligned}
b_0 = -0.346\, {\rm GeV}^{-1} \,, \quad b_D = 0.061\, {\rm GeV}^{-1} \,, \quad b_F =-0.195 \,{\rm GeV}^{-1} \,,
\label{b-result}\end{aligned}$$ are rather close to values expected from the baryon octet mass splitting (\[mass-splitting\]). The pion-nucleon sigma term $\sigma_{\pi N}$ if evaluated at leading chiral order $Q^2$ (see (\[spin:naive\])) would be $\sigma_{\pi N} \simeq 32$ MeV. That value should not be compared directly with $\sigma_{\pi N}$ as extracted usually from pion-nucleon scattering data at the Cheng-Dashen point. The required subthreshold extrapolation involves further poorly convergent expansions [@pin-news]. Here we do not attempt to add anything new to this ongoing debate. We turn to the analogous symmetry-breaking parameters $d_0$ and $d_D$ for the baryon decuplet states. Like for the baryon octet states we use the isospin averaged empirical values for the baryon masses in any u-channel exchange contribution. That means we use $m^{(\Delta )}_{[10]} = 1232.0$ MeV, $m^{(\Sigma )}_{[10]} = 1384.5$ MeV and $m^{(\Xi)}_{10} = 1533.5$ MeV in the decuplet exchange expressions. In the s-channel decuplet expressions we use the slightly different values $m^{(\Delta )}_{10} = 1223.2$ MeV and $m^{(\Sigma )}_{10} = 1374.4$ MeV to compensate for a small mass renormalization induced by the unitarization. Those values are rather consistent with $d_D \simeq -0.49$ GeV$^{-1}$ (see (\[mass-splitting\])). Moreover note that all values used are quite compatible with the large $N_c$ sum rule $$b_D+b_F = d_D/3 \,.$$ The parameter $d_0$ is not determined by our analysis. Its determination required the study of the meson baryon-decuplet scattering processes.
=1.1mm
$h_F^{(1)}$\[GeV$^{-3}$\] -0.129 $h_D^{(1)}$\[GeV$^{-3}$\] -0.548 $h_{F}^{(2)}$\[GeV$^{-2}$\] 0.174 $h_{F}^{(3)}$\[GeV$^{-2}$\] -0.221
--------------------------- -------- --------------------------- -------- ----------------------------- ------- ----------------------------- -------- -- --
: Chiral $Q^3$-parameters resulting from a fit to low-energy meson-baryon scattering data. Further parameters at this order are determined by the large $N_c$ sum rules.[]{data-label="q3param:tab"}
At chiral order $Q^3$ the number of parameters increases significantly unless further constraints from QCD are imposed. Recall for example that [@q3-meissner] presents a large collection of already 102 chiral $Q^3$ interaction terms. A systematic expansion of the interaction kernel in powers of $1/N_c$ leads to a much reduced parameter set. For example the $1/N_c$ expansion leads to only four further parameters $h^{(1)}_{F}$, $h^{(1)}_{D}$, $h^{(2)}_{F}$ and $h^{(3)}_{F}$ describing the refined symmetry-conserving two-body interaction vertices. This is to be compared with the ten parameters established in Appendix B, which were found to be relevant at order $Q^3$ if large $N_c$ sum rules are not imposed. In our global fit we insist on the large $N_c$ sum rules $$h_1^{(1)}=2\,h_0^{(1)}= 4\,h_D^{(1)}/3 \;, \quad h^{(2)}_1 = h^{(2)}_D=0
\;, \quad h^{(3)}_1 = h^{(3)}_D=0 \,.$$ Note that at order $Q^3$ there are no symmetry-breaking 2-body interaction vertices. To that order the only symmetry-breaking effects result from the refined 3-point vertices. Here a particularly rich picture emerges. At order $Q^3$ we established 23 parameters describing symmetry-breaking effects in the 3-point meson-baryon vertices. For instance, to that order the baryon-octet states may couple to the pseudo-scalar mesons also via pseudo-scalar vertices rather than only via the leading axial-vector vertices. Out of those 23 parameters 16 contribute at the same time to matrix elements of the axial-vector current. Thus in order to control the symmetry breaking effects, it is mandatory to include constraints from the weak decay widths of the baryon octet states also. A detailed analysis of the 3-point vertices in the $1/N_c$ expansion of QCD reveals that in fact only ten parameters $c_{1,2,3,4,5}$, $\delta c_{1,2}$ and $\bar c_{1,2}$ and $a$, rather than the 23 parameters, are needed at leading order in that expansion. Since the leading parameters $F_R, D_R$ together with the symmetry-breaking parameters $c_i$ describe at the same time the weak decay widths of the baryon octet and decuplet ground states (see Tab. \[weak-decay:tab\],\[weak-decay:tabb\]), the number of free parameters does not increase significantly at the $Q^3$ level if the large $N_c$ limit is applied.
=1.2mm
$c_1 $ -0.0707 $c_2 $ -0.0443 $c_3 $ 0.0624 $c_4 $ 0.0119 $c_5 $ -0.0434
--------------- --------- --------------- --------- --------------- -------- --------------- --------- -------- --------- -- --
$\bar c_1 $ 0.0754 $\bar c_2 $ 0.1533 $\delta c_1 $ 0.0328 $\delta c_2 $ -0.0043 $ a $ -0.2099
: Chiral $Q^3$-parameters, which break the $SU(3)$ symmetry explicitly, resulting from a fit to low-energy meson-baryon scattering data. []{data-label="q3param:tab"}
We conclude that the parameter reduction achieved in this work by insisting on chiral and large $N_c$ sum rules is significant. It is instructive to recall that for instance the analysis by Kim [@kim], rather close in spirit to modern effective field theories, required already 44 parameters in the strangeness minus one sector only. As was pointed out in [@Hurtado] that analysis, even though troubled with severe shortcomings [@piN:Hurtado], was the only one so far which included s- and p-waves and still reproduced the most relevant features of the subthreshold $\bar K N$ amplitudes. We summarize that a combined chiral and large $N_c$ analysis leads to a scheme with a reasonably small number of parameters at the $Q^3$ level.
Axial-vector coupling constants
-------------------------------
The result of our global fit for the axial-vector coupling constants of the baryon octet states are presented in Tab. \[weak-decay:tab\]. The six data points, which strongly constrain the parameters $F_R, D_R$ and $c_{1,2,3,4}$, are well reproduced. Note that the recent measurement of the decay process $\Xi^0 \to \Sigma^+ \,e^-\,\bar \nu_e$ by the KTeV experiment does not provide a further stringent constraint so far [@KTeV:1; @KTeV:2]. The axial-vector coupling constant of that decay $g_A(\Xi^0 \to \Sigma^+ \,e^-\,\bar \nu_e) = \sqrt{2}\,g_A (\Xi^- \to \Sigma^0 \,e^-\,\bar \nu_e )$ is related to the decay process $\Xi^- \to \Sigma^0 \,e^-\,\bar \nu_e$ included in Tab. \[weak-decay:tabb\] by isospin symmetry. The value given in [@KTeV:2] is $g_A(\Xi^0 \to \Sigma^+ \,e^-\,\bar \nu_e) = 1.23 \pm 0.44 $. As emphasized in [@KTeV:2] it would be important to reduce the uncertainties by more data taking. We confirm the result of [@Dai] which favors values for $g_A (\Xi^- \to \Sigma^0 \,e^-\,\bar \nu_e )$ and $g_A (\Xi^- \to \Lambda \,e^-\,\bar \nu_e )$ which are somewhat smaller than the central values given in Tab. \[weak-decay:tab\]. This is a non-trivial result because in our approach the parameters $c_{1,2,3,4}$ are constrained not only by the weak decay processes of the baryon octet states but also by the meson-baryon scattering data.
=1.4mm
$g_A $ (Exp.) $\chi$-BS(3) SU(3)
--------------------------------------------- -------------------- -------------- ------- -- -- -- -- --
$ n \to p\,e^-\,\bar \nu_e $ $1.267 \pm 0.004$ 1.26 1.26
$ \Sigma^- \to \Lambda \,e^-\,\bar \nu_e $ $0.601 \pm 0.015$ 0.58 0.65
$ \Lambda \to p \,e^-\,\bar \nu_e $ $-0.889 \pm 0.015$ -0.92 -0.90
$ \Sigma^- \to n \,e^-\,\bar \nu_e $ $0.342 \pm 0.015$ 0.33 0.32
$ \Xi^- \to \Lambda \,e^-\,\bar \nu_e $ $0.306 \pm 0.061 $ 0.19 0.25
$ \Xi^- \to \Sigma^0 \,e^-\,\bar \nu_e $ $0.929 \pm 0.112 $ 0.79 0.89
: Axial-vector coupling constants for the weak decay processes of the baryon octet states. The empirical values for $g_A$ are taken from [@Dai]. Here we do not consider small $SU(3)$ symmetry-breaking effects of the vector current. The column labelled by SU(3) shows the axial-vector coupling constants as they follow from $F_R=0.47$ and $D_R=0.79$ and $c_{i}=0$.[]{data-label="weak-decay:tabb"}
Meson-baryon coupling constants
-------------------------------
We turn to the meson-baryon coupling constants. To subleading order the Goldstone bosons couple to the baryon octet states via axial-vector but also via suppressed pseudo-scalar vertices (see (\[chi-sb-3:p\])). In Tab. \[meson-baryon:tab\] we collect the results for all the axial-vector meson-baryon coupling constants, $A^{(B)}_{\Phi B}$, and their respective pseudo-scalar parts $P^{(B)}_{\Phi B}$. The $SU(3)$ symmetric part of the axial-vector vertices is characterized by parameters $F_A$ and $D_A$ $$\begin{aligned}
F_A &=& F_R -\frac{\beta}{\sqrt{3}}\,\Big({\textstyle{2\over 3}}\,\delta c_1 +\delta c_2 -a\Big) =
0.270\,,
\quad \!
\nonumber\\
D_A &=& D_R - \frac{\beta}{\sqrt{3}}\,\delta c_1 = 0.726\,,
\label{FDA:def}\end{aligned}$$ where $\beta \simeq 1.12$. To subleading order the parameters $F_A$ and $D_A$ differ from the corresponding parameters $F_R = 0.418$ and $D_R =0.748$ relevant for matrix elements of the axial-vector current (see Tab. \[weak-decay:tab\]) by a sizeable amount. The $SU(3)$ symmetry-breaking effects in the axial-vector coupling constants $A$ are determined by the parameters $c_i$, which are already tightly constrained by the weak decay widths of the baryon octet states, $\delta c_{1,2}$ and $a$. Similarly the four parameters $\bar c_{1,2}$ and $a$ characterize the pseudo-scalar meson-baryon 3-point vertices. Their symmetric contributions are determined by $F_P$ and $D_P$ $$\begin{aligned}
F_P = -\frac{\beta}{\sqrt{3}}\,\Big({\textstyle{2\over 3}}\,\bar c_1 +\bar c_2 +a\Big) = 0.004\,,
\quad \!
D_P = - \frac{\beta}{\sqrt{3}}\,\bar c_1 = -0.049\,.
\label{FDP:def}\end{aligned}$$ Note that in the on-shell coupling constants $G=A+P$ the parameter $a$ drops out. In that sense that parameter should be viewed as representing an effective quasi-local 2-body interaction term, which breaks the $SU(3)$ symmetry explicitly.
=0.9mm
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$A^{(N)}_{\pi N} $ $A^{(N)}_{\eta N} $ $A^{(\Lambda)}_{\bar $A^{(\Lambda)}_{\pi \Sigma } $ $A^{(\Lambda)}_{\eta $A^{(\Lambda)}_{K $A^{(\Sigma)}_{\bar K N} $ $A^{(\Sigma)}_{\pi \Sigma} $ $A^{(\Sigma)}_{\eta \Sigma} $ $A^{(\Sigma)}_{K \Xi} $ $A^{(\Xi)}_{\pi \Xi} $ $A^{(\Xi)}_{\eta \Xi} $
K N} $ \Lambda} $ \Xi} $
-------------- -------------------- --------------------- ----------------------------- -------------------------------- ----------------------- -------------------- ----------------------------- ------------------------------ ------------------------------- ------------------------- ------------------------ -------------------------
$\chi$-BS(3) 2.15 0.20 -1.29 1.41 -0.64 0.12 0.73 -1.02 1.09 1.24 -0.59 -0.32
SU(3) 1.73 0.05 -1.25 1.45 -0.84 -0.07 0.65 -0.76 0.84 1.41 -0.79 -0.89
$P^{(N)}_{\pi N} $ $P^{(N)}_{\eta N} $ $P^{(\Lambda)}_{\bar K N} $ $P^{(\Lambda)}_{\pi \Sigma $P^{(\Lambda)}_{\eta $P^{(\Lambda)}_{K $P^{(\Sigma)}_{\bar K N} $ $P^{(\Sigma)}_{\pi \Sigma} $ $P^{(\Sigma)}_{\eta \Sigma} $ $P^{(\Sigma)}_{K \Xi} $ $P^{(\Xi)}_{\pi \Xi} $ $P^{(\Xi)}_{\eta \Xi} $
} $ \Lambda} $ \Xi} $
$\chi$-BS(3) -0.01 0.16 0.04 -0.01 0.20 -0.07 -0.11 -0.00 -0.02 -0.09 0.01 0.13
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Axial-vector (A) and pseudo-scalar (P) meson-baryon coupling constants for the baryon octet states. The row labelled by SU(3) gives results excluding SU(3) symmetry-breaking effects with $F_A =0.270$ and $D_A= 0.726$. The total strength of the on-shell meson-baryon vertex is determined by $G=A+P$.[]{data-label="meson-baryon:tab"}
We emphasize that our meson-baryon coupling constants are strongly constrained by the axial-vector coupling constants. In particular we reproduce a sum rule derived first by Dashen and Weinstein [@Dashen:Weinstein] $$\begin{aligned}
G_{\pi N}^{(N)}-\sqrt{3}\,g_A (n \to p\,e^-\,\bar \nu_e )
&=& -\frac{m_\pi^2}{\sqrt{6}\,m_K^2}\,\Big( G_{\bar K N }^{(\Sigma)}
-g_A(\Sigma^-\!\to n \,e^-\,\bar \nu_e) \Big)
\nonumber\\
&-& \frac{m_\pi^2}{\sqrt{2}\,m_K^2}\,\Big( G_{\bar K N}^{(\Lambda)}
-2\,g_A (\Lambda \to p \,e^-\,\bar \nu_e ) \Big) \;.
\label{DW-sum}\end{aligned}$$ We observe that, given the expected range of values for $g_{\pi NN}$, $g_{\bar K N \Lambda }$ and $g_{\bar K N \Sigma }$ together with the empirical axial-vector coupling constants, the Dashen-Weinstein relation strongly favors a small $f$ parameter value close to $f_\pi$.
=1.4mm
$\chi $-BS(3) [@Stoks] [@Juelich:2; @Juelich:1; @Juelich:3] [@keil] [@A.D.Martin] [@Dalitz:g]
------------------------------ --------------- ---------- -------------------------------------- --------- --------------- ------------- -- --
$|g_{\pi NN }|$ 12.9 13.0 13.5 13.1 - -
$|g_{\bar K N \Lambda}|$ 10.1 13.5 14.0 10.1 13.2 16.1
$|g_{\pi \Sigma\, \Lambda}|$ 10.4 11.9 9.3 6.4 - -
$|g_{\bar K N \Sigma }|$ 5.2 4.1 2.7 2.4 6.4 3.5
$|g_{\pi \Sigma \,\Sigma}|$ 9.6 11.8 10.8 0.7 - -
: On-shell meson-baryon coupling constants for the baryon octet states (see (\[trans:tab\])). We give the central values only because reliable error analyses are not available in most cases.[]{data-label="meson-baryon:tab:2"}
In Tab. \[meson-baryon:tab:2\] we confront our results with a representative selection of published meson-baryon coupling constants. For the clarity of this comparison we recall here the connection with our convention $$\begin{aligned}
&& g_{\pi NN} = \frac{m_N}{\sqrt{3}\,f}\,G^{(N)}_{\pi N} \,, \! \quad
g_{\bar K N \Lambda } = \frac{m_N+m_\Lambda}{\sqrt{8}\,f}\,G^{(\Lambda)}_{\bar KN} \,, \!\quad
g_{\pi \Lambda \Sigma }= \frac{m_\Lambda+m_\Sigma }{\sqrt{12}\,f }\,G^{(\Lambda)}_{\pi \Sigma }\, ,
\nonumber\\
&& g_{\bar K N \Sigma }= \frac{m_N+m_\Sigma}{\sqrt{8}\,f}\,G^{(\Sigma)}_{\bar K N} \,,\! \quad
g_{\pi \Sigma \Sigma} = \frac{m_\Sigma }{\sqrt{2}\,f}\,G^{(\Sigma)}_{\pi \Sigma } \,.
\label{trans:tab}\end{aligned}$$ Further values from previous analyses can be found in [@Dumbrajs]. Note also the interesting recent results within the QCD sum rule approach [@Kim:Doi:Oka:Lee; @Doi:Kim:Oka] and also [@Buchmann:Henley]. We do not confront our values with those of [@Kim:Doi:Oka:Lee; @Doi:Kim:Oka] and [@Buchmann:Henley] because the $SU(3)$ symmetry-breaking effects are not yet fully under control in these works. The analysis [@Stoks] is based on nucleon-nucleon and hyperon-nucleon scattering data where $SU(3)$ symmetry breaking effects in the meson-baryon coupling constants are parameterized according to the model of [@p3-model]. The values given in [@Juelich:2; @Juelich:1; @Juelich:3] do not allow for $SU(3)$ symmetry-breaking effects and moreover rely on $SU(6)$ quark-model relations. Particularly striking are the extreme $SU(3)$ symmetry-breaking effects claimed in [@keil]. The parameters result from a K-matrix fit to the phase shifts of $K^-N$ scattering as given in [@gopal]. We do not confirm these results. Note also the recent analysis [@Loiseau:Wycech] which deduces the value $g_{\pi \Lambda \Sigma } = 12.9 \pm 1.2$ from hyperonic atom data, a value somewhat larger than our result of $10.4$. For the most recent and accurate pion-nucleon coupling constant $g_{\pi NN} = 13.34 \pm 0.09$ we refer to [@gpinn:best]. We do not compete with the high precision and elaborate analyses of this work.
In Tab. \[meson-baryon-decuplet:tab\] we collect our results for the meson-baron coupling constants of the decuplet states. Again we find only moderate $SU(3)$ symmetry-breaking effects in the coupling constants. This is demonstrated by comparing the two rows of Tab. \[meson-baryon-decuplet:tab\]. The SU(3) symmetric part is determined by the parameter $C_A$ $$\begin{aligned}
C_A = C_R -2\,\frac{\beta}{\sqrt{3}}\,\Big(\bar c_1 +\delta c_1 \Big) = 1.593 \,.
\label{}\end{aligned}$$ We find that the coupling constants given in (\[meson-baryon-decuplet:tab\]) should not be used in the simple expressions (\[decuplet-decay\]) for the decuplet widths. For example with $A_{\pi N}^{(\Delta )} \simeq 2.62 $ one would estimate $\Gamma_\Delta \simeq $ 102 MeV not too close to the empirical value of $\Gamma_\Delta \simeq $ 120 MeV [@fpi:exp]. It will be demonstrated below, that nevertheless the $P_{33}$ phase shift of the pion-nucleon scattering process, which probes the $\Delta $ resonance width, is reproduced accurately. This reflects an important energy dependence in the decuplet self energy. =1.2mm
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$A^{(\Delta )}_{\,\pi\, N} $ $A^{(\Delta )}_{\,K\, \Sigma} $ $A^{(\Sigma )}_{\,\bar $A^{(\Sigma)}_{\,\pi \,\Sigma } $ $A^{(\Sigma )}_{\,\pi \,\Lambda} $ $A^{(\Sigma )}_{\,\eta \,\Sigma } $ $A^{(\Sigma )}_{\,K \,\Xi} $ $A^{(\Xi )}_{\,\bar K \,\Lambda} $ $A^{(\Xi )}_{\,\bar K \,\Sigma} $ $A^{(\Xi )}_{\,\eta \,\Xi} $ $A^{(\Xi )}_{\,\pi \,\Xi} $
K \,N} $
-------------- ------------------------------ --------------------------------- ------------------------- ----------------------------------- ------------------------------------ ------------------------------------- ------------------------------ ------------------------------------ ----------------------------------- ------------------------------ -----------------------------
$\chi$-BS(3) 2.62 -2.03 1.54 -1.40 -1.64 1.55 -0.96 1.67 1.75 -1.29 -1.46
SU(3) 2.25 -2.25 1.30 -1.30 -1.59 1.59 -1.30 1.59 1.59 -1.59 -1.59
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Meson-baryon coupling constants for the baryon decuplet states. The row labelled by SU(3) gives results obtained with $C_A =1.593 $ excluding all symmetry-breaking effects.[]{data-label="meson-baryon-decuplet:tab"}
We summarize the main findings of this section. All established parameters prove the $SU(3)$ flavor symmetry to be an extremely useful and accurate tool. Explicit symmetry breaking effects are quantitatively important but sufficiently small to permit an evaluation within the $\chi$-BS(3) approach. This confirms a beautiful analysis by Hamilton and Oades [@Hamilton:Oades] who strongly supported the $SU(3)$ flavor symmetry by a discrepancy analysis of kaon-nucleon scattering data.
Pion-nucleon scattering
-----------------------
We begin with a detailed account of the strangeness zero sector. For a review of pion-nucleon scattering within the conventional meson-exchange picture we refer to [@EW]. The various chiral approaches will be discussed more explicitly below. Naively one may want to include the pion-nucleon threshold parameters in a global $SU(3)$ fit. In conventional chiral perturbation theory the latter are evaluated perturbatively to subleading orders in the chiral expansion [@Bernard; @pin-q4]. The small pion mass justifies the perturbative treatment. Explicit expressions for the threshold parameters are given in Appendix H and confirm the results [@Bernard]. Unfortunately there is no unique set of threshold parameters available. This is due to difficulties in extrapolating the empirical data set down to threshold, subtle electromagnetic effects and also some inconsistencies in the data set itself [@GMORW]. A collection of mutually contradicting threshold parameters is collected in Tab. 8. In order to obtain an estimate of systematic errors in the various analyses we confront the threshold values with the chiral sum rules: $$\begin{aligned}
&&4\,\pi \left( 1+\frac{m_\pi}{m_N}\right)
a^{(\pi N)}_{[S_-]}=\frac{m_\pi}{2\,f^2}+{\mathcal O}\left(Q^3 \right)\; ,
\nonumber\\
&&4\,\pi \left( 1+\frac{m_\pi}{m_N}\right)
b^{(\pi N)}_{[S_-]}=\frac{1}{4\,f^2\,m_\pi}
-\frac{2\,g_A^2+1}{4\,f^2\,m_N}
\nonumber\\
&&\quad \quad \quad \quad \quad \quad \quad \quad
+\frac{C^2}{18\,f^2}\,\frac{m_\pi}{m_N\,(\mu_\Delta+m_\pi)}
+{\mathcal O}\left(Q \right)
\nonumber\\
&&4\,\pi \left( 1+\frac{m_\pi}{m_N}\right)
\Big( a^{(\pi N)}_{[P_{13}]}-a^{(\pi N)}_{[P_{31}]} \Big)
=\frac{1}{4\,f^2\,m_N}+{\mathcal O}\left( Q\right)\;,
\nonumber\\
&& 4\,\pi\,\left( 1+\frac{m_\pi}{m_N}\right)
\,a^{(\pi N)}_{SF}
= -\frac{3\,g_A^2}{2\,f^2\,m_\pi}\,\left(1+\frac{m_\pi}{m_N}\right)
\nonumber\\
&& \quad \quad \quad \quad \quad \quad \quad \quad
-\frac23\,\frac{C^2}{f^2}\,\frac{m_\pi\,m_N }
{m_\Delta\,(\mu_\Delta^2-m_\pi^2)} +{\mathcal O}\left( Q\right)\;,
\label{sum-rules}\end{aligned}$$ where $\mu_\Delta = m_\Delta -m_N $. We confirm the result of [@Bernard] that the spin-flip scattering volume $a_{SF}^{(\pi N)}=a^{(\pi N)}_{[P_{11}]}+2\,a^{(\pi N)}_{[P_{31}]}-a^{(\pi N)}_{[P_{13}]}
-2\, a^{(\pi N)}_{[P_{33}]}$ and the combination $a^{(\pi N)}_{[P_{13}]}-a^{(\pi N)}_{[P_{31}]}$ in (\[sum-rules\]) are independent of the quasi-local 4-point interaction strengths at leading order. Confronting the analyses in Tab. \[pin:th\] with the chiral sum rules quickly reveals that only the EM98 analysis [@EM98] appears consistent with the sum rules within 20 $\%$. The analysis [@GMORW] and [@SP98] badly contradict the chiral sum rules (\[sum-rules\]), valid at leading chiral orders, and therefore would require unnaturally large correction terms, possibly discrediting the convergence of the chiral expansion in the pion-nucleon sector. The KA86 analysis [@Koch86] is consistent with the two p-wave sum rules but appears inconsistent with the s-wave range parameter $b_{[S_-]}$. The recent $\pi^-$ hydrogen atom experiment [@Schroeder] gives rather precise values for the $\pi^-$-proton scattering lengths $$\begin{aligned}
&& a_{\pi^-p\to \pi^- p} =a_{S_-}^{(\pi N)} +a_{S_+}^{(\pi N)}= ( 0.124\pm 0.001 )\;{\rm fm} \;,\qquad
\nonumber\\
&& a_{\pi^-p\to \pi^0 n} = -\sqrt{2}\,a_{S_-}^{(\pi N)}=(-0.180\pm 0.008)\; {\rm fm} \;.
\label{pi-atoms}\end{aligned}$$ These values are in conflict with the s-wave scattering lengths of the EM98 analysis. For a comprehensive discussion of further constraints from the pion-deuteron scattering lengths as derived from recent pionic atom data we refer to [@gpinn:best]. All together the emerging picture is complicated and inconclusive at present. Related arguments are presented by Fettes and Meißner in their work [@pin-q4] which considers low-energy pion-nucleon phase shifts at chiral order $Q^4$. The resolution of this mystery may be found in the most recent work of Fettes and Meißner [@pin-em] where they consider the electromagnetic correction terms within the $\chi $PT scheme to order $Q^3$.
=1.3mm
$\chi $-BS(3) KA86[@Koch86] EM98[@EM98] SP98[@SP98] GMORW[@GMORW]
------------------------------------------- --------------- --------------- ------------------- ------------------- -------------------
$a_{[S_-]}^{(\pi N)}$ \[fm\] 0.124 0.130 0.109$\pm$ 0.001 0.125$\pm$ 0.001 0.116$\pm$ 0.004
$a_{[S_+]}^{(\pi N)}$ \[fm\] -0.014 -0.012 0.006$\pm$ 0.001 0.000$\pm$ 0.001 0.005$\pm$ 0.006
$b_{[S_-]}^{(\pi N)}$ \[$m_\pi^{-3}]$ -0.007 0.008 0.016 0.001$\pm$ 0.001 -0.009$\pm$ 0.012
$b_{[S_+]}^{(\pi N)}$ \[$m_\pi^{-3}]$ -0.028 -0.044 -0.045 -0.048$\pm$ 0.001 -0.050$\pm$ 0.016
$a_{[P_{11}]}^{(\pi N)}$ \[$m_\pi^{-3}$\] -0.083 -0.078 -0.078$\pm$ 0.003 -0.073$\pm$ 0.004 -0.098$\pm$ 0.005
$a_{[P_{31}]}^{(\pi N)}$ \[$m_\pi^{-3}$\] -0.045 -0.044 -0.043$\pm$ 0.002 -0.043$\pm$ 0.002 -0.046$\pm$ 0.004
$a_{[P_{13}]}^{(\pi N)}$ \[$m_\pi^{-3}$\] -0.038 -0.030 -0.033$\pm$ 0.003 -0.013$\pm$ 0.004 0.000$\pm$ 0.004
$a_{[P_{33}]}^{(\pi N)}$ \[$m_\pi^{-3}$\] 0.198 0.214 0.214$\pm$ 0.002 0.211$\pm$ 0.002 0.203 $\pm$ 0.002
: Pion-nucleon threshold parameters. The reader not familiar with the common definition of the various threshold parameters is referred to Appendix H.[]{data-label="pin:th"}
We turn to another important aspect to be discussed. Even though the EM98 analysis is rather consistent with the chiral sum rules (\[sum-rules\]), does it imply background terms of natural size? This can be addressed by considering a further combination of p-wave scattering volumes $$\begin{aligned}
&&4\,\pi \left( 1+\frac{m_\pi}{m_N}\right)\left( a^{(\pi N)}_{[P_{11}]}
-4\,a^{(\pi N)}_{[P_{31}]}\right) =
\frac{3}{2\,f^2\,m_N} +B+{\mathcal O}\left(Q \right)\;,
\nonumber\\
&&B=-\frac{5}{12 f^2}\,\Big( 2\,\tilde g_0^{(S)}+\tilde g_D^{(S)}+\tilde g_F^{(S)} \Big)
-\frac{1}{3 f^2}\,\Big( \tilde g_D^{(T)}+\tilde g_F^{(T)} \Big) \,,
\label{}\end{aligned}$$ where we absorbed the Z-dependence into the tilde couplings for simplicity (see (\[Z-absorb\])). The naturalness assumption would lead to $B \sim 1/(f^2\,m_\rho)$, a typical size which is compatible with the background term $B\simeq 0.92\,m_\pi^{-3}$ of the EM98 solution.
In order to avoid the ambiguities of the threshold parameters we decided to include the single energy pion-nucleon phase shifts of [@SP98] in our global fit. The phase shifts are evaluated in the $\chi$-BS(3) approach including all channels suggested by the $SU(3)$ flavor symmetry. The single energy phase shifts are fitted up to $\sqrt{s} \simeq 1200$ MeV. In Fig. \[fig:pionphases\] we confront the result of our fit with the empirical phase shifts. All s- and p-wave phase shifts are well reproduced up to $\sqrt{s} \simeq 1300$ MeV with the exception of the $S_{11}$ phase for which our result agrees with the partial-wave analysis less accurately. We emphasize that one should not expect quantitative agreement for $\sqrt{s} > m_N+2\,m_\pi \simeq 1215$ MeV where the inelastic pion production process, not included in this work, starts. The missing higher order range terms in the $S_{11}$ phase are expected to be induced by additional inelastic channels or by the nucleon resonances $N(1520)$ and $N(1650)$. We confirm the findings of [@Kaiser; @new-muenchen] that the coupled $SU(3)$ channels, if truncated at the Weinberg-Tomozawa level, predict considerable strength in the $S_{11}$ channel around $\sqrt{s} \simeq 1500$ MeV where the phase shift shows a resonance-like structure. Note, however that it is expected that the nucleon resonances $N(1520)$ and $N(1650)$ couple strongly to each other [@Sauerman] and therefore one should not expect a quantitative description of the $S_{11}$ phase too far away from threshold. Similarly we observe considerable strength in the $P_{11}$ channel leading to a resonance-like structure around $\sqrt{s} \simeq 1500 $ MeV. We interpret this phenomenon as a precursor effect of the p-wave $N(1440)$ resonance. We stress that our approach differs significantly from the recent work [@new-muenchen] in which the coupled SU(3) channels are applied to pion induced $\eta$ and kaon production which require much larger energies $\sqrt{s} \simeq m_\eta +m_N \simeq$ 1486 MeV or $\sqrt{s} \simeq m_K +m_\Sigma \simeq $ 1695 MeV. We believe that such high energies can be accessed reliably only by including more inelastic channels. It may be worth mentioning that the inclusion of the inelastic channels as required by the $SU(3)$ symmetry leaves the $\pi N$ phase shifts basically unchanged for $\sqrt{s}< 1200$ MeV. Our discussion of the pion-nucleon sector is closed by returning to the threshold parameters. In Tab. \[pin:th\] our extracted threshold parameters are presented in the second row. We conclude that all threshold parameters are within the range suggested by the various analyses.
![S- and p-wave pion-nucleon phase shifts. The single energy phase shifts are taken from [@pion-phases].[]{data-label="fig:pionphases"}](piphase.eps){width="14cm"}
$K^+$-nucleon scattering
------------------------
We turn to the strangeness plus one channel. Since it is impossible to give here a comprehensive discussion of the many works dealing with kaon-nucleon scattering we refer to the review article by Dover and Walker [@Dover] which is still up-to-date in many respects. The data situation can be summarized as follows: there exist precise low-energy differential cross sections for $K^+ p$ scattering but no scattering data for the $K^+$-deuteron scattering process at low energies. Thus all low-energy results in the isospin zero channel necessarily follow from model-dependent extrapolations. We include the available differential cross section in our global fit. They are nicely reproduced as shown in Fig. \[fig:Kpdif\]. We include Coulomb interactions which are sizeable in the forward direction with $\cos \theta > 0$.
![Differential cross section of $K^+$-proton scattering. The data points are taken from [@kplscat]. The solid lines give the result of the $\chi$-BS(3) analysis including Coulomb effects. The dashed lines follow for switched off Coulomb interaction.[]{data-label="fig:Kpdif"}](difcrossKpl3x3.eps){width="13cm"}
It is instructive to consider the threshold amplitudes in detail. In the $\chi$-BS(3) approach the threshold parameters are determined by the threshold values of the effective interaction kernel $V^{(\pm)}_{KN}(m_N\!+\!m_K; n)$ and the partial-wave loop function $J^{(\pm)}_{KN}(m_N\!+\!m_K; n)$ (see (\[result-loop:ab\],\[M:kplus\])). Since all loop functions vanish at threshold but the one for the s-wave channel, all p-wave scattering volumes remain unchanged by the unitarization and are directly given by the threshold values of the appropriate effective interaction kernel $V_{K N}$. The explicit expressions for the scattering volumes to leading order can be found in Appendix H. In contrast the s-wave scattering lengths are renormalized strongly by the loop function $J^{(+)}_{KN}(m_N\!+\!m_K; 0) \neq 0$. At leading order the s-wave scattering lengths are $$\begin{aligned}
4\,\pi\left( 1+\frac{m_K}{m_N}\right) a^{(KN)}_{S_{21}} &=&
-m_K \left( f^2+\frac{m^2_K}{8\,\pi}
\left(1-\frac{1}{\pi}\,\ln \frac{m_K^2}{m_N^2} \right)\right)^{-1} \;,
\nonumber\\
4\,\pi\left( 1+\frac{m_K}{m_N}\right) a^{(KN)}_{S_{01}} &=&0 \;,
\label{}\end{aligned}$$ which lead to $a^{(KN)}_{S_{21}} \simeq -0.22$ fm and $a^{(KN)}_{S01} =0$ fm, close to our final values to subleading orders given in Tab. \[tab-r-kp\]. In Tab. \[tab-r-kp\] we collected typical results for the p-wave scattering volumes also. The large differences in the isospin zero channel reflect the fact that this channel is not constrained by scattering data directly [@Dover]. We find that some of our p-wave scattering volumes, also shown in Tab. \[tab-r-kp\], differ significantly from the values obtained by previous analyses. Such discrepancies may be explained in part by important cancellation mechanisms among the u-channel baryon octet and decuplet contributions (see Appendix H). An accurate description of the scattering volumes requires a precise input for the meson-baryon 3-point vertices. Since the $\chi$-BS(3) approach describes the 3-point vertices in accordance with all chiral constraints and large $N_c$ sum rules of QCD we believe our values for the scattering volumes to be rather reliable.
=0.5mm
$a^{(K N)}_{S_{01}}$ \[fm\] $a^{(K N)}_{S_{21}}$ \[fm\] $a^{(K N)}_{P_{01}}$ $[m_\pi^{-3}]$ $a^{(K N)}_{P_{21}}$ $[m_\pi^{-3}]$ $a^{(K N)}_{P_{03}}$ $[m_\pi^{-3}]$ $a^{(K N)}_{P_{23}}$ $[m_\pi^{-3}]$
-------------- ----------------------------- ----------------------------- ------------------------------------- ------------------------------------- ------------------------------------- ------------------------------------- --
$\chi$-BS(3) 0.06 -0.30 0.033 -0.017 -0.003 0.012
[@Hyslop] 0.0 -0.33 0.028 -0.056 -0.046 0.025
[@BR:Martin] -0.04 -0.32 0.030 -0.011 -0.007 0.007
: $K^+$-nucleon threshold parameters. The values of the $\chi$-BS(3) analysis are given in the first row. The last two rows recall the threshold parameters as given in [@Hyslop] and [@BR:Martin].[]{data-label="tab-r-kp"}
In Fig. \[fig:K+phases\] we confront our s- and p-wave $K^+$-nucleon phase shifts with the most recent analyses by Hyslop et al. [@Hyslop] and Hashimoto [@Hashimoto]. We find that our partial-wave phase shifts are reasonably close to the single energy phase shifts of [@Hyslop] and [@Hashimoto] except the $P_{03}$ phase for which we obtain much smaller strength. Note however, that at higher energies we smoothly reach the single energy phase shifts of Hashimoto [@Hashimoto]. A possible ambiguity in that phase shift is already suggested by the conflicting scattering volumes found in that channel by earlier works (see Tab. \[tab-r-kp\]). The isospin one channel, on the other hand, seems well-established even though the data set does not include polarization measurements close to threshold, which are needed to unambiguously determine the p-wave scattering volumes.
![S- and p-wave $K^+$-nucleon phase shifts. The solid lines represent the results of the $\chi$-BS(3) approach. The open circles are from the Hyslop analysis [@Hyslop] and the open triangles from the Hashimoto analysis [@Hashimoto].[]{data-label="fig:K+phases"}](kplphase.eps){width="14cm"}
$K^-$-nucleon scattering
------------------------
We now turn to our results in the strangeness minus one sector. The antikaon-nucleon scattering process shows a large variety of intriguing phenomena. Inelastic channels are already open at threshold leading to a rich coupled-channel dynamics. Also the $\bar K N$ state couples to many of the observed hyperon resonances for which competing dynamical scenarios are conceivable. We fit directly the available data set rather than any partial wave analysis. Comparing for instance the energy dependent analyses [@gopal] and [@garnjost] one finds large uncertainties in the s- and p-waves in particular at low energies. This reflects on the one hand a model dependence of the analysis and on the other hand an insufficient data set. A partial wave analysis of elastic and inelastic antikaon-nucleon scattering data without further constraints from theory is inconclusive at present [@Dover; @Gensini]. For a detailed overview of former theoretical analyses, we refer to the review article by Dover and Walker [@Dover].
As motivated above we include the d-wave baryon resonance nonet field. An update of the analysis [@Plane] leads to the estimates $F_{[9]} \simeq 1.8$, $D_{[9]} \simeq 0.84$ and $C_{[9]}
\simeq 2.5$ for the resonance parameters. The singlet-octet mixing angle $\vartheta \simeq 28^\circ $ confirms the finding of [@Tripp:2], that the $\Lambda(1520)$ resonance is predominantly a flavor singlet state. By analogy with the expressions for the decuplet decay widths (\[decuplet-decay\]) of section 2 we apply the simple expressions $$\begin{aligned}
&& \Gamma_{N(1520)} = \frac{E_N-m_N}{16\,\pi \,f^2}\,\frac{p_{\pi N}^3}{m_{N(1520)}}
\,\Big( F_{[9]}+ D_{[9]} \Big)^2
\nonumber\\
&& \qquad \quad \;\,+ \frac{E_N-m_N}{16\,\pi \,f^2}\,\frac{p_{\eta N}^3}{m_{N(1520)}}
\,\Big( F_{[9]}- {\textstyle{1\over \sqrt{3}}}\,D_{[9]} \Big)^2 \,,
\nonumber\\
&& \Gamma_{\Sigma (1680)} = \frac{E_N -m_N}{24\,\pi \,f^2}\,\frac{p_{\bar K N}^3}{m_{\Sigma (1680)}}
\,\Big(F_{[9]}- D_{[9]} \Big)^2
+ \frac{E_\Sigma -m_\Sigma}{6\,\pi \,f^2}\,\frac{p_{\pi \Sigma}^3}{m_{\Sigma (1680)}}
\,F_{[9]}^2
\nonumber\\
&& \qquad \quad \;\, +\frac{E_\Lambda -m_\Lambda}{36\,\pi \,f^2}\,\frac{p_{\pi \Lambda}^3}{m_{\Sigma (1680)}}
\,D_{[9]}^2 \,,
\nonumber\\
&& \Gamma_{\Xi (1820)} = \frac{E_\Lambda-m_\Lambda}{16\,\pi \,f^2}\,
\frac{p_{\bar K \Lambda}^3}{m_{\Xi(1820)}}
\,\Big( F_{[9]}- {\textstyle{1\over \sqrt{3}}}\,D_{[9]} \Big)^2
\nonumber\\
&& \qquad \quad \;\, +\frac{E_\Sigma-m_\Sigma }{16\,\pi \,f^2}\,
\frac{p_{\bar K \Sigma}^3}{m_{\Xi(1820)}}
\,\Big( F_{[9]}+ D_{[9]} \Big)^2
\nonumber\\
&& \qquad \quad \;\, +\frac{E_\Xi-m_\Xi }{16\,\pi \,f^2}\,
\frac{p_{\pi \Xi}^3}{m_{\Xi(1820)}}
\,\Big( F_{[9]}- D_{[9]} \Big)^2 \,,
\label{nonet-decay}\end{aligned}$$ where for example $E_N= \sqrt{m_N^2+p_{\pi N}^2}$ with the relative momentum $p_{\pi N}$ defined in the rest frame of the resonance. Our values for $F_{[9]}$ and $D_{[9]} $ describe the decay widths and branching ratios of the $\Sigma (1670)$ and $\Xi (1820)$ reasonably well within their large empirical uncertainties. Note that we put less emphasis on the properties of the $N(1520)$ resonance since that resonance is strongly influenced by the $\pi \Delta_\mu $ channel not considered here. In our global fit we take $F_{[9]}$ and $D_{[9]}$ fixed as given above but fine-tune the mixing angle $\vartheta =27.74^\circ$ and $C_{[9]} =2.509 $. To account for further small inelastic three-body channels we assign the ’bare’ $\Lambda(1520)$ resonance an energy independent decay width of $\Gamma_{\Lambda(1520)}^{(3-{\rm body})}\simeq $ 1.4 MeV. The total cross sections are included in the fit for $p_{{\rm lab.}}< $ 500 MeV. For the bare masses of the d-wave resonances we use the values $m_{\Lambda (1520)} \simeq 1528.2$ MeV, $m_{\Lambda (1690)}\simeq 1705.3$ MeV and $m_{\Sigma (1680)} \simeq 1690.7 $ MeV.
![$K^-$-proton elastic and inelastic cross sections. The data are taken from [@mast-pio; @sakit; @Evans; @Ciborowski; @mast-ko; @bangerter-piS; @Armenteros; @old-scat]. The solid lines show the results of our $\chi$-BS(3) theory including all effects of s-, p- and d-waves. The dashed lines represent the s-wave contributions only. We fitted the data points given by open circles [@mast-pio; @sakit; @Evans; @Ciborowski; @mast-ko; @bangerter-piS; @Armenteros]. Further data points represented by open triangles [@old-scat] were not considered in the global fit.[]{data-label="fig:totcross"}](totcross.eps){width="13cm"}
In Fig. \[fig:totcross\] we present the result of our fit for the elastic and inelastic $K^-p$ cross sections. The data set is nicely reproduced including the rather precise data points for laboratory momenta 250 MeV$<p_{\rm lab}<$ 500 MeV. In Fig. \[fig:totcross\] the s-wave contribution to the total cross section is shown with a dashed line. Important p-wave contributions are found at low energies only in the $\Lambda \pi^0$ production cross section. Note that the $\Lambda \pi^0$ channel carries isospin one and therefore provides valuable constraints on the poorly known $K^-$-neutron interaction. The deviation of our result from the empirical cross sections above $p_{\rm lab} \simeq 500$ MeV in some channels may be due in part to the fact that we do not consider the p-wave $\Lambda(1600)$ and $\Sigma(1660) $ resonances quantitatively in this work. As will be demonstrated below when presenting the partial-wave amplitudes, there is, however, a strong tendency that those resonances are generated in the $\chi$-BS(3) scheme. We checked that, by giving up some of the large $N_c$ sum rules and thereby increasing the number of free parameters we can easily obtain a fit with much improved quality beyond $p_{lab} = 500$ MeV. We refrained from presenting those results because it is not clear that this procedure leads to the correct partial wave interpretation of the total cross sections. Note that the inelastic channel $K^-p \to \Lambda \pi \pi$, not included in this work, is no longer negligible at a quantitative level for $p_{\rm lab} > 300$ MeV [@watson].
![$K^-p \to K^-p,\,\bar K^0\,n $ differential cross sections at $p_{\rm lab}=245$ MeV, $265$ MeV, $285$ MeV and $305$ MeV. The data are taken form [@mast-ko]. The solid lines represent our $\chi$-BS(3) theory including s-, p- and d-waves as well as Coulomb effects. The dashed lines follow if Coulomb interactions are switched off. []{data-label="fig:difkm"}](difcrossKmKo.eps){width="10cm"}
In Fig. \[fig:difkm\] we confront our result with a selection of differential cross sections from elastic $K^-$-proton scattering [@mast-ko]. The angular distribution patterns are consistent with weak p-wave interactions. The almost linear slope in $\cos \theta $ reflects the interference of the p-wave contribution with a strong s-wave. Coulomb effects are small in these channels. The differential cross sections provide valuable constraints on the p-wave interaction strengths.
![Coefficients $A_1$ and $A_2$ for the $K^-p\to \pi^0 \Lambda$, $K^-p\to \pi^\mp \Sigma^\pm$ and $K^-p\to \pi^0 \Sigma$ differential cross sections. The data are taken from [@mast-pio; @bangerter-piS]. The solid lines are the result of the $\chi$-BS(3) approach with inclusion of the d-wave resonances. The dashed lines show the effect of switching off d-wave contributions.[]{data-label="fig:a"}](a12.eps){width="14.0cm"}
![Coefficients $B_1$ and $B_2$ for the $K^-p\to \pi^\mp \Sigma^\pm$, $K^-p\to \pi^0 \Sigma$ and $K^-p\to \pi^0 \Lambda$ differential cross sections. The data are taken from [@mast-pio; @bangerter-piS]. The solid lines are the result of the $\chi$-BS(3) approach with inclusion of the d-wave resonances. The dashed lines show the effect of switching off d-wave contributions.[]{data-label="fig:b"}](b12.eps){width="11.0cm"}
Further important information on the p-wave dynamics is provided by angular distributions for the inelastic $K^-p$ reactions. The available data are represented in terms of coefficients $A_n$ and $B_n$ characterizing the differential cross section $d\sigma(\cos \theta , \sqrt{s}\,) $ and the polarization $P(\cos \theta , \sqrt{s}\,)$ as functions of the center of mass scattering angle $\theta $ and the total energy $\sqrt{s}$: $$\begin{aligned}
\frac{d\sigma (\sqrt{s}, \cos \theta )}{d\cos \theta } &=&
\sum_{n=0}^\infty A_n(\sqrt{s}\,)\,P_n(\cos \theta ) \,,
\nonumber\\
\frac{d\sigma (\sqrt{s}, \cos \theta )}{d\cos \theta } \,P (\sqrt{s}, \cos \theta )
&=& \sum_{n=1}^\infty B_l(\sqrt{s}\,)\,P^1_n(\cos \theta ) \;.
\label{a-b-def}\end{aligned}$$ In Fig. \[fig:a\] we compare the empirical ratios $A_1/A_0$ and $A_2/A_0$ with the results of the $\chi$-BS(3) approach. Note that for $p_{\rm lab} < 300$ MeV the empirical ratios with $n\geq 3$ are compatible with zero within their given errors. A large $A_1/A_0$ ratio is found only in the $K^-p\to \pi^0 \Lambda$ channel demonstrating again the importance of p-wave effects in the isospin one channel. The dashed lines of Fig. \[fig:a\], which are obtained when switching off d-wave contributions, confirm the importance of this resonance for the angular distributions in the isospin zero channel. The fact that the $\Lambda(1520)$ resonance appears more important in the differential cross sections than in the total cross sections follows simply because the tail of the resonance is enhanced if probed via an interference term. In the differential cross section the $\Lambda(1520)$ propagator enters linearly whereas the total cross section probes the squared propagator only. Note also the sizeable p-wave contributions at somewhat larger momenta seen in the charge-exchange reaction of Fig. \[fig:a\] and also in Fig. \[fig:totcross\].
The constraint from the ratios $B_1/A_0$ and $B_2/A_0$, presented in Fig. \[fig:b\], is weak due to rather large empirical errors. New polarization data, possibly with polarized hydrogen targets, would be highly desirable.
=0.9mm
$a_{K^-p }$ \[fm\] $a_{K^-n }$ \[fm\] $\gamma $ $R_c$ $R_n $
-------------- ------------------------------------------ ------------------------------------------ ---------------- ------------------------------------------ ------------------------------------------
Exp. -0.78$\pm$0.18 - 2.36$\pm$ 0.04 0.664$\pm$0.011 0.189$\pm$0.015
+$i$(0.49$\pm$0.37)
$\chi$-BS(3) -1.09+$i$ 0.82 0.29+$i$ 0.54 2.42 0.65 0.19
SU(2) -0.79+$i$ 0.95 0.30+$i$ 0.49 4.58 0.63 0.32
$a^{(\bar K N)}_{P_{01}}$ $[m_\pi^{-3}]$ $a^{(\bar K N)}_{P_{03}}$ $[m_\pi^{-3}]$ $a^{(\bar K N)}_{P_{21}}$ $[m_\pi^{-3}]$ $a^{(\bar K N)}_{P_{23}}$ $[m_\pi^{-3}]$
$\chi$-BS(3) 0.025+$i$ 0.001 0.002+$i$ 0.001 -0.004+$i$ 0.001 -0.055+$i$ 0.021
: $K^-$-nucleon threshold parameters. The row labelled by $SU(2)$ gives the results in the isospin limit with $m_{K^-}=m_{\bar K^0}= 493.7$ MeV and $m_p = m_n = 938.9$ MeV. []{data-label="tab-r-km"}
We turn to the threshold characteristics of the $K^-p$ reaction which is constrained by experimental data for the threshold branching ratios $\gamma, R_c$ and $R_n$ where $$\begin{aligned}
\gamma &=& \frac{\sigma (K^-\,p\rightarrow \pi^+\,\Sigma^-)}
{\sigma (K^-\,p\rightarrow \pi^-\,\Sigma^+)} \;, \quad
R_c = \frac{\sigma (K^-\,p\rightarrow \mbox{charged\,particles })}
{\sigma (K^-\,p\rightarrow \mbox{all})} \;,
\nonumber\\
R_n &=& \frac{\sigma (K^-\,p\rightarrow \pi^0\,\Lambda)}
{\sigma (K^-\,p\rightarrow \mbox{all\,neutral \,channels})} \;.
\label{km:thres}\end{aligned}$$ A further important piece of information is provided by the recent measurement of the $K^-$ hydrogen atom state which leads to a value for the $K^-p$ scattering length [@Iwasaki]. In Tab. \[tab-r-km\] we confront the empirical numbers with our analysis. All threshold parameters are well described within the $\chi$-BS(3) approach. We confirm the result of [@Kaiser; @Ramos] that the branching ratios are rather sensitive to isospin breaking effects. However, note that it is sufficient to include isospin breaking effects only in the $\bar
KN$ channel to good accuracy. The empirical branching ratios are taken from [@branch-rat]. The real part of our $K^-n$ scattering length with $\Re \,a_{K^-n \to K^- n} \simeq 0.29$ fm turns out considerably smaller than the value of $ 0.53$ fm found in the recent analysis [@Ramos]. In Tab. \[tab-r-km\] we present also our results for the p-wave scattering volumes. Here isospin breaking effects are negligible. All scattering volumes but the one in the $P_{23}$ channel are found to be small. The not too small and repulsive scattering volume $a_{P_{23}} \simeq (-0.16+i\,0.06)$ fm$^3$ reflects the presence of the $\Sigma (1385)$ resonance just below the $\bar K N$ threshold. The precise values of the threshold parameters are of crucial importance when describing $K^-$-atom data which constitute a rather sensitive test of the in-medium dynamics of antikaons. In particular one expects a strong sensitivity of the level shifts to the s-wave scattering lengths.
In Fig. \[fig:massspec\] we show the $\Lambda(1405)$ and $\Sigma (1385)$ spectral functions measured in the reactions $K^-p\rightarrow \Sigma^+\pi^-\pi^+\pi^-$ [@lb-spec] and $K^-p\rightarrow \Lambda \pi^+\pi^-$ [@sig-spec] respectively. We did not include the $\Lambda(1405)$ spectrum of [@lb-spec] in our global fit. Since the $\Lambda(1405)$-spectrum shows a strong energy dependence, incompatible with a Breit-Wigner resonance shape, the spectral form depends rather strongly on the initial and final states through which it is measured. The empirical spectrum of [@lb-spec] describes the reaction $\Sigma^+(1600) \,\pi^- \to \Lambda(1405) \to \Sigma^+ \,\pi^-$ rather than the reactions $\Sigma^\pm \,\pi^\mp \to \Lambda(1405) \to \Sigma^+ \,\pi^-$ accessible in our present scheme. In Fig. 6 the spectral form of the $\Lambda(1405)$ resulting from two different initial states $\Sigma^\pm \,\pi^\mp$ are confronted with the empirical spectrum of [@lb-spec]. While the spectrum defined with respect to the initial state $\Sigma^+ \,\pi^-$ represents the empirical spectrum reasonably well the other choice of initial state $\Sigma^+ \,\pi^-$ leads to a significantly altered spectral form. We therefore conclude that in a scheme not including the $\Sigma(1600) \pi$ state explicitly it is not justified to use the $\Lambda(1405)$ spectrum of [@lb-spec] as a quantitative constraint for the kaon-nucleon dynamics.
![$\Lambda(1405)$ and $\Sigma (1385)$ resonance mass distributions in arbitrary units.[]{data-label="fig:massspec"}](mass-spectra.eps){width="12cm"}
We turn to the mass spectrum of the decuplet $\Sigma (1385)$ state. The spectral form, to good accuracy of Breit-Wigner form, is reproduced reasonably well. Our result for the ratio of $\Sigma (1385) \to \pi \Lambda $ over $\Sigma (13 85) \to \pi \Sigma $ of about $17\, \% $ compares well with the most recent empirical determination. In [@Dionisi] that branching ratio was extracted from the $K^-p \to \Sigma (1385) \,K\,\bar K$ reaction and found to be $20 \pm 6 \,\%$. Note that we obtained our ratio from the two reaction amplitudes $\pi \Lambda \to \pi \Lambda$ and $\pi \Lambda \to \pi \Sigma $ evaluated at the $\Sigma (1385)$ pole. The schematic expression (\[decuplet-decay\]) would give a smaller value of about $ 15 \, \%$. Finally we mention that the value for our $\Xi (1530)$ total width of $10.8 $ MeV comfortably meets the empirical value of $9.9^{+1.7}_{-1.9} $ MeV given in [@fpi:exp].
Analyticity and crossing symmetry
---------------------------------
It is important to investigate to what extent our multi-channel scattering amplitudes are consistent with the expectations from analyticity and crossing symmetry. As discussed in detail in section 4.3 the crossing symmetry constraints should not be considered in terms of partial-wave amplitudes, but rather in terms of the forward scattering amplitudes only. We expect crossing symmetry to be particularly important in the strangeness sector, because that sector has a large subthreshold region not directly accessible and constrained by data. In the following we reconstruct the forward $\bar KN$ and $KN$ scattering amplitudes in terms of their imaginary parts by means of dispersion integrals. We then confront the reconstructed scattering amplitudes with the original ones. It is non-trivial that those amplitudes match even though our loop functions and effective interaction kernels are analytic functions. One can not exclude that the coupled channel dynamics generates unphysical singularities off the real axis which would then spoil the representation of the scattering amplitudes in terms of dispersion integrals. Note that within effective field theory unphysical singularities are acceptable, however, only far outside the applicability domain of the approach.
Our analysis is analogous to that of Martin [@A.D.Martin]. However, we consider the dispersion integral as a consistency check of our theory rather than as a predictive tool to derive the low-energy kaon-nucleon scattering amplitudes in terms of the more accurate scattering data at $E_{\rm lab}>300$ MeV [@Queen]. This way we avoid any subtle assumptions on the number of required subtractions in the dispersion integral. Obviously the dispersion integral, if evaluated for small energies, must be dominated by the low-energy total cross sections which are not known empirically too well. We write a subtracted dispersion integral $$\begin{aligned}
&& \! \!T^{(0)}_{\bar K N}(s) = \frac{f^2_{KN \Lambda }}
{s-m^2_\Lambda} + \sum_{k=1}^n\,c_{\bar K N}^{(0,k)}\,(s-s_0)^k
+\!\!\!\int_{(m_\Sigma+m_\pi)^2}^\infty \!\!\! \! \frac{d \,s'}{\pi }\,
\frac{(s-s_0)^n}{(s'-s_0)^n}\,\frac{\Im \,T^{(0)}_{\bar K N} (s')}{s'-s -i\,\epsilon}\;,
\nonumber\\
&& \! \! T^{(1)}_{\bar K N}(s) = \frac{f_{KN \Sigma }^2}
{s-m^2_\Sigma} + \sum_{k=1}^n\,c_{\bar K N}^{(1,k)}\,(s-s_0)^k
+\!\!\! \int_{(m_\Lambda+m_\pi)^2}^\infty \!\!\! \!\frac{d \,s'}{\pi }\,
\frac{(s-s_0)^n}{(s'-s_0)^n}\,\frac{\Im \,T^{(1)}_{\bar K N} (s')}{s'-s -i\,\epsilon} \;,
\nonumber\\
&&\qquad \quad f_{KN Y } = \sqrt{\frac{m_K^2-(m_Y-m_N)^2}{2\,m_N}}\,
\frac{m_N+m_Y}{2\,f}\,G^{(Y)}_{\bar K N } \,,
\label{disp-check}\end{aligned}$$ where we identify $s_0 = \Lambda_{\rm opt.}^2= m_N^2+m_K^2$ with the optimal matching point of (\[opt-match\]). We recall that the kaon-hyperon coupling constants $G^{(Y)}_{\bar K N}=A^{(Y)}_{\bar K N}+P^{(Y)}_{\bar K N}$ with $Y=\Lambda, \Sigma $ receive contributions from pseudo-vector and pseudo-scalar vertices as specified in Tab. \[meson-baryon:tab\]. The values $f_{K N \Lambda } \simeq -12.8\,m^{1/2}_{\pi^+}$ and $f_{K N \Sigma } \simeq 6.1\,m^{1/2}_{\pi^+}$ follow.
The forward scattering amplitude $T_{\bar KN}^{}(s)$ reconstructed in terms of the partial-wave amplitudes $f_{\bar K N, J}^{(L)}(\sqrt{s}\,)$ of (\[match\]) reads $$\begin{aligned}
T^{}_{\bar KN}(s) &=&4\,\pi\,\frac{\sqrt{s}}{m_N}\,
\sum_{n=0}^\infty \,\Big(n+1 \Big)\,\Big( f_{\bar K N,n+\half}^{(n)}(\sqrt{s}\,)+
f_{\bar K N,n+\half}^{(n+1)}(\sqrt{s}\,)
\Big) \;.
\label{forward-amplitude}\end{aligned}$$ The subtraction coefficients $c_{\bar K N}^{(I,k)}$ are adjusted to reproduce the scattering amplitudes close to the kaon-nucleon threshold. One must perform a sufficient number of subtractions so that the dispersion integral in (\[disp-check\]) is dominated by energies still within the applicability range of our theory. With $n=4$ in (\[disp-check\]) we indeed find that we are insensitive to the scattering amplitudes for $\sqrt{s} > 1600$ MeV to good accuracy. Similarly we write a subtracted dispersion integral for the amplitudes $T^{(I)}_{K N}(s)$ of the strangeness plus one sector $$\begin{aligned}
T^{(0)}_{K N}(s) &=& -\frac{1}{2}\,\frac{f^2_{KN \Lambda }}{2\,s_0-s-m^2_\Lambda}
+\frac{3}{2}\,\frac{f^2_{KN \Sigma }}{2\,s_0-s-m^2_\Sigma}
\nonumber\\
&+&\sum_{k=1}^n\,c_{K N}^{(0,k)}\,(s-s_0)^k+\int_{(m_N+m_K)^2}^\infty \frac{d \,s'}{\pi }\,
\frac{(s-s_0)^n}{(s'-s_0)^n}\,\frac{\Im \,T^{(0)}_{K N} (s')}{s'-s -i\,\epsilon}\;,
\nonumber\\
T^{(1)}_{K N}(s) &=& \frac{1}{2}\, \frac{f^2_{KN \Lambda }}{2\,s_0-s-m^2_\Lambda}
+\frac{1}{2}\, \frac{f^2_{KN \Sigma }}{2\,s_0-s-m^2_\Sigma}
\nonumber\\
&+&\sum_{k=1}^n\,c_{K N}^{(1,k)}\,(s-s_0)^k+\int_{(m_N+m_K)^2}^\infty \frac{d \,s'}{\pi }\,
\frac{(s-s_0)^n}{(s'-s_0)^n}\,\frac{\Im \,T^{(1)}_{K N} (s')}{s'-s -i\,\epsilon} \;.
\label{disp-check-2}\end{aligned}$$ The subtraction coefficients $c_{K N}^{(I,k)}$ are adjusted to reproduce the scattering amplitudes close to the kaon-nucleon threshold. The choice $n=4$ leads to a sufficient emphasis of the low-energy $KN$-amplitudes.
![Causality check of the pole-subtracted kaon-nucleon scattering amplitudes $f^2\,\Delta T^{(I)}_{K N}$ and $f^2\,\Delta T^{(I)}_{\bar K N}$. The full lines represent the real part of the forward scattering amplitudes. The dotted lines give the amplitudes as obtained from their imaginary parts (dashed lines) in terms of the dispersion integrals (\[disp-check\]) and (\[disp-check-2\]). The dashed-dotted lines give the s-wave contribution to the real part of the forward scattering amplitudes only.[]{data-label="fig:causality"}](disprelat.eps){width="13cm"}
In Fig. \[fig:causality\] we compare the real part of the pole-subtracted amplitudes $\Delta T^{(I)}_{\bar KN}(s)$ and $\Delta T^{(I)}_{KN}(s)$ with the corresponding amplitudes reconstructed via the dispersion integral (\[disp-check\]) $$\begin{aligned}
\Delta T_{\bar KN}^{(I)}(s) &=& T_{\bar KN}^{(I)}(s)- \frac{f^2_{K N Y}}{s-m_H^2}
-\left( A_{\bar K N}^{(Y)}\right)^2 \frac{2\,m_N^2+m_K^2-s-m_H^2}{8\,f^2\,m_N}
\nonumber\\
&+&\left( P_{\bar K N}^{(Y)}\right)^2 \frac{(m_N+m_H)^2}{8\,f^2\,m_N}
+ P_{\bar K N}^{(Y)}\,A_{\bar K N}^{(Y)}\frac{m^2_H-m_N^2}{4\,f^2\,m_N} \,,
\label{pole-subtr}\end{aligned}$$ with $Y=\Lambda$ for $I=0$ and $Y= \Sigma $ for $I=1$. Whereas it is straightforward to subtract the complete hyperon pole contribution in the $\bar K N$ amplitudes (see (\[pole-subtr\])), it is less immediate how to do so for the $KN$ amplitudes. Since we do not consider all partial wave contributions in the latter amplitudes the u-channel pole must be subtracted in its approximated form as given in (\[u-approx-1\]). The reconstructed amplitudes agree rather well with the original amplitudes. For $\sqrt{s} > 1200$ MeV the solid and dotted lines in Fig. \[fig:causality\] can hardly be discriminated. This demonstrates that our amplitudes are causal too good accuracy. Note that the discrepancy for $\sqrt{s} < 1200$ MeV is a consequence of the approximate treatment of the non-local u-channel exchanges which violates analyticity at subthreshold energies to some extent (see (\[prescription\])). With Fig. \[fig:causality\] we confirm that such effects are well controlled for $\sqrt{s} > 1200$ MeV. In any case, close to $\sqrt{s}\simeq 1200$ MeV the complete forward scattering amplitudes are largely dominated by the s- and u-channel hyperon pole contributions absent in $\Delta T_{\bar KN}^{(I)}(s)$.
As can be seen from Fig. \[fig:causality\] also, we find sizeable p-wave contributions in the pole-subtracted amplitudes at subthreshold energies. This follows from comparing the dashed-dotted lines, which give the s-wave contributions only, with the solid lines which represent the complete real part of the pole-subtracted forward scattering amplitudes. The p-wave contributions are typically much larger below threshold than above threshold. The fact that this is not the case in the isospin zero $KN$ amplitude reflects a subtle cancellation mechanism of hyperon exchange contributions and quasi-local two-body interaction terms. In the $\bar K N$ amplitudes the subthreshold effects of p-waves are most dramatic in the isospin one channel. Here the amplitude is dominated by the $\Sigma(1385)$ resonance. Note that p-wave channels contribute with a positive imaginary part for energies larger than the kaon-nucleon threshold but with a negative imaginary part for subthreshold energies. A negative imaginary part of a subthreshold amplitude is consistent with the optical theorem which only relates the imaginary part of the forward scattering amplitudes to the total cross section for energies above threshold. Our analysis may shed some doubts on the quantitative results of the analysis by Martin, which attempted to constrain the forward scattering amplitude via a dispersion analysis [@A.D.Martin]. An implicit assumption of Martin’s analysis was, that the contribution to the dispersion integral from the subthreshold region, which is not directly determined by the data set, is dominated by s-wave dynamics. As was pointed out in [@Hirschegg] strong subthreshold p-wave contributions should have an important effect for the propagation properties of antikaons in dense nuclear matter.
We turn to the approximate crossing symmetry of our scattering amplitudes. Crossing symmetry relates the subthreshold $\bar K N$ and $K N$ scattering amplitudes. As a consequence the exact amplitude ${\rm T}^{(0)}_{\bar K N}(s)$ shows unitarity cuts not only for $\sqrt{s}>m_\Sigma+m_\pi$ but also for $\sqrt{s}<m_N-m_K$ representing the elastic $K N$ scattering process. Consider for example the isospin zero amplitude for which one expects the following representation:
![Approximate crossing symmetry of the pole subtracted kaon-nucleon forward scattering amplitudes. The lines in the left hand parts of the figures result from the $KN$ amplitudes. The lines in the right hand side of the figures give the $\bar K N$ amplitudes.[]{data-label="fig:crossing"}](cross.eps){width="12cm"}
$$\begin{aligned}
\!\! {\rm T}^{(0)}_{\bar K N}(s)-{\rm T}^{(0)}_{\bar K N}(s_0) &=&
\frac{f^2_{KN \Lambda }}{s-m^2_\Lambda}-\frac{f^2_{KN \Lambda }}{s_0-m^2_\Lambda} +
\! \! \!\int_{-\infty}^{(m_N-m_K)^2} \frac{d \,s'}{\pi }\,\frac{s-s_0}{s'-s_0}\,
\frac{\Im \,{\rm T}^{(0)}_{\bar K N} (s')}{s'-s -i\,\epsilon}
\nonumber\\
&+&\int_{(m_\Sigma+m_\pi)^2}^{+\infty} \frac{d \,s'}{\pi }\,\frac{s-s_0}{s'-s_0}\,
\frac{\Im \,{\rm T}^{(0)}_{\bar K N} (s')}{s'-s -i\,\epsilon}
\;,
\label{cross-disp-check}\end{aligned}$$
where we performed one subtraction to help the convergence of the dispersion integral. Comparing the expressions for $T^{(0)}_{\bar K N}(s)$ in (\[disp-check\]) and ${\rm T}^{(0)}_{\bar K N}(s)$ in (\[cross-disp-check\]) demonstrates that the contribution of the unitarity cut at $\sqrt{s}< m_N-m_K$ in (\[cross-disp-check\]) is effectively absorbed in the subtraction coefficients $c_{\bar K N}^{(I,k)}$ of (\[disp-check\]). Similarly the subtraction coefficients $c_{K N}^{(I,k)}$ in (\[disp-check-2\]) represent the contribution to $T^{(I)}_{K N}(s)$ from the inelastic $\bar K$N scattering process. Thus both model amplitudes $T^{(I)}_{K N}(s)$ and $T^{(I)}_{\bar K N}(s)$ represent the exact amplitude ${\rm T}^{(I)}_{\bar K N}(s)$ within their validity domains and therefore we expect approximate crossing symmetry $$\begin{aligned}
&& \Delta T^{(0)}_{\bar K N}(s) \simeq
-\frac{1}{2}\, \Delta T^{(0)}_{K N}(2\,s_0-s)+\frac{3}{2}\, \Delta T^{(1)}_{ K N}(2\,s_0-s) \;,
\nonumber\\
&& \Delta T^{(1)}_{\bar K N}(s) \simeq
+ \frac{1}{2}\, \Delta T^{(0)}_{K N}(2\,s_0-s)+\frac{1}{2}\, \Delta T^{(1)}_{ K N}(2\,s_0-s) \;,
\label{exp-cross}\end{aligned}$$ close to the optimal matching point $s_0=m_N^2+m_K^2$ only. In Fig. \[fig:crossing\] we confront the pole-subtracted $ \Delta T^{(I)}_{K N}$ and $ \Delta T^{(I)}_{\bar KN} $ amplitudes with the expected approximate crossing identities (\[exp-cross\]). Since the optimal matching point $s_0=m_N^2+m_K^2 \simeq (1068)^2$ MeV$^2$ is slightly below the respective validity range of the original amplitudes, we use the reconstructed amplitudes of (\[disp-check\]) and (\[disp-check-2\]) shown in Fig. \[fig:causality\]. This is justified, because the reconstructed amplitudes are based on the imaginary parts of the amplitude which have support within the validity domain of our theory only. Fig. \[fig:crossing\] indeed confirms that the kaon-nucleon scattering amplitudes are approximatively crossing symmetric. Close to the the point $s\simeq m_N^2+m_K^2$ the $K N$ and $\bar K N$ amplitudes match. We therefore expect that our subthreshold kaon-nucleon scattering amplitudes are determined rather reliably and well suited for an application to the nuclear kaon dynamics.
![Real and imaginary parts of the s- and p-wave partial-wave amplitudes $f^2\,M$ in the isospin zero channel. The labels (ab) refer to our channel convention of (\[r-def\]) with (11): $\bar K N \to \bar K N$, (12): $\bar K N \to \pi \Sigma $ and (22): $\pi \Sigma \to \pi \Sigma $.[]{data-label="fig:amplitudes:zero"}](ampI0ml-long.eps){width="14cm"}
Scattering amplitudes
---------------------
We discuss now the s- and p-wave partial-wave amplitudes for the various $\bar K N, \pi \Sigma $ and $\pi \Lambda$ reactions. Since we are interested in part also in subthreshold amplitudes we decided to present the invariant amplitudes $M^{(\pm )}(\sqrt{s},n)$ rather than the more common $f^{(l)}_{J}(\sqrt{s}\,)$ amplitudes. Whereas the latter amplitudes lead to convenient expressions for the cross sections $$\begin{aligned}
\sigma_{i\,\to f} (\sqrt{s}\,) &=& 4\,\pi \,\frac{p^{(f)}_{\rm cm}}{p^{(i)}_{\rm cm}}\,
\sum_{l=0}^\infty \,\Bigg(
l\,|f^{(l)}_{J=l-\frac{1}{2}}(\sqrt{s}\,) |^2
+ (l+1)\,|f^{(l)}_{J=l+\frac{1}{2}}(\sqrt{s}\,)|^2
\Bigg)\;,
\label{}\end{aligned}$$ the former amplitudes $M^{(\pm )}(\sqrt{s},n)$ provide a more detailed picture of the higher partial-wave amplitudes, simply because the trivial phase-space factor $(p^{(f)}_{\rm cm}\,p^{(i)}_{\rm cm})^{l}$ is taken out
![Real and imaginary parts of the s- and p-wave partial-wave amplitudes $f^2\,M$ in the isospin one channel. The labels (ab) refer to our channel convention of (\[r-def\]) with (11): $\bar K N \to \bar K N$, (12): $\bar K N \to \pi \Sigma $ and (13): $\bar K N \to \pi \Lambda $.[]{data-label="fig:amplitudes:one"}](ampI1ml-long.eps){width="13cm"}
$$\begin{aligned}
f^{(l)}_{J=l\pm \frac{1}{2}}(\sqrt{s}\,) &=&
\frac{(p^{(f)}_{\rm cm}\,p^{(i)}_{\rm cm})^{J-\frac{1}{2}}}{8\,\pi\,\sqrt{s}} \;
\sqrt{E_i\pm m_i}\,\sqrt{E_f\pm m_f}\,M^{(\pm )}(\sqrt{s},J-{\textstyle{1\over 2}})\;.
\label{}\end{aligned}$$
Here $p^{(i,f)}_{\rm cm}$ denote the relative momenta and $E_{i,f}$ the baryon energies in the center of mass frame. Also $m_{i,f}$ are the baryon masses of the initial and final baryons.
In Fig. \[fig:amplitudes:zero\] and in Fig. \[fig:amplitudes:one\] the isospin zero and isospin one amplitudes are shown up to rather high energies $\sqrt{s} =2$ GeV. We emphasize that we trust our amplitudes quantitatively only up to $\sqrt{s} \simeq 1.6$ GeV. Beyond that energy one should consider our results as qualitative only. The amplitudes reflect the presence of the s-wave $\Lambda(1405)$ and p-wave $\Sigma (1385)$ resonances. From the relative height of the peak structures in the figures one can read off the branching ratios of those resonances. For instance it is clearly seen from Fig. \[fig:amplitudes:zero\] that the $\Lambda (1405)$ has a rather small coupling to the $\pi \Sigma $ channel. It is gratifying to find precursor effects for the p-wave $\Lambda (1600)$ and $\Lambda (1890)$ resonances in that figure also. Note that the estimates for the widths of those resonances range up to 200 MeV. Similarly Fig. \[fig:amplitudes:one\] indicates attractive strength in the s- and p- wave channel where one would expect the s-wave $\Sigma (1750)$ and p-wave $\Sigma (1660)$ resonances.
It is a highly non-trivial but nevertheless expected result, that all resonances but the p-wave baryon decuplet and the d-wave baryon nonet resonances were generated dynamically by the chiral coupled channel dynamics once agreement with the low-energy data set with $p_{\rm lab.}< 500$ MeV was achieved. A more accurate description of the latter resonances requires the extension of the $\chi$-BS(3) approach including more inelastic channels. These finding strongly support the conjecture that all baryon resonances but the decuplet ground states are a consequence of coupled-channels dynamics.
Predictions for cross sections
------------------------------
We close the result section by a presentation of total cross sections relevant for transport model calculation of heavy-ion reactions. We believe that the $\chi$-BS(3) approach is particularly well suited to determine some cross sections not directly accessible in scattering experiments. Typical examples would be the $\pi \Sigma \to \pi \Sigma , \pi \Lambda$ reactions. Here the quantitative realization of the chiral SU(3) flavor symmetry including its important symmetry breaking effects are an extremely useful constraint when deriving cross sections not accessible in the laboratory directly. It is common to consider isospin averaged cross sections [@Cugnon; @Brown-Lee] $$\begin{aligned}
\bar \sigma_{}(\sqrt{s}\,) = \frac{1}{N}\,\sum_{I} \,(2\,I+1)\,\sigma_{I}(\sqrt{s}\,) \,.
\label{}\end{aligned}$$ The reaction dependent normalization factor is determined by $ N= \sum \,(2\,I+1)$ where the sum extends over isospin channels which contribute in a given reaction. In Fig. \[fig:cross-pred\] we confront the cross sections of the channels $\bar K N, \pi \Sigma $ and $\pi \Lambda$. The results in the first row repeat to some extent the presentation of Fig. \[fig:totcross\] only that here we confront the cross sections with typical parameterizations used in transport model calculations. The cross sections in the first column are determined by detailed balance from those of the first row. Uncertainties are present nevertheless, reflecting the large empirical uncertainties of the antikaon-nucleon cross sections close to threshold. The remaining four cross sections in Fig. \[fig:cross-pred\] are true predictions of the $\chi$-BS(3) approach. Again we should emphasize that we trust our results quantitatively only for $\sqrt{s}< 1600$ MeV. It is remarkable that nevertheless our cross sections agree with the parameterizations in [@Brown-Lee] qualitatively up to much higher energies except in the $\bar K N \leftrightarrow \pi \Sigma $ reactions where we overshoot those parameterizations somewhat. Besides some significant deviations of our results from [@Cugnon; @Brown-Lee] at $\sqrt{s}-\sqrt{s_{\rm th}} <$ 200 MeV, an energy range where we trust our results quantitatively, we find most interesting the sizeable cross section of about 30 mb for the $\pi \Sigma \to \pi \Sigma $ reaction. Note that here we include the isospin two contribution as part of the isospin averaging. As demonstrated by the dotted line in Fig. \[fig:cross-pred\], which represent the $\chi$-BS(3) approach with s-wave contributions only, the p- and d-wave amplitudes are of considerable importance for the $\pi \Sigma \to \pi \Sigma $ reaction.
![Total cross sections $\bar K N \to \bar K N$, $\bar K N \to \pi \Sigma $, $\bar K N \to \pi \Lambda $ etc relevant for subthreshold production of kaons in heavy-ion reactions. The solid and dashed lines give the results of the $\chi$-BS(3) approach with and without p- and d-wave contributions respectively. The dotted lines correspond to the parameterizations given in [@Brown-Lee].[]{data-label="fig:cross-pred"}](hic-cros.eps){width="14cm"}
Summary and outlook
===================
In this work we successfully used the relativistic chiral SU(3) Lagrangian to describe meson-baryon scattering. Within our $\chi$-BS(3) approach we established a unified description of pion-nucleon, kaon-nucleon and antikaon-nucleon scattering describing a large amount of empirical scattering data including the axial vector coupling constants for the baryon octet ground states. We derived the Bethe-Salpeter interaction kernel to chiral order $Q^3$ and then computed the scattering amplitudes by solving the Bethe-Salpeter equation. This leads to results consistent with covariance and unitarity. Moreover we consider the number of colors ($N_c$) in QCD as a large parameter performing a systematic $1/N_c$ expansion of the interaction kernel. This establishes a significant reduction of the number of parameters. Our analysis provides the first reliably estimates of previously poorly known s- and p-wave parameters. It is a highly non-trivial and novel result that the strength of all quasi-local 2-body interaction terms are consistent with the expectation from the large $N_c$ sum rules of QCD. Further intriguing results concern the meson-baryon coupling constants. The chiral $SU(3)$ flavor symmetry is found to be an extremely useful and accurate tool. Explicit symmetry breaking effects are quantitatively important but sufficiently small to permit an evaluation within chiral perturbation theory. We established two essential ingredients in a successful application of the chiral Lagrangian to the meson-baryon dynamics. First it is found that the explicit s- and u-channel decuplet contributions are indispensable for a good fit. Second, we find that it is crucial to employ the relativistic chiral Lagrangian. It gives rise to well defined kinematical structures in the quasi-local 4-point interaction terms which leads to a mixing of s-wave and p-wave parameters. Only in the heavy-baryon mass limit, not applied in this work, the parameters decouple into the s-wave and p-wave sector. In the course of developing our scheme we constructed a projector formalism which decouples in the Bethe-Salpeter equation covariant partial wave amplitudes and also suggested a minimal chiral subtraction scheme within dimensional regularization which complies manifestly with the chiral counting rules. An important test of our analysis could be provided by new data on kaon-nucleon scattering from the DA$\Phi$NE facility [@DAPHNE]. In particular additional polarization data, possibly with a polarized hydrogen or deuteron target, would be extremely useful.
We performed a consistency check of our forward scattering amplitudes by confronting them with their dispersion-integral representations. Our analysis shows that the scattering amplitudes are compatible with their expected analytic structure. Moreover we demonstrate that the kaon-nucleon and antikaon-nucleon scattering amplitudes are approximatively crossing symmetric in the sense that the $K N$ and $\bar K N$ amplitudes match at subthreshold energies. Our results for the $\bar K N$ amplitudes have interesting consequences for kaon propagation in dense nuclear matter as probed in heavy ion collisions [@Senger]. According to the low-density theorem [@dover; @njl-lutz] an attractive in-medium kaon spectral function probes the kaon-nucleon scattering amplitudes at subthreshold energies. The required amplitudes are well established in our work. In particular we find sizeable contributions from p-waves not considered systematically so far [@Kolomeitsev; @ml-sp; @ramos-sp].
We expect our scattering amplitudes to lead to an improved description of the spectral functions of antikaons in nuclear matter and pave the way for a microscopic description of kaonic atom data. The latter are known to be a rather sensitive test of the antikaon-nucleon dynamics [@Gal].
[**Appendices**]{}
Isospin in $SU(3)$
===================
The $SU(3)$ meson and baryon octet fields $\Phi=\sum \Phi_i\,\lambda_i$ and the baryon octet field $B= \sum B_i\,\lambda_i/\sqrt{2}$ with the Gell-Mann matrices $\lambda_i$ normalized by $\tr \lambda_i \,\lambda_j = 2\,\delta_{ij}$ are decomposed into their isospin symmetric components $$\begin{aligned}
\Phi &=& \tau \cdot \pi
+ \alpha^\dagger \cdot K + K^\dagger \cdot \alpha +
\eta \,\lambda_8 \;,
\nonumber\\
B &=& {\textstyle{1\over\sqrt{2}}}\,
\left( \tau \cdot \Sigma +
\alpha^\dagger \cdot N +\Xi^t\,i\,\sigma_2 \cdot \alpha +\lambda_8 \,\Lambda \right) \, ,
\nonumber\\
\alpha^\dagger &=&
{\textstyle{1\over\sqrt{2}}}\left( \lambda_4+i\,\lambda_5 ,
\lambda_6+i\,\lambda_7 \right)
\;,\;\;\;\tau = (\lambda_1,\lambda_2,\lambda_3)\;,
\label{field-decomp-app}\end{aligned}$$ with the isospin singlet fields $\eta, \Lambda$, the isospin doublet fields $K =(K^+,K^0)^t $, $N=(p,n)^t$, $\Xi = (\Xi^0,\Xi^-)^t$ and the isospin triplet fields $\vec \pi = (\pi^{(1)},\pi^{(2)},\pi^{(3)})$, $\vec \Sigma =(\Sigma^{(1)},\Sigma^{(2)},\Sigma^{(3)})$. Similarly we derive the isospin decomposition of $(\bar \Delta_\mu \cdot \Phi)$, $(\Phi \cdot \Delta_\mu )$ and $(\bar \Delta_\mu \cdot \Delta_\nu)$ as defined in (\[dec-prod\]). The latter objects are expressed in terms of the isospin singlet field $\Omega^-$, the isospin doublet fields $\Xi_\mu = (\Xi^0_\mu, \Xi^-_\mu)^t$, the isospin triplet field $\vec \Sigma_\mu =(\Sigma^{(1)}_\mu,\Sigma^{(2)}_\mu,\Sigma^{(3)}_\mu)$ and the isospin 3/2 field $\underline \Delta_\mu = (\Delta_\mu^{++},\Delta_\mu^+,\Delta_\mu^0,\Delta_\mu^-)^t$. We find $$\begin{aligned}
\Big(\bar \Delta_\mu \cdot \Phi \Big)_{b}^{a}
&=&\Big(\Big(\bar
{\underline \Delta}_\mu \, S \, K \Big)\,\cdot \tau
-\Big(\bar {\underline \Delta}_\mu
\, S \, \alpha \Big)\,\cdot \pi
-\frac{1}{\sqrt{6}}\,K^\dagger
\,\Big(\bar \Sigma_\mu \cdot \sigma \Big) \,\alpha
\nonumber\\
&+&\frac{1}{\sqrt{6}}\,\alpha^\dagger\, \Big(\bar \Sigma_\mu \cdot
\sigma \Big)\,K
+\frac{1}{\sqrt{2}}\,\Big(\bar \Sigma_\mu \cdot \pi \Big)\,\lambda_8
-\frac{1}{\sqrt{2}}\,\Big(\bar
\Sigma_\mu \cdot \tau \Big)\,\eta
\nonumber\\
&-&\frac{i}{\sqrt{6}}\, \Big( \bar \Sigma_\mu \times \pi \Big)\,\cdot \tau
+\frac{1}{\sqrt{2}}\,\Big( K^\dagger\,i\,\sigma_2\,\bar
\Xi_\mu ^t\Big) \,\lambda_8
- \frac{1}{\sqrt{6}}\,\Big(K^\dagger \,\sigma\,i\,\sigma_2\,\bar
\Xi^t_\mu\Big)\,\cdot \tau
\nonumber\\
&+&\frac{1}{\sqrt{6}}\,\alpha^\dagger
\,\Big(\sigma\cdot \pi \Big)\,i\,\sigma_2\,\bar \Xi^t_\mu
-\frac{1}{\sqrt{2}}\,\Big(\alpha^\dagger\,i\,\sigma_2\,\bar
\Xi^t_\mu
\Big)\,\eta
\nonumber\\
&-&\bar \Omega^-_\mu\,\Big(\alpha^\dagger \,i\,\sigma_2\,\Big(K^\dagger \Big)^t
\Big)
\Big)_{b}^{a} \;,
\nonumber\\
\Big(\Phi \cdot \Delta_\mu \Big)_{b}^a
&=&\Big(\Big( K^\dagger \, S^\dagger \, {\underline \Delta}_\mu \Big)\,\cdot
\tau
-\Big(\alpha^\dagger
\, S^\dagger \,{\underline \Delta}_\mu \Big)\, \cdot \pi
-\frac{1}{\sqrt{6}}\,\alpha^\dagger
\,\Big( \Sigma_\mu \cdot \sigma \Big) \,K
\nonumber\\
&+&\frac{1}{\sqrt{6}}\, K^\dagger\,\Big(\Sigma \cdot \sigma \Big)
\,\alpha+\frac{1}{\sqrt{2}}\,
\Big( \Sigma_\mu \cdot \pi \Big)\,\lambda_8
-\frac{1}{\sqrt{2}}\,\Big(\Sigma \cdot \tau \Big)\,\eta
\nonumber\\
&+&\frac{i}{\sqrt{6}}\, \Big( \Sigma_\mu \times \pi \Big)\,\cdot \tau
-\frac{1}{\sqrt{2}}\Big( \Xi^t_\mu \,i\,\sigma_2\,K
\Big) \, \lambda_8
+\frac{1}{\sqrt{6}}\,\Big(\Xi^t_\mu \,i\,\sigma_2\,\sigma \,K\Big)\,\cdot \tau
\nonumber\\
&-& \frac{1}{\sqrt{6}}\,\Xi^t\,\,i\,\sigma_2
\,\Big(\sigma\cdot \pi \Big)\,\alpha
+\frac{1}{\sqrt{2}}\,\Big(\Xi^t_\mu \,i\,\sigma_2\,\alpha
\Big)\,\eta
\nonumber\\
&+&\Omega^-_\mu \,\Big(K^t\,i\,\sigma_2\, \alpha \Big)
\Big)_{b}^a
\label{dec-ap-1}\end{aligned}$$ and $$\begin{aligned}
\Big(\bar \Delta_\mu \cdot \Delta_\nu \Big)_{b}^a
&=&
\Big(\frac{1}{2}\,\Big( \sum_j\,\bar {\underline \Delta}_\mu \,S_j\,\sigma\, S^\dagger_j\,{\underline \Delta}_\nu \Big)\,\cdot \tau
+\frac{1}{6}\,\bar {\underline \Delta}_\mu \cdot {\underline \Delta}_\nu \,\Big(
2+\sqrt{3}\,\lambda_8
\Big)
\nonumber\\
&+&\frac{1}{3}\,\bar
\Sigma_\mu \cdot \Sigma_\nu
+\frac{i}{3}\, \tau \cdot \Big( \bar \Sigma_\mu \times \Sigma_\nu \Big)
\nonumber\\
&+& \frac{1}{6}\,\bar \Xi_\mu \,\Xi_\nu \,\Big( 2-\sqrt{3}\,\lambda_8\Big)
+\frac{1}{6}\,\Big(\bar \Xi_\mu \,\sigma \, \Xi_\nu \Big)\,\cdot \tau
\nonumber\\
&+&
\frac{1}{3}\,\bar \Omega^-_\mu\,\Omega^-_\nu \,\Big(1-\sqrt{3}\,\lambda_8
\Big)
\nonumber\\
&+&\frac{1}{\sqrt{6}}\,\bar {\underline \Delta}_\mu \,\Big(S \cdot \Sigma_\nu
\Big)\,\alpha + \frac{1}{\sqrt{6}}\,
\alpha^\dagger \,\Big(\bar \Sigma_\mu \cdot S^\dagger \Big)\,{\underline \Delta}_\nu
\nonumber\\
&-&\frac{1}{3}\,
\bar \Sigma_\mu \cdot \, \Big( \Xi^t_\nu \,i\,\sigma_2\,\sigma \,\alpha \Big)
+\frac{1}{3}\, \Big( \alpha^\dagger \,\sigma \,i\,\sigma_2\,\bar
\Xi^t_\mu\Big) \,\cdot \Sigma_\nu
\nonumber\\
&+& \frac{1}{\sqrt{6}}\,\bar \Omega^-_\mu\,\Big(\alpha^\dagger \,\Xi_\nu
\Big) +\frac{1}{\sqrt{6}}\,
\Big(\bar \Xi_\mu \, \alpha \Big) \, \Omega^-_\nu \Big)_{b}^a \;.
\label{dec-ap-2}\end{aligned}$$ The isospin transition matrices $S_i$ are normalized by $S^\dagger_i\,S_j = \delta_{ij}-\sigma_i\,\sigma_j/3$. Note that the isospin Pauli matrices $\sigma_i$ act exclusively in the space of isospin doublet fields $K,N,\Xi,\Xi_\mu$ and the matrix valued isospin doublet $\alpha$. Expressions analogous to (\[dec-ap-1\]) hold for $(\bar \Delta_\mu \cdot B)$ and $(\bar B \cdot \Delta_\mu )$.
The isospin reduction of the $SU(3)$ symmetric interaction terms is most conveniently derived applying the set of identities $$\begin{aligned}
\tau \cdot \alpha^\dagger &=& \alpha^\dagger \cdot \sigma
\;,\;\;\;\;\;\alpha^\dagger \cdot \tau =0\;,\;\;\;\;\;
\alpha \cdot \tau = \sigma \cdot \alpha
\;,\;\;\;\;\; \tau \cdot \alpha =0 \;,
\nonumber\\
\tau \, \lambda_8 &=& \lambda_8\, \tau
\;,\;\;\;\;\;
\tr \big(\tau_i\,\tau_j \big)= 2\,\delta_{ij}
\;,\;\;\;\;
\tr \big(\alpha_i\,\alpha^\dagger_j \big)= 2\,\delta_{ij} \,,
\label{isospin-reduction-1}\end{aligned}$$ where the $SU(2)$ Pauli matrices $ \vec \sigma $ act exclusively on the isospin doublet fields. For example $\big(\vec \sigma \cdot \alpha \big)_a=\sum_b\,\vec \sigma_{ab}\,\alpha_b$. The algebra (\[isospin-reduction-1\]) is completed with $$\begin{aligned}
\Big[ \alpha , \alpha^\dagger \Big]_- &=&-\tau \cdot \sigma
-\sqrt{3}\,\lambda_8
\;,\;\;\;\;\;\alpha \cdot \alpha=0\;,
\nonumber\\
\Big[ \alpha \,,\alpha^\dagger \Big]_+ &=& {\textstyle{4\over 3}}\,1
-{\textstyle{1\over \sqrt{3}}}\, \lambda_8 + \tau \cdot \sigma
\;,\;\;\;\;\;
\alpha^\dagger \cdot \alpha^\dagger =0 \;,
\nonumber\\
\Big[ \tau_i\,, \tau_j \Big]_- &=&2\,i\,\epsilon_{ijk}\,\tau_k
\;,\;\;\;
\Big[ \tau_i\,, \tau_j \Big]_+ =
\left( {\textstyle{2\over \sqrt{3}}}\,\lambda_8
+{\textstyle{4\over 3}}\,1 \right) \delta_{ij}\;,
\nonumber\\
\Big[ \lambda_8\,, \alpha^\dagger \Big]_- &=&
\textstyle{\sqrt{3}}\,\alpha^\dagger
\;,\;\;\;\;
\Big[ \lambda_8 \, , \alpha^\dagger \Big]_+
=-{\textstyle{1\over \sqrt{3}}}\,\alpha^\dagger
\;,\;\;\;\;
\Big[ \lambda_8\, , \lambda_8 \Big]_+ = {\textstyle{4\over 3}}\,1
-{\textstyle{2\over \sqrt{3}}}\,\lambda_8 \;,
\nonumber\\
\Big[ \lambda_8\,, \alpha \,\Big]_- &=&-\textstyle{\sqrt{3}}\,\alpha
\;,\;\;\;
\Big[ \lambda_8\,, \alpha \,\Big]_+ =-{\textstyle{1\over \sqrt{3}}}\,\alpha
\;,\;\;\;
\Big[\lambda_8 \,, \tau \Big]_+ ={\textstyle{2\over \sqrt{3}}}\,\tau \;
\label{isospin-reduction-2}\end{aligned}$$ where $[A,B]_\pm= A\,B\pm B\,A$. By means of (\[isospin-reduction-1\], \[isospin-reduction-2\]) it is straightforward to derive the isospin structure of the chiral interaction terms. Note a typical intermediate result $$\begin{aligned}
\Big[\Phi ,B\Big]_+
&=&\sqrt{\frac{2}{3}}\, \Big(\lambda_8 + \frac{2}{\sqrt{3}}\Big)
\,\Big( \pi \cdot \Sigma \Big)
+\frac{1}{\sqrt{2}}\, \pi \cdot \Big( \alpha^\dagger \, \sigma \,N \Big)
+\sqrt{\frac{2}{3}}\, \tau \cdot \Big( \pi \,\Lambda \Big)
\nonumber\\
&-&\frac{1}{\sqrt{6}}\, \Big(\lambda_8 - \frac{4}{\sqrt{3}}\Big)\,
\Big( K^\dagger \,N \Big) +\frac{1}{\sqrt{2}}\, \tau\cdot \Big( K^\dagger \,
\sigma \,N \Big)
-\frac{1}{\sqrt{6}}\,\Big( K^\dagger\,\alpha \Big)\,\Lambda
\nonumber\\
&+&\frac{1}{\sqrt{2}}\,\Big( K^\dagger \sigma \,\alpha \Big) \cdot \Sigma
-\frac{1}{\sqrt{6}}\,\Big(\alpha^\dagger\,K \Big) \,\Lambda
+\frac{1}{\sqrt{2}}\,\Big( \alpha^\dagger \, \sigma \,K \Big) \cdot \Sigma
\nonumber\\
&-&\sqrt{\frac{2}{3}}\, \Big(\lambda_8 - \frac{2}{\sqrt{3}}\Big)\,
\,\Big( \eta \,\Lambda \Big)
-\frac{1}{\sqrt{6}}\,\eta\,\Big( \alpha^\dagger\,N \Big)
+\sqrt{\frac{2}{3}}\, \tau \cdot \Big( \eta \,\Sigma \Big)
\nonumber\\
&+&\frac{1}{\sqrt{2}}\, \pi \cdot \Big( \Xi^t\,i\,\sigma_2\,\sigma
\,\alpha \Big)
-\frac{1}{\sqrt{6}}\, \eta \,\Big(\Xi^t\,i\,\sigma_2\,\alpha\Big)
\nonumber\\
&+& \frac{1}{\sqrt{2}}\, \tau \cdot \Big(\Xi^t\,i\,\sigma_2\,
\sigma \,K \Big)
-\frac{1}{\sqrt{6}}\, \Big(\lambda_8 - \frac{4}{\sqrt{3}}\Big)\, \Big(\Xi^t\,i\,\sigma_2\,
K \Big) \;.
\label{ex-app}\end{aligned}$$
Local interaction terms of chiral order $Q^3$
=============================================
For heavy-baryon chiral $SU(3)$ perturbation theory 102 terms of chiral order $Q^3$ are displayed in [@q3-meissner]. We find that only 10 chirally symmetric terms are relevant for elastic meson-baryon scattering. Following the constructing rules of Krause [@Krause] one writes down 19 interaction terms for the relativistic chiral Lagrangian where terms which are obviously redundant by means of the equation of motion or $SU(3)$ trace identities [@Fearing] are not displayed. The Bethe-Salpeter interaction kernel (see (\[BS-eq\])) receives the following contributions $$\begin{aligned}
&&K^{(I)}_{[8][8]}(\bar k, k; w)=
\frac{1}{8\,f^2}\,\Big( \big( \bar p \cdot \bar q \big)\big( p \cdot q\big)
+\big( \bar p \cdot q \big)\big( p \cdot \bar q\big)\Big)\,C^{(I)}[h^{(1)}]
\\
&&\qquad +\frac{1}{16\,f^2}\,\Big( \qslash\,\big( p-\bar p\big) \cdot \bar q
-\barqslash \,\big(p- \bar p\big) \cdot q\Big)\,\bar C^{(I)}[h^{(2)}]
\nonumber\\
&&\qquad +\frac{1}{8\,f^2}\,\big( \barqslash +\qslash \big)\,(\bar q \cdot q) \,
\,\bar C^{(I)}[h^{(3)}]
\nonumber\\
&&\qquad +\frac{1}{8\,f^2}\,i\,\gamma_5\,\gamma^\mu \, (p+\bar p)^\nu\,\epsilon_{\mu \nu \alpha \beta}\,
\bar q^\alpha\,q^\beta \,\bar C^{(I)}[h^{(4)}]
\nonumber\\
&&\qquad +\frac{1}{16\,f^2}\,i\,\gamma_5\,\gamma^\mu\,
\Big((\bar p \cdot (\bar q-q) )\, p^\mu +(p \cdot (\bar q-q) )\,\bar p^\mu \Big)\,
\epsilon_{\mu \nu \alpha \beta}\,\bar q^\alpha\,q^\beta \,\bar C^{(I)}[h^{(5)}]
\nonumber\\
&&\qquad +\frac{1}{16\,f^2}\,i\,\gamma_5\,\gamma^\mu\,
\Big((\bar p \cdot (\bar q+q) )\, p^\mu -(p \cdot (\bar q+q) )\,\bar p^\mu \Big)\,
\epsilon_{\mu \nu \alpha \beta}\,\bar q^\alpha\,q^\beta \,\bar C^{(I)}[h^{(6)}]
\nonumber \;,
\label{k-q3}\end{aligned}$$ where the interaction terms are already presented in momentum space for notational convenience. Their SU(3) structure is expressed in terms of the matrices $C_{0,1,D,F}$ and $\bar C_{1,D,F}$ introduced in (\[def-local-1\]). In (\[k-q3\]) we applied the convenient notation: $$\begin{aligned}
\bar C^{(I)}[ h^{(i)}]&=&h_{1}^{(i)}\,\bar C_1^{(I)}
+h_{D}^{(i)}\,C_D^{(I)}+h_{F}^{(i)}\,C_F^{(I)} \,,
\nonumber\\
C^{(I)}[h^{(i)}]&=&h_{0}^{(i)}\,C_0^{(I)}+h_{1}^{(i)}\,C_1^{(I)}
+h_{D}^{(i)}\,C_D^{(I)}+h_{F}^{(i)}\,C_F^{(I)} \,.
\label{}\end{aligned}$$ Explicit evaluation of the terms in (\[k-q3\]) demonstrates that in fact only 10 terms contribute to the s and p- wave interaction kernels to chiral order $Q^3$. Obviously there are no quasi-local counter terms contributing to higher partial waves to this order. The terms of chiral order $Q^2$ are $$\begin{aligned}
V^{(I,+)}_{[8][8],2}(s;0)&=& \frac{s}{4\,f^2}\,
\Big( \sqrt{s}-M^{(I)}\Big)\,C^{(I)}[h^{(1)}]\,\Big( \sqrt{s}-M^{(I)}\Big)\,,
\nonumber\\
V^{(I,-)}_{[8][8],2}(s;0)&=& \frac{1}{6\,f^2}\,
M^{(I)}\,\Big[ \,\bar C^{(I)}[h^{(4)}]\, , M^{(I)} \Big]_+\,M^{(I)}\,,
\nonumber\\
&-& \frac{1}{8\,f^2}\,
\Big( \sqrt{s}+M^{(I)}\Big)\,\Big[\, \bar C^{(I)}[h^{(4)}] \,, M^{(I)} \Big]_+ \,
\Big( \sqrt{s}+M^{(I)}\Big)\,,
\nonumber\\
V^{(I,+)}_{[8][8],2}(s;1)&=& \frac{1}{24\,f^2}\,
\Big[ \,\bar C^{(I)}[h^{(4)}] \,, M^{(I)} \Big]_+ \, .
\label{}\end{aligned}$$ We observe that the $h^{(1)}_{0,1,D,F}$ parameters can be absorbed into the $g^{(V)}_{0,1,D,F}$ to order $Q^2$ by the replacement $ g^{(V)} \to g^{(V)}- M \, h^{(1)}$. Similarly the replacement $g^{(T)} \to g^{(T)} + M \,h^{(4)} $ cancels the dependence on $h^{(4)}$ at order $Q^2$ (see (\[local-v\])). This mechanism illustrates the necessary regrouping of interaction terms required for the relativistic chiral Lagrangian as discussed in [@nn-lutz]. We turn to the quasi-local interaction terms of chiral order $Q^3$: $$\begin{aligned}
\nonumber\\
V^{(I,+)}_{[8][8],3}(s;0)&=& \frac{1}{8\,f^2}\,
\,\Bigg(
\Big(\phi^{(I)}+m_{\Phi(I)}^{2}\Big)\, \bar C^{(I)}[h^{(3)}]\,\Big( \sqrt{s}-M^{(I)}\Big)
\nonumber\\
&& \qquad +
\Big( \sqrt{s}-M^{(I)}\Big)\,\bar C^{(I)}[h^{(3)}]\,
\Big(\phi^{(I)}+m_{\Phi(I)}^{2}\Big)\Bigg)
\nonumber\\
-\frac{1}{4\,f^2} \!\!\!&&\!\!\!
\Bigg( \frac{\phi^{(I)}}{2 \,\sqrt{s}}\,\Big( C^{(I)}[ g^{(S)}]
+{\textstyle{1\over 2}} \,\sqrt{s}\,\bar C^{(I)}[ h^{(2)}+2\,h^{(4)}] \Big)
\,\Big( \sqrt{s}-M^{(I)}\Big)
\nonumber\\
+ \!\!\!&&\!\!\!\Big( \sqrt{s}-M^{(I)}\Big)\,\Big( C^{(I)}[ g^{(S)}]
+{\textstyle{1\over 2}} \,\sqrt{s}\,\bar C^{(I)}[ h^{(2)}+2\,h^{(4)}] \Big)
\,\frac{\phi^{(I)}}{2 \,\sqrt{s}} \Bigg)
\nonumber\\
-\frac{1}{4\,f^2} \!\!\!&&\!\!\!
\Bigg( \frac{\phi^{(I)}}{2 \,\sqrt{s}}\,\Big( \bar C^{(I)}[ g^{(T)}]
-{\textstyle{1\over 2}} \,\Big[\, \bar C^{(I)}[ h^{(4)}]\,, M^{(I)} \Big]_+ \Big)
\,\Big( \sqrt{s}-M^{(I)}\Big)
\nonumber\\
+ \!\!\!&&\!\!\! \Big( \sqrt{s}-M^{(I)}\Big)\,\Big( \bar C^{(I)}[ g^{(T)}]
-{\textstyle{1\over 2}} \,\Big[\, \bar C^{(I)}[ h^{(4)}]\,, M^{(I)} \Big]_+ \Big)
\,\frac{\phi^{(I)}}{2 \,\sqrt{s}} \Bigg)
\nonumber\\
&-&\frac{1}{8\,f^2}\,
\,\Bigg(
\Big(\phi^{(I)}+m_{\Phi(I)}^{2}\Big)\, C^{(I)}[g^{(V)}\!+\!2\,\sqrt{s}\,h^{(1)}]\,\Big( \sqrt{s}-M^{(I)}\Big)
\nonumber\\
&& \qquad +
\Big( \sqrt{s}-M^{(I)}\Big)\,C^{(I)}[g^{(V)}\!+\!2\,\sqrt{s}\,h^{(1)}]\,
\Big(\phi^{(I)}+m_{\Phi(I)}^{2}\Big)\Bigg) \,,
\nonumber\\
V^{(I,-)}_{[8][8],3}(s;0)&=&
\frac{1}{8\,f^2} \,\sqrt{s}\,\Bigg(
\Big( \sqrt{s}+M^{(I)}\Big)\,C^{(I)}[g^{(V)}]\,\Big( \sqrt{s}-M^{(I)}\Big)
\nonumber\\
&& \qquad \quad +\Big( \sqrt{s}-M^{(I)}\Big)\,C^{(I)}[g^{(V)}]\,\Big( \sqrt{s}+M^{(I)}\Big) \Bigg)
\nonumber\\
&+&\frac{1}{12\,f^2}\,M^{(I)}\,
\Big[\,C^{(I)}[g^{(V)}+2\,\sqrt{s}\,h^{(1)}]\,,\sqrt{s}-M^{(I)}\Big]_+\,M^{(I)}
\nonumber\\
&+&\frac{1}{12\,f^2}\,M^{(I)}\,
\Big[\,\bar C^{(I)}[h^{(2)}+2\,h^{(4)}-2\,h^{(3)}]\,,\sqrt{s}-M^{(I)}\Big]_+\,M^{(I)}
\,,
\nonumber\\
V^{(I,+)}_{[8][8],3}(s;1)&=& \frac{1}{48\,f^2}\,
\Big[\,C^{(I)}[g^{(V)}+2\,\sqrt{s}\,h^{(1)}]\,,\sqrt{s}-M^{(I)}\Big]_+
\nonumber\\
&+&\frac{1}{48\,f^2}\,
\Big[\,\bar C^{(I)}[h^{(2)}+2\,h^{(4)}-2\,h^{(3)}]\,,\sqrt{s}-M^{(I)}\Big]_+\,.
\label{local-v-q3}\end{aligned}$$ We observe that neither the $h^{(5)}$ nor the $h^{(6)}$ coupling constants enter the interaction kernel to chiral order $Q^3$. Furthermore the structure $h^{(4)}$ is redundant, because it disappears with the replacements to $g^{(T)} \to g^{(T)} + M \,h^{(4)} $ and $h^{(2)} \to h^{(2)} -2\,h^{(4)} $. Thus at chiral order $Q^3$ we find 10 relevant chirally symmetric parameters $h^{(1)}$, $h^{(2)}$ and $h^{(3)}$.
Projector algebra
=================
We establish the loop-orthogonality of the projectors ${Y}^{(\pm )}_n(\bar q,q;w)$ introduced in (\[cov-proj\]). In order to facilitate our derivations we rewrite the projectors in terms of the convenient building objects $P_\pm$ and $V_\mu$ as $$\begin{aligned}
&&{Y}^{(\pm )}_n(\bar q,q;w)=\pm \,P_\pm\,\bar Y_{n+1}(\bar q,q;w)
\pm 3\,(\bar q \cdot V)\,P_\mp\, (V \cdot q)\,\bar Y_{n}(\bar q,q;w)\;,
\nonumber\\
&&\bar Y_{n}(\bar q,q;w)= \sum_{k=0}^{[(n-1)/2]}\,\frac{(-)^k\,(2\,n-2\,k) !}{2^n\,k !\,(n-k)
!\,(n-2\,k -1) !}\,Y_{\bar q \bar q}^{k}\,Y_{\bar q q}^{n-2\,k-1}\,Y_{q q}^{k}\;,
\nonumber\\
&&Y_{a b}=\frac{(w\cdot a )\,(b\cdot w)}{w^2} -a \cdot b \,, \quad
\label{cov-proj:a}\end{aligned}$$ where $$\begin{aligned}
&& P_\pm = \frac{1}{2} \left(1\pm \frac{\wslash}{\sqrt{w^2}} \right)\,,\quad
V_\mu = \frac{1}{\sqrt{3}}\left(\gamma_\mu - \frac{\wslash }{w^2}\,w_\mu \right)\,,
\quad P_\pm \,P_\pm =P_\pm \,,
\nonumber\\
&& P_\pm \,P_\mp =0\,,\quad P_\pm \,\lslash = \lslash \,P_\mp
\pm (l \cdot w)/\sqrt{w^2}\,,\quad
P_\pm \,V_\mu = V_\mu \,P_\mp \,.
\label{cov-proj:b}\end{aligned}$$ In this appendix we will derive the identities: $$\begin{aligned}
&&\Im \ll n \,\pm | m \,\pm \gg (\bar q,q;w) = \delta_{nm}\,Y^{(\pm)}_n(\bar q,q;w)
\,\Im J_{ab}^{(\pm)}(w;n) \;,
\nonumber\\
&&\Im \ll n \,\pm | m \,\mp \gg (\bar q,q;w) =0 \,,
\label{des-result}\end{aligned}$$ with the convenient notation $$\begin{aligned}
&&\ll n \,\pm | m \,\pm \gg (\bar q,q;w)=
\int \frac{d^4l}{(2\pi)^4}\,
Y^{(\pm)}_n(\bar q,l;w)\,G(l;w)\,Y^{(\pm)}_m(l,q;w)\,,
\nonumber\\
&&G(l;w)=\frac{-i}{(w-l)^2-m_a^2+i\,\epsilon}\,\frac{\lslash +m_b}{l^2-m_b^2+i\,\epsilon}\,,
\nonumber\\
&& \Im\,J^{(\pm)}_{ab}(w;n) = \frac{p^{2\,n+1}_{ab}}{8\,\pi\,\sqrt{w^2}}
\left( \frac{\sqrt{w^2}}{2}+ \frac{m_b^2-m_a^2}{2\,\sqrt{w^2}}\pm m_b \right) \,,
\nonumber\\
&& \sqrt{w^2}= \sqrt{m_a^2+p^2}+\sqrt{m_b^2+p^2}\,.
\label{im-loop}\end{aligned}$$ The real parts of the loop functions $J_{ab}^{(\pm )}(w;n)$ are readily reconstructed by means of a dispersion integrals in terms of their imaginary parts. Since the loop functions are highly divergent they require a finite number of subtractions which are to be specified by the renormalization scheme. These terms are necessarily real and represent typically power divergent tadpole terms (see (\[tadpole:a\])). According to our renormalization condition (\[ren-v\]) such terms must be moved into the effective interaction kernel.
In order to arrive at the desired result (\[des-result\]) we introduce further notation streamlining our derivation. For any covariant function $F(\bar q,l,q;w)$ we write $$\begin{aligned}
&&\langle F \rangle_{n,m} (\bar q,q;w)=
\int \frac{d^4l}{(2\pi)^4}\,F(\bar q,l,q;w)\,
\bar Y_n(\bar q,l;w)\,\bar G(l;w)\,\bar Y_m(l,q;w)\,,
\nonumber\\
&& \bar G(l;w)= \frac{1}{(w-l)^2-m_a^2+i\,\epsilon}\,
\frac{-i}{l^2-m_b^2+i\,\epsilon}\,.
\label{f-def}\end{aligned}$$ The result (\[des-result\]) is now derived in two steps. First the expressions in (\[des-result\]) are simplified by standard Dirac algebra methods. In the convenient notation of (\[f-def\]) we find $$\begin{aligned}
\ll n \,\pm | m \,\pm \gg &=&
\langle m_N \pm (l \cdot \hat w)\rangle_{n+1,m+1}\,P_\pm
\nonumber\\
&+&3\, \langle \big(m_N \pm (l \cdot \hat w)\big) \,
(l \cdot V)\rangle_{n+1,m}\,P_\mp\,(V \cdot q)
\nonumber\\
&+& 3\,(\bar q \cdot V )\,P_\mp \,\langle (V \cdot l)\,\big(m_N \pm (l \cdot \hat w)\big)
\rangle_{n,m+1}
\nonumber\\
&-& 3\,(\bar q \cdot V )\,P_\mp \,\langle Y_{ll}\,\big(m_N \pm (l \cdot \hat w)\big)
\rangle_{n,m}\,(V \cdot q)\,,
\nonumber\\
\ll n \,\pm | m \,\mp \gg &=& -P_\pm
\langle \lslash \rangle_{n+1,m+1}\,P_\mp
+ 3\,(\bar q \cdot V )\,P_\mp \,\langle \lslash\,Y_{ll} \rangle_{n,m}\,P_\pm\,(V \cdot q)
\nonumber\\
&+& P_\pm\,\langle \barqslash \,Y_{ll} \rangle_{n,m+1}\,P_\mp
+P_\pm\, \langle \qslash \, Y_{ll} \rangle_{n+1,m}\,P_\mp \,,
\label{first-step}\end{aligned}$$ where $\hat w_\mu = w_\mu/\sqrt{w^2}$. Next we observe that the terms $2\,(l \cdot w)=l^2-m_b^2-(w-l)^2+m_a^2+m_b^2-m_a^2+w^2$ and $Y_{ll}$ in (\[first-step\]) can be replaced by $$\begin{aligned}
l \cdot \hat w \to \frac{\sqrt{w^2}}{2}+ \frac{m_b^2-m_a^2}{2\,\sqrt{w^2}} \,, \quad
Y_{ll} \to p^2 \,,
\label{rep}\end{aligned}$$ if tadpole contributions are neglected. The first replacement rule in (\[rep\]) generates the typical structure occurring in (\[im-loop\]). The final step consists in evaluating the remaining integrals. Consider for example the identities: $$\begin{aligned}
&&\langle 1 \rangle_{n,m} (\bar q,q;w) = \langle 1 \rangle_{m,n} (q,\bar q;w)=
Y_{\bar q \bar q}^{(n-m)/2}\,\bar Y_{m}(\bar q,q;w)\,J_{n+m-1}(w)\,,
\nonumber\\
&&\langle 1 \rangle_{n+1,m} (\bar q,q;w) = \langle 1 \rangle_{m,n+1} (q,\bar q;w)= 0\,,
\quad
\Im \,J_n(w) = \frac{p^{n}}{8\pi \,\sqrt{w^2}}\,,
\label{ex:1}\end{aligned}$$ which hold modulo some subtraction polynomial for $n\geq m$ and both $n,m$ either even or odd. The result (\[ex:1\]) is readily confirmed in the particular frame where $\vec w =0 $ $$\bar Y_{n}(l,q;w) = \big(|\vec l\, |\,|\vec q\,|\big)^{n-1}\,
P_n'\big(\cos (\vec l\,,\vec q\,)\big)\,,
\label{intp}$$ by applying the Cutkosky cutting rule in conjunction with standard properties of the Legendre polynomials[^14]. Here it is crucial to observe that the object $\bar Y_{n}(q,l;w)=\bar Y_{n}(l,q;w)$ does not exhibit any singularity in $l_\mu$. For example a square root term $\sqrt{Y_{\bar q l}}$ in $\bar Y(\bar q, l;w)$ would invalidate our derivation. We point out that this observation leads to the unique interpretation of $P_n'$ in terms of $Y_{qq}$, $Y_{\bar q \bar q}$ and $Y_{\bar q q}$ (see (\[intp\])) and thereby defines the unambiguous form of our projectors (\[cov-proj:a\]). Similarly one derives the identities: $$\begin{aligned}
&&\langle Y_{\bar q l} \rangle_{n+1,m} (\bar q,q;w) =
\langle Y_{\bar q l } \rangle_{m,n+1} (q, \bar q;w)
\nonumber\\
&& \qquad \qquad \qquad \quad \quad = Y_{\bar q \bar q}\,
Y_{\bar q \bar q}^{(n-m)/2}\,\bar Y_{m}(\bar q,q;w)\,J_{n+m+1}(w)\,,
\nonumber\\
&&\langle Y_{l q} \rangle_{n+1,m} (\bar q,q;w) =
\langle Y_{l q } \rangle_{m,n+1} (q, \bar q;w)
\nonumber\\
&& \qquad \qquad \qquad \quad \quad = Y_{\bar q q}\,
Y_{\bar q \bar q}^{(n-m)/2}\,\bar Y_{m}(\bar q,q;w)\,J_{n+m+1}(w)\,,
\nonumber\\
&&\langle Y_{\bar q l} \rangle_{n,m} (\bar q,q;w) =
\langle Y_{\bar q l } \rangle_{m,n} (q, \bar q;w) =0 \,,
\nonumber\\
&&\langle Y_{l q} \rangle_{n,m} (\bar q,q;w) =
\langle Y_{l q } \rangle_{m,n} (q, \bar q;w) =0\,,
\label{ex:2}\end{aligned}$$ which again hold modulo some subtraction polynomial for $n\geq m$ and both $n,m$ either even or odd. Our proof of (\[des-result\]) is completed with the convenient identities: $$\begin{aligned}
&&\langle l_\mu \rangle_{n+1,m} =
\hat w_\mu\,\langle Y_{ll}\,(l-\bar q )\cdot \hat w \rangle_{n,m}
+\bar q_\mu \,\langle Y_{ll} \rangle_{n,m} \quad
{\rm if\;n\geq m } \;,
\nonumber\\
&&Y_{qq}\,\langle l_\mu \rangle_{n+1,m} =
\hat w_\mu\,\langle (l-q )\cdot \hat w \rangle_{n+1,m+1}
+q_\mu \,\langle 1 \rangle_{n+1,m+1} \quad
{\rm if\;n < m } \;,
\nonumber\\
&&\langle l_\mu \rangle_{n,m+1} =
\hat w_\mu\,\langle Y_{ll}\,(l-q )\cdot \hat w \rangle_{n,m}
+q_\mu \,\langle Y_{ll} \rangle_{n,m} \quad
{\rm if\;n\leq m } \;,
\nonumber\\
&&Y_{\bar q \bar q}\,\langle l_\mu \rangle_{n,m+1} =
\hat w_\mu\,\langle (l-\bar q )\cdot \hat w \rangle_{n+1,m+1}
+\bar q_\mu \,\langle 1 \rangle_{n+1,m+1} \quad
{\rm if\;n > m } \;,
\label{help:a}\end{aligned}$$ which follow from (\[ex:2\]) and covariance which implies the replacement rule $$\begin{aligned}
&&l_\mu \to \frac{Y_{q \bar q}\,Y_{\bar q l}-Y_{\bar q \bar q}\,Y_{q l}}
{Y_{q \bar q}^2-Y_{q q}\,Y_{\bar q \bar q}}\,\left( q_\mu-\frac{w \cdot q}{w^2}\,w_\mu \right)
\nonumber\\
&&\quad +\frac{Y_{q \bar q}\,Y_{q l}-Y_{q q}\,Y_{\bar q l}}
{Y_{q \bar q}^2-Y_{q q}\,Y_{\bar q \bar q}}\,
\left( \bar q_\mu-\frac{w \cdot \bar q}{w^2}\,w_\mu \right)+ \frac{l \cdot w}{w^2}\,w_\mu \,.
\label{}\end{aligned}$$ The identities (\[des-result\]) follow now from (\[first-step\]) and (\[help:a\]). For example, the first identity follows for $n=m$, because the first two terms in (\[first-step\]) lead to the loop function $J_{ab}^{(\pm)}(w;n)$ and the last two terms cancel. Similarly for $n> m$ the first and second terms are canceled by the third and fourth terms respectively whereas for $n< m$ both, the first two and last two terms in (\[first-step\]) cancel separately.
Isospin breaking effects
========================
Isospin breaking effects are easily incorporated by constructing super matrices $V^{(I'I)}, J^{(I'I)}$ and $T^{(I'I)}$ which couple different isospin states. Here we only consider isospin breaking effects induced by the loop functions, i.e. the interaction kernel $V^{(I'I)} = \delta_{I'I}\,V^{(I)}$ is assumed isospin diagonal.
Furthermore we neglect isospin breaking effects in all but the s-wave $\bar KN$-channels. This leads to $$\begin{aligned}
J_{\bar K N}^{(00)} =J_{\bar K N}^{(11)} = \frac{1}{2}\Big(J_{K^- p }+ J_{\bar{K^0} n }
\Big)
\,,\,\,
J_{\bar K N}^{(01)} =J_{\bar K N}^{(10)} = \frac{1}{2}\Big(J_{K^- p }-J_{\bar{K^0} n }
\Big)
\,.
\label{}\end{aligned}$$ The remaining channels $X$ are defined via $J^{(I'I)}_X=\delta_{I'I}\,J^{(I)}_X$ with isospin averaged masses in the loop functions. We use also isospin averaged meson and baryon masses in $V^{(I)}$. Note that there is an ambiguity in the subtraction point of the isospin transition loop function $J_{\bar KN}^{(01)} $. We checked that taking $m_\Lambda $ as used for $J_{\bar KN}^{(00)} $ or $m_\Sigma $ as used for $J_{\bar K N}^{(11)} $ makes little difference. We use the average hyperon mass $(m_\Lambda+m_\Sigma)/2$. The $K^-p$-reaction matrices can now be linearly combined in terms of appropriate matrix elements of $M^{(I'I)}$ $$\begin{aligned}
M_{K^-\,p \,\rightarrow K^-\, p\;} &=& \frac{1}{2} \,M^{I=(0 ,0)}_{\bar K N
\rightarrow \bar K N} +\frac{1}{2}\,M^{I=(1,1)}_{\bar K N \rightarrow \bar K N}
+\frac{1}{2}\,M^{I=(0,1)}_{\bar K N \rightarrow \bar K N}
+\frac{1}{2}\,M^{I=(1,0)}_{\bar K N \rightarrow \bar K N} \;,
\nonumber\\
M_{K^-\,p \,\rightarrow \bar K^0\; n\;} &=& \frac{1}{2} \,M^{I=(0,0)}_{\bar K N
\rightarrow \bar K N} -\frac{1}{2}\,M^{I=(1,1)}_{\bar K N \rightarrow \bar K N}
-\frac{1}{2}\,M^{I=(0,1)}_{\bar K N \rightarrow \bar K N}
+\frac{1}{2}\,M^{I=(1,0)}_{\bar K N \rightarrow \bar K N}\;,
\nonumber\\
M_{K^-\,p \,\rightarrow \pi^- \Sigma^+\!} &=& \frac{1}{\sqrt{6}}
\,M^{I=(0,0)}_{\bar K N
\rightarrow \pi \Sigma } +\frac{1}{2}\,M^{I=(1,1)}_{\bar K N \rightarrow \pi \Sigma}
+\frac{1}{\sqrt{6}} \,M^{I=(1,0)}_{\bar K N
\rightarrow \pi \Sigma } +\frac{1}{2}\,M^{I=(0,1)}_{\bar K N \rightarrow \pi \Sigma} \;,
\nonumber\\
M_{K^-\,p \,\rightarrow \pi^+ \Sigma^-\!} &=& \frac{1}{\sqrt{6}}
\,M^{I=(0,0)}_{\bar K N \rightarrow \pi \Sigma }
-\frac{1}{2}\,M^{I=(1,1)}_{\bar K N \rightarrow \pi \Sigma}
+\frac{1}{\sqrt{6}} \,M^{I=(1,0)}_{\bar K N \rightarrow \pi \Sigma }
-\frac{1}{2}\,M^{I=(0,1)}_{\bar K N \rightarrow \pi \Sigma} \;,
\nonumber\\
M_{K^-\,p \,\rightarrow \pi^0\, \Sigma\;} &=&
\frac{1}{\sqrt{6}} \,M^{I=(0,0)}_{\bar K N \rightarrow \pi \Sigma }
+\frac{1}{\sqrt{6}} \,M^{I=(1,0)}_{\bar K N \rightarrow \pi \Sigma } \;,
\nonumber\\
M_{K^-\,p \,\rightarrow \,\pi^0\, \Lambda\;} &=& \frac{1}{\sqrt{2}}
\,M^{I=(0,1)}_{\bar K N \rightarrow \pi \Lambda }
+\frac{1}{\sqrt{2}}\,M^{I=(1,1)}_{\bar K N \rightarrow \pi \Lambda } \;.
\label{}\end{aligned}$$
The $K^+p \to K^+p$ reaction of the strangeness +1 channel remains a single channel problem due to charge conservation. One finds $$\begin{aligned}
M_{K^+\,p \,\rightarrow K^+\, p\;} &=&
M^{I=(1,1)}_{K N \rightarrow K N} \;, \quad
J_{K N}^{(11)} =J_{K^+ p} \;.
\label{}\end{aligned}$$ The charge exchange reaction $K^+n \to K^0 p$, on the other hand, turns into a 2 channel problem with ($K^+n$, $K^0 p$). The proper matrix structure in the isospin basis leads to $$\begin{aligned}
&&M_{K^+\,n \,\rightarrow K^0\, p\;} = \frac{1}{2} \,M^{I=(1 ,1)}_{K N
\rightarrow K N} -\frac{1}{2}\,M^{I=(0,0)}_{K N \rightarrow K N}
+\frac{1}{2}\,M^{I=(0,1)}_{K N \rightarrow K N}
+\frac{1}{2}\,M^{I=(1,0)}_{K N \rightarrow K N} \;,
\nonumber\\
&&J_{K N}^{(00)} =J_{K N}^{(11)}=
\frac{1}{2}\,\Big( J_{K^+ n}+ J_{K^0 p} \Big) \;,\;\;
J_{K N}^{(01)} =J_{K N}^{(10)} =\frac{1}{2}\,\Big( J_{K^+ n}- J_{K^0 p} \Big)\;.
\label{}\end{aligned}$$ The subtraction point $\mu^{(I)}$ in the loop functions $J_{K^+ p}, J_{K^+ n}$ and $J_{K^0 p}$ is identified with the average hyperon mass $\mu^{(0)}=\mu^{(1)}=(m_\Lambda+m_\Sigma)/2$.
Strangeness minus one channel
=============================
We collect the coefficients $C^{(I)}_{...}$ defined in (\[k-nonlocal\],\[G-explicit\],\[def-local-1\]) in Tab. \[tabkm-1\] and Tab. \[tabkm-2\]. For completeness we also include the $I=2$ channel. All coefficients are defined within the natural extension of our notation in (\[r-def\]). The appropriate isospin states $R^{(2)}$ are introduced as $$\begin{aligned}
R^{(2)}_{[n]} = \pi_c \cdot S_{[n]}\cdot \Sigma \,, \qquad
\label{I=2:def}\end{aligned}$$ where the matrix valued vector $S_{[n]}$ satisfies $$\begin{aligned}
\sum_{n=1}^5\,S^\dagger_{[n],\,ac}\,S^{\,}_{[n],\,bd} ={\textstyle{1\over 2}}\, \delta_{ab}\,\delta_{cd}+
{\textstyle{1\over 2}}\,\delta_{ad}\,\delta_{cb} -{\textstyle{1\over 3}}\,\delta_{ac}\,\delta_{bd} \,.
\label{}\end{aligned}$$
=0.7mm
$ C_{WT}^{(0)} $ $ C_{N_{[8]}}^{(0)}$ $C_{\Lambda_{[8]}}^{(0)}$ $C_{\Sigma_{[8]}}^{(0)}$ $C_{\Delta_{[10]}}^{(0)}$ $C_{\Sigma_{[10]} }^{(0)}$ $\widetilde{C}_{N_{[8]}}^{(0)}$ $\widetilde{C}_{\Lambda_{[8]} }^{(0)}$ $\widetilde{C}_{\Sigma_{[8]}}^{(0)}$ $\widetilde {C}_{\Xi_{[8]} }^{(0)}$ $\widetilde {C}_{\Delta_{[10]} }^{(0)}$ $\widetilde {C}_{\Sigma_{[10]}}^{(0)}$ $\widetilde C_{\Xi_{[10]}}^{(0)}$
------ ----------------------- ---------------------- --------------------------- -------------------------- --------------------------- ---------------------------- --------------------------------- ---------------------------------------- -------------------------------------- ------------------------------------- ----------------------------------------- ---------------------------------------- ----------------------------------- --
$11$ $3$ 0 $1$ $0$ 0 0 0 0 0 0 0 0 0
$12$ $\sqrt{\frac{3}{2}}$ 0 $1$ $0$ 0 0 $\sqrt{\frac{2}{3}}$ 0 0 0 $2\sqrt{\frac{2}{3}}$ 0 0
$13$ $\frac{3}{\sqrt{2}}$ 0 $1$ $0$ 0 0 $\sqrt{2}$ 0 0 0 0 0 0
$14$ $0$ 0 $1$ $0$ 0 0 0 $\frac{1}{2}$ -$\frac{3}{2}$ 0 0 -$\frac{3}{2}$ 0
$22$ $4$ 0 $1$ $0$ 0 0 0 $\frac{1}{3}$ -1 0 0 -1 0
$23$ $0$ 0 $1$ $0$ 0 0 0 0 $\sqrt{3}$ 0 0 $\sqrt{3}$ 0
$24$ -$\sqrt{\frac{3}{2}}$ 0 $1$ $0$ 0 0 0 0 0 -$\sqrt{\frac{2}{3}}$ 0 0 -$\sqrt{\frac{2}{3}}$
$33$ $0$ 0 $1$ $0$ 0 0 0 1 0 0 0 0 0
$34$ -$\frac{3}{\sqrt{2}}$ 0 $1$ $0$ 0 0 0 0 0 -$\sqrt{2}$ 0 0 -$\sqrt{2}$
$44$ $3$ 0 $1$ $0$ 0 0 0 0 0 0 0 0 0
$ C_{WT}^{(1)} $ $ C_{N_{[8]}}^{(1)}$ $C_{\Lambda_{[8]}}^{(1)}$ $C_{\Sigma_{[8]}}^{(1)}$ $C_{\Delta_{[10]}}^{(1)}$ $C_{\Sigma_{[10]} }^{(1)}$ $\widetilde{C}_{N_{[8]}}^{(1)}$ $\widetilde{C}_{\Lambda_{[8]} }^{(1)}$ $\widetilde{C}_{\Sigma_{[8]}}^{(1)}$ $\widetilde {C}_{\Xi_{[8]} }^{(1)}$ $\widetilde {C}_{\Delta_{[10]} }^{(1)}$ $\widetilde {C}_{\Sigma_{[10]}}^{(1)}$ $\widetilde C_{\Xi_{[10]}}^{(1)}$
$11$ 1 0 0 1 0 1 0 0 0 0 0 0 0
$12$ 1 0 0 1 0 1 -$\frac{2}{3}$ 0 0 0 $\frac{2}{3}$ 0 0
$13$ $\sqrt{\frac{3}{2}}$ 0 $0$ $1$ 0 1 $\sqrt{\frac{2}{3}}$ 0 0 0 0 0 0
$14$ $\sqrt{\frac{3}{2}}$ 0 $0$ $1$ 0 1 $\sqrt{\frac{2}{3}}$ 0 0 0 0 0 0
$15$ $0$ 0 $0$ $1$ 0 1 0 -$\frac{1}{2}$ -$\frac{1}{2}$ 0 0 -$\frac{1}{2}$ 0
$22$ $2$ 0 $0$ $1$ 0 1 0 -$\frac{1}{3}$ $\frac{1}{2}$ 0 0 $\frac{1}{2}$ 0
$23$ $0$ 0 $0$ $1$ 0 1 0 0 -1 0 0 -1 0
$24$ $0$ 0 $0$ $1$ 0 1 0 0 1 0 0 1 0
$25$ $-1$ 0 $0$ $1$ 0 1 0 0 0 $\frac{2}{3}$ 0 0 $\frac{2}{3}$
$33$ $0$ 0 $0$ $1$ 0 1 0 0 1 0 0 1 0
$34$ $0$ 0 $0$ $1$ 0 1 0 $\frac{1}{\sqrt{3}}$ 0 0 0 0 0
$35$ $\sqrt{\frac{3}{2}}$ 0 $0$ $1$ 0 1 0 0 0 -$\sqrt{\frac{2}{3}}$ 0 0 -$\sqrt{\frac{2}{3}}$
$44$ $0$ 0 $0$ $1$ 0 1 0 0 1 0 0 1 0
$45$ $\sqrt{\frac{3}{2}}$ 0 $0$ $1$ 0 1 0 0 0 -$\sqrt{\frac{2}{3}}$ 0 0 -$\sqrt{\frac{2}{3}}$
$55$ $1$ 0 $0$ $1$ 0 1 0 0 0 0 0 0 0
$ C_{WT}^{(2)} $ $ C_{N_{[8]}}^{(2)}$ $C_{\Lambda_{[8]}}^{(2)}$ $C_{\Sigma_{[8]}}^{(2)}$ $C_{\Delta_{[10]}}^{(2)}$ $C_{\Sigma_{[10]} }^{(2)}$ $\widetilde{C}_{N_{[8]}}^{(2)}$ $\widetilde{C}_{\Lambda_{[8]} }^{(2)}$ $\widetilde{C}_{\Sigma_{[8]}}^{(2)}$ $\widetilde {C}_{\Xi_{[8]} }^{(2)}$ $\widetilde {C}_{\Delta_{[10]} }^{(2)}$ $\widetilde {C}_{\Sigma_{[10]}}^{(2)}$ $\widetilde C_{\Xi_{[10]}}^{(2)}$
$11$ -2 0 0 0 0 0 0 $\frac{1}{3}$ $\frac{1}{2}$ 0 0 $\frac{1}{2}$ 0
: Weinberg-Tomozawa interaction strengths and baryon exchange coefficients in the strangeness minus one channels as defined in (\[k-nonlocal\]) and (\[I=2:def\]).[]{data-label="tabkm-1"}
=1.05mm
[|r||c||c|c||c|c|c||c|c|c|c||c|c|c|]{}\
\
& $C_{\pi,0}^{(0)}$ & $C_{\pi,D}^{(0)}$ & $C_{\pi,F}^{(0)}$ & $C_{K,0}^{(0)}$ & ${C}_{K,D}^{(0)}$ & ${C}_{K,F}^{(0)}$ & ${C}_{0}^{(0)}$ & ${C}_{1}^{(0)}$ & ${C}_{D}^{(0)}$ & ${C}_{F}^{(0)}$ & $\bar C_{1}^{(0)}$ & $\bar C_{D}^{(0)}$ & $\bar C_{F}^{(0)}$\
$11$& 0 & 0 & 0 & -4 &-6 & -2 & 2 & 2 & 3& 1 & 2 &1& 3\
$12$& 0& -$\sqrt{\frac32}$ & $\sqrt{\frac32}$ & 0& -$\sqrt{\frac32}$ & $\sqrt{\frac32}$ & 0 & $\sqrt 6$ & $\sqrt{\frac32}$ &-$\sqrt{\frac32}$ & $\sqrt 6$ & -$\sqrt{\frac32}$ &$\sqrt{\frac32}$\
$13$& 0& $\frac{1}{\sqrt 2}$ & $\frac{3}{\sqrt 2}$ & 0& -$\frac{5}{3\,\sqrt 2}$ & -$\frac{5}{\sqrt 2}$ & 0 & $\sqrt 2$ & $\frac{1}{3\sqrt 2}$ & $\frac{1}{\sqrt 2}$ & $\sqrt 2$ &$\frac{1}{\sqrt 2}$&$\frac{3}{\sqrt 2}$\
$14$& 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3& 0 & 0 & -1& 0 & 0\
$22$& -4 & -4 & 0 & 0 & 0 & 0 & 2 & 4& 2 & 0 & 2 & 0& 4\
$23$& 0 & -$\frac{4}{\sqrt 3}$ & 0 & 0 & 0 & 0 & 0 & $\sqrt 3$& $\frac{2}{\sqrt 3}$ & 0 & $\sqrt 3$ & 0 & 0\
$24$& 0& $\sqrt{\frac32}$ & $\sqrt{\frac32}$ & 0& $\sqrt{\frac32}$ & $\sqrt{\frac32}$ & 0 & -$\sqrt 6$& -$\sqrt{\frac32}$ & -$\sqrt{\frac32}$ & -$\sqrt 6$ &-$\sqrt{\frac32}$& -$\sqrt{\frac32}$\
$33$& $\frac43$ & $\frac{28}{9}$ & 0 & -$\frac{16}{3}$ & -$\frac{64}{9}$& 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0\
$34$& 0& -$\frac{1}{\sqrt 2}$ & $\frac{3}{\sqrt{2}}$ & 0& $\frac{5}{3\sqrt 2}$ & -$\frac{5}{\sqrt 2}$ & 0 & -$\sqrt 2$ & -$\frac{1}{3\sqrt 2}$ & $\frac{1}{\sqrt 2}$ & -$\sqrt 2$ &$\frac{1}{\sqrt 2}$ & -$\frac{3}{\sqrt 2}$\
$44$& 0 & 0 & 0 & -4 &-6 & 2 & 2 & 2 & 3&-1 & 2 & -1 &3\
& $C_{\pi,0}^{(1)}$ & $C_{\pi,D}^{(1)}$ & $C_{\pi,F}^{(1)}$ & $C_{K,0}^{(1)}$ &${C}_{K,D}^{(1)}$ & ${C}_{K,F}^{(1)}$ & ${C}_{0}^{(1)}$ & ${C}_{1}^{(1)}$& ${C}_{D}^{(1)}$ & ${C}_{F}^{(1)}$ & $\bar C_{1}^{(1)}$ & $\bar C_{D}^{(1)}$ & $\bar C_{F}^{(1)}$\
$11$&0 & 0 & 0 & -4 & -2& 2 & 2 & 0 & 1& -1 & 0& -1 & 1\
$12$& 0& -1 & 1 & 0& -1 & 1 & 0 & 0 & 1& -1 & 0 &-1& 1\
$13$& 0 & $\frac{1}{\sqrt 6}$ & $\sqrt{\frac32}$ & 0 & $\frac{1}{\sqrt 6}$& $\sqrt{\frac32}$ & 0 & 0 & -$\frac{1}{\sqrt 6}$ &-$\sqrt{\frac32}$ & 0 &$\frac{1}{\sqrt 6}$& $\sqrt{\frac32}$\
$14$& 0 & -$\sqrt{\frac32}$& $\sqrt{\frac32}$ & 0 & $\frac{5}{\sqrt 6}$ &-$\frac{5}{\sqrt 6}$ & 0 & 0 & -$\frac{1}{\sqrt 6}$ & $\frac{1}{\sqrt 6}$ &0 & -$\sqrt{\frac32}$& $\sqrt{\frac32}$\
$15$& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0\
$22$& -4&-4 & 0 & 0& 0 & 0 & 2 & -1 & 2 & 0 & 1 & 0 & 2\
$23$& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -$2\sqrt{\frac23}$& 0\
$24$& 0& 0 & $4\sqrt{\frac23}$ & 0 & 0 & 0 & 0 & 0 & 0 & -$2\sqrt{\frac23}$ & 0 & 0 &0\
$25$& 0 & 1& 1 & 0 & 1 & 1 &0 & 0 & -1 & -1 &0 & -1 & -1\
$33$& -4 & -$\frac43$ & 0 & 0 & 0 & 0 & 2 & 0 & $\frac23$& 0 &0 & 0 & 0\
$34$& 0 & -$\frac43$ & 0 & 0 &0 & 0 & 0 & 1 & $\frac23$ & 0 &-1 & 0 & 0\
$35$& 0 & $\frac{1}{\sqrt 6}$ & -$\sqrt{\frac32}$ & 0 & $\frac{1}{\sqrt 6}$ &-$\sqrt{\frac32}$ & 0 & 0 &-$\frac{1}{\sqrt 6}$& $\sqrt{\frac32}$ & 0 & -$\frac{1}{\sqrt 6}$ & $\sqrt{\frac32}$\
$44$& $\frac43$ & -$\frac43$ & 0 & -$\frac{16}{3}$ & 0 & 0 & 2 & 0 & $\frac23$ & 0 & 0 & 0 & 0\
$45$& 0 & -$\sqrt{\frac32}$ & -$\sqrt{\frac32}$ & 0 & $\frac{5}{\sqrt 6}$ & $\frac{5}{\sqrt 6}$ & 0 & 0 & -$\frac{1}{\sqrt 6}$ & -$\frac{1}{\sqrt 6}$ & 0 & $\sqrt{\frac32}$ & $\sqrt{\frac32}$\
$55$& 0 & 0 & 0 & -4 & -2 & -2 & 2 & 0 & 1 & 1 & 0 & 1 & 1\
& $C_{\pi,0}^{(2)}$ & $C_{\pi,D}^{(2)}$ & $C_{\pi,F}^{(2)}$ & $C_{K,0}^{(2)}$ &${C}_{K,D}^{(2)}$ & ${C}_{K,F}^{(2)}$ & ${C}_{0}^{(2)}$ & ${C}_{1}^{(2)}$& ${C}_{D}^{(2)}$ & ${C}_{F}^{(2)}$ & $\bar C_{1}^{(2)}$ & $\bar C_{D}^{(2)}$ & $\bar C_{F}^{(2)}$\
$11$ &-4 & -4 & 0 & 0 & 0& 0 & 2 & 1 & 2 & 0 & -1& 0 & -2\
Differential cross sections
===========================
In this appendix we provide the expressions needed for the evaluation of cross sections as required for the comparison with available empirical data. Low-energy data are available for the reactions $K^-p \to K^-p, \bar K^0 n, \pi^\mp\Sigma^\pm, \pi^0 \Sigma^0$, $\pi^0 \Lambda$ and $K^-p \to \pi^0 \pi^- p, \pi^+ \pi^- n $ in the strangeness minus one channel and for the reactions $K^+p \to K^+p $ and $K^+ n \to K^0p $ in the strangeness +1 channel. The differential cross sections with a two-body final state can be written in the generic form $$\begin{aligned}
\frac{d\,\sigma (\sqrt{s}, \cos \theta )}{d\,\cos \theta } &=&
\frac{1}{32\,\pi\,s}\,\frac{p^{(f)}_{\rm cm}}{p^{(i)}_{\rm cm}}\,\Bigg(
| F_+ (\sqrt{s},\theta )|^2\,\Big(m_i+E_i\Big) \,\Big(m_f+E_f\Big)
\nonumber\\
&+&| F_-(\sqrt{s},\theta )|^2\Big(m_i-E_i\Big)
\,\Big(m_f-E_f\Big)
\nonumber\\
&+&2\,\Re \,\Big( F_+(\sqrt{s},\theta )\,F_-^\dagger(\sqrt{s},\theta )
\Big)\,p_{\rm cm}^{(i)}\,p_{\rm cm}^{(f)}\,\cos
\theta \Bigg)
\label{cross-section}\end{aligned}$$ where $p_{\rm cm}^{(i)}$ and $p_{\rm cm}^{(f)}$ is the relative momentum in the center of mass system of the initial and final channel respectively. The angle $ \theta $ denotes the scattering angle in the center of mass system. Also, $m_{i}$ and $m_{f}$ are the masses of incoming and outgoing fermions, respectively, and $$\begin{aligned}
E_i=\sqrt{m^2_i+\Big(p_{\rm cm}^{(i)}\Big)^2}\,, \qquad
E_f=\sqrt{m^2_f+\Big(p_{\rm cm}^{(f)}\Big)^2}
\,.
\label{}\end{aligned}$$ The amplitudes $F_\pm(\sqrt{s},\theta )$ receive contributions from the amplitudes $M^{(\pm)}(\sqrt{s},0)$ and $M^{(\pm)}(\sqrt{s},1)$ as introduced in (\[t-sum\]). We derive $$\begin{aligned}
F_+(\sqrt{s},\theta ) &=& M^{(+)}(\sqrt{s}; 0)
+3 \, p_{\rm cm}^{(f)}\,p_{\rm cm}^{(i)}\,M^{(+)}(\sqrt{s};1)\,\cos \theta
\nonumber\\
&-& \Big(E_f-m_f\Big)\,\Big(E_i-m_i\Big)\,M^{(-)}(\sqrt{s};1) \,,
\nonumber\\
F_- (\sqrt{s},\theta )&=&M^{(-)}(\sqrt{s}; 0)
+3 \, p_{\rm cm}^{(f)}\,p_{\rm cm}^{(i)}\,M^{(-)}(\sqrt{s};1)\,\cos \theta
\nonumber\\
&-&\Big(E_f+m_f\Big)\,\Big(E_i+m_i\Big)\,M^{(+)}(\sqrt{s};1) \;,
\label{mplus-minus}\end{aligned}$$ where the appropriate element of the coupled channel matrix $M^{(I,\pm)}_{ab}$ is assumed. For further considerations it is convenient to introduce reduced partial-wave amplitudes $f^{(l)}_{J=l\pm \frac{1}{2}}(s)$ with $$\begin{aligned}
f^{(l)}_{J=l\pm \frac{1}{2}}(\sqrt{s}\,) &=&
\frac{(p^{(f)}_{\rm cm}\,p^{(i)}_{\rm cm})^{J-\frac{1}{2}}}{8\,\pi\,\sqrt{s}} \;
\sqrt{E_i\pm m_i}\,\sqrt{E_f\pm m_f}\,M^{(\pm )}(\sqrt{s},J-{\textstyle{1\over 2}})\;,
\label{}\end{aligned}$$ in terms of which the total cross section reads $$\begin{aligned}
\sigma_{i\,\to f} (\sqrt{s}\,) &=& 4\,\pi \,\frac{p^{(f)}_{\rm cm}}{p^{(i)}_{\rm cm}}\,
\sum_{l=0}^\infty \,\Bigg(
l\,|f^{(l)}_{J=l-\frac{1}{2}}(\sqrt{s}\,) |^2
+ (l+1)\,|f^{(l)}_{J=l+\frac{1}{2}}(\sqrt{s}\,)|^2
\Bigg)\;.
\label{}\end{aligned}$$
The measurement of the three-body final states $\pi^0 \pi^- p$ and $\pi^+\pi^- n$ provides more detailed constraints on the kaon induced $\Lambda$ and $\Sigma$ production matrix elements due to the self-polarizing property of the hyperons. Particularly convenient is the triple differential cross section $$\begin{aligned}
\frac{k_{\pi^- n}\,d\sigma_{K^-p\,\to \pi^+ \pi^-n}}{dm_{\Sigma^- }^2\,
d\! \cos \theta \,d k_\perp}
&=&\frac{\Gamma_{\Sigma^{-}\to \pi^- n}}{2\pi\,\Gamma_{\Sigma^-}^{(tot.)}}\,
\frac{d\sigma_{K^-p\,\to \pi^+ \Sigma^- } }{d\!\cos \theta }\,\Big( 1
+\alpha_{\Sigma^-}\,\frac{k_\perp}{k_{\pi- n}} \,P_{K^-p\,\to \pi^+ \Sigma^- }\Big)\;,
\nonumber\\
\frac{k_{\pi^0p}\,d\sigma_{K^-p\,\to \pi^0 \pi^-p}}{dm_{\Sigma^+}^2\,d\! \cos \theta \,
d k_\perp}
&=&\frac{\Gamma_{\Sigma^+\to \pi^0 p}}{2\pi\,\Gamma_{\Sigma^+}^{(tot.)}}\,
\frac{d\sigma_{K^-p\,\to \pi^-\Sigma^+ } }{d\!\cos \theta }\,\Big( 1
+\alpha_{\Sigma^+}\,\frac{k_\perp}{k_{\pi^0 p}}\,P_{K^-p\,\to \pi^-\Sigma^+ }\Big)
\nonumber\\
\frac{k_{\pi^-p}\,d\sigma_{K^-p\,\to \pi^0 \pi^-p}}{dm_\Lambda^2\,d\! \cos \theta \,d k_\perp}
&=&\frac{\Gamma_{\Lambda\to \pi^- p}}{2\pi\,\Gamma_\Lambda^{(tot.)}}\,
\frac{d\sigma_{K^-p\,\to \pi^0\Lambda } }{d\!\cos \theta }\,\Big( 1
+\alpha_\Lambda\,\frac{k_\perp}{k_{\pi^- p}}\,P_{K^-p\,\to \pi^0\Lambda }\Big)\;,
\label{}\end{aligned}$$ which determines the polarizations $P_{K^-p\to \pi^0\Lambda }$ and $P_{K^-p\to \pi\mp \Sigma^\pm }$ $$\begin{aligned}
\frac{d\sigma (\sqrt{s},\cos \theta )}{d\!\cos \theta }\,P(\sqrt{s},\theta )
&=& \frac{\Big(p^{(f)}_{\rm cm}\Big)^2\,\sin \theta }{32\,\pi\,s}
\,\Im \,\Big( F_+(\sqrt{s},\theta )\,F_-^\dagger(\sqrt{s},\theta ) \Big)\;.
\label{}\end{aligned}$$ Here $\alpha_{\Sigma^-} =-0.068\pm$0.013, $\alpha_{\Sigma^+} =-0.980\pm$0.017 and $\alpha_\Lambda =0.642\pm$0.013 are the hyperon polarizeabilties. The variable $k_\perp$ is defined in the $K^-p$ center of mass frame via $k_\perp\,p_{\rm cm }^{(i)}\,p_{\rm cm }^{(f)}\sin \theta =
{\vec k} \cdot ({\vec p}^{\,(i)}_{\rm cm}\! \times {\vec p}^{\,(f)}_{\rm cm})$. In the hyperon rest frame it represents the decay angle $\beta $ of the $\Sigma $ and $\Lambda $ with $k_\perp =k_{\pi^\pm n}\,\cos \beta$ and $k_\perp =k_{\pi^- p}\,\cos \beta$ relative to the final neutron and proton three momentum ${\vec k}$ respectively. Furthermore $m_{\Sigma^\pm}^2= (m_{\pi^\pm}^2+k_{\pi^\pm n}^2)^{1/2}+(m_n^2+k_{\pi^\pm n}^2)^{1/2}$ and $m_\Lambda^2= (m_{\pi^-}^2+k_{\pi^- p}^2)^{1/2}+(m_p^2+k_{\pi^- p}^2)^{1/2}$.
Differential cross sections and polarizations are conveniently parameterized in terms of moments $A_n(\sqrt{s}\,)$ and $B_n(\sqrt{s}\,)$ [@mast-pio; @mast-ko; @bangerter-piS] defined as the nth order Legendre weights $$\begin{aligned}
\sum_{n=0}^\infty A_n(\sqrt{s}\,)\,P_n(\cos \theta )&=&
\frac{d\sigma (\sqrt{s}, \cos \theta )}{d\cos \theta } \,,
\nonumber\\
-\sum_{l=1}^\infty B_l(\sqrt{s}\,)\,P^1_l(\cos \theta )&=&
\frac{\Big(p^{(f)}_{\rm cm}\Big)^2\,\sin \theta }{32\,\pi\,s}\,
\Im \,\Big( F_+(\sqrt{s},\theta )\,F_-^\dagger(\sqrt{s},\theta ) \Big)\;,
\label{a-distributions}\end{aligned}$$ where $P_l^1(\cos \theta)=-\sin \theta \,P_l'(\cos \theta)$. One derives $$\begin{aligned}
A_1(\sqrt{s}\,)&=& 4\,\pi\,\frac{p^{(f)}_{\rm cm}}{p^{(i)}_{\rm cm}}\,
\Re \Bigg( f^{(S)}_{J=\frac{1}{2}}(\sqrt{s}\,)\,\Big(
f^{(P,*)}_{J=\frac{1}{2}}(\sqrt{s}\,)+2\,f^{(P,*)}_{J=\frac{3}{2}}(\sqrt{s}\,)
\Big)\,
\nonumber\\
&& \qquad \quad +
\Big(2\,f^{(P)}_{J=\frac{1}{2}}(\sqrt{s}\,)+\frac{2}{5}\, f^{(P)}_{J=\frac{3}{2}}(\sqrt{s}\,)
\Big)\,f^{(D,*)}_{J=\frac{3}{2}}(\sqrt{s}\,) \Bigg)\;,
\nonumber\\
A_2(\sqrt{s}\,)&=&4\,\pi\, \frac{p^{(f)}_{\rm cm}}{p^{(i)}_{\rm cm}}\,
\Re \Bigg(| f^{(P)}_{J=\frac{3}{2}}(\sqrt{s}\,)|^2
+ |f^{(D)}_{J=\frac{3}{2}}(\sqrt{s}\,)|^2
\nonumber\\
&& \qquad \quad +2\,
f^{(S)}_{J=\frac{1}{2}}(\sqrt{s}\,)\,f^{(D,*)}_{J=\frac{3}{2}}(\sqrt{s}\,)
+2\,f^{(P)}_{J=\frac{1}{2}}(\sqrt{s}\,)\,f^{(P,*)}_{J=\frac{3}{2}}(\sqrt{s}\,)
\Bigg) \;,
\nonumber\\
B_1(\sqrt{s}\,)&=& 2\,\pi\,\frac{p^{(f)}_{\rm cm}}{p^{(i)}_{\rm cm}}\,
\Im \Bigg(
f^{(S)}_{J=\frac{1}{2}}(\sqrt{s}\,)\,
\Big(f^{(P,*)}_{J=\frac{1}{2}}(\sqrt{s}\,)- f^{(P,*)}_{J=\frac{3}{2}}(\sqrt{s}\,)
\Big)
\nonumber\\
&& \qquad \quad
- f^{(D)}_{J=\frac{3}{2}}(\sqrt{s}\,)
\,\Big(f^{(P,*)}_{J=\frac{1}{2}}(\sqrt{s}\,)- f^{(P,*)}_{J=\frac{3}{2}}(\sqrt{s}\,)
\Big) \Bigg)\;,
\nonumber\\
B_2(\sqrt{s}\,)&=& 2\,\pi\,\frac{p^{(f)}_{\rm cm}}{p^{(i)}_{\rm cm}}\,
\Im \Bigg(
f^{(S)}_{J=\frac{1}{2}}(\sqrt{s}\,)\,f^{(D,*)}_{J=\frac{3}{2}}(\sqrt{s}\,)
- f^{(P)}_{J=\frac{1}{2}}(\sqrt{s}\,)\,f^{(P,*)}_{J=\frac{3}{2}}(\sqrt{s}\,) \Bigg) \;.
\label{}\end{aligned}$$
We use the empirical mass values for the kinematical factors in (\[cross-section\]) and include isospin breaking effects in the matrix elements $M_\pm $ as described in the Appendix D. Coulomb effects which are particularly important in the $K^+ p \to K^+ p$ reaction are also considered. They can be generated by the formal replacement rule [@Coulomb] $$\begin{aligned}
&&F_{+}(\sqrt{s},\theta) \to F_{+}(\sqrt{s},\theta )
-\frac{\alpha }{v}\,\frac{4 \pi \,\sqrt{s}}{E+m}\,\frac{
\exp \left( -i\,\frac{\alpha}{v}\,\ln \left(\sin^2 (\theta/2) \right)
\right)}{p^{}_{\rm cm}\,\sin^2 (\theta/2)}\,,
\nonumber\\
&& \quad \quad \quad \quad \quad
v= \frac{p^{}_{\rm cm}\,\sqrt{s}}{E\,(\sqrt{s}-E)} \;,
\quad \quad \quad
\alpha = \frac{e^2\,Z^{}_1\,Z^{}_2}{4\,\pi} \;
\label{Coulomb}\end{aligned}$$ for the amplitude $F_{+}(\sqrt{s},\theta )$ in (\[cross-section\]). Here $E_i=E_f=E$, $m_f=m_i=m $, $e^2/(4\pi) \simeq 1/137$ and $e\,Z_{1,2}^{}$ are the charges of the particles.
Chiral expansion of baryon exchange
===================================
In this appendix we provide the leading and subleading terms of the baryon exchange contributions not displayed in the main text. We begin with the s-channel exchange contributions for which only the baryon decuplet states induce terms of chiral order $Q^3$: $$\begin{aligned}
V^{(I,+)}_{s-[10],3}(\sqrt{s};0)
&=& \sum_{c=1}^2\,\frac{(2-Z_{[10]})\,\sqrt{s}+3\,m^{(c)}_{[10]}}{3\,(m^{(c)}_{[10]})^2}\,
\Bigg(
\frac{\phi^{(I)}(s)}{2 \,\sqrt{s}}\,\frac{C^{(I,c)}_{[10]}}{4\,f^2 }\,
\Big( \sqrt{s}-M^{(I)}\Big)
\nonumber\\
&& \qquad \quad +
\Big( \sqrt{s}-M^{(I)}\Big)\,\frac{C^{(I,c)}_{[10]}}{4\,f^2 }\,
\frac{\phi^{(I)}(s)}{2 \,\sqrt{s}} \Bigg)\,,
\nonumber\\
V^{(I,-)}_{s-[10],3}(\sqrt{s};0)&=&\sum_{c=1}^2\,
\frac{Z_{[10]}\,\sqrt{s}}{3\,(m^{(c)}_{[10]})^2}\,\Bigg(
\Big( \sqrt{s}-M^{(I)}\Big)\,\frac{C^{(I,c)}_{[10]}}{4\,f^2 }\,
\Big( \sqrt{s}+M^{(I)}\Big)
\nonumber\\
&& \qquad \quad +\Big( \sqrt{s}+M^{(I)}\Big)\,\frac{C^{(I,c)}_{[10]}}{4\,f^2 }\,
\Big( \sqrt{s}-M^{(I)}\Big) \Bigg) \,,
\nonumber\\
V^{(I,+)}_{s-[10],3}(\sqrt{s};1)&=&
V^{(I,\pm)}_{s-[8],3}(\sqrt{s};0)=V^{(I,+)}_{s-[8],3}(\sqrt{s};1)=0 \,,
\label{v-result-1:q3}\end{aligned}$$ where the index ’3’ in (\[v-result-1:q3\]) indicates that only the terms of order $Q^3$ are shown. Next we give the $Q^3$-terms of the u-channel baryon exchanges characterized by the subleading moment of the functions $h_{n,\pm}^{(I)}(\sqrt{s},m)$ $$\begin{aligned}
\Big[V^{(I,\pm )}_{u-[8]}(\sqrt{s};n)\Big]_{ab} &=& \sum_{c=1}^4\,
\frac{1}{4\,f^2 }\,\Big[\widetilde C^{(I,c)}_{[8]}\Big]_{ab}\,
\Big[{ h}^{(I)}_{n \pm }(\sqrt{s},m^{(c)}_{[8]})\Big]_{ab} \;.
\label{u-result-8:appendix}\end{aligned}$$ We derive $$\begin{aligned}
&&\Big[{h}_{0+}^{(I)} (\sqrt{s},m)\Big]_{ab,3}=
\frac{m+M_{ab}^{(L)}}{\mu_{+,ab}}\,\Bigg(
\frac{2}{3}\,\frac{\phi_a\,\phi_b}{s\,\mu_{-,ab}}
+\frac43\,\frac{\phi_a\,
\Big(\omx{a}+\omx{b}\Big)\,\phi_b }
{\sqrt{s}\,\mu_{+,ab}\,\big(\mu_{-,ab}\big)^2}
\Bigg)\,
\frac{m+M_{ab}^{(R)}}{\mu_{+,ab}}\,,
\nonumber\\
&&\Big[{h}_{0-}^{(I)} (\sqrt{s},m)\Big]_{ab,3}=\sqrt{s}-m
-\tilde R^{(I)}_{L,ab}-\tilde R^{(I)}_{R,ab}
-\frac{m+M_{ab}^{(L)}}{\mu_{+,ab}}\,
\Bigg( \frac23\,\frac{\phi_a\,M_b+M_a\,\phi_b }
{\sqrt s \,\mu_{-,ab}}
\nonumber\\
&& \qquad \quad
+\frac{16}{3}\,\frac{\phi_a\,M_a\,M_b\,\omx{b}}
{\sqrt{s}\,\mu_{+,ab}\,(\mu_{-,ab})^2}
+\frac{16}{3}\,\frac{\omx{a}\,M_a\,M_b\,\phi_b}
{\sqrt{s}\,\mu_{+,ab}\,(\mu_{-,ab})^2} -\frac{8}{3}\,\frac{\phi_a\,\sqrt{s}\,\phi_b}{\mu_{-,ab}^3\,\mu_{+,ab}}
\nonumber\\
&& \qquad \quad
-2\,\frac{\phi_a\,\omx{b}+\omx{a}\,\phi_b}{(\mu_{-,ab})^2}
+\frac{32}{5}\,\frac{\phi_a\,M_a\,M_b\,\phi_b\,}
{(\mu_{+,ab})^2\,(\mu_{-,ab})^3}
\Bigg)\,\frac{m+M_{ab}^{(R)}}{\mu_{+,ab}} \,,
\nonumber\\
&&\Big[{h}_{1+}^{(I)} (\sqrt{s},m)\Big]_{ab,3}=
-\frac{m+M_{ab}^{(L)}}{\mu_{+,ab}}\,
\Bigg( \frac{8}{5}\,\frac{\phi_a\,\phi_b\,}
{(\mu_{+,ab})^2\,(\mu_{-,ab})^3} \left(1-\frac{\mu_{+,ab}}{6\,\sqrt{s}} \right)
\nonumber\\
&& \qquad \quad
+\frac{4}{3}\,\frac{\phi_a\,\omx{b}}
{\sqrt{s}\,\mu_{+,ab}\,(\mu_{-,ab})^2}
+\frac{4}{3}\,\frac{\omx{a}\,\phi_b}
{\sqrt{s}\,\mu_{+,ab}\,(\mu_{-,ab})^2}
\Bigg)\,\frac{m+M_{ab}^{(R)}}{\mu_{+,ab}}
\nonumber\\
&&\Big[{h}_{1-}^{(I)} (\sqrt{s},m)\Big]_{ab,3}=
\frac23\,\frac{(m+M_{ab}^{(L)})\,(m+M_{ab}^{(R)}) }{\mu_{-,ab}\,
\big(\mu_{+,ab}\big)^2}
\label{u-approx-1:q3}\;,\end{aligned}$$ where we introduced the short hand notations $M_a=m_{B(I,a)}$ and $ m_a=m_{\Phi(I,a)}$. Also, $\omx{a}= \sqrt{s}-M_a$ and $\phi_a= \omx{a}^2-m_a^2$. We recall here $M_{ab}^{(L)}=m_{B(I,a)}+\tilde R^{(I,c)}_{L,ab}$ and $M_{ab}^{(R)}=m_{B(I,b)}+\tilde R^{(I,c)}_{R,ab}$ with $\tilde R $ specified in (\[P-result\]).
We turn to the decuplet functions ${p}^{(I)}_{n \pm}(\sqrt{s},m)$ $$\begin{aligned}
\Big[V^{(I,\pm )}_{u-[10]}(\sqrt{s};n)\Big]_{ab} &=& \sum_{c=1}^2\,
\frac{1}{4\,f^2 }\,\Big[\widetilde C^{(I,c)}_{[10]}\Big]_{ab}\,
\Big[{ p}^{(I)}_{n \pm }(\sqrt{s},m^{(c)}_{[10]})\Big]_{ab} \;,
\label{u-result-8:appendix}\end{aligned}$$ for which we provide the leading moments $$\begin{aligned}
&&\left[p_{0+}^{(I)}(s,m)\right]_{ab}=
\frac{\omx{a}\, \omx{b}}{\mu_{-,ab}}\,
\left(1-\frac{s}{m^2}+\frac{\sqrt s}{m^2}\, \Big(\omx{a}+\omx{b}\Big)\right)
+ \frac{s}{3\,m^2}\, \frac{\omx{a}\, \omx{b}}{\mu_{+,ab}}
\nonumber\\
&&\qquad+ \frac{(M_a+m)\,(M_b+m)}{3\,\mu_{+,ab}}\,
\Bigg(
1+\frac{\omx{a}\, \phi_b+\omx{b}\,
\phi_a}{\sqrt s\, \mu_{+,ab}\,\mu_{-,ab}}
+ \frac{\phi_a\,(4\,\sqrt{s}-\mu_{+,ab})\,\phi_b}
{3\,\sqrt{s}\,\mu_{+,ab}^2\,\mu_{-,ab}^2} \Bigg)
\nonumber\\
&&\qquad -\frac13\, (\sqrt s +m)
+\left(\frac{M_a+m}{3\,m\, \mu_{+,ab}}-\frac{1}{3\,m}\right)
\left(\sqrt s\, \omx{a}-\frac12\,
\phi_a-\omx{a}\,\omx{b} -m_a^2\right)
\nonumber\\
&&\qquad +\left(\frac{M_b+m}{3\,m\, \mu_{+,ab}}-\frac{1}{3\,m}\right)
\left(\sqrt s\, \omx{b}-\frac12\,\phi_b-\omx{a}\,\omx{b}- m_b^2\right)
-\frac{2}{3}\,\frac{\phi_a\,\phi_b}{\mu_+ \,\mu_{-,ab}^2}
\nonumber\\
&&\qquad
+ \sqrt{s}\,\frac{ \omx{a} \, m_b^2 +\omx{b} \, m_a^2}{m^2\, \mu_{-,ab}}
-\frac{\omx{a}\,\omx{b}}{6\,m^2} \,\Big(
M_a+M_b-\sqrt{s}-2\,m \Big)\,Z_{[10]}^2
\nonumber\\
&&\qquad +\frac{\omx{a}\,\omx{b}}{3\,m^2} \, \Big(2\,\sqrt{s}-m \Big)\,Z_{[10]}
+{\mathcal O}\left(Q^3 \right)
-\frac{ \omx{a}\,\phi_b+\phi_a\,\omx{b}}{2\,\sqrt{s}\,\mu_{-,ab}} \left( 1-\frac{s}{m^2}\right)
\nonumber\\
&& \qquad
+\frac{\phi_a\,\phi_b}{4\,s\mu_{-,ab}}\,\Bigg( 1-\frac{7\,s}{3\,m^2} \Bigg)
+\frac{4}{3}\,\frac{ \omx{a}\,\phi_a\,\phi_b\, \omx{b}}{\mu_{+,ab}^2\,\mu_{-,ab}^3}
-\omx{a}\,\omx{b}\,\frac{(\omx{a}+\omx{b})^2}{m^2\, \mu_{-,ab}}
\nonumber\\
&&\qquad
-\frac{2}{3}\,\frac{\phi_a\,\phi_b}{ \mu^2_{-,ab}}\,\Bigg(
\frac{\sqrt{s}\,(\omx{a}+\omx{b})}{m^2\,\,\mu_{+,ab}}
+\frac{2\,s}{m^2}\,\frac{\omx{a}\,\omx{b}}{ \mu^2_+\,\mu_{-,ab}}
\Bigg)
-\frac{\sqrt{s}}{m^2}\,
\frac{\omx{a}^2\,\phi_b\,\omx{b} +\omx{a}\, \phi_a\,\omx{b}^2}{ \mu_{+,ab}\,\mu_{-,ab}^2}
\nonumber\\
&&\qquad
+\frac{1}{6}\left(\frac{\phi_a\,\omx{b}+\omx{a}\,\phi_b}{\sqrt{s}\,m\,\mu_{+,ab}}
+\frac{\phi_a\,\big(4\,\sqrt{s}-\mu_{+,ab}\big)\phi_b}{3\,\sqrt{s}\,m\,\mu_{+,ab}^2\,\mu_{-,ab}} \right)
\Big(M_a+M_b+2\,m \Big)
\nonumber\\
&&\qquad
+\frac{2}{3}\,\frac{\phi_a\,\phi_b}{\sqrt{s}\,\mu_{-,ab}^2}\,\frac{\chi_a+\chi_b}{\mu_{+,ab}}
+\frac{\chi_a\,\chi_b}{\mu_{+,ab}}\left(\frac{\chi_a\,\phi_b+\chi_b\,\phi_a}{\sqrt{s}\,\mu_{-,ab}^2}
-\frac{\sqrt{s}}{m}\,\frac{\chi_a+\chi_b}{3\,m} \right)
\nonumber\\
&&\qquad+
\,\frac{\omx{a}\,(M_a+m )+\omx{b}\,(M_b+m )}{3\,m\,\mu^2_{+,ab}}\, \Bigg(
\frac{\omx{a}\, \phi_b+\omx{b}\, \phi_a}{\mu_{-,ab}}
+\frac{\phi_a\,\big(4\,\sqrt{s}-\mu_{+,ab}\big)\,\phi_b}{3\,\mu_{+,ab}\,\mu^2_{-,ab}}
\Bigg)
\nonumber\\
&&\qquad
-\frac{\sqrt{s}}{m}\,\frac{\chi_a\,m_b^2+\chi_b\,m_a^2}{3\,m\,\mu_{+,ab}}
-\frac{\sqrt{s}}{m}\,\frac{\chi_a\,\phi_b+\chi_b\,\phi_a}{6\,m\,\mu_{+,ab}}
+\frac{\phi_a\,\phi_b}{\mu_{-,ab}} \,\frac{2\,\sqrt{s}+\mu_{+,ab}}{12\,s\,\mu_{+,ab}}
\nonumber\\
&&\qquad-\frac{2}{9}
\frac{\phi_a\,\phi_b}{\mu_{-,ab}} \,
\frac{(M_a+m)\,(M_b+m)}{s\,\mu^2_{+,ab}}\Bigg(
1+ \frac{2\,\sqrt{s}}{\mu_{+,ab}}\frac{\chi_a+\chi_b}{\mu_{-,ab}} \Bigg)
\nonumber\\
&&\qquad
+\frac{1}{3\,\sqrt{s}\, m}\, \Big(\omx{a}\,\phi_b+\phi_a \,\omx{b}\Big)\,
\Bigg(Z_{[10]}-1-Z_{[10]}^2\,\left(1+\frac{\sqrt{s}}{2\,m} \right) \Bigg)
\nonumber\\
&&\qquad
-\frac{Z_{[10]}}{3\,m^2} \,\Bigg(
\Big({\textstyle{3\over 2}}\,\phi_a+2\,m_a^2 \Big) \,\omx{b}
+\omx{a}\,\Big({\textstyle{3\over 2}}\,\phi_b+2\,m_b^2 \Big)
\Bigg)+{\mathcal O}\left(Q^4 \right) \,,\end{aligned}$$ and $$\begin{aligned}
&&\Big[{ p}_{0-}^{(I)} (\sqrt{s},m)\Big]_{ab} =
\frac{M_a\,M_b}{3} \,
\Bigg( 4\,\frac{M_a+M_b+2\,m}{3\,m\,\mu_{+,ab}}
-\frac{4}{\mu_{-,ab}} -\frac{8}{3\,m} \Bigg)
\nonumber\\
&&\qquad
- \Big(M_a+m \Big)\,
\Bigg(\frac{1}{3\,\mu_{+,ab}} +\frac{2\,\sqrt s}{3\,\mu_{-,ab}\,\mu_{+,ab}}
-\frac{8\,M_a\,M_b} {9\,\mu_{-,ab}\,\mu^2_{+,ab}}
\Bigg)\,\Big( M_b+m\Big)
\nonumber\\
&&\qquad -\frac{\omx{a}}{3\,\mu_{-,ab}}
\Bigg(\frac{2\,s\,(M_a+m)}{m\,\mu_{+,ab}}
+4\,\frac{\sqrt s \,M_a\,M_b}{m^2}
-\frac{8}{3}\,\frac{\sqrt s \,(M_a+m)}{m\,\mu_{+,ab}}
\,\frac{M_a\,M_b}{\mu_{+,ab}}
\Bigg)
\nonumber\\
&&\qquad -\frac{\omx{b}}{3\,\mu_{-,ab}}
\Bigg(\frac{2\,s\,(M_b+m)}{m\,\mu_{+,ab}}
+4\,\frac{\sqrt s \,M_a\,M_b}{m^2}
-\frac{8}{3}\,\frac{\sqrt s \,(M_b+m)}
{m\,\mu_{+,ab}}
\,\frac{M_a\,M_b}{\mu_{+,ab}}
\Bigg)
\nonumber\\
&&\qquad -8\,Z_{[10]}\,\frac{M_a\,M_b}{9\,m} \,\Bigg(
Z_{[10]}-1-3\,Z_{[10]} \,\frac{M_a+M_b}{16\,m} \Bigg)
+\frac{Z_{[10]}^2\,\sqrt s }{6\,m^2}\,
\left(s-\frac{5}{3}\,M_a\,M_b\right)
\nonumber\\
&& \qquad
+\frac{Z_{[10]}\,(Z_{[10]}-1)}{3\,m}\,\Big( \sqrt s +M_a\Big)\,
\Big( \sqrt s +M_b\Big)
+{\mathcal O}\left( Q\right)
\nonumber\\
&& \qquad
+\Bigg( \frac{2}{9}\,\frac{(M_a+m)\,(M_b+m)}{\mu_{+,ab}^2}-\frac{1}{3}
\Bigg)\,\frac{\phi_a\,M_b+M_a\,\phi_b}{\sqrt{s}\,\mu_{-,ab}}
\nonumber\\
&&\qquad
-\frac{2}{3}\,\frac{(M_a+m)\,(M_b+m)}{\mu_{-,ab}\,\mu_{+,ab}}\,
\left(
\frac{\omx{a}\, \phi_b+\omx{b}\, \phi_a}{ \mu_{+,ab}\,\mu_{-,ab}}
+ \frac43\, \frac{\phi_a\,\sqrt{s}\,\phi_b}{\mu_{+,ab}^2\,\mu_{-,ab}^2} \right)
\nonumber\\
&&\qquad + \frac{M_a\, M_b}{\mu_{+,ab}} \,\Bigg(
\frac83\,\frac{\omx{a}\,\omx{b}}{\mu_{-,ab}^2} \left(1-\frac{s}{m^2} \right)
-\frac{16\,\phi_a\,\phi_b}{5\,\mu_{+,ab}\,\mu_{-,ab}^3}
-\frac{8}{3}\,\frac{\omx{a}\,\phi_b+\phi_a\,\omx{b}}{\sqrt{s}\,\mu^2_{-,ab}}
\Bigg)
\nonumber\\
&& \qquad +\left(\frac{2}{3}\,\sqrt{s}-\frac{8}{9}\,\frac{M_a\,M_b}{\mu_{+,ab}}\right)
\frac{M_a+m}{\mu_{+,ab}}\,\frac{\frac12\,
\phi_a+\omx{a}\,\omx{b}+m_{a}^2}{m\,\mu_{-,ab}}
\nonumber\\
&& \qquad +\left(\frac{2}{3}\,\sqrt{s}-\frac{8}{9}\,\frac{M_a\,M_b}{\mu_{+,ab}}\right)
\frac{M_b+m}{\mu_{+,ab}}\,\frac{\frac12\,
\phi_b+\omx{a}\,\omx{b}+m_{b}^2}{m\,\mu_{-,ab}}
\nonumber\\
&& \qquad
-\left(\frac{2}{3}\,\sqrt{s}-\frac{8}{9}\,\frac{M_a\,M_b}{\mu_{+,ab}}\right)
\frac{s}{m^2}\,\frac{\omx{a}\,\omx{b}}{\mu_{+,ab}\,\mu_{-,ab}}
+\frac{4}{9}\,\frac{M_a\,\sqrt{s}\,M_b}{m^2 }\,\frac{\omx{a}+\omx{b}}{\mu_{+,ab}}
\nonumber\\
&& \qquad
+\frac{m+M_a}{\mu_{+,ab}}\,\Bigg(
\frac{16}{9}\,\frac{M_a\,M_b}{\sqrt{s}\,\mu_{+,ab}}\,\frac{\chi_a\,\phi_b+\chi_b\,\phi_a}{\mu_{-,ab}^2}
+\frac{32}{15}\,\frac{M_a\,M_b}{\mu_{+,ab}^2}\,\frac{\phi_a\,\phi_b}{\mu_{-,ab}^3}
\Bigg)\,\frac{m+M_b}{\mu_{+,ab}}
\nonumber\\
&& \qquad
-\frac{1}{3}\,\frac{\sqrt{s}}{m}\left( \frac{m+M_a}{\mu_{+,ab}}\,\chi_a
+\frac{m+M_b}{\mu_{+,ab}}\,\chi_b \right)
+\frac{4}{3}\,\frac{M_a\,\sqrt{s}\,M_b}{m^2\,\mu_{+,ab}}\,\frac{\chi_a\,m_b^2+\chi_b\,m_a^2}{\mu_{-,ab}^2}
\nonumber\\
&& \qquad
+\left( \frac{8}{3}\,\frac{\chi_a\,\chi_b}{\mu_{-,ab}}\,
+\frac{4}{3}\,\frac{m_a^2+m_b^2}{\mu_{-,ab}}+\frac{2}{3}\,\frac{\phi_a+\phi_b}{\mu_{-,ab}}
+\frac{4\,\sqrt{s}}{\mu_{+,ab}}\,\frac{\chi_a^2\,\chi_b+\chi_b^2\,\chi_a}{\mu_{-,ab}^2}\
\right) \frac{M_a\,M_b}{m^2}
\nonumber\\
&& \qquad
-\frac{1}{3}\,\Big( \sqrt{s}-m \Big) +\frac{\sqrt{s}}{3\,m}\,
\Bigg(1-\frac{\sqrt{s}}{m}\,Z_{[10]}^2+\frac{4}{3}\,\frac{M_a\,M_b}{\sqrt{s}\,m}\,Z_{[10]} \Bigg)
\,\Big( \omx{a}+\omx{b}\Big)
\nonumber\\
&& \qquad
+\frac{2\,\sqrt{s}}{3\,m^2} \,\Big( s-M_a\,M_b \Big)\,Z_{[10]}
+{\mathcal O}\left( Q^2\right)\;,
\label{}\end{aligned}$$ and $$\begin{aligned}
&&\Big[{ p}_{1+}^{(I)} (\sqrt{s},m)\Big]_{ab} =
-\frac{1}{3\,\mu_{-,ab}}
+ \frac29\,
\frac{(m+M_a)\,(m+M_b)}
{\mu_{+,ab}^2\,\mu_{-,ab}}
+\frac19\, \frac{M_a+M_b}{m\,\mu_{+,ab}}
\nonumber\\
&& \qquad
-\frac{\sqrt s }{3\,m}\,\frac{\omx{a}}
{\mu_{-,ab}}\,
\Bigg(\frac{1}{m}-\frac23\,\frac{m+M_a}
{\mu^2_{+,ab}}
\Bigg)
-\frac{\sqrt s }{3\,m}\,\frac{\omx{b}}
{\mu_{-,ab}}\,
\Bigg(\frac{1}{m}-\frac{2}{3}\,\frac{m+M_b}
{\mu_{+,ab}^2} \Bigg)
\nonumber\\
&&\qquad
+\frac{2}{9\,\mu_{+,ab}}-\frac{2}{9\,m}
-\frac{Z_{[10]}^2}{9\,m^2}\,\Big(\sqrt s +2\,m \Big)
+\frac{2\,Z_{[10]}}{9\,m}+{\mathcal O}\left( Q\right)
\nonumber\\
&& \qquad
+\frac{2}{3}\,\frac{{\textstyle{3\over 2}}\,\phi_a+2\,m_a^2 }{m^2\,\mu_{-,ab}^2}\,
\frac{\sqrt{s}\,\omx{b}}{\mu_{+,ab}}
+\frac{2}{3}\,\frac{\sqrt{s}\,\omx{a}}{\mu_{+,ab}}\,
\frac{{\textstyle{3\over 2}}\,\phi_b+2\,m_b^2 }{m^2\,\mu_{-,ab}^2}
-\frac{4}{5}\,\frac{\phi_a\,\phi_b}{\mu_{+,ab}^2\,\mu_{-,ab}^3}
\nonumber\\
&& \qquad
+\frac{2}{3}\, \frac{\omx{a}\,\omx{b}}{\mu_{+,ab}\,\mu_{-,ab}^2}\left( 1-\frac{s}{m^2}\right)
+\frac{2}{9}\,\frac{\omx{a}\,s\,\omx{b}}{m^2\,\mu^2_{+,ab}\,\mu_{-,ab}}
+\frac{\sqrt{s}\,(\omx{a}+\omx{b})}{9\,m^2\,\mu_{+,ab}}
\nonumber\\
&& \qquad -\frac{2\,(M_a+m)}{9\,m}\,
\frac{\frac12\,
\phi_a+\omx{a}\,\omx{b}+m_a^2}{\mu_{-,ab}\,\mu_{+,ab}^2}
-\frac{2\,(M_b+m)}{9\,m}\,
\frac{\frac12\,
\phi_b+\omx{a}\,\omx{b}+m_b^2}{\mu_{-,ab}\,\mu_{+,ab}^2}
\nonumber\\
&& \qquad
-\frac{2}{3}\,
\frac{\omx{a}\,\phi_b+\phi_a\,\omx{b}}{\sqrt{s}\,\mu_+\,\mu^2_-}
+\frac{\phi_a\,\phi_b}{\mu_{-,ab}^3}\,
\frac{(m+M_a)\,(m+M_b)}{\mu_{+,ab}^4}\,\Bigg( \frac{8}{15}-
\frac{4}{45}\,\frac{\mu_{+,ab}}{\sqrt{s}}\Bigg)
\nonumber\\
&& \qquad
+\frac{2}{3}\,\frac{\chi_a\,\chi_b}{m^2\,\mu_{-,ab}}
+\frac{m_a^2+m_b^2}{3\,m^2\,\mu_{-,ab}}
+\frac{1}{6}\,\frac{\phi_a+\phi_b}{m^2\,\mu_{-,ab}}
\nonumber\\
&& \qquad
+\frac{4}{9}\,\frac{(m+M_a)\,(m+M_b)}{\sqrt{s}\,\mu_{-,ab}^2\,\mu_{+,ab}^3}\,
\Big(\chi_a\,\phi_b+\chi_b\,\phi_a \Big)
+ \frac{\omx{a}+\omx{b}}{9\,m^2} \,Z_{[10]}+{\mathcal O}\left( Q^2\right)\;,
\label{u-approx-2}\end{aligned}$$ and $$\begin{aligned}
&&\Big[{ p}_{1-}^{(I)} (\sqrt{s},m)\Big]_{ab} =
-\frac{16}{15}\, \frac{M_a\,M_b}{\big(\mu_{-,ab}\big)^2\,\mu_{+,ab}}
+\frac{32}{45}\, \frac{(m+M_a)\,M_a\,M_b\,(m+M_b)}
{\big(\mu_{-,ab}\big)^2\,\big(\mu_{+,ab}\big)^3}
\nonumber\\
&&\qquad -\frac{4}{9}\, \frac{(m+M_a)\,\sqrt{s}\,(m+M_b)}
{\big(\mu_{-,ab}\big)^2\,\big(\mu_{+,ab}\big)^2}
+{\mathcal O}\left( Q^{-1}\right)
\nonumber\\
&&\qquad +\frac{1}{3}\,\frac{\mu_{+,ab}+2\,\sqrt{s}}{\mu_{+,ab}\,\mu_{-,ab}}
-\frac{8}{15}\,\frac{M_a\,M_b}{m^2\,\mu_{-,ab}}
-\frac{2}{9}\,\frac{(m+M_a)\,(m+M_b)}{\mu_{-,ab}\,\mu_{+,ab}^2}
\nonumber\\
&& \qquad -\frac{16}{15}\,\frac{M_a\,\sqrt{s}\,M_b}{m^2}\,
\frac{\omx{a}+\omx{b}}{\mu_{+,ab}\,\mu_{-,ab}^2}
+\frac{16}{45}\,\frac{M_a\,M_b}{\mu_{-,ab}}\,\frac{M_a+M_b+2\,m}{m\,\mu_{+,ab}^2}
\nonumber\\
&& \qquad -\frac{4}{9}\,\frac{\omx{a}\,s\,(m+M_a)}{m\,\mu_{-,ab}^2\,\mu_{+,ab}^2}
-\frac{4}{9}\,\frac{(m+ M_b)\,s\,\omx{b}}{m\,\mu_{-,ab}^2\,\mu_{+,ab}^2}
-\frac{2}{9}\,\frac{\sqrt{s}}{m } \,\frac{2\,m +M_a+M_b}{\mu_{+,ab}\,\mu_{-,ab}}
\nonumber\\
&& \qquad +\frac{32}{45}\,\frac{M_a\,\sqrt{s}\,M_b}{m\,\mu_{+,ab}^3\,\mu_{-,ab}^2}
\Big( (m +M_a)\,\omx{a}+(m+ M_b)\,\omx{b}\Big)
+{\mathcal O}\left( Q^{0}\right) \;.
\label{u-approx-2:b}\end{aligned}$$ This appendix ends with the d-wave resonance functions $q_{n\pm}^{(I)}(\sqrt{s},m)$ $$\begin{aligned}
\Big[V^{(I,\pm )}_{u-[9]}(\sqrt{s};n)\Big]_{ab} &=& \sum_{c=1}^4\,
\frac{1}{4\,f^2 }\,\Big[\widetilde C^{(I,c)}_{[9]}\Big]_{ab}\,
\Big[{ q}^{(I)}_{n \pm }(\sqrt{s},m^{(c)}_{[9]})\Big]_{ab} \;.
\label{u-result-8:appendix}\end{aligned}$$ One may or may not apply the questionable formal rule $\sqrt{s}-m_{[9]}\sim Q$. The explicit expressions below, which rely on $\sqrt{s}-m_{[9]}\sim Q$ and $\mu_- \sim Q$, demonstrate that our total result to order $Q^3$ are basically independent on this assumption. Modified results appropriate for an expansion with $\sqrt{s}-m_{[9]}\sim Q^0$ and $\mu_- \sim Q^0$ follow upon dropping some terms proportional to $(1/\mu_-)^n$. We derive $$\begin{aligned}
&& [q_{0+}^{(I)}(\sqrt{s},m)]_{ab}=
-\frac{ \omx{a}\,\omx{b}}{3\,m^2} \, \Big(2\,\sqrt{s}+m \Big)\,Z_{[9]}
\nonumber\\
&&\qquad
+\frac{\omx{a}\,\omx{b}}{6\,m^2} \,\Big(
M_a+M_b-\sqrt{s}+2\,m \Big)\,Z_{[9]}^2
+{\mathcal O}\left(Q^3 \right)
\nonumber\\
&& \qquad
-\frac{\omx{a}\,\omx{b}}{\mu_{+,ab}}\, \left(1-\frac{s}{m^2}\right)
-\frac{\sqrt{s}\,\omx{a}}{m^2}\,
\frac{{\textstyle{3 \over 2}}\,\phi_b+2\,m_b^2}{\mu_{+,ab}}
-\frac{{\textstyle{3 \over 2}}\,\phi_a+2\,m_a^2 }{\mu_{+,ab}}\,
\frac{\sqrt{s}\,\omx{b}}{m^2}
\nonumber\\
&&\qquad
+\frac{\omx{a}\,\phi_b+\phi_a\,\omx{b}}{6\,\sqrt{s}\,\mu_{+,ab}}
+\frac{2}{9}\,\frac{\phi_a\,\phi_b}{\mu^2_+\,\mu_{-,ab}}
\left(1+2\,\frac{\mu_{+,ab}}{m} \right)
-\frac{1}{9}\,\frac{\phi_a\,\phi_b}{m^2\,\mu_{-,ab}}
\nonumber\\
&&\qquad
-\frac{1}{3\,m^2\mu_{-,ab}}\,\left(\omx{a}\,\mu_{-,ab}+{\textstyle{1 \over 2}}\,\phi_a \right)
\left(\omx{b}\,\mu_{-,ab}+{\textstyle{1 \over 2}}\,\phi_b \right)
\nonumber\\
&&\qquad
+\frac{1}{3\,m^2} \,\left(
2\,m_a^2\,\omx{b}+2\,\omx{a}\,m_b^2
+\Big(\omx{a}\,\phi_b+\phi_a\,\omx{b}\Big)\left(\frac{3}{2}+
\frac{m}{\sqrt{s}} \right)\right)Z_{[9]}
\nonumber\\
&&\qquad
+\frac{1}{6\,m^2} \,
\Big(\omx{a}\,\phi_b+\phi_a\,\omx{b}\Big)\,\left(1 -2\,\frac{m}{\sqrt{s}} \right)
Z_{[9]}^2
+{\mathcal O}\left(Q^4 \right)\;,\end{aligned}$$ and $$\begin{aligned}
&& [q_{0-}^{(I)}(\sqrt{s},m)]_{ab}= \frac{1}{3}\,(\sqrt{s}+m)
+\frac{4\,M_a\,M_b}{9\,\mu_{+,ab}}
\nonumber\\
&& \qquad
+\frac{Z_{[9]}^2}{6\,m^2}\,\Big(
M_a\,\Big({\textstyle{5\over3}}\,\sqrt{s}-{\textstyle{16\over3}}\,m-M_a-M_b\Big)\,M_b
-s\,\sqrt{s} \Big)
\nonumber\\
&& \qquad +\frac{Z_{[9]}\,(Z_{[9]}-1)}{3\,m} \,
(\sqrt{s}+M_a) \,(\sqrt{s}+M_b)+\frac{8\,Z_{[9]}}{9\,m} \,M_a\,M_b
+{\mathcal O}\left(Q \right)
\nonumber\\
&& \qquad
+\frac{2\,\omx{a}\,\sqrt{s}\,\omx{b}}{\mu_{+,ab}\,\mu_{-,ab}} \left(1-\frac{s}{m^2}\right)
+\frac{1}{3}\,\left(1+2\,\frac{\sqrt{s}}{\mu_{+,ab}}\right)\,\mu_{-,ab}
+\sqrt{s}\,\frac{\omx{a}+\omx{b}}{3\,m}
\nonumber\\
&& \qquad
+4\,M_a\,M_b\,\frac{ \omx{a}+\omx{b}}{m\,\mu_{+,ab}}\left(
\frac{\sqrt{s}}{3\,m}-\frac{1}{9}\,\frac{m-\sqrt{s}}{m\,\mu_{-,ab}}\,\Big(2\,m-\mu_{+,ab} \Big) \right)
\nonumber\\
&& \qquad
-\frac{4\,M_a\,\big(2\,m-\mu_{+,ab}\big)\,M_b}{9\,m^2\,\mu_{+,ab}\,\mu_{-,ab}}
\Bigg(2\,\omx{a}\,\omx{b}
+m_a^2+m_b^2+{\textstyle{1\over 2}} \,(\phi_a+\phi_b)\Bigg)
\nonumber\\
&& \qquad
+\frac{\omx{a}+\omx{b}}{3\,m^2}
\left(s\,Z_{[9]}-\frac{4}{3}\,M_a\,M_b \right)\,Z_{[9]}
-\frac{2\,\sqrt{s}}{3\,m^2} \,\Big( s-M_a\,M_b\Big)\,Z_{[9]}
+{\mathcal O}\left(Q^2 \right) \;,
\nonumber\\\end{aligned}$$ and $$\begin{aligned}
&&[q_{1+}^{(I)}(\sqrt{s},m)]_{ab}
=\frac{1}{9\,\mu_{+,ab}} +\frac{Z_{[9]}^2}{9\,m^2} \,\Big( \sqrt{s} -2\,m\Big)
+\frac{2\,Z_{[9]}}{9\,m}
+{\mathcal O}\left( Q\right)
\nonumber\\
&& \qquad-\frac{2\,m-\mu_{+,ab}}{9\,m^2\,\mu_{+,ab}\,\mu_{-,ab}}
\Bigg( 2\,\omx{a}\,\omx{b}
+m_a^2+m_b^2+{\textstyle{1\over 2}} \,(\phi_a+\phi_b)\Bigg)
-\frac{\sqrt{s}}{m}\,\frac{\chi_a+\chi_b}{9\,m\,\mu_{-,ab}}
\nonumber\\
&& \qquad
+\frac{ \omx{a}+\omx{b}}{m\,\mu_{+,ab}}\left(
\frac{\sqrt{s}}{3\,m}-\frac{2}{9}\,\frac{m-\sqrt{s}}{\mu_{-,ab}} \right)
-\frac{\omx{a}+\omx{b}}{9\,m^2} \,Z_{[9]}
+{\mathcal O}\left( Q^2\right)\;,\end{aligned}$$ and $$\begin{aligned}
&&[q_{1-}^{(I)}(\sqrt{s},m)]_{ab}=0+{\mathcal O}\left( Q^{-1}\right)
-\frac{2\,\sqrt{s}}{3\,\mu_{+,ab}\,\mu_{-,ab}}
+\frac{4}{15}\,\frac{4\,M_a\,M_b}{\mu_{-,ab}\,\mu_{+,ab}^2}
\nonumber\\
&& \qquad
+\frac{8}{45}\,\frac{4\,M_a\,M_b}{\mu_{-,ab}\,\mu_{+,ab}\,m}
-\frac{2}{45}\,\frac{4\,M_a\,M_b}{\mu_{-,ab}\,m^2}
-\frac{32}{45}\,\frac{M_a\,M_b}{\mu_{-,ab}\,\mu_{+,ab}^2}
+{\mathcal O}\left( Q^{0}\right)\,.\end{aligned}$$
Perturbative threshold analysis
===============================
Close to threshold the scattering phase shifts $\delta^{(L)}_{J=L\pm \frac{1}{2}} (s)$ are characterized by the scattering length or scattering volumes $a_{[L_{2I,2J}]}$ $$\begin{aligned}
\Re \,f^{(L)}_{I,J=L\pm \frac{1}{2}} (s) &=&
q^{2\,L}\,\Big(
a_{[L_{2I,2J}]}+b_{[L_{2I,2J}]}\,q^2+{\mathcal Q}\left( q^{4}\right)
\Big) \;,
\nonumber\\
q^{2\,L+1}\,\cot \delta^{(L)}_{I,J=L\pm \frac{1}{2}} (s) &=&
\frac{1}{a_{[L_{2I,2J}]}}+\frac{1}{2}\,r_{[L_{2I,2J}]}\, q^2
+{\mathcal Q}\left( q^{4}\right) \,,
\label{}\end{aligned}$$ where we include the isospin in the notation for completeness. For s-wave channels one finds $b=-r\,a^2/2$. The threshold parameters, $a^{(I,\pm)}_{\pi N,n}$, can equivalently be extracted from the threshold values of the reduced amplitudes $$\begin{aligned}
4\,\pi\,\left(1+\frac{m_\pi}{m_N}\right) a^{(I,+)}_{\pi N,n} &=&
M_{\pi N}^{(I,+ )}(m_N+m_\pi;n) \;,
\nonumber\\
4\,\pi\,\left(1+\frac{m_\pi}{m_N}\right) a^{(I,-)}_{\pi N,n} &=& \frac{1}{4\,m_N^2}\,
M_{\pi N}^{(I,-)}(m_N+m_\pi;n) \;.
\label{}\end{aligned}$$ We identify the s and p-wave threshold parameters $a^{(I,+)}_{\pi N,0}=a^{(\pi N)}_{[S_{2I,1}]}$, $a^{(I,-)}_{\pi N,0}=a^{(\pi N)}_{[P_{2I,1}]}$ and $a^{(I,+)}_{\pi N,1}=a^{(\pi N)}_{[P_{2I,3}]}$ for which all terms up to chiral order $Q^2$ are collected. The s-wave pion-nucleon scattering lengths are $$\begin{aligned}
&&4\,\pi \left( 1+\frac{m_\pi}{m_N}\right)
a^{(\pi N)}_{[S_-]}=\frac{m_\pi}{2\,f^2}+{\mathcal O}\left(Q^3 \right)\; ,
\nonumber\\
&&4\,\pi \left( 1+\frac{m_\pi}{m_N}\right)
a^{(\pi N)}_{[S_+]}=-\frac{2}{9}\,\frac{C_{[10]}^2}{f^2}\,
\frac{m_\pi^2}{m_{\Delta}}\,
\,\big(2-Z_{[10]} \big) \left(1+Z_{[10]}+\big(2-Z_{[10]}\big)\,\frac{m_N}{2\,m_\Delta
}\right)
\nonumber\\
&&\quad \quad \quad
+\Big(2\,g^{(S)}_0+g^{(S)}_D+g^{(S)}_F \Big)\, \frac{m_\pi^2}{4 f^2}
+m_N\,\Big(2\,g^{(V)}_0+g^{(V)}_D+g^{(V)}_F \Big)\,\frac{m_\pi^2 }{4 f^2}
\nonumber\\
&&\quad \quad \quad
- \frac{g_A^2\,m_\pi^2}{4\,f^2\,m_N}
-2\,\Big(2\,b_0+b_D+b_F\Big)\frac{m_\pi^2}{f^2}
+{\mathcal O}\left(Q^3 \right) \,,
\label{piN-s}\end{aligned}$$ where $3\,a_{[S_+]}=a_{[S_{11}]}+2\,a_{[S_{31}]}$ and $3\,a_{[S_-]}=a_{[S_{11}]}-a_{[S_{31}]}$ and $F_{[8]}+D_{[8]}=g_A$. For the s-wave range parameter we find $$\begin{aligned}
&&4\,\pi \left( 1+\frac{m_\pi}{m_N}\right)
b^{(\pi N)}_{[S_-]}=\frac{1}{4\,f^2\,m_\pi}
-\frac{2\,g_A^2+1}{4\,f^2\,m_N}
\nonumber\\
&&\quad \quad \quad
+\frac{C_{[10]}^2}{18\,f^2}\,\frac{m_\pi}{m_N\,(\mu_\Delta+m_\pi)}
+{\mathcal O}\left(Q \right)\; ,
\nonumber\\
&&4\,\pi \left( 1+\frac{m_\pi}{m_N}\right)
b^{(\pi N)}_{[S_+]}= \frac{g_A^2}{4\,f^2\,m_N}
+\frac{C_{[10]}^2}{9\,f^2}\,\frac{m_\pi}{m_N\,(\mu_\Delta+m_\pi)}
\nonumber\\
&&\quad \quad \quad
+\frac{1}{4 f^2}\,\Big( 2g^{(S)}_0+g^{(S)}_D +g^{(S)}_F \Big)
+\frac{m_N}{4 f^2}\,\Big(2\,g^{(V)}_0+g^{(V)}_D+g^{(V)}_F \Big)
\nonumber\\
&&\quad \quad \quad
-\frac{2}{9}\,\frac{C_{[10]}^2}{f^2}\,
\frac{1}{m_{\Delta}}\,
\,\big(2-Z_{[10]} \big) \left(1+Z_{[10]}+\big(2-Z_{[10]}\big)\,\frac{m_N}{2\,m_\Delta
}\right)+{\mathcal O}\left(Q \right)
\label{piN-s-range}\end{aligned}$$
The p-wave scattering volumes are $$\begin{aligned}
&&4\,\pi \left( 1+\frac{m_\pi}{m_N}\right)
a^{(\pi N)}_{[P_{11}]}= -\frac{2\,g_A^2}{3\,m_\pi\,f^2}
+\frac{8\,C_{[10]}^2}{27\,f^2}\,\frac{1}{\mu_\Delta+m_\pi}
+\frac{3-4\,g_A^2}{6\,f^2\,m_N}
\nonumber\\
&&\quad \quad \quad+\frac{16}{27}\,\frac{C_{[10]}^2}{f^2}\,
\frac{m_\pi}{m_\Delta\,(\mu_\Delta+m_\pi)}
-\frac{C_{[10]}^2}{f^2}\,\frac{2\,Z_{[10]}}{27\,m_{\Delta}}\,
\left(1-Z_{[10]}-Z_{[10]}\,\frac{m_N}{2\,m_\Delta}\right)
\nonumber\\
&&\quad \quad \quad
-\frac{1}{12 f^2}\,\Big( 2\,g_0^{(S)}+g_D^{(S)}+g_F^{(S)} \Big)
+\frac{1}{3 f^2}\,\Big( g_D^{(T)}+g_F^{(T)} \Big)
+{\mathcal O}\left(Q \right)
\nonumber\\
&&4\,\pi \left( 1+\frac{m_\pi}{m_N}\right)
a^{(\pi N)}_{[P_{31}]}= -\frac{g_A^2}{6\,m_\pi\,f^2}
+\frac{2\,C_{[10]}^2}{27\,f^2}\,\frac{1}{\mu_\Delta+m_\pi}
-\frac{2\,g_A^2+3}{12\,f^2\,m_N}
\nonumber\\
&&\quad \quad \quad+\frac{4}{27}\,\frac{C_{[10]}^2}{f^2}\,
\frac{m_\pi} {m_\Delta\,(\mu_\Delta+m_\pi)}
+\frac{C_{[10]}^2}{f^2}\,\frac{4\,Z_{[10]}}{27\,m_{\Delta}}\,
\left(1-Z_{[10]}-Z_{[10]}\,\frac{m_N}{2\,m_\Delta}\right)
\nonumber\\
&&\quad \quad \quad
-\frac{1}{12 f^2}\,\Big( 2\,g_0^{(S)}+g_D^{(S)}+g_F^{(S)} \Big)
-\frac{1}{6 f^2}\,\Big( g_D^{(T)}+g_F^{(T)} \Big)
+{\mathcal O}\left(Q \right) \,,
\label{piN-pi1}\end{aligned}$$ and $$\begin{aligned}
&&4\,\pi \left( 1+\frac{m_\pi}{m_N}\right)
a^{(\pi N)}_{[P_{13}]}= -\frac{g_A^2}{6\,m_\pi\,f^2}
+\frac{2\,C_{[10]}^2}{27\,f^2}\,\frac{1}{\mu_\Delta+m_\pi}
-\frac{g_A^2}{6\,f^2\,m_N}
\nonumber\\
&&\quad \quad \quad+\frac{4}{27}\,\frac{C_{[10]}^2}{f^2}\,
\frac{m_\pi}{m_\Delta\,(\mu_\Delta+m_\pi)}
+\frac{C_{[10]}^2}{f^2}\,\frac{4\,Z_{[10]}}{27\,m_{\Delta}}\,
\left(1-Z_{[10]}-Z_{[10]}\,\frac{m_N}{2\,m_\Delta}\right)
\nonumber\\
&&\quad \quad \quad
-\frac{1}{12 f^2}\,\Big( 2\,g_0^{(S)}+g_D^{(S)}+g_F^{(S)} \Big)
-\frac{1}{6 f^2}\,\Big( g_D^{(T)}+g_F^{(T)} \Big)
+{\mathcal O}\left(Q \right)
\nonumber\\
&&4\,\pi \left( 1+\frac{m_\pi}{m_N}\right)
a^{(\pi N)}_{[P_{33}]}= \frac{g_A^2}{3\,m_\pi\,f^2}
+\frac{C_{[10]}^2}{54\,f^2}\,\left( \frac{1}{\mu_\Delta+m_\pi}
+\frac{9}{\mu_\Delta-m_\pi}\right)
\nonumber\\
&&\quad \quad \quad+\frac{1}{27}\,\frac{C_{[10]}^2}{f^2}\,
\frac{m_\pi}{m_\Delta\,(\mu_\Delta+m_\pi)}
+\frac{C_{[10]}^2}{f^2}\,\frac{Z_{[10]}}{27\,m_{\Delta}}\,
\left(1-Z_{[10]}-Z_{[10]}\,\frac{m_N}{2\,m_\Delta}\right)
\nonumber\\
&&\quad \quad \quad
+\frac{g_A^2}{3\,f^2\,m_N}
-\frac{1}{12 f^2}\,\Big( 2\,g_0^{(S)}+g_D^{(S)}+g_F^{(S)} \Big)
+\frac{1}{12 f^2}\,\Big( g_D^{(T)}+g_F^{(T)} \Big)
\nonumber\\
&&\quad \quad \quad
+{\mathcal O}\left(Q \right) \,,
\label{piN-pi3}\end{aligned}$$ where $\mu_\Delta =m_\Delta-m_N\sim Q$. Note the Z-dependence in (\[piN-s\],\[piN-pi1\],\[piN-pi3\]) can be completely absorbed into the quasi-local 4-point coupling strength. Expressing the threshold parameters (\[piN-s\],\[piN-pi1\],\[piN-pi3\]) in terms of renormalized coupling constants $\tilde g$ as $$\begin{aligned}
2\,g^{(V)}_0+g^{(V)}_D+g^{(V)}_F&=&
2\,\widetilde{g}^{(V)}_0+\widetilde{g}^{(V)}_D+
\widetilde{g}^{(V)}_F
\nonumber
\\ &+& \frac{8}{9}\,
\frac{2-Z_{[10]}}{m_{\Delta}\,m_N}\,
\left(1+\frac{m_N}{m_\Delta} +Z_{[10]} \left( 1-\frac{m_N}{2\,m_\Delta}\right) \right) C_{[10]}^2\;,
\nonumber\\
2\,g^{(S)}_0+g^{(S)}_D+g^{(S)}_F&=&
2\,\widetilde{g}^{(S)}_0+\widetilde{g}^{(S)}_D
\nonumber
\\ &+&
\widetilde{g}^{(S)}_F+\frac89\,\frac{Z_{[10]}}{m_{\Delta}}\,
\left(1-Z_{[10]}\left(1+\frac{m_N}{2\,m_\Delta}\right) \right)C_{[10]}^2 \;,
\nonumber\\
g^{(T)}_D+g^{(T)}_F&=&\widetilde{g}^{(T)}_D+
\widetilde{g}^{(T)}_F + \frac49\,\frac{Z_{[10]}}{m_{\Delta}}\,
\left(1-Z_{[10]}\left(1+\frac{m_N}{2\,m_\Delta}\right)\right) C_{[10]}^2\;,
\label{Z-absorb}\end{aligned}$$ leads to results which do not depend on $Z_{[10]}$ explicitly. Similarly one can absorb the decuplet pole terms $1/(\mu_\Delta \pm m_\pi)$ by expanding in the ratio $m_\pi/ \mu_\Delta $.
We turn to the strangeness channels. It only makes sense to provide the p-wave scattering volumes of the strangeness plus channel, because all other threshold parameters are non-perturbative. The leading orders expressions are $$\begin{aligned}
&&4\,\pi\, \left( 1+\frac{m_K}{m_N}\right)\, a^{(KN)}_{[P_{01}]}
=\frac{1}{36\,f^2}\,\left(
\frac{(3\,F_{[8]}+D_{[8]})^2}{\mu^{(\Lambda )}_{[8]}+m_K}
-9\,\frac{(D_{[8]}-F_{[8]})^2}{\mu_{[8]}^{(\Sigma )}+m_K}\right)\,\Big(1+\frac{m_K}{m_N}\Big)
\nonumber\\
&&\qquad\quad+\frac{C_{[10]}^2}{9\,f^2\,m^{(\Sigma )}_{[10]}}
\left( \frac{m_{[10]}^{(\Sigma )}+2\,m_K}{\mu^{(\Sigma )}_{[10]}+m_K}\,
-\frac{Z_{[10]}}{4}\left(1-Z_{[10]}-Z_{[10]}\,\frac{m_N+m_K}{2\,m^{(\Sigma )}_{[10]}}\right) \right)
\nonumber \\
&&\qquad\quad -\frac{1}{12 f^2}\,\Big(2\,g_0^{(S)}-g_1^{(S)}-2\,g_F^{(S)}\Big)
+\frac{1}{6 f^2}\,\left(1+\frac32\,\frac{m_K}{m_N}\right)\,
\Big(g_1^{(T)}+2\,g_D^{(T)}\Big)
\nonumber\\
&&\qquad\quad+ {\mathcal O}\left(Q \right)\,,
\nonumber\\
&&4\,\pi\, \left( 1+\frac{m_K}{m_N}\right)\, a^{(KN)}_{[P_{21}]}
=
-\frac{1}{36\,f^2}\,\left(
\frac{(3\,F_{[8]}+D_{[8]})^2}{\mu_{[8]}^{(\Lambda )}+m_K}
+3\,\frac{(D_{[8]}-F_{[8]})^2}{\mu_{[8]}^{(\Sigma )}+m_K}\right)\,\Big(1+\frac{m_K}{m_N}\Big)
\nonumber\\
&&\qquad\quad+\frac{C_{[10]}^2}{27\,f^2\,m^{(\Sigma )}_{[10]}}
\left( \frac{m^{(\Sigma )}_{[10]}+2\,m_K}{\mu_{[10]}^{(\Sigma )}+m_K}\,
-\frac{Z_{[10]}}{4}\left(1-Z_{[10]}-Z_{[10]}\,\frac{m_N+m_K}{2\,m^{(\Sigma )}_{[10]}}\right) \right)
\nonumber \\
&&\qquad\quad -\frac{1}{12 f^2}\,\Big(2\,g_0^{(S)}+g_1^{(S)}+2\,g_D^{(S)}\Big)
-\frac{1}{6 f^2}\,\left(1+\frac32\,\frac{m_K}{m_N}\right)\,
\Big(g_1^{(T)}+2\,g_F^{(T)}\Big)
\nonumber\\
&&\qquad\quad-\frac{1}{2\,m_N\,f^2}+ {\mathcal O}\left(Q \right) \,,
\label{kn-p-wave-1}\end{aligned}$$ and $$\begin{aligned}
&&4\,\pi\, \left( 1+\frac{m_K}{m_N}\right)\, a^{(KN)}_{[P_{03}]}
=\frac{1}{18\,f^2}\,\left(9\,\frac{(D_{[8]}-F_{[8]})^2}{\mu_{[8]}^{(\Sigma )}+m_K}
-\frac{(3\,F_{[8]}+D_{[8]})^2}{\mu_{[8]}^{(\Lambda )}+m_K}
\right)\,\Big(1+\frac{m_K}{m_N}\Big)
\nonumber\\
&&\qquad\quad+\frac{C_{[10]}^2}{36\,f^2\,m^{(\Sigma )}_{[10]}}\left(
\frac{m^{(\Sigma )}_{[10]}+2\,m_K}{\mu_{[10]}^{(\Sigma )}+m_K}
+2\,Z_{[10]}\left(1-Z_{[10]}\left( 1+ \frac{m_N+m_K}{2\,m^{(\Sigma )}_{[10]}}\right) \right)
\right)
\nonumber \\
&&\qquad\quad -\frac{1}{12 f^2}\,\Big(2\,g_0^{(S)}-g_1^{(S)}-2\,g_F^{(S)}\Big)
-\frac{1}{12 f^2}\,\Big(g_1^{(T)}+2\,g_D^{(T)}\Big)
+{\mathcal O}\left(Q \right)\,,
\nonumber\\
&&4\,\pi\, \left( 1+\frac{m_K}{m_N}\right)\, a^{(KN)}_{[P_{23}]}
=\frac{1}{18\,f^2}\,\left(
\frac{(3\,F_{[8]}+D_{[8]})^2}{\mu_{[8]}^{(\Lambda )}+m_K}
+3\,\frac{(D_{[8]}-F_{[8]})^2}{\mu_{[8]}^{(\Sigma )}+m_K}\right)\,\Big(1+\frac{m_K}{m_N}\Big)
\nonumber\\
&&\qquad\quad+\frac{C_{[10]}^2}{108\,f^2\,m^{(\Sigma )}_{[10]}}\left(
\frac{m^{(\Sigma )}_{[10]}+2\,m_K}{\mu_{[10]}^{(\Sigma )}+m_K}\,
+2\,Z_{[10]} \left(1-Z_{[10]}\left( 1+\frac{m_N+m_K}{2\,m^{(\Sigma )}_{[10]}}\right) \right) \right)
\nonumber \\
&&\qquad\quad -\frac{1}{12 f^2}\,\Big(2\,g_0^{(S)}+g_1^{(S)}+2\,g_D^{(S)}\Big)
+\frac{1}{12 f^2}\,
\Big(g_1^{(T)}+2\,g_F^{(T)}\Big)
+{\mathcal O}\left(Q \right)\,,
\label{kn-p-wave-2}\end{aligned}$$ where $\mu_{[8,10]}^{(Y)}=m_{[8,10]}^{(Y)}-m_N$ with $H=\Lambda ,\, \Sigma $. Note that we included in (\[kn-p-wave-1\],\[kn-p-wave-2\]) large kinematical correction terms $\sim m_K$ of formal order $Q$ induced by the covariant chiral counting assignment scheme at leading order.
Note that the p-wave scattering volumes of the kaon-nucleon sector probe four independent combinations of background terms as compared to the p-wave scattering volumes of the pion-nucleon sector which probe only two combinations. It is possible to form a particular combination which does not depend on the hyperon u-channel exchange dynamics $$\begin{aligned}
&&4\,\pi\, \left( 1+\frac{m_K}{m_N}\right)\, \left(
2\,a^{(KN)}_{[P_{01}]} +a^{(KN)}_{[P_{03}]}-6\,a^{(KN)}_{[P_{21}]}-3\,a^{(KN)}_{[P_{23}]}\right)
\nonumber\\
&&\quad \quad =2\,\Big(2\,g_0^{(S)}-g_1^{(S)}-2\,g_F^{(S)}\Big)
+4\,\Big(g_1^{(T)}+2\,g_D^{(T)}\Big)
+{\mathcal O}\left(Q \right) \;.
\label{}\end{aligned}$$
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[^1]: The parameters of [@Dai] are obtained with $c_i \to -\sqrt{3}\,c_i/2$ and $a= D_R+(c_1+3\,c_3)/\sqrt{3}$ and $b=F_R-2\,a/3+(2\,c_1/3+c_2+2\,c_3+c_4)/\sqrt{3}$. Note that the analysis of [@Dai] does not determine the parameter $c_5$.
[^2]: The corresponding counter terms in the chiral Lagrangian (\[chi-sb\]) are linear combinations of $b_0,b_D$ and $b_F$ and $\zeta_0$, $\zeta_D$ and $\zeta_F$.
[^3]: This aspect was not addressed satisfactorily in [@nn-lutz; @Gegelia].
[^4]: Note that (\[algebra-1\]) leads to a well-behaved loop function $J_{\pi N}(w)$ at $w=0$ (see (\[jpin-def\])).
[^5]: In the scheme of Becher and Leutwyler [@Becher] a pion-mass dependent subtraction scheme for the scalar one-loop functions is suggested. To be specific the master-loop function $I_{\pi N}(\sqrt{s})$ is subtracted by a regular polynomial in $s, m_N$ and $m_\pi$ of infinite order. That may lead to a pion-mass dependence of the counter terms in the chiral Lagrangian. In our scheme we subtract only a constant which agrees with the non-relativistic limit of the suggested polynomial of Becher and Leutwyler. Our subtraction constant does not depend on the pion mass.
[^6]: The renormalized loop function $J_{\pi N}(w)$ is no longer well-behaved at $w=0$. This was expected and does not cause any harm, because the point $\sqrt{s}=0$ is far outside the applicability domain of our effective field theory.
[^7]: One may make contact with the non-relativistic so-called PDS-scheme of [@KSW] by slightly modifying the replacement rule for the pole at $d=3$. With $$\frac{1}{d-3} \to 1-\frac{m_N}{2\pi\,\mu} \;, \qquad
I_{\pi N}(m_N) \to \frac{1}{16\,\pi}\,\frac{2\,\mu-m_\pi}{m_N}
+{\mathcal O}\left( \frac{m_\pi^2}{m_N^2}\right)\,,$$ power counting is manifest if one counts $\mu \sim Q$. This is completely analogous to the PDS-scheme
[^8]: Note that it is unclear how to generalize our prescription in the presence of two massive fields with respective masses $m_1$ and $m_2$ and $m_1 \gg m_2$. For the decuplet baryons one may count $m_{[8]}-m_{[9]} \sim Q $ as suggested by large $N_c$ counting arguments and therefore start with a common mass for the baryon octet and decuplet states. The presence of the $B_\mu^*$ field in the chiral Lagrangian does cause a problem. Since we include the $B_\mu^*$ field only at tree-level in the interaction kernel of pion-nucleon and kaon-nucleon scattering, we do not further investigate the possible problems in this work. Formally one may avoid such problems all together if one counts $m_{[8]}-m_{[9]} \sim Q$ even though this assignment may not be effective.
[^9]: We discuss here the most general case not necessarily imposing the minimal chiral subtraction scheme (\[def-sub\]).
[^10]: There is a further strong indication that the expansion of the inverse effective potential indeed requires a reorganization. The effective p-wave interaction kernel is troublesome, because the nucleon-pole term together with a smooth background term will lead necessarily to a non-trivial zero.
[^11]: Note that consistency with the renormalization condition (\[ren-cond\]) requires a further subtraction in the loop function $J^{(-)}_{\pi N}(\sqrt{s},0)$ if the potential $V^{(-)}_{\pi N}(\sqrt{s},0) \sim 1/(s-m_N^2+i\,\epsilon)$ exhibits the s-channel nucleon pole (see (\[pole-protection\])).
[^12]: For a neutral scalar field $\phi(x)=\phi_c(x)+\phi^\dagger_c(x)$ with mass $m$ we write $$\phi_c(0,\vec x)= \int \frac{d^3 k}{(2\pi)^3} \,\frac{e^{i\,\vec k \cdot \vec x}}{2\,\omega_k}\,a(\vec k)
\;, \qquad \phi^\dagger_c(0,\vec x)= \int \frac{d^3 k}{(2\pi)^3} \,
\frac{e^{-i\,\vec k \cdot \vec x}}{2\,\omega_k}\,a^\dagger(\vec k)\,,$$ where $\omega_k=(m^2+\vec k^2\,)^{\frac{1}{2}}$ and $[a(k),a^\dagger(k')]_-= (2\pi)^3\,2\,\omega_k\,\delta^3(k-k')$. In (\[r-def\]) we suppress terms which do not contribute to the two-body scattering process at tree-level. For example terms like $\bar N\,\eta_c \,N\,\eta_c$ or $\bar N \,\eta_c^\dagger\,N\,\eta_c^{\dagger }$ are dropped.
[^13]: Note that in the minimal chiral subtraction scheme the counter terms $\zeta_{0,D,F}$ are already renormalization scale independent. The consistency of this procedure follows from the symmetry conserving property of dimensional regularization.
[^14]: Note the convenient identities: $P_{n+1}'(x)=\sum_{l={\rm even}}^n\,(2\,l+1)\,P_l(x)$ for n even and $P_{n+1}'(x)=\sum_{l={\rm odd}}^n\,(2\,l+1)\,P_l(x)$ for n odd.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The core of quantum tomography is the possibility of writing a generally unbounded complex operator in form of an expansion over operators that are generally nonlinear functions of a generally continuous set of spectral densities—-the so-called [*quorum*]{} of observables. The expansion is generally non unique, the non unicity allowing further optimization for given criteria. The mathematical problem of tomography is thus the classification of all such operator expansions for given (suitably closed) linear spaces of unbounded operators—e.g. Banach spaces of operators with an appropriate norm. Such problem is a difficult one, and remains still open, involving the theory of general basis in Banach spaces, a still unfinished chapter of analysis. In this paper we present new nontrivial operator expansions for the quorum of quadratures of the harmonic oscillator, and introduce a first very preliminary general framework to generate new expansions based on the Kolmogorov construction. The material presented in this paper is intended to be helpful for the solution of the general problem of quantum tomography in infinite dimensions, which corresponds to provide a coherent mathematical framework for operator expansions over functions of a continuous set of spectral densities.'
address:
- |
[*QUIT*]{} Group, Dipartimento di Fisica “A. Volta” and INFN Sezione di Pavia, via A. Bassi 6, 27100 Pavia, Italy\
and\
Center for Photonic Communication and Computing, Department of Electrical and Computer Engineering, Northwestern University, Evanston, IL 60208-3118, USA
- '[*QUIT*]{} Group, CNR-INFM and CNISM Unità di Pavia, via A. Bassi 6, I-27100 Pavia, Italy'
author:
- 'Giacomo Mauro D’Ariano'
- Massimiliano Federico Sacchi
bibliography:
- '/Users/dariano/BOOK/book.bib'
title: Renormalized quantum tomography
---
[*2000 Mathematics Subject Classification.*]{} 47N30,47N40,47N50. [*Keywords and phrases.*]{} Quantum tomography, operator expansions.
Introduction {#sec:intro}
============
The state of a physical system is the mathematical description that provides a complete information on the system. Its knowledge is equivalent to know the result of any possible measurement on the system. In classical mechanics it is always possible, at least in principle, to devise a procedure made of multiple measurements which fully recovers the state of a single system. In quantum mechanics, on the contrary, there is no way, not even in principle, to infer the quantum state of a single system without having some prior knowledge on it [@imposs]. It is however possible to estimate the quantum state of a system when many identical copies are available prepared in the same state, so that a different measurement can be performed on each copy. Such a procedure is called [*quantum tomography*]{}.
The problem of finding a strategy for determining the state of a system from multiple copies dates back to 1957 by Fano [@fano], who called [*quorum*]{} a set of observables sufficient for a complete determination of the density matrix. However, quantum tomography entered the realm of experiments more recently, with the pioneering experiments by Raymer’s group [@raymer] in the domain of quantum optics. In quantum optics, in fact, using a balanced homodyne detector one has the unique opportunity of measuring all possible linear combinations of position and momentum—the so-called [*quadratures*]{}—of the harmonic oscillator representing a single mode of the radiation field.
The first technique to reconstruct the density matrix from homodyne measurements — so called [*homodyne tomography*]{} — originated from the observation by Vogel and Risken [@vogel] that the collection of probability distributions achieved by homodyne detection is just the Radon transform of the Wigner function $W$. Therefore, similarly to classical imaging, one can obtain $W$ by inverting the Radon transform, and then from $W$ one can recover the matrix elements of the density operator. This original method, however, works well only in a semi-classical regime, whereas generally for small photon numbers it is affected by an unknown bias caused by the smoothing procedure needed for the analytical inversion of the Radon transform. The solution to the problem is to bypass the evaluation of the Wigner function, and to evaluate the matrix elements of the density operator by simply averaging suitable functions of homodyne data: this is the basis of the first unbiased topographic technique presented in Ref. [@dmp]. Clearly the state is perfectly recovered in principle only in the limit of infinitely many measurements: however, for finitely many measurements one can estimate the statistical error affecting each matrix element. For infinite dimensions there is the further problem that the propagation of statistical errors of the density matrix elements make them useless for estimating the ensemble average of some operators (unbounded), and a method for estimating the ensemble average is needed, which bypasses the evaluation of the density matrix itself, as was first suggested in Ref. [@tokio]. For a brief historical excursus on quantum tomography, along with a review on the generalization to any number of radiation modes, arbitrary quantum systems, noise deconvolution, adaptive methods, and maximum-likelihood strategies the reader is addressed to Ref. [@revt].
The most comprehensive theoretical approach to quantum tomography uses the concept of [ *frame of observables*]{}, i. e. a set of observables spanning the linear space of operators, from which one derives contextually the [*quorum*]{} of observables and the estimation rule. The ensemble average $\< H\>$ of any arbitrary operator $H$ on a Hilbert space $\sH$ is estimated using measurement outcomes of the quorum $\{ O_l\}$ upon expanding $H$ over a set of functions $f_n(O_l)$ of the observables $\{ O_l\}$. What makes the general theory nontrivial in infinite dimensions is the crucial role of the non linear functions $f_n(O_l)$ in making the infinite expansion convergent. Let’s denote by $P_j:=f_n(O_l)$, $j=(n,l)$, such complete set of operators. Once you have the $P_j$, then the problem is reduced to the [*linear*]{} problem of expanding an operator as $H=\sum_j
\< Q_j^\dag, H\> P_j$, for a suitable “dual” set of operators $\{Q_j\}$. (Notice that generally the index $l$ is continuous, whence also the operator expansion.) The scalar product in the expansion is generally not simply the Frobenius’s when we need to expand operators that are not Hilbert-Schmidt. The mathematical [*theory of frames*]{} [@Christiansen; @Casazza; @Han; @Li] is the perfect tool for establishing completeness of $\{P_j\}$ and for finding dual sets $\{Q_j\}$. In most practical situations the set $\{P_j\}$ is over-complete, and there are many alternate dual sets $\{Q_j\}$, the non unicity providing room for optimization. A general theory should also classify the operators $H$ having bounded scalar product with $\{Q_l\}$ and expansion $H=\sum_j \< Q_j^\dag,
H\> P_j$ converging in average for a given class of quantum states. In infinite dimensions—e. g. for homodyne tomography—the easy known approach works only for Hilbert-Schmidt operators, whence including trace-class ones—e. g. for estimating the matrix elements of the density operator over an orthonormal basis. For unbounded operators, however, such operator expansion becomes an infinite sum of unbounded terms. On the other hand, converging (and even finite) alternate expansions are known to exist for various unbounded operators [@Rich]. As we will see in this paper, the mechanism allowing “renormalization” of the expansion relies on the existence of [ *null-estimators*]{}—namely operator-valued functions that have zero mean over the quorum—their existence being related to a group of symmetries of the quorum. The notion of null-estimator was first introduced in Ref. [@null], in the context of of homodyne tomography, where the quorum is made of the quadratures of the field mode—the quantum analogue of the Radon transform. Here the symmetry group of the quorum is the group $U(1)$ of rotations of the quadrature phase. The existence of null-estimators leads to infinitely many alternate expansions of the same operator over the quorum, allowing cancellations of the infinities in the expansion—a kind of “renormalization” procedure.
The problem of classifying all operator expansions for a given quorum in infinite dimensions for given spaces of unbounded operators is a difficult one, and remains still open. It involves the theory of [*frames*]{} or even more general notions of basis in Banach spaces [@Christiansen; @Casazza; @Han; @Li], a still unfinished chapter of analysis. In this paper we present new nontrivial operator expansions for the quorum of quadratures of the harmonic oscillator, and introduce a first preliminary general framework to generate and classify new expansions, based on the Kolmogorov construction. We hope that the material presented here will open the way to the solution of the problem of quantum tomography in infinite dimensions, leading to a general mathematical framework for operator expansions over functions of a set of spectral densities.
Quantum tomography and quorum of observables
============================================
The general idea of quantum tomography is that there is a set of observables $\{X_\xi\}_{\xi\in\Klm}$ on the Hilbert space of the system $\sH $—called “quorum”—by which one can estimate any desired ensemble average by measuring the observables of the quorum, each at the time, in a scheme of a repeated measurements. The observables of the quorum must be independent each other, namely they are not commuting: $$[X_{\xi'},X_{\xi''}]=0\;\Longleftrightarrow\; \xi'=\xi''.$$ Generally the set $\Klm$ parameterizing the quorum is infinite, and most commonly, is a continuum. In these cases, since clearly one can measure only a finite number of observables, these are randomly picked out according to a given probability measure on $\Klm$, which, therefore, must be a probability space. In the following, for simplicity, we will also assume a probability density over $\Klm$ and denote it with the symbol $\d \mu(\xi)$. It follows that the ensemble average of a (generally complex) operator is written in the form of double expectation $$\< X\>=\int_\Klm\d\mu(\xi)\<f_\xi(X_\xi|X)\>,\label{doubleav}$$ where the generally nonlinear function $f_\xi(x|X)$ of the variable $x$ has an analytic form which depends on the particular operator $X$. We will call the function $f_\xi(x|X)$ the [*tomographic estimator*]{} for $X$ with quorum $\{X_\xi\}_{\xi\in\Klm}$. If we want to achieve the estimation of $\< X\>=\Tr[\rho X]$—the expectation being supposedly bounded on the state $\rho$—by averaging the estimator $f_\xi(x|X)$ over both quorum and measurement outcomes with a bounded variance, we need to have the function $f_\xi(x|X)$ square-summable over $x$ and $\xi$, more precisely $$\int_\Klm\d\mu(\xi)\int_{\mathcal{X}_\xi}\<\d E_\xi(x)\>|f_\xi(x|X)|^2 <\infty,$$ where $\mathcal{X}_\xi$ denotes the spectrum of $X_\xi$, and $\d E_\xi(x)$ its spectral measure. In the following, for simplicity, we will consider the spectrum $\mathcal{X}_\xi\equiv\mathcal{X}$ independent on $\xi$. Clearly, the above square-summability will depend again on the state $\rho$ and on the operator $X$.
We first want to notice two main features of estimators:
1. The estimator $f_\xi(x|X)$ is generally not unique, namely there can be many different estimators for the same operator $X$. This is equivalent to the existence of [*null estimators*]{}, namely functions $n_\xi(x)$ such that $$\int_\Klm\d\mu(\xi)\int_{\mathcal{X}}\d E_\xi(x)n_\xi(x)=0.$$ Accordingly, the estimators are grouped into equivalence classes, each class corresponding to an operator $X$. For such equivalence we will use the notation $\simeq$, i. e. we will write $f\simeq
g$ or $f-g\simeq 0$ to denote that the two estimators are equivalent, namely they differ by a null estimator.
2. For fixed $x$ and $\xi$ the estimator $f_\xi(x|X)$ must be a linear functional of $X$, namely $$f_\xi(x|aX+bY)=af_\xi(x|X)+bf_\xi(x|Y),\qquad f_\xi(x|X^\dag)=f_\xi(x|X)^*.$$
[@revt] The quorum is given by $\{X_\phi\}_{[0,\pi)}$ with $X_\phi$ denoting the quadrature $X_\phi\doteq\tfrac{1}{2}(a^\dag e^{i\phi}+ae^{-i\phi})$, $a
,a^\dag $ being the annihilation and creation operators of the harmonic oscillator with commutator $[a, a^\dag ]=1$. Estimators for the dyads $|n\>\< m|$ made with the orthonormal basis $\{|n\>\}$, $n=0,\ldots,\infty$ are $$\begin{aligned}
f_\phi (x||n\>\< m|) &=&
\int_{-\infty}^{+\infty}\frac
{dk\,|k|}{4}\, e^{\frac{1-\eta}{8\eta}k^2-ikx}\< n+d| e^{ik
X_{\phi}}| n\> \label{estimat} \nonumber \\&= &
e^{id(\phi+\frac{\pi}{2})}\sqrt{\frac{n!}{(n+d)!}}\int_{-\infty}^{+\infty}
dk\,|k| e^{\frac{1-2\eta}{2\eta}k^2-i2kx} k^d L_n^d(k^2)\;,\end{aligned}$$ where $L_n^d(x)$ denotes the generalized Laguerre polynomials. For the unbounded operators $a$ and $a^\dag a$ one can check that the following are unbiased estimators $$\begin{aligned}
f_\phi (x|a)&=&2e^{i\phi}x, \nonumber \\
f_\phi (x|a^\dag a)&=&2x^2-\tfrac{1}{2}.
\;\end{aligned}$$
The problem of quantum tomography is to establish the general rule for estimation, namely
Given the quorum $\{X_\xi\}_{\xi\in\Klm}$ find the correspondence: $$f_\xi(x|X)\Longleftrightarrow X,\qquad\text{for every operator}\, X
\text{on}\,\sH,$$
where we possibly mean to find the whole equivalence class of estimators $f_\xi (X_\xi |X )$.
Before solving this task, first one needs to know that the set of observables $\{X_\xi\}_{\xi\in\Klm}$ is actually a quorum. The easiest thing to do, however, is to derive both the quorum and the estimation rule contextually, starting from a spanning-set of observables—shortly [*observable spanning-set*]{}—namely a set of observables $\{F_\omega\}_{\omega\in\mathfrak{O}}$ in terms of which we can linearly expand operators as follows $$X=\int_\mathfrak{O}\d\omega\; c_\omega(X) F_\omega.\label{obsframe}$$ Notice that the notion of operator spanning-set used here generalizes the notion of frames for Banach spaces to unbounded operators (see also the following), and is generally not strictly a frame according to the definition of Refs. [@Christiansen; @Casazza]. In the following throughout the paper we will always assume probability distributions admitting densities. Generally, the set $\mathfrak{O}$ is unbounded, and the measure $\d\omega$ is not normalizable, whence, as such, the expansion (\[obsframe\]) cannot be used for quantum tomography. However, generally this feature is related to the redundancy of the observable spanning-set, which includes many observables $F_\omega$ that are just different functions of the same observable. Then, collecting the observables of the spanning-set into functional equivalence classes $\mathfrak{K}_\xi$, each corresponding to an observable of the quorum $\{X_\xi\}_{\xi\in\Klm}$, one can relabel the observable spanning-set as $F_{\kappa,\xi}\doteq
f_{\kappa}(X_\xi)$ with $\kappa\in\mathfrak{K}_\xi$, writing $$X=\int_\Klm\d\mu(\xi)\int_{\mathfrak{K}_\xi}\d\nu(\kappa)\; c_{\kappa,\xi}(X) f_{\kappa}(X_\xi)=
\int_\Klm\d\mu(\xi)f_{\xi}(X_\xi|X),$$ where the function $f_{\xi}(x|X)$ is the integral over the observables equivalent to $X_\xi$, namely $$f_{\xi}(X_\xi|X)\doteq\int_{\mathfrak{K}_\xi}\d\nu(\kappa)\; c_{\kappa,\xi}(X) f_{\kappa}(X_\xi).$$ Notice that in terms of the spectral decomposition of $X_\xi$ we can write $$X=\int_\Klm\d\mu(\xi)\int_{\mathcal{X}_\xi}\d E_\xi(x) f_{\xi}(x|X),\label{XEx}$$ and since this expansion is linear in the spectral measure $\d E_\xi(x)$, the latter can be regarded itself as an observable spanning-set. Indeed, upon introducing the spectral density $\d
E_\xi(x)\doteq Z_{\xi,x}\d x$, and the density $\d\mu(\xi)\doteq m(\xi)\d\xi$, and renaming $\zeta=(\xi,x)$ and $\mathfrak{Z}\doteq\{(\xi,x),x\in\mathcal{X}_\xi,\xi\in\Klm\}$ Eq. (\[XEx\]) can be rewritten in the same form of Eq. (\[obsframe\]), namely $$X=\int_\mathfrak{Z}\d\zeta c'_\zeta(X) Z_\zeta,$$ where the new expansion coefficients are now given by $$c'_{\xi,x}(X)=m(\xi)f_\xi(x|X)=m(\xi)\int_{\mathfrak{K}_\xi}\d\nu(\kappa)\; c_{\kappa,\xi}(X) f_{\kappa}(x).
\label{cc}$$ For homodyne tomography the above quantities are explicitly given in Table \[t:tomo\].
For $F_\omega $ an operator frame, the coefficients of the expansion (\[obsframe\]) can be rewritten in form of a pairing $(\cdot|\cdot)$ with a dual frame $G_\omega$ as follows $$c_\omega(X)=(G_\omega|X),$$ in terms of which Eq. (\[cc\]) rewrites in the pairing form $$c'_{\xi,x}(X)=m(\xi)(W_{\xi,x}|X),$$ with dual frame $$W_{\xi,x}=\int_{\mathfrak{K}_\xi}\d\nu(\kappa)\, G_{\xi,x}f_{\kappa}^*(x).$$ From the last equation it follows that the estimator itself can be written using the pairing $$f_\xi(x|X)=(W_{\xi,x}|X),\label{pairingW}$$ or, in terms of the original observable frame $$f_\xi(x|X)=\int_{\mathfrak{K}_\xi}\d\nu(\kappa)\, (G_{\xi,x}|X) f_{\kappa}(x).$$
general homodyne
-------------- ------------------------
$X_\xi$ $X_\phi$
$\Klm$ $[0,\pi)$
$\xi$ $\phi$
$\d\mu(\xi)$ $\tfrac{\d\phi}{\pi} $
$\omega$ $\alpha $
$F_\omega$ $D(\alpha)$
: Table of correspondence for homodyne tomography.[]{data-label="t:tomo"}
general homodyne
--------------------- ------------------------
$c_\omega(X)$ $\Tr[D^\dag(\alpha)X]$
$\omega$ $\alpha $
$\mathfrak{K}_\xi$ ${\mathbb R}$
$\d\nu(k)$ $\frac{1}{4}\d k |k|$
$c_{\xi,\kappa}(X)$ $\Tr[e^{-ikX_\phi} X]$
$f_\kappa(X_\xi|X)$ $e^{ikX_\phi}$
: Table of correspondence for homodyne tomography.[]{data-label="t:tomo"}
general homodyne
------------------- --------------------------------------------------------
$\d\mu(\xi)$ $\tfrac{\d\phi}{\pi} $
$Z_{\xi,x}$ $|x\>_\phi{}_\phi\<x|$
$\mathcal{X}_\xi$ ${\mathbb R}$
$\mathfrak{Z}$ $[0,\pi)\times{\mathbb R}$
$m(\xi)$ $\frac{1}{\pi}$
$W_{\xi,x}$ $-\frac{1}{4\pi}\frac{\operatorname{P}}{(x-X_\phi)^2}$
: Table of correspondence for homodyne tomography.[]{data-label="t:tomo"}
Unbiasing noise
---------------
It is possible to estimate the [*ideal*]{} ensemble average $\<
X\>$ by measuring the quorum in the presence of instrumental noise, when the noise map $\map{N}$ is invertible, or, more generally, if there exists the right inverse of ${\map N}$. In terms of observable frames this just corresponds to using a different dual frame. More precisely, one has: $$\begin{split}
X=\map{N}\map{N}^{-1}(X)=&\int_\Klm\d\mu(\xi) \map{N}[f_\xi(X_\xi|\map{N}^{-1}(X))]\\=&
\int_\Klm\d\mu(\xi)\int_{\mathcal{X}_\xi}\map{N}(\d E_\xi(x))
f_{\xi}(x|\map{N}^{-1}(X)).
\end{split}
\label{unbias}$$ Notice that Eq. (\[unbias\]) means that $$\<X\>=\int_\Klm\d\mu(\xi)\int_{\mathcal{X}_\xi}\<\d E_\xi(x)\>_\map{N} f_{\xi}(x|\map{N}^{-1}(X)),$$ where $\<\,\cdot\,\>\doteq\Tr[\rho\,\cdot\,]$ denotes the ideal ensemble average, whereas $\<\,\cdot\,\>_{\map{N}} \doteq\Tr[\dual{\map{N}}(\rho)\,\cdot\,]$ denotes the experimental ensemble average, ${\map {N}}_*$ being the predual map of $\map {N}$ (Schroedinger versus Heisenberg picture). This also means that for left invertible map $\map{N}$ the noisy spectral measures $\map{N}(\d
E_\xi(x))$ are still a quorum. In terms of the pairing in Eq. (\[pairingW\]) unbiasing the noise is equivalent to use the new dual frame $\map{N}^{-1}{}^\dag(W_{\xi,x})$. When $\map{N}$ is not right-invertible one can still estimate the ensemble average of operators in the range of the map. Moreover, in infinite dimension, when the noise CP map $\map{N}$ is compact its inverse map is unbounded, and one generally cannot unbias the noise without restricting the space of reconstructed operators. Otherwise, one has a Hadamard ill-posed problem, for which there are biased compromises, such as putting a cutoff on the vanishing singular values of $\map{N}$.
[@revt] The operators $[\sigma_\alpha/\sqrt2]$ make an observable orthonormal basis for $\mathbb{C}^{\otimes 2}$. We consider now the noise described by the depolarizing Pauli channel $${\map N}=(1-p)\map{I}+\frac p2 \map{T},$$ where $\map{I}$ denotes the identity map and $\map{T}$ the trace map $\map{T}(X)\doteq I\Tr(X)$. This noise can be simply unbiased via noise-map inversion: $$\map{N}^{-1\dag}=\frac{1}{1-p}\map{I}-\frac{p}{2(1-p)}\map{T}.$$
[@revt] The set of displacements operators $D(\alpha):=e^{\alpha a^\dag -\alpha ^* a}\,,\quad\alpha\in\mathbb C$ make an observable Dirac-orthonormal frame for $\set{T}_{1/2}$, where $$\set{T}_s=\{X\in\set{T},\;
X=\: f(a,a^\dag)\:,\;\mbox{s.t.} \;
\lim_{\alpha\to\infty}f(\alpha,\bar{\alpha})e^{s|\alpha|^2}=0\}$$ where $\: \ \ \:$ denotes normal ordering. In the presence of noise from nonunit quantum efficiency $\eta$, the unbiased reconstruction is possible for operators in $\set{T}_s$ if $\eta\ge (2s)^{-1}$. In fact, one uses the new dual: $$D(\alpha)\to\map{N}^{-1\dag}(D(\alpha))=\eta
D(\eta^{1/2}\alpha)e^{\frac{1-\eta}{2}|\alpha|^2}.$$
[@revt] As for quantum efficiency, Gaussian noise can be unbiased for mean thermal photon number $\bar n\le s-\frac 12$. One has the new dual: $$D(\alpha)\to\map{N}^{-1\dag}(D(\alpha))=D(\alpha)e^{\bar n|\alpha|^2}.$$
The case of homodyne tomography
===============================
Before addressing the general problem of deriving a general tomographic rule for unbounded operators, in this section we will re-derive the known pattern function of homodyne tomography in order to illustrate the general concepts introduced in the previous section.
The starting point is the observable frame $\{D(\alpha)\}_{\alpha\in\Cmplx}$, in terms of which the decomposition (\[obsframe\]) for Hilbert-Schmidt operators rewrites as follows $$X=\int_\Cmplx
\frac{\d^2\alpha}{\pi}\Tr[D^\dag(\alpha)X]D(\alpha).\label{glauber}$$ By changing to polar variables $\alpha = (-i/2)k e^{i\phi}$, Eq. (\[glauber\]) becomes $$X= \int^{\pi}_0\frac{d\phi}{\pi}\int^{+\infty}_{-\infty}
\frac{d k\, |k|}{4}\,\Tr [ X\; e^{ik X_{\phi}}]\,
e^{-ik X_{\phi}}.\label{op}$$ Eq. (\[op\]) can be used only for Hilbert-Schmidt operators, for which the trace under the integral in Eq. (\[op\]) exists. In terms of the quadrature spectral measure, one has $$X= \int^{\pi}_0\frac{d\phi}{\pi}\int^{+\infty}_{-\infty}
\frac{d k\, |k|}{4}\,\Tr [ X\; e^{ik X_{\phi}}]\, e^{-ik X_{\phi}}
=\int^{\pi}_0\frac{d\phi}{\pi}\int^{+\infty}_{-\infty}\d E_\phi(x)
\Tr[X W_{\phi,x}],$$ where $$W_{\phi,x}=e^{-i\phi a^\dag a}D(x)W_{0,0}D^\dag(x)e^{i\phi a^\dag a},\qquad
W_{0,0}= -\frac 12\operatorname{P}\frac 1{X_0^2},\label{W}$$ $\operatorname{P}$ denoting the Cauchy principal value. On the other hand, in the next section we will show that for unbounded operators we also have the expansion $$\begin{split}
X=&\int_0^{\pi}\frac{\d\phi}{\pi} \int_{-\infty}^\infty\d t
\Tr[G^\dag(t,\phi)X]F(t,\phi),\\
F(t,\phi)=&\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}(t-i2X_\phi)^2},\quad
G(t,\phi)= \frac{\d}{\d t}t e^{\frac{t^2}{2}}
\int_0^1\d\theta | i(1-\theta)te^{i\phi}\>\< -i\theta te^{i\phi}|.
\end{split}\label{Gexp}$$ which in terms of the the quadrature spectral measure reads $$X=\int_0^{\pi}\frac{\d\phi}{\pi}\int_{-\infty}^\infty
\d E_\phi(x)\Tr[XW'_{\phi,x}],\label{frm1}$$ where now $$W'_{\phi,x}= \int_{-\infty}^\infty\d t
\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}(t-i2x)^2} \frac{\d}{\d t}t
e^{\frac{t^2}{2}} \int_0^1\d\theta | i(1-\theta)te^{i\phi}\>\<
-i\theta te^{i\phi}|.\label{W1}$$
Alternatively, as shown in Sec. (\[versus\]) by means of the frame of normal-ordered moments, one has the expansion $$\begin{split}
&X=\sum_{n,m=0}^\infty a^{\dag n}a^m \Tr [g_{n,m}^\dag X] \\
&= \int_{0}^\pi \frac{\d \phi }{\pi }\int_{-\infty }^{+ \infty }\d
E_\phi (x)\, \Tr \left [X \sum_{n,m=0 }^\infty g^\dag _{n,m}
{ n+m \choose n}^{-1} \frac{H_{n+m}(\sqrt 2\, x)}{\sqrt {2^{n+m}}}
\right ]\;\label{frm2}
\end{split}$$ where $$\begin{aligned}
g_{n,m}=\sum_{j=0}^{\min{(n,m)}}
\frac{(-1)^{j}}{j!\sqrt {(n-j)!(m-j)!}}|n-j \> \< m-j |
\;.\end{aligned}$$ The above expansions in Eqs. (\[frm1\]) and (\[frm2\]) are just examples of alternate expansions which are equivalent for the estimation of the expectation values of (even unbounded) observables, but can be very different as regards the statistical noise affecting such estimation. As a matter of fact, the problem of classifying all possible expansions has been never solved, and, hopefully, the results of the present paper may suggest a unifying approach to the solution of such a difficult problem. As we will see in the next section, the existence of many alternate expansions is due to the symmetry of the quorum of quadrature operators, and the resulting properties of null estimator functions.
Calculus with null functions
============================
We first notice that to a null estimator function $n_\xi\simeq 0$ it corresponds a null expansion over the quorum, namely $$\int_\Klm\d\mu(\xi)\int_\mathcal{X}\d E_\xi(x)n_\xi(x)=0\Longleftrightarrow
\int_\Klm\d\mu(\xi) n_\xi(X_\xi)=0.$$ Let us recall the ordering relation [@wun3] $$\begin{aligned}
\: a^{\dag k} a^l \: _s
=\sum _{j=0}^{\{k,l\}} \frac{k ! l!}{j! (k-j)!
(l-j)!}\left ( \frac {s-r}{2}\right )^j \: a^{\dag k-j}a^{l-j} \: _r\;,\label{arb}\end{aligned}$$ where $\{k,l\}:=\min (k,l)$, and $s=-1,0,\hbox{ and }1$ correspond to normal, symmetrical, and anti-normal ordering, respectively. We will also write the symmetrical ordering as $S\{ a^{\dag k} a^l\}:=\: a^{\dag k} a^l \: _0 $, and the normal ordering as $\: a^{\dag k} a^l \: := \: a^{\dag k} a^l \: _{-1}
$.
Then we have:
[@null] The following equivalence relation holds $$x^ke^{\pm i(k+2n+2)\phi}\simeq 0,\qquad \forall k,n\geq 0.\label{mainid}$$
Since $X_\phi ^k = \frac{1}{2^k}\sum _{l=0}^k {k \choose l} S\{a^{\dag
l} a^{k-l}\}\, e^{i\phi (2l-k)}$, one has $$\int_0^{\pi}\!\!\frac{d\phi}{\pi} e^{\pm i(k+2+2n)\phi}
X_\phi^k=0,\qquad\forall k,n\geq 0, \label{mainidop}$$ which is equivalent to (\[mainid\]).
Stated differently:
The following equivalence relation holds $$H_k(\sqrt 2 \,x)e^{\pm i(k+2n+2)\phi}\simeq 0,\qquad \forall k,n\geq
0, \label{mainid2}$$ where $H_k(x)$ denote the $k$-th Hermite polynomial.
From the definition of Hermite polynomials one has $$\begin{split}
\frac{1}{\sqrt {2^n}}
&H_n (\sqrt 2 X_\phi ) =
\frac{1}{\sqrt {2^n}}\left. \frac {\partial ^n}{\partial t^n}\right | _{t=0}
e^{-t^2 +\sqrt 2 t (a^\dag e^{i\phi }+a e^{-i\phi })} \\&=
\frac{1}{\sqrt {2^n}}\left. \frac {\partial ^n}{\partial t^n}\right | _{t=0}
e^{\sqrt 2 t a^\dag e^{i\phi }}e^{\sqrt 2 t a e^{-i\phi }} =
\sum _{k=0}^n {n \choose k} a^{\dag k} a^{n-k} e^{i\phi (2k-n)}= 2
^n {\bf :}X^n_\phi {\bf :}\;.\label{defh}
\end{split}$$ Then, it follows that $$\int_0^{\pi}\!\!\frac{d\phi}{\pi} e^{\pm i(k+2+2n)\phi} H_k({\sqrt 2} X_\phi )=
0,\qquad\forall k,n\geq 0,$$ which is equivalent to (\[mainid2\]).
Moreover, we also have
\[l:Her\] The following equivalence relations hold: $$e^{\pm in\phi}H_{2l+n}^{(l)}(\kappa x)\simeq e^{\pm in\phi}H_{2l+n}(\kappa x),$$ where we introduced the [*truncated*]{} Hermite polynomial $$H_n^{(l)}(z)=\sum_{m=0}^l \frac{(-)^m n! (2z)^{n-2m}}{m!(n-2m)!}. \qquad n\ge 2l.$$
The two identities are just the complex conjugated of each other. Therefore, it is sufficient to prove the identity with the plus sign. The latter is a simple consequence of identity (\[mainid\]), namely $x^ke^{\pm i(k+2n+2)\phi}\simeq 0$. By using the un-truncated Hermite polynomial, we have $$\begin{split}
e^{in\phi}H_{2l+n}(\kappa x)=&
e^{in\phi}\sum_{m=0}^{l+[[n/2]]}\frac{(-)^m (2l+n)! (2\kappa)^{2l+n-2m}}{m!(2l+n-2m)!}\\
\times &x^{2l+n-2m}e^{i[n+2(l-m)+2(m-l)]\phi},
\end{split}$$ from which it follows that all terms with $m>l$ are equivalent to zero.
Finally, one can show the Poisson identities (whose proof can be found in the Appendix)
The following identities hold $$\begin{split}
f(x^2)\delta_\pi(\phi)&\simeq
\frac{1}{\pi}\left[\frac{f(x^{2}e^{2i\phi})}{1-e^{-2i\phi}}+\frac{f(x^{2}e^{-2i\phi})}{1-e^{2i\phi}}\right],\\
xf(x^2)\delta_\pi(\phi)&\simeq \frac{1}{\pi}\left[\frac{xe^{i\phi}f(x^{2}e^{2i\phi})}{1-e^{-2i\phi}}+\frac{xe^{-i\phi}f(x^{2}e^{-2i\phi})}{1-e^{2i\phi}}\right],
\end{split}$$ In particular, we have the identity $$\delta_\pi(\phi)\simeq \frac{1}{\pi}.\label{dicomb}$$
The Kolmogorov construction
===========================
In this Section we present the Kolmogorov construction, and its relation with the fundamental identity of quantum tomography.
In the following, by $\ltwo$ we denote the Hilbert space of square summable sequences of complex numbers, and by $\Ltwo(\Klm)$ we denote the Hilbert space of square summable functions over the space $\Klm$. For example, $\Klm=\Reals$, and $\Ltwo(\Klm)$ the Hilbert space of square summable functions on the real axis, or $\Klm=S^1$, and $\Ltwo(\Klm)$ is the Hardy space of square-summable complex functions on the circle. In the following we will focus attention only to the case $\Klm\equiv\Reals$. Consider now a complete orthonormal set of functions $[\upsilon_n(x)]$ for $\Ltwo(\Klm)$. The completeness of the set corresponds to the following distribution identity $$\sum_n \upsilon_n(x)^*\upsilon_n(y)=\delta(x-y),$$ where $\delta$ denotes the usual Dirac-delta. Consider now a (infinite-dimensional) Hilbert space $\sH$ and denote by $[w_n]$ an orthonormal basis for it. The following vector $$|\upsilon(x)\>=\sum_n\upsilon_n(x)|w_n\>,$$ is Dirac-normalizable, in the sense that $$\<\upsilon(y)|\upsilon(x)\>=\delta(x-y).$$ Consider now another (infinite-dimensional) Hilbert space $\sK\simeq\sH$. To the orthonormal basis $[\upsilon_n(x)]$ for $\Ltwo(\Klm)$ and $[z_n]$ for $\sK$ we associate a map from the observables $\set{O}_\Klm$ with spectrum $\Klm$ on $\sH$ to operators in $\Bnd{\sH,\sH\otimes\sK}$ given by $$\upsilon(X)=\sum_n\upsilon_n(X)\otimes|z_n\>,$$ where, as usual, we define the operators $\upsilon_n(X)$ in terms of the spectral resolution of $X$ as follows $$\upsilon_n(X)=\int_\Klm\d E_X(x)\upsilon_n(x),$$ where $\d E_X(x)$ denotes the spectral measure of $X$.
For $X,Y\in\set{O}_\Klm$ formally we write $$\upsilon(X)^\dag\upsilon(Y)=\sum_n \upsilon_n(X)^\dag \upsilon_n(Y),$$ Consider the integral kernel $K(x,y)$, $x,y\in\Klm$ corresponding to the positive operator $K\in\Bnd{H}$ $$K(x,y)=\<\upsilon(x)|K|\upsilon(y)\>.$$ For any two self-adjoint operators $X,Y$ on $\Ltwo(\Reals)$, the expression $K(X,Y)$ is well defined in the following sense $$K(X,Y)\doteq \upsilon(X)^\dag (I\otimes\tilde{K}) \upsilon(Y),$$ where $\tilde{K}\in\Bnd{K}$ is given by $$\tilde{K}=\sum_{n,m} |z_n\>\<w_n|K|w_m\>\<z_m|,$$ namely we can also write $$K(X,Y)=\sum_{n,m}\upsilon_n(X)^\dag \<w_n|K|w_m\> \upsilon_m(Y)
=\int_\Klm\d E_X(x) \int_\Klm\d E_X(y) K(x,y).\label{pro}$$ This is also equivalent to say that for any expansion of $K(x,y)$ in series of products of functions of single variable, $K(X,Y)$ is defined as the same expansion, ordered with the functions of $X$ on the left and the functions of $Y$ on the right. For commuting $X,Y$, then $K(X,Y)$ simply represents the same analytic expression of $K(x,y)$, now substituting the operators in place of the variables. As an example, the identity operator $K=I$ corresponds to the Dirac-delta kernel, and for commuting $X,Y\in\set{O}_\Klm$ we have $$\upsilon(X)^\dag\upsilon(Y)=\delta(X-Y).$$ By replacing now $\sH\to\sH^{\otimes 2}$, even for non commuting $X$ and $Y$, one has $$(\upsilon(X)^\dag \otimes I )(I \otimes \upsilon(Y))=\delta(X\otimes I-I\otimes Y).$$ Moreover, similarly to Eq. (\[pro\]), one has $$\begin{aligned}
K(X \otimes I ,I \otimes Y) &\doteq &
(\upsilon(X)^\dag \otimes I )
(I_\sH^{\otimes 2}\otimes \tilde{K}) (I \otimes \upsilon(Y))
\nonumber \\&= &
\sum_{n,m}\upsilon_n(X)^\dag \otimes \upsilon_m(Y) \<w_n|K|w_m\>
\nonumber \\&= &
\int_\Klm\d E_X(x) \otimes \int_\Klm\d E_X(y) K(x,y).\label{pro2}\end{aligned}$$ The fundamental identities of quantum tomography are obtained as an expansion of the swap operator $E$ over the [*quorum*]{}, since for any state $\rho $ and observable $A$ one has $\Tr[\rho A]=\Tr[(\rho
\otimes A) E]$, where $E|\psi \> \otimes |\phi \> =|\phi
\> \otimes |\psi \> $.
Homodyne tomography
-------------------
From Eq. (\[op\]), it is clear the swap operator can be written as $$E= \int^{\pi}_0\frac{d\phi}{\pi}\int^{+\infty}_{-\infty}
\frac{d k\, |k|}{4}\,e^{-ik X_{\phi}} \otimes
e^{ik X_{\phi}}.\label{op2}$$ Then, the usual homodyne tomographic formula can be obtained by the Kolmogorov construction in writing the swap operator as follows $$E=\int_0^\pi\frac{\d\phi}{\pi} (\upsilon(X_\phi)^\dag \otimes I)
(I_\sH^{\otimes 2}\otimes \tilde {K})
I \otimes \upsilon(X_\phi),$$ corresponding to the positive kernel $$K(x,x')=\frac{\pi}{2}\< x||Y||x'\>=\int_\Reals\frac{\d k}{4}|k|e^{ik(x'-x)}
=-\frac{\operatorname{P}}{2} \frac{1}{(x-x')^2},$$ where $Y$ is the quadrature conjugated to $X$, with $[X,Y]=\frac i
2$. The kernel is clearly positive, since one has $$\sum_{i,j}K(x_i,x_j)\xi_i^*\xi_j=
\int_\Reals\frac{\d k}{4}|k|\left|\sum_je^{ix_j}
\xi_j\right|^2$$ The tomographic formula consists in the following identity $$E=\int_0^\pi\frac{\d\phi}{\pi} \, K(X_\phi\otimes I,I\otimes X_\phi)$$ Using Eq. (\[pro2\]), one can also write $$\begin{aligned}
E= \int_0^\pi\frac{\d\phi}{\pi} \sum _n \upsilon _n (X_\phi)^\dag \otimes
u_n (X_\phi)\;,\end{aligned}$$ with $$\begin{aligned}
u_n(X_\phi)=\sum _m \upsilon _m (X_\phi) \< n | |Y| |m \>
\;. \end{aligned}$$ The existence of null estimator functions can be taken into account by considering any operator $N_{n,\phi}$ such that $$\begin{aligned}
\int_0^\pi\frac{\d\phi}{\pi} \upsilon _n (X_\phi)^\dag \otimes N_{n,\phi}=0
\;,\end{aligned}$$ and any estimation rule can be obtained by the swap operator $$\begin{aligned}
E= \int_0^\pi\frac{\d\phi}{\pi} \sum _n \upsilon _n (X_\phi)^\dag \otimes
D_{n,\phi}\;,\end{aligned}$$ with $D_{n,\phi}=u_n(X_\phi)+ N_{n,\phi}$, as follows $$\begin{aligned}
\Tr[\rho X] =\int_0^\pi\frac{\d\phi}{\pi} \sum _n \Tr [\rho \upsilon
_n (X_\phi)^\dag ]\Tr [D_{n,\phi} X]\;.
\label{rull}\end{aligned}$$
Spin tomography
---------------
For spin tomography the swap operator writes as follows $$E=\frac{2J+1}{2\pi}\int\frac{\d\vec n}{4\pi}\int_0^{2\pi}\d\psi
\sin^2 \tfrac{\psi}{2} e^{i(\vec J_1-\vec J_2)\cdot\vec n\psi}=
\int\frac{\d\vec n}{4\pi} K(\vec J_1\cdot\vec n \otimes I, I\otimes
\vec J_2\cdot\vec n),$$ and we immediately see that the kernel can be written as follows $$K(r,s)=\tfrac{1}{2}\left(J+\tfrac{1}{2}\right)\< r+J|2-e_+-e_-|s+J\>,$$ where $r,s=-J,-J+1\ldots J$, and $\{|n\> \}$ denote any orthonormal basis of the infinite dimensional Hilbert space $\sH$. Such a basis can be conveniently regarded as the Hardy space of functions on the unit circle, with $\<n|z\>=z^n$, $|z|=1$, and $$\oint \frac{\d z}{2\pi i z} |z\>\< z|\equiv
\int_0^{2\pi}\frac{\d\psi}{2\pi} |e^{i\psi}\>\< e^{i\psi}|,$$ $e_-$ denoting the shift operator $e_-|n\>=|n-1\>$, and $e_+=e_-^\dag$. By introducing the vectors $|\upsilon(m)\>\doteq |m+J\>$, we can write $$E=\int\frac{\d\vec n}{4\pi} (\upsilon(\vec J_1\cdot\vec n)^\dag \otimes
I)(I_{\sH^{\otimes2}}
\otimes K)(I\otimes \upsilon(\vec J_2\cdot\vec n)),$$ where the operator $K\in\Bnd{K}$ is given by $$K=\left(J+\tfrac{1}{2}\right)(1-C),$$ where $C=\frac{1}{2}(e_++e_-)$ is the .
Alternate expansions
--------------------
The general form of the swap operator is $$E=\sum_\nu (\upsilon(X_\nu )^\dag \otimes I)
(I_{\sH^{\otimes 2}}\otimes K)(I \otimes \upsilon(X_\nu)).$$ Introducing an invertible operator $L\in\Bnd{K}$, we can write $$\begin{aligned}
E&=&\sum_\nu (\upsilon(X_\nu )^\dag \otimes I)
(I_{\sH^{\otimes 2}}\otimes K^{\frac{1}{2}}L^{-1}LK^{\frac{1}{2}})(I \otimes \upsilon(X_\nu)),
\nonumber \\&= &
\sum_\nu \sum_{n,m,l}
\upsilon_n(X_\nu)^\dag\otimes\upsilon_m(X_\nu)
\<n|K^{\frac{1}{2}}L^{-1}|z(l)\>\<z(l)|LK^{\frac{1}{2}}|m\>,
\;\end{aligned}$$ where $\{|z(l)\>\}$ is any orthonormal basis for $\sK$. Therefore, we have all the alternate expansions on the quorum $$Z=\sum_\nu \sum_l \Tr[L_l(X_\nu)^\dag Z]M_l(X_\nu),\label{Zexp}$$ where $$\begin{split}
M_l(x)&=\sum_m\upsilon_m(x)\<z(l)|LK^{\frac{1}{2}}|m\>
=\<z(l)|LK^{\frac{1}{2}}|\upsilon(x)\>,\\
L_l(x)^*&=\sum_m\upsilon_m^*(x)\<m|K^{\frac{1}{2}}L^{-1}|z(l)\>=
\<\upsilon(x)|K^{\frac{1}{2}}L^{-1}|z(l)\>.
\end{split}$$
Expansion of unbounded operators over the quadratures
=====================================================
As already noticed, the swap operator in Eq. (\[op2\]) provides the estimation rule just for trace-class operators. However, it is known since Richter [@Rich] the following formula $$\begin{aligned}
a^{\dag n} a^m=
{n+m \choose n}^{-1}\,\int_{0}^\pi \frac {\d\phi }{\pi }
\,\frac {1}{\sqrt {2^{n+m}}} \,H_{n+m}(\sqrt 2 X_\phi )
\,e^{i\phi (m-n)}
\;.\label{ric}\end{aligned}$$ Eq. (\[ric\]) was originally derived by using nontrivial identities involving trilinear products of Hermite polynomials.
Here, we provide a much simpler derivation as follows. Using the definition of Hermite polynomials in Eq. (\[defh\]), one has $$\begin{split}
a^{\dag n} a^m&={n+m \choose n}^{-1}\,\sum _{k=0}^{n+m}{n+m\choose
k}a^{\dag k} a^{n+m-k} \,\delta _{k,n} \\&=
{n+m \choose n}^{-1}\,\int_{0}^\pi \frac {\d\phi }{\pi }
\sum _{k=0}^{n+m}{n+m\choose
k}a^{\dag k} a^{n+m-k} \,e^{i\phi (2k-n-m)}e^{i\phi (m-n)}
\\&=
{n+m \choose n}^{-1}\,\int_{0}^\pi \frac {\d\phi }{\pi }
\,{\bf :}X^{n+m}_\phi {\bf :}
\,e^{i\phi (m-n)}
\\&=
{n+m \choose n}^{-1}\,\int_{0}^\pi \frac {\d\phi }{\pi }
\,\frac {1}{\sqrt {2^{n+m}}} \,H_{n+m}(\sqrt 2 X_\phi )
\,e^{i\phi (m-n)}
\;\label{otto}
\end{split}$$ Similarly, for the symmetrical ordering, one derives the identity $$\begin{aligned}
S\{ a^{\dag n} a^m\} &=& {n+m \choose m }^{-1}\sum _{l=0}^{n+m} {n+m
\choose l} S\{ a^{\dag n+m-l} a^l\} \delta _{lm}
\nonumber \\&= &
{n+m \choose m }^{-1}\int _{0} ^\pi \frac {d\phi }{\pi} e^{i\phi
(m-n)}
\sum _{l=0}^{n+m} {n+m
\choose l} S\{ (a^{\dag }e^{i\phi })^ {n+m-l} (ae^{-i\phi
})^l\}\nonumber \\&= &
{n+m \choose m }^{-1}\int _{0} ^\pi \frac {d\phi }{\pi}
e^{i\phi
(m-n)} \, 2^{n+m}\, X_\phi ^{n+k}
\;.\label{simm}\end{aligned}$$ For arbitrary ordering, using Eq. (\[arb\]), one obtains $$\begin{aligned}
:a^{\dag k} a^l: _s
&=&\sum _{j=0}^{\{k,l\}} \frac{k ! l!}{j! (k-j)!
(l-j)!}\left ( \frac {s}{2}\right )^j S\{a^{\dag k-j}a^{l-j} \}
\nonumber \\&= &
\int _{0} ^\pi \frac {d\phi }{\pi} e^{i\phi (l-k)}
\sum _{j=0}^{\{k,l\}} \frac{k ! l!}{j! (k+l-2j)!} \left (\sqrt {\frac
{s}{2}}\right )^{k+l}
\left( 2\sqrt {\frac 2
s} X_\phi \right )^{k+l-2j}\nonumber \\&= &
{ k+l \choose k }^{-1}
\int _{0} ^\pi \frac {d\phi }{\pi} e^{i\phi (l-k)}
\left (\sqrt {\frac
{s}{2}}\right )^{k+l}
H_{k+l}^{(k,l)}\left (\sqrt {\frac 2s} X_\phi \right )
\;.\end{aligned}$$ Using Lemma 3, one has the equivalent identity $$\begin{aligned}
:a^{\dag k} a^l: _s =
{ k+l \choose k }^{-1}
\int _{0} ^\pi \frac {d\phi }{\pi} e^{i\phi (l-k)}
\left (\sqrt {\frac
{s}{2}}\right )^{k+l}
H_{k+l}\left(\sqrt {\frac 2s} X_\phi \right)
\;.\end{aligned}$$ In a similar way, one can derive the useful relation $$\begin{aligned}
(\mu a + \nu a^\dag )^n =\int _0 ^\pi \frac{d\phi }{\pi} (2 X_\phi )^n \frac{(\nu
e^{-i\phi})^{n+1}- (\mu e^{i\phi})^{n+1}}{\nu e^{-i\phi }-\mu e^{i\phi}}
\;, \end{aligned}$$ whence $$\begin{aligned}
X_\varphi ^n =\int _0 ^\pi \frac{d\phi }{\pi} X_\phi ^n
\frac{\sin [(\phi -\varphi)(n+1)]}{\sin (\phi -\varphi)}
\;.\end{aligned}$$ Using Eq. (\[simm\]), for the displacement operator one obtains $$\begin{aligned}
D(\alpha )&=&\sum _{n=0}^\infty \frac {1}{n !}(\alpha a^\dag -\alpha ^*
a)^n = \sum _{n=0}^\infty \sum _{k=0}^n {n\choose k} \alpha
^{n-k}(-\alpha ^*)^k S\{a^{\dag n-k} a^k\}
\nonumber \\&= &
\int _{0} ^\pi \frac {d\phi }{\pi}
\sum _{k=0}^\infty \sum _{n=0}^\infty \frac {(2 \alpha e^{-i\phi
}X_\phi)^n (-2 \alpha ^* e^{i\phi} X_\phi)^k}{(n+k)!}
\;.\label{estr}\end{aligned}$$ The last equation can be summed using the identity $$\begin{split}
&\sum_{n,m=0}^\infty \frac{z^n(-z^*)^m}{(n+m)!}=\sum_{s=0}^\infty \frac{z^s}{s!}\sum_{d=0}^s\left(
-\frac{z^*}{z}\right)^d=\sum_{s=0}^\infty\frac{z^s}{s!}
\frac{1-\left(-\frac{z^*}{z}\right)^{s+1}}{1+\frac{z^*}{z}}\\
=&\sum_{s=0}^\infty\frac{1}{s!}\frac{z^{s+1}-(-z^*)^{s+1}}{z+z^*} =\frac{ze^z+z^*e^{-z^*}}{z+z^*}
\end{split}$$ which gives the estimation rule $$f_\phi(X_\phi|D(\alpha))=\frac{e^{-i\phi}\alpha e^{2X_\phi e^{-i\phi}\alpha}
+e^{i\phi}\alpha^*e^{-2X_\phi
e^{i\phi}\alpha^*}}{e^{-i\phi}\alpha+e^{i\phi}\alpha^*} \;.
\label{estr2}$$ Identity (\[estr2\]) should be compared with the equivalent estimator given in Ref. [@tokio]. The identity in Eq. (\[estr\]) can be also derived by explicitly using the properties of null estimator functions, as shown in the Appendix.
All estimation rules $f_\phi(X_\phi | X)$ given in the present section do not correspond to an expectation of $X$ as in Eq. (\[rull\]). However, we can suitably recover an expectation rule—which is generally not unique—for any observable. Consider, for example, Eq. (\[estr\]). Using the following identity [@gradshtein] $$\int_0^1\d\theta\theta^n\,(1-\theta)^m=B(n+1,m+1)=\frac{n!m!}{(n+m+1)!},$$ along with $$(m+n+1)=\left.\frac{\d}{\d t} t^{m+n+1}\right|_{t=1},$$ one obtains the integral form for the inverse binomial coefficient $${ m+n \choose m}^{-1}
=\left.\frac{\d}{\d
t}\right|_{t=1}t\,\int_0^1\d\theta(t\theta)^n\,(t(1-\theta))^m .
\label{invbin}$$ Then, the estimator (\[estr\]) becomes $$f_\phi(X_\phi|D(\alpha))=\left.\frac{\d}{\d t}\right|_{t=1}t\, \int_0^1\d\theta
\exp(-2 X_\phi e^{i\phi}\alpha^*t\theta)
\exp(2 X_\phi e^{-i\phi}\alpha t(1-\theta)),$$ and for its spectral kernel one has $$f_\phi(x|D(\alpha))=\frac{\d}{\d x}x\, \int_0^1\d\theta\exp(-2 x e^{i\phi}\alpha^*\theta)
\exp(2 x e^{-i\phi}\alpha (1-\theta)).$$ We can rewrite the estimator in form of expectation $$f_\phi(x|D(\alpha))=\frac{\d}{\d x}x\, \int_0^1\d\theta
\< 2 x e^{i\phi}(1-\theta)|e^{\alpha a^\dag}e^{-\alpha^* a}|2 x e^{i\phi}\theta\>,\label{form}$$ where the vectors are coherent states. Eq. (\[form\]) corresponds to the functional form $$f_\phi(x|D(\alpha))=\Tr[W_{\phi,x}\:D(\alpha)\:],$$ where $$W_{\phi,x}=\frac{\d}{\d x}x\, \int_0^1\d\theta|2 x e^{i\phi}\theta\>\< 2 x e^{i\phi}(1-\theta)|.$$ In order to give an expectation rule corresponding to Eq. (\[otto\]), we can proceed as follows. From the integral representation [@gradshtein] $$H_n(x) = (-2i)^n\int_{-\infty}^\infty \frac{\d t}{\sqrt{\pi}}
e^{-(t-ix)^2}\,t^n\;,
$$ using identity (\[invbin\]), Eq. (\[otto\]) for normal ordering rewrites as follows $$\begin{split}
a^{\dag m} a^n = &\int_0^{\pi}\frac{\d\phi}{\pi}
\int_{-\infty}^\infty\frac{\d t}{\sqrt{2\pi}}
e^{-\frac{1}{2}(t-i2X_\phi)^2}
\frac{\d}{\d t} t \int_0^1\d\theta (-i\theta te^{i\phi})^n
[-i(1-\theta)te^{-i\phi}]^m\\ =&
\int_0^{\pi}\frac{\d\phi}{\pi} \int_{-\infty}^\infty\frac{\d t}{\sqrt{2\pi}}
e^{-\frac{1}{2}(t-i2X_\phi)^2}\\ \times &
\frac{\d}{\d t}t \int_0^1\d\theta \Tr [a^{\dag m} a^n | -i\theta
te^{i\phi}\>\< i(1-\theta)te^{i\phi}|]e^{\frac{t^2}{2}},\label{boring}
\end{split}$$ where we have used matrix elements on coherent states $$\<\beta| a^{\dag m} a^n |\alpha\> =\alpha ^n{\beta ^* m}
e^{-\frac{1}{2}(|\alpha|^2+|\beta|^2-2\alpha\beta^*)}.$$ Eq. (\[boring\]) is equivalent to the following expansion for operators $X$ admitting normal ordered form $$\begin{split}
X=&\int_0^{\pi}\frac{\d\phi}{\pi} \int_{-\infty}^\infty\d t
\Tr[G^\dag(t,\phi)X]F(t,\phi),\\
F(t,\phi)=&\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}(t-i2X_\phi)^2},\quad
G(t,\phi)= \frac{\d}{\d t} t e^{\frac{t^2}{2}}
\int_0^1\d\theta | i(1-\theta)te^{i\phi}\>\< -i\theta te^{i\phi}|.
\end{split}$$
Canonical dual for homodyne tomography
======================================
The frame theory approach to quantum homodyne tomography gives further insight to the structure of the [*quorum*]{} of quadrature observables. Given a set of vectors ${|v_n \> }$ in a Hilbert space, if the positive operator $F=\sum
_{n}|v_n \> \< v_n |$ is invertible, then the scalar product between two arbitrary vectors can be written as $$\begin{aligned}
\< \psi | \eta \> = \sum _n \< \psi | u_n \>
\< v_n | \eta \> \;,\end{aligned}$$ where the set of vectors $\{u_n \equiv F^{-1}|v_n \> \}$ is called “canonical dual” of the set $\{ |v_n \> \}$, and $F$ is denoted as “frame operator”. In other words, the set $\{|v_n \> \}$, along with its dual $\{|u_n \> \}$, is a spanning set for the Hilbert space, and provide a generalized resolution of the identity. In this section we show that the set of (generalized) projectors $|x \> _\phi {}_\phi \< x| $ over the quadratures $X_\phi $ give a frame when varying $\phi $, and the expansion for trace-class operators in Eq. (\[op\]) corresponds to using the canonical dual for the estimation rule.
In the following, we will make extensive use of the isomorphism between the Hilbert space of the Hilbert-Schmidt operators $A,B$ on $\sH$, with scalar product $\< A,B\>
=\hbox{Tr}[A^\dag B]$, and the Hilbert space of bipartite vectors $|A{\>\!\>},|B{\>\!\>}\in {\sH}\otimes{\sH}$, with ${\<\!\<} A|B{\>\!\>}\equiv\<
A,B\> $, and $$\begin{aligned}
|A{\>\!\>}=\sum_{n}
\sum_{m}
A_{nm}|n\> \otimes|m\> \;,\label{iso} \end{aligned}$$ where $|n\>$ and $|m\>$ are fixed orthonormal bases for $\sH$, and $A_{nm}=\< n|A|m \>
$. Notice the identities $$\begin{aligned}
A\otimes B{| \, C \rangle\!\rangle}={| \, ACB^\tau \rangle\!\rangle}\,,\qquad A\otimes B^\dag {| \, C \rangle\!\rangle}={| \, ACB^* \rangle\!\rangle}\,,
\label{ids}\end{aligned}$$ where $\tau$ and $*$ denote transposition and complex conjugation with respect to the fixed bases in Eqs. (\[iso\]).
By taking $\< n|x \> _0$ as real, in the $| \ \kk $ notation the spanning set $|x \> _\phi {}_\phi \< x|$ corresponds to the vectors on $\sH _a \otimes \sH _b$ of modes $a$ and $b$ $$| (|x \> _\phi {}_\phi \<
x|) \kk = e^{i\phi (a^\dag a -b^\dag b)}| x \> _0 |x \> _0
\;.$$ Notice the identities $$\bb D(z)| x \> _0 |x \> _0 =
\exp (2i x \hbox{Im} z) \,\delta
(\hbox{Re} z) \;,$$ and $$e^{i \phi (a^\dag a -b^\dag b)}|D(z) \kk =|D(z e^{i\phi}) \kk \;.$$ The frame operator can be evaluated as follows $$\begin{split}
F&=\int _0^\pi \frac{\d\phi }{\pi }\int _{-\infty}^{\infty} \d x\,
e^{i\phi (a^\dag a -b^\dag b)}| x \> _0 |x \> _0 {}_0\<
x |{}_0 \< x | e^{-i\phi (a^\dag a -b^\dag b)} \\ &=
\int _0^\pi \frac{\d\phi }{\pi }\int _{-\infty}^{\infty} \d x\, \int
\frac{\d^2 z }{\pi }\int \frac{\d^2 w}{ \pi} |D(z) \kk \bb
D(ze^{-i\phi})| x \> _0 |x \> _0
{}_0\< x |{}_0 \< x | D(we^{-i\phi }) \kk \bb D(w)|
\\ & = \int _0^\pi \frac{\d\phi }{\pi }\int
_{-\infty}^{\infty} \d x\, \int \frac{\d^2 z }{\pi }\int \frac{\d^2 w}{
\pi} |D(ze^{i\phi}) \kk \bb D(we^{i\phi }) | e^{2ix
\hbox{\scriptsize Im}z - 2ix
\hbox{\scriptsize Im}w } \delta
(\hbox{Re}z) \delta (\hbox{Re}w) \\ &= \frac 1 \pi \int
_0^\pi \frac{\d\phi }{\pi }\int _{-\infty}^{\infty} dt\, |D(it e^{i\phi
}) \kk \bb D(ite^{i\phi })| = \frac {1}{\pi} \int\frac{\d^2 z}{\pi}\frac {1}{|z|}
|D(z) \kk
\bb D(z)|\\&= \frac{1}{\pi|a-b^\dag|} \;, \label{4}
\end{split}$$ where we used the eigenvalue equation $(a- b^\dag )|D(z) \kk =z |D(z)
\kk $. The inverse of $F$ is simply given by $$F^{-1}=\pi|a-b^\dag|=\int \d^2 z \,|z|\,
|D(z) \kk \bb D(z)|\;.\label{5}$$ The canonical dual is obtained as follows $$\begin{split}
&F^{-1} | (|x \> _\phi {}_\phi \< x|) \kk =
\int \d^2 z \,|z| |D(z) \kk \bb D(z)| x \> _\phi |x \>
_\phi \\&=\int \d^2 z \,|z|\, |D(ze^{i\phi}) \kk
e^{2i x \hbox{\scriptsize Im}z }\delta (\hbox{Re}z) =
\int_{-\infty }^\infty \d k\,|k|\,|D(ike^{i\phi}) \kk e^{2ikx}
\\ & =\frac 14 \int_{-\infty }^\infty
\d k\,|k|\,|e^{ik(X_\phi -x)} \kk \;.\label{6}
\end{split}$$ Hence, it follows that the usual Kernel operator corresponds to the canonical dual.
Alternate dual frames
---------------------
The dual of the quadrature projectors is not unique. However, the formula of Li [@Li] for characterize all possible alternate duals for bounded frames and discrete indexes cannot provide any new dual set. By denoting the frame as $\{|\Xi(x,\phi)\>\!\>\}$ with $\Xi(x,\phi) = \delta (X_\phi
-x)$, such a formula can be formally written in the form $$|\Theta(x,\phi)\>
\!\>=F^{-1}|\Xi(x,\phi)\>\!\>+|f(x,\phi)\>\!\>-
\int_0^\pi \frac{\d\phi'}{\pi} \int_{-\infty}^\infty d x'
\<\!\<\Xi(x',\phi')|F^{-1}|\Xi(x,\phi)\>\!\>
|f(x',\phi')\!\>\,,
\label{allduals}$$ where $\{F^{-1}|\Xi(x,\phi)\>\!\>\}$ is the canonical dual, and $\{f(x,\phi)\}$ is an arbitrary Bessel set, namely $$\int_{-\infty}^\infty \d x\int_0^\pi \frac{\d\phi}{\pi} |f(x,\phi)|^2\le \infty.$$ The scalar product that appears in the integral of Eq. (\[allduals\]) writes $$\begin{aligned}
&&\<\!\<\Xi(x',\phi')|F^{-1}|\Xi(x,\phi)\>\!\>
\\& & =
\int d^2 z \,|z| {}_0 \< x' | {}_0 \< x'|
|D(z) \kk \bb D(ze^{i(\psi -\phi)})| x \> _0 |x \> _0
\nonumber \\& & =\int d^2 z \,|z|\,
e^{-i \hbox{\scriptsize Im}z (\hbox{\scriptsize Re}z+2x')}\,\delta
(\hbox{Re}z)
e^{i \hbox{\scriptsize Im}(ze^{i(\psi -\phi)})(\hbox{\scriptsize
Re}(ze^{i(\psi -\phi)}+2x)}\,\delta (\hbox{Re}(ze^{i(\psi -\phi)}))
\nonumber \\& &=
\int _{-\infty}^{\infty }dk\,|k|\,
e^{2i k[x\cos (\psi -\phi )-x']}
\,\delta(k\sin (\psi -\phi))
\nonumber \\& &=
\int _{-\infty}^{\infty }dk\,
e^{2i k[x\cos (\psi -\phi )-x']}
\,\delta_\pi (\psi -\phi)
\nonumber \\& & =
\pi \,\delta(x-x')\,\delta_\pi (\psi -\phi)
\;.\label{wr}\end{aligned}$$ This bi-orthogonality relation implies that the formula (\[allduals\]) cannot reveal any new dual set.
Frames of normal-ordered moments
================================
In this Section, by simply applying the frame theory, we recover some results of Refs. [@Wun1; @Lee], where the set of normally ordered moments $\{a^{\dag k}a^l\}$ is shown to be complete, and related to a biorthogonal set given on the basis of Fock states.
From the set $\{a^{\dag k}a^l\}$ we immediately write the frame operator $$\tilde F=\sum_{k,l=0}^\infty {| \, a^{\dag k}a^l \rangle\!\rangle} {\<\!\< a^{\dag k}a^l \, |}
\;.\label{fnorm}$$ On the Fock basis one has $$\begin{split}
\tilde F&=
\sum_{k,l,n,j=0}^\infty \frac{\sqrt{(k+n)!(l+n)!(k+j)!(l+j)!}}{n! j!}
|k+n \> |l+n \> \< k+j |\< l+j | \\ &=\sum_{n=0}^\infty \frac{a^nb^n }{n!}
(\sum_{k,l=0}^\infty k! l!|k \> \< k|\otimes |l \>
\< l |) \sum_{j=0}^\infty \frac{a^{\dag j}b^{\dag j}}{j!} \\&=
e^{ab} (a^\dag a !\otimes b^\dag b !) e^{a^{\dag }b^{\dag}}.
\end{split}$$ The inverse of $\tilde F$ simply writes $$\tilde F^{-1}=e^{-a^\dag b^\dag}
\left(\frac {1}{a^\dag a !}\otimes \frac{1}{b^\dag b!} \right) e^{-ab}
\;.$$ The dual set $g_{k,l}$ can be obtained as follows $$\begin{split}
{| \, g_{k,l} \rangle\!\rangle}&\equiv \tilde F^{-1}{| \, a^{\dag k}a^l \rangle\!\rangle}
=\sum_{n,m,t,j=0}^{\infty }
|n \> |m \> \frac{(-1)^{t+j}}{t!j!} \< n|\< m|
a^t b^t \frac{1}{a^\dag a!}\otimes \frac{1}{b^\dag b!} a^{\dag j}b^{\dag
j}{| \, a^{\dag k}a^l \rangle\!\rangle}
\\& = \sum_{n,m,t,j,s=0}^{\infty }
|n \> |m \> \frac{(-1)^{t+j}}{t!j!} \< n+t|k+j+s
\> \< m+t| l+j+s \>
\frac{\sqrt{(k+j+s)!(l+j+s)!}}{s!\sqrt{n!m! (n+t)!(m+t)!}}
\\& = \sum_{n,m,t,j,s=0}^{\infty }
|n \> |m \> \frac{(-1)^{t+j}}{t!j!} \delta
_{n+t,k+j+s}\delta _{m+t,l+j+s}
\frac{\sqrt{(k+j+s)!(l+j+s)!}}{s!\sqrt{n!m! (n+t)!(m+t)!}}
\\&= \sum_{n,m,t=0}^{\infty }
\frac{1}{\sqrt {n!m!}}|n \> |m \> \frac{(-1)^{t}}{t!}
\delta_{m,l+n-k} \frac{1}{(n+t-k)!}
\sum _{j=0}^{n+t-k}\frac{(-1)^{j}}{j!}\frac{(n+t-k)!}{(n+t-k-j)!}
\\& =\sum_{n,m,t=0}^{\infty }
\frac{1}{\sqrt {n!m!}}|n \> |m \> \frac{(-1)^{t}}{t!}
\delta _{n,k-t}\delta_{m,l-t} \frac{1}{(n+t-k)!}
\\&= \sum_{t=0}^{\min{(k,l)}}
\frac{(-1)^{t}}{t!\sqrt {(k-t)!(l-t)!}}|k-t \> | l-t \>
\;.
\end{split}$$ The dual set is then given by $$g_{k,l}=\sum_{t=0}^{\min{(k,l)}}
\frac{(-1)^{t}}{t!\sqrt {(k-t)!(l-t)!}}|k-t \> \< l-t | =\sum_{t=0}^\infty
\frac{(-1)^t}{t!} a^\dag{}^{k-t}|0\> \< 0|a^{l-t}.$$ The dual set is unique, and in fact one has the biorthogonal relation $$\hbox{Tr}[g^{\dag} _{k',l'}a^{\dag k}a^l]=\delta _{k,k'}\delta _{l,l'}.
\;$$
Frame of moments versus quadrature distribution {#versus}
===============================================
The frame of moments allows to recover the estimation rule for unbounded operators as an expectation rule with a dual operator of the quadrature projectors $|x \> _\phi {}_\phi \< x|$. In other words, by inspecting Eq. (\[otto\]), we would like to write an operator $G(x,\phi
)$ such that $$\hbox{Tr}[G^\dag (x,\phi )a^{\dag n}a^m]=
{n+m \choose n}^{-1}
\,\frac {1}{\sqrt {2^{n+m}}} \,H_{n+m}(\sqrt 2 x)
\,e^{i\phi (m-n)}
\;.$$ In this case, the operator $G(x,\phi ) $ is a dual of the quadrature projectors $|x \> _\phi {}_\phi \< x|$, but is different from the canonical dual (\[6\]), which is divergent for unbounded operators. One has $$\begin{split}
{| \, G^\dag (x, \phi ) \rangle\!\rangle}
&= \tilde F^{-1}\tilde F {| \, G^\dag (x, \phi ) \rangle\!\rangle}=
\tilde F^{-1} \sum_{k,l=0}^\infty
{| \, a^{\dag k}a^l \rangle\!\rangle}{\<\!\< a^{\dag k}a^l \, |}
{| \, G^\dag (x,\phi) \rangle\!\rangle} \\& =
\sum_{k,l=0}^\infty
{| \, g_{k,l} \rangle\!\rangle} \hbox{Tr}[a^{\dag l}a^k G^\dag (x,\phi)]\;.
\end{split}$$ Then we obtain $$G^\dag (x,\phi)= e^{i\phi a^\dag a } G^{\dag }(x,0) e^{-i\phi
a^\dag a}\;,$$ with $$\begin{split}
G^\dag (x,0)&= \sum_{k,l=0}^\infty
g_{k,l} {k+l \choose k}^{-1}
\,\frac {1}{\sqrt {2^{k+l}}} \,H_{k+l}(\sqrt 2 x)
\\&= \sum_{k,l=0}^\infty \sum_{t=0}^{\min (k,l)}
\frac{(-1)^t}{t! \sqrt {k! l!}}a^t |k \> \< l| a^{\dag t}
{k+l \choose k}^{-1}
\,\frac {1}{\sqrt {2^{k+l}}} \,H_{k+l}(\sqrt 2 x)\\&=
\sum_{k,l=0}^\infty \sum_{t=0}^{\min (k,l)}
\frac{(-1)^t}{t! \sqrt {(k-t)! (l-t)!}} |k -t\> \< l-t|\,
{k+l \choose k}^{-1}
\,\frac {1}{\sqrt {2^{k+l}}} \,H_{k+l}(\sqrt 2 x)
\;.
\end{split}$$ The dual $G(x,\phi )$ provides also new pattern functions for the matrix elements, as previously noticed in Refs. [@Rich; @Wun2]. For example, for the vacuum component one has $$\hbox{Tr}[G^\dag (x,\phi ) |0 \> \< 0| ]=
\sum_{k=0}^\infty \left ( - \frac 12\right )^k \frac{k!}{(2k)!}
H_{2k}(\sqrt 2 x)\;.$$ Notice that such pattern functions are generally no longer bounded for $x \to \pm \infty $, even for $\eta > 0.5$.
One can check that the set $G(x,\phi )$ is dual to the set of quadrature projectors $|x \> _\phi {}_\phi \<
x|$ as follows $$\begin{split}
&\int _{-\infty}^{+\infty }\d x \int_{0}^\pi \frac{\d\phi }{\pi }
{| \, \delta (X_\phi -x) \rangle\!\rangle} {\<\!\< G(x,\phi ) \, |} \\&
=\int _{-\infty}^{+\infty }\d x \int _{-\infty}^{+\infty }\d x'
\int_{0}^\pi \frac{\d\phi }{\pi }
{| \, |x' \> _\phi {}_\phi \< x'| \rangle\!\rangle} {\<\!\< G(x,\phi ) \, |}
\,\delta (x-x') \\ &
=\int_{0}^\pi \frac{\d\phi }{\pi }\sum_{k,l=0}^\infty
{k+l \choose k}^{-1}
\,\frac {1}{\sqrt {2^{k+l}}}
e^{i\phi (k-l)}\,{| \, H_{k+l}(\sqrt 2 X_\phi ) \rangle\!\rangle} {\<\!\< g^\dag
_{k,l} \, |}
\\ &
=\sum_{k,l=0}^\infty {| \, a^{\dag l}a^k \rangle\!\rangle} {\<\!\< g^\dag _{k,l} \, |}
=\sum_{k,l=0}^\infty {| \, a^{\dag l}a^k \rangle\!\rangle} {\<\!\< g_{l,k} \, |}= \tilde F \tilde
F^{-1}=I\otimes I\;.
\end{split}$$
Generating new frames
=====================
We can generate different frames by changing the function of $a-b^\dag$ which gives the frame operator in Eq. (\[4\]). Explicitly, we have $$\begin{split}
&f(|a-b^\dag|)=\int \frac {\d^2 z}{\pi}
f(|z|) |D(z) \kk \bb D(z)|= \int _0^\pi \frac{\d\phi }{\pi }
\int _{-\infty}^{\infty} dt\, |t|\,f(|t|)\, |D(it e^{i\phi
}) \kk \bb D(ite^{i\phi })| \\ =&
\int _0^\pi \frac{\d\phi }{\pi }\int
_{-\infty}^{\infty}\frac{\d x}{\pi}\, \int \d^2 z\int \d^2 w
|D(ze^{i\phi}) \kk \bb D(we^{i\phi }) |
\\ \times &
g(|z|)g^*(|w|) e^{2ix \hbox{\scriptsize Im}z - 2ix
\hbox{\scriptsize Im}w } \delta (\hbox{Re}z) \delta (\hbox{Re}w)\\
=&\pi \int _0^\pi \frac{\d\phi }{\pi }\int _{-\infty}^{\infty} \d x\, \int
\frac{\d^2 z }{\pi }\int \frac{\d^2 w}{ \pi} |D(z) \kk \bb
D(ze^{-i\phi})|g(|a^\dag -b|)| x \> _0 |x \> _0
\\ \times &
{}_0\< x |{}_0 \< x |g^*(|a^\dag-b|)| D(we^{-i\phi }) \kk \bb D(w)|\\
=&\int _0^\pi \frac{\d\phi }{\pi }\int _{-\infty}^{\infty} \d x\,
e^{i\phi (a^\dag a -b^\dag b)}\sqrt{\pi}g(|a^\dag-b|)| x \> _0 |x \> _0 {}_0\<
x |{}_0 \< x |\sqrt{\pi}g^*(|a^\dag-b|) e^{-i\phi (a^\dag a -b^\dag b)},
\end{split}$$ where the functions $g$ and $f$ are related as follows $$|t| f(|t|)=|g(t)|^2.$$ Using the identity $$(a-b^\dag)|A\kk=(|aA\kk-|Aa^*\kk)=|[a,A]\kk,$$ for reference basis such that $a^*\equiv a$, e. g. the photon number basis, we have $$|a-b^\dag|^2|A\kk=(a^\dag-b)|[a,A]\kk=|[a^\dag,[a,A]]\kk.$$ But the double commutator can be written in terms of the (dual) Lindblad super-operator $$[a^\dag,[a,A]]=-(\map{L}[a]+\map{L}[a^\dag])A,$$ where $L[W]A\doteq W^\dag A W-\frac{1}{2}(W^\dag W A +AW^\dag W)$. Remarkably, this is exactly the dissipative super-operator of the displacement Gaussian noise, corresponding to a distributed loss compensated by a phase-insensitive amplification.
With the aid of the following commutator rule $$[a^\dag,[a,e^{i\lambda X}]]=\tfrac{1}{4}\lambda^2e^{i\lambda X},$$ we easily obtain $$|a^\dag -b|^2|x\>_0|x\>_0=|a -b^\dag|^2|x\>_0|x\>_0=-\tfrac{1}{4}\partial_x^2
|\delta(x-X)\kk=|\mathcal{F}[\tfrac{1}{4}\lambda^2](X-x)\kk ,$$ where $\mathcal{F}$ denotes the Fourier transform $$\mathcal{F}[f](x)=\int_{-\infty}^\infty \frac{d \lambda}{2\pi}e^{i\lambda x}f(\lambda).$$ Therefore, we have $$g(|a^\dag -b|)|x\>_0|x\>_0=|\mathcal{F}[g(\tfrac{1}{2}\lambda)](X-x)\kk,$$ corresponding to the frame $$\Xi(x,\phi)=\sqrt{\pi}\mathcal{F}[g(\tfrac{1}{2}\bullet)](X_\phi-x).$$ For example, if we choose $g(x)=\sqrt{\frac{\sigma^2}{2\pi}}\exp\left(-\frac{\sigma^2}{2}x^2\right)$, we have $$\Xi(x,\phi)=\sqrt{2\pi}\exp\left[-\frac{1}{\sigma^2}(X_\phi-x)^2\right].$$ Notice that a function $$\Xi(x,\phi)=h(X_\phi-x)$$ corresponds to a frame if the function $h$ has Fourier transform which is invertible and bounded. Moreover two functions $h$ and $h'$ will correspond to the same frame operator if their Fourier transform have the same module. More precisely, the frame operator will be given by $$F=f(|a-b^\dag|),\qquad f(t)=\frac{1}{\pi t
}\left|\mathcal{F}^{-1}[h](2t)\right|^2 \;.$$ The canonical dual frame operator can be obtained by inverting the frame operator $$\begin{split}
&F^{-1} | (|x \> _\phi {}_\phi \< x|) \kk =
\int\frac{\d^2 z}{\pi} \,\frac{1}{f(|z|)}|D(z) \kk \bb D(z)| x \> _\phi |x \>
_\phi \\ & =\int\frac{\d^2 z}{\pi} \,\frac{1}{f(|z|)} |D(ze^{i\phi}) \kk
e^{2i x \hbox{\scriptsize Im}z }\delta (\hbox{Re}z) =
\int_{-\infty }^\infty \d k\,\frac{1}{f(|k|)}\,|D(-ike^{i\phi}) \kk e^{2ikx}
\\ &=\pi\int_{-\infty }^\infty \frac{d k}{2\pi}\,\frac{1}{f(|k|/2)}\,| e^{-ikX_\phi}\kk e^{ikx}
=|h^\vee(x-X_\phi)\kk,
\end{split}$$ where $$h^\vee(x)=\pi \mathcal{F}\left[\frac{1}{f(|k|/2)}\right](x).$$ Hence, it follows that the usual Kernel operator corresponds to the canonical dual.
Other frames
============
Using frame calculus, it is easy to show that the following sets of operators are spanning sets $$\begin{aligned}
&&A_{n,\phi }=e^{-\frac 12 X_\phi ^2}\, X_\phi ^n
\nonumber \\& &
B_{n,\phi }=\left( \frac{2}{\pi}\right )^{\frac 14}{\frac {1}{\sqrt {2^n
n!}}}\,e^{-X_\phi ^2}H_n (\sqrt 2 X_\phi)
\;,\end{aligned}$$ with corresponding frame operators $$\begin{aligned}
&&\int_{0}^\pi \frac{\d\phi }{\pi }\sum _{n=0}^\infty |A_{n,\phi} \kk
\bb A_{n,\phi}|=\frac{e^{-|Z|^2}}{|Z|}\;,
\nonumber \\& &
\int_{0}^\pi \frac{\d\phi }{\pi }\sum _{n=0}^\infty |B_{n,\phi} \kk
\bb B_{n,\phi}|=\frac{1}{\pi |Z|}\;, \end{aligned}$$ where $Z=a-b^\dag$.
Appendix {#appendix .unnumbered}
========
Proof of Lemma 4
----------------
Consider the Poisson form of the Dirac delta for the $2\pi$-interval $$\delta_{2\pi}(\phi)=\lim_{\epsilon\to 1^-}\frac{1}{2\pi}\sum_{n=-\infty}^{+\infty}\epsilon^{|n|}e^{in\phi}.$$ In the following we will use $\epsilon$ to mean $\epsilon=1_-$. Rescaling $\phi$ by a factor 2 we obtain $$\delta_\pi(\phi)=\frac{1}{\pi}\sum_{n=-\infty}^{+\infty}\epsilon^{|n|}e^{i2n\phi}
\equiv\frac{1}{\pi}\sum_{n=-\infty}^{+\infty}\epsilon^{2|n|}e^{i2n\phi}\doteq\delta_\pi^{(+)}(\phi),$$ which is also equivalent to the [*even folding*]{} relation $$\delta_\pi^{(+)}(\phi)=\delta_{2\pi}(\phi)+\delta_{2\pi}(\phi+\pi),$$ since $$\begin{split}
\delta_{2\pi}(\phi)+\delta_{2\pi}(\phi+\pi)=&
\frac{1}{2\pi}\sum_{n=-\infty}^{+\infty}\epsilon^{|n|}\left[e^{in\phi}+(-)^ne^{in\phi}\right]\\
=&\frac{1}{\pi}\sum_{n=-\infty}^{+\infty}\epsilon^{2|n|}e^{i2n\phi}
\end{split}$$ On the other hand, we have the [*odd folding*]{} relation $$e^{i\phi}\delta_\pi^{(-)}(\phi)=\delta_{2\pi}(\phi)-\delta_{2\pi}(\phi+\pi)$$ since $$\begin{split}
&\delta_{2\pi}(\phi)-\delta_{2\pi}(\phi+\pi)=
\frac{1}{2\pi}\sum_{n=-\infty}^{+\infty}\epsilon^{|n|}\left[e^{in\phi}-(-)^ne^{in\phi}\right]\\
=&\frac{1}{\pi}\sum_{n=-\infty}^{+\infty}\epsilon^{|2n+1|}e^{i(2n+1)\phi}\doteq
e^{i\phi}\delta_\pi^{(-)}(\phi)
\end{split}$$ From the equivalence relations (\[mainid\]), we immediately derive the equivalence $$\begin{split}\label{x2kd}
&x^{2k}\delta_\pi(\phi)\simeq\frac{1}{\pi} x^{2k}\sum_{n=-k}^k\epsilon^{|n|}e^{2in\phi}
=\frac{1}{\pi} x^{2k}\left[\frac{1-(\epsilon e^{2i\phi})^{k+1}}{1-\epsilon e^{2i\phi}}+
\frac{1-(\epsilon e^{-2i\phi})^{k+1}}{1-\epsilon e^{-2i\phi}}-1\right]\\
&=\frac{1}{\pi} x^{2k}\left[\frac{(\epsilon e^{2i\phi})^{k+1}}{\epsilon e^{2i\phi}-1}+
\frac{(\epsilon e^{-2i\phi})^{k+1}}{\epsilon e^{-2i\phi}-1}+\kappa_\epsilon(\phi)\right],
\end{split}$$ where the distribution $$\kappa_\epsilon(\phi)\doteq\frac{1-\epsilon^2}{1+\epsilon^2-\epsilon(e^{2i\phi}+e^{-2i\phi})}$$ with support in $\phi=0$ gives $$\begin{split}
&\int_0^\pi \frac{\d\phi}{\pi}\kappa_\epsilon(\phi)e^{i2n\phi}=
\int_0^{2\pi}\frac{\d\phi}{2\pi}\frac{1-\epsilon^2}{1+\epsilon^2-\epsilon(e^{i\phi}+e^{-i\phi})}
e^{in\phi}\\=&\frac{1}{2\pi i}\oint\frac{\d z}{z}\frac{1-\epsilon^2}{(1-\epsilon z)(1-\epsilon z^{-1})}z^n=
\frac{1}{2\pi i}\oint\d
z\frac{\epsilon-\epsilon^{-1}}{(z-\epsilon^{-1})(z-\epsilon)}z^n= : \vartheta(n)
\end{split}$$ where $\vartheta(n)=1$ for $n\ge 0$ and $\vartheta(n)=0$ for $n< 0$, and the integral is performed on the unit circle. Eq. (\[x2kd\]), which contains identity (\[dicomb\]) as a special case, generalizes as follows $$f(x^2)\delta_\pi(\phi)\simeq
\frac{1}{\pi}\left[\frac{f(\epsilon x^{2}e^{2i\phi})}{1-\epsilon^{-1}e^{-2i\phi}}+
\frac{f(\epsilon x^{2}e^{-2i\phi})}{1-\epsilon^{-1}e^{2i\phi}}+\kappa_\epsilon(\phi)f(x^2)\right],$$ where $f(z)$ denotes any analytic function in $z$. In a similar way we calculate $$\begin{split}
&x^{2k+1}e^{i\phi}\delta_\pi(\phi)\simeq\frac{1}{\pi} x^{2k+1}\sum_{n=-k-1}^k\epsilon^{|n|}
e^{i(2n+1)\phi}\\
=&\frac{1}{\pi} x^{2k+1}e^{i\phi}\left[\frac{1-(\epsilon e^{2i\phi})^{k+1}}{1-\epsilon e^{2i\phi}}+
\frac{1-(\epsilon e^{-2i\phi})^{k+2}}{1-\epsilon e^{-2i\phi}}-1\right]\\
&=\frac{1}{\pi} x^{2k+1}e^{i\phi}\left[\frac{(\epsilon e^{2i\phi})^{k+1}}{\epsilon e^{2i\phi}-1}+
\frac{(\epsilon e^{-2i\phi})^{k+2}}{\epsilon e^{-2i\phi}-1}+\kappa_\epsilon(\phi)\right],
\end{split}$$ which generalizes as follows $$xf(x^2)e^{i\phi}\delta_\pi(\phi)\simeq \frac{1}{\pi}x
\left[\frac{e^{i\phi}f(\epsilon x^2e^{2i\phi})}{1-\epsilon^{-1} e^{-2i\phi}}+\frac{e^{-i\phi}
f(\epsilon x^2e^{-2i\phi})}{1-\epsilon^{-1}e^{2i\phi}}+\kappa_\epsilon(\phi)e^{i\phi}f(x^2)
\right].$$
Alternative derivation of identity (\[estr\])
---------------------------------------------
By posing $\alpha=\frac{i}{2}re^{i\phi}$, we have $$\begin{split}
D(\alpha)=&\int_{-\pi}^\pi \frac{\d\phi}{2\pi}2\pi\delta_{2\pi}(\phi-\phi) e^{ir X_\phi}\\
=&\int_0^\pi \frac{\d\phi}{\pi}\pi\left[\delta_{2\pi}(\phi-\phi)e^{ir X_\phi}+\delta_{2\pi}
(\phi-\phi-\pi)e^{-ir X_\phi}\right]\\
=&\int_0^\pi \frac{\d\phi}{\pi}\pi\left[\cos rX_\phi(\delta_{2\pi}(\phi-\phi)+\delta_{2\pi}
(\phi-\phi-\pi))\right.\\
+&\left.i\sin rX_\phi(\delta_{2\pi}(\phi-\phi)-\delta_{2\pi}(\phi-\phi-\pi))\right]\\
=&\int_0^\pi \frac{\d\phi}{\pi}\pi\left[\cos rX_\phi\delta_\pi^{(+)}(\phi-\phi)
+i\sin rX_\phi\delta_\pi^{(-)}(\phi-\phi)\epsilon e^{i(\phi-\phi)}\right].
\end{split}$$ Now, we evaluate separately the cosine and the sine terms. In the following, we will denote $\psi=\phi-\phi$. The cosine term can be transformed as follows $$\begin{split}
&\cos rX_\phi\pi\delta_\pi^{(+)}(\psi)
\simeq\sum_{k=0}^\infty \frac{(-)^k}{(2k)!}r^{2k}X_\phi^{2k}\sum_{n=-k}^k
\epsilon^{2|n|}e^{2i n\psi}\\=&\sum_{k=0}^\infty \frac{(-)^k}{(2k)!}r^{2k}X_\phi^{2k}
\left[\sum_{n=0}^k\epsilon^{2n}\left(e^{2in\psi}+e^{-2in\psi}\right)-1\right]\\
=&\sum_{n=0}^\infty\epsilon^{2n}\left(e^{2in\psi}+e^{-2in\psi}\right)
\sum_{k=n}^\infty \frac{(-)^k}{(2k)!}r^{2k}X_\phi^{2k}-\cos(rX_\phi)\\
=&\sum_{n=0}^\infty\epsilon^{2n}\left(e^{2in\psi}+e^{-2in\psi}-\delta_{n0}\right)
(-)^nr^{2n}X_\phi^{2n}\sum_{k=0}^\infty \frac{(-)^kr^{2k}X_\phi^{2k}}{(2k+2n)!}
\end{split}$$ On the other hand, the sine term transforms as follows $$\begin{split}
&\sin rX_\phi\pi\delta_\pi^{(-)}(\psi)\epsilon e^{i\psi}\simeq\sum_{k=0}^\infty \frac{(-)^kr^{2k+1}X_\phi^{2k+1}}{
(2k+1)!}\sum_{n=-k-1}^k\epsilon^{|2n+1|}e^{i(2n+1)\psi}\\=&
\sum_{k=0}^\infty \frac{(-)^kr^{2k+1}X_\phi^{2k+1}}{(2k+1)!}\left(\sum_{n=0}^k\epsilon^{2n+1}
e^{i(2n+1)\psi}+\sum_{n=1}^{k+1}\epsilon^{2n-1} e^{-i(2n-1)\psi}\right)\\=&
\sum_{k=0}^\infty \frac{(-)^kr^{2k+1}X_\phi^{2k+1}}{(2k+1)!}\left(\sum_{n=0}^k\epsilon^{2n+1}
e^{i(2n+1)\psi}+\sum_{n=0}^k\epsilon^{2n+1} e^{-i(2n+1)\psi}\right)\\=&
\sum_{n=0}^\infty\epsilon^{2n+1}\left(e^{i(2n+1)\psi}+e^{-i(2n+1)\psi}\right)
\sum_{k=n}^\infty\frac{(-)^kr^{2k+1}X_\phi^{2k+1}}{(2k+1)!}\\=&
\sum_{n=0}^\infty\epsilon^{2n+1}\left(e^{i(2n+1)\psi}+e^{-i(2n+1)\psi}\right)r^{2n}X_\phi^{2n}
\sum_{k=0}^\infty\frac{(-)^kr^{2k+1}X_\phi^{2k+1}}{(2k+2n+1)!}
\end{split}$$ It is convenient now to make the following substitutions $$r=2|\alpha|,\qquad e^{-i\phi }=i \frac{\alpha^*}{|\alpha|},\qquad e^{i\psi}=
i\frac{\alpha^*}{|\alpha|}e^{i\phi},\qquad e^{i\psi}r =2i\alpha e^{i\phi}.$$ In this way, the cosine term becomes $$\begin{split}
&\cos rX_\phi\pi\delta_\pi^{(+)}(\psi)\\ \simeq &
\sum_{n=0}^\infty\left((\epsilon\alpha^*)^{2n}e^{i2n\phi}+(\epsilon\alpha)^{2n}e^{-i2n\phi}-\delta_{n0}
\right)(2X_\phi)^{2n}\sum_{k=0}^\infty \frac{(-)^k(2|\alpha|X_\phi)^{2k}}{(2k+2n)!},
\end{split}$$ and the sine term $$\begin{split}
&i\sin rX_\phi\pi\delta_\pi^{(-)}(\psi)\epsilon e^{i\psi}\\ \simeq &
\sum_{n=0}^\infty\left(-(\epsilon\alpha^*)^{2n+1}e^{i(2n+1)\phi}+(\epsilon\alpha)^{2n+1}e^{-i(2n+1)\phi}
\right)(2X_\phi)^{2n+1}\sum_{k=0}^\infty\frac{(-)^k(2|\alpha|X_\phi)^{2k}}{(2k+2n+1)!},
\end{split}$$ and putting the two terms together we have $$D(\alpha)=\int_0^\pi \frac{\d\phi}{\pi}f_\phi^\epsilon(X_\phi|D(\alpha)),$$ with $$\begin{split}
&f_\phi^\epsilon(X_\phi|D(\alpha))=\sum_{n=0}^\infty
\left((-\epsilon\alpha^*)^n e^{in\phi}+(\epsilon\alpha)^ne^{-in\phi}-\delta_{n0}
\right)(2X_\phi)^n\sum_{k=0}^\infty\frac{(-)^k(2|\alpha|X_\phi)^{2k}}{(2k+n)!}\\
=& \sum_{n=0}^\infty\sum_{k=0}^\infty
\left(\epsilon^ne^{in\phi}(-\alpha^*)^{n+k}\alpha^k +\epsilon^ne^{-in\phi}
(-\alpha^*)^k\alpha^{n+k}-\delta_{n0}
\right)\frac{(2 X_\phi)^{2k+n}}{(2k+n)!}
\end{split}$$ In the limit $\epsilon\to 1^-$ the last expression can be simplified using the reordering rule $$\begin{split}
&\sum_{n=0}^\infty\sum_{k=0}^\infty a_{n+2k}(z^{n+k}w^kt^n+ z^kw^{n+k}t^{-n}-\delta_{n0}z^kw^k)\\
=&\sum_{k=0}^\infty\sum_{h=k}^\infty a_{h+k}(z^hw^kt^{h-k}+ z^kw^ht^{k-h}-\delta_{hk}z^kw^k)\\
=&\sum_{h=0}^\infty\sum_{k=0}^h a_{h+k}(tz)^h(t^{-1}w)^k+\sum_{k=0}^\infty\sum_{h=k}^\infty
a_{h+k}(tz)^k(t^{-1}w)^h-\sum_{h=0}^\infty a_{2h}z^hw^h\\
=&\sum_{h=0}^\infty\sum_{k=0}^\infty a_{h+k}(tz)^h(t^{-1}w)^k,
\end{split}$$ corresponding to $$f_\phi(X_\phi|D(\alpha))=\sum_{h=0}^\infty\sum_{k=0}^\infty
\frac{(-2 X_\phi e^{i\phi}\alpha^*)^h(2 X_\phi e^{-i\phi}\alpha)^k}{(h+k)!}\;.$$
[99]{} G. M. D’Ariano and H. P. Yuen, Phys. Rev. Lett. [**76**]{}, 2832 (1996). U. Fano, Rev. Mod. Phys. [**29**]{}, 74 (1957), Sec. 6. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. [**70**]{}, 1244 (1993); M. G. Raymer, M. Beck, and D. F. McAlister, Phys. Rev. Lett. [**72**]{},1137 (1994); D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, Phys. Rev. A [**48**]{}, 3159 (1993). K. Vogel and H. Risken, Phys. Rev. A, [**40**]{}, 2847 (1989). G. M. D’Ariano, C. Macchiavello and M. G. A. Paris, Phys. Rev. A [**50**]{}, 4298 (1994). G. M. D’Ariano, in [*Quantum Communication, Computing, and Measurement*]{}, ed. by O. Hirota, A. S. Holevo and C. M. Caves, Plenum Publishing (New York and London 1997), p. 253. G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, [*Quantum Tomography*]{}, Advances in Imaging and Electron Physics [**128**]{}, p. 205-308 (2003); also available at quant-ph/0302028. O. Christensen, [*An Introduction to Frames and Riesz basis*]{} (Birkha̋user, Boston, Basel, Berlin 1996) P. Casazza, Taiwanese J. Math. [**4**]{}, 129 (2000). D. Han and D. R. Larson, Mem. Amer. Math. Soc. [ **147**]{}, n. 697, 0 ( AMS, Providence, Rhode Island, 2000). S. Li, Numer. Funct. Anal. and Optimiz. [**16**]{}, 1181 (1995). Th. Richter, Phys. Lett. A [**211**]{}, 327 (1996). G. M. D’Ariano and M. G. A. Paris, Phys. Rev. A [ **60**]{}, 518 (1999). A. Wünsche, J. Opt. B: Quantum Semiclass. Opt. [ **1**]{}, 264 (1999). I. S. Gradshteyn and I. M. Ryzhik, [*Table of integrals, series, and products*]{}, (Academic Press, New York, 1980). A. Wünsche, Quantum Opt. [**2**]{}, 453 (1990). C. T. Lee, Phys. Rev. A [**46**]{}, 6097 (1992). A. Wünsche, J. Mod. Opt. [**44**]{}, 2293 (1990).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Connes formula giving the dual description for the distance between points of a Riemannian manifold is extended to the Lorentzian case. It resulted that its validity essentially depends on the global structure of spacetime. The duality principle classifying spacetimes is introduced. The algebraic account of the theory is suggested as a framework for quantization along the lines proposed by Connes. The physical interpretation of the obtained results is discussed.'
author:
- 'G.N.Parfionov, R.R.Zapatrin[^1]'
title: Connes duality in Lorentzian geometry
---
\[th\][Proposition]{} \[th\][Lemma]{}
Introduction {#introduction .unnumbered}
============
The mathematical account of general relativity is based on the Lorentzian geometry being a model of spacetime. As any model, it needs identification for a physicist in terms of measurable values. In this paper we focus on evaluations of intervals between events, and the measurable entities will be the values of scalar fields.
This work was anticipated by the Connes distance formula for Riemannian manifolds
$${\rm Dist}\, (x,y) = \sup\{ f(y) - f(x) \}$$
where the supremum is taken over the smooth functions whose gradient does not exceed 1.
This formula gives rise to a new paradigm in the account of differential geometry being more sound from the physicist’s point of view since it expresses the distance through the values of scalar fields on the manifold. Our goal was to investigate to what extent this formula is applicable in Lorentzian manifolds.
The first observation was that even in the Minkowskian space this formula is no longer valid in its literal form. The reason is that the Cauchy inequality on which the Connes formula is based, does not hold in the Minkowskian space. In section \[sdulor\] following the Connes’ guidelines we managed to obtain an [*evaluation*]{} rather than the [*expression*]{} for the distance. In ’good’ cases, in particular, in the Minkowski spacetime, this evaluation is exact and gives an analog of the Connes formula.
The attempt to generalize it to arbitrary Lorentzian manifolds resulted in building of counterexamples which show the drastic difference between the Riemannian and Lorentzian manifolds. In section \[sdupr\] the duality principle was introduced in order to point out the class of Lorentzian manifolds being as ’good’ as Riemannian ones. It turned out that one can find a ’bad’ spacetime even among those conformally equivalent to the Minkowskian one. An example is provided in section \[sdupr\].
The Connes duality principle play an important rôle in the framework of the so-called ’spectral paradigm’ in the account of non-commutative differential geometry [@connes]. However, the correspondence principle for this theory is corroborated on Riemannian manifolds. Following these lines, in section \[salg\] we show a way to introduce non-comutative Lorentzian geometry.
Connes formula {#scd}
==============
Both the Riemannian distance and Lorentzian interval are based on calculation of the same integral:
$$\int_\gamma \sqrt{ds^2}$$
which is always referred to a pair of points. The intervals (resp., distances) as functions of two points are obtained as extremal values of this integral over all appropriate curves connecting the two points.
There is a remarkable duality to evaluate this integral suggested by Connes for the Riemannian case. Consider it in more detail.
For any two points $x,y$ of a Riemannian manifold ${{\cal M}}$ connected by a smooth curve $\gamma$ the following evaluation of its length $\ell (\gamma)$ takes place:
f(y) - f(x) \_[[[M]{}]{}]{}[| f |]{}()
based on the Cauchy inequality:
$$(\nabla f, \dot{\gamma}) \le |\nabla f| \cdot |\dot{\gamma}|$$
So, the distance $\rho(x,y)$ between the points of the manifold satisfies the following inequality
$$\rho(x,y) \ge \sup (f(y) - f(x)),$$
where $f$ ranges over all functions whose gradient does not exceed 1: $| \nabla f | \le 1$.
It was shown by Connes [@connesreal], that as a matter of fact no curves are needed to determine the distance, which may be obtained directly as:
(x,y) = \_[| f | 1]{}(f(y) - f(x))
The physical meaning of this result is the following: we can evaluate the distance between the points measuring the difference of potentials of a scalar field whose intensity is not too high. So, the following [*duality principle*]{} takes place in Riemannian geometry:
$$\sup (f(y) - f(x)) = \inf \ell (\gamma)$$
Note that this formula is valid even for non-connected spaces: in this case both sides of the above equality are equal $+\infty$, if we assume, as usually, the infinite value of the infimum when the ranging set is void.
The question arises: can we write down a similar evaluation for the Lorentzian case?
Duality inequality in Lorentzian geometry {#sdulor}
=========================================
The Cauchy inequality from which the Riemannian duality principle was derived is no longer valid in the Lorentzian case. Instead, we have the following:
(f, )\^2 (f)\^2 ()\^2
where both $\nabla f$, $\dot{\gamma}$ are non-spacelike:
$$(\nabla f)^2 \ge 0, \qquad (\dot{\gamma})^2 \ge 0$$
If under these circumstances we also have $(\nabla f,
\dot{\gamma}) \ge 0$, the inequality (\[eanticauchy\]) reduces to
$$(\nabla f, \dot{\gamma}) \ge |\nabla f| \cdot |\dot{\gamma}|$$
Now let $x,y$ be two points of a Lorentzian manifold ${{\cal M}}$ such there is a causal curve $\gamma$ going from $x$ to $y$. Then for any global time function $f$ on ${{\cal M}}$ we immediately obtain the analog of the inequality (\[eriem\])
$$f(y) - f(x) \ge \inf_{{{\cal M}}}{| \nabla f |}\cdot \ell (\gamma)$$
Introduce the class ${{\cal F}}({{\cal M}})$ of global time functions satisfying the following condition:
(f)\^2 1
Then the Lorentzian interval between $x$ and $y$
l(x,y) = \_
can be evaluated as follows:
$$f(y) - f(x) \ge l(\gamma)$$
provided the class ${{\cal F}}$ (\[eclassf\]) is not empty. Introducing the value
$$L(x,y) = \inf_{f \in {{\cal F}}} (f(y) - f(x))$$
we obtain the following [*duality inequality*]{}:
l(x,y) L(x,y)
It is worthy to mention that this inequality is still meaningful when the points $x$, $y$ can not be connected by a causal curve. In this case the supremum $l(x,y)$ is taken over the empty set of curves and its value is, as usually, taken to be $-\infty$, that is why the inequality (\[eduineq\]) trivially holds.
Let us thoroughly describe this construction in the case when ${{\cal M}}$ is the Minkowskian spacetime. The following proposition holds:
#### Proposition.
Let ${{\cal M}}$ be Minkowskian spacetime. Then $L(a,b) = l(a,b)$ for any pair of points $a,b \in {{\cal M}}$.
#### Proof.
Assume with no loss of generality that $a=0$. If $b$ is in the future cone of $a=0$, then $l = (b,b)$ is realized on the segment $[0,b]$. The value of $L$ is achieved on the function $f(x) = (b,x)/\sqrt{(b,b)}$. Let $b$ be a future-directed isotropic vector, then $l = 0$. For any $\epsilon$ such that $0< \epsilon <
1$ consider the function
$$f_\epsilon (x) =
\frac{((1-\epsilon)b + \epsilon v,
x)}{\sqrt{\epsilon(1-\epsilon)(b,v)}}$$
where $v$ a vector defining the time orientation. The direct calculation shows that $(\nabla f)^2 \ge 1$ and $(\nabla
f,v)\ge 0$, that is, $f_\epsilon \in {{\cal F}}$. In the meantime $f(b) =
\sqrt{\epsilon(v,b)/(1-\epsilon)}$ which can be made arbitrarily close to zero by appropriate choice of $\epsilon$.
Now let the point $b$ be beyond the future cone of the point $0$, therefore they can be separated by a spacelike hyperplane $f_k(x) = (k,x) = 0$ and choose the vector $k$ to be future-directed. Then $f_k(b) < 0$. Since $f_{\lambda k} (b) =
\lambda f_k \in {{\cal M}}$, the infimum is $-\infty$. In the meantime $l(0,b) = -\infty$ as well since there are no future-directed non-spacelike curves connecting $0$ with $b$.
[$\Box$]{}
#### Remark.
Note that if we borrow the definition of $l(a,b)$ from [@bimerl], namely $l(a,b)=0$ for $b\not\in J^+(a)$, then the duality principle will not hold even for the Minkowskian case: this was the reason for us to introduce the definition (\[elab\]).
Consider one more example. Let ${{\cal M}}= S^1 \times {{\bf R}}^3$ be a Minkowskian cylinder where $S^1$ is the time axis. In this case any two points $x,y \in {{\cal M}}$ can be connected by an arbitrary long timelike curve, therefore $l(x,y)=+\infty$. In the meantime the class ${{\cal F}}({{\cal M}})$ is empty (since there is no global time functions), and therefore $L(x,y)=+\infty$. So we see that even in this “pathological” case the duality principle is still valid.
Note that the class ${{\cal F}}({{\cal M}})$ itself characterizes spacetimes. In general, if the class ${{\cal F}}({{\cal M}})$ is not empty, the spacetime ${{\cal M}}$ is chronological (it follows immediately from that ${{\cal F}}({{\cal M}})$ consists of global time functions).
Now we may inquire whether all Lorentzian manifolds are as ’good’ as Minkowskian? In the next section we show that the answer is no.
Duality principle {#sdupr}
=================
In the previous section we have proved the duality inequality (\[eduineq\]) which is always true in any Lorentzian manifold. However, unlike the case of Riemannian spaces, this inequality may be strict, which is corroborated by the following example. Let ${{\cal M}}$ be a Minkowskian plane from which a closed segment connecting the points $(1, -1)$ and $(-1, 1)$ is cut out (Fig. \[fig1\]).
(60,40) (25,25)[(1,-1)[10]{}]{} (0,20)[(1,0)[60]{}]{} (30,0)[(0,1)[40]{}]{} (30,10) (31.5,10) (30,30) (31.5,30) (25,25) (35,15)
Consider two points $a=(-2.0)$ and $b=(2,0)$. They can not be linked by a timelike curve in ${{\cal M}}$, therefore $l(a,b) = -\infty$. Meanwhile the class ${{\cal F}}({{\cal M}})$ is not empty: it contains at least the restrictions of all such functions defined on the whole Minkowskian plane, thus $L(a,b)<+\infty$. Let us prove that the value $L(a,b)$ is finite, supposing the opposite. If $L(a,b)$ would be equal to $-\infty$, a function $f\in {{\cal F}}({{\cal M}})$ should exist for which $f(b)<f(a)$. Consider the behavior of the level line $l_b$ of $f$ passing through the point $b$. Being spacelike, it can not enter the cone $(-\infty < t < b; |x| \le |t-b|)$ which contains the point $a$. From the other hand, the point $a$ must lie in the causal future cone $J_+(l_b)$, which is not the case. So, $L(a,b) \neq l(a,b)$.
Now, specifying the notion of ’good’ spacetime, introduce the duality principle.
A Lorentzian manifold ${{\cal M}}$ is said to satisfy the [*duality principle*]{} if for any its points $x,y$
$$L(x,y) = l(x,y)$$
This characterization is global. The example we presented above show that this notion is not hereditary: if we take an open subset of a ’good’ manifold it may happen that it will no longer enjoy the duality principle.
Contemplating the above mentioned examples may lead us to an erroneous conclusion that the reason for the duality principle to be broken is when the spacetime manifold is not simply connected. The next example [@krasnprivate] shows that there are manifolds which are simply connected, geodesically convex, admit global chronology but do not enjoy the duality principle.
Let ${{\cal M}}$ be a right semiplane $(-\infty < t < +\infty ; x > 0)$ with the metric tensor conformally equivalent to Minkowskian. It is defined in coordinates $t,x$ as follows:
$$g_{ik} = \frac{1}{x}
\left(
\begin{array}{cc}
1 & 0 \cr
0 &-1
\end{array}
\right)$$
The example illustrates the problems related with the dual evaluations: it shows that the existence of a global time function does not guarantee the class ${{\cal F}}$ to be non-empty. The spacetime ${{\cal M}}$ evidently admits global time functions such as, for instance, $f(t,x) = t$. However, the following proposition can be proved.
#### Proposition.
The class ${{\cal F}}({{\cal M}})$ is empty.
#### Proof.
Suppose there is a function $f(x,t)$ satisfying (\[eclassf\]) and consider two values $A, B$ ($A<B$) of the function $f$. The appropriate lines of constant level of $f$ are the graphs of functions $t_A(x), t_B(x)$. Since the derivative $f_t>0$, we have $t_A(x)<t_B(x)$. These functions are differentiable and their derivatives are bounded: $t_A', t_B' \le
1$ (because these lines are always spacelike), therefore they have limits when $x \to 0$. Let us show that these limits are equal.
Consider the difference $B-A$ and evaluate it:
$$B-A = f(t_B(x),x) - f(t_A(x),x) =$$ $$= \int_{t_A(x)}^{t_B(x)} f_t(t,x) dt \ge
\frac{1}{\sqrt{x}} \cdot (t_B(x) - t_A(x))$$
where the first factor $1/\sqrt{x}$ is directly obtained from the condition $(\nabla f)^2 \ge 1/x$. So, the limit of $t_B(x) - t_A(x)$ is to be equal 0. Since the values $A,B$ were taken arbitrary, we conclude that all the lines of levels of the global time function $f$ come together to a certain point. Therefore these lines (being spacelike) cannot cover all the manifold ${{\cal M}}$.
[$\Box$]{}
This proposition shows that the space ${{\cal M}}$ does not support duality principle: we can take two points $a,b$ on a timelike geodesic and calculate the interval $l(a,b)$, while $L(a,b)=+\infty$.
Algebraic aspects and quantization {#salg}
===================================
Let us study the dual evaluations from the algebraic point of view. It was pointed out yet by Geroch [@geroch] that the geometrical framework of general relativity can be reformulated in a purely algebraic way. Recall the basic ingredients of Geroch’s approach.
The starting object is the algebra ${{\cal A}}= {\mbox{$ {\cal C}^{\infty} $}}({{\cal M}})$, then the vector fields on ${{\cal M}}$ are the derivations of ${{\cal A}}$, that is, the linear mappings $v: {{\cal A}}\to {{\cal A}}$ satisfying the Leibniz rule:
$$v(a \cdot b) = a\,v\cdot b + va\cdot b$$
Denote by ${{\cal V}}$ the set of vector fields on ${{\cal M}}$ ($=$ derivations of ${{\cal A}}$). It is possible to develop tensor calculus along these lines: like in differential geometry, tensors are appropriate polylinear forms on ${{\cal V}}$. In particular, the metric tensor can be introduced in mere algebraic terms.
The Geroch’s viewpoint is in a sense ’pointless’ [@ps]: it contains no points given [*ab initio*]{}. However the points are immediately restored as one-dimensional representations of ${{\cal A}}$. For any $x\in {{\cal M}}$ the appropriate representation $\hat{x}$ reads:
$$\hat{x}(a) = a(x) \qquad a\in {{\cal A}}$$
Now let ${{\cal M}}$ be a Riemannian manifold. If we then decide to calculate the distance between two representation $x,y$ in a ’traditional’ way we have to introduce such a cumbersome object as continuous curve in the space of representations. It is the result of Connes (\[parfeq\]) which lets us stay in the algebraic environment:
$$\rho(x,y) = \sup_{f\in {{\cal F}}}(f(y) - f(x))$$
and the problem now reduces to an algebraic description of the class ${{\cal F}}$ of suitable elements $f$ of the algebra ${{\cal A}}$. The initial Connes’ suggestion still refers to points: ${{\cal F}}= \{ a\in {{\cal A}}|\, \forall m\in {{\cal M}}\: |\nabla
a(m)| \le 1 \}$.
Connes’ intention was to build a quantized theory which could incorporate non-commutative algebras as well. For that, the construction of [*spectral triple*]{} was suggested [@connes].
A spectral triple $({{\cal A}}, {{\cal H}}, D)$ is given by an involutive algebra of operators ${{\cal A}}$ in a Hilbert space ${{\cal H}}$ and a self-adjoint operator $D$ with compact resolvent in ${{\cal H}}$ such that the commutator $[D,a]$ is bounded for any $a\in {{\cal A}}$ (note that $D$ is not required to be an element of ${{\cal A}}$).
Then for any pair $(x,y)$ of states ($=$ non-negative linear functionals) on ${{\cal A}}$ the distance $d(x,y)$ between $x$ and $y$ may be introduced:
$$d(x,y) = \{ |x(a) - y(a)| \,:\, a\in {{\cal F}}\}$$
with the following class of ’test elements’ of the algebra ${{\cal A}}$
[[F]{}]{}= { a[[A]{}]{}: ||\[D,a\]|| 1 }
The suggested construction satisfies the correspondence principle with the Riemannian geometry. Namely, we form the spectral triple with ${{\cal A}}= {\mbox{$ {\cal C}^{\infty} $}}({{\cal M}})$, ${{\cal H}}= {{\cal L}}^2({{\cal M}}, S)$ — the Hilbert space of square integrable sections of the irreducible spinor bundle over ${{\cal M}}$ and $D$ is the Dirac operator associated with the Levi-Cività connection on ${{\cal M}}$ [@semaz]. Then $d(x,y)$ recovers the Riemannian distance on ${{\cal M}}$ (see, e.g. [@connes]).
Comparing the definition (\[efnonc\]) of the class ${{\cal F}}$ with that used in section \[scd\]: ${{\cal F}}= \{ a\in {{\cal A}}|\, \forall
m\in {{\cal M}}\: |\nabla a(m)| \le 1 \}$ we see that the operator $D$ is a substitute of the gradient. Following [@gianni; @froh] the gradient condition (\[efnonc\]) can be written in terms of the Laplace operator taking into account that:
$$(\nabla f)^2 =
\frac{1}{2}\Delta(f^2) - f\Delta f$$
which restores the metric on ${{\cal M}}$ according to the Connes’ duality principle (\[parfeq\]) for Riemannian manifolds. However this condition is still checked at every point of ${{\cal M}}$.
We suggest an equivalent algebraic reformulation of (\[efnonc\]) with no reference to points. Starting from the notion of the spectrum of an element of algebra [@dix]:
$$\mbox{\rm spec} (a) = \{\lambda \in {{\bf C}}|\, a-\lambda \cdot {{\bf 1}}\mbox{ is not invertible} \}$$
and taking into account that the spectrum of the multiplication operator coincides with the domain of the multiplicator we reformulate the Connes’ condition $f\in {{\cal F}}$ as
([[**1**]{}]{} - (f)\^2 )
Within this framework, to pass to Lorentzian case, we simply substitute the Laplacian $\Delta$ by the D’Alembertian $\Box$, and the spectral condition (\[espectr\]) is changed to
$$\mbox{\rm spec} ((\nabla f)^2 - {{\bf 1}}) \quad \mbox{ is non-negative}$$
which makes it possible to recover the Lorentzian interval provided the duality principle holds.
So we see that the notion of spectral triple is well applicable to develop quantized Lorentzian geometry along the lines of Connes’ theory.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work was supported by the RFFI research grant (97-14.3-62). One of us (R.R.Z.) acknowledges the financial support from the Soros foundation (grant A97-996) and the research grant “Universities of Russia”.
We would like to thank the participants of the research seminar of the Friedmann Laboratory for theoretical physics (headed by A.A.Grib), especially S.V.Krasnikov and R.Saibatalov, for profound discussions.
[99]{}
Beem, J., Erlich, P., Global Lorentzian geometry, Marcel Dekker, New York, 1981
Connes, A., Non-commutative geometry, Academic Press, San Francisco, 1994
Connes, A., Non-commutative geometry and reality, Journal of Mathematical Physics, [**36**]{} (1995), 6194
Dixmier, J., Les $C^*$-algèbres et leurs reprśentation, Gauthier-Villars, Paris, 1964
Fröhlich, J., K.Gawedzki, A.Recknagel, Supersymmetric quantum theory and (non-commutative) differential geometry, eprint hep-th/9612205
R.Geroch, Einstein Algebras, Communications in Mathematical Physics, [**26**]{} (1972), 271
Landi, G., Introduction to noncommutative spaces and their geometry, Springer-Verlag, Berlin, 1997
Krasnikov, S.V., private communication
Palais, R.S., Seminar on the Atiyah-Singer index theorem, Princeton, New Jersey, 1965
Parfionov, G.N., Zapatrin, R.R., Pointless spaces in general relativity, International Journal of Theoretical Physics, [**34**]{} (1995), 737
[^1]: Friedmann Laboratory for theoretical physics, SPb UEF, Griboyedova, 30-32, 191023, St.Petersburg, Russia
|
{
"pile_set_name": "ArXiv"
}
|
Introduction
============
Considerable confusion currently exists, both theoretically and experimentally, regarding the dynamical universality class of the zero-field superconducting phase transition in high temperature superconductors. The short coherence length in these materials leads to large, non-gaussian fluctuations, and there is some experimental evidence that the static behavior is that of the 3-dimensional XY model[@ffh91; @friesen-muzikar; @3DXY1; @3DXY2; @3DXY3; @moloni-prl; @moloni-prb; @ginsberg], as in the lambda transition in superfluid helium. The dynamical universality class of the lambda transition in superfluid helium is that of a two component order parameter coupled to a conserved density which gives a propagating mode (second sound in $^4$He; a spin wave in the XY spin model) in the broken symmetry phase. This is model E in the notation of Hohenberg and Halperin.[@hohenberg77] However, the dynamical behavior of a superconductor could be very different, since the lattice acts as a momentum sink destroying Galilean invariance making the system more like helium in a porous medium. The existence of disorder does not destroy the Goldstone mode in the ordered phase but it does affect the vortices and the normal-fluid quasiparticles which tend to equilibrate with the lattice rather than co-moving with the condensate.
Furthermore, in d-wave superconductors, scalar disorder causes pair-breaking and quasiparticle branch recombination which may make it inappropriate to assume particle number conservation in the dynamics. There does not appear to have been any work to date on this question.
A related issue is that of the role of the long-range Coulomb interaction, which severely suppresses longitudinal current fluctuations, leaving only the transverse currents associated with vortices. However it is also possible that the low superfluid density and screening from the high normal fermion density will turn on the low energy longitudinal Carlson-Goldman fluctuations[@carlson-goldman] of the order parameter as $T_c$ is approached. These microscopic fermionic effects may further confuse the data analysis and affect the width of the critical regime.
The issues raised above remain largely unresolved. The particular issues that we will explicitly discuss here are the role of magnetic screening and the role of disorder. In an extreme type-II, system, coupling to gauge fluctuations is weak,[@ffh91] but nonetheless, in principle, magnetic screening becomes important extremely close to the critical point where the system crosses over to inverted XY behavior.[@herbut; @dasgupta81; @inverted] The static correlations related to this issue have recently been studied numerically by Olsson and Teitel.[@olsson-teitel] Here we will address the dynamics.
Most theoretical studies of critical dynamics in XY like spin systems have focussed on Landau-Ginsburg representations of the problem involving the [*phase*]{} (i.e. angle) of the spin. For static properties, there also exist equivalent dual representations[@dasgupta81; @villain75; @jose77; @kleinert89] in terms of interacting [*vortex*]{} degrees of freedom. Although the static properties of the phase and vortex representations are the same, there is no reason, [*a priori*]{}, why the dynamical universality classes should be the same.
For the reasons discussed above it may be more appropriate, for superconductors, to consider a model with overdamped dissipative dynamics of the topological defects (the vortices). This is the approach that we will take here.
To add dynamics to the the phase representation, one can either include just dissipative dynamics, model A,[@hohenberg77] in which case the dynamical exponent, $z$, is close to 2, or one can incorporate the propagating (spin wave) modes, model E,[@hohenberg77] for which $z$ is exactly 3/2 in three dimensions, ($d/2$ in $d$-dimensions). For the vortex representation, which has [*discrete*]{} variables, the natural dynamics is purely dissipative, such as that generated by Monte Carlo simulations (in which Monte Carlo time is equated with real time). Naively, it would seem unlikely that the [*dissipative*]{} dynamics of the vortex representation would be in the same universality class as the dynamics of model E (phase representation), which has [*propagating*]{} modes. Surprisingly, recent results by Weber and Jensen[@weber97] come to the opposite conclusion. They find, for unscreened vortex interactions ($\kappa=\infty$) and overdamped dynamics, that the dynamical exponent is $z=d/2=1.5$, precisely the value one expects in model E dynamics, and considerably [*less*]{} than the value generally found with dissipative dynamics ($z \approx 2$).
In this paper we present results of Monte Carlo calculations of the dynamical critical exponents for the 3-dimensional XY model, in the vortex representation, with and without magnetic screening. For no screening, we confirm the unexpected result of Weber and Jensen[@weber97], while by contrast, for strong magnetic screening, we find a rather large [*enhancement*]{} of $z$. Although it is known from the Harris criterion and verified numerically[@moon] that uncorrelated disorder is weakly irrelevant at the 3-dimensional XY critical point, the effect of such disorder on the dynamical properties is unknown.[@columnar] We therefore also investigate the effects of disorder on the model with strong screening.
The models
==========
The model under investigation here is the XY–model with a fluctuating vector potential, $$\label{ham-phase}
{\cal H} = -J \sum_{\langle i,j \rangle} \cos(\phi_i - \phi_j
-\lambda_0^{-1 } a_{ij}) + \frac{1}{2} \sum_\Box [{\bf \nabla}
\times {\bf a}]^2,$$ where $J$ is the coupling constant (set to unity in the simulations), the $\phi_i$ denote the phases of the condensate on the sites $i$ of a simple cubic lattice of size $N=L^3$ with periodic boundary conditions. The sum is taken over all nearest neighbors $\langle i,j \rangle$. Additionally, we have a fluctuating gauge potential $a_{ij}$ with the gauge constraint $[{\bf \nabla}\cdot {\bf a}]=0$, and $\lambda_0$ denotes the bare screening length. The last term describes the magnetic energy, where the sum runs over all elementary plaquettes on the lattice and the curl is the directed sum of the gauge potential round a plaquette.
In our work, the main focus is equilibrium vortex dynamics. However, vortices are only [*implicitly*]{} present in the above model through the relation $$\oint \nabla \phi({\bf r})\cdot {\rm d} {\bf r}=2\pi n ,$$ where $n=0,\pm 1,\pm 2,...$ denotes the net vorticity encircled by the integration contour. In order to analyze the critical behavior of this model due to vortex fluctuations [*explicitly*]{}, it is easier to go from the above [*phase*]{} representation of the XY–model to its [*vortex*]{} representation. This is achieved by replacing the cosine in Eq. (\[ham-phase\]) with the periodic Villain function and performing fairly standard manipulations[@dasgupta81; @villain75; @jose77; @kleinert89] to obtain $$\label{ham-vortex}
{\cal H}_V = \frac{1}{2} \sum_{i,j} {\bf n}_i \cdot {\bf n}_j\,
G_{ij}[\lambda_0].$$ Here, the ${\bf n}_i$ are vortex variables which sit on the links of the [*dual lattice*]{} (which is also a simple cubic lattice) and $G_{ij}$ is the screened lattice Green’s function $$\label{greens}
G_{ij}[\lambda_0] = J\frac{(2\pi)^2}{L^3} \sum_{\bf k} \frac{
\exp[{\rm i }\, {\bf k}\cdot({\bf r}_i - {\bf r}_j)]}
{2\sum_{m=1}^3 [1- \cos(k_m)] + \lambda_0^{-2}}.$$ In the long range case, $\lambda_0 \to \infty$, the divergent ${\bf
k=0}$ contribution has to be excluded from the sum and the constraint $\sum_i {\bf n}_i = {\bf 0}$ has to be imposed. In the short range case (finite $\lambda_0$) this is not necessary. The transformations involved in going from the phase to the vortex representation also yield the local constraints $[{\bf \nabla}\cdot
{\bf n}]_i=0$, i.e., there are no magnetic monopoles (zero divergence constraint). It is important to note, that the vortex representation does not contain spin wave degrees of freedom anymore, since they have been “integrated out” in the transformation procedure.
We will be interested in two limits of the above model: (i) no screening ($\lambda_0\to \infty$), corresponding to the extreme type–II limit ($\kappa\to \infty$), where the individual vortex lines have long range interactions; (ii) strong screening ($\lambda_0\to 0$), i.e., short range interactions, which is supposed to be the correct description of a superconductor extremely close to the critical point,[@ffh91] though the size of the critical region where such screening is relevant may be too small to be observable in practice. In this limit the interaction reduces to $G_{ii} = J(2 \pi
\lambda_0)^2$ and $G_{i \ne j} = 0$ (plus exponentially small corrections of order $\exp(-r/\lambda_0)$). We will in this case use units where $J(2 \pi \lambda_0)^2 = 1$. The resulting Hamiltonian is of the very simple form $$\label{ham-vortex-screen}
{\cal H}_V = \frac{1}{2} \sum_i {\bf n}_i\cdot{\bf n}_i,
\quad (\lambda_0\to 0).$$ Note, however, that this Hamiltonian is not trivial, since the constraints on the local divergence effectively generate interactions among the ${\bf n}_i$. Note further, that this is also the dual representation of the XY model [*without*]{} screening, in which the temperature scale is inverted.[@dasgupta81; @kleinert89; @wengel96] The static universality class of Eq. (\[ham-vortex-screen\]) is then the same as that of Eq. (\[ham-vortex\]) with $\lambda_0=\infty$, and is given by XY exponents.[@yeomans] The dynamical universality class, however, may be different and determining it is one of the goals of our study.
In addition to the pure short-range model, we also study the short-range model with quenched random local $T_c$. In this case we replace Eq. (\[ham-vortex-screen\]) by $$\label{ham-vortex-screen-disorder}
{\cal H}_V = \frac{1}{2} \sum_{i,\mu} \xi_{i\mu} n_{i\mu}^2, \quad
(\lambda_0\to
0,\; \xi_{i\mu}\ \mbox{random}).$$ where $\xi_{i\mu}$ is uniformly distributed in the interval $[0.5,1.5]$.
Monte Carlo simulation and finite size scaling
==============================================
We simulate the following model Hamiltonians in the vortex representation:
1. Eq. (\[ham-vortex\]) with $\lambda_0\to \infty$
2. Eqs. (\[ham-vortex-screen\]) and (\[ham-vortex-screen-disorder\]), which corresponds to $\lambda_0\to 0$.
We take simple cubic lattices of size $L^3$ where $4 \le L \le $12–64. Periodic boundary conditions are imposed. We start with configurations with all ${\bf n}_i = {\bf 0}$, which clearly satisfies the constraints, and a Monte Carlo (MC) move consists of trying to create a closed vortex loop around a plaquette.[@global] This trial state is accepted according to the heat bath algorithm with probability $1/[1+\exp(\beta \Delta E)]$, where $\Delta E$ is the change of energy and $\beta=1/T$.
Each time a loop is formed it generates a voltage pulse $\Delta Q =
\pm 1$ perpendicular to its plane, the sign depending on the orientation of the loop. This leads to a net electric field[@hyman95] $$E(t) = \frac{h}{2e} J^V(t)
\quad \mbox{with} \quad
J^V(t) = \frac{\Delta Q}{\Delta t},
\label{voltage}$$ where $J^V(t)$ is the vortex current density, and $\Delta t = 1$ for one full sweep through the system, where, on average, an attempt is made to create or destroy one vortex loop per plaquette.
The nonlinear I–V characteristics of the inverted XY model can be modeled as the electric field $E$, due to vortex current response in the presence of a uniform Lorentz force on the vortex lines, proportional to the applied current density $J$.[@caveat1] In addition, the linear response resistance can be calculated from the equilibrium voltage–voltage fluctuations via the Kubo formula[@young94] $$\label{kubo}
R = \frac{1}{2T} \sum_{t=-\infty}^{\infty} \Delta t
\langle V(t)V(0) \rangle$$ Here, $\langle \cdots \rangle$ denotes the thermal average, and the voltage across the sample is $V(t) = L E(t)$. The resistivity is $\rho = L^{d-2}R$.
Since we are working with lattices of finite length $L$, one has to employ finite size scaling techniques to extract the critical behavior. A detailed scaling theory has been developed for superconductors by Fisher et al.[@ffh91] and we now summarize the results from it which will be needed for our data analysis.
Near a second order phase transition the linear resistivity obeys the scaling law $$\rho_{\rm lin}(T,L)=L^{-(2-d+z)}\, \tilde{\rho}\left[
L^{1/\nu}(T-T_c)
\right ],
\label{rho_lin_scale}$$ where $\nu$ is the correlation length exponent, $z$ is the dynamical exponent and $\tilde{\rho}$ is a scaling function. At the critical temperature, $\tilde{\rho}(0)$ becomes a constant and therefore $\rho_{\rm lin}(T_c,L) \sim L^{-(2-d+z)}$. If we plot the ratio of $\rho_{\rm lin}$ for different system sizes against $T$, then $$\label{intersect}
\frac{\ln[\rho_{\rm lin}(L)/\rho_{\rm lin}(L^{\prime})]}
{\ln[L/L^{\prime}]} = d-2-z \quad\mbox{at}\;\; T_c,$$ i.e., all curves for different pairs $(L,L^{\prime})$ should intersect and one can read off the values of $T_c$ and $z$. We will refer to this kind of data plot as the [*intersection method*]{}. With the values of $T_c$ and $z$ determined by the intersection method we can then use a scaling plot according to Eq. (\[rho\_lin\_scale\]) to obtain the value of $\nu$.
A similar analysis[@ffh91; @columnar] shows that, above a characteristic current scale $J_{\rm nl}$, which varies as $\sim L^{-2}$ at the critical point, non-linear response sets in and the electric field varies as $$E\sim J^{(1+z)/2}.
\label{Enonlin}$$
It is useful to locate the critical temperature from equilibrium properties instead of the dynamic scaling of the linear resistivity, since such simulations are easier to converge. In the short-range case we do this by considering an ensemble with a fluctuating winding number $W$, defined by $$W_\mu = \frac{1}{L} \sum_i { n}_{i\mu}.$$ From the finite size scaling relation $$\left< W_\mu^2 \right> = f(L^{1/\nu}(T-T_c))$$ we can locate the critical point in the short range case, both with and without disorder using the fact that the winding number is scale invariant at the critical point. Note that the simulations of this quantity require global moves, where vortex lines going all the way through the system are created and destroyed, and therefore do not represent the dynamics of the system in a realistic way. This does not pose a problem since we use this calculation solely as a way to accurately locate the critical point and not to follow the dynamics.
Because different winding number classes are difficult to equilibrate efficiently in large systems in this model, we use an (exact) duality transformation from the short-range vortex models, Eqs. (\[ham-vortex-screen\]) and (\[ham-vortex-screen-disorder\]), back to an XY phase model with a Villain interaction:[@villain75; @dasgupta81] $$\begin{aligned}
\label{villain}
Z_V&=&\sum_{\{{\bf n}_{i\mu}\}} \delta_{\nabla\cdot {\bf n}_i,0}
e^{-\frac{1}{2T}\sum_{i\mu}\xi_{i\mu}{\bf n}_{i\mu}^2 } \nonumber \\
&=& \int\limits_0^{2\pi} \left[ \prod_i d\theta_i \right]
\sum_{\{{ m}_{i\mu}\}} e^{-\sum_{i\mu}{T \over 2\xi_{i\mu}} \left(
\theta_i-\theta_{i+{\bf e}_\mu} -2\pi { m}_{i\mu}\right)^2}\end{aligned}$$ (a constant prefactor was suppressed in the last equality). Performing the sum over the integer dummy variables ${m}_{i\mu}$ leaves a phase-only model. This phase representation allows us to take advantage of the Wolff algorithm in the simulation which largely circumvents problems of critical slowing down and equilibration in the limit of large system sizes.[@wolff] The spin-wave stiffness of the dual model is given by $\rho_s = \partial^2 f / \partial
\tilde{A}_\mu^2|_{\tilde{A}=0}$, where $\tilde{A}_\mu$ is a constant vector potential added to the phase gradient in Eq. (\[villain\]). This is related to the winding number fluctuations in the vortex model by $\rho_s = T
L^{2-d} \left< W_\mu^2 \right>$.
Data analysis
=============
In this section we analyze our simulation data for the XY-model in the vortex representation, starting with the model in Eq. (\[ham-vortex\]) with long range interactions, i.e., neglecting screening ($\lambda_0\to \infty$).
Long range interactions
-----------------------
In Fig. (\[purel\_pic\]) we show the data for the linear resistivity $\rho_{\rm lin}$ plotted vs. $T$ obtained from a simulation of the Hamiltonian (\[ham-vortex\]) with $\lambda_0=\infty$. One observes that at high temperatures ($T=3.1$) there is hardly any size dependence in the data, consistent with a correlation length which is smaller than the system size. As one goes to lower temperature, the size dependence becomes stronger. This indicates critical behavior, since in the limit $L\to\infty$ the linear resistivity should go to zero below $T_c$.
However, since it is impossible to locate the critical temperature by this kind of plot, we show, in Fig. (\[purel-is\_pic\]), the same data of Fig. (\[purel\_pic\]) plotted according to the intersection method. The curves intersect at approximately $T_c=3.01\, (\pm 0.01)$ and $y\approx -0.45$ corresponding to $z=1.45\, (\pm 0.05)$, very similar to the result of Weber and Jensen, who find $z=1.51\, (\pm 0.03)$.[@weber97] Also, the value of $T_c$ agrees nicely with the one obtained earlier from simulations by Dasgupta and Halperin.[@dasgupta81]
Having established $T_c$ and $z$ we can now perform a scaling plot of the data according to Eq. (\[rho\_lin\_scale\]). In Fig. (\[purel-scale\_pic\]) we plot $\rho_{\rm lin} L^{2-d+z}$ vs. $L^{1/\nu}\,(T - T_c)$ and find that the data collapses best with $T_c=3.01\pm 0.01$, $z=1.5 \pm 0.05$ and $\nu=0.66 \pm0.01$. This independent result confirms that the dynamical scaling ansatz for $\rho_{\rm lin}$ yields the expected value of the correlation length exponent $\nu$ as well as a consistent value for $z$.
Short-Range Interactions
------------------------
In this section we will first describe how we locate the critical point of the pure and disordered short range models defined by Eqs. (\[ham-vortex-screen\]) and (\[ham-vortex-screen-disorder\]), using finite size scaling analysis of Monte Carlo data for static quantities. Then, using dynamic scaling analysis we determine the dynamical exponents both from equilibrium vortex dynamics simulations, and from driven nonequilibrium simulations.
In our simulations of Eq. (\[ham-vortex-screen\]), we used $10^6-10^7$ sweeps to calculate averages, and discard the initial $10\%$ of the data for equilibration. For the disordered case we average over $10-100$ samples to obtain small fluctuations.
As noted earlier, for the purpose of locating the critical point as precisely as possible for both the pure and random short-range models, we found it convenient to perform an exact transformation from the inverted XY model back to the phase representation. We compute the spin stiffness $\rho_s$ using the Wolff algorithm to overcome the critical slowing down and the difficult problem of equilibrating different winding number classes.[@wolff; @moon] From hyperscaling, $\langle W^2\rangle = L^{d-2}\rho_s/T$ is scale invariant at the critical point.[@rhos] Thus curves for different $L$ all cross at $T=T_c$ as shown in Fig. \[winding\]. Using this method, we find for the pure model $T_c = 0.333\pm0.001$ (which agrees nicely with the [*inverse*]{} of $T_c$ for the long range model) and $T_c =
0.313\pm0.001$ for the disordered model. Of course, this technique cannot be used to speed up the determination of $z$ since the Wolff algorithm intentionally does not represent local relaxation dynamics. Note also that the values of $\langle W^2\rangle $ at the crossing points in Fig. \[winding\] in the pure (a) and disordered (b) cases agree within the error bars, which is what we expect from two-scale factor universality[@two-scale-factor] and the fact that disorder is weakly irrelevant to the statics.
=
Simulations were then carried out at the measured critical temperatures to compute the dynamics in the vortex representations. In Fig. (\[rho\]) the dynamical exponent $z$ of the 3D loop model is determined from Monte Carlo data for the linear resistivity computed from the Kubo formula, Eq. (\[kubo\]).[@young94] Using $\rho_{\rm lin} \sim L^{1-z}$ at $T=T_c$, and from a power law fit to the data we get $z=2.7 \pm 0.1$. This value is consistent with, but more accurate than, the result of Wengel and Young[@wengel96]. We also verified that is possible to collapse the data for different sizes and temperatures using Eq. (\[rho\_lin\_scale\]) with $\nu
\approx {2/3}$.
In the case of the disordered vortex model, it is convenient to determine $z$ from the nonlinear I-V characteristics instead of the linear resistivity. Driving the system out of equilibrium makes it cheaper to converge the simulations, which is desirable for the disorder averaging. Fig. (\[iv\]) shows data for the nonlinear I-V characteristics. For the largest size studied, $L=64$, it is reasonable to assume that the data is in the range where $J > J_{\rm nl} \sim 1/L^2$. Hence, according to Eq. (\[Enonlin\]), the dynamical exponents can be obtained from power law fits of the form $E \sim
J^{(1+z)/2}$ to the data at $T_c$. The fits were done in the current interval where a power law best describes the data, leaving out the highest currents where saturation artifacts in the simulation limit the voltage. This gives $z_{\rm pure} = 2.60 \pm 0.1$ and $z_{\rm
dirty} = 2.69 \pm 0.1$. The small discrepancy between these values is within the statistical uncertainty of the simulations, and the values are consistent with the exponent obtained from the calculation of the linear resistivity. The coincidence of $z$ for the pure model with the result of linear resistivity gives a consistency test showing that the nonlinear I-V characteristics correctly determines $z$. In particular, our model assumption of a uniform current driving the vortices in the presence of the disorder does not seem to introduce any errors.
=
=
Summary and conclusion
======================
We have performed simulations of the dynamics of the 3-dimensional XY model in a vortex representation with and without magnetic screening. Without screening, we find $z\approx 3/2$, in agreement with earlier work of Weber and Jensen[@weber97], who note that this result agrees with the dynamical critical exponent of the phase model with spin wave degrees of freedom, model E[@hohenberg77]. However, the spin wave degrees of freedom are [*separated*]{} from the vortex degrees of freedom when going to the vortex representation[@villain75; @jose77; @kleinert89] and the remaining vortex degrees of freedom have only dissipative motion. Hence we find it quite surprising, that the exponents from spin wave and vortex dynamics agree numerically. We note that there is some experimental evidence for super-diffusive 3-dimensional XY dynamics in superconductors with $z\sim 1.25-1.5$, but only for the case of finite magnetic fields.[@ginsberg; @moloni-prl]
One can try to argue that it is the long-range forces among the vortices that accounts for the super-diffusive (i.e. $z<2$) behavior. Indeed one can even argue that the spin waves are implicitly present since they are what mediate the long-range forces. However, we note that, in this model, the long-range forces are [*instantaneous*]{} and not retarded. Hence it is still a mystery to us why the vortex model appears to be consistent numerically with Model E dynamics. Note that Lee and Stroud,[@lee92] who measured $I-V$ characteristics on the resistively shunted junction model (which is described by Model E dynamics) using Langevin (as opposed to Monte Carlo) dynamics, find $z=1.5\,(\pm 0.5)$, as expected for this model.
The value of $z$ for the short-range case does not seem to be significantly affected by the disorder, and so disorder appears to be irrelevant dynamically as well as statically. In both the pure and dirty case however, $z$ is significantly larger than is usually seen in relaxational dynamics where $z$ is typically only slightly larger than two.[@hohenberg77] There is some experimental evidence for such enhanced values of $z$. Anlage’s group[@3DXY2] finds $\nu=1.0\pm
0.2$, and $z=2.65\pm 0.3$. The value of $z$ is measured directly at the critical point from the resistivity scaling and is probably more reliable than the value of $\nu$ which is measured somewhat more indirectly (necessarily) using data away from the critical point.[@anlage-private-comm] It is possible therefore that the value of $\nu$ is actually consistent with the 3DXY value of 0.667. Moloni et al.[@moloni-prb] find $z=2.3\pm 0.2$.
These experimental values are consistent with the value we obtain here. However, in the extreme type-II limit, the inverted XY critical regime is expected to be very narrow and difficult to access experimentally. It is therefore unclear at this point how significant this agreement is. Further work is needed to estimate more precisely the crossover point to inverted XY behavior in real materials.
To conclude, we have raised a number of issues about experimental and theoretical uncertainties regarding the dynamical universality class of the superconducting phase transition in high $T_{\rm c}$ superconductors. Clearly considerably more work needs to be done to address this problem. One aspect that we have not yet addressed is the effect of disorder in the case of long-ranged unscreened interactions. A second problem worth pursuing is the precise theoretical relationship between the various possible spin wave dynamics in the phase representation and the corresponding dynamics of the same model in the various dual (vortex) representations.
C. W. and A. P. Y. wish to thank Hemant Bokil for useful discussions and the Maui High Performance Computing Center for an allocation of computer time. C. W. and A. P. Y. are supported by NSF DMR 94-11964. S. M. G. is supported by DOE MISCON DE-FG02-90ER45427 and NSF CDA-9601632 and thanks N. Goldenfeld and S. Anlage for illuminating discussion. J. L. and M. W. are supported by the Swedish Natural Science Research Council. A. P. Y. and S. M. G.acknowledge the support of the Aspen Center for Physics. J. L. and M. W. are supported by the Swedish Natural Science Research Council, by the Swedish Foundation for Strategic Research (SSF), and by the Swedish Council for High Performance Computing (HPDR) and Parallelldatorcentrum (PDC), Royal Institute of Technology.
D. S. Fisher, M. P. A. Fisher, and D. A. Huse, , 130 (1991).
Mark Friesen and Paul Muzikar, (unpublished).
M. B. Salamon et al., , 5520 (1993).
Steven M. Anlage et al., , 2792 (1996); James C. Booth et al., , 4438 (1996).
S. Kamal et al., 73, 1845 (1996).
Katerina Moloni, Mark Friesen, Shi Li, Victor Souw, P. Metcalf Lifang Hou, and M. McElfresh, Phys. Rev. Lett. [**78**]{}, 3173, (1997). (The low value of $z$ obtained here may be due to background effects. Ref.() uses a different analysis method to avoid this complication. Mark Friesen, private communication.)
Katerina Moloni, Mark Friesen, Shi Li, Victor Souw, P. Metcalf and M. McElfresh, Phys. Rev. B (in press, Dec. 1997).
Jin-Tae Kim, Nigel Goldenfeld, J. Giapintzakis, and D. M. Ginsberg, Phys. Rev. B [**56**]{}, 118 (1997).
P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. [**49**]{}, 435 (1977).
A. V. Carlson and A. M. Goldman, Phys. Rev. Lett. [**31**]{}, 880 (1973); [*Nonequilibrium Superconductivity, Phonons, and Kapitza Boundaries*]{}, ed. Kenneth E. Gray (Plenum, New York, 1980).
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H. Weber and H. J. Jensen, , 2620 (1997).
K. Moon and S. M. Girvin, , 1328 (1995).
In the vortex glass, where disorder is essential for the very existence of the phase transition, the dynamical exponent $z\sim4.7$ is quite large. See for example, J. D. Reger, T. A. Tokuyasu, A. P. Young and M. P. A. Fisher, Phys. Rev. B [**44**]{}, 7147 (1991). The effect of [*columnar*]{} disorder on the dynamics has been investigated and was found to be very significant. See, Mats Wallin and S. M. Girvin, , 14462 (1993); D. R. Nelson and Leo Radzihovksy, , R6845, (1996).
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See, e.g., J. M. Yeomans, [*Statistical Mechanics of Phase Transitions*]{}, (Oxford University Press, New York, 1992), p. 46.
In principle, global vortex line fluctuations, extending across the entire sample, are not energetically excluded in the short range case. These would change the winding number. However, since we are only interested in the local time evolution of the system, we use no such MC moves, thus restricting the MC trial moves to local plaquette moves also for the vortex model with screening.
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This overly-simplified model neglects possible inhomogeneous current distributions and Amp[è]{}res law. It is not known if this affects the critical properties we will consider. However, note that our result for $z$ for the short range model for the case of finite currents is consistent with the linear response result at zero applied current, suggesting that the critical dynamics is unaffected.
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S. Anlage (private communication).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $\matX$ be a data matrix of rank $\rho$, whose rows represent $n$ points in $d$-dimensional space. The linear support vector machine constructs a hyperplane separator that maximizes the 1-norm soft margin. We develop a new oblivious dimension reduction technique which is precomputed and can be applied to any input matrix ${\matX}$. We prove that, with high probability, the margin and minimum enclosing ball in the feature space are preserved to within -relative error, ensuring comparable generalization as in the original space in the case of classification. For regression, we show that the margin is preserved to $\epsilon$-relative error with high probability. We present extensive experiments with real and synthetic data to support our theory.'
author:
- 'SAURABH PAUL CHRISTOS BOUTSIDIS MALIK MAGDON-ISMAIL PETROS DRINEAS'
bibliography:
- 'references.bib'
title: Random Projections for Linear Support Vector Machines
---
A short version of this paper appeared in the 16th International Conference on Artificial Intelligence and Statistics (AISTATS 2013) [@PBMD13]. Note, that the short version of our paper [@PBMD13] does not include the details of the proofs, comparison of random projections with principal component analysis, extension of random projections for SVM regression in terms of both theory and experiments and experiments with fast SVM solver on RCV1 and Hapmap-HGDP datasets.
Christos Boutsidis acknowledges the support from XDATA program of the Defense Advanced Research Projects Agency (DARPA), administered through Air Force Research Laboratory contract FA8750-12-C-0323; Petros Drineas and Malik Magdon-Ismail are supported by NSF CCF-1016501 and NSF DMS-1008983; Saurabh Paul is supported by NSF CCF-916415.
Author’s addresses: S. Paul [and]{} M. Magdon-Ismail [and]{} P. Drineas, Computer Science Department, Rensselaer Polytechnic Institute, [email protected] [and]{} {magdon, drinep}@cs.rpi.edu ; C. Boutsidis, Mathematical Sciences Department, IBM T.J. Watson Research Center, [email protected].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper studies algorithms similar to the Gaussian elimination algorithm in symplectic and split orthogonal groups. We discuss two applications of this algorithm in computational group theory. One computes the spinor norm and the other computes the double coset decomposition with respect to Siegel maximal parabolic subgroup.'
address: 'IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, INDIA.'
author:
- 'Sushil Bhunia, Ayan Mahalanobis and Anupam Singh'
bibliography:
- 'paper.bib'
nocite: '[@praeger; @kantor]'
title: Gaussian Elimination in symplectic and split orthogonal groups
---
[^1]
Introduction
============
Gaussian elimination is a very old theme in Mathematics. It appeared in print as chapter eight in a Chinese mathematical text called, “The nine chapters of the mathematical art”. It is believed, a part of that book was written as early as 150 BCE. For a historical perspective on Gaussian elimination, we refer to a nice work by Grcar [@grc].
Due to many reasons [@ob3], computational group theorists became interested in the *constructive group recognition project*. We will not go in any details of this project, but will refer an interested reader to the works of Leedham-Green and O’Brein [@lo] and O’Brein [@ob1 Section 9]. One can also read a nice but slightly outdated review by Seress [@seress Matrix Groups] in this context. In dealing with constructive group recognition, one needs to solve the *word problem* in some generating set. As we know, in the special linear group SL$(d,k)$, the word problem has an efficient solution in elementary transvections – Gaussian elimination.
In this paper, we work with Chevalley generators [@ca §11.3]. Chevalley generators for the special linear group are elementary transvections. These Chevalley generators for other classical groups are known for a very long time. However, its use in row-column operations in symplectic and split orthogonal groups is new. We develop row-column operations, very similar to the Gaussian elimination algorithm for special linear groups. We call our algorithms Gaussian elimination in symplectic and split orthogonal groups respectively. Similar algorithm for twisted orthogonal groups and unitary groups [@aa] are being developed.
The current trend in *computational group theory* is to use *standard generators*. Using standard generators, Brooksbank [@brooksbank], Costi [@costi] solves the word problem in classical groups and Ambrose et. al. [@csaba] solves the membership problem in black-box groups. One advantage of using standard generators is that they are few in number. However, there is a disadvantages in working with them – they only work in finite fields. While working with Chevalley generators, our algorithms have two advantages:
- It works for arbitrary fields.
- It is much more efficient, as we demonstrate with an actual implementation in Magma [@magma], see Figures 1 & 2.
From our algorithm, one can compute the spinor norm easily, see Section \[spinornorm\]. Murray and Roney-Dougal [@mr] studied computing spinor norm earlier. Our algorithm can also be used to compute the double coset decomposition corresponding to the Siegel maximal parabolic subgroup, see Section \[parabolicdecomp\].
Algorithms that we develop in this paper work only for a given bilinear form $\beta$(see Equations \[beta2\], \[beta1\]). Our algorithm work well on all characteristics for symplectic groups. However, for orthogonal groups our algorithms work only for odd characteristic. Henceforth, by **suitable characteristics** we mean all characteristics for the symplectic groups and zero or odd characteristic for orthogonal groups. It is known, in a field of suitable characteristics, all non-degenerate skew-symmetric bilinear forms are equivalent and all split (maximal Witt index) symmetric bilinear forms are equivalent. These equivalent bilinear forms are obtained by a base-change matrix and gives rise to conjugate linear groups. Though in our algorithm, we work with only one bilinear form $\beta$, given by a fixed basis, with a suitable change of basis matrix our algorithm works on any equivalent bilinear forms.
Another way to look at this paper, we have an algorithmic proof of this well-known theorem. For definitions of elementary matrices, one can look ahead to Section \[chgenerators\].
\[thma\] Let $k$ be a field of suitable characteristic. For $d\geq 4$ or $l\geq 2$ following holds:
- Every element of the split orthogonal group $\text{O}(d,k)$ can be written as a product of elementary matrices and a diagonal matrix. Furthermore, the diagonal matrix is of the form $$\begin{aligned}
{\textup{diag}}(1,\ldots,1, \lambda,1,\ldots,1,\lambda^{-1}) & \lambda\in
k^\times &
\text{whenever}\;d=2l\\
{\textup{diag}}(\vartheta,1,\ldots,1,\lambda,1,\ldots,1,\lambda^{-1})&\lambda\in k^\times\; \text{and}\;
\vartheta=\pm 1& \text{whenever}\; d=2l+1.\end{aligned}$$
- Every element of the symplectic group $\text{Sp}(2l,k)$ can be written as a product of elementary matrices.
This theorem has a surprising corollary. It follows that:
In an orthogonal group, the image of $\lambda$ in $k^\times/k^{\times 2}$ is the spinor norm.
This means that we have an efficient algorithm to compute the spinor norm. Since the commutator subgroup of the orthogonal group is the kernel of the spinor norm restricted to special orthogonal group, the above corollary is a membership test for the commutator subgroup in the orthogonal group. In other words, an element $g$ in the special orthogonal group belongs to the commutator subgroup if and only if the $\lambda$ it produces in the Gaussian elimination algorithm is a square in the field, see Equation \[spin\].
The bilinear form that we use and the generators that we define have its roots in the abstract root system of a semisimple Lie algebra and Chevalley groups defined by Chevalley and Steinberg [@st2; @ch]. However we assume no knowledge of Lie theory or Chevalley groups in this paper.
Orthogonal and Symplectic Groups {#des2}
================================
We begin with a brief introduction of orthogonal and symplectic groups. We follow Carter [@ca], Taylor [@ta] and Grove [@gr] in our introduction. In this section, we **fix some notations** which will be used throughout this paper. We denote the transpose of a matrix $X$ by ${\ensuremath{{}^T\!\!}}{X}$.
Let $V$ be a vector space of dimension $d$ over a field $k$ of suitable characteristic. Let $\beta\colon V\times V \rightarrow k$ be a bilinear form. By fixing a basis of $V$ we can associate a matrix to $\beta$. We shall abuse the notation slightly and denote the matrix of the bilinear form by $\beta$ itself. Thus $\beta(x,y)={\ensuremath{{}^T\!\!}}x\beta y$ where $x,y$ are column vectors. We will work with non-degenerate bilinear forms, which implies, $\det\beta\neq 0$. A symmetric or skew-symmetric bilinear form $\beta$ satisfies $\beta={\ensuremath{{}^T\!\!}}\beta$ or $\beta=-{\ensuremath{{}^T\!\!}}\beta$ respectively. In even order, for characteristic 2, the symmetric and skew-symmetric forms are the same.
A square matrix $X$ of size $d$ is called orthogonal if ${\ensuremath{{}^T\!\!}}X\beta X=\beta$ where $\beta$ is symmetric. The set of orthogonal matrices form the orthogonal group.
In this paper, we deal with the split orthogonal group defined by one particular bilinear form defined over a field of zero or odd characteristics in Equations \[beta2\]. So any mention of orthogonal group means this one particular orthogonal group, unless stated otherwise.
A square matrix $X$of size $d$ is called symplectic if ${\ensuremath{{}^T\!\!}}X\beta
X=\beta$ where $\beta$ is skew-symmetric. The set of symplectic matrices form the symplectic group.
In this paper, we deal with the symplectic group defined by the bilinear form defined by Equation \[beta1\]. So any mention of symplectic group means this one particular symplectic group, unless stated otherwise.
We write the dimension of $V$ as $d$ where $d=2l+1$ or $d=2l$ and $l\geq 1$. In the case $\beta$ is symmetric we define the corresponding quadratic form $Q\colon V\rightarrow k$ by $Q(v)=\frac{1}{2}\beta(v,v)$. Up to equivalence, there is an unique non-degenerate skew-symmetric bilinear form over a field $k$ of suitable characteristics. Furthermore a skew-symmetric bilinear form exists only in even dimension. We fix a basis of $V$ as $\{e_1,\ldots, e_l, e_{-1},\ldots, e_{-l}\}$ so that the matrix $\beta$ is: $$\label{beta1}
\beta=\begin{pmatrix}0&I_l\\ -I_l&0\end{pmatrix}.$$ The symplectic group with this $\beta$ is denoted by $\text{Sp}(2l,k)$.
Up to equivalence, there is a unique non-degenerate symmetric bilinear form of maximal Witt index over a field $k$ of suitable characteristics. This is also called the split form. We fix a basis $\{e_0,e_1,\ldots,e_l,e_{-1},\ldots,e_{-l}\}$ for odd dimension and $\{e_1,\ldots, e_l,e_{-1},\ldots, e_{-l}\}$ for even dimension so that the matrix $\beta$ is: $$\label{beta2}
\beta=\left\{\begin{array}{ll}
\begin{pmatrix}2&0&0\\ 0&0&I_l\\ 0&I_l&0\end{pmatrix} & \text{when}\; d=2l+1\\
\begin{pmatrix}0&I_l\\ I_l&0\end{pmatrix} & \text{when}\; d=2l.
\end{array}\right.$$ The orthogonal group corresponding to this form is a *split orthogonal group*. In this paper, we will simply call it *the orthogonal group* and this group will be denoted by $\text{O}(d,k)$. If we need to emphasize parity of the dimension, we will write $\text{O}(2l+1,k)$ or $\text{O}(2l,k)$. We denote by $\Omega(d,k)$ the commutator subgroup of the orthogonal group $\text{O}(d,k)$ which is equal to the commutator subgroup of $\text{SO}(d,k)$. There is a well known exact sequence $$\label{spin}
1\longrightarrow \Omega(d,k)\longrightarrow
\text{SO}(d,k)\overset{\Theta}{\longrightarrow}
k^{\times}/{k^{\times}}^2\longrightarrow 1$$ where $\Theta$ is the spinor norm. The spinor norm is defined as $\Theta(g)=\underset{i=1}{\overset{m}{\prod}} Q(v_i)$ where $g=\rho_{v_1}\cdots\rho_{v_m}$ is written as a product of reflections. Since the group $\text{SO}(d,k)$ is of index $2$ in $\text{O}(d,k)$, we fix a generator for the quotient as $w_l=I-e_{l,l}-e_{-l,-l}-e_{l,-l}-e_{-l,l}$.
Elementary Matrices and Elementary Operations {#chgenerators}
=============================================
In what follows, the scalar $t$ varies over the field $k$ and $1\leq
i,j\leq l$. Furthermore, $l\geq 2$ which means $d\geq 4$. We define $te_{i,j}$ as the matrix unit with $t$ in the $(i,j)$ position and zero everywhere else. We use $e_{i,j}$ to denote $1e_{i,j}$. We often use the well known identity $e_{i,j}e_{k,l}=\delta_{j,k}e_{i,l}$ where $\delta_{i,j}$ is the Kronecker delta.
Elementary Matrices for $\text{O}(2l,k)$
----------------------------------------
We index rows by $1,2,\ldots,l,-1,-2,\ldots,-l$. The elementary matrices are defined as follows: $$\begin{aligned}
x_{i,j}(t)=&I+t(e_{i,j}-e_{-j,-i}) &\text{for}\; i\neq j,\\
x_{i,-j}(t)=&I+t(e_{i,-j}-e_{j,-i}) &\text{for}\;i<j,\\
x_{-i,j}(t)=&I+t(e_{-i,j}-e_{-j,i})&\text{for}\; i<j,\\
w_l=&I-e_{l,l}-e_{-l,-l}-e_{l,-l}-e_{-l,l}\end{aligned}$$ and in matrix format
1. $\begin{pmatrix}R&0\\0 & {\ensuremath{{}^T}\!}{R}^{-1}\end{pmatrix}$ where $R=I+te_{i,j}$; $i\neq j $
2. $\begin{pmatrix} I& R \\ 0&I\end{pmatrix}$ where $R$ is $t(e_{i,j}-e_{j,i})$ for $i<j$
3. $\begin{pmatrix} I& 0 \\ R &I\end{pmatrix}$ where $R$ is $t(e_{i,j}- e_{j,i})$ for $i<j$.
The row and column operations: $$\begin{aligned}
ER1:&\begin{pmatrix}R&0\\0 & {\ensuremath{{}^T}\!}{R}^{-1}\end{pmatrix}\begin{pmatrix}A&B\\C&D\end{pmatrix}&=\begin{pmatrix}RA&RB \\ {\ensuremath{{}^T}\!}{R}^{-1}C & {\ensuremath{{}^T}\!}{R}^{-1} D\end{pmatrix}\\
EC1:&\begin{pmatrix}A&B\\C&D\end{pmatrix}\begin{pmatrix}R&0\\0 & {\ensuremath{{}^T}\!}{R}^{-1}\end{pmatrix}&=\begin{pmatrix} AR&B{\ensuremath{{}^T}\!}{R}^{-1} \\ CR & D{\ensuremath{{}^T}\!}{R}^{-1}\end{pmatrix}.\end{aligned}$$ $$\begin{aligned}
ER2: & \begin{pmatrix}I&R\\0 &I\end{pmatrix}\begin{pmatrix}A&B\\C&D\end{pmatrix}&=\begin{pmatrix}A+RC&B+RD \\ C & D\end{pmatrix} \\
EC2: &\begin{pmatrix}A&B\\C&D\end{pmatrix}\begin{pmatrix}I&R\\0 &I \end{pmatrix}&=\begin{pmatrix} A&AR+B \\ C & CR+ D\end{pmatrix}.\end{aligned}$$ $$\begin{aligned}
ER3: &\begin{pmatrix}I&0\\R &I\end{pmatrix}\begin{pmatrix}A&B\\C&D\end{pmatrix}&=\begin{pmatrix}A&B \\ RA+ C & RB+D\end{pmatrix}\\
EC3: &\begin{pmatrix}A&B\\C&D\end{pmatrix}\begin{pmatrix}I&0\\ R&I \end{pmatrix}&=\begin{pmatrix} A+BR&B \\ C+DR & D\end{pmatrix}.\end{aligned}$$
Elementary Matrices for $\text{O}(2l+1,k)$
------------------------------------------
We index rows by $0, 1, \ldots,l,-1,\ldots,-l$. The elementary matrices are: $$\begin{aligned}
x_{i,j}(t)=&I+t(e_{i,j}-e_{-j,-i})& \text{for}\; i\neq j,\\
x_{i,-j}(t)=&I+t(e_{i,-j}-e_{j,-i}),&\text{for}\; i<j,\\
x_{-i,j}(t)=&I+t(e_{-i,j}-e_{-j,i})&\text{for}\; i<j ,\\ x_{i,0}(t)=&I+t(2e_{i,0}-e_{0,-i})-t^2e_{i,-i},&\\
x_{0,i}(t)=&I+t(-2e_{-i,0}+e_{0,i})-t^2e_{-i,i}.&\\\end{aligned}$$
Written in matrix format, these four kind of elementary matrices are:
1. $\begin{pmatrix}1&0&0\\0 &R&0\\ 0&0& {\ensuremath{{}^T}\!}R^{-1}\end{pmatrix}$ where $R=I+te_{i,j}$; $i\neq j$.
2. $\begin{pmatrix} 1&0&0\\ 0&I& R \\ 0&0&I\end{pmatrix}$ where $R$ is $t(e_{i,j}-e_{j,i})$; $i < j$.
3. $\begin{pmatrix} 1&0&0\\ 0&I&0 \\0 &R &I\end{pmatrix}$ where $R$ is $t(e_{i,j}-e_{j,i})$; $i < j$.
4. $\begin{pmatrix} 1&0&R\\-2R &I&-{\ensuremath{{}^T\!\!}}RR \\0 & 0 &I\end{pmatrix}\ \text{where}\; R=te_i $
5. $\begin{pmatrix} 1&R&0\\0&I&0 \\-2R &-{\ensuremath{{}^T\!\!}}RR &I\end{pmatrix}\ \text{where}\; R=te_i $
Here $e_i$ is the row vector with $1$ at $i^{\text{th}}$ place and zero elsewhere.
Elementary operations for O$(2l+1,k)$
-------------------------------------
Let $g=\begin{pmatrix}\alpha&X&Y\\ E& A&B\\F&C&D\end{pmatrix}$ be a $(2l+1)\times (2l+1)$ matrix where $A,B,C,D$ are $l\times l$ matrices. The matrices $X=(X_1,X_2,\ldots,X_l)$, $Y=(Y_1,Y_2,\ldots,Y_l)$, $E={\ensuremath{{}^T}\!}(E_1,E_2,\ldots,E_l)$ and $F={\ensuremath{{}^T}\!}(F_1,F_2,\ldots,F_l)$. Let $\alpha\in k$. Let us note the effect of multiplication by elementary matrices from above. $$\begin{aligned}
ER1:&\begin{pmatrix}1&0&0\\ 0&R&0\\0 &0& {\ensuremath{{}^T}\!}{R}^{-1}\end{pmatrix}\begin{pmatrix}\alpha & X&Y\\E& A&B\\F& C&D\end{pmatrix}&=\begin{pmatrix}\alpha &X&Y\\ RE&RA&RB \\ {\ensuremath{{}^T}\!}{R}^{-1}F&{\ensuremath{{}^T}\!}{R}^{-1}C & {\ensuremath{{}^T}\!}{R}^{-1} D\end{pmatrix}\\
EC1:&\begin{pmatrix}\alpha & X&Y\\ E&A&B\\F&C&D\end{pmatrix}\begin{pmatrix}1&0&0\\ 0&R&0\\ 0&0& {\ensuremath{{}^T}\!}{R}^{-1}\end{pmatrix}&=\begin{pmatrix} \alpha& XR & Y{\ensuremath{{}^T}\!}{R}^{-1}\\ E&AR&B{\ensuremath{{}^T}\!}{R}^{-1} \\ F& CR & D{\ensuremath{{}^T}\!}{R}^{-1}\end{pmatrix}.\end{aligned}$$ $$\begin{aligned}
ER2: & \begin{pmatrix}1&0&0\\0 &I&R\\ 0&0&I\end{pmatrix}\begin{pmatrix}\alpha &X&Y\\ E&A&B\\F&C&D\end{pmatrix}&=\begin{pmatrix}\alpha &X&Y\\ E+RF& A+RC&B+RD \\ F& C & D\end{pmatrix} \\
EC2:&\begin{pmatrix}\alpha& X&Y\\ E& A&B\\F& C&D\end{pmatrix}\begin{pmatrix}1&0&0\\ 0&I&R\\ 0&0&I \end{pmatrix}&=\begin{pmatrix} \alpha& X& XR+Y\\ E&A&AR+B \\ F& C & CR+ D\end{pmatrix}.\end{aligned}$$ $$\begin{aligned}
ER3: &\begin{pmatrix}1&0&0\\ 0&I&0\\0&R &I\end{pmatrix}\begin{pmatrix}\alpha & X&Y\\ E&A&B\\F&C&D\end{pmatrix}&=\begin{pmatrix}\alpha&X&Y\\ E&A&B \\ RE+F&RA+ C & RB+D\end{pmatrix}\\
EC3:&\begin{pmatrix}\alpha& X&Y\\ E&A&B\\F&C&D\end{pmatrix}\begin{pmatrix}1&0&0\\ 0&I&0\\ 0&R&I \end{pmatrix}&=\begin{pmatrix}\alpha& X+YR&Y\\E& A+BR&B \\ F&C+DR & D\end{pmatrix}.\end{aligned}$$ For E4 we only write the equations that we need later.
- Let the matrix $g$ has $C={\textup{diag}}(d_1,\ldots,d_l)$. $$ER4:\;[(I+te_{0,-i}-2te_{i,0}-t^2e_{i,-i})g]_{0,i}= X_i+td_i$$ $$EC4:\;[g(I+te_{0,-i}-2te_{i,0}-t^2e_{i,-i})]_{-i,0}= F_i-2td_i.$$
- Let the matrix $g$ has $A={\textup{diag}}(d_1,\ldots,d_l)$. $$ER4:\;[(I+te_{0,i}-2te_{-i,0}-t^2e_{-i,i})g]_{0,i}= X_i+td_i$$ $$EC4:\;[g(I+te_{0,-i}-2te_{i,0}-t^2e_{i,-i})]_{i,0}= E_i-2 td_i.$$
Elementary Matrices for $\text{Sp}(2l,k)$
-----------------------------------------
$$\begin{aligned}
x_{i,j}(t)=&I+t(e_{i,j}-e_{-j,-i}) &\text{for}\; i\neq j,\\
x_{i,-j}(t)=&I+t(e_{i,-j}+e_{j,-i})&\text{for}\; i<j,\\
x_{-i,j}(t)=&I+t(e_{-i,j}+e_{-j,i})& \text{for}\; i<j,\\
x_{i,-i}(t)=&I+te_{i,-i}&\\
x_{-i,i}(t)=&I+te_{-i,i}.&\end{aligned}$$
There are three kinds of elementary matrices.
1. $\begin{pmatrix}R&0\\0 & {\ensuremath{{}^T}\!}{R}^{-1}\end{pmatrix}$ where $R=I+te_{i,j}$; $i\neq j $.
2. $\begin{pmatrix} I& R \\0 &I\end{pmatrix}$ where $R$ is either $t(e_{i,j}+e_{j,i})$; $i<j$ or $te_{i,i}$.
3. $\begin{pmatrix} I& 0 \\ R &I\end{pmatrix}$ where $R$ is either $t(e_{i,j}+ e_{j,i})$; $i<j$ or $te_{i,i}$.
Elementary Operations for $\text{Sp}(2l,k)$
-------------------------------------------
Let $g=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ be a $2l\times 2l$ matrix written in block form of size $l\times l$. Then the row and column operations are as follows:
$$\begin{aligned}
ER1:&\begin{pmatrix}R&0\\ 0& {\ensuremath{{}^T}\!}{R}^{-1}\end{pmatrix}\begin{pmatrix}A&B\\C&D\end{pmatrix}&=\begin{pmatrix}RA&RB \\ {\ensuremath{{}^T}\!}{R}^{-1}C & {\ensuremath{{}^T}\!}{R}^{-1} D\end{pmatrix}\\
EC1:&\begin{pmatrix}A&B\\C&D\end{pmatrix}\begin{pmatrix}R&0\\0 & {\ensuremath{{}^T}\!}{R}^{-1}\end{pmatrix}&=\begin{pmatrix} AR&B{\ensuremath{{}^T}\!}{R}^{-1} \\ CR & D{\ensuremath{{}^T}\!}{R}^{-1}\end{pmatrix}.\end{aligned}$$
$$\begin{aligned}
ER2: & \begin{pmatrix}I&R\\ 0&I\end{pmatrix}\begin{pmatrix}A&B\\C&D\end{pmatrix}&=\begin{pmatrix}A+RC&B+RD \\ C & D\end{pmatrix} \\
EC2: &\begin{pmatrix}A&B\\C&D\end{pmatrix}\begin{pmatrix}I&R\\ 0&I \end{pmatrix}&=\begin{pmatrix} A&AR+B \\ C & CR+ D\end{pmatrix}.\end{aligned}$$
$$\begin{aligned}
ER3: &\begin{pmatrix}I&0\\R &I\end{pmatrix}\begin{pmatrix}A&B\\C&D\end{pmatrix}&=\begin{pmatrix}A&B \\ RA+ C & RB+D\end{pmatrix}\\
EC3: &\begin{pmatrix}A&B\\C&D\end{pmatrix}\begin{pmatrix}I&0\\ R&I \end{pmatrix}&=\begin{pmatrix} A+BR&B \\ C+DR & D\end{pmatrix}.\end{aligned}$$
**Remark:** We would require row-interchange of $i^{\text{th}}$ row with $-i^{\text{th}}$ row in our algorithms. In the case of symplectic and odd-order orthogonal groups this row-interchange is a product of elementary matrices. In the case of even-order orthogonal groups, we need to add to generators a row-interchange matrix $w_{l}$. For more see Lemma \[lemmaD\].
Gaussian Elimination in orthogonal and symplectic groups {#wordproblem}
========================================================
Recall the field $k$ is of suitable characteristic. Cohen, Murray and Taylor [@CMT] proposed a generalized row-column operations, using a representation of Chevalley groups. The key idea there was to bring down an element to a maximal parabolic subgroup and repeat the process inductively. The emphasis there was to represent generators as symbols so that it takes less memory to store. Here we use the natural matrix representation of these groups.
Gaussian Elimination for $\text{Sp}(2l,k)$ and $\text{O}(2l,k)$ {#GEsp}
---------------------------------------------------------------
The algorithm is as follows.
- **Input**: A matrix $g=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ which belongs to $\text{Sp}(2l,k)$ or $\text{O}(2l,k)$.
**Output**: The matrix $g_1=\begin{pmatrix}A_1&B_1\\C_1&D_1\end{pmatrix}$ is one of the following kind:
1. The matrix $A_1$ is a diagonal matrix ${\textup{diag}}(1,\ldots, 1, \lambda)$ with $\lambda\neq 0$ and $C_1$ is $\begin{pmatrix} C_{11}&C_{12}\\ C_{21}&c_{22}\end{pmatrix}$ where $C_{11}$ is symmetric when $g$ is in $\text{Sp}(2l,k)$ and skew-symmetric when $g$ is in $\text{O}(2l,k)$ and is of size $l-1$. Furthermore, $C_{12} = \lambda {\ensuremath{{}^T}\!}C_{21}$ when $g$ is in $\text{Sp}(2l,k)$ and $C_{12} = -\lambda {\ensuremath{{}^T}\!}C_{21}$, $c_{22}=0$ when $g$ is in $\text{O}(2l,k)$.
2. The matrix $A_1$ is a diagonal matrix ${\textup{diag}}(1,\ldots,1,0,\ldots,0)$ with number of $1$s equal to $m$ and $C_1$ is of the form $\begin{pmatrix}C_{11}&0\\ C_{21}& C_{22}\end{pmatrix}$ where $C_{11}$ is an $m\times m$ symmetric matrix when $g$ is in $\text{Sp}(2l,k)$ and skew-symmetric when $g$ is in $\text{O}(2l,k)$.
[**Justification**]{}: Observe the effect of ER1 and EC1 on the block $A$. This amounts to Gaussian elimination on a $l\times l$ matrix. Thus we can reduce $A$ to a diagonal matrix and Corollary \[corA\] makes sure that $C$ has required form.
- **Input:** matrix $g_1=\begin{pmatrix}A_1&B_1\\C_1&D_1\end{pmatrix}$.
**Output:** matrix $g_2=\begin{pmatrix}A_2&B_2\\0&{\ensuremath{{}^T}\!}A_2^{-1}\end{pmatrix}$; $A_2$ is a diagonal matrix ${\textup{diag}}(1,\ldots,1,\lambda)$.
**Justification**: Observe the effect of ER3. It changes $C_1$ to $RA_1+C_1$. Using Lemma \[lemmaC\] we can make the matrix $C_1$ the zero matrix in the first case and $C_{11}$ the zero matrix in the second case. Furthermore, in the second case, we use Lemma \[lemmaD\] to interchange the rows so that we get the zero matrix in the place of $C_1$. If required use ER1 and EC1 to make $A_1$ a diagonal matrix. Lemma \[lemmaB\] ensures that $D_1$ becomes ${\ensuremath{{}^T}\!}A_2^{-1}$.
- **Input:** matrix $g_2=\begin{pmatrix}A_2&B_2\\0&{\ensuremath{{}^T}\!}A_2^{-1}\end{pmatrix}$; $A_2$ is a diagonal matrix ${\textup{diag}}(1,\ldots,1,\lambda)$.
**Output:** Matrix $g_3=\begin{pmatrix}A_2&0\\0&{\ensuremath{{}^T}\!}A_2^{-1}\end{pmatrix}$; $A_2$ is diagonal matrix ${\textup{diag}}(1,\ldots,1,\lambda)$.
**Justification**: Using Corollary \[corB\] we see that the matrix $B_2$ has certain form. We can use ER2 to make the matrix $B_2$ a zero matrix because of Lemma \[lemmaC\].
The algorithm terminates here for $\text{O}(2l,k)$. However for $\text{Sp}(2l,k)$ there is one more step.
- **Input:** matrix $g_3={\textup{diag}}(1,\ldots,1,\lambda,1,\ldots,1,\lambda^{-1})$.
**Output:** Identity matrix
**Justification**: This can be written as a product of elementary matrices by the first part of Lemma \[lemmaE\].
Gaussian Elimination for $\text{O}(2l+1,k)$ {#GEo}
-------------------------------------------
An overview of the algorithm is as follows:
- **Input:** matrix $g=\begin{pmatrix}\alpha&X&Y\\ E&A&B\\F&C&D\end{pmatrix}$ which belongs to $\text{O}(2l+1,k)$;
**Output:** matrix $g_1=\begin{pmatrix}\alpha & X_1 &Y_1 \\ E_1& A_1 &B_1\\ F_1& C_1&D_1\end{pmatrix}$ of one of the following kind:
1. The matrix $A_1$ is a diagonal matrix ${\textup{diag}}(1,\ldots, 1, \lambda)$ with $\lambda\neq 0$.
2. The matrix $A_1$ is a diagonal matrix ${\textup{diag}}(1,\ldots,1,0,\ldots,0)$ with number of $1$s equal to $m$ and $m<l$.
**Justification**: Using ER1 and EC1 we do the classical Gaussian elimination on a $l\times l$ matrix $A$.
- **Input:** matrix $g_1=\begin{pmatrix}\alpha & X_1 & Y_1 \\ E_1& A_1 &B_1\\ F_1& C_1&D_1 \end{pmatrix}$.
**Output:** matrix $g_2=\begin{pmatrix} \alpha_2 & X_2 & Y_2 \\ E_2& A_2 &B_2\\ F_2& C_2& D_2\end{pmatrix}$ of one of the following kind:
1. The matrix $A_2$ is ${\textup{diag}}(1,1,\ldots, 1, \lambda)$ with $\lambda\neq 0$, $X_2=0=E_2$ and $C_2$ is of the form $\begin{pmatrix} C_{11}&-\lambda {\ensuremath{{}^T}\!}C_{21}\\ C_{21}& 0 \end{pmatrix}$ where $C_{11}$ is skew-symmetric of size $l-1$.
2. The matrix $A_2$ is ${\textup{diag}}(1,\ldots,1,0,\ldots,0)$ with number of $1$s equal to $m$; $X_2$ and $E_2$ have first $m$ entries $0$, and $C_2$ is of the form $\begin{pmatrix}C_{11}&0\\ C_{21}& C_{22}\end{pmatrix}$ where $C_{11}$ is an $m\times m$ skew-symmetric.
**Justification**: Once we have $A_1$ in diagonal form we use ER4 and EC4 to change $X_1$ and $E_1$ in the required form. Then Lemma \[lemmaF\] makes sure that $C_1$ has required form.
- **Input:** matrix $g_2=\begin{pmatrix} \alpha_2 & X_2 & Y_2 \\ E_2& A_2 &B_2\\ F_2& C_2& D_2\end{pmatrix}$.
**Output:**
1. matrix $g_3=\begin{pmatrix}\alpha_3 & 0 & Y_3 \\ 0& A_3 &B_3\\ F_3& 0 & D_3 \end{pmatrix}$ where $A_3$ is ${\textup{diag}}(1,1,\ldots, 1, \lambda)$.
2. matrix $g_3=\begin{pmatrix}\alpha_3 & X_3 & Y_3 \\ E_3& A_3 &B_3\\ F_3& C_3& D_3 \end{pmatrix}$ where $A_3$ is ${\textup{diag}}(1,\ldots,1,0,\ldots,0)$ with number of $1$s equal to $m$; $X_3$ and $E_3$ have first $m$ entries $0$, and $C_3$ is of the form $\begin{pmatrix}0&0\\ C_{21}& C_{22}\end{pmatrix}$.
**Justification**: Observe the effect of ER3 and the Lemma \[lemmaC\] ensures the required form.
- **Input:** $g_3=\begin{pmatrix}\alpha_3 & X_3 & Y_3 \\ E_3& A_3 &B_3\\ F_3& C_3& D_3 \end{pmatrix}$.
**Output:** $g_4=\begin{pmatrix}\vartheta &0&0\\ 0&A_4& B_4 \\0&0&A_4^{-1}\end{pmatrix}$ with $A_4$ diagonal matrix ${\textup{diag}}(1,\ldots,1,\lambda)$.
**Justification**: In the first case, Lemma \[lemmaG\] ensures the result. In the second case we interchange $i\textsuperscript{th}$ with $-i\textsuperscript{th}$ for $m+1\leq i
\leq l$. This will make $C_3=0$. Then if needed we use ER1 and EC1 on $A_3$ to make it a diagonal. Lemma \[lemmaI\] ensures that $A_3$ has full rank. Furthermore, we can use ER4 and EC4 to make $X_3=0$ and $E_3=0$. Lemma \[lemmaG\] gives the required form.
- **Input:** $g_4=\begin{pmatrix}\vartheta&0&0\\ 0&A_4&B_4\\0&0&A_4^{-1}\end{pmatrix}$ with $A_4={\textup{diag}}(1,\ldots,1,\lambda)$.
**Output:** $g_5={\textup{diag}}(\vartheta, 1\ldots, 1, \lambda, 1, \ldots, 1,\lambda^{-1})$.
**Justification**: Lemma \[lemmaG\] ensures that $B_4$ is of certain kind. Use ER2 to make $B_4=0$. Lemma \[lemmaD\] ensures that the first diagonal entry is $\vartheta$.
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- Let $g\in \text{O}(d,k)$. Using the above algorithm we can reduce $g$ to a diagonal matrix of the required form.
- If $g\in\text{Sp}(2l,k)$ using above algorithm and Lemma \[lemmaE\] we can reduce $g$ to the identity. Thus $g$ is a product of elementary matrices.
This algorithm can be used to compute the determinant of an orthogonal group. In the even-order case, refer to Algorithm 4.1. After Step 1, either $A_1$ has full rank, or the rank of $A_1$ is $m$. In the first case the determinant of $g$ is 1 and in the second case it is $(-1)^{l-m}$. It is fairly straightforward to see this. Observe that all elementary matrix other than the row-interchange matrix has determinant one and the number of row-interchange that we do is $l-m$, where $m$ is the rank of $A_1$.
In the odd-order case, as we have observed that the row-interchange matrix is a product of elementary matrices and all elementary matrices have determinant one. It then follows clearly that the determinant of $g$ is $\vartheta$.
Time-complexity of the above algorithm
--------------------------------------
We establish that the worst case time-complexity of the above algorithm is $\mathrm{O}(l^3)$.
- In Step 1, we make $A$ a diagonal matrix by row-column operations. That has complexity $\mathrm{O}(l^3)$.
- In Step 2, $C_1+RA_1$ is multiplying two rows by a field element and two additions. In the worst case, it has to be done $O(l)$ times and done $O(l^2)$ many times. So the complexity is $\mathrm{O}(l^3)$.
- Step 3 is similar to Step 2 above and has complexity $\mathrm{O}(l^3)$.
- Step 4 has only a few steps that is independent of $l$.
Then clearly, the time-complexity of our algorithm is $\mathrm{O}(l^3)$.
Lemmas used in the justification of the Gaussian elimination
------------------------------------------------------------
To justify the steps of Gaussian algorithm we need several lemmas. Some of these might be well known to experts but we include them here for the convenience of the reader.
\[lemmaA\] Let $Y={\textup{diag}}(1,\ldots,1,\lambda,\ldots,\lambda)$ be of size $l$ with number of $1$s equal to $m<l$. Let $X$ be a matrix of size $2l$ such that $YX$ is symmetric (skew-symmetric) then $X$ is of the form $\begin{pmatrix}X_{11}& \lambda {\ensuremath{{}^T\!\!}}X_{21}\\ X_{21}&X_{22}\end{pmatrix}$ where $X_{11}$ is symmetric (skew symmetric) and $X_{12}=\lambda {\ensuremath{{}^T\!\!}}X_{21}$ ($X_{12}=- \lambda {\ensuremath{{}^T\!\!}}X_{21}$). Furthermore, if $\lambda\neq 0$ then $X_{22}$ is symmetric (skew-symmetric).
We observe that the matrix $YX=\begin{pmatrix}X_{11}& X_{12}\\\lambda X_{21}&\lambda X_{22}\end{pmatrix}$. The condition that $YX$ is symmetric implies $X_{11}$ (and $X_{22}$ if $\lambda\neq 0$) is symmetric and $X_{12}=\lambda {\ensuremath{{}^T}\!}X_{21}$.
\[corA\] Let $g=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ be either in $\text{Sp}(2l,k)$ or $\text{O}(2l,k)$.
1. If $A$ is a diagonal matrix ${\textup{diag}}(1,\ldots,1,0,\ldots,0)$ with number of $1$s equal to $m(<l)$ then the matrix $C$ is $\begin{pmatrix}C_{11}&0\\ C_{21}& c_{22}\end{pmatrix}$ where $C_{11}$ is an $m\times m$ symmetric if $g$ is symplectic and is skew-symmetric if $g$ is orthogonal.
2. If $A$ is a diagonal matrix ${\textup{diag}}(1,1,\ldots,1,\lambda)$ then the matrix $C$ is $\begin{pmatrix}C_{11}&\lambda {\ensuremath{{}^T\!\!}}C_{21}\\ C_{21}& c_{22}\end{pmatrix}$ where $C_{11}$ is an $(l-1)\times (l-1)$ symmetric if $g$ is symplectic and $C_{11}$ is skew-symmetric with $c_{22}=0$ if $g$ is orthogonal.
We use the condition that $g$ satisfies ${\ensuremath{{}^T}\!}g\beta g=\beta$ and get $AC$ is symmetric (using $A={\ensuremath{{}^T}\!}A$ as $A$ is diagonal) when $g$ is symplectic and skew-symmetric when $g$ is orthogonal. The Lemma \[lemmaA\] gives the required form for $C$.
\[corB\] Let $g=\begin{pmatrix}A&B\\0&A^{-1}\end{pmatrix}$ where $A={\textup{diag}}(1,\ldots,1,\lambda)$ be an element of either $\text{Sp}(2l,k)$ or $\text{O}(2l,k)$ then the matrix $B$ is of the form $\begin{pmatrix}B_{11}&\pm\lambda^{-1} {\ensuremath{{}^T\!\!}}B_{21}\\ B_{21}& B_{22}\end{pmatrix}$ where $B_{11}$ is a symmetric matrix of size $l-1$ if $g$ is symplectic and is skew-symmetric along with $B_{22}=0$ if $g$ is orthogonal.
Yet again, we use the condition that $g$ satisfies ${\ensuremath{{}^T}\!}g\beta g=\beta$ and $A={\ensuremath{{}^T\!\!}}A$ to get $A^{-1}B$ is symmetric if $g$ is symplectic and is skew-symmetric if $g$ is orthogonal. Then Lemma \[lemmaA\] gives the required form for $B$.
\[lemmaB\] Let $g=\begin{pmatrix}A&B\\0&D\end{pmatrix}\in \text{GL}(2l,k)$. Then,
1. the element $g$ belongs to $\text{Sp}(2l,k)$ if and only if $D={\ensuremath{{}^T}\!}A^{-1}$ and $BA=A{\ensuremath{{}^T\!\!}}B$ and
2. the element $g$ belongs to $\text{O}(2l,k)$ if and only if $D={\ensuremath{{}^T}\!}A^{-1}$ and $BA=-A{\ensuremath{{}^T\!\!}}B$.
This follows by simple computation using ${\ensuremath{{}^T}\!}g \beta g=\beta$.
\[lemmaC\] Let $Y={\textup{diag}}(1,1,\ldots,1,\lambda)$ be of size $l$ where $\lambda\neq 0$ and $X=(x_{ij})$ be a matrix such that $YX$ is symmetric (skew-symmetric). Then $X=(R_1+R_2+\ldots)Y$ where each $R_m$ is of the form $t(e_{i,j}+e_{j,i})$ for some $i < j$ or of the form $te_{i,i}$ for some $i$ (in the case of skew-symmetric each $R_m$ is of the form $t(e_{i,j}-e_{j,i})$ for some $i < j$).
Since $YX$ is symmetric, the matrix $X$ is of the form $\begin{pmatrix}X_{11}&X_{12}\\ X_{21} & x_{ll}\end{pmatrix}$ where $X_{11}$ is symmetric and $X_{21}$ is a row of size $l-1$ and $X_{12} = \lambda {\ensuremath{{}^T\!\!}}X_{21}$. Clearly any such matrix is sum of the matrices of the form $R_mY$. A similar calculation proves the result for the skew-symmetric case.
We need certain Weyl group elements which can be used for interchanging rows. We use a formula $w_{r}=x_r(1)x_{-r}(-1)x_r(1)$ from the theory of Chevalley groups [@ca Lemma 6.4.4] and construct elements $w_{i,j}$ and $w_{i,-j}$. In our algorithm, we need elements which interchanges $i\textsuperscript{th}$ row with $-i\textsuperscript{th}$ row for any $i$. The element $w_{i,-i}$ that we create when multiplied to a matrix $g$ interchanges its rows while simultaneously multiplying some rows by $-1$. However that does no harm to our algorithm.
\[lemmaD\] For $1\leq i \leq l$,
1. the element $w_{i,-i}=I +e_{i,-i}- e_{-i,i}-e_{i,i} - e_{-i,-i}\in\text{Sp}(2l,k)$ is a product of elementary matrices.
2. the element $w_{i,-i}=I-2e_{0,0}-e_{i,i}-e_{i,-i}-e_{-i,-i}-e_{-i,i}\in \text{O}(2l+1,k)$ is a product of elementary matrices.
3. The element $w_{i,-i}=I -e_{i,-i}- e_{-i,i}-e_{i,i} - e_{-i,-i}\in\text{O}(2l,k)$ is a product of elementary matrices.
For the symplectic group $\text{Sp}(2l,k)$ we have $w_{i,-i}= x_{i,-i}(1)x_{-i,i}(-1)x_{i,-i}(1)$. For the orthogonal group $\text{O}(2l+1,k)$ we have $w_{i,-i}=x_{0,i}(-1)x_{i,0}(1)x_{0,i}(-1)$.
For the orthogonal group $\text{O}(2l,k)$ we inductively produce these elements. First we get $w_{i,-j}= (I+e_{i,-j}-e_{j,-i})(I+e_{-i,j}-e_{-j,i})(I+e_{i,-j}-e_{j,-i}) = I-e_{i,i}-e_{j,j}-e_{-i,-i}-e_{-j,-j}+e_{i,-j}-e_{j,-i}+e_{-i,j}-e_{-j,i}$ and $w_{i,j}=I-e_{i,i}-e_{j,j}+e_{i,j}-e_{j,i}-e_{-i,-i}-e_{-j,-j}+e_{-i,-j}-e_{-j,-i}$. Now we set $w_{l,-l}=w_l$. Then compute $w_{(l-1),-(l-1)}=w_l w_{l,l-1}w_{l,-(l-1)}= I-e_{l-1,l-1}-e_{-(l-1),-(l-1)}-e_{(l-1),-(l-1)}-e_{-(l-1),(l-1)}$.
\[lemmaE\]
1. In the case of $\text{Sp}(2l,k)$, the element ${\textup{diag}}(1,\ldots,1\lambda,1,\ldots,1,\lambda^{-1})$ is a product of elementary matrices.
2. In the case of $\text{O}(2l+1,k)$ the diagonal element ${\textup{diag}}(-1,1,\ldots,1,\lambda^2,1,\ldots,1,\lambda^{-2})$ is a product of elementary matrices.
In the case of $\text{Sp}(2l,k)$, we compute $w_{l,-l}(t)=(I+te_{l,-l})(I-t^{-1}e_{-l,l})(I+te_{l,-l}) = I-e_{l,l}-e_{-l,-l}+te_{l,-l}-t^{-1}e_{-l,l}$ and then compute $h_l(\lambda)=w_{l,-l}(\lambda)w_{l,-l}(-1)$ which is the required element.
In the case of $\text{O}(2l+1,k)$ we compute $w_{l,0}(t)=x_{0,l}(-t)x_{l,0}(t^{-1})x_{0,l}(-t) = I-e_{-l,-l}-t^2e_{-l,l}-e_{l,l}-2e_{0,0}-t^{-2}e_{l,-l}$ and multiply it with $w_{l,0}(1)$ to get the required matrix.
**Remark :** In the case of $\text{O}(2l,k)$, the element ${\textup{diag}}(1,\ldots,1\lambda^2,1,\ldots,1,\lambda^{-2})$ and in the case of $\text{O}(2l+1,k)$ the element ${\textup{diag}}(1,1,\ldots,1\lambda^2,1,\ldots,1,\lambda^{-2})$ is a product of elementary matrices.
\[lemmaF\] Let $g=\begin{pmatrix}\alpha&X&*\\ *&A&*\\ *&C&*\end{pmatrix}$ be in $\text{O}(2l+1,k)$.
1. If $A={\textup{diag}}(1,\ldots,1,\lambda)$ and $X=0$ then $C$ is of the form $\begin{pmatrix}C_{11}&-\lambda {\ensuremath{{}^T}\!}C_{21}\\ C_{21}& 0 \end{pmatrix}$ with $C_{11}$ skew-symmetric.
2. If $A={\textup{diag}}(1,\ldots,1,0,\ldots,0)$ with number of $1$s equal $m<l$ and $X$ has first $m$ entries $0$ then $C$ is of the form $\begin{pmatrix}C_{11}& 0 \\ *&*\end{pmatrix}$ with $C_{11}$ skew-symmetric.
We use the equation ${\ensuremath{{}^T}\!}g \beta g=\beta$ and get $2{\ensuremath{{}^T}\!}XX= -({\ensuremath{{}^T}\!}CA+{\ensuremath{{}^T}\!}AC)$. In the first case $X=0$, so use \[corA\] to get the required form for $C$. In the second case, we note that ${\ensuremath{{}^T}\!}XX$ has top-left block $0$ and get the required form.
\[lemmaI\] Let $g=\begin{pmatrix}\alpha&X&Y\\ *&A&*\\ *&0&D\end{pmatrix}$ be in $\text{O}(2l+1,k)$ then $X=0$ and $D={\ensuremath{{}^T}\!}A^{-1}$.
We compute ${\ensuremath{{}^T}\!}g\beta g=\beta$ and get $2{\ensuremath{{}^T}\!}XX=0$ and $2{\ensuremath{{}^T}\!}XY+{\ensuremath{{}^T}\!}AD=I$. This gives the required result.
\[lemmaG\] Let $g=\begin{pmatrix}\alpha& 0&Y\\ 0&A&B\\F&0&D\end{pmatrix}$, with $A$ an invertible diagonal matrix. Then, $g\in \text{O}(2l+1,k)$ if and only if $\alpha^2=1, F=0=Y$, $D=A^{-1}$ and ${\ensuremath{{}^T}\!}DB+{\ensuremath{{}^T}\!}BD=0$.
$$\begin{aligned}
{\ensuremath{{}^T}\!}g\beta g &=& \begin{pmatrix}\alpha& 0&{\ensuremath{{}^T}\!}F\\ 0&{\ensuremath{{}^T}\!}A& 0 \\{\ensuremath{{}^T}\!}Y& {\ensuremath{{}^T}\!}B&{\ensuremath{{}^T}\!}D\end{pmatrix} \begin{pmatrix}2& 0&0\\ 0&0&I\\0&I&0\end{pmatrix}\begin{pmatrix}\alpha& 0&Y\\ 0&A&B\\F&0&D\end{pmatrix}\\
&=& \begin{pmatrix}2\alpha^2& {\ensuremath{{}^T}\!}F A &2\alpha Y+{\ensuremath{{}^T}\!}FB\\ {\ensuremath{{}^T}\!}AF &0 &{\ensuremath{{}^T}\!}AD\\2\alpha {\ensuremath{{}^T}\!}Y +{\ensuremath{{}^T}\!}BF& {\ensuremath{{}^T}\!}DA & 2{\ensuremath{{}^T}\!}YY +{\ensuremath{{}^T}\!}DB+{\ensuremath{{}^T}\!}BD\end{pmatrix}.\end{aligned}$$
Equating this with $\beta$ gives us the required result.
Computing Spinor Norm for orthogonal groups {#spinornorm}
===========================================
In this section, we show how we can use Gaussian elimination to compute spinor norm for orthogonal groups. The classical way to define spinor norm is via Clifford algebras [@gr Chapters 8 & 9]. Spinor norm is a group homomorphism $\Theta\colon \text{O}(d,k)\rightarrow k^{\times}/{k^{\times}}^2$, restriction of which to $\text{SO}(d,k)$ gives $\Omega(d,k)$ as kernel. However, in practice, it is difficult to use that definition to compute the spinor norm. Wall [@wa], Zassenhaus [@za] and Hahn [@ha] developed a theory to compute the spinor norm. For our exposition, we follow [@ta Chapter 11].
Let $g$ be an element of the orthogonal group. Consider $g$ as a linear transformation. Futhermore, denote $\tilde g=I-g$ and $V_g=\tilde g(V)$ and $V^g=ker(\tilde g)$. Using $\beta$ we define Wall’s bilinear form $[\ ,\ ]_g$ on $V_g$ as follows: $$[u,v]_g=\beta(u,y),\ \text{where},\ v=\tilde g(y).$$ This bilinear form satisfies following properties:
1. $[u,v]_g+[v,u]_g=\beta(u,v)$ and $[u,u]_g=Q(u)$ for all $u,v\in V_g$.
2. $g$ is an isometry on $V_g$ with respect to $[\ ,\ ]_g$.
3. $[v,u]_g=-[u,gv]$ for all $u,v\in V_g$.
4. $[\ ,\ ]_g$ is non-degenerate.
Then the **spinor norm** is $$\Theta(g)= \overline{\text{disc}(V_g,[\ ,\ ]_g)}\; \text{if}\; g\neq I$$ extended to $I$ by defining $\Theta(I)=\overline 1$. An element $g$ is called regular if $V_g$ is non-degenerate subspace of $V$ with respect to the form $\beta$. Hahn [@ha Proposition 2.1] proved that for a regular element $g$ the spinor norm is $\Theta(g)=\overline{\det(\tilde g|_{V_g})\text{disc}(V_g)}$. This gives,
\[propospinor\]
1. For a reflection $\rho_v$, $\Theta(\rho_v)=\overline{Q(v)}$.
2. $\Theta(-1)=\overline{\text{disc}(V,\beta)}$.
3. For a unipotent element $g$ the spinor norm is trivial, i.e., $\Theta(g)=\overline{1}$.
Murray and Roney-Dougal [@mr] used the formula of Hahn to compute spinor norm. However, we show (Corollary B) that the Gaussian elimination developed in Section \[wordproblem\] ouptuts the spinor norm quickly. First we observe the following:
\[spinornorm\] For the group $\text{O}(d,k)$, $d\geq 4$,
1. $\Theta(x_{i,j}(t))=\Theta(x_{-i,j}(t))=\Theta(x_{i,-j}(t))=\overline{1}$. Furthermore, in odd case we also have $\Theta(x_{i,0}(t))=\overline 1=\Theta(x_{0,i}(t))$.
2. $\Theta(w_l)=\overline{1}$.
3. $\Theta({\textup{diag}}(1,\ldots,1,\lambda,1,\ldots,1,\lambda^{-1}))=\overline{\lambda}$.
We use Proposition \[propospinor\]. The first claim follows from the fact that all elementary matrices are unipotent. The element $w_l=\rho_{(e_l+e_{-l})}$ is a reflection thus $\Theta(w_l)=\overline{Q(e_l+e_{-l})}=\overline{1}$.
For the third part we note that ${\textup{diag}}(1,\ldots,1,\lambda,1,\ldots,1,\lambda^{-1}) = \rho_{(e_l+e_{-l})} \rho_{(e_l+\lambda e_{-l})}$ and hence the spinor norm $\Theta({\textup{diag}}(1,\ldots,1,\lambda,1,\ldots,1,\lambda^{-1}) )=\Theta(\rho_{(e_l+\lambda e_{-l})}) =\overline{Q(e_l+\lambda e_{-l})}=\overline{\lambda}$.
Let $g\in\text{O}(d,k)$. From Theorem A, we write $g$ as a product of elementary matrices and a diagonal matrix ${\textup{diag}}(1,\ldots,1,\lambda,1,\ldots,1,\lambda^{-1})$. Furthermore, the spinor norm of an elementary matrix is $\overline 1$. Thus $\Theta(g)=\Theta({\textup{diag}}(1,\ldots,1,\lambda,1,\ldots,1,\lambda^{-1}))=\overline{\lambda}$.
Double coset decomposition for Siegel maximal parabolic {#parabolicdecomp}
=======================================================
In this section, we compute the double coset decomposition with respect to Siegel maximal parabolic subgroup using our algorithm. Let $P$ be the Siegel maximal parabolic of $G$ where $G$ is either $\text{O}(d,k)$ or $\text{Sp}(2l,k)$. In Lie theory, a parabolic is obtained by fixing a subset of simple roots [@ca Section 8.3]. Siegel maximal parabolic corresponds to the subset consisting of all but the last simple root. Geometrically, a parabolic subgroup is obtained as fixed subgroup of a totally isotropic flag [@mt Proposition 12.13]. The Siegel maximal parabolic is the fixed subgroup of following isotropic flag (with the basis in Section \[des2\]): $$\{0\}\subset \{e_1,\ldots,e_l\}\subset V.$$ Thus $P$ is of the form $\begin{pmatrix} \alpha&0&Y\\ E&A&B\\F&0&D\end{pmatrix}$ in $\text{O}(2l+1,k)$ and $\begin{pmatrix} A&B\\0&D\end{pmatrix}$ in $\text{Sp}(2l,k)$ and $\text{O}(2l,k)$.
The problem is to get the double coset decomposition $P\backslash G/ P$. That is, we want to write $G=\underset{\omega\in\hat W}{\bigsqcup} P\omega P $ as disjoint union where $\hat W$ is a finite subset of $G$. Equivalently, given $g\in G$ we need an algorithm to determine the unique $\omega\in \hat W$ such that $g\in
P\omega P$. If $G$ is connected with Weyl group $W$ and suppose $W_P$ is the Weyl group corresponding to $P$ then [@ca2 Proposition 2.8.1] $$P\backslash G/P \longleftrightarrow W_P\backslash W/W_P.$$ We need a slight variation of this as the orthogonal group is not connected.
In the case of $\text{Sp}(2l,k)$, the Weyl group $W=N(T)/T$ where $T$ is a diagonal maximal torus and $T=\{{\textup{diag}}(\lambda_1,\ldots,\lambda_l,\lambda_1^{-1},\ldots,\lambda_l^{-1})\mid
\lambda_i\in k^{\times}\}$. The group $W$ is isomorphic to a subgroup of $S_{2l}$, the symmetric group on $2l$ symbols $\{1,\ldots, l,
-1,\ldots,-l\}$ and is generated by elements $w_{i,i+1}$ and $w_{i,-i}$ which map to permutations $(i,i+1)(-i,-(i+1))$ and $(i,-i)$ respectively. Thus $W$ is isomorphic to $S_l\rtimes (\mathbb
Z/2\mathbb Z)^l$ and the subgroup $W_P$ is generated by $w_{i,i+1}$ which proves that the subgroup $\{(i,i+1)(-i,-(i+1))\mid 1\leq i\leq
l\}$ is isomorphic to $S_l$.
For Sp$(2l,k)$, we set $\hat W=\{\omega_0=I, \omega_i=w_{1,-1}\cdots w_{i,-i}\mid 1\leq i\leq l\}$ and note that $W={\underset{i=0}{\overset{l}\bigsqcup}} W_P\omega_iW_P$.
In the case of $\text{O}(d,k)$, we set $\hat W=\{\omega_0=I,
\omega_i=w_{1,-1}\cdots w_{i,-i}\mid 1\leq i\leq l\}$ where $w_{i,-i}$ is inductively produced (see Lemma \[lemmaD\]).
\[thmbruhatdecomposition\] Let $P$ be the Siegel maximal parabolic subgroup in $G$, where $G$ is either $\text{O}(d,k)$ or $\text{Sp}(d,k)$. Let $g\in G$. Then there is an efficient algorithm to determine $\omega$ such that $g\in P\omega P$. Furthermore, $\hat
W$ the set of all $\omega$s is a finite set of $l+1$ elements where $d=2l$ or $2l+1$.
In this proof we proceed with a similar but slightly different Gaussian elimination algorithm. Recall that $g=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ whenever $g$ belongs to $\text{Sp}(2l,k)$ or $\text{O}(2l,k)$ or $g=\begin{pmatrix}\alpha&X&Y\\ E&A&B\\F&C&D\end{pmatrix}$ whenever $g$ belongs to $\text{O}(2l+1,k)$. In our algorithm, we made $A$ into a diagonal matrix. Instead of that, we can use elementary matrices ER1 and EC1 to make $C$ into a diagonal matrix and then do the row interchange to make $A$ into a diagonal matrix and $C$ a zero matrix. If we do that, we note that elementary matrices E1 and E2 are in $P$. The proof is just keeping track of elements of $P$ in this Gaussian elimination algorithm. The step 1 in the algorithm says that there are elements $p_1, p_2\in P$ such that $p_1gp_2 = \begin{pmatrix} A_1&B_1\\C_1&D_1\end{pmatrix}$ where $C_1$ is a diagonal matrix with $m$ non-zero entries. Clearly $m=0$ if and only if $g\in P$. In that case $g$ is in the double coset $P\omega_0P=P$. Now suppose $m\geq 1$. Then in Step 2 we multiply by E2 to make the first $m$ rows of $A_1$ zero, i.e., there is a $p_3\in P$ such that $p_3p_1gp_2 = \begin{pmatrix} \tilde A_1 & \tilde B_1\\C_1&D_1\end{pmatrix}$ where first $m$ rows of $\tilde A_1$ are zero. After this we interchange rows $i$ with $-i$ for $1\leq i\leq m$ which makes $C_1$ zero, i.e., multiplying by $\omega_m$ we get $\omega_m p_3p_1gp_2=\begin{pmatrix} A_2&B_2\\0&D_2\end{pmatrix}\in P$. Thus $g\in P\omega_m P$.
For $\text{O}(2l+1,k)$ we note that the elementary matrices E1, E2 and E4a are in $P$. Rest of the proof is similar to the earlier case and follows by carefully keeping track of elementary matrices used in our algorithm in Section \[GEo\].
Conclusions
===========
We conclude this paper with some implementation results of our algorithm and some comparisons with an existing algorithm. Before we state those, we wander a little towards a direction for further research. Chevalley generators are known for a long time. However, its use in row-column operations is new. Earlier, in computational group theory, while working with quasisimple groups, the generators of choice were always the standard generators. This use of Chevalley generaotrs can bring in a paradigm shift with algorithms in matrix groups. It is now a very interesting project to redo the constructive group recognition project [@lo] with our algorithms and Chevalley generators.
Now some implementation results, we implemented our algorithm in magma [@magma]. We found our implementation to be fast and stable. In magma, Costi and C. Schneider installed a function *ClassicalRewriteNatural*. It is the row-column operation developed by Costi [@costi] in natural representation. We tested the time taken by our algorithm and the one taken by the Magma function. To do this test, we followed Costi [@costi Table 6.1] as closely as possible. Two kind of simulations were done. In one case, we fixed the size of the field at $7^{10}$ and varied the size of the matrix from $20$ to $60$. To time both these algorithms for any particular input, we took one thousand random samples from the group and run the algorithm for each one of them. Then the final time was the average of this one thousand random repetitions. The times were tabulated and presented below.
In the other case, we kept the size of the matrix fixed at $20$ and we varied the size of the field, keeping the characteristic fixed at $7$. In many cases the magma computation for the function ClassicalRewriteNatural will not stop in a reasonable amount of time or will give an error and not finish computing. In those cases, though our algorithm worked perfectly, we were unable to get adequate data to plot and are represented by gaps in the graph drawn. Here also the times are the average of one thousand random repetitions.
 
 
It seems that our algorithms perform better than that of Costi’s on all fronts.
[^1]: This work is supported by a SERB research grant.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Many embryonic deformations during development are the global result of local cell shape changes and other local active cell sheet deformations. Morphogenesis does not only therefore rely on the ability of the tissue to produce these active deformations, but also on the ability to regulate them in such a way as to overcome the intrinsic variability of and geometric constraints on the tissue. Here, we explore the interplay of regulation and variability in the green alga *Volvox*, whose spherical embryos turn themselves inside out to enable motility. Through a combination of light sheet microscopy and theoretical analysis, we quantify the variability of this inversion and analyse its mechanics in detail to show how shape variability arises from a combination of geometry, mechanics, and active regulation.'
author:
- 'Pierre A. Haas'
- 'Stephanie S. M. H. Höhn'
- 'Aurelia R. Honerkamp-Smith'
- 'Julius B. Kirkegaard'
- 'Raymond E. Goldstein'
bibliography:
- 'inv2.bib'
title: 'Mechanics and Variability of Cell Sheet Folding in the Embryonic Inversion of *Volvox*'
---
Introduction
============
Julian Huxley’s pronouncement, “In some colony like \[the green alga\] *Volvox*, there once lay hidden the secret to the body and shape of \[humans\]” [@huxley], emphasises that morphogenesis across kingdoms relies on the fundamental ability of organisms to, firstly, produce active forces that drive the deformations of cell sheets underlying the development of many organs and tissues and, secondly, regulate these active deformations in such a way to complete morphogenesis. Unravelling the biomechanics of these processes is therefore of crucial importance to understand pathological errors and foster bioengineering to address these errors [@sasai12]. Local cellular changes can produce forces that are transmitted along the cell sheet to drive its global deformations [@lecuit07; @lecuit11]. Simple events of cell sheet folding such as ventral furrow formation in *Drosophila* can be driven primarily by cell shape changes [@sweeton91]. In more complex metazoan developmental processes such as gastrulation [@leptin05; @wang09], optic cup formation [@fuhrmann10; @chauhan15], neurulation [@lowery04; @vija17] and related processes [@sherrard10], the effect of such cell shape changes is overlaid by that of other cellular changes such as cell migration, cell intercalation, cell differentiation, and cell division.
In all of these processes however, these local cellular changes occur in specific regions of the cell sheet and at specific stages of morphogenesis. On the one hand, the spatio-temporal distribution of these local cellular changes affects the global tissue shape. On the other hand, a certain amount of noise is unavoidable in biological systems; indeed, it may even be necessary for robust development, as demonstrated for example by [@hong16], who showed that variability in cell growth is necessary for reproducible sepal size and shape in *Arabidopsis*. While some processes may be subject to less intrinsic variability than others, one must therefore ask: how are these processes orchestrated so that development can complete despite the intrinsic biological variability? Differences in the observed shapes of organisms at certain stages of development (i.e. what one might term their geometric variability) stem from a combination of mechanical variability (i.e. differences in mechanical properties or mechanical state) and active variability (i.e. differences in the active forces generated by individual cells). What experimental data there are suggest that the mechanical properties are subject to a large amount of variability [@vondassow07 and references therein]. Finally, differences in the mechanical stress state of the tissue are another facet of mechanical variability that is induced by active variability.
The first mechanical models of morphogenesis [@odell81] represented cells as discrete collections of springs and dashpots; they were soon followed by elastic continuum models [@hardin86; @hardin88]. Notable among this early modelling of morphogenesis is for example the work of @davidson95 (***[-@davidson95]***, ***[-@davidson99]***), who combined models of several mechanisms of sea urchin gastrulation with measurements of mechanical properties to test the plausibility of these different mechanisms. These models heralded the emergence of a veritable plethora of mechanical modelling approaches over the subsequent decades [@fletcher17], though the choice of model must ultimately be informed by the questions one seeks to answer [@rauzi13]. More recent endeavours were directed at deriving models that can represent the chemical and mechanical contributions to morphogenesis and their interactions [@howard11] and at establishing the continuum laws that govern these out-of-equilibrium processes [@prost15].
There is, however, a rather curious gap in the study of the variability of development: the importance of quantifying the morphogenesis and its variability has been recognised [@cooper08; @oates09], yet accounts of the variability of development, e.g. in the loach [@cherdantsev05; @cherdantsev16], have often been merely descriptive. For this reason, the interplay between mechanics and active variability has seemingly received little attention and hence a question we believe to be fundamental appears to lie in uncharted waters: how does active variability lead to geometric variability? Conversely, what does geometric variability tell us about active variability?
![Habitus of and embryonic inversion in *Volvox globator*. (a) Adult spheroid with somatic cells and one embryo labelled. Scale bar: $50\,\text{\textmu m}$. (b) Schematic drawing of *Volvox* globator parent spheroid with embryos. ECM: extracellular matrix. CB: cytoplasmic bridges. (c) Schematic drawing of *Volvox* embryo before inversion, with anterior and posterior poles, and phialpore labelled. Cells are teardrop-shaped \[T\]. (d) *Volvox* invagination: the formation of wedge-shaped cells \[W\] in the bend region initiates inversion. At the same time, cells in the posterior become spindle-shaped \[S\], while cells in the anterior close to the anterior cap become disc-shaped \[D\]. (d) At the end of posterior inversion, cells in the whole of the anterior hemisphere are disc-shaped, while cells in the bend region are pencil-shaped \[P\]. (f) As the anterior hemisphere peels over the inverted posterior, more and more cells become pencil-shaped. Red lines in panels (c–f) mark position of cytoplasmic bridges. Panels (c–f) adapted from [@hohn11].[]{data-label="fig:volvox"}](inv_Fig1)
![Cell shape changes in *Volvox globator* [@hohn11] associated with bending and stretching of the cell sheet. Cell shape changes (black arrows) (a) from teardrop-shaped to paddle-shaped cells in combination with movement relative to the cytoplasmic bridges (CBs), associated with invagination of the bend region; (b) from teadrop-shaped to spindle-shaped cells, associated with contraction of the posterior hemisphere; (c) from teardrop-shaped to disc-shaped cells, associated with expansion of the anterior hemisphere (before opening of the phialopore) and (d) from disc to pencil-shaped, associated with contraction of the anterior hemisphere and involution over the anterior cap. Red line: position of the CBs, blue arrows: direction of view of cell groups shown.[]{data-label="fig:cellshapechanges"}](inv_Fig2)
This is the question that we explore in this paper in the context of the development of the multicellular green alga *Volvox* (Fig. \[fig:volvox\]a). *Volvox* and the related Volvocine algal genera have been recognised since the work of [@weismann] as model organisms for the evolution of multicellularity [@kirkbook; @kirkessay; @herron16], spawning more recent investigations of kindred questions in fluid dynamics and biological physics [@goldstein15]. The cells of *Volvox* (Fig. \[fig:volvox\]b) are differentiated into biflagellated somatic cells and a small number of germ cells, or gonidia, that will form daughter colonies [@kirkbook]. The somatic cells in the adult are embedded in a glycoprotein-rich extracellular matrix [@kirk86; @hallmann03]. The germ cells undergo several rounds of cell division, after which each embryo consists of several thousand cells arrayed to form a thin spherical sheet confined to a fluid-filled vesicle. Cells are connected to their neighbours by cytoplasmic bridges (Fig. \[fig:volvox\]b), thin membrane tubes resulting from incomplete cell division [@green81a; @green81b; @hoops06]. Those cell poles whence will emanate the flagella however point into the sphere at this stage, and so the embryos must turn themselves inside out through an opening at the anterior pole of the cell sheet (the phialopore), to enable motility and thus complete their development [@kirkbook]. Because of this process of inversion, *Volvox* has become a model organism for the study of cell sheet deformations, too [@kirkchapter; @kirkreview; @matt16].
Inversion in *Volvox* [@viamontes77; @hohn11] and in related species [@hallmann06; @iida11; @iida13; @hohn16] results from cell shape changes only, without the complicating additional processes found in metazoan development discussed above. This simplification facilitates the study of morphogenesis. While different species of *Volvox* have developed different ways of turning themselves inside out [@hallmann06], here, we focus on the so-called type-B inversion arising, for example, in *Volvox globator* [@zimmermann25; @hallmann06; @hohn11]. This shares features such as invagination and involution with developmental events in metazoans [@keller11; @feroze15; @czerniak16]. This inversion scenario is distinct from type-A inversion, in which four lips open at the anterior of the shell and peel back to achieve inversion [@viamontes77]. Type-B inversion begins with the appearance of a circular bend region at the equator of the embryo (Fig. \[fig:volvox\]c,d, Fig. \[fig:cellshapechanges\]a): cells there become wedge-shaped by developing narrow basal stalks [@hohn11]. At the same time, the cells move relative to the cytoplasmic bridges so as to be connected at their thin stalks, thus splaying the cells and bending and, eventually, invaginating the cell sheet [@hohn11]. [@nishii03] showed that inversion is arrested in the absence of analogous motion of cells relative to the cytoplasmic bridges in type-A inversion in *Volvox carteri*. The relative motion results from a kinesin associated to the microtubule cytoskeleton (Fig. \[fig:cellshapechanges\], figure supplement 1); orthologues of this kinesin are found throughout the Volvocine algae [@kirkessay]. After invagination, the posterior hemisphere moves into the anterior (Fig. \[fig:volvox\]e), the phialopore widens and the anterior hemisphere moves over the subjacent posterior (Fig. \[fig:volvox\]f) while ‘rolling’ over a second circular bend region, the anterior cap [@hohn11]. Additional cell shape changes (Fig. \[fig:volvox\]d–f, Fig. \[fig:cellshapechanges\]b–d) in the anterior and posterior hemispheres are implicated in the relative contraction and expansion of either hemisphere with respect to the other [@hohn11]. This plethora of cell shape changes is possible as *Volvox* cells do not have a cell wall [@kirkbook].
In a previous study [@hohn15], we combined light sheet microscopy and theory to analyse the early stages of inversion, showing that only a combination of active bending and active stretching (i.e. expansion or contraction) can account for the cell sheet deformations observed during invagination. The crucial role of active stretching was also highlighted by [@nishii99] who showed that type-A inversion in *Volvox carteri* cannot complete if acto-moysin mediated contraction is inhibited chemically. We later analysed the mechanics of this competition between bending and stretching in more detail [@haas15]. Here, we analyse experimentally the variability of the shapes of inverting *Volvox globator* at consecutive stages of inversion. We refine our theoretical model to capture later stages of the inversion process, and finally combine theory and experiment to untangle the geometric, mechanical, and active contributions to the observed spatial structure of the shape variations.
![**Video 1.** Timelapse video of inverting *Volvox globator* embryo from selective-plane illumination imaging of chlorophyll autofluorescence. Left: maximum intensity projection of z-stacks. Right: tracing of midsagittal cross-section (Methods). Scale bar: $50\,\text{\textmu m}$.[]{data-label="video:inversion"}](Video1_thumbnail.png){width="70.00000%"}
Results
=======
We acquired three-dimensional time-lapse visualisations of inverting *Volvox globator* embryos (Video 1) using a selective-plane-illumination-microscopy setup (Methods) based on the <span style="font-variant:small-caps;">OpenSPIM</span> system [@OpenSPIM]. Data were recorded for 13 parent spheroids containing, on average, 6 embryos. Summary statistics for 33 embryos were obtained from the recorded z-stacks and, for a more quantitative analysis of inversion, embryo outlines were traced on midsagittal sections of 11 of the recorded inversion processes, selected for optimal image quality (Methods).
![Quantification of 11 inversion processes. (a) Distance $e$ from the posterior pole to the plane of the circular bend region, normalised by its initial value $e_0$; (b) Surface area $A$ computed from a surface of revolution obtained from each half of traced outlines, scaled by its maximal value $A_{\max}$; (c) Minimum (most negative) value $\kappa_*$ of the curvature in the bend region; (d) Diameter of the phialopore $d$ normalised with its maximal value $d_{\max}$; (e) Width $w$ of the bend region (region of negative curvature), normalised by arclength of the traced shape; (f) Position $p$ of the bend region defined as the distance from the posterior pole to the centre of the bend region, normalised by arclength of the traced shape. Measurements (blue lines) on midsagittal embryo outlines are aligned at the time where $e$ reaches half of its initial value. Averages (red lines) and standard deviations thereof (red shaded areas) are shown for timepoints for which measurements were obtained for at least half of the quantified inversion processes. Insets: cartoons of definitions of $e$, $d$, $w$, $p$.[]{data-label="fig:measurements"}](inv_Fig3)
In our previous work [@hohn15], we discussed in detail three geometric descriptors of the traced embryo outlines, which we have reproduced for this dataset (Methods):
1. the distance $e$ (Fig. \[fig:measurements\]a) from the posterior pole to the plane of the circular bend region; this serves as an indicator of the progress of the ‘upwards’ movement of the posterior hemisphere;
2. the embryonic surface area $A$ (Fig. \[fig:measurements\]b), which was computed by determining a surface of revolution from each half of the midsagittal slice and averaging the two values for each timepoint;
3. the minimal (most negative) value $\kappa_\ast$ of the meridional curvature in the bend region (Fig. \[fig:measurements\]c).
We have computed three additional descriptors associated with the progress of later of inversion:
1. the diameter $d$ of the phialopore (Fig. \[fig:measurements\]d) as an indicator of progress of inversion of the anterior hemisphere;
2. the width $w$ of the bend region (Fig. \[fig:measurements\]e), where the bend region is defined as the region of negative curvature;
3. the position of the bend region (Fig. \[fig:measurements\]f), measured along the arclength of the deformed shell from the posterior pole to the midpoint of the bend region.
The computation of these descriptors is discussed in the Methods section. Each of these descriptors evolves in qualitatively similar ways in individual embryos, yet their evolution occurs over different timescales in different embryos and the local maxima in surface area (Fig. \[fig:measurements\]b) and in phialopore width (Fig. \[fig:measurements\]d) occurs at different relative times in different embryos. This initial impression of the variability of inversion is confirmed by the analysis of three summary statistics: (1) the duration of inversion, from appearance of the bend region to closure of the phialopore, (2) the diameter of the embryos post-inversion, (3) the relative time during inversion at which the phialopore starts to open. Histograms of these quantities in Fig. \[fig:summarystats\]a–c reveal considerable variability, thus showing that the noise does not only affect the global duration of inversion, but also the relative timing of parts of it. We additionally note that there is no correlation between the size of an embryo and the duration of its inversion (Fig. \[fig:summarystats\]d), not even between embryos from the same parent spheroid.
![Summary statistics for inversion from $N=33$ embryos. Histograms of (a) duration of inversion, (b) embryo diameter (after inversion), and (c) relative time of phialopore opening. (d) Duration of inversion plotted against embryo diameter (after inversion). Data points corresponding to embryos from the same parent spheroid are shown in the same colour.[]{data-label="fig:summarystats"}](inv_Fig4)
It is natural to ask to what extent the different deformations of inversion must arise in a particular order: while invagination occurs before phialopore opening in all our samples, analysis of characteristic ‘checkpoints’ of inversion (Fig. \[fig:summarystats\], figure supplement 1) reveals that there is still considerable leeway in the timing of posterior inversion and phialopore opening. To further quantify the variability of inversion, we must define an average inversion sequence; our averaging approach must take into account these different types of variability.
The Local Variability of Inversion
----------------------------------
To define an average inversion sequence and analyse its mechanics, we compare the local geometry of the traced curves. The question of how to define an appropriate metric for this kind of comparison goes back at least to the work of D’Arcy Thompson [@thompson], and is altogether a rather philosophical one, to which there is no unique answer. Thompson showed for example how the outlines of fish of different species could be mapped onto one another by dilations, shears, and compositions thereof. In *Volvox* inversion, these shape differences are likely to arise from variations in cell shape and variations in the positions of cell shape changes. Our averaging method must therefore allow for these local variations as well as for differences in the timing of the cell shape changes (as suggested by the analysis of the summary statistics), while recognising that the posterior poles and the rims of the phialopores of the different embryos must correspond to each other. Our approach is therefore based on minimising the Euclidean distance between individual embryo shapes and their averages, with alignments obtained using dynamic time warping (Methods and Fig. \[fig:averages\], figure supplement 1). Results are shown in Fig. \[fig:averages\].
Averaging approaches that do not consider both stretching in time of individual inversions and local stretching of corresponding points of individual shapes tend to give unsatisfactory results: the simplest averaging approach is to align the inversion sequences by a single time point, say when the posterior-to-bend distance reaches half of its initial value (Methods and Fig. \[fig:averages\], figure supplement 2). The absence of time stretching, however, means that large variations arise at later stages of inversion. (Given the dramatic embryonic shape changes during inversion, it is not suprising that there should be no single parameter that could be used to align inversions of different embryos.) A better alignment is obtained if we allow stretching in time (Methods and Fig. \[fig:averages\], figure supplement 3), but this method, without local stretching of individual shapes relative to each other, produces unsatisfactory kinks in the bend region of the average shapes (Fig. \[fig:averages\], figure supplement 3).
![Average Stages of Inversion. $N=22$ overlaid and scaled embryo halves from experimental data (lines in shades of blue), and averages thereof (red lines), for ten stages of inversion.[]{data-label="fig:averages"}](inv_Fig5)
The averages reveal that inversion seems to proceed at an approximately constant speed relative to the average inversion sequence (Fig. \[fig:average\_stats\]a,b). However, the alignment reveals that different stages of inversion take different times in different embryos (Fig. \[fig:average\_stats\]a), with some embryos seeming to linger in certain stages. This is the same non-linearity that we already saw earlier on the timelines in Fig. \[fig:summarystats\], figure supplement 1 obtained from the measurements in Fig. \[fig:measurements\].
To analyse the local variations of the embryo shapes, we define, at each point of the average shapes, a covariance ellipse. The curves that are parallel to the average shape and tangent to the covariance ellipse define what we shall term the standard deviation shape. These standard deviation shapes measure the variability of the average shapes and are shown in Fig. \[fig:stdshapes\]. The variations they represent naturally divide into two components: first, those variations that are parallel to the average shape, and second, those perpendicular to the average shape. The former represent mere local stretches of the average shapes, while the latter correspond to actual variations of the shapes; we shall therefore refer to the thickness of the standard deviation shapes as ‘shape variation’ in what follows. We report the mean shape variation and its standard error in Fig. \[fig:average\_stats\]c. This plot shows that the mean shape variation reaches a maximal value around the stages in Fig. \[fig:stdshapes\]g–i: different embryos start from the same shape and reach the same inverted shape after inversion (up to a scaling), but may take different inversion paths. Plotting the mean shape variation for different averaging methods (Fig. \[fig:average\_stats\], figure supplement 1), we confirm that the present averaging method yields a better alignment than the alternative methods discussed earlier.
![Alignment Statistics. (a) Timepoints $t$ for $N=22$ embryo halves (relative to first fitted timepoint) plotted against the mean values $\langle t\rangle$ of these times. Red line: time evolution illustrating non-linear progression of inversion. Insets: average embryo shapes at earliest and latest fitted times. (b) Histogram of $\mathrm{R}^2$ statistic for fits of the time evolutions in the first panel to a model of constant inversion speed. (c) Mean shape variation (in arbitrary units), and standard errors thereof, against mean time $\langle t\rangle$. Corresponding panels in Fig. \[fig:stdshapes\] are marked for some data points.[]{data-label="fig:average_stats"}](inv_Fig6)
![Local Variations of *Volvox* Shapes during Inversion. Average shapes from Fig. \[fig:averages\] (red lines) and corresponding standard deviation shapes (shaded areas).[]{data-label="fig:stdshapes"}](inv_Fig7)
It is intriguing, however, to note the spatial structure of the local shape variations. In particular, during the early stages of posterior inversion (Fig. \[fig:stdshapes\]d–f), the shape variation is smaller in the active bend region than in the adjacent anterior cap (Fig. \[fig:volvox\]e, the second bend region of increased positive curvature). As the phialopore opens and the anterior begins to peel back over the partially inverted posterior (Fig. \[fig:stdshapes\]h) the relative shape variations becomes smaller in the anterior cap. The initially small variation in the bend region is especially intriguing since this is where cells become wedge-shaped to drive invagination, while the anterior cap bends passively [@hohn15]. In other words, the shape variation is reduced in the part of the cell sheet where the active cell shape changes driving inversion arise. This correspondence characterises what one might term, from a teleological point of view, a ‘good’ inversion. We shall focus on a less exalted question, the answer to which will be falsifiable, however: how is this spatial structure of the variability related to the mechanics of inversion? Before addressing this question, we need to analyse the mechanics of inversion in some more detail.
Active Bending and Stretching during Inversion
----------------------------------------------
Which active deformations are required for inversion? In our previous work [@hohn15; @haas15], we addressed this question for the early stages of inversion: at a mechanical level of description, invagination arises from an interplay of active bending and stretching [@hohn15; @haas15] associated with different types of cell shape changes . A key role is played by the cells close to the equator of the cell sheet (Fig. \[fig:volvox\]d, Fig. \[fig:cellshapechanges\]a), which become wedge-shaped [@hohn11], thus splaying the cells and hence imparting intrinsic curvature to the cell sheet [@haas15]. Yet no such cell wedging has been reported at the anterior cap at later stages of inversion, when the anterior hemisphere peels back over the partly inverted posterior (Fig. \[fig:volvox\]f, Fig. \[fig:cellshapechanges\]d).
To resolve this conundrum, we ask whether the additional cell shape changes observed during type-B inversion [@hohn11] are sufficient to explain anterior peeling: cells in the anterior hemisphere have flattened, ellipsoidal shapes, while cells on the posterior side of the anterior cap are pencil-shaped (Fig. \[fig:volvox\]e, Fig. \[fig:cellshapechanges\]d). We have previously described the early stages of inversion using a mathematical model [@hohn15] in which cell shape changes appear as local variations of the intrinsic (meridional and circumferential) curvatures $\kappa_s^0,\kappa_\phi^0$ and stretches $f_s^0,f_\phi^0$ of an elastic shell. We recall the difference between open, one-dimensional elastic filaments and two-dimensional elastic shells in this context: the former can simply adopt a shape in which the curvature and stretch are everywhere equal to their intrinsic values. For the latter, by contrast, the intrinsic curvatures and stretches may not be compatible with the global geometry, causing the shell to deform elastically and adopt actual (meridional and circumferential) curvatures $\kappa_s,\kappa_\phi$ and stretches $f_s,f_\phi$ different from the imposed intrinsic curvatures and stretches. In order to address these later stages of inversion, we must first generalise our previous mathematical model using ideas from morphoelasticity (Methods). Indeed, that model was derived under the assumption of small strains. While the *elastic* strains are small indeed (since the metric tensor, which describes the deformed shape, is close to the intrinsic tensor defined by the cell shape changes), the *geometric* strains are large: both the metric tensor of the deformed shell and the intrinsic tensor differ considerably from the metric tensor of the undeformed sphere.
![Mechanics of Anterior Peeling. Functional form of (a) the meridional intrinsic stretch $f_s^0$ and (b) the circumferential intrinsic stretch $\smash{f_\phi^0}$, with position $X(t)$ of the peeling front indicated. (c) Definition of the position $X$ of the peeling front and its initial value $X_0$ at the equator of the undeformed shell. The shaded area indicates the posterior hemisphere in which the intrinsic curvatures are equal and opposite to those of the undeformed sphere. Inset: definitions of the intrinsic stretches $\smash{f_s^0,f_\phi^0}$. (d) Shape before peeling, with inverted posterior hemisphere. (e) Resulting shape after anterior peeling, just before phialopore closure, with $X_0$ and $X$ indicated.[]{data-label="fig:antpeeling"}](inv_Fig8)
To test whether anterior peeling can be achieved by contraction of the cell sheet alone, we impose functional forms for the intrinsic stretches of the shell (Fig. \[fig:antpeeling\]a–c) representing these cell shape changes, but we do not modify the intrinsic curvatures in the anterior hemisphere (Fig. \[fig:antpeeling\]c). In particular, the linear variation of the circumferential stretch in the anterior hemisphere represents the different orientations of the ellipsoidal cells at the phialopore (Fig. \[fig:cellshapechanges\]c), where the long axis is the circumferential axis, and at the anterior cap, where the long axis is the meridional axis [@hohn11]. In our quasi-static simulation, we approximate the shape in Fig. \[fig:averages\]h by a configuration with inverted posterior hemisphere (Fig. \[fig:antpeeling\]d), and displace the intrinsic ‘peeling front’ (Fig. \[fig:antpeeling\]a,b). The shell responds by peeling (Fig. \[fig:antpeeling\]e), with shapes in qualitative agreement with the experimentally observed shapes. Since the peeling front is located at the anterior cap, where the shape variation is reduced during anterior peeling as discussed previously, we again see a correlation between reduced shape variations and the location of the active cell shape changes driving inversion.
These considerations suggest that contraction is sufficient to drive the peeling stage of inversion, even without changes in intrinsic curvature. Although the position of the cytoplasmic bridges (Fig. \[fig:cellshapechanges\]d), on the inside end of the cells at the end of inversion [@hohn11], suggests that the intrinsic curvature may change sign in the anterior hemisphere, too, this appears to be a seconday effect. Hence intrinsic bending complements intrinsic stretching. By contrast, our previous work [@hohn15] revealed that stretching complements bending during invagination. The roles of stretching and bending are thus interchanged during inversion of the posterior and anterior hemispheres, and the embryo uses these two different deformation modes for different tasks during inversion.
### Analysis of Cell Shape Changes
For a more quantitative analysis of the data and to validate our model, we proceed to fit the elastic model to the experimental average shapes (Methods). In the model, we impose a larger extent of the phialopore than in the biological system, where the phialopore is initially very small (Fig. \[fig:averages\]a). This is an important simplification to deal with the discrete nature of the few cells that meet up at the phialopore. Nonetheless, using fifteen fitting parameters to represent previously observed cell shape changes [@hohn11] in terms of the intrinsic stretches and curvatures (Methods), the model captures the various stages of inversion (Fig. \[fig:fittedshapes\]). This supports our interpretation of the observed cell shape changes (Fig. \[fig:cellshapechanges\]) and their functions. Comparing the geometric descriptors discussed previously (Fig. \[fig:measurements\]) for the experimental averages and the fitted shapes (Fig. \[fig:fittedshapes\], figure supplement 1), we notice that the fitted shapes underestimate the width of the bend region. Because curvature is a second derivative of shape, it is not surprising that larger differences arise in the minimal bend region curvature of the average and fitted shapes (Fig. \[fig:fittedshapes\], figure supplement 1).
![Average Embryo Shapes reproduced by the elastic model. In each panel, the left half shows average shapes from Fig. \[fig:averages\] (thick red line) and corresponding fits (black line) from the elastic model for different stages of inversion. The right half shows colour-coded representations of the meridional curvature $\kappa_s$ and stretches $f_s$ and $f_\phi$ in the fitted shapes.[]{data-label="fig:fittedshapes"}](inv_Fig9)
Nonetheless, the fitted values of the intrinsic curvature of the cell sheet also resolve a cell shape conundrum: during invagination, the curvature in the bend region increases (Fig. \[fig:fittedshapes\]), yet [@hohn11] reported similar wedge-shaped cells in the bend region at early and late invagination stages, although the number of wedge-shaped cells in the bend region increases as invagination progresses [@hohn11]. The fitted parameters indeed suggest a constant value of the intrinsic curvature at early stages of inversion, while the actual curvature in the bend region increases (Fig. \[fig:params\]a). This serves to illustrate that the intrinsic parameters cannot simply be read off the deformed shapes and confirms that there is but a single type of cell change, expanding in a wave to encompass more cells, and thus driving invagination. It is only at later stages of inversion, when the wedge-shaped cells in the bend region become pencil-shaped [@hohn11], that both the intrinsic curvature and the actual curvature in the bend region decrease (Fig. \[fig:params\]a). The fitted shapes also yield the posterior and anterior limits of the bend region (Fig. \[fig:params\]b), i.e. the original positions, relative to the undeformed sphere, of the corresponding cells. Because of the varying spatial stretches of the shell, these positions cannot simply be read off the deformed shapes, but must be inferred from the fits. The fitted data suggest that invagination results from an intrinsic bend region of constant width, complemented by other cell shape changes (Fig. \[fig:volvox\]d, Fig. \[fig:cellshapechanges\]b,c). The region of wedge-shaped cells (and, by implication, of negative intrinsic curvature) starts to expand into the posterior at constant speed (i.e. at a constant number of cell shape changes per unit of time) between the stages in Fig. \[fig:fittedshapes\]e,f. Anterior inversion starts about five minutes later when this region begins to expand into the anterior just after the stage in Fig. \[fig:fittedshapes\]g.
![Analysis of fitted parameters. (a) Plot of most negative values of the intrinsic and actual meridional curvatures, $\kappa^0_\ast=-\min{\kappa_s^0}$ and $\kappa_\ast=-\min{\kappa_s}$ against mean time $\langle t\rangle$. (b) Positions of posterior and anterior limits of the bend region relative to the undeformed sphere, plotted against mean time $\langle t\rangle$. Thick lines indicate straight line fits. Corresponding panels in Fig. \[fig:fittedshapes\] are marked for some data points.[]{data-label="fig:params"}](inv_Fig10)
Fig. \[fig:fittedshapes\] also shows the stretches $f_s,f_\phi$ in the fitted shapes. It is particularly interesting to relate the values of $f_s,f_\phi$ in the fitted shapes to the measurements of individual cells by [@hohn11]: before inversion starts, the cells are teardrop-shaped, and measure $3-5\,\text{\textmu m}$ in the plane of the cell sheet. As invagination starts, the cells in the posterior hemisphere become spindle-shaped, measuring $2-3\,\text{\textmu m}$. This suggests values $f_s,f_\phi\approx 0.6-0.66$ in the posterior hemisphere during invagination, in agreement with the fitted data (Fig. \[fig:fittedshapes\]d). At later stages of inversion, the cells in the bend region become pencil-shaped, measuring $1.5-2\,\text{\textmu m}$ in the meridional direction, suggesting smaller values $f_s\approx 0.4-0.5$ there, again in agreement with the fitted data (Fig. \[fig:fittedshapes\]h). The large stretches $f_s>2$ seen in the anterior cap during inversion of the posterior hemisphere (Fig. \[fig:fittedshapes\]f) cannot be accounted for by the disc-shaped cells in the anterior (which only measure $4-6\,\text{\textmu m}$) in the meridional direction). While examination of the thin sections of [@hohn11] does suggest, in qualitative agreement with the fits, that the largest meridional stretches arise in the anterior cap, the fact that the model overestimates the actual values of these stretches may stem from the simplified modelling of the phialopore. Further, at the very latest stages of inversion (Fig. \[fig:fittedshapes\]j), the fitted shapes suggest very small values $f_s<0.3$ and corresponding values $f_\phi>3$ that are not borne out by the cell measurements.
### Phialopore Opening and Cell Rearrangement
To understand how these values of the stretches at odds with the observed cell shape changes arise in the fitted shapes, we must analyse the opening of the phialopore in more detail. The observations of [@hohn11] show that the cytoplasmic bridges stretch considerably, to many times their initial length, as the phialopore opens. Circumferential elongation of cells as a means to increase effective radius was discussed in some detail by [@viamontes79], but is not sufficient to explain the circumferential stretches observed at the phialopore. Additional elongation of cytoplasmic bridges as a means to further increase the effective radius (Fig. \[fig:phopen\]) may suffice to produce the large circumferential stretches, but does not explain the small values of meridional stretch at the phialopore in the fitted shapes. For this reason, we additionally imaged the opening of the phialopore using confocal laser scanning microscopy (Methods) to resolve single cells close to the phialopore (Video 2).
[L]{}[6.3cm]{} 
The data reveal that cells rearrange near the phialopore, suggesting an additional mechanism to stretch the phialopore sufficiently for the anterior to be able to peel over the inverted posterior (Fig. \[fig:phopen\]). Video 2 shows how, initially, only a small number of cells form a ring at the anterior pole. When the phialopore widens, cells that were initially located away from this initial ring come to be positioned at the rim of the phialopore. It is unclear whether the cytoplasmic bridges between these cells stretch or break, or whether these cells were not connected by cytoplasmic bridges in the first place. While such cell rearrangement is beyond the scope of the current model, it is nevertheless captured qualitatively by the small values of $f_s$ near the phialopore. [@kelland77] observed elongation of cytoplasmic bridges near the phialopore of *Volvox aureus*, but not in small fragments of broken-up embryos, and concluded that the elongation of cytoplasmic bridges was the result of passive mechanical forces. By contrast, in our model, the opening of the phialopore is the result of active cell shape changes there. This discrepancy may herald a breakdown of the approximations made to represent the phialopore. The data also hint that there may be a different mechanical contribution at later stages of inversion (Fig. \[fig:averages\]i), where the rim of the phialopore may be in contact with the inverted posterior. Since the model does not resolve the rim of the phialpore in the first place, we do not pursue this further here. For completeness of the mechanical analysis, we analyse such a contact configuration in Appendix \[app:1\], where we also discuss a toy problem to highlight the intricate interplay of mechanics and geometry in the contact configuration.
![**Video 2.** Timelapse video of the phialopore opening obtained from confocal laser scanning microscopy of chlorophyll autofluorescence and manual tracing of selected cells (Methods). Scale bar: $20\,\text{\textmu m}$. The video shows a rearrangement of cells surrounding the phialopore.[]{data-label="video:phialopore"}](Video2_thumbnail.png){width="50.00000%"}
Mechanics and Regulation of Local Shape Variations
--------------------------------------------------
We now return to the spatial structure of the shape variations discussed previously. It is clear that some of this structure is geometrical: since the shapes are aligned so that the positions of their centres of mass along the axis coincide, the shape variations accumulate, and are thus expected to e.g. increase in the anterior hemisphere, towards the phialopore, as at the stage in Fig. \[fig:stdshapes\]c. At the same stage however, the shape variation is smaller in the bend region than in the adjacent anterior cap. Both of these regions are, however, close to the centre of mass, and so we do not expect this difference to arise from mere geometric accumulation of shape variations. We must therefore ask: can this structure arise purely mechanically (i.e. from a uniform distribution of the intrinsic parameters), but possibly as a statistical fluke, or must there be some regulation (i.e. non-uniform variation of the intrinsic parameters)?
To answer this question, we analyse random perturbations of the fitted intrinsic parameters of the inversion stage in Fig. \[fig:stdshapes\]c. We observe that, if the relative size of perturbations (the ‘noise level’) exceeds about $4\%$ at this stage of inversion, computation of the perturbed shapes fails for some parameter choices. This mechanical effect is not surprising: our previous analysis of invagination [@haas15] revealed strong shape non-linearities and the possibility of bifurcations as the magnitude of the intrinsic curvature in the bend region is increased. While we may therefore expect more leeway in some parameters than in others, we shall simply discard those perturbations for which the computation fails; further estimation of the distribution of possible perturbations is beyond the scope of the present discussion. We now estimate, for each noise level, the mean shape variation from 1000 perturbations of the fitted shape. By comparing this to the mean shape variation estimated from the $N=22$ embryo halves in Fig. \[fig:averages\]c, we roughly estimate a noise level of $7.5\%$ (Fig. \[fig:shapevar\]a). At this noise level, about $15\%$ of perturbations fail; while the non-uniformities are small, they are statistically significant (Methods).
With this noise level, we obtain 10000 samples of $N=22$ perturbations to the fitted shape each (Fig. \[fig:shapevar\]b), and we compute their averages in the same way as for the experimental samples. While these samples qualitatively capture the spatial structure of the shape variation, they overestimate the shape variation at the poles. More strikingly, they feature a local maximum of the shape variation in the bend region, rather than in the anterior cap. From the sample distribution of the position of these local maxima (Fig. \[fig:shapevar\]c), it is clear that the experimental distribution with the local maximum in the anterior cap, is very unlikely to arise under this model. We make this statement more precise statistically in the Methods section. To explain the observed structure of the shape variation, we therefore allow more variability in the meridional stretch in the anterior cap (with a noise level of $80\%$, compared to $2.5\%$ for the remaining parameters to reproduce the mean shape variation). The resulting distribution is consistent with the experimentally observed position of the local maximum of shape variation in the anterior cap (Fig. \[fig:shapevar\]b,c). While still overestimating the variability near the posterior pole, this modified distribution of the parameter variability captures the magnitude of the variability in the anterior cap much better than the original one.
Thus, at this early stage of inversion (Fig. \[fig:stdshapes\]c), the observed embryo shapes are consistent with an increased variation of the intrinsic meridional stretch in the anterior cap. We can take the interpretation of this active regulation (or lack thereof) further by relating it to the observed cell shape changes: at the stage of Fig. \[fig:stdshapes\]c, the variations of the meridional stretch in the anterior cap correspond to the formation of disc-shaped cells there (Fig. \[fig:volvox\]d). This indicates that invagination and initiation of the expansion of the anterior hemisphere (via the formation of disc-shaped cells) are really two separate processes the relative timing of which is not crucial. (The formation of disc-shaped cells starting at different times also explains the large noise level in the meridional stretch under the modified model, although there is no fitting involved, here.) This adds to our earlier point, that these processes rely on different deformation modes (active bending for invagination and active contraction and stretching for inversion of the anterior hemisphere). These considerations also rationalise our second observation concerning the spatial structure of shape variations, that the variation in the anterior cap is reduced as inversion of the posterior hemisphere ends (Fig. \[fig:stdshapes\]h): there are no longer two separate processes at work. We finally point to a purely mechanical aspect of the structure of the shape variations: despite the increased variability in the anterior cap, the mechanics ensure that the variability is lowest in the bend region, where the main cell shape changes driving invagination take place.
![Analysis of Shape Variations. (a) Mean shape variation (in arbitrary units) against magnitude of uniform perturbations to the fitted shape of the stage in Fig. \[fig:stdshapes\]c. Each data point was obtained from 1000 perturbations of the fitted shape. Horizontal line: mean shape variation obtained from the experimental data. (b) Magnitude of shape variations against (deformed) arclength. Thick blue line: experimental average from $N=22$ embryo halves. Thin gray lines: distributions of $N=22$ perturbations each under the uniform model. Thin orange lines: distributions of $N=22$ perturbations each under the modified model. Shape variations are scaled so that each curve has the same mean value. Inset: average shape of Fig. \[fig:stdshapes\]c, with bend region (BR) and anterior cap (AC) marked; these positions are marked by dotted lines in the main diagram. (c) Cumulative distribution function (CDF) of the positions of the peak (local maximum) of shape variation under the uniform (blue lines) and modified (orange lines) models, with positions of the bend region and of the maximum (M) of experimental distribution from panel (b) labelled. Dashed lines show distributions from all random perturbations; solid lines show those from shape variations with a single local maximum. We consider a maximum to lie in the anterior cap if it falls within the hatched region, which is used for the statistical estimates in the Methods section.[]{data-label="fig:shapevar"}](inv_Fig12)
Discussion
==========
In this paper, we have combined experiment and theory to analyse the variability of *Volvox* inversion and obtain a detailed mechanical description of this process. From observations of the structure of the variability of the shapes of inverting *Volvox* embryos, we showed, using our mathematical model, that this structure results from a combination of geometry, mechanics, and active regulation. The simplest scenario with which the observed shape variations are consistent is that type-B inversion in *Volvox globator* results from two separate processes, with most of the variability at the invagination stage attributed to the relative timing of these processes in individual embryos. The difference between these processes is mirrored, at a mechanical level, by the different types of deformations driving them: the first process, to invert the posterior hemisphere, mainly relies on active bending, whereas the second process, to invert the anterior hemisphere, is mainly driven by active expansion and contraction. We anticipate that these ideas and methods can be applied to other morphogenetic events in other model organisms to add to our understanding of the regulation of morphogenesis: what amount of regulation, be it spatial or temporal, of the cell-level processes is there, and how does it relate to the amount required mechanically for the processes to be able to complete? Additionally, [@houchmandzadeh05] showed that diffusion of two morphogens with inhibition *à la* [@turing] has error-correcting properties that can explain the precise domain specification that is observed in *Drosophila* embryos in spite of the huge variability of morphogen gradients [@houchmandzadeh02]. Does the interplay of geometry and mechanics yield analogous error-correcting properties?
While we have begun to analyse the mechanical regulation of development in the context of *Volvox* inversion, our answers this far have been either negative (excluding certain mechanisms of regulation) or of what one might term the Occam’s razor variety (invoking the law of parsimony to find the simplest modification of the model that can explain the observations). This approach of testing falsifiable hypotheses mitigates the risk of drawing conclusions that are mere teleology [@goldstein16]. Nonetheless, a fuller answer to the questions above requires estimation of the variability of the model parameters from the experimental data, yet that endeavour entails significant statistical, computational, and experimental difficulties: to estimate the variability with statistical signficance we need a large number of experimental samples to estimate the experimental distribution; for each step of the optimisation algorithm used to estimate the large number of variability parameters, a large number of computational samples must be computed to estimate the distribution under the model. Similar difficulties arise when estimating the variability allowed mechanically. While we have previously noted [@haas15] that the dynamic data for type-B inversion suggest that invagination proceeds without a ‘snap-through’ bifurcation, there is no general requirement for individual developmental paths to lie on one and the same side of a mechanical bifurcation boundary. This poses an additional challenge for modelling approaches.
After this discussion of general challenges for a mechanobiological analysis of morphogenesis and its regulation, we mention some of the remaining questions specific to *Volvox* inversion: our model does not resolve the details of the phialopore, and hence does not describe the closure of the phialopore at the end of inversion, which remains a combined challenge for experiment and theory: as discussed above, the cytoplasmic bridges elongate drastically at the phialopore [@hohn11], and confocal imaging has revealed the possibility of rearrangements within the cell sheet at the phialopore. Do some cytoplasmic bridges rend to make such rearrangements possible? Understanding the details of the opening of the phialopore may also require answering a more fundamental question the answer to which has remained elusive [@green81b; @nishii03]: what subcellular structures are located within the cytoplasmic bridges and how is it possible for them to stretch to such an extent? At the theoretical level, rearrangements of cells near the phialopore raise more fundamental questions of morphoelasticity [@goriely]: in particular, how does one describe the evolution of the boundary of the manifold underlying the elastic description? Cytoplasmic bridges rending next to the phialopore would lead to the formation of lips similar to those seen in type-A inversion [@viamontes77; @hallmann06]. Is there a simple theory to describe the elasticity of this non-axisymmetric setup?
At the close of this discussion, it is meet to briefly dwell on a question of more evolutionary flavour: how did different species of *Volvox* evolve different ways of turning themselves inside out? Mapping inversion types to a phylogenetic tree of Volvocine algae shows shows that different inversion types evolved several times independently in different lineages [@hallmann06]. Additionally, [@pocock33] reported that in *Volvox rousseletii* and *Volvox capensis*, inversion type depends on the (sexual or asexual) reproduction mode. This may be a manifestation of the poorly understood role of environmental and evolutionary cues in morphogenesis [@vondassow11], but it is natural to wonder whether there is a mechanical side to this issue. Ultimately, this is another incentive to study the mechanics of type-A inversion in more detail.
Methods and Materials
=====================
Acquisition of Experimental Data
--------------------------------
Wild-type strain *Volvox globator* Linné (SAG 199.80) was obtained from the Culture Collection of Algae at the University of Göttingen, Germany [@SAG], and cultured as previously described [@brumley14] with a cycle of 16h light at 24C and 8h dark at 22C.
### <span style="font-variant:small-caps;">OpenSPIM</span> Imaging
A selective plane illumination microscope (SPIM) was assembled based on the <span style="font-variant:small-caps;">OpenSPIM</span> setup [@OpenSPIM], with modifications to accommodate a <span style="font-variant:small-caps;">Stradus</span> <span style="font-variant:small-caps;">Versalase</span> laser system with multiple wavelengths (Vortran Laser Technology, Inc., Sacramento, CA, USA) and a Cool<span style="font-variant:small-caps;">Snap Myo</span> CCD camera ($1940\times 1460$ pixels; Photometrics, AZ, USA). Moreover, to decrease the loss of data due to shadowing a second illumination arm was added to the setup (Fig. \[fig:SPIM\]). Illumination from both sides improved the image quality and enabled re-slicing of the z-stacks when embryos began to spin during anterior inversion.
*Volvox globator* parent spheroids were mounted in a column of low-melting-point agarose and suspended in fluid medium in the sample chamber. To visualise the cell sheet deformations of inverting *Volvox globator* embryos, chlorophyll-autofluorescence was excited at $\lambda=561\,\text{nm}$ and detected at $\lambda>570\,\text{nm}$. Z-stacks were recorded at intervals of 60s over 4$-$6 hours to capture inversion of all embryos in a parent spheroid. We acquired time-lapse data of 13 different parent spheroids each containing 4$-$7 embryos.
[L]{}[.46]{} 
### Confocal Laser Scanning Microscopy
Samples were immobilised on glass-bottom dishes by embedding them in low-melting-point agarose and covered with fluid medium. Chlorophyll-autofluorescence was excited at and detected at $\lambda>647\,\text{nm}$. Z-stacks were recorded at intervals of 30s over 1$-$2 hours to capture inversion of a single embryo. Trajectories of individual cells close to the phialopore were obtained using <span style="font-variant:small-caps;">Fiji</span> [@Fiji]. Experiments were carried out using a Observer Z1 spinning-disk microscope (Zeiss, Germany).
### Image Tracing
To ensure optimal image quality (traceability) for the quantitative analyses of inversion, from the inversion processes recorded with the SPIM, we selected 11 inversions (in 6 different parent spheroids) in which the acquisition plane was initially approximately parallel to the midsagittal plane of the embryos. Midsagittal cross-sections were obtained using <span style="font-variant:small-caps;">Fiji</span> [@Fiji] and <span style="font-variant:small-caps;">Amira</span> (FEI, OR, USA).
Splines were fitted to these cross-sections using the following semi-automated approach implemented in Python/C++: in a preprocessing step, images were bandpass-filtered to remove short-range noise and large-range intensity correlations. Low-variance Gaussian filters were applied to smooth out the images slightly. Splines were obtained from the pre-processed images $I({\boldsymbol{x}})$ using the active contour model [@kass88], with modifications to deal with intensity variations and noise in the image: the spline ${\boldsymbol{x_{\text{s}}}}(s)$, where $s$ is arclength, minimises an energy $$\mathcal{E}[{\boldsymbol{x}}_{\text{s}}]=\mathcal{E}_{\text{image}}[{\boldsymbol{x}}_{\text{s}}]+\mathcal{E}_{\text{spline}}[{\boldsymbol{x}}_{\text{s}}]+\mathcal{E}_{\text{skel}}[{\boldsymbol{x}}_{\text{s}}],$$ where
$$\begin{aligned}
\mathcal{E}_{\text{image}}[{\boldsymbol{x}}_{\text{s}}] &=-\alpha\int{I\bigl({\boldsymbol{x_{\text{s}}}}(s)\bigr)\,\mathrm{d}s},\\
\mathcal{E}_{\text{spline}}[{\boldsymbol{x}}_{\text{s}}] &= \beta\int{\biggl\|\dfrac{\partial^2{\boldsymbol{x_{\text{s}}}}}{\partial s^2}\biggr\|^2\mathrm{d}s}+\gamma\left(\int{\mathrm{d}s}-L_0\right)^2,\\
\mathcal{E}_{\text{skel}}[{\boldsymbol{x}}_{\text{s}}]&=\delta\int{I_{\text{skel}}\bigl({\boldsymbol{x_{\text{s}}}}(s)\bigr)\,\mathrm{d}s},\end{aligned}$$
wherein $\alpha,\beta,\gamma,\delta$ are parameters, $L_0$ is the estimated length of the shape outline, and $I_{\text{skel}}$ is obtained by skeletonising $I$ using the algorithm of [@zhang84] to minimise the number of branches.
The energy $\mathcal{E}$ was minimised using stochastic gradient descent. Initial guesses for the splines were obtained by manually initialising about 15 timepoints for each inversion using a few guidepoints and polynomial interpolation. An initial guess for other frames was obtained from these frames by interpolation; these interpolated shapes were used to estimate $L_0$.
With $\delta = 0$, the standard active contour model of [@kass88] is recovered. We found that this model was not sufficient to yield fits of sufficient quality, because of the existence of local minima at small values of $\alpha$, while larger values of $\alpha$ lead to noisy splines. Thresholding methods on their own were not sufficient either, because of branching and, in particular, since they failed to capture the bend region properly. Dynamic thresholding methods [e.g. @otsu79] are not applicable either because of the fast variations of the brightness of the images. The modified active contour model did however produce good fits when we progressively reduced $\delta$ to zero with increasing iteration number of the minimisation scheme, yielding smooth splines, while overcoming the local minima (or, from the point of view of the skeletonisation method, choosing the correct, branchless part of the skeleton). All outlines obtained from this algorithm were manually checked and corrected.
Analysis of Traced Embryo Shapes
--------------------------------
From the traced cell sheet outlines, anterior-posterior axes of the embryos were determined as follows: for shapes for which the bend region was visible on either side of the cross-section, the embryo axis was defined to be the line through the centre of mass of the shape that is perpendicular to the common tangent to the two bend regions (the apex line). Shapes were then rotated and translated manually so that their axes coincided. Since embryos do not rotate much before the flagella grow, the orientation of the axes of the earliest traces (for which the bend regions are not apparent) were taken to be the same as that of the earliest timepoint for which two bend regions were visible. The intersection of the embryo trace and axis defines the posterior pole. After manually recentring some embryos with more pronounced asymmetry, embryos were halved to obtain $N=22$ embryo halves.
### Computation of Inversion Descriptors
From the aligned shapes, the geometric descriptors of inversion reported in Fig. \[fig:measurements\] were computed as follows: the posterior-to-bend distance $e$ was computed as the distance from the apex line to the posterior pole. The maximal surface area $A_{\max}$ and the most negative value of curvature $\kappa_\ast$ in the bend region were computed as described previously [@hohn15]; traces were smoothed before computing the curvature. The phialopore width $d$ was computed as the absolute distance between the two ends of a complete embryo trace. The bend region was defined as the region of negative curvature; the distance between the first and last points of negative curvature defined the bend region width. The bend region position is defined by the distance, along the embryo trace, between the posterior pole and the midpoint of the bend region. (The latter may differ from the point where the most negative value of curvature is attained.) The values of bend region width and position obtained for each embryo half were averaged to yield the reported values.
### Aligning and Averaging Embryo Shapes
To align embryos to each other, one embryo half was arbitrarily taken as the reference shape, and $T=10$ regularly spaced timepoints were chosen for fitting. (These timepoints were chosen to be well after invagination had started and before the phialopore had closed, so that defining the start and end of inversion was not required.) For each of the remaining $N-1$ embryo halves, a scale and $T$ corresponding timepoints were then sought, with shapes being (linearly) interpolated at intermediate timepoints. The interpolated and scaled shapes were centred so that the centres of mass of the cross-sections coincided. This fixes the degree of freedom of translation parallel to the embryo axis; the position perpendicular to the axis is fixed by requiring that the embryo axes coincide (Fig. \[fig:averages\], figure supplement 1a). The motivation for using the centres of mass of the cross-sections (rather than of that of the embryos, which assigns the same mass to each cell by assigning more mass to those points of the cross-section that are farther away from the embryo axis) is a biological one: because of the cylindrical symmetry of the cell shape changes, this average assigns the same mass to each cell shape change.
For aligning embryo shapes, we distribute $M=100$ averaging points uniformly along the (possibly different) arclength of each of the embryo halves. Corresponding points were determined using dynamic time warping (DTW) as described by e.g. [@dtw], and the distances between these shapes and their averages were minimised as explained in what follows. The parameters describing the alignment are thus the scale factors $S_1 = 1, S_2,\dots S_N$ and the averaging time points ${\boldsymbol{\tau}}_1 = (\tau_{11},\tau_{12},\dots,\tau_{1T}),{\boldsymbol{\tau}}_2,\dots,{\boldsymbol{\tau}}_N$, where ${\boldsymbol{\tau}}_1$ is fixed. Each choice of these parameters yields a set of shapes ${\boldsymbol{X}}_1=(x_{11},\dots,x_{1M}),{\boldsymbol{X}}_2,\dots,{\boldsymbol{X}}_N$ with points matched up by maps $\sigma_1,\sigma_2,\dots,\sigma_N$ obtained from the DTW algorithm. The effect of the local stretching allowed by the DTW algorithm is illustrated in Fig. \[fig:averages\], figure supplement 1b,c. The mean shapes having been determined, the sum of Euclidean distances between shapes of individual embryos and the mean, $$\sum_{t=1}^T{\left\{\sum_{n=1}^N{\sum_{m=1}^M}{\left(x_{n\sigma_n(m)}-\overline{x}_m\right)^2}\right\}^{1/2}},\qquad\mbox{where }\overline{x}_m=\dfrac{1}{N}\sum_{n=1}^N{x_{n\sigma_n(m)}},\label{eq:score}$$ was minimised over the space of all these alignment parameters using the <span style="font-variant:small-caps;">Matlab</span> (The MathWorks, Inc.) routine `fminsearch`, modified to incorporate the variant of the Nelder–Mead algorithm suggested by [@gao12] for problems with a large number of parameters. After the algorithm had converged, each of the alignment parameters was modified randomly, and the algorithm was run again. This was repeated until the alignment score defined by (\[eq:score\]) did not decrease further. The means $\overline{x}_1,\overline{x}_2,\dots,\overline{x}_M$ for the alignment minimising (\[eq:score\]) define the average embryo shapes.
Aligning shapes in this way using dynamic time warping requires a considerable amount of computer time. To make the problem computationally tractable, we invoked the usual heuristics of only computing pairwise DTW distances, and reducing the size of the DTW matrix by only computing a band centred on the diagonal. To verify the algorithm, we also ran several instantiations of the alignment algorithm without DTW (i.e. with $\sigma_n=\mathrm{id}$) and with larger parameter randomisations, confirming that the modified Nelder–Mead algorithm finds an appropriate alignment. This also enabled us to verify that results do not change qualitatively if the centres of mass of the cross-sections are replaced with those of the embryo halves (even though, as noted in the main text, the shapes without DTW are unsatisfactory since they have kinks in the bend region that are not seen in individual embryo shapes).
For the simple alternative averaging method in Fig. \[fig:averages\], figure supplement 2, different numbers of averaging points were distributed at equal arclength spacing along all individual shapes. Differences in arclengths of individual embryos mean that the rims of the phialopores of individual embryo halves are not necessarily matched up (Fig. \[fig:averages\], figure supplement 1c). No time stretching was applied. The averaging method in Fig. \[fig:averages\], figure supplement 3, is the method discussed above, without DTW (i.e. with $\sigma_n=\mathrm{id}$).
Elastic Model
-------------
![Geometry of the problem. (a) Undeformed geometry: a spherical shell of radius $R$ and thickness $h\ll R$ is characterised by its arclength $s$ and distance from the axis of revolution $\rho(s)$. (b) Deformed configuration, characterised by its arclength $S(s)$ and distance $r(s)$ from the axis of revolution. Intrinsic volume conservation sets the thickness $H(s)$ of the sheet. A local basis $({\boldsymbol{u_r}},{\boldsymbol{u_\phi}},{\boldsymbol{u_z}})$ describes the deformed surface. (c) Cross-section of the shell under the Kirchhoff hypothesis, with a coordinate $\zeta$ across the thickness of the shell, parallel to the normal ${\boldsymbol{n}}$ to the midsurface.[]{data-label="figS1"}](inv_Fig14)
We consider a spherical shell of radius $R$ and uniform thickness $h\ll R$ (Fig. \[figS1\]a), characterised by its arclength $s$ and distance from the axis of revolution $\rho(s)$, to which correspond arclength $S(s)$ and distance from the axis of revolution $r(s)$ in the axisymmetric deformed configuration (Fig. \[figS1\]b). We define the meridional and circumferential stretches $$\begin{aligned}
&f_s(s)=\dfrac{\mathrm{d}S}{\mathrm{d}s},&& f_\phi(s)=\dfrac{r(s)}{\rho(s)}.\end{aligned}$$ The position vector of a point on the midsurface of the deformed shell is thus $${\boldsymbol{r}}(s,\phi)=r(s){\boldsymbol{u_r}}(\phi)+z(s){\boldsymbol{u_z}},$$ in a right-handed set of axes $({\boldsymbol{u_r}},{\boldsymbol{u_\phi}},{\boldsymbol{u_z}})$ and so the tangent vectors to the deformed midsurface are $$\begin{aligned}
{\boldsymbol{e_s}}&=r'{\boldsymbol{u_r}}+z'{\boldsymbol{u_z}},&&{\boldsymbol{e_\phi}}=r{\boldsymbol{u_\phi}},\end{aligned}$$ where dashes denote differentiation with respect to $s$. By definition, $r'^2+z'^2=f_s^2$, and so we may write $$\begin{aligned}
&r'=f_s\cos{\beta},&& z'=f_s\sin{\beta}.\end{aligned}$$ Hence the normal to the deformed midsurface is $${\boldsymbol{n}}= \dfrac{r'{\boldsymbol{u_z}}-z'{\boldsymbol{u_r}}}{f_s}=\cos{\beta}\,{\boldsymbol{u_z}}-\sin{\beta}\,{\boldsymbol{u_r}}.$$ We now make the Kirchhoff ‘hypothesis’ [@audolypomeau], that the normals to the undeformed midsurface remain normal to the deformed midsurface (Fig. \[figS1\]c). Taking a coordinate $\zeta$ across the thickness $h$ of the undeformed shell, the position vector of a general point in the shell is $${\boldsymbol{r}}(s,\phi,\zeta)=r{\boldsymbol{u_r}}+z{\boldsymbol{u_z}}+\zeta{\boldsymbol{n}}=(r-\zeta\sin{\beta}){\boldsymbol{u_r}}+(z+\zeta\cos{\beta}){\boldsymbol{u_z}}.\label{eq:ds}$$ The tangent vectors to the shell are thus $$\begin{aligned}
{\boldsymbol{e_s}}&= f_s(1-\kappa_s\zeta)(\cos{\beta}\,{\boldsymbol{u_r}}+\sin{\beta}\,{\boldsymbol{u_z}}),&{\boldsymbol{e_\phi}}&= \rho f_\phi(1-\kappa_\phi\zeta){\boldsymbol{u_\phi}},\end{aligned}$$ where $\kappa_s=\beta'/f_s$ and $\kappa_\phi=\sin{\beta}/r$ are the curvatures of the deformed midsurface. The metric of the deformed shell under the Kirchhoff hypothesis accordingly takes the form $$\mathrm{d}\boldsymbol{r}^2 = f_s^2(1-\kappa_s\zeta)^2\mathrm{d}s^2+f_\phi^2(1-\kappa_\phi\zeta)^2\rho^2\mathrm{d}\phi^2.$$ The geometric and intrinsic deformation gradient tensors are thus $$\begin{aligned}
&\mathbfsfit{F}^{\mathbfsf{g}}=\left(\begin{array}{cc}
f_s(1-\kappa_s\zeta)&0\\
0&f_\phi(1-\kappa_\phi\zeta)
\end{array}\right), &&\mathbfsfit{F^0}=\left(\begin{array}{cc}
f_s^0(1-\kappa_s^0\zeta)&0\\
0&f_\phi^0(1-\kappa_\phi^0\zeta)
\end{array}\right),\label{eq:defgrad}\end{aligned}$$ where $f_s^0,f_\phi^0$ and $\kappa_s^0,\kappa_\phi^0$ are the intrinsic stretches and curvatures of the shell. Thence, invoking the standard multiplicative decomposition of morphoelasticity [@goriely], the elastic deformation gradient tensor is $$\mathbfsfit{F}=\mathbfsfit{F}^{\mathbfsf{g}}\bigl(\mathbfsfit{F^0}\bigr)^{-1}=\left(\begin{array}{cc}
\dfrac{f_s(1-\kappa_s\zeta)}{f_s^0(1-\kappa_s^0\zeta)}&0\\
0&\dfrac{f_\phi(1-\kappa_\phi\zeta)}{f_\phi^0(1-\kappa_\phi^0\zeta)}
\end{array}\right).$$ While we do not make any assumption about the *geometric* or *intrinsic* strains derived from $\mathbfsfit{F}^{\mathbfsf{g}}$ and $\mathbfsfit{F^0}$, respectively, we assume that the *elastic* strains derived from $\mathbfsfit{F}$ remain small; we may thus approximate $$\begin{aligned}
&\varepsilon_{ss}\approx \dfrac{f_s(1-\kappa_s\zeta)}{f_s^0(1-\kappa_s^0\zeta)}-1, &&\varepsilon_{\phi\phi}\approx \dfrac{f_s(1-\kappa_s\zeta)}{f_s^0(1-\kappa_s^0\zeta)}-1,\end{aligned}$$ with the off-diagonal elements vanishing, $\varepsilon_{s\phi}=\varepsilon_{\phi s}=0$. For a Hookean material with elastic modulus $E$ and Poisson’s ratio $\nu$ [@libai; @audolypomeau], the elastic energy density (per unit extent in the meridional direction) is found by integrating across the thickness of the shell: $$\begin{aligned}
\dfrac{\mathcal{E}}{2\pi\rho}&=\dfrac{E}{2(1-\nu^2)}\int_{-h/2}^{h/2}{\Bigl(\varepsilon_{ss}^2+\varepsilon_{\phi\phi}^2+2\nu\varepsilon_{ss}\varepsilon_{\phi\phi}\Bigr)\mathrm{d}\zeta}\nonumber\\
&=\dfrac{Eh}{2(1-\nu^2)}\biggl\{\left(1+\dfrac{h^2}{4}{\kappa_s^0}^2\right)E_s^2+\left(1+\dfrac{h^2}{4}{\kappa_\phi^0}^2\right)E_\phi^2+2\nu\left(1+\dfrac{h^2}{12}\left({\kappa_s^0}^2+\kappa_s^0\kappa_\phi^0+{\kappa_\phi^0}^2\right)\right)E_sE_\phi\biggr\}\nonumber\\
&\qquad+\dfrac{Eh^3}{24(1-\nu^2)}\biggl\{K_s^2+K_\phi^2+2\nu K_sK_\phi-4\kappa_s^0E_sK_s-4\kappa_\phi^0E_\phi K_\phi-2\nu\bigl(\kappa_s^0+\kappa_\phi^0\bigr)\bigl(E_\phi K_s+E_sK_\phi\bigr)\biggr\},\label{eq:E}\end{aligned}$$ where we have expanded the energy up to third order in the thickness, and where we have defined the shell strains and curvature strains $$\begin{aligned}
&E_s=\dfrac{f_s-f_s^0}{f_s^0},&&E_\phi=\dfrac{f_\phi-f_\phi^0}{f_\phi^0}, &&K_s=\dfrac{f_s\kappa_s-f_s^0\kappa_s^0}{f_s^0},&&K_\phi=\dfrac{f_\phi\kappa_\phi-f_\phi^0\kappa_\phi^0}{f_\phi^0}. \end{aligned}$$ As in our previous work [@hohn15; @haas15], the elastic modulus is an overall constant that ensures that $\mathcal{E}$ has units of energy, but does not otherwise affect the shapes. We shall also assume that $\nu=1/2$ for incompressible biological material; the cell size measurements of [@viamontes79] for type-A inversion in *Volvox carteri* support this assumption qualitatively. (These considerations also explain why we do not perturb these mechanical parameters in our analysis of the shape variations.) We finally set $h/R=0.15$ as in our previous work.
### Derivation of the Governing Equations
The derivation of the governing equations proceeds similarly to standard shell theories [@libai; @audolypomeau; @knoche11]. In fact, the resulting equations turn out to have a form very similar to those of standard shell theories, but a host of extra terms arise in the expressions for the shell stresses and moments due to the assumptions of morphoelasticity. The variation of the elastic energy takes the form $$\dfrac{\delta\mathcal{E}}{2\pi\rho}=n_s\delta E_s+n_\phi\delta E_\phi+m_s\delta K_\phi+m_\phi\delta K_\phi,$$ with
$$\begin{aligned}
\delta E_s&=\dfrac{\delta f_s}{f_s^0}=\dfrac{1}{f_s^0}\Bigl(\sec{\beta}\,\delta r'+f_s\tan{\beta}\,\delta\beta\Bigr),&\delta E_\phi&=\dfrac{\delta f_\phi}{f_\phi^0}=\dfrac{\delta r}{f_\phi^0\rho},\\
\delta K_s&=\dfrac{\delta(f_s\kappa_s)}{f_s^0}=\dfrac{\delta\beta'}{f_s^0},&\delta K_\phi&=\dfrac{\delta(f_\phi\kappa_\phi)}{f_\phi^0}=\dfrac{\cos{\beta}}{f_\phi^0\rho}\delta\beta,\end{aligned}$$
wherein dashes again denote differentiation with respect to $s$, and where the shell stresses and moments are defined by
$$\begin{aligned}
n_s&=\dfrac{Eh}{1-\nu^2}\left\{E_s+\nu E_\phi+\dfrac{h^2}{12}\left(3{\kappa_s^0}^2E_s+\nu\left({\kappa_s^0}^2+\kappa_s^0\kappa_\phi^0+{\kappa_\phi^0}^2\right)E_\phi-2\kappa_s^0K_s-\nu\bigl(\kappa_s^0+\kappa_\phi^0\bigr)K_\phi\right)\right\},\label{eq:ns}\\
n_\phi&=\dfrac{Eh}{1-\nu^2}\left\{E_\phi+\nu E_s+\dfrac{h^2}{12}\left(3{\kappa_\phi^0}^2E_\phi+\nu\left({\kappa_s^0}^2+\kappa_s^0\kappa_\phi^0+{\kappa_\phi^0}^2\right) E_s-2\kappa_\phi^0K_\phi-\nu\bigl(\kappa_s^0+\kappa_\phi^0\bigr)K_s\right)\right\},\end{aligned}$$
and
$$\begin{aligned}
m_s&=\dfrac{Eh^3}{12(1-\nu^2)}\biggl\{K_s+\nu K_\phi-2\kappa_s^0 E_s-\nu\bigl(\kappa_s^0+\kappa_\phi^0\bigr)E_\phi\biggr\},\label{eq:ms}\\
m_\phi&=\dfrac{Eh^3}{12(1-\nu^2)}\biggl\{K_\phi+\nu K_s-2\kappa_\phi^0 E_\phi-\nu\bigl(\kappa_s^0+\kappa_\phi^0\bigr)E_s\biggr\}.\end{aligned}$$
Defining $$\begin{aligned}
&N_s=\dfrac{n_s}{f_s^0f_\phi},&& N_\phi=\dfrac{n_\phi}{f_\phi^0f_s},&&M_s=\dfrac{m_s}{f_s^0f_\phi},&& M_\phi=\dfrac{m_\phi}{f_\phi^0f_s},\end{aligned}$$ the variation becomes $$\begin{aligned}
\dfrac{\delta\mathcal{E}}{2\pi}&=\Bigl\llbracket r N_s\sec{\beta}\,\delta r+r M_s\,\delta\beta\Bigr\rrbracket\nonumber\\
&\hspace{10mm}-\int{\biggl\{\biggl(\dfrac{\mathrm{d}}{\mathrm{d}s}\Bigl(r N_s\sec{\beta}\Bigr)-f_sN_\phi\biggr)\delta r - \biggl(r f_sN_s\tan{\beta}}+f_sM_\phi\cos{\beta}-\dfrac{\mathrm{d}}{\mathrm{d}s}\Bigl(r M_s\Bigr)\biggr)\delta\beta\biggr\}\,\mathrm{d}s. \label{eq:var}\end{aligned}$$ The Euler–Lagrange equations of (\[eq:E\]) are thus $$\begin{aligned}
&\dfrac{\mathrm{d}}{\mathrm{d}s}\Bigl(rN_s\sec{\beta}\Bigr)-f_sN_\phi=0, &&\dfrac{\mathrm{d}}{\mathrm{d}s}\Bigl(rM_s\Bigr)-f_sM_\phi\cos{\beta}-rf_sN_s\tan{\beta}=0.\label{eq:E2}\end{aligned}$$ To remove the singularity that arises in the second of (\[eq:E2\]) when $\beta=\pi/2$, we define the transverse shear tension $T=-N_s\tan{\beta}$ as in standard shell theories. The governing equations can then be rearranged to give $$\begin{aligned}
\dfrac{\mathrm{d}N_s}{\mathrm{d}s}&=f_s\left(\dfrac{N_\phi-N_s}{r}\cos{\beta}+\kappa_s T\right),&\dfrac{\mathrm{d}M_s}{\mathrm{d}s}&=f_s\left(\dfrac{M_\phi-M_s}{r}\cos{\beta}-T\right).\label{eq:Eb}\end{aligned}$$ By differentiating the definition of $T$ and using the first of (\[eq:Eb\]), one finds that $$\dfrac{\mathrm{d}T}{\mathrm{d}s}=-f_s\left(\kappa_sN_s+\kappa_\phi N_\phi+\dfrac{T}{r}\cos{\beta}\right). \label{eq:ET}$$ Together with the geometrical equations $r'=f_s\cos{\beta}$ and $\beta'=f_s\kappa_s$, equations (\[eq:Eb\]) and (\[eq:ET\]) describe the deformed shell. The five required boundary conditions can be read off the variation (\[eq:var\]) and the definition of $T$,
$$\begin{aligned}
\beta &= 0, &r&= 0, &&T = 0&&\text{at the posterior pole},\\
N_s &= 0, &M_s &= 0&&&&\text{at the phialopore}.\end{aligned}$$
We solve these equations numerically using the boundary value-problem solver `bvp4c` of <span style="font-variant:small-caps;">Matlab</span> (The MathWorks, Inc.).
For completeness, we note that if external forces are applied to the shell, and $\delta\mathcal{W}$ is the variation of the work done by these forces, then the variational condition is $\delta\mathcal{E}+\delta\mathcal{W}=0$. In that case, it is useful to write the variation (\[eq:var\]) in terms of $\delta r$ and $\delta z$. We note that $\delta r'=-f_s\sin{\beta}\,\delta \beta$ and $\delta z'=f_s\cos{\beta}\,\delta \beta$, and so $$f_s\delta\beta=\cos{\beta}\,\delta z'-\sin{\beta}\,\delta r'.$$ Using this geometric relation and integrating by parts, we obtain $$\begin{aligned}
\dfrac{\delta\mathcal{E}}{2\pi}&=\Biggl\llbracket r M_s\,\delta\beta+\left\{rN_s\cos{\beta}-\dfrac{\sin{\beta}}{f_s}\left(M_\phi\cos{\beta}-\dfrac{\mathrm{d}}{\mathrm{d}s}\Bigl(rM_s\Bigr)\right)\right\}\delta r\nonumber\\
&\hspace{45mm}+\left\{rN_s\sin{\beta}+\dfrac{\cos{\beta}}{f_s}\left(M_\phi\cos{\beta}-\dfrac{\mathrm{d}}{\mathrm{d}s}\Bigl(rM_s\Bigr)\right)\right\}\delta z\Biggr\rrbracket\nonumber\\
&\hspace{15mm}+\int{\Biggl\{f_sN_\phi-\dfrac{\mathrm{d}}{\mathrm{d}s}\left(rN_s\cos{\beta}-\dfrac{\sin{\beta}}{f_s}\left(M_\phi\cos{\beta}-\dfrac{\mathrm{d}}{\mathrm{d}s}\Bigl(rM_s\Bigr)\right)\right)\Biggr\}\,\delta r\,\mathrm{d}s}\nonumber \\
&\hspace{30mm}-\int{\dfrac{\mathrm{d}}{\mathrm{d}s}\left(rN_s\sin{\beta}+\dfrac{\cos{\beta}}{f_s}\left\{M_\phi\cos{\beta}-\dfrac{\mathrm{d}}{\mathrm{d}s}\Bigl(rM_s\Bigr)\right\}\right)\delta z\,\mathrm{d}s}.\label{eq:var2}\end{aligned}$$
### Limitations of the Theory
The theory presented here has a singularity in a biologically relevant limit: the intrinsic deformation gradient $\mathbfsfit{F^0}$ becomes singular at $|\kappa_s^0|=(h/2)^{-1}$ or $|\kappa_\phi^0|=(h/2)^{-1}$. This value corresponds precisely to the case of cells that are constricted to a point at one cell pole.
The way around this issue would presumably involve writing down an energy directly relative to the (possibly incompatible) intrinsic configuration of the shell. Working in the intrinsic configuration of the shell raises another issue to contend with, however: intrinsic volume conservation, which implies that the thickness $H$ of the intrinsically deformed shell, which is close to the thickness of the deformed shell by assumption, differs from the thickness $h$ of the undeformed shell. For a doubly curved shell, the relative thickness $\eta=H/h$ is a function of both the intrinsic stretches $f_s^0,f_\phi^0$ and the intrinsic curvatures $\kappa_s^0,\kappa_\phi^0$. The volume of an element of shell is $$\int_{-H/2}^{H/2}{f_s^0f_\phi^0\bigl(1-\kappa_s^0\zeta\bigr)\bigl(1-\kappa_\phi^0\zeta\bigr)\rho\,\mathrm{d}s\,\mathrm{d}\phi\,\mathrm{d}\zeta}= f_s^0f_\phi^0H\left(1+\dfrac{H^2}{12}\kappa_s^0\kappa_\phi^0\right)\rho\,\mathrm{d}s\,\mathrm{d}\phi.$$ It follows that $\eta$ satisfies satisfies the cubic equation $$\left(\dfrac{h^2}{12}f_s^0f_\phi^0\kappa_s^0\kappa_\phi^0\right)\eta^3+f_s^0f_\phi^0\eta-\left(1+\dfrac{h^2}{12R^2}\right)=0, \label{eq:eta}$$ the solution of which can be expressed in closed form. It is clear that this equation always has a solution if $\kappa_s^0\kappa_\phi^0>0$. If $\kappa_s^0\kappa_\phi^0<0$, there is a solution if and only if $$\bigl|\kappa_s^0\kappa_\phi^0\bigr|<\left(\dfrac{4f_s^0f_\phi^0}{3h}\right)^2\left(1+\dfrac{h^2}{12R^2}\right)^{-2}.$$ Since $16/9<4$, this condition may fail before the intrinsic geometry becomes singular, so this additional condition is not vacuous. This brief discussion therefore points to some interesting, more fundamental problems in the theory of morphoelastic shells.
There is an additional subtlety associated with the geometric and intrinsic deformation gradient tensors in Eq. (\[eq:defgrad\]): the components of $\mathbfsfit{F}^{\mathbfsf{g}}$ are expressed in (\[eq:defgrad\]) relative to the (natural) mixed basis $\bigl\{{\boldsymbol{\hat{e}_s}},{\boldsymbol{\hat{e}_\phi}}\bigr\}\otimes\bigl\{{\boldsymbol{\hat{E}_s}},{\boldsymbol{\hat{E}_\phi}}\bigr\}$, where ${\boldsymbol{\hat{e}_s}},{\boldsymbol{\hat{e}_\phi}}$ are the unit vectors tangent to the deformed configuration of the shell and ${\boldsymbol{\hat{E}_s}},{\boldsymbol{\hat{E}_\phi}}$ are defined analogously for the undeformed configuration. We have implicitly written down the components of $\mathbfsfit{F^0}$ relative to the same basis. In general however, the components of $\mathbfsfit{F^0}$ in (\[eq:defgrad\]) are those relative to the basis $\bigl\{{\boldsymbol{\hat{e}^0_s}},{\boldsymbol{\hat{e}^0_\phi}}\bigr\}\otimes\bigl\{{\boldsymbol{\hat{E}_s}},{\boldsymbol{\hat{E}_\phi}}\bigr\}$, where the unit basis $\bigl\{{\boldsymbol{\hat{e}^0_s}},{\boldsymbol{\hat{e}^0_\phi}}\bigr\}$ can *a priori* be specified freely. We have neglected these additional degrees of freedom in the above derivation; the question of how to define a natural intrinsic tangent basis $\bigl\{{\boldsymbol{\hat{e}^0_s}},{\boldsymbol{\hat{e}^0_\phi}}\bigr\}$ is however an interesting one, since the intrinsic stretches and curvatures need not be compatible.
Fitting Embryo Shapes
---------------------
For the purpose of fitting the model to the observed averages shapes, we define a family of piecewise constant or linear functional forms for the intrinsic stretches and curvatures, shown in Fig. \[figSfit\]. This family of intrinsic stretches and curvatures is defined in terms of fifteen parameters, which are to be fitted for. Their functional forms are based on observations of cell shape changes by [@hohn11] summarised below:
- The intrinsic stretches $f_s^0,f_\phi^0$ vary in the both hemispheres (Fig. \[figSfit\]a): in the posterior hemisphere, the initially teardrop-shaped cells thin into spindle-shaped cells (Fig. \[fig:volvox\]c,d, Fig. \[fig:cellshapechanges\]b), while, in the anterior hemisphere, they flatten into disc-shaped (‘pancake-shaped’) cells (Fig. \[fig:volvox\]d,e, Fig. \[fig:cellshapechanges\]c). While the evolution towards spindle-shaped cells appears to occur at the same time all over the posterior hemisphere, the data from thin sections suggest that the transition to disc-shaped cells starts at the bend region and progresses towards the phialopore (Fig. \[fig:volvox\]d,e). Moreover, the spindle-shaped cells are isotropic, $f_s^0\approx f_\phi^0$, while the pancake-shaped cells are markedly anisotropic: next to the bend region, the long axis of their elliptical cross-section is the meridional one; next to the phialopore, it is the circumferential axis (Fig. \[fig:cellshapechanges\]c).
- The meridional intrinsic curvature $\kappa_s^0$ (Fig. \[figSfit\]b) is expected to vary most drastically in the region where paddle-shaped cells with thin wedge ends form (Fig. \[fig:volvox\]d, Fig. \[fig:cellshapechanges\]a). Because of the motion of cytoplasmic bridges relative to the cells, some additional, yet slighter, variation may be expected.
- The variations of the circumferential intrinsic curvature $\kappa_\phi^0$ are less clear: on the one hand, $\kappa_\phi^0$ does not vary as drastically as the meridional one, because of the anisotropy of the paddle-shaped cells. This is a marked difference to type-A inversion, where the flasks-shaped cells are isotropic [@viamontes79], and both intrinsic curvatures therefore vary more dramatically in the bend region. On the other hand, some variation of the circumferential intrinsic curvature may be expected because of the motion of cytoplasmic bridges (Fig. \[figSfit\]c). We impose a continuous functional form for $\kappa_\phi^0$, regularising a step function over a distance $\Delta s$ in arclength (Fig. \[figSfit\]c), but we do not fit for $\Delta s$ since we lack detailed information about the cell shape changes that define it.
The other geometrical parameter of the shell, the angular extent $P$ of the phialopore, is not fitted for. We arbitrarily set $P=0.3$. The reasons for this simplification are discussed in the main text.
Numerical shapes were fitted to the average shapes by distributing $M=100$ points uniformly along the arclength of the numerical and average shapes, and minimising a Euclidean distance between them using <span style="font-variant:small-caps;">Matlab</span> (The MathWorks, Inc.) routine `fminsearch`, modified as discussed above. A custom-written adaptive stepper was used to move about in parameter space and select the initial guess for the Nelder–Mead simplex. For each shape, the fit for the previous stage of inversion was used as the initial guess for the optimisation.
![Shape Fitting. Piecewise constant or linear functional forms of (a) the intrinsic stretches $\smash{f_s^0,f_\phi^0}$, (b) the meridional intrinsic curvature $\kappa_s^0$, and (c) the circumferential intrinsic curvature $\smash{\kappa_\phi^0}$, plotted against the arclength $s$ of the undeformed shell. Labels define fifteen fitting parameters. The constant $\Delta s = 0.05$ is set arbitrarily for continuity.[]{data-label="figSfit"}](inv_Fig15)
Shape Perturbations and Statistical Statements
----------------------------------------------
To define perturbations for the $F=15$ fitted model parameters ${\boldsymbol{P_0}}\in\mathbb{R}^F$ at noise level $\delta$, we draw independent $N$ uniform random samples ${\boldsymbol{X}}\sim\mathcal{U}[0,1]^F$ on the unit interval and define the perturbed parameters ${\boldsymbol{P}}={\boldsymbol{P_0}}\bigl(1+2\delta({\boldsymbol{X}}-1)\bigr)$.
### Uniformity of the Distribution of Perturbations
As discussed in the main text, some of these perturbed parameters must be discarded. As a result, the samples that are retained are uniform on an unknown set $\mathcal{A}\subseteq[0,1]^F$ with means ${\boldsymbol{\mu}}$. To establish that these means are not all the same, we derive confidence intervals for $\mu_i-\mu_j$. Since $|X_i-X_j|\leqslant 1$, we may bound the variance of these differences by $\mathrm{Var}(X_i-X_j)\leqslant 1$, and hence, by the central limit theorem, a $100(1-p)\%$ confidence interval is $$\langle X_i\rangle-\langle X_j\rangle\pm\dfrac{z}{\sqrt{N}},\quad\mbox{where }z=\Phi^{-1}\left(1-\dfrac{p/2}{\binom{F}{2}}\right),$$ wherein $\Phi^{-1}$ is the inverse of the cumulative distribution function of the $\mathcal{N}(0,1)$ distribution, and where have included a multiple-testing correction. At noise level $\delta=0.075$, we have run 10000 perturbations, finding $M=\max{\langle{\boldsymbol{X}}\rangle}\approx 0.526$ and $m=\min{\langle{\boldsymbol{X}}\rangle}\approx 0.485$. With $M-m\approx 0.041$ and , we infer that the 99% confidence interval for the maximum difference of the means does not contain zero, and hence that the means are not all the same. We notice however that these deviations of the means are small, in that they are not statistically signficantly different from $0.5$.
### Position of the Maxima of Shape Variation
We now make quantitative our statement, based on the cumulative distributions in Fig. \[fig:shapevar\]c, that the experimental distribution of shape variation (with a maximum in the anterior cap) is very unlikely to arise under the uniform model. We ask: what is the probability $p$, under the uniform model, for the maximum in shape variation to lie in the anterior cap (Fig. \[fig:shapevar\]c)? For 10000 perturbations, we found that 757 had a maximum in the anterior cap. Among these perturbations, 2345 yielded a single maximum in shape variation, with 60 of these maxima in the anterior cap. With 99% confidence, we therefore have upper bounds $p<0.0757+0.0129<0.09$ from all perturbations, and $p<0.0256+0.0266<0.06$ if we restrict to shape variations with a single maximum.
Acknowledgements
================
We are grateful to D. Page-Croft and C. Hitch for instrument fabrication. We thank S. Hilgenfeldt for asking a question at the right time, T. B. Berrett for a conversation on matters statistical, and the Engineering and Physical Sciences Research Council, the Schlumberger Chair Fund, and the Wellcome Trust for partially funding this work.
\[app:1\] In this appendix, we analyse the configuration where the rim of the phialopore is in contact with the inverted posterior for completeness of the mechanical analysis. We also analyse a toy problem to illustrate the intricate interplay of geometry and mechanics during contact.
Elastic Model in the Contact Configuration
------------------------------------------
Let $P$ be the angular extent of the axisymmetric phialopore at the anterior pole of the shell. Here, we discuss the contact problem where the shell has deformed in such a way that the rim of the phialopore (at $\theta=\pi-P=Q$, where $\theta=s/R$ is the polar angle) is in contact with the shell at some as yet unknown position $\theta=C$, as shown in Fig. \[figS2\]a,b.
 \[figS2\]
As in the derivation of the governing equations without contact (Methods), we shall express the variations in terms of $\delta r$ and $\delta\beta$. The third variation, $\delta z$, is not independent of the former two, and so the condition that the vertical positions of the shell at the point of contact and at the phialopore match must be incorporated via a Lagrange multiplier, $U$. [@julicher94] raised a related issue in the derivation of the shape equations for vesicles. The Lagrangian for the problem is therefore $$\mathcal{L}=\mathcal{E}-2\pi U\int_C^Q{f_s\sin{\beta}\,\mathrm{d}\theta}, \label{eq:L5}$$ where the prefactor has been introduced for mere convenience. We note the variation of Eq. (\[eq:L5\]), $$\dfrac{\delta\mathcal{L}}{2\pi}=\dfrac{\delta\mathcal{E}}{2\pi}-\Bigl\llbracket U\tan{\beta}\,\delta r\Bigr\rrbracket_{C_+}^Q+Uf_s(C_+)\sin{\beta(C)}\,\delta C+U\int_C^Q{\Bigl\{f_s\kappa_s\sec^2{\beta}\,\delta r-f_s\sec{\beta}\,\delta\beta\Bigr\}\,\mathrm{d}\theta}.$$ Next, expanding the condition $\beta(C_-)=\beta(C_+)$ of geometric continuity that we have already implicitly applied in the above, we note that $$\delta\beta(C_-)+f_s(C_-)\kappa_s(C_-)\,\delta C=\delta\beta(C_+)+f_s(C_+)\kappa_s(C_+)\,\delta C.$$ Since the outer part of the shell can rotate freely with respect to the inner part at the points of contact, the variations $\delta\beta(C_\pm)$ and $\delta\beta(Q)$ are, by contrast, independent. This is not true of the variations $\delta r(C_\pm)$ and $\delta r(Q)$, however: $$\delta r(Q) = \delta r(C_-)+f_s(C_-)\cos{\beta(C)}\,\delta C=\delta r(C_+)+f_s(C_+)\cos{\beta(C)}\,\delta C.$$ Analogous expansions were used by [@seifert91] for discussing an adhesion problem for vesicles. Next, a straightforward calculation reveals that the governing equations (\[eq:Eb\]) and (\[eq:ET\]) remain unchanged if we define $T=-N_s\tan{\beta}+U\sec{\beta}/r$ for $C\leqslant \theta\leqslant Q$. For convenience, we adjoin the equation $\mathrm{d}z/\mathrm{d}s=f_s\sin{\beta}$ to the system (thereby fixing the degree of freedom of vertical translation). The system thus becomes a system of six first-order differential equations on two regions, with two unknown parameters (the contact position $C$ and the Lagrange multiplier $U$). We thus have to impose fourteen boundary conditions:
$$\begin{aligned}
r(0)&=0,&z(0)&=0,&\beta(0)&=0,&T(0)&=0,\\
r(Q)&= r(C),&z(Q)&= z(C),&N_s(Q)&=0,&M_s(Q)&=0,\end{aligned}$$
as well as the continuity conditions at $\theta=C$, $$\begin{aligned}
\llbracket\beta\rrbracket&=0,&\llbracket r\rrbracket&=0,&\llbracket z\rrbracket&=0,&\llbracket M_s\rrbracket&=0, \end{aligned}$$ and $$\begin{aligned}
&r(C)\llbracket N_s\rrbracket\sec{\beta(C)}-U\tan{\beta(C)}=r(Q)N_s(Q)\sec{\beta(Q)}-U\tan{\beta(Q)},\\
&\llbracket T\rrbracket = -\llbracket N_s\rrbracket\tan{\beta(C)} + \dfrac{U\sec{\beta(C)}}{r(C)},\qquad\dfrac{\llbracket\mathcal{E}\rrbracket}{2\pi}=r(C)\bigl\llbracket f_sN_s\bigr\rrbracket+r(C)M_s(C)\llbracket f_s\kappa_s\rrbracket.\end{aligned}$$
We also note that the conditions $r(Q)=r(C)$ and $z(C)=z(Q)$ do not take into account the finite, but small, thickness of the shell. A more detailed condition would require knowledge of the nature of the contact (and is anyway beyond the remit of a thin shell theory).
We briefly explore shapes in the contact configuration in what follows. We start from a configuration where the posterior hemisphere has inverted, as in Fig. \[fig:antpeeling\]d, and advance the peeling front, but now without increasing the intrinsic circumferential stretch $f_\phi^0$ at the phialopore. As the peeling front advances, the circumferential stretch at the phialopore increases (Fig. \[figS2\]c) at constant $f_\phi^0$, showing how the phialopore is pushed open by the posterior hemisphere. The procession of the point of contact between the posterior and the phialopore along the inverted posterior speeds up with advancing peeling front position (Fig. \[figS2\]d) because the closer the point of contact is to the posterior, the more the latter resists the progression of the contact point because of the changing tangent angle.
The inset configurations in Fig. \[figS2\]c,d also suggest that, as the peeling front advances, the regime of contact at a point discussed here gives way to a second contact regime, where the contact is over a finite extent of the meridian of the shell. We do not pursue this further.
Asymptotic Analysis of a Toy Problem
------------------------------------
Some analytic progress can be made and additional insight into the contact configuration can be gained by asymptotic analysis of a toy problem: two elastic spherical shells, an inner shell of radius $R_1$ and an outer, open shell of radius $R_2>R_1$, touch at the respective angular positions $\mathit{\Theta}_1$ and $\mathit{\Theta}_2<\mathit{\Theta}_1$ (Fig. \[figS3\]a), so that $R_2/R_1=\sin{\mathit{\Theta}_1}/\sin{\mathit{\Theta}_2}$. The intrinsic stretches and curvatures are those of the undeformed shells. For the remainder of this section, we non-dimensionalise distances with respect to the radius $R_1$ of the inner shell; stresses we non-dimensionalise with $Eh$.
If the outer shell is moved relative to the inner shell by a distance $d$ (Fig. \[figS3\]b), the two shells deform in asymptotically small regions near the point of contact. This point of contact moves a distance $d\mathit{\Xi}$ down along the inner shell, determined by matching the displacements of the contact point and the forces exerted by one shell on the other. We assume in particular that the nature of the contact is such that the shells do not exert torques on each other. Since we have non-dimensionalised distances with $R_1$, our asymptotic small parameter is $$\varepsilon^2=\dfrac{1}{12(1-\nu^2)}\dfrac{h^2}{R_1^2}\ll 1.$$
 \[figS3\]
The classical leading-order scalings for this problem are discussed by [@haas15], for example: deformations are localised to asymptotic inner regions of width $\delta\sim\varepsilon^{1/2}$, in which deviations of the tangent angle from its equilibrium value are of order $d/\delta$, and we assume that $d\ll \delta$. We introduce an inner coordinate $\xi$, and write the polar angles as $\theta_1=\mathit{\Theta}_1+\delta\xi+\mathcal{O}(d)$, $\theta_2=\mathit{\Theta}_2+\delta \xi$. We thus expand $$\begin{aligned}
\beta_1(\theta_1)&=\mathit{\Theta}_1+(d/\delta)b_1(\xi), &\beta_2(\theta_2)&=\mathit{\Theta}_2+(d/\delta)b_2(\xi).\end{aligned}$$ Assuming that $\delta^2\ll d\ll\delta$, we then have the leading-order expansions
$$\begin{aligned}
N_s^{(1)}&\stackrel{\hphantom{(\ast)}}{=}Eh\,\delta d\,\sigma_1(\xi), &N_s^{(2)}&\stackrel{\hphantom{(\ast)}}{=}Eh\,\delta d\,\sigma_2(\xi),\\
N_\phi^{(1)}&\stackrel{(\ast)}{=}Eh\,E_\phi^{(1)}+\nu N_s^{(1)}=Eh\,\delta\,a_1(\xi),&N_\phi^{(2)}&\stackrel{(\ast)}{=}Eh\,E_\phi^{(2)}+\nu N_s^{(2)}=Eh\,\delta\,a_2(\xi),\end{aligned}$$
where $a_1,a_2$ are hoop strains. We note that the relations marked $(\ast)$ are only valid at leading order, where we may approximate $f_s\approx f_\phi\approx 1$. Let $F_r$ and $F_z$ denote the (suitably scaled) radial and vertical forces exerted by the outer shell on the inner shell. We obtain the leading-order force balances from the energy variation (\[eq:var2\]): using dashes to denote differentiation with respect to $\xi$, $$\begin{aligned}
&\sigma_1'\sin^2{\mathit{\Theta}_1}-b_1'''\cos{\mathit{\Theta}_1}\sin{\mathit{\Theta}_1}=F_z\delta(\xi),&&\sigma_1'\sin{\mathit{\Theta}_1}\cos{\mathit{\Theta}_1}-a_1+b_1'''\sin^2{\mathit{\Theta}_1}=F_r\delta(\xi).\end{aligned}$$ This system is closed, at leading order, by the geometric relation $a_1'=-b_1$, as in [@haas15]. Eliminating $\sigma_1$, we obtain $$b_1''''+b_1=\bigl(F_r-F_z\cot{\mathit{\Theta}_1}\bigr)\delta'(\xi).$$ The matching conditions $b_1\rightarrow 0$ as $\xi\rightarrow\pm\infty$ reduce the number of undetermined constants to four, which are determined by the jump conditions at the contact point $\xi=0$.
The asymptotic balance for the outer shell is of course the same, but we must remember that the system has been non-dimensionalised with the radius of the inner shell, for which reason a geometric factor arises in the equations. Thus $$b_2''''+\left(\dfrac{\sin{\mathit{\Theta}_1}}{\sin{\mathit{\Theta}_2}}\right)^4b_2=0,$$ with the matching condition $b_2\rightarrow 0$ as $\xi\rightarrow\infty$, leaving two boundary conditions to be imposed on this equation. Since the shells do not exert any moments on each other, $b_2'(0)=0$. The second condition is obtained from the force balance: the vertical force balance can be integrated once to yield $$\sin{\mathit{\Theta}_1}\sin{\mathit{\Theta}_2}\left\{\sigma_2-\cot{\mathit{\Theta}_2}\left(\dfrac{\sin{\mathit{\Theta}_2}}{\sin{\mathit{\Theta}_1}}\right)^4b_2''\right\}=F_z. \label{eq:vert}$$ Matching to the undeformed, unstressed shell as $\xi\rightarrow\infty$ implies $F_z=0$. The radial force boundary condition resulting from (\[eq:var2\]) is $$\sin{\mathit{\Theta}_1}\cos{\mathit{\Theta}_2}\left\{\sigma_2(0)+\tan{\mathit{\Theta}_2}\left(\dfrac{\sin{\mathit{\Theta}_2}}{\sin{\mathit{\Theta}_1}}\right)^4b_2''(0)\right\}=F_r,$$ which, upon imposing (\[eq:vert\]), reduces to $$b_2''(0)=\left(\dfrac{\sin{\mathit{\Theta}_1}}{\sin{\mathit{\Theta}_2}}\right)^3F_r.$$ Let $U_r^{(1)},U_z^{(1)}$ and $U_r^{(2)},U_z^{(2)}$ denote the respective (non-dimensional) displacements of the contact point $\xi=0$, scaled with $d$. Then
$$\begin{aligned}
U_r^{(1)}&=\sin{\mathit{\Theta}_1}\int_0^\infty{b_1\,\mathrm{d}\xi}=-\dfrac{F_r}{2\sqrt{2}}\sin{\mathit{\Theta}_1},&U_z^{(1)}&=-\cos{\mathit{\Theta}_1}\int_0^\infty{b_1\,\mathrm{d}\xi}=\dfrac{F_r}{2\sqrt{2}}\cos{\mathit{\Theta}_1},\\
U_r^{(2)}&=\sin{\mathit{\Theta}_1}\int_0^\infty{b_2\,\mathrm{d}\xi}=-\sqrt{2}F_r\sin{\mathit{\Theta}_1},&U_z^{(2)}&=-\sin{\mathit{\Theta}_1}\cot{\mathit{\Theta}_2}\int_0^\infty{b_2\,\mathrm{d}\xi}=\sqrt{2}F_r\sin{\mathit{\Theta}_1}\cot{\mathit{\Theta}_2}.\end{aligned}$$
In particular, these expressions once again contain additional geometric factors resulting from the non-dimensionalisation.
The values of the two remaining undetermined constants, $F_r$ and $\mathit{\Xi}$, are finally obtained by imposing continuity of the displacement of the contact point, i.e. $$\begin{aligned}
U_r^{(1)}+\mathit{\Xi}\cos{\mathit{\Theta}_1}&=U_r^{(2)},&U_z^{(1)}+\mathit{\Xi}\sin{\mathit{\Theta}_1}&=U_z^{(2)}+1.\end{aligned}$$ Notice that arclength is computed here from the anterior pole of the shell to match the asymptotic setup of [@haas15], and so the ‘vertical’ axis is pointing downwards in Fig. \[figS3\], giving rise to some sign changes. In particular, we obtain $$\mathit{\Xi}=\dfrac{3\sin{\mathit{\Theta}_1}}{1+2\operatorname{cosec}{\mathit{\Theta}_2}\sin{\bigl(2\mathit{\Theta}_1-\mathit{\Theta}_2\bigr)}}.$$ The contours of this expression are plotted in Fig. \[figS3\]c. The very non-linear nature of this expression illustrates that the contact geometry is quite intricate; in particular, $\mathit{\Theta}_2(\mathit{\Theta}_1)$ at fixed $\mathit{\Xi}$ is not a monotonic function, but, as expected (since it is easier for the the contact point to slide along the inner shell the more parallel it is to the axis of symmetry), at fixed $\mathit{\Theta}_1$, $\mathit{\Xi}$ increases with $\mathit{\Theta}_2$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We dynamically simulate fractionation (partitioning of particle species) during spinodal gas-liquid separation of a size-polydisperse colloid, using polydispersity up to $\sim 40\%$ and a skewed parent size distribution. We introduce a novel coarse-grained Voronoi method to minimise size bias in measuring local volume fraction, along with a variety of spatial correlation functions which detect fractionation without requiring a clear distinction between the phases. These can be applied whether or not a system is phase separated, to determine structural correlations in particle size, and generalise easily to other kinds of polydispersity (charge, shape, etc.). We measure fractionation in both mean size and polydispersity between the phases, its *direction* differing between model interaction potentials which are identical in the monodisperse case. These qualitative features are predicted by a perturbative theory requiring only a monodisperse reference as input. The results show that intricate fractionation takes place almost from the start of phase separation, so can play a role even in nonequilibrium arrested states. The methods for characterisation of inhomogeneous polydisperse systems could in principle be applied to experiment as well as modelling.'
author:
- 'J. J. Williamson'
- 'R. M. L. Evans'
title: 'Measuring local volume fraction, long-wavelength correlations and fractionation in a phase-separating polydisperse fluid'
---
\[sec:intro\]Introduction
=========================
Polydispersity is pervasive in soft matter. Systems which are polydisperse exhibit a continuous variation among constituent particles in e.g. size, charge, or shape – this is common even in nominally ‘pure’ systems, where one would like to think of the particles all belonging to the same overall species. For example, *length* polydispersity of skin ceramide lipids is thought to be key to their function [@Das2013]. Polydispersity of colloids strongly affects phase behaviour [@Liddle2011], and acts to significantly degrade the quality of photonic crystals [@Allard2004]. In colloid literature, the polydispersity of some property is $\sigma$, the standard deviation of the distribution in units of its mean. Polydispersity of polymers [@Rogosic1996; @Warren1999] (usually quantified by a ‘polydispersity index’ which is unity in the monodisperse case) is ubiquitous and can be significantly larger than typical values in colloids. Polydispersity is also present – intentionally – in virtually all studies of colloidal glasses [@Zargar2013], but relatively little attention has so far been paid to its role beyond the pragmatic necessity of preventing crystallisation (one recent exception being Ref. [@Zaccarelli2013]). In general the presence of a continuum of different particle species leads to greatly increased complexity relative to the monodisperse case, where each particle is strictly identical.
Polydispersity in particle size is particularly common and perhaps the most widely studied form – the present work continues in this tradition, focusing on size-polydisperse model colloids. We note from the outset that much of the conceptual apparatus is common to other kinds of polydispersity [@Evans2000].
By now there is a reasonably clear picture of how mild polydispersity affects the phase equilibria of hard spheres and related systems [@Evans1998; @Evans2000; @Sollich2009; @Sollich2011; @Fasolo2004; @Wilding2004; @Fantoni2006; @Bartlett1999]. The kinetics by which polydisperse systems *approach* equilibrium are, however, just as complex and far less well understood [@Zaccarelli2009; @Williams2008; @Martin2003; @Martin2005; @Schope2007; @Liddle2011; @Warren1999; @Evans2001].
Fractionation (partitioning between phases) of the polydisperse property is a key aspect of polydisperse phase separation and has typically been measured either in equilibrium simulations [@Wilding2004] or in experiment after long equilibration time [@Evans1998; @Erne2005; @Fairhurst2004]. There is scant data on how these systems behave whilst evolving towards their fractionated equilibria [@Williamson2012; @Williamson2012a; @Leocmach2013]. This may be especially important where phase separation serves as a route to some nonequilibrium arrested state [@Varrato2012; @DiMichele2013] – in such cases, the true compositional equilibrium (in terms of fractionation) may never be reached. In other cases, fractionation is required in order to even access the equilibrium phase [@Sollich2011], so that the dynamics of fractionation directly influences whether the system can equilibrate in any meaningful sense [@Liddle2011]. As we will show, actually measuring fractionation in a highly inhomogeneous multiple-phase system that is far from equilibrium (so does not contain macroscopic chunks of each phase) is not a trivial task.
In this work we dynamically simulate gas-liquid separation of a polydisperse fluid and measure in detail the effects of fractionation, building significantly on the findings of Ref. [@Williamson2012]. The use of a skewed size distribution and large polydispersity ($\sigma \sim 40\,\%$) motivates development of a coarse-grained Voronoi method for determining local volume fraction with minimal bias with respect to particle size. We also develop a suite of spatial correlation functions which can test for fractionation effects when the distinction between phases is unclear or absent.
All these methods can be easily applied to simulations and in principle to experimental data too, given the availability of high-quality microscopy [@Zargar2013] and recent developments in particle-sizing [@Leocmach2013]. They are not specific to size polydispersity, so can be used in cases such as charge or shape polydispersity. The results shed light on polydispersity’s influence in an inhomogeneous, phase-separating fluid. The correlation functions introduced, which do not assume distinct phases, may be useful in determining structural effects of polydispersity even in systems that are *prima facie* homogeneous, such as glasses [@Zargar2013].
\[sec:model\]Model and theory
=============================
Model\[sec:modmod\]
-------------------
The simulation (described in detail elsewhere [@Williamson2012]) contains spherical model colloids with hard cores of mean diameter $\langle d \rangle_\textrm{p} \equiv 1$, and attractive square wells of range $\lambda = 1.15$ and depth $u = 1.82\, k_\textrm{B}T$ (a subcritical dimensionless temperature $T_\textrm{eff} \approx 0.55$ [@Liu2005]). The parent volume fraction is set to $\phi_{p} = 0.229$, its critical value in the monodisperse limit. In this limit the fluid is in a region of instability to gas-liquid separation [@Pagan2005] – due to the short attraction range, the resulting coexistence is in turn metastable with respect to crystallisation [@Williamson2012a] which is not observed in the simulation timescale. In practice, it is found [@Williamson2012] that the polydisperse case to be studied here exhibits spinodal gas-liquid separation just as in the monodisperse limit. This is expected from equilibrium work [@Wilding2004]; even for rather large polydispersity, the gas-liquid binodals (more accurately the cloud and shadow curves into which each binodal splits) are in roughly similar positions in the $\phi$ axis to the monodisperse case as long as volume fraction $\phi$ (as opposed to number density $\rho$) is used as the order parameter.
The diameters of the hard particle cores are taken from a truncated Schulz distribution of parent polydispersity $\sigma_\textrm{p}$. The normalised deviation of particle $i$’s diameter $d_i$ from the mean $\langle d \rangle_\textrm{p} \equiv 1$ is given by $\epsilon_i = (d_i - \langle d \rangle_\textrm{p} )/\langle d \rangle_\textrm{p} $. In contrast to the pseudo-Gaussian particle size distribution used in Ref. [@Williamson2012], the Schulz has a skew, i.e. a nonzero third central moment $\langle \epsilon^3 \rangle_\textrm{p}$. We have employed an upper cutoff at $d=2$ for the distributions, in order to avoid large particles which significantly slow the simulation. The total number of particles is $N= 10000$ and the time unit is the mean time for a free particle of mean diameter to diffuse a unit distance.
We have modelled two possible choices for how the pairwise interaction potential $V(r_{ij})$ depends on the particular sizes of two particles $i$ and $j$ and the distance $r_{ij}$ between their centres [@Williamson2012]. In the ‘scalable’ case, the square well range depends multiplicatively on the hard core size: [$$V_\textrm{{scal}}(r_{ij}) =
\begin{cases}
\infty & \text{if } r_{ij}\leq d_{ij} \\
-u & \text{if } d_{ij} < r_{ij}\leq \lambda d_{ij}\\
0 & \text{if } r_{ij} > \lambda d_{ij}
\end{cases}~,
\label{eqn:scalablesquarewell}$$]{} where $d_{ij} = (d_{i} + d_{j})/2$. In the ‘nonscalable’ case the attraction range depends on the hard core size via an additive constant: [$$V_\textrm{{non-scal}}(r_{ij}) =
\begin{cases}
\infty & \text{if } r_{ij}\leq d_{ij} \\
-u & \text{if } d_{ij} < r_{ij}\leq d_{ij} + (\lambda - 1)\\
0 & \text{if } r_{ij} > d_{ij} + (\lambda - 1)\\
\end{cases}~.
\label{eqn:nonscalablesquarewell}$$]{} In the monodisperse case, $d_{ij} = 1$ for all particle pairs and the definitions become *strictly identical*. In the following we also consider for comparison a hard sphere (HS) fluid; in that case, $u = 0$ and again there is no distinction between the potentials.
Theory
------
As described in Ref. [@Williamson2012], a perturbative theory for polydispersity can be used to predict fractionation at steady state (metastable) gas-liquid coexistence. The relevant thermodynamic potential for fractionation is $A(\rho) = \rho {d\mu ^{\textrm{ex}} (\epsilon)} / {d\epsilon}$, quantifying the variation in excess chemical potential $\mu ^{\textrm{ex}}$ with scaled particle size deviation $\epsilon$. This tells us ‘how costly it is’ (free-energetically) to increase particle size at a given density $\rho$, in the monodisperse reference case. Then, the fractionation of the $n$th moment between the phases depends on the parent value of the $n+1$th moment like so: $$[\langle\epsilon ^n \rangle]^\textrm{l}_\textrm{g} = - [A/\rho ]^\textrm{l}_\textrm{g} \langle \epsilon^{n+1}\rangle_\textrm{p} + \mathcal{O}(\epsilon^{n+2})~,
\label{eqn:momentfractionation}$$ where for any quantity $x$ we write $[x]^{\rm l}_{\rm g}\equiv x_{\rm l}-x_{\rm g}$. Subscripts $\textrm{l}$, $\textrm{g}$ indicate evaluation in the liquid and gas phases respectively, and $\textrm{p}$ a quantity evaluated for the parent (overall) distribution.
### Qualitative predictions: scalable\[sec:qualscal\]
The distinction between the scalable and nonscalable model potentials (Eqs. \[eqn:scalablesquarewell\] and \[eqn:nonscalablesquarewell\]) was found, for the current parameters, to switch the sign of $ [A/\rho ]^\textrm{l}_\textrm{g}$ [@Williamson2012]. For the scalable potential, $- [A/\rho ]^\textrm{l}_\textrm{g} \sim 5.3$ so the gas is predicted to prefer smaller particles than the liquid ($n=1$ in Eq. \[eqn:momentfractionation\]). This was confirmed via simulation in the near-monodisperse regime in Ref. [@Williamson2012]. Eq. \[eqn:momentfractionation\] predicts this to hold also for fractionation of variance $\sigma^2 \approx \langle \epsilon^2 \rangle$ induced by nonzero *skew* $\langle \epsilon^3 \rangle_\textrm{p}$ of the parent distribution. Thus the truncated Schulz distribution used here, for which $\langle \epsilon^3 \rangle_\textrm{p} > 0$, should cause the gas phase also to have lower *variance* (hence polydispersity) in the scalable case.
### Qualitative predictions: nonscalable\[sec:qualnonscal\]
For the nonscalable potential, $- [A/\rho ]^\textrm{l}_\textrm{g} \sim -2.2$ so the gas phase should prefer *larger* particles, and be of higher polydispersity.
The two model potentials used here provide a ‘switch’ to control the equilibrium direction of fractionation – at least to the extent that the perturbative theory of polydispersity [@Evans2000] and the approximate square well free energy [@Williamson2012] remain valid. In a physical system, interactions will generally be polydisperse in *depth*, not just range. For example, although the depletion attraction in colloid-polymer mixtures somewhat resembles our nonscalable interaction viz. the dependence of its *range* on particle size, its attraction strength is also size-dependent, with larger particles experiencing a stronger attraction. In Ref. [@Evans2000] the overall dependence of the depletion potential on particle size led to $- [A/\rho ]^\textrm{l}_\textrm{g} > 0$, so that the liquid prefers larger particles, as for the *scalable* potential in our model. Experimental work on a different colloid-polymer system qualitatively confirms this fractionation direction for the depletion potential [@Liddle2014]. Thus, fractionation in any given physical system is sensitive to all details of the polydispersity in the inter-particle potential. The value of the model potentials used here is that they should exhibit opposite fractionation directions to one another, providing a good test-bed for the methods we will develop.
 Average particle diameter $\langle d_i \rangle$ (black crosses) and number of particles $\langle n\rangle$ (red circles) in bins of the local volume fraction $\phi_i$ in a HS fluid of polydispersity $\sigma_\textrm{p} = 0.0998$. (b) Using the locally coarse-grained volume fraction $\phi_i ^\textrm{CG}$. ](naivevsneigh.eps){width="8.6cm"}
{width="18.6cm"}
\[sec:results\]Results
======================
The system described in Section \[sec:modmod\] is simulated up to $t=8000$, using three independent trajectories for each choice of parameters. We now introduce methods to detect fractionation by locally distinguishing the phases, or examining long-wavelength particle size correlations.
Measuring local volume fraction in a polydisperse system\[sec:measure\]
-----------------------------------------------------------------------
The obvious way to detect fractionation is by identifying the distinct phases and measuring their properties, requiring a per-particle distinction between the phases. This was done in Ref. [@Williamson2012] by performing a simple neighbour count within a fixed arbitrary range, with a threshold defining which particles are ‘gas’ and which are ‘liquid’. For larger polydispersities such a method is not satisfactory since, e.g., very large particles may exclude neighbours from their surroundings, leading to an anomalously low neighbour count and tending to cause large particles to be recognised as ‘gas’. It is not clear how one should adjust the definition to compensate while avoiding simply introducing some other arbitrary bias.
Therefore one seeks a measure of local density avoiding an arbitrary cutoff. The standard Voronoi cell method is a widely-used solution [@Slotterback2008; @Voro; @Fern2007] but, as we will show, can be significantly improved upon for the present purposes.
The Voronoi cell is a polyhedron containing all the space that is closer to particle $i$ than any other, and its volume $V^\textrm{Voronoi}_i$ leads to a commonly used definition for $i$’s local volume fraction: $$\begin{aligned}
\phi_i = \frac{V_i}{V^\textrm{Voronoi}_i}~,\label{eqn:naive}\end{aligned}$$ where $V_i$ is the volume of particle $i$. As well as its widespread use in granular media, the Voronoi cell method has been applied to phase separation of thermal systems, e.g. the so-called $2\phi\textrm{MD}$ method in which coexistence conditions are extracted via direct simulation of a two-phase system [@Fern2007].
 Average particle diameter (black crosses) and number of particles (red circles) in bins of $\phi_i^\textrm{CG}$ in a phase-separating fluid of polydispersity $\sigma_\textrm{p} = 0.0998$ using the scalable potential. The number peaks corresponding to the gas and liquid are seen, and the gas prefers smaller particles (cf. Table \[table:scalfrac\]). (b) Using the nonscalable potential, the fractionation is reversed (cf. Table \[table:nonscalfrac\]). Error bars are approximately the symbol size. These data are averaged over $t=7200-8000$. ](svsns.eps){width="8.6cm"}
![\[40\_svsns\]As Fig. \[svsns\] but for larger polydispersity $\sigma_\textrm{p} = 0.37$. ](40_svsns.eps){width="8.6cm"}
{width="18.6cm"}
Given the presence of polydispersity, we use the radical Voronoi tessellation, where boundaries between Voronoi cells are weighted according to relative particle radius, avoiding the problem of Voronoi cell faces ‘cutting’ particles [^1] (we employ C. H. Rycroft’s Voro++ library [@Voro]). However, a difficulty still remains in applying this approach to measure *fractionation*. In Fig. \[naivevsneigh\]a we plot the dependence of mean particle diameter on local volume fraction measured with the naive implementation of Eq. \[eqn:naive\], in a hard sphere (HS) fluid of polydispersity $\sigma_\textrm{p} = 0.0998$. Since the HS fluid at this overall volume fraction does not phase separate, a good protocol for our purposes should i) exhibit a narrow-peaked distribution of particle volume fractions, since all particles are in the same thermodynamic phase; ii) exhibit minimal correlation between particle size and volume fraction, since our purpose later will be to use such a correlation as an indicator of fractionation driven by phase separation, which is absent in the HS fluid. In Fig. \[naivevsneigh\]a (using Eq. \[eqn:naive\]), we firstly note a wide spread of measured particle volume fractions, reflecting the wide variability in the values of $\phi_i$ generated by Eq. \[eqn:naive\], despite the actual homogeneity of the single-phase system. Secondly, a strong positive correlation is present between particle size and $\phi_i$. This correlation by definition does not represent the phase separation driven fractionation that we intend to study, but a separate ‘geometric’ signal arising from the particular dependence of the denominator versus the numerator of Eq. \[eqn:naive\] on particle size in this HS fluid. This presents a significant background bias which interferes with attempting to measure fractionation in a phase-separating system using Eq. \[eqn:naive\] – again it is not clear how to safely subtract this signal out, which would presume existence and knowledge of a consistent relationship between $V_i$ and $V_i^\textrm{Voronoi}$ in the absence of fractionation.
Instead, we revisit the definition of $\phi_i$. Eq. \[eqn:naive\] is suitable for getting detailed statistics on the local free volume of particles, but this is not the information one has in mind when speaking about ‘gas’ and ‘liquid’ phases. A particle in a region of liquid randomly experiencing a large upward fluctuation in its own free volume $V^\textrm{Voronoi}_i$ does not thereby become a member of its own one-particle gas phase. A phase in the usual sense is a connected region of space with certain *average* properties, and particles in that region belong to that phase.
Hence, some local coarse-graining is appropriate, but this risks the re-introduction of an arbitrary parameter to set e.g. the size of a coarse-graining box. We circumvent this by introducing a coarse-grained volume fraction based on the topology of the Voronoi tessellation: $$\begin{aligned}
\phi^\textrm{CG}_i = \frac{V_i + \sum_j V_j}{V^\textrm{Voronoi}_i + \sum_j V^\textrm{Voronoi}_j}~.\label{eqn:neigh}\end{aligned}$$ Here, $j$ ranges over neighbours of $i$ where neighbours are those that share a Voronoi cell face. Hence the volume fraction is coarse-grained, but the coarse-graining box is a complex polyhedron defined naturally by the neighbour environment in the vicinity of $i$, avoiding introduction of a new parameter. Fig. \[naivevsneigh\]b shows a much more narrow-peaked distribution of volume fractions using this method. The background ‘geometric’ signal correlating particle $i$’s size and local volume fraction is significantly reduced. This is because the local averaging includes the surrounding neighbour region which – in the absence of fractionation – will on average contain an unbiased, representative sample of the HS fluid irrespective of $i$’s size.
Comparison of simulation snapshots shows the more uniform local volume fraction when $\phi^\textrm{CG}_i $ is used (Fig. \[HS\_combined\]b). On the other hand, close inspection of Fig. \[HS\_combined\]a reveals that those particles of particularly high $\phi_i$ are also the larger ones, reflecting the large bias introduced if Eq. \[eqn:naive\] is employed. As the size of relevant features in a fluid becomes large compared to the particle size (e.g. through domain coarsening) and less spatial resolution is required, the averaging could be systematically refined by including neighbours of neighbours (etc.) in the sums in Eq. \[eqn:neigh\]. Then, for instance, the proportional contribution of $V_i$ and $V^\textrm{Voronoi}_i$ in Eq. \[eqn:neigh\] would be further decreased (thus further reducing the residual size-volume fraction correlation in Fig. \[naivevsneigh\]b), at the cost of lower spatial resolution.
As far as we are aware, this is a novel method of defining a locally coarse-grained volume fraction. In the present context it is motivated by the need for a definition that minimises particle size bias present in the absence of fractionation, but the method should be useful in other applications – including monodisperse systems – where a local volume fraction measure is required.
Fractionation of mean size and polydispersity
---------------------------------------------
Having chosen $\phi^\textrm{CG}_i $ as a coarse-grained per-particle volume fraction, fractionation can be measured by applying a suitable definition of the gas and liquid phases (dependent on $\phi^\textrm{CG}_i $) and calculating the particle size statistics of the phases. Although the definition of $\phi^\textrm{CG}_i $ is parameter-free, subsequently defining the phases still introduces arbitrary threshold(s).
In this section we perform fractionation measurements on phase-separating square well fluids, using a simple criterion that assigns $i$ liquid if $\phi^\textrm{CG}_i > 0.229$. Later we will study independent measurements that eliminate even this parameter from the determination of fractionation. The mean diameter and polydispersity of each phase are averaged over $t = 7200-8000$ using truncated Schulz distributions of parent polydispersity $\sigma_\textrm{p} = 0.0998,\, 0.37$. Both the scalable and nonscalable square well potentials are used, which should give opposite fractionation directions (Sections \[sec:qualscal\], \[sec:qualnonscal\]).
The results are summarised in Tables \[table:scalfrac\] and \[table:nonscalfrac\]. The direction of fractionation is reversed by choosing the nonscalable versus scalable potential. The positive skew $\langle \epsilon^3 \rangle_\textrm{p} > 0 $ of the size distribution leads to fractionation in variance $\sigma^2$ and thus *polydispersity* $\sigma$ between the phases. It is interesting that these important qualitative features hold at large polydispersity, being predicted by a theory that is strictly applicable only in the monodisperse limit.
We have also expressed the results in the language of Eq. \[eqn:momentfractionation\], which implies $ [\langle \epsilon^2 \rangle ]^\textrm{l}_\textrm{g} / \langle \epsilon^3 \rangle_\textrm{p} = [\langle \epsilon \rangle ]^\textrm{l}_\textrm{g} / \langle \epsilon^2 \rangle_\textrm{p} = - [A/\rho ]^\textrm{l}_\textrm{g} $ at metastable steady state (although we should be careful in placing too much confidence in this prediction far from the monodisperse limit). It appears that the fractionation achieved in the simulation time is weaker than at steady state. Further, we can resolve $[\langle \epsilon^2 \rangle ]^\textrm{l}_\textrm{g} / \langle \epsilon^3 \rangle_\textrm{p} \neq [\langle \epsilon \rangle ]^\textrm{l}_\textrm{g} / \langle \epsilon^2 \rangle_\textrm{p} $ in some cases, perhaps implying that the first and second moments are fractionating at proportionally different speeds. This opens up the interesting possibility of understanding fractionation dynamics in terms of moment relaxation rates [@Warren1999].
Fractionation of polydispersity (as opposed to mean particle size) was detected in experiment [@Evans1998] and equilibrium simulation [@Wilding2004] but has not previously been measured in the early stages of spinodal decomposition. It may be especially important in cases where the system must form crystals to equilibrate [@Liddle2011]; for instance a liquid phase purified (i.e. of reduced $\sigma$) via fractionation may be better able to subsequently nucleate a crystal phase.
Plots of mean diameter and number versus $\phi_i^\textrm{CG}$ (Figs. \[svsns\] and \[40\_svsns\]) are in agreement with Tables \[table:scalfrac\] and \[table:nonscalfrac\]. The dominant local volume fractions corresponding to the gas and liquid phases are apparent, and the preference of the gas for smaller particles (scalable potential) or larger particles (nonscalable) can also be seen. Interestingly, at $\sigma_\textrm{p} = 0.37$ nonscalable (Fig. \[40\_svsns\]b) the distinct peaks in $\langle n \rangle (\phi_i^\textrm{CG})$ are no longer apparent. This is consistent with the less sharply defined phases in Fig. \[40poly\_combined\]b resulting from slower coarsening, leaving more ‘interfacial’ particles of intermediate $\phi_i^\textrm{CG}$. There may also be an effect of the nonscalable potential on the phase diagram (critical temperature, binodal positions) this far from the monodisperse limit. Fig. \[40poly\_combined\]b indicates that phase separation is taking place, while Figs. \[quadratic\] and \[linear\] confirm the associated fractionation, but this large polydispersity causes the choice of potential (viz. its scaling with polydispersity) to strongly affect structure during phase separation.
$\mathbf{\sigma_\textrm{p}\!=\!0.0998\!\pm\!0.0001}$ Mean diameter Polydispersity
---------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------
Gas $ \langle d_i \rangle_\textrm{g} = 0.994 \pm 0.001$ $\sigma_\textrm{g} = 0.0989 \pm 0.0008$
Liquid $ \langle d_i \rangle_\textrm{l} = 1.0023 \pm 0.0007$ $\sigma_\textrm{l} = 0.1000 \pm 0.0003$
$- [A/\rho ]^\textrm{l}_\textrm{g} \sim 5.3$ [@Williamson2012] $ {\frac{[\langle \epsilon \rangle ]^\textrm{l}_\textrm{g}} { \langle \epsilon^2 \rangle_\textrm{p} }}= 0.84 \pm 0.03$ ${ \frac{[\langle \epsilon^2 \rangle ]^\textrm{l}_\textrm{g} }{ \langle \epsilon^3 \rangle_\textrm{p} }} =0.8 \pm 0.8 $
$\mathbf{\sigma_\textrm{p}\!=\!0.37\!\pm\!0.0008}$ Mean diameter Polydispersity
Gas $ \langle d_i \rangle_\textrm{g} = 0.935 \pm 0.004$ $\sigma_\textrm{g} = 0.341 \pm 0.002$
Liquid $ \langle d_i \rangle_\textrm{l} = 1.024 \pm 0.003$ $\sigma_\textrm{l} = 0.377 \pm 0.001$
$- [A/\rho ]^\textrm{l}_\textrm{g} \sim 5.3 $ ${ \frac{[\langle \epsilon \rangle ]^\textrm{l}_\textrm{g} }{\langle \epsilon^2 \rangle_\textrm{p} }}= 0.65 \pm 0.02$ $ {\frac{[\langle \epsilon^2 \rangle ]^\textrm{l}_\textrm{g} }{ \langle \epsilon^3 \rangle_\textrm{p} }}= 1.04 \pm 0.07$
: Gas-liquid fractionation results using the scalable square well potential (Eq. \[eqn:scalablesquarewell\]). The gas prefers smaller particles and lower polydispersity. Eq. \[eqn:momentfractionation\] is tested by comparing ${[\langle \epsilon \rangle ]^\textrm{l}_\textrm{g} /\langle \epsilon^2 \rangle_\textrm{p} }$ and ${[\langle \epsilon^2 \rangle ]^\textrm{l}_\textrm{g} / \langle \epsilon^3 \rangle_\textrm{p} }$ which, within the perturbative theory [@Evans2000], should both equal $- [A/\rho ]^\textrm{l}_\textrm{g}$ at steady state. It appears that the fractionation measured is weaker than its steady state magnitude. The data are averaged over $t = 7200-8000$.[]{data-label="table:scalfrac"}
$\mathbf{\sigma_\textrm{p}\!=\!0.0998\!\pm\!0.0001}$ Mean diameter Polydispersity
----------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------
Gas $ \langle d_i \rangle_\textrm{g} = 1.005 \pm 0.001$ $\sigma_\textrm{g} = 0.1002 \pm 0.0007$
Liquid $ \langle d_i \rangle_\textrm{l} = 0.9978 \pm 0.0007$ $\sigma_\textrm{l} = 0.0995 \pm 0.0004$
$- [A/\rho ]^\textrm{l}_\textrm{g} \sim -2.2$ [@Williamson2012] $ {\frac{[\langle \epsilon \rangle ]^\textrm{l}_\textrm{g} }{ \langle \epsilon^2 \rangle_\textrm{p} }}= -0.75 \pm 0.04$ ${ \frac{[\langle \epsilon^2 \rangle ]^\textrm{l}_\textrm{g} }{ \langle \epsilon^3 \rangle_\textrm{p} }}=-0.7 \pm 0.8$
$\mathbf{\sigma_\textrm{p}\!=\!0.37\!\pm\!0.0008}$ Mean diameter Polydispersity
Gas $ \langle d_i \rangle_\textrm{g} = 1.023 \pm 0.004$ $\sigma_\textrm{g} = 0.373 \pm 0.002$
Liquid $ \langle d_i \rangle_\textrm{l} = 0.987 \pm 0.003$ $\sigma_\textrm{l} = 0.368 \pm 0.001$
$- [A/\rho ]^\textrm{l}_\textrm{g} \sim -2.2$ $ {\frac{[\langle \epsilon \rangle ]^\textrm{l}_\textrm{g} }{ \langle \epsilon^2 \rangle_\textrm{p} }} = -0.26 \pm 0.02 $ ${\frac{ [\langle \epsilon^2 \rangle ]^\textrm{l}_\textrm{g} }{ \langle \epsilon^3 \rangle_\textrm{p} }} = -0.19 \pm 0.04$
: Gas-liquid fractionation results using the nonscalable square well potential (Eq. \[eqn:nonscalablesquarewell\]). The gas now prefers larger particles and higher polydispersity.[]{data-label="table:nonscalfrac"}
Fig. \[40poly\_combined\] shows visualisations at $\sigma_\textrm{p} =0.37$ for the two potentials. The coarse-grained volume fraction $\phi_i^\textrm{CG}$ correctly picks out liquid and gas-like regions. Although pictures cannot on their own provide evidence of fractionation, it is satisfying that e.g. Fig. \[40poly\_combined\]a is consistent with the measured fractionation of smaller particles into the gas and larger ones into the liquid.
The results in this section demonstrate that thermodynamically driven fractionation of mean size and also polydispersity (in the presence of a skewed parent distribution) begins during the early stages of gas-liquid separation. The coarse-grained Voronoi method Eq. \[eqn:neigh\] provides an improved measure of local volume fraction which substantially reduces particle size bias relative to a naive approach Eq. \[eqn:naive\], enabling definitions of the phases from which fractionation can safely be measured.
Real space correlation functions: $\xi_2(r)$
--------------------------------------------
{width="12.6cm"}
![\[time\]Evolution of $\xi_2(r)$ at $\sigma_\textrm{p} = 0.37 $ for the scalable potential. The arrow signifies later time intervals: $t=0-800,\,800-1600,\,1600-2400,\,2400-3200,\,7200-8000$. ](40scal_quadratic_time.eps){width="8.0cm"}
{width="12.6cm"}
Although the coarse-grained volume fraction employed in the previous section was defined in a parameter-free way, subsequently defining the phases inevitably introduces one or more parameters. We are able to corroborate the results with parameter-free methods that detect fractionation not by identifying distinct phases, but by measuring long-wavelength correlations.
If fractionation takes place, we can expect particle size correlations on the lengthscale of the domain size, because particles in the same domain will tend to deviate from $\langle d \rangle_\textrm{p}$ in the same direction – the signs of their $\epsilon$ will be the same. Such correlations can be tested for using a real space correlation function that resembles a weighted radial distribution function (RDF): $$\begin{aligned}
\xi_2(r) &= \langle \epsilon_i \epsilon_j \rangle_r - \langle \epsilon_i \rangle^2 \\ \notag
&= \left( \frac{\sum_i \sum_{j \neq i} \delta(r_{ij} - r) \epsilon_i \epsilon_j\, }{ \, \sum_k\sum_{l \neq k} \delta(r_{kl} - r)}\right)~\end{aligned}$$ where $r_{ij}$ is the centre-to-centre distance between particles $i$ and $j$. Note that $ \langle \epsilon_i \rangle \equiv 0 $ since the average is performed over the entire system. $\xi_2(r)$ will be positive whenever particles separated by a distance $r$ are positively correlated in terms of their deviation $\epsilon_i$ from mean diameter. Normalisation by the unweighted $\delta(r_{kl} - r)$ ensures that the structure of the usual RDF is factored out, so $\xi_2(r)$ is only sensitive to correlations in particle size, not in density.
The behaviour of $\xi_2(r)$ is shown in Fig. \[quadratic\]. In the HS case, there is only a short-wavelength component reflecting local packing considerations; e.g. the smallest separations $r$ can only be achieved if both particles involved are small, leading to $\epsilon_i \epsilon_j > 0$ at low $r$. This effect (previously observed in Refs. [@Williamson2012a; @Pagonabarraga2000]) is suppressed here due to the large bin size, but we resolve that it is enhanced if the volume fraction of the HS fluid is increased ($\phi_\textrm{p} = 0.4$), as for a standard RDF. Beyond a few particle diameters, particle size is uncorrelated within error, i.e. the HS fluids are homogeneous mixtures of size species.
In contrast, measurements in the phase separating square well fluids reveal an additional long-wavelength component. Particles within the same domain are correlated in particle size, signifying fractionation. For $\sigma_\textrm{p} = 0.0998$ this signal is almost too weak to detect within error, but is far stronger for $\sigma_\textrm{p} = 0.37$. At $\sigma_\textrm{p} =0.37$, the wavelength of the signal is slightly smaller for the nonscalable potential – this reflects a smaller characteristic domain size which can also be seen in Figs. \[40poly\_combined\]b and \[unweighted\]. Fig. \[time\] shows the evolution through time of $\xi_2(r)$, demonstrating growing wavelength of size correlation as the gas and liquid domains coarsen. Finally, we note that the long-range component is absent in the HS case even when volume fraction is increased, demonstrating that it is not simply a result of the high volume fraction of the liquid phase.
Real space correlation functions: $\xi_1(r)$
--------------------------------------------
The function $\xi_2(r)$ detects the presence of fractionation, but not its direction. For that purpose we introduce an alternative function which is linear, not quadratic, in the size deviations $ \epsilon $: $$\begin{aligned}
\xi_1(r) &= \langle \epsilon_i + \epsilon_j \rangle_r -2 \langle \epsilon_i \rangle \\ \notag
&= \left( \frac{\sum_i \sum_{j \neq i} \delta(r_{ij} - r) (\epsilon_i +\epsilon_j)\, }{ \, \sum_k \sum_{k \neq l} \delta(r_{kl} - r)}\right)~.\end{aligned}$$ The ‘linear’ function $\xi_1(r)$ detects fractionation in a different manner to $\xi_2(r)$. Note that the sum over bonds ($\sum_i \sum_{j \neq i}$) will be dominated by the higher density phase, in this case the liquid. If the liquid contains on average larger particles, then for particles in the same domain, $ \epsilon_i +\epsilon_j$ will tend to be positive since particles $i$ and $j$ will, on average, be in the liquid phase. However, if the liquid contains smaller particles, then $ \epsilon_i +\epsilon_j$ will tend to be negative. Hence $\xi_1(r)$ unlike $\xi_2(r)$ is sensitive to the *direction* of fractionation, but relies on a difference in number density between the phases which $\xi_2(r)$ does not.
As shown in Fig. \[linear\], $\xi_1(r)$ like $\xi_2(r)$ decays quickly to zero for HS fluids. The short-wavelength component described in $\xi_2(r)$ is present but manifests differently; now the smallest separations $r$ (associated with a pair of small particles) lead to negative $\xi_1(r)$ since $\epsilon_i +\epsilon_j < 0$. In the phase separated case, this dependence on the sign of the $\epsilon$ allows the opposite fractionation directions in the scalable and nonscalable potentials to be resolved. Fractionation of smaller particles into the liquid (nonscalable) results in negative $\xi_1(r)$ for $r$ less than the domain size, as the sum is dominated by liquid particle pairs for which $\epsilon_i +\epsilon_j < 0$. We note also that the relative size of error bars for $\xi_1(r)$ is smaller than for $\xi_2(r)$.
Weighted structure factor: $S_\xi(q)$
-------------------------------------
Since $\xi_2(r)$ and $\xi_1(r)$ contain a short-wavelength component for local packing in addition to the long-wavelength component that signifies fractionation, it is useful to analyse the size correlations in Fourier space using a weighted structure factor: $$\begin{aligned}
S_\xi(\bm{q}) = \frac{1}{N} \langle \sum_{i,j} \epsilon_i \epsilon_j \exp{-i \bm{q} \cdot (\bm{r}_i - \bm{r}_j)}\rangle~.\label{eqn:S}\end{aligned}$$ In practice we employ the radially-averaged $S_\xi(q)$, and the average over microstates implies, for the phase-separating fluids, an average over the independent trajectories and over suitably short time periods (as for Fig. \[time\]). The lowest value of $q$ in each dimension is excluded, to avoid artefacts from the periodic boundary conditions.
In contrast to the real space functions, $S_\xi(q)$ represents particle size correlations in frequency space, so any fractionation signal should be manifest as a size correlation at some low $q$ corresponding to the domain size.
{width="12.6cm"}
![\[unweighted\]The standard (unweighted) structure factor $S(q)$ at different values of the parent polydispersity $\sigma_\textrm{p}$ for scalable (black) and nonscalable (red) potentials. Error bars are less than the symbol size.](unweighted_prune.eps){width="7.5cm"}
The results are shown in Fig. \[weighted\] and are in agreement with the real space measurements in Fig. \[quadratic\]. Particularly for $\sigma_\textrm{p} = 0.37$, peaks at low $q$ which are absent in the homogeneous HS case become clear in the phase separating cases. The benefit of $S_\xi(q)$ is that long-wavelength size correlations can be clearly distinguished from short-wavelength packing effects via their separation in $q$.
Comparison of the standard (unweighted) $S(q)$ (Fig. \[unweighted\]) reveals interesting differences between the two potentials used. At $\sigma_\textrm{p} = 0.0998$ the two are almost indistinguishable. At $\sigma_\textrm{p} = 0.37$ the low-$q$ peak corresponding to domain formation is at higher $q$ for the nonscalable potential, reflecting the smaller domain size reached on the simulated timescale (Fig. \[40poly\_combined\]b). Also, the peak corresponding to near-neighbours occurs at higher $q$ for the nonscalable potential. This may be a structural effect of fractionation, the partitioning of smaller particles into the liquid (which dominates the structure factor) causing it to have a smaller inter-particle spacing. Finally, in respect of short-wavelength correlations, the inter-particle peak in $S(q)$ occurs around $q \sim 6$ while its corresponding peak in $S_\xi(q)$ occurred at around $q \sim 4$. This indicates that the size correlations for local packing occur over slightly longer lengthscales than the typical inter-particle distance, and may be related to the phase quadrature (in real space) between size correlation functions and RDFs seen in previous studies of these short-range correlations [@Williamson2012a; @Pagonabarraga2000].
\[sec:conclusion\]Conclusions
=============================
We have investigated ways of robustly measuring structure and correlations in phase separating polydisperse fluids, using them to study fractionation during gas-liquid separation with large polydispersity and a skewed particle size distribution. Fractionation of polydisperse systems can strongly influence the properties of resultant phases [@Wilding2004; @Sollich2011; @Evans1998] and is a necessary process in crystallisation [@Sollich2011; @Leocmach2013], the frustration of which is important in glass formation [@Zargar2013; @Zaccarelli2013]. Like many aspects of polydisperse systems, fractionation is typically studied at equilibrium, and until recently not much was known about the kinetic path towards fractionated equilibrium [@Williamson2012; @Williamson2012a; @Leocmach2013]. Understanding the dynamics of fractionation requires measurement methods which can be applied to inhomogeneous multiple-phase systems, without macroscopic regions of the coexisting phases that can be easily isolated [@Evans1998].
We have employed a number of such methods to give a comprehensive and consistent picture of fractionation during the early stages of spinodal decomposition, when domains are only a few particle diameters in size. The methods are applicable in principle to experiment as well as simulation, given recent advances in *in situ* particle size measurements [@Leocmach2013]. Further, they are easily generalised to polydispersity in properties other than size, in which case $\epsilon_i$ represents an appropriately normalised deviation of the polydisperse property from its mean.
Firstly we introduced a Voronoi method which coarse-grains local volume fraction according to the topology of the Voronoi network. This minimises biasing with respect to particle size, which would otherwise interfere with fractionation measurements. This method can be used wherever a local volume fraction is required (not just in polydisperse systems) such as $2\phi\textrm{MD}$ simulations [@Fern2007], and allows systematic coarsening of the average by including neighbours of neighbours (etc.) without introducing any free parameters.
We then introduced an alternative approach in which fractionation is detected via associated long-range correlations in particle size (separate from short-range correlations which are sensitive to local packing efficiency [@Williamson2012a; @Pagonabarraga2000]). In real space, a pair of ‘weighted RDFs’ $\xi_2(r)$ and $\xi_1(r)$ can be used (a related idea occurs in the use of ‘RDF descriptors’ for chemical structure [@Hemmer1999; @Fernandez2013]). The first shows up fractionation irrespective of how the phases differ, but does not show the direction of fractionation. The second relies on a difference in number density between the phases, but *is* sensitive to the direction of fractionation – in this case, which phase prefers bigger/smaller particles. It also has smaller characteristic error bars for the system studied here. Finally, it is possible to use a structure factor $S_\xi(q)$ weighted by particle size (or whatever the polydisperse property is) to represent the correlations of $\xi_2(r)$ in Fourier space. This has the advantage of clearly distinguishing the component due to short-range correlations from the long-range one due to fractionation.
With these methods, we measured fractionation of mean diameter and polydispersity, with its direction dependent on fine details of the interaction potential. The important qualitative features are predicted correctly by a theory which only requires as input properties of the monodisperse reference system [@Williamson2012; @Evans2000]. Given that the system has not reached steady state, it is difficult to judge the quantitative predictions of Eq. \[eqn:momentfractionation\] beyond noting that fractionation on the simulated timescale appears not as strong as at predicted steady state; one would expect this part-way through phase separation. A direct comparison with fractionation in an equilibrium simulation of the system would shed light on this issue. However, determining polydisperse phase equilibria – the object of almost all previous work polydispersity – is itself extremely difficult, requiring highly specialised simulation techniques [@Wilding2005] which so far have not been applied to the system studied here.
Although the fractionation measured here is quite weak, we note that i) polydispersity of interaction *depth* as well as range will generally exist in a physical system, and can serve to enhance fractionation; ii) our simulations are only in the very earliest stages of phase separation, so we expect that the phases will ripen compositionally on longer timescales [@Warren1999]. This latter point could be quantitatively addressed in future work by applying both equilibrium and dynamical simulations to the same system, and comparing the results.
To avoid significantly slowing the simulation, we truncated the Schulz size distributions at $2 \langle d \rangle _\textrm{p}$. Thus the distributions have lower skew than a true Schulz of the same polydispersity (particularly for the high polydispersity $\sigma_\textrm{p} = 0.37$ case), and we would thus expect stronger fractionation of polydispersity (setting $n=2$ in Eq. \[eqn:momentfractionation\]) if the cutoff were removed. Also, we note equilibrium work has shown particle size cutoffs *per se* can have important effects on phase behaviour [@Wilding2005]. Fractionation of polydispersity may be very important where crystals nucleate from a metastable liquid [@Fortini2008]; depending on the shape of the parent distribution and the interaction potential, fractionation can reduce or increase the polydispersity of the liquid, thus promoting or suppressing subsequent crystal nucleation. Where crystals are involved, even small changes in polydispersity can have strong effects [@Williamson2012a; @Martin2003].
The results show emphatically that complex fractionation is involved right from the beginning of polydisperse phase separation, local composition relaxing alongside local density rather than long after it. It can therefore play a role in the formation of nonequilibrium structures which arrest before coming to equilibrium [@Varrato2012; @DiMichele2013], such as when gels form from a polydisperse colloid [@Liddle2011].
As well as fractionation, the correlation functions introduced could be of general use in characterising structural effects of polydispersity. For instance, polydispersity is ubiquitous in colloidal glasses (in order to prevent crystal growth) [@Zargar2013], but its detailed effects in that context are only just beginning to be understood [@Zaccarelli2013]. It would be interesting to study high density amorphous states from the point of view of particle size correlations.
We are grateful for discussions with Dan Blair and Emanuela Del Gado. JJW sincerely acknowledges Chris Rycroft (developer of Voro++ [@Voro]) and Alexander Stukowski (OVITO visualisation software [@OVITO]) for provision of software and helpful advice. This work was funded by an EPSRC DTG award and by Georgetown University.
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[^1]: It is arguably most natural to first consider the Voronoi ‘S cell’ tessellation, in which the Voronoi cell is defined as the space closer to particle $i$’s *surface* than any other particle’s surface, but this causes the cell faces to be curved, considerably increasing computational complexity. The radical tessellation avoids this issue while still closely approximating the structure of the S cells. See Ref. [@Pinheiro2013] for a comprehensive discussion.
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---
abstract: 'We give the following version of Fatou’s theorem for distributions that are boundary values of analytic functions. We prove that if $f\in\mathcal{D}^{\prime}\left( a,b\right) $ is the distributional limit of the analytic function $F$ defined in a region of the form $\left(a,b\right) \times\left( 0,R\right) ,$ if $\ $the one sided distributional limit exists, $f\left( x_{0}+0\right) =\gamma,$ and if $f$ is distributionally bounded at $x=x_{0},$ then the Łojasiewicz point value exists, $f\left( x_{0}\right) =\gamma$ distributionally, and in particular $F(z)\to \gamma$ as $z\to x_{0}$ in a non-tangential fashion.'
address:
- |
Mathematics Department\
Louisiana State University\
Baton Rouge, LA 70803, USA
- 'Department of Mathematics, Ghent University, Krijgslaan 281 Gebouw S22, B 9000 Gent, Belgium'
author:
- Ricardo Estrada
- Jasson Vindas
title: On distributional point values and boundary values of analytic functions
---
Introduction\[Intro\]
=====================
The study of boundary values of analytic functions is an important subject in mathematics. In particular, it plays a vital role in the understanding of generalized functions [@beltrami; @bremermann; @c-k-p]. As well known, the behavior of an analytic function at the boundary points is intimately connected with the pointwise properties of the boundary generalized function [@EstradaComVar; @estrada-vindasT2010; @vindas-estradaT2008; @vindas-estrada2008; @vladimirov-d-z] and the study of this interplay has often an Abelian-Tauberian character. There is a vast literature on Abelian and Tauberian theorems for distributions (see the monographs [@estrada; @ML; @p-s-v; @vladimirov-d-z] and references therein).
In this article we present sufficient conditions for the existence of Łojasiewicz point values [@lojasiewicz] for distributions that are boundary values of analytic functions. The pointwise notions for distributions used in this paper are explained in Section \[Prelim\]. The following result by one of the authors is well known [@EstradaComVar]:
*Suppose that* $f\in\mathcal{D}^{\prime}\left( \mathbb{R}\right)
$ *is the boundary value of a function* $F$, *analytic in the upper half-plane, that is,* $f\left( x\right) =F\left( x+i0\right)
;$ *if the distributional lateral limits* $f\left( x_{0}\pm0\right)
=\gamma_{\pm}$ *both exist, then* $\gamma_{+}=\gamma_{-}=\gamma
,$ *and the distributional limit* $f\left( x_{0}\right)
$* exists and equals* $\gamma.$
On the other hand, the results of [@EstradaChina] imply that there are distributions $f\left( x\right) =F\left( x+i0\right) $ for which one distributional lateral limit exits but not the other. In Theorem \[TeoremaCaso2\] we show that the existence of one the distributional lateral limits may be removed from the previous statement if an additional Tauberian-type condition is assumed, namely, if the distribution is distributionally bounded at the point. We also show that when the distribution $f$ is a bounded function near the point, then the distributional point value is of order 1. Furthermore, we give a general result of this kind for analytic functions that have distributional limits on a contour.
As an immediate consequence of our results, we shall obtain the following version of Fatou’s theorem [@koosis; @Pom] for distributions that are boundary values of analytic functions.
Let $F$ be analytic in a rectangular region of the form $\left( a,b\right) \times\left( 0,R\right) .$ Suppose that $f\left(
x\right) =\lim_{y\rightarrow0^{+}}F\left( x+iy\right) $ in $\mathcal{D}^{\prime}\left( a,b\right)$, that $f$ is a bounded function near $x_{0}\in\left(a,b\right)$, and that the following average lateral limit exists $$\lim_{x\to x^{+}_{0}}\frac{1}{(x-x_{0})}\int_{x_{0}}^{x} f(t)\:\mathrm{d}t=\gamma\ .$$Then, $$\lim_{z\to x_{0}}F(z)=\gamma \ \ \ (\mbox{angularly}).$$
Finally, we remark that Theorem \[Thm:Cor\] below generalizes some of our Tauberian results from [@vindas-estradaT2008].
Preliminaries\[Prelim\]
=======================
We explain in this section several pointwise notions for distributions. There are several equivalent ways to introduce them. We start with the useful approach from [@CF]. Define the operator $\mu_{a}$ on locally integrable complex valued functions in $\mathbb{R}$ as $$\mu_{a}\left\{ f\left( t\right) ;x\right\} =\frac{1}{x-a}\int_{a}^{x}f\left( t\right) \,\mathrm{d}t\,,\ \ x\neq a\,, \label{4.1}$$ while the operator $\partial_{a}$ is the inverse of $\mu_{a},$$$\partial_{a}\left( g\right) =\left( \left( x-a\right) g\left( x\right)
\right) ^{\prime}\,. \label{4.2}$$ Suppose first that $f_{0}=f$ is *real.* Then if it is bounded near $x=a,$ we can define $$\overline{f_{0}}\left( a\right) =\limsup_{x\rightarrow a}f\left( x\right)
\,,\;\;\;\;\;\;\underline{f_{0}}\left( a\right) =\liminf_{x\rightarrow
a}f\left( x\right) \,. \label{4.3}$$ Then $f_{1}=\mu_{a}\left( f\right) $ will be likewise bounded near $x=a$ and actually $$\underline{f_{0}}\left( a\right) \leq\underline{f_{1}}\left( a\right)
\leq\overline{f_{1}}\left( a\right) \leq\overline{f_{0}}\left( a\right)$$ and, in particular, if $f\left( a\right) =f_{0}\left( a\right) $ exists, then $f_{1}\left( a\right) $ also exists and $f_{1}\left( a\right)
=f_{0}\left( a\right) .\smallskip$
A distribution $f\in\mathcal{D}^{\prime}\left( \mathbb{R}\right) $ is called distributionally bounded at $x=a$ if there exist $n\in\mathbb{N}$ and $f_{n}\in\mathcal{D}^{\prime}\left( \mathbb{R}\right) ,$ continuous and bounded in a pointed neighborhood $\left( a-\varepsilon,a\right) \cup\left(
a,a+\varepsilon\right) $ of $a,$ such that $f=\partial_{a}^{n}f_{n}.\smallskip$
If $f_{0}$ is distributionally bounded at $x=a,$ then there exists a *unique* distributionally bounded distribution near $x=a,$ $f_{1},$ with $f_{0}=\partial_{a}f_{1}.$ Therefore, $\partial_{a}$ and $\mu_{a}$ are isomorphisms of the space of distributionally bounded distributions near $x=a.$ Given $f_{0}$ we can form a sequence of distributionally bounded distributions $\{f_{n}\}_{n=-\infty}^{\infty}$ with $f_{n}=\partial_{a}f_{n+1}$ for each $n\in\mathbb{Z}.$
We say that $f$ has the distributional point value $\gamma$ in the sense of Łojasiewicz [@lojasiewicz; @estrada-vindasIntegral] and write $$f\left( a\right) =\gamma\;\;\;\left( \mathrm{L}\right) \,, \label{4.7}$$ if there exists $n\in\mathbb{N},$ the order of the point value, such that $f_{n}$ is continuous near $x=a$ and $f_{n}\left( a\right) =\gamma.$
It can be shown [@CF; @estrada; @lojasiewicz; @p-s-v] that $f\left( a\right) =\gamma$ $\left( \mathrm{L}\right) $ if and only if $$\lim_{\varepsilon\rightarrow0}f\left( a+\varepsilon x\right) =\gamma\,,$$ distributionally, that is, if and only if $$\lim_{\varepsilon\rightarrow0^{+}}\left\langle f\left( a+\varepsilon x\right)
,\phi\left( x\right) \right\rangle =\gamma\int_{-\infty}^{\infty}\phi\left(
x\right) \,\mathrm{d}x\,, \label{4.9}$$ for each $\phi\in\mathcal{D}\left( \mathbb{R}\right) .$ On the other hand, if $f$ is distributionally bounded at $x=a$ then $\left\langle f\left( a+\varepsilon x\right)
,\phi\left( x\right) \right\rangle $ is bounded as $\varepsilon
\rightarrow0.$
We can also consider distributional lateral limits [@lojasiewicz; @vindas2007]. We say that the distributional lateral limit $f\left( a+0\right) $ $\left( \mathrm{L}\right) $ as $x\rightarrow a$ from the right exists and equals $\gamma,$ and write$$f\left( a+0\right) =\gamma\;\;\;\left( \mathrm{L}\right) \,, \label{4.10}$$ if (\[4.9\]) holds for all $\phi\in\mathcal{D}\left( \mathbb{R}\right) $ with support contained in $(0,\infty).$ The distributional lateral limit from the left $f\left( a-0\right) $ $\left( \mathrm{L}\right) $ is defined in a similar fashion.
Observe also that if $f=\partial_{a}^{n}f_{n},$ and $f_{n}$ is bounded near $x=a,$ then $f\left( a+0\right) $ $\left( \mathrm{L}\right) $ exists, and equals $\gamma$, if and only if $f_{n}\left(a+0\right) =\gamma$ $\left(
\mathrm{L}\right).$
These notions have straightforward extensions to distributions defined in a smooth contour of the complex plane. A natural extension of this pointwise notions for distributions is the so called quasiasymptotic behavior of distributions, explained, e.g., in [@p-s-v; @vindas2010; @vladimirov-d-z].
Boundary values \[tauberianPV\] and distributional point values
===============================================================
We shall need the following well known fact [@beltrami]. We shall use the notation $\mathbb{H}$ for the half plane $\left\{ z\in\mathbb{C}:\Im
m\,z>0\right\}.$
Let $F$ be analytic in the half plane $\mathbb{H},$ and suppose that the distributional limit $f\left( x\right) =F\left( x+i0\right) $ exists in $\mathcal{D}^{\prime}\left( \mathbb{R}\right) .$ Suppose that there exists an open, non-empty interval $I$ such that $f$ is equal to the constant $\gamma$ in $I.$ Then $f=\gamma$ and $F=\gamma.\smallskip$
Actually using the theorem of Privalov [@Pom Cor 6.14] it is easy to see that if $F$ is analytic in the half plane $\mathbb{H},$ $f\left( x\right)
=F\left( x+i0\right) $ exists in $\mathcal{D}^{\prime}\left( \mathbb{R}\right) ,$ and there exists a subset $X\subset\mathbb{R}$ of non-zero measure such that the distributional point value $f\left( x_0\right) $ exists and equals $\gamma$ if $x_0\in X,$ then $f=\gamma$ and $F=\gamma.$
Our first result is for *bounded* analytic functions.
\[TeoremaCaso1\]Let $F$ be analytic and bounded in a rectangular region of the form $\left( a,b\right) \times\left( 0,R\right) .$ Set $f\left(
x\right) =\lim_{y\rightarrow0^{+}}F\left( x+iy\right) $ in $\mathcal{D}^{\prime}\left( a,b\right)$, so that $f\in L^{\infty}(a,b)$. Let $x_{0}\in\left( a,b\right) $ be such that $$f\left( x_0+0\right) =\gamma\;\;\;\left( \mathrm{L}\right)
\label{Ta.1}$$ exists. Then the distributional point value $$f\left( x_0\right) =\gamma\;\;\;\left( \mathrm{L}\right)
\label{Ta.2}$$ also exists. In fact, the point value is of the first order, and thus $$\lim_{x\rightarrow x_{0}}\frac{1}{x-x_{0}}\int_{x_{0}}^{x}f\left( t\right)
\,\mathrm{d}t=\gamma\,. \label{Ta.3}$$
We shall first show that it is enough to prove the result if the rectangular region is the upper half-plane $\mathbb{H}.$ Indeed, let $\mathsf{C}$ be a smooth simple closed curve contained in $\left( a,b\right) \times\lbrack0,R)$ such that $\mathsf{C}\cap\left( a,b\right) =[ x_{0}-\eta,x_{0}+\eta] ,$ and which is symmetric with respect to the line $\Re e\,z=x_{0}.$ Let $\varphi$ be a conformal bijection from $\mathbb{H}$ to the region enclosed by $\mathsf{C}$ such that the image of the line $\Re e\,z=x_{0}$ is contained in $\Re
e\,z=x_{0},$ so that, in particular, $\varphi\left( x_{0}\right) =x_{0}.$ Then (\[Ta.1\])–(\[Ta.3\]) hold if and only if the corresponding equations hold for $f\circ\varphi$.
Therefore we may assume that $a=-\infty,$ and $b=R=\infty.$ In this case, $f$ belongs to the Hardy space $H^{\infty},$ the closed subspace of $L^{\infty}\left(
\mathbb{R}\right) $ consisting of the boundary values of bounded analytic functions on $\mathbb{H}.$ Let $f_{\varepsilon}\left( x\right) =f\left(
x_{0}+\varepsilon x\right) .$ Clearly, the set $\left\{ f_{\varepsilon
}:\varepsilon>0\right\} $ is weak\* bounded (as a subset of the dual space $\left( L^{1}\left( \mathbb{R}\right) \right) ^{\prime}=L^{\infty}\left(
\mathbb{R}\right) $) and, consequently, a relatively weak\* compact set. If $\left\{ \varepsilon_{n}\right\} _{n=0}^{\infty}$ is a sequence of positive numbers with $\varepsilon_{n}\rightarrow0$ such that the sequence $\left\{
f_{\varepsilon_{n}}\right\} _{n=0}^{\infty}$ is weak\* convergent to $g\in
L^{\infty}\left( \mathbb{R}\right) ,$ then $g\equiv\gamma,$ since $g\in
H^{\infty},$ and $g\left( x\right) =\gamma$ for $x>0.$ In fact, the condition (\[Ta.1\]) means that $$\int_{0}^{\infty}g(x)\psi(x) \mathrm{d}x = \lim_{n\to\infty} \int_{0}^{\infty}f_{\varepsilon_{n}}(x)\psi(x) \mathrm{d}x= \gamma \int_{0}^{\infty}\psi(x) \mathrm{d}x\ ,$$ for all $\psi\in \mathcal{D}(0,\infty)$, which yields the claim. Since any sequence $\left\{ f_{\varepsilon_{n}}\right\} _{n=0}^{\infty}$ with $\varepsilon
_{n}\rightarrow0$ has a weak\* convergent subsequence, and since that subsequence converges to the constant function $\gamma,$ we conclude that $f_{\varepsilon}\rightarrow\gamma$ in the weak\* topology of $L^{\infty}\left(
\mathbb{R}\right) .$ Furthermore, (\[Ta.3\]) follows by taking $x=x_{0}+\varepsilon$ and $\phi\left( t\right) =\chi_{\left[ 0,1\right] }\left( t\right) ,$ the characteristic function of the unit interval, in the limit $\lim
_{\varepsilon\rightarrow0}\left\langle f_{\varepsilon}\left( t\right)
,\phi\left( t\right) \right\rangle =\gamma\int_{-\infty}^{\infty}\phi\left(
t\right) \,\mathrm{d}t.$
We can now prove our main result, a distributional extension of Theorem \[TeoremaCaso1\].
\[TeoremaCaso2\]Let $F$ be analytic in a rectangular region of the form $\left( a,b\right) \times\left( 0,R\right) .$ Suppose $f\left( x\right)
=\lim_{y\rightarrow0^{+}}F\left( x+iy\right) $ in the space $\mathcal{D}^{\prime}\left( a,b\right) .$ Let $x_{0}\in\left( a,b\right) $ such that $f\left( x_{0}+0\right) =\gamma$ $\left( \mathrm{L}\right) .$ If $f$ is distributionally bounded at $x=x_{0}$ then $f\left( x_{0}\right) =\gamma$ $\left( \mathrm{L}\right) .$ Furthermore, $F(z)\to \gamma$ as $z\to x_{0}$ in an angular fashion.
There exists $n\in\mathbb{N}$ and a function $f_{n}$ bounded in a neighborhood of $x_{0}$ such that $f=\partial_{x_{0}}^{n}f_{n};$ notice that $f\left(
x_{0}\right) =\gamma$ $\left( \mathrm{L}\right) $ if and only if $f_{n}\left( x_{0}\right) =\gamma$ $\left( \mathrm{L}\right) .$ But $f_{n}\left( x\right) =F_{n}\left( x+i0\right) $ distributionally, where $F_{n}$ is analytic in $\left( a,b\right) \times\left( 0,R\right) ;$ here $F_{n}$ is the only angularly bounded solution of $F\left( z\right)
=\partial_{x_{0}}^{n}F_{n}\left( z\right) $ (derivatives with respect to $z$). Clearly, $f_{n}(x)=F_{n}(x+i0)$. Since $f_{n}$ is bounded near $x=x_{0},$ $F_{n}$ is also bounded in a rectangular region of the form $\left( a_{1},b_{1}\right) \times\left(
0,R_{1}\right) ,$ where $x_{0}\in\left( a_{1},b_{1}\right) .$ Clearly $f_{n}\left( x_{0}+0\right) =\gamma$ $\left( \mathrm{L}\right),$ so the Theorem \[TeoremaCaso1\] yields $f_{n}\left( x_{0}\right) =\gamma$ $\left( \mathrm{L}\right) ,$ as required. Finally, the fact that $F(z)\to\gamma$ as $z\to x_{0}$, angularly, is a consequence of the existence of the distributional point value, as shown in [@estrada2005; @vindas2010].
Observe that in general the result (\[Ta.3\]) does not follow if $f$ is not bounded but just distributionally bounded near $x_{0}.$
We may use a conformal map to obtain the following general form of the Theorem \[TeoremaCaso2\].
\[TeoremaCaso3\]Let $\mathsf{C}$ be a smooth part of the boundary $\partial\Omega$ of a region $\Omega$ of the complex plane. Let $F$ be analytic in $\Omega,$ and suppose that $f\in\mathcal{D}^{\prime}\left(
\mathsf{C}\right) $ is the distributional boundary limit of $F.$ Let $\xi
_{0}\in\mathsf{C}$ and suppose that the distributional lateral limit $f\left(
\xi_{0}+0\right) =\gamma$ $\left( \mathrm{L}\right) $ exists and $f$ is distributionally bounded at $\xi=\xi_{0},$ then $f\left( \xi_{0}\right)
=\gamma$ $\left( \mathrm{L}\right) $ and $F(z)$ has non-tangential limit $\gamma$ at the boundary point $\xi_{0}$.
We also immediately obtain the following Tauberian theorem. As mentioned at the Introduction, it generalizes some Tauberian results by the authors from [@vindas-estradaT2008].
\[Thm:Cor\]Let $F$ be analytic in a rectangular region of the form $\left( a,b\right) \times\left( 0,R\right) .$ Suppose $f\left( x\right)
=\lim_{y\rightarrow0^{+}}F\left( x+iy\right) $ in the space $\mathcal{D}^{\prime}\left( a,b\right) .$ Let $x_{0}\in\left( a,b\right) $ such that the distributional limit $\lim_{y\rightarrow0^{+}}F\left( x_{0}+iy\right)
=\gamma$ $\left( \mathrm{L}\right) $ exists. If $f$ is distributionally bounded at $x=x_{0}$ then $f\left( x_{0}\right) =\gamma$ $\left(
\mathrm{L}\right) $ and the angular (ordinary) limit exists: $\lim_{z\to x_{0}}F\left( z\right) =\gamma.$
If we consider the curve $\mathsf{C}$ to be the union of the segments $(a,x_{0}]$ and $[x_{0},iR),$ then the distributional lateral limit of the boundary value of $F$ on $\mathsf{C}$ exists and equals $\gamma$ as we approach $x_{0}$ from the right along $\mathsf{C}$ and so the Theorem \[TeoremaCaso3\] yields that the distributional limit from the left, which is nothing but $f\left( x_{0}-0\right) $ $\left( \text{L}\right) ,$ also exists and equals $\gamma.$ Then the Theorem \[TeoremaCaso2\] gives us that $f\left( x_{0}\right) =\gamma$ $\left( \mathrm{L}\right) .$ The existence of the angular limit of $F\left( z\right) $ as $z\rightarrow x_{0}$ then follows.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Wolfgang Reich, Patricia Reich and Xiaohui Sun'
bibliography:
- 'bbfile.bib'
date: 'Received; accepted'
title: 'Long, depolarising H$\alpha$-filament towards the Monogem ring[^1]'
---
[In soft X-rays, the Monogem ring is an object with a diameter of $25\degr$ located in the Galactic anti-centre. It is believed to be a faint, evolved, local supernova remnant. The ring is also visible in the far-ultraviolet, and a few optical filaments are related. It is not seen at radio wavelengths, as other large supernova remnants are. ]{} [We study a narrow about $4\fdg5$ long, faint H$\alpha$-filament, G203.7+11.5, that is seen towards the centre of the Monogem ring. It causes depolarisation and excessive Faraday rotation of radio polarisation data. ]{} [Polarisation observations at $\lambda$11 cm and $\lambda$21 cm with the Effelsberg 100-m telescope were analysed in addition to $WMAP$ data, extragalactic rotation measures, and H$\alpha$ data. A Faraday-screen model was applied.]{} [From the analysis of the depolarisation properties of the H$\alpha$ filament, we derived a line-of-sight magnetic field, $B_{||}$, of 26$\pm5\mu$G for a distance of 300 pc and an electron density, $n_\mathrm{e}$, of 1.6 cm$^{-3}$. The absolute largest rotation measure of G203.7+11.5 is -86$\pm3$ rad m$^{-2}$, where the magnetic field direction has the opposite sign from the large-scale Galactic field. We estimated the average synchrotron emissivity at $\lambda$21 cm up to 300 pc distance towards G203.7+11.5 to about 1.1 K $T_\mathrm{b}$/kpc, which is higher than typical Milky Way values.]{} [The magnetic field within G203.7+11.5 is unexpected in direction and strength. Most likely, the filament is related to the Monogem-ring shock, where interactions with ambient clouds may cause local magnetic field reversals. We confirm earlier findings of an enhanced but direction-dependent local synchrotron emissivity. ]{}
Introduction
============
X-rays from the Monogem ring were first detected during a rocket flight in 1969 [@Bunner70]. It is seen as a bright soft X-ray object with a diameter of $25\degr$ in the [*ROSAT*]{} All-sky survey [@Voges99], where its centre is at $l,b \sim 201\degr, +12\degr$. The Monogem ring was discussed in some detail by @Plucinsky96, who concluded that it is a large evolved supernova remnant (SNR) in the adiabatic phase of evolution that expands in a very low-density interstellar medium (ISM). More recent X-ray observations of the Monogem ring with [*SUZAKU*]{} led to a refined analysis of the SNR parameters [@Knies18].
Far-ultraviolet (FUV) emission was detected by @Kim07, where C IV and other lines indicate interaction details of the Monogem ring with the ambient ISM. Two optical filaments associated with the Monogem ring have been identified by @Reimers84 and by @Weinberger06. The spectra of both filaments show excitation by a slow shock, as expected from an evolved SNR. @Thorsett03 showed that the pulsar PSR B0656+14 is located close to the centre of the Monogem ring, and the authors concluded that both objects result from a supernova explosion that occurred about one hundred thousand years ago at a distance of about 300 pc.
The Monogem ring shows no radio emission, as expected for older SNRs in the metre or decimetre range, where synchrotron emission is the dominating process. This was explained by @Plucinsky96 by the unusually low ISM density. However, @Vallee93 pointed out that the Monogem ring and similar large local features may significantly contribute to the observed rotation measure ($RM$) of extragalactic sources, which are the basis for magnetic field models of the Milky Way.
During the observations for the $\lambda$21 cm Effelsberg Medium Latitude Survey (EMLS) [@Uyaniker98; @Uyaniker99; @Reich04], we found numerous depolarised canal-like structures in the polarisation maps. For some of them, we made follow-up observations at $\lambda$11 cm to study the origin of the canals. A prominent long canal is seen close to the direction of the centre of the Monogem ring. The filament has a clear counterpart in the all-sky H$\alpha$ map compiled by @Finkbeiner03. Its properties are discussed in the following. We show ROSAT data of the Monogem ring in Fig. \[ROSAT\], where we indicate the position of PSR B0656+14 and the area we observed at radio frequencies.
![Overview of the Monogem ring in soft-X-rays as observed by ROSAT (0.1- 0.4 keV band, [@Voges99]) with the related PSR B0656+14. The observed radio field is indicated.[]{data-label="ROSAT"}](Reich-1.eps){width="49.00000%"}
In Sect. 2 we describe the radio and H$\alpha$ data that we used and the zero-level calibration of the $\lambda$11 cm polarisation data. Section 3 presents the maps of our observations. A Faraday-screen analysis is given in Sect. 4. We discuss the physical properties of G203.7+11.5 in Sect. 5, where we also derive the local synchrotron emissivity in the direction of the Monogem ring. Section 6 gives a summary.
Data
====
Effelsberg $\lambda$21 cm data
------------------------------
G203.7+11.5 is visible on the EMLS $\lambda$21 cm maps as a long, narrow polarisation depression (canal) that extends from [*[l,b]{}*]{} $\sim 204\degr, +12\degr$ to [*[l,b]{}*]{} $\sim 203\fdg3, +10\degr$. We used total-intensity and linear-polarisation data from an unpublished section of the EMLS. When completed, the EMLS will cover the northern Galactic plane $\pm20\degr$ in latitude at an angular resolution of $9\farcm4$. Table \[ObsTab\] lists some technical details. Missing large-scale emission components and the absolute zero-levels of the EMLS were provided by the Stockert 25-m telescope $\lambda$21 cm total-intensity survey of the northern sky [@Reich82; @Reich86] and the DRAO 26-m linear-polarisation survey [@Wolleben06]. Adding large-scale structures from these surveys provides an absolute zero-level for total intensities and linear polarisation. The $\lambda$21 cm polarised-intensity (${PI}$) map with overlaid total-intensity contours is shown in Fig. \[21cm\]. $PI$ is calculated from the observed Stokes parameter $U$ and $Q$ as $PI = \sqrt{U^2+Q^2-1.2\sigma_{U,Q}^{2}}$, including zero-bias correction. $\sigma$ is listed in Table \[ObsTab\].
----------------------------------------- --------------------------- --------------------- --
Data Effelsberg $\lambda$11 cm EMLS $\lambda$21 cm
Frequency \[MHz\] 2639 1408
Bandwidth \[MHz\] 80 20
HPBW\[$\arcmin$\] 4.3 9.4
Main Calibrator 3C286 3C286
Flux Density of 3C286 \[Jy\] 11.5 14.4
Polarisation Percentage of 3C286 \[%\] 9.9 9.3
Polarisation Angle of 3C286 \[$\degr$\] 33 32
Number of Coverages $<$16 min. 1(B)+1(L)
Integration Time \[s\] $<$16 $\geq$ 2
r.m.s. ($I/U,Q$)\[mK $T_\mathrm{b}$\] 4/2 15/8
----------------------------------------- --------------------------- --------------------- --
\[ObsTab\]
New Effelsberg $\lambda$11 cm observations
------------------------------------------
Radio continuum and linear polarisation observations of G203.7+11.5 were made at $\lambda$11 cm with the Effelsberg 100-m radio telescope. The observations started in 1997/1999 and were completed with an improved $\lambda$11 cm receiver in 2007. The layout of the Effelsberg $\lambda$11 cm receiving system has been described by @Uyaniker04. The receiving system available in 1997/1999 was upgraded in 2005 with new low-noise amplifiers and an eight-channel IF-polarimeter. Its channels are 10 MHz wide, and a broad-band channel provides 80 MHz bandwidth in addition, which we used when no radio interference (RFI) was visible in the observing band.
A field of $2\fdg33 \times 3\fdg33$ was mapped by raster scans in Galactic longitude and latitude direction. Altogether, eight maps in each direction were observed. However, about 50% of the observations could not or only partly be used because of RFI, solar side-lobe distortions, or bad weather. Some observational parameters are listed in Table \[ObsTab\]. The raw data were reduced and calibrated by standard NOD2-based methods used for continuum and polarisation observations with the Effelsberg 100-m telescope (e.g. @Reich9011). We present the Effelsberg map of the $\lambda$11 cm observations in Fig. \[11cm\].
H$\alpha$ emission
------------------
There is a clear H$\alpha$ counterpart of the polarised $\lambda$11 cm emission in the all-sky H$\alpha$ map with 6$\arcmin$ angular resolution combined by @Finkbeiner03, which is a combination of various H$\alpha$ surveys. The H$\alpha$ map (Fig. \[Ha\]) shows that the faint filament extends from [*l,b*]{} $\sim
204\fdg5, +13\degr$ to $\sim 203\fdg1, +10\degr$ with intensity maxima around 6 to 7 Rayleigh that become slightly fainter at its ends. The intensity gradient is moderate and increases towards the upper right area of the map. The H$\alpha$ filament exceeds the diffuse large-scale emission by about 3 to 4 Rayleigh and has a width of about $30\arcmin$ (HPBW).
Absolute zero-level of the $\lambda$11 cm polarisation data
-----------------------------------------------------------
Interferometric data miss short spacings, and single-dish maps are usually set to an arbitrary zero-level. As discussed by @Reich06 and others, polarisation data without restored large-scale emission make any interpretation of polarisation features unreliable when they result from radiation transfer and not from emitting sources. Missing large-scale structures of polarisation maps from the magnetised interstellar medium will otherwise cause misinterpretations.
The observed $\lambda$11 cm Effelsberg maps of Stokes parameters $U$ and $Q$ were set to zero at their boundaries and thus miss polarised emission from components exceeding about $2\degr$ to $3\degr$ in extent. The zero-level problem of single-dish telescopes was solved for the Sino-German $\lambda$6 cm polarisation survey of the Galactic plane by @Sun07, where polarisation data on an absolute zero-level were not available, by adding scaled large-scale components from the $WMAP$ K-band ($\lambda$1.3 cm) polarisation data [@Page07]. The $WMAP$ polarisation data are at an absolute zero-level, as required for this purpose. This procedure assumes that Faraday rotation of the large-scale emission in the Galactic plane has a negligible influence on $\lambda$6 cm polarisation angles ${PA}$s ($PA = \frac{1}{2} $atan$({U/Q})$ ), and thus the ratio of $U$ and $Q$ for large scales remains unchanged. This assumption seems to be valid for most regions of the Galactic plane at $\lambda$6 cm, except for some emission from the innermost Galaxy. The Monogem ring is located in the Galactic anti-centre direction and well outside of the Galactic plane, so that applying the same correction method for the Effelsberg $\lambda$11 cm polarisation data seems to be justified although the Faraday rotation is about three times higher than at $\lambda$6 cm.
We used the $\lambda$21 cm and the $\lambda$1.3 cm $WMAP$ absolute polarisation data (nine-year release, @Hinshaw09) to calculate the spectral index $\beta$ of the diffuse large-scale $PI$ outside of the area of the thermal filament from the mean $U$ and $Q$ values. We found $\beta$ = -3.1 ($T_\mathrm{b}$ $\propto \nu^{+\beta}$). Between $\lambda$21 cm and $\lambda$1.3 cm, the mean angle difference for the large-scale emission is about 44$\pm5\degr$. With $PA_{\lambda}$ = $RM \times \lambda^{2}$ + $PA_{0}$, this angle difference corresponds to $RM$ = 1.7$\pm1.9$ rad m$^{-2}$ and implies a negligible correction for the $\lambda$11 cm polarisation data. It proves that the method we used for zero-level correction can be applied to the $\lambda$11 cm map. A recent spectral study of polarised emission observed with $PLANCK$ between 30 GHz and 44 GHz by @Jew revealed similar spectral-index values in the range $\beta$ = -2.99 to $\beta$ = -3.12, depending on the method that was applied.
Based on the spectral extrapolation with $\beta$ = -3.1 from the $\lambda$21 cm or the $\lambda$1.3 cm data, we added zero-level offsets of +20.5 mK and +11.2 mK to the originally observed $\lambda$11 cm $U$ and $Q$ values. The effect on the resulting $PI$ emission is shown in Fig. \[11cm+K9\]. The morphology changed drastically compared to the $PI$ distribution in Fig. \[11cm\], where the apparent $PI$ emission along the H$\alpha$ filament is in emission, into a depression (Fig. \[11cm+K9\]). This result clearly demonstrates the importance of absolute zero-levels for mapping Galactic polarised emission and its analysis. A depression is expected when the H$\alpha$ filament causes sufficiently high Faraday rotation on the polarised background, which then adds to the foreground emission. When the spectral index varies by $\pm$ 0.1, the quoted zero-level offsets for $U$ and $Q$ vary by 20$\%$. This does not change the morphology, but $PI$ and $PA$ values are slightly different. We take the influence of offset variations into account when we discuss the Faraday-screen model in Sect. 4.
Total intensities and polarised emission
========================================
The $\lambda$21 cm and $\lambda$11 cm total-intensity contours of Fig. \[21cm\] and Fig. \[11cm\] show several compact sources in the field and small fluctuations of the diffuse emission. No significant intensity gradient or any filamentary structure is visible in total intensities. The polarisation maps at an absolute zero-level (Fig. \[21cm\] and Fig. \[11cm+K9\]), however, show filamentary features that have no counterpart in total intensities and thus result from Faraday-rotation effects along the line of sight. The narrow depolarisation canal as seen in the $\lambda$21 cm polarised-intensity map (Fig. \[21cm\]) described above is located at the gradient of the broader $\lambda$11 cm depolarisation filament (Fig. \[21cmPI+11cm\]) that extends from [*[l,b]{}*]{} $\sim 204\fdg5, +13\degr$ to [*[l,b]{}*]{} $\sim 203\fdg1, +10\degr$, which itself coincides very well with the faint H$\alpha$ filament (Fig. \[Ha\]).
Faraday-screen model for G203.7+11.5
====================================
@Sun07 have described a method for calculating the properties of a discrete Faraday screen located somewhere along the line of sight. The modified polarised emission $PI_{\mathrm{on}}$ in the direction of the Faraday screen (on-position) is compared with the emission outside of the screen $PI_{\mathrm{off}}$ (off-position). In addition, the difference of the polarisation angles $PA_{\mathrm{on}} - PA_{\mathrm{off}}$ is required. The parameter $c$ is the ratio $PI_{\mathrm{fg}}$/$(PI_{\mathrm{fg}}+PI_{\mathrm{bg}})$, where fg and bg indicate the polarised foreground and background components, and $\psi_{s}$ is the angle rotation caused by the Faraday screen. The parameter $f$ describes the depolarisation of the Faraday screen, where 1 stands for no and 0 for total depolarisation,
$$\centering
\displaystyle{
\left\{
\begin{array}{cc}
\displaystyle
\frac{PI_{\mathrm{on}}}{PI_{\mathrm{off}}}=\sqrt{\mathit{f}^2(1-c)^2+c^2+2\mathit{f}c(1-c)\cos2\psi_s}\ , \\ \displaystyle
PA_{\mathrm{on}} - PA_{\mathrm{off}}=\frac{1}{2}\arctan\left(\frac{\mathit{f}(1-c)\sin2\psi_s}{c+\mathit{f}(1-c)\cos2\psi_s}\right).
&
\end{array}
\right.
}
\label{eq1}$$
For G203.7+11.5, $PA_{\mathrm{off}}$ is not around zero, as is typical of $PA$s in the Galactic plane [@Sun07], but at about $20\degr$, which is an offset to be subtracted from the $PA$ map, so that Eq. \[eq1\] can be directly applied. At $\lambda$11 cm, we found for the two required observables $PI_{\mathrm{on}}/PI_{\mathrm{off}}$ around 0.5 with variations, where the depolarisation maximum at [$\it{l,b}$]{} $\sim 203\fdg7, +11\fdg9$ was used for $PI_{\mathrm{on}}$. We measured for $PA_{\mathrm{on}}-PA_{\mathrm{off}} \sim -15\degr$ to $\sim -20\degr$. $PI_{\mathrm{on}}/PI_{\mathrm{off}}$ increases at the gradients of the filament.
There are considerable fluctuations in $PI$ and $PA$. They prevent a very precise estimate of the differences between on- and off-positions that indicate that the physical properties within the filament vary. In the following, we therefore calculate representative parameters. Based on Eq. \[eq1\], we obtained an angle rotation caused by the filament in the range of $\psi_{s} \sim -62\degr$ to $ \sim -66\degr$ for $f$ = 1. Then $c$ is calculated as $c$ = 0.63$\pm$0.03, which is the portion of the foreground polarised emission. For a wavelength of $\lambda$11 cm, the corresponding $RM$ is -86$\pm3$ rad m$^{-2}$. When the internal depolarisation increases, $c$ and/or parameter $f$ decrease to $ c$ = 0.61$\pm$0.03 for $f$ = 0.9 and $c$ = 0.58$\pm$0.03 for $f$ = 0.8, where the corresponding $RM$ changes slightly to -82$\pm3$ rad m$^{-2}$.
When we take the spectral uncertainties of $\Delta\beta \pm0.1$ into account when we calculate the $U$ and $Q$ offsets at $\lambda$11 cm, we see almost no effect on the angle differences $PA_{\mathrm{on}} - PA_{\mathrm{off}}$, but the ratio $PI_{\mathrm{on}}/PI_{\mathrm{off}}$ changes from about 0.5 for a spectral index of $\beta$ = -3.1 to about 0.6 or 0.4 in the case of a steeper or a flatter spectrum, respectively. The corresponding foreground portions $c$ do not change, but the $RM$ values are then calculated as $RM$ = -96$\pm2$ rad m$^{-2}$ for $\beta$ = -3.2 and $RM$ = -76$\pm2$ rad m$^{-2}$ for $\beta$ = -3.0.
With decreasing depolarisation at the gradients of the filament, the absolute $RM$ values will also decrease and are expected to affect the $\lambda$21 cm data, where the maximum depolarisation is expected at about $\pm 35$ rad m$^{-2}$, corresponding to an angle rotation of $\pm90\degr$ of the polarised background. Figure \[21cmPI+11cm\] shows that the narrow depolarisation canal runs parallel along the eastern gradient of the $\lambda$11 cm depolarised filament. The $RM$ value of the $\lambda$21 cm canal of around -35 rad m$^{-2}$ at the outer gradient of the filament is as expected, when the high absolute $RM$ values decrease and match the low $RM$ of the diffuse offset-emission. Along the $\lambda$11 cm depolarised filament, the $\lambda$21 cm $PI$ is at a similar level compared to the off-area, which implies that the depolarisation factor $f$ must be close to 1. The depolarisation from the Faraday screen is low and the assumption of $f$ = 0.8 we quoted above seems to be a lower limit.
The slope of the $PI$ gradients of the filament towards longer and lower Galactic longitudes are different, as can be clearly seen in the $\lambda$11 cm $PI$ map at $9\farcm4$ resolution in Fig. \[21cmPI+11cm\]. The $\lambda$21 cm canal is visible at the steeper gradient, but on the other side of the filament, there is a more extended $PI$ minimum, as expected from the shallow $\lambda$11 cm $PI$ and $RM$ gradients, which cannot be clearly separated from the general $PI$ fluctuations in this area. Thus the maximum depolarisation at around -35 rad m$^{-2}$ is smoothed out.
Discussion
==========
Thermal filament
----------------
To calculate the physical parameters of G203.7+11.5, we have to know its distance, size, and the electron temperature of the thermal gas. We can only make estimates of these parameters, which is reflected in the result. We assume that G203.7+11.5 is located at the distance of the Monogem ring, which is about 300 pc. The filament extends slightly to the north and south of the area shown in Fig. \[Ha\] as seen in the @Finkbeiner03 H$\alpha$ map. Figure \[Ha\] corresponds to the area of the $\lambda$11 cm observations. We estimated a total projected length of the filament of about $4\fdg5$, which corresponds to 24 pc. However, the apparent size of the Monogem ring is about 25$\degr$, which means a diameter of about 130 pc if it is symmetric in 3D. If the G203.7+11.5 filament is not located near the Monogem ring centre, as suggested by its coordinates, but instead in the SNR shell, its distance then is about 235 pc or 365 pc. Its projected length in that case is about 19 pc or 29 pc, respectively.
The depolarising H$\alpha$ filament has no counterpart in total intensities, which means that the thermal emission must be very low. From the H$\alpha$ excess, $I_{\mathrm{H\alpha}}$, of 3 to 4 Rayleigh, we may calculate the emission measure $EM$\[pc cm$^{-6}$\] from Eq. \[eq2\], where $T{_4}$ is the electron temperature in units of 10${^4}$ K. An $E(B-V)$ extinction correction in the direction of G203.7+11.5 raises $EM$ by about 2% [@GGreen19] and can be disregarded in view of all other uncertainties,
$$\centering
EM = 2.75~T_{4}^{0.9}~I_{\mathrm{H\alpha}}~exp[2.44E(B-V)].
\label{eq2}$$
The range of $EM$ values in the central area of the filament is between 3.6 and 11 pc cm$^{-6}$ for electron temperatures of 4000 K and 10000 K and H$\alpha$ intensities of 3 to 4 Rayleigh. We adopt an $EM$ of 7 pc cm$^{-6}$ in the following.
The width of the depolarising filament is about 30$\arcmin$, which corresponds to 2.6 pc for a distance of 300 pc. For a cylindrical morphology, we obtain the same size $L$ along the line of sight, but if G203.7+11.5 is a sheet-like structure seen edge-on, $L$ may be larger. In the following, we assume the cylindrical case, which means $L$ = 2.6 pc. With $EM$ \[pc cm$^{-6}$\] = $n_{\mathrm{e}}^{2}$ \[cm$^{-6}]\times$ $L$ \[pc\], the average electron density $n_{\mathrm{e}}$ is about 1.6 cm$^{-3}$ for $EM$ = 7 pc cm$^{-6}$. As discussed above, with the range of possible distances of 235 pc to 365 pc and the range of $EM$ as a function of the electron temperature, we calculated $n_{\mathrm{e}}$ between 1.1 cm$^{-3}$ and 2.3 cm$^{-3}$. However, for a clumpy thermal gas, the influence of the filling-factor $f_\mathrm{{n_e}}$ for $n_{\mathrm{e}}$ has to be taken into account. $n_{\mathrm{e}}$ depends on $f_\mathrm{{n_e}}$ as
$$\centering
n_{e} = \sqrt{\frac{EM}{\mathit{f_\mathrm{{n_{e}}}}L}}~ {\rm cm^{-3}}.
\label{eq5}$$
Thus the electron density scales with the filling factor $f_\mathrm{{n_e}}$ as $1 / \sqrt{f_\mathrm{{n_e}}}$. It is unclear what a reasonable filling factor for G203.7+11.5 might be. In any case, values of $f_\mathrm{{n_e}}$ below 1 will increase $n_{\mathrm{e}}$. Recent discussions of the filling factor were reported by @Harvey-Smith11 for -regions and by @Gao15 for the W4 Super-Bubble.
Magnetic field strength and pressure of G203.7+11.5
---------------------------------------------------
From the Faraday-screen model, we derived an $RM$ of about $-86$ rad m$^{-2}$ for the central part of G203.7+11.5. $RM$ depends on the regular magnetic field strength along the line of sight, the electron density, and the thickness $L$ of the filament as $RM$ = 0.81 $n_{\mathrm{e}}$\[cm${^{-3}}$\] $B{_{||}} [\mu$G\] $L$\[pc\]. We calculated a magnetic field strength along the line of sight of about 26 $\mu$G with an uncertainty of about 20$\%$. 26 $\mu$G is a lower limit because the orientation of the filament is most likely inclined with respect to the line of sight. A filling factor $f_\mathrm{{n_e}}$ that is most likely below 1 will increase $B{_{||}}$ by $B{_{||}} / \sqrt{f_\mathrm{{n_e}}}$. This high magnetic field strength largely exceeds that of the local Galactic magnetic field in the ISM in any case. This local Galactic magnetic field is typically a few $\mu$G, see for example @Sun08, @Sun10, @Ferriere11, @vanEck11, and @Farrar12, by an order of magnitude, but is not unusual for the magnetic field strength expected in SNR shock fronts. Beyond the compression of the ambient Galactic magnetic field in the adiabatic SNR expansion phase by a factor of four, further magnetic field amplification effects may increase the magnetic field strength in SNR shock fronts up to 100 $\mu$G or more, see for example @Reynolds12 and @Dubner15.
The magnetic field strength in thermal filaments traced by H$\alpha$ emission is not known, but may be compared with filaments found for other Faraday screens. The study of magnetic fields of large regions based on $RM$s of extragalactic sources by @Harvey-Smith11 revealed values of the regular magnetic field component along the line of sight, $B_{||}$, of between 2 and 6 $\mu$G. This clearly is lower than the values we found for G203.7+11.5 and similar to Galactic magnetic field strengths observed in the disc.
Several Faraday screens were detected and discussed in the $\lambda$6 cm Urumqi survey publications by @Sun07, @Sun11a, @Gao10, and @Xiao09. High $RM$ values were found for some regions, and in particular, from a few nearly spherical Faraday screens with sizes of up to several degrees. For most of these Faraday screens, the thermal electron density must be very low because the thermal emission is not visible in H$\alpha,$ and moreover, its radio continuum emission is too faint to be detected. Thus, the electron density could not be precisely determined and just lower limits for $B_{||}$ can be quoted, which reach values up to 10 $\mu$G.
From a Faraday-screen analysis of the large W4 Super-Bubble, @Gao15 derived $B_{||}$ of 5 $\mu$G and estimated that the total magnetic field strength will exceed 12 $\mu$G when its geometry is taken into account. @Wolleben04 found excessive Faraday rotation towards the ionised rims of some local Taurus molecular clouds and derived values for $B_{||}$ that exceed 20 $\mu$G. The last two results for $B_{||}$ are close to what we found for G203.7+11.5. All Faraday-screen results, with the exception of those for regions, indicate regions or objects in the Galaxy with a significantly enhanced regular magnetic field strength when compared to typical Galactic values. In most cases, the origin is not clear, although for G203.7+11.5, the old SNR shock-front of the Monogem ring seems to be a good candidate for having caused its strong magnetic field.
We calculated the magnetic pressure $P_\mathrm{{mag}} = B_\mathrm{{tot}}^{2}/8\pi$ for G203.7+11.5 as $P_\mathrm{{mag}}$ =2.7${\pm}0.5{\times}$10$^{-11}$ dyn cm$^{-2}$. The thermal pressure $P_\mathrm{{ther}} = 2n_{0}kT_{e}$, where $n_{0} = n_{e}$, in the case of total ionisation, and $T_{e}$ is taken as 7000 K. We determined $P_\mathrm{{ther}}$ = 3.1${\pm}1.2{\times}$10$^{-12}$ dyn cm$^{-2}$. Clearly, the magnetic pressure largely exceeds the thermal pressure and thus determines the shape and evolution of G203.7+11.5.
$RM$s of extragalactic sources and pulsars compared to the $RM$ of G203.7+11.5
------------------------------------------------------------------------------
$RM$s of extragalactic sources in the G203.7+11.5 area were selected from the catalogue by [@Xu14]. In general, the $RM$s in the field are positive. Seven listed $RM$s are in the range +72 to +100 rad/m$^2$. For the source at ${\it l,b} = 204\fdg8,10\fdg1$, four $RM$s with +7.0,+7.0,-7.7 and -74.4 rad m$^{-2}$ were listed, and the source at ${\it l,b} = 205\fdg1,12\fdg98$ has $RM$ = +3.5 rad m$^{-2}$. However, these two sources have the largest distances from the map centre, so that most of the sources probably indicate that the large-scale magnetic field direction points towards us. The nearby pulsar PSR B0656+14 in the centre of the Monogem ring also has a positive but lower $RM$ of $RM$ = +22.73 rad m$^{-2}$ [@Sobey19]. @Vallee84 have modelled the $RM$ distribution of the Monogem ring. They calculated a positive $RM$ excess for a thick-shell object when compared with its surroundings. The $RM$ value we found above from comparing the diffuse polarised emission at $\lambda$21 cm and at $\lambda$1.3 cm (K band) of about $RM$ = 1.7$\pm1.9$ rad m$^{-2}$ is much lower and indicates that very local diffuse synchrotron emission dominates. Thus, the $RM$s of extragalactic sources in general trace the magnetic field beyond the Monogem ring.
Origin of the magnetic field of G203.7+11.5
-------------------------------------------
The G203.7+11.5 filament has an $RM$ with a negative sign, and thus its magnetic field points in the opposite direction compared to the large-scale Galactic field traced by extragalactic sources. This means that the magnetic field of G203.7+11.5 does not result from a simple compression of the large-scale field. Shock-excited optical filaments in the Monogem area have been identified previously, and G203.7+11.5 may have the same origin, although optical spectra to prove its shock excitation are missing so far. The opposite magnetic field direction may result from instabilities that arise from the interaction of the SNR shell with interstellar clouds or from the reverse-shock interaction with material from the SN explosion if the filament is located close to the centre of the Monogem ring. Investigations of Rayleigh-Taylor instabilities by @Jun95 showed that filaments that lie orthogonal to the direction of the expanding shock-wave may be formed in this way, and magnetic fields will also be amplified by this process. It is important to find signs of shock excitation for G203.7+11.5, which will strongly support the suggested association with the Monogem ring from the present study.
Local synchrotron emissivity
----------------------------
We can use the separation of the foreground and background polarised emission to constrain the local synchrotron emissivity in the direction of the Monogem ring. Its small distance allows us to check previous claims of a local excess of synchrotron emissivity [@Fleishman95; @Wolleben04]. The origin of the excess is not understood, in particular if it is caused by an enhancement of the local cosmic-ray electron density or by a stronger magnetic field, when compared to more distant emissivity values. The 3D structure of the local excess is not known either, but more data from different directions will provide key information to determine its location.
We calculated the $\lambda$21 cm synchrotron emissivity up to the 300 pc distance of G203.7+11.5 in the following way: The percentage of polarised foreground emission at $\lambda$11 cm was determined from the Faraday-screen model to be about 63$\%$ of the total polarised emission. We assumed the same percentage at $\lambda$21 cm, which gives about 70 mK $T_\mathrm{b}$ polarised foreground emission. For the maximum possible degree of polarisation of 70$\%$, the total intensity is about 100 mK $T_\mathrm{b}$ or 330 mK $T_\mathrm{b}$/kpc. A more realistic polarisation of 20$\%$ increases the intensity to about 350 mK $T_\mathrm{b}$ or 1.1 K $T_\mathrm{b}$/kpc. The total-intensity emission along the line of sight at $\lambda$21 cm is about 0.7 K $T_\mathrm{b}$ and the polarisation is about 16$\%$, so that the assumption of 20$\%$ polarised emission for the local emission seems realistic. The value of 1.1 K $T_\mathrm{b}$/kpc is below the value of 1.7 K $T_\mathrm{b}$/kpc derived by @Wolleben04 for the foreground synchrotron emission based on Faraday-screen effects at the rims of the local Taurus molecular clouds at a distance of 140pc, but confirms that the local synchrotron emissivity is higher than more distant Galactic values. @Wolleben04 have discussed available Galactic synchrotron emissivity data, which span a range from 0.14 K $T_\mathrm{b}$ to 0.9 K $T_\mathrm{b}$ at $\lambda$21 cm. The recent study of low-frequency absorption data from regions by @Poldermann19 confirmed an increased local synchrotron emissivity.
Summary
=======
Radio continuum and polarisation observations at $\lambda$11 cm and $\lambda$21 cm were used to study the physical properties and the magnetic field of the thermal filament G203.7+11.5. G203.7+11.5 acts as a Faraday screen and is most likely related to the Monogem ring, which is classified as an evolved SNR. For a distance of about 300 pc and an electron density of 1.6$\pm0.6$ cm$^{-3}$, we found a regular magnetic field component along the line of sight of about 26$\pm5$ $\mu$G, which is a lower limit for the total regular field in the filament. The resulting $RM$ of -86$\pm3$ rad m$^{-2}$ is similar to the $RM$s of extragalactic sources observed in this direction, but with an opposite sign. The magnetic field direction of G203.7+11.5 and its unusual strength most likely originate in shock interaction of the Monogem ring with clumpy interstellar material, where instabilities cause locally enhanced magnetic fields.\
This research is based on observations with the Effelsberg 100-m telescope of the MPIfR, Bonn. X.S. is supported by the National Natural Science Foundation of China (Grant No. 11763008).
[^1]: The FITS images of the radio maps are available in electronic form at the CDS via anonymous ftp to ...
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report a theoretical study of the optical centrifuge acceleration of an asymmetric top molecule interacting with an electric static field by solving the time-dependent Schrödinger equation in the rigid rotor approximation. A detailed analysis of the mixing of the angular momentum in both the molecular and the laboratory fixed frames allow us to deepen the understanding of the main features of the acceleration process, for instance, the effective angular frequency of the molecule at the end of the pulse. In addition, we prove numerically that the asymmetric superrotors rotate around one internal axis and that their dynamics is confined to the plane defined by the polarization axis of the laser, in agreement with experimental findings. Furthermore, we consider the orientation patterns induced by the dc field, showing the characteristics of their structure as a function of the strength of the static field and the initial configuration of the fields.'
author:
- 'Juan J. Omiste'
title: 'Theoretical study of asymmetric superrotors: alignment and orientation'
---
Introduction {#sec:introduction}
============
The control of molecular dynamics using external fields has reached an unprecedent degree of precision in the last decades [@Lemeshko2013]. Specifically, the possibility of aligning and orienting molecular ensembles in space has significant implications in many areas, including the study of chemical reactions [@brooks:science; @brooks:jcp45; @Loesch1994; @Loesch1994a], steric effects in collisions [@Janssen1991] or the description of molecules using X-ray and electron diffraction [@Nakajima2015], among others.
One of the most prominent techniques to constrain the motion of molecules in space relies on the application of non-resonant nano- to femtosecond laser pulses, which allow to align a molecule along one axis of the laboratory [@sakai:jcp110; @poulsen:jcp121] or, furthermore, along the three axes using circular or elliptically polarized laser fields [@larsen:phys_rev_lett_85_2470]. On the other hand, the orientation adds a direction to the alignment. The simplest method to orient an ensemble of polar molecules is the application of a static field [@bulthuis:jpca101; @loesch:jcp93; @friedrich:zpd18]. The combination of both, a dc field and a non-resonant laser pulse allows for a large degree of orientation and alignment in the adiabatic limit for linear [@friedrich:jcp111; @Sakai2003; @omiste:pra2012; @Nielsen2012] and asymmetric molecules [@kupper:jcp131; @Omiste2011; @Omiste2013; @Hansen2013; @Omiste2016_asymmetric_molecules; @Thesing2017], but it is extremely difficult to reach in general [@Omiste2016_asymmetric_molecules; @Thesing2017]. Many other experimental setups have been proposed to enhance the alignment of a molecular ensemble using external fields, such as the application of two laser pulses with different polarizations [@Bisgaard2006; @lee:phys_rev_lett_97; @viftrup:pra79], the application of single cycle pulses [@Ortigoso2012] and single THz pulses [@Damari2016; @Zhang2017], or a combination of THz and femtosecond laser [@Kitano2011a; @Zai2015] to efficiently achieve both orientation and alignment.
A different insight to the control of the molecular motion using non-resonant laser fields is achieved by means of the optical centrifuge, an IR linearly polarized laser pulse whose polarization axis rotates with linearly increasing angular velocity [@Karczmarek1999; @Villeneuve2000]. This type of laser fields are able to drive the molecular rotation, which follows the laser’s polarization axis up to high angular velocities [@Korobenko2014], leading the molecules to superrotor states. These states are characterized not only by their fast rotation, but also by the constrain of their dynamics to the plane defined by the centrifuge. For highly rotating states, the centrifugal force may lead to the stretching of bonds [@Korobenko2015a] or even to dissociation [@Karczmarek1999]. Furthermore, it has been shown that an ensemble of superrotors is stable against collisions [@Forrey2001; @Tilford2004; @Korobenko2014], which allows to measure the spin-rotation coupling [@Milner2014], the interaction with a magnetic field [@Floss2015; @Milner2015b; @Korobenko2015b] or may also cause magnetization in an ensemble of paramagnetic superrotors [@Milner2017].
Many properties of the superrotors can be understood classically, for instance the constraining of the molecular motion to the plane defined by the polarization axis of the laser induced by the centrifugal force [@Karczmarek1999]. However, phenomena as the revivals in the alignment observed in the experiment [@Korobenko2015a] demand a full quantum approach [@Hartmann2012]. In this work, we tackle a full quantum mechanical description of the alignment and orientation of an asymmetric molecule prototype, SO$_2$, interacting with an optical centrifuge and a static electric field by solving the time-dependent Schrödinger equation (TDSE).
This paper is organized as follows: In Sec. \[sec:system\_and\_numerical\_methods\] the system under study, its full Hamiltonian and the numerical techniques to solve the TDSE of the system are presented. In Sec. \[sec:results\] we discuss the numerical results for the optical centrifuge interacting with SO$_2$. Specifically, we describe in Sec. \[sec:centrifugal\] the centrifugal dynamics in terms of the mixing of the angular momentum along different directions. In Sec. \[sec:alignment\_polarization\_plane\], we analyze the alignment of the SO$_2$ for several optical centrifuge parameters and in Sec. \[sec:orienting\_superrotors\] we study the orientation induced by the electric static field. In Sec. \[sec:conclusions\], we summarize the main conclusions of this work and the outlook for future projects. Finally, we include in the Appendices \[sec:derivation\_of\_the\_laser\_term\] and \[sec:coupling\_wigner\] the derivation of the Hamiltonian corresponding to the interaction with the fields and a summary of the Wigner D-matrix elements, respectively.
The system and numerical methods {#sec:system_and_numerical_methods}
================================
We analyze the impact of a static dc field and an intense non-resonant centrifuge laser pulse on the rotational dynamics of SO$_2$. We work in the Born-Oppenheimer and the rigid rotor approximations, neglecting transitions among electronic or vibrational levels. In this framework, the wavefunction is written in terms of the Euler angles $\Omega=(\phi,\,\theta,\,\chi)$, which determine the relative orientation of the $xyz$ molecular fixed frame (MFF) with respect to the $XYZ$ laboratory fixed frame (LFF) [@Zare1988]. In the Born-Oppenheimer and the rigid rotor approximation, the Hamiltonian reads as $$\label{eq:hamiltonian}
\mathbf{H}(t)=\mathbf{H}_{rot}+\mathbf{H}_{S}+\mathbf{H}_{L}(t),$$ where $\mathbf{H}_{rot}$ is the rotational kinetic term $$\label{eq:hrot}
\mathbf{H}_{rot}=\hbar^{-2}\left(C\mathbf{J_x}^2+A\mathbf{J_y}^2+B\mathbf{J_z}^2\right),$$ $\mathbf{J}_k$ being the projection of the angular momentum operator along the $k$ axis of the MFF. The rotational constants are $A=\SI{2.028}{\energy}, B=\SI{.3442}{\energy}~\text{and}~C=\SI{.2935}{\energy}$ [@Kallush2015], where the $a$ axis of the MFF is parallel to the line which contains the oxygen atoms, $b$ lies in the plane of the molecule and contains the sulfur atom and $c$ is perpendicular to the molecular plane \[see Fig. \[fig:fig1\](a)\]. Note that we identify the axes $a,\, b$ and $c$ with $y,\, z$ and $x$, respectively, throughout the paper.
The molecule interacts with the static electric field by means of its permanent dipole moment $$\label{eq:hs}
\mathbf{H}_S=-\vec\mu \cdot \vec{E}_S=-\mu E_S\cos\theta_{Zz},$$ where $\vec{E}_S=E_S\hat{Z}$ is the electric field, the dipole moment $\mu=1.62$ D points to the sulfur atom [@Kallush2015] \[see Fig. \[fig:fig1\](a)\] and $\theta_{Pq}$ is the angle formed by the $P$ and $q$ axes of the LFF and the MFF, respectively. Without loss of generality, we only consider electric fields parallel to the $Z$ axis of the LFF, because the inclination angle can be included in the angle formed by the polarization of the laser field and the $Z$ axis of the LFF.
The coupling with the non-resonant laser field, $\mathbf{H}_L(t)$ is obtained after averaging over rapid oscillations of its electric field [@stapelfeldt:rev_mod_phys_75_543], $$\begin{aligned}
\label{eq:hl}
\nonumber
&&\mathbf{H}_{L}(t)=-\frac{1}{4}\vec{E}_L^\dagger(t) {\ensuremath{\underline{\underline{\alpha}}}}\vec{E}_L(t)=-\frac{E_L(t)^2}{4}\left\{\alpha_{xx}+\right.\\
\nonumber
&& \cos^2\beta(t)\left[\left(\alpha_{zz}-\alpha_{xx}\right)\cos^2\theta_{Zz}+\left(\alpha_{yy}-\alpha_{xx}\right)\cos^2\theta_{Zy}\right]+\\
\nonumber
&& \sin^2\beta(t)\left[\left(\alpha_{zz}-\alpha_{xx}\right)\cos^2\theta_{Xz}+\left(\alpha_{yy}-\alpha_{xx}\right)\cos^2\theta_{Xy}\right]+\\
\nonumber
&& \sin 2\beta(t) \left[\left(\alpha_{zz}-\alpha_{xx}\right)\cos\theta_{Xz}\cos\theta_{Zz}+\right.\\
\label{eq:coupling_laser_expanded_def}
&&\left.\left.\left(\alpha_{yy}-\alpha_{xx}\right)\cos\theta_{Xy}\cos\theta_{Zy}\right]\right\}\end{aligned}$$ where the polarizability tensor, ${\ensuremath{\underline{\underline{\alpha}}}}$, is diagonal in the MFF with components $\alpha_{xx}=\SI{2.756}{\pol},\,\alpha_{yy}=\SI{4.638}{\pol}~\text{and}~\alpha_{zz}=\SI{3.082}{\pol}$ [@Xenides2000] and $\vec{E}_L(t)$ is the envelope of the pulse. In Appendix \[sec:derivation\_of\_the\_laser\_term\] we describe in detail the expansion of $\mathbf{H}_{L}(t)$ in terms of the relative orientation between the MFF and the LFF.
![\[fig:fig1\] Sketches of (a) the SO$_2$ molecule referred to the molecular fixed frame; and (b) the polarization axis of the electric field of the laser pulse, $\vec{E}_L$, with respect to the axes of the LFF.](fig1.eps){width=".7\linewidth"}
A field-dressed state is labeled as the field-free state $J_{K_a,K_c}M$ adiabatically connected with it [@king_jcp11], where $J$ is the total angular momentum, $M$ the magnetic quantum number and $K_a$ and $K_c$ the projection of the total angular momentum along the $a$ and $c$ axis in the prolate and oblate limiting cases, respectively. We consider the following process: Initially a SO$_2$ molecule interacts with a static dc field parallel to the $Z$ axis of the LFF. The field is switched on adiabatically until a maximum $E_S$ at $t=0$, and is kept constant. We assume that this process is adiabatic, hence, the wavefunction at $t=0$ is an eigenstate of the field-dressed Hamiltonian $\mathbf{H}_{rot}-\mathbf{H}_S$. In a second step, at $t=0$ an optical centrifuge pulse contained in the $XZ$ plane of the LFF is switched on. Its analytical form is $\vec{E}_L(t)=E_{L,\text{max}}g(t)\left[\hat{X}\sin \beta(t)+\hat{Z}\cos\beta(t)\right]$, where $\beta(t)=\frac{\gamma}{2}t^2+\delta$, $\gamma$ is the angular acceleration of the polarization axis, $\delta$ the angular initial phase and the envelope $g(t)$ reads $$\label{eq:el_envelope}
g(t)=\left\{
\begin{array}{ll}
\sin^2\left[\cfrac{\pi t}{2 t_\text{on}}\right],& 0\le t \le t_\text{on},\\
&\\
1,& t_\text{on}\le t \le t_\text{p}-t_\text{off},\\
&\\
\sin^2\left[\cfrac{\pi (t-t_\text{p})}{2 t_\text{off}}\right],& t_\text{p}-t_\text{off}\le t \le t_\text{p},\\
0,& t_\text{p}<t.
\end{array}
\right.$$ In the present work, we consider a pulse with an intensity ${\textup{I}_{0}}=\SI{5e12}{\intensity}$, turning on/off times $t_\text{on}=t_\text{off}=\SI{2}{\pico\second}$ and a duration of $t_\text{p}=\SI{32}{\pico\second}$, for several values of $\gamma$. Finally, we also analyze the rotational dynamics in the static electric field once the pulse is switched off.
To investigate the rotational dynamics, we calculate the time dependent wavefunction by solving the TDSE using the short iterative Lanczos method [@mctdh; @Omiste2016_asymmetric_molecules]. For the angular degrees of freedom we use a basis set expansion in terms of the symmetrized field-free symmetric top eigenstates ${\left|JKMs\right\rangle}$ [@Omiste2016_asymmetric_molecules], where $J$ is the total angular momentum quantum number, $K$ and $M$ are its projections along the $z$ axis of the MFF and the $Z$ axis of the LFF, respectively, and $s=0,\, 1$ is the parity under reflections on the polarization plane $XZ$ of the LFF, $\sigma_{XZ}$. The total Hamiltonian $\mathbf{H}(t)$ commutes with $\sigma_{XZ}$ and two-fold rotation around the $z$-axis of the MFF ($C_2^z$), therefore $s$ and the parity of $K$ are preserved and define the four irreducible representations of the system [@Omiste2011a; @Omiste2016_asymmetric_molecules]. See Appendix \[sec:coupling\_wigner\] for further details on ${\left|JKMs\right\rangle}$ and its relation with the Wigner elements, $D_{M,K}^J(\Omega)$ [@Zare1988].
Results {#sec:results}
=======
In this section we investigate in detail the rotational dynamics induced in a SO$_2$ molecule by a centrifuge laser pulse and a static dc field. Throughout this work, we only consider even wavefunctions under the symmetry operations $C_{2}^z$ and $\sigma_{XZ}$. The calculations are converged for basis set functions with $0\le J\le J_\text{max}=90$.
Centrifugal dynamics {#sec:centrifugal}
--------------------
Here we analyze the centrifugal dynamics of the molecule induced by the optical centrifuge with $\delta=0$. The slow rotation of the polarization axis during the turning-on of the laser allows the most polarizable axis (MPA) of the SO$_2$ molecule to align along it. Next, the accelerated rotation of the laser polarization axis is followed by the MPA. After the pulse is over, the molecule continues rotating, ideally, at the final angular frequency of the pulse. This rotation leads to the mixing of the angular momentum of the system. To illustrate this effect, we show in Fig. \[fig:fig2\] the expectation value ${\left\langle \mathbf{J}^2\right\rangle}$ for the rotational groundstate, $0_{00}0$, and ${\textup{I}_{0}}=\SI{5e12}{\intensity}$, $E_\text{S}=\SI{300}{\fieldstrength}$ and $\gamma=0,\,5~\text{and}~\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$. For $\gamma=0$, ${\left\langle \mathbf{J}^2\right\rangle}$ increases until $66.93\hbar^2$ and oscillates around this value until the pulse is off, remaining almost constant.
Just after the turning-on of the laser, the optical centrifuge is still slow, therefore, ${\left\langle \mathbf{J}^2\right\rangle}$ almost coincides for $\gamma=\SI{0}{},\,\SI{5}{}~\text{and}~\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$, up to the first peak of approximately $105\hbar^2$ around 2.17 ps. However, as the polarization axis accelerates, the angular momentum increases linearly in time, tending to be proportional to the angular velocity. This effect can be understood classically for a linear rotor, where the classical angular momentum is $J_\text{cl}=I\omega$, being $I$ the inertia constant and $\omega$ the angular velocity which, in this setup, increases linearly in time. Under this assumption, we may approximate at the end of the pulse $\frac{J^2_\text{cl}{(\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2})}}{J^2_\text{cl}{(\gamma=\SI{5}{{\ensuremath{^\circ}}/\pico\second^2})}}\approx\frac{10^2}{5^2}=4$, which is close to the result of the time propagation $\frac{{\left\langle \mathbf{J^2}\right\rangle}_{\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}}}{{\left\langle \mathbf{J^2}\right\rangle}_{\gamma=\SI{5}{{\ensuremath{^\circ}}/\pico\second^2}}}=\frac{1662.41}{468.92}\approx 3.54$.
![\[fig:fig2\] For the initial state $0_{00}0$, expectation value ${\left\langle \mathbf{J}^2\right\rangle}$ for an electric field with strength $E_S=\SI{300}{\fieldstrength}$ and $\gamma=\SI{0}{}~\text{(solid red)},~\SI{5}{}$ and $\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$ (dash-dotted orange). The envelope of the centrifuge is also shown (gray).](fig2){width=".95\linewidth"}
To obtain a better physical insight of the rotational dynamics of the wavefunction $\Psi(t)$ we analyze the population of all the components with the same rotational quantum number $J$ but different values of $M$ and $K$, which is defined as $P_J(t)=\sum_{KMs}|{\left\langle JKMs|\Psi(t)\right\rangle}|^2$. In Fig. \[fig:fig3\] we show the distribution of $P_J(t)$ during and after the pulse for the initial states ${J}_{{K_a},{K_c}}{M}=0_{00}0$, $2_{02}2$ and $6_{06}6$. First note that the weights of even and odd $J$ contributions to $P_J(t)$ are comparable due to the mixing induced by the dc electric field during the turning-on of the laser, as has already been proven for excitations by a non-resonant linearly polarized laser [@Nielsen2012; @omiste:pra2012; @Omiste2016_asymmetric_molecules]. However, there is still a predominance of the even contributions over the odd. For each initial state, we observe that $P_J(t=20\text{~ps})$ is formed by two distributions with a large overlap, centered approximately around $J=8\text{~and~}J=28$, being more remarkable for $2_{02}2$ and $6_{06}6$. After the pulse, the distribution at low $J$ has slightly changed not only the shape, but also the population. This part of the distribution is constituted by the components which are unable to follow the rotation of the laser polarization axis. However, the other part is pushed to larger $J$ values during the acceleration, increasing the net rotation of the molecule. For the initial state $6_{06}6$ the population of the components which does not contribute to the rotation is larger, since the higher angular momentum of the initial state implies that more counter rotating components play a role during the dynamics. For the groundstate as initial state ${\left\langle \vec{\mathbf{J}}^2\right\rangle}\approx 0$ before the turning-on of the laser, which ensures that the initial population is not corotating or counterrotating.
![\[fig:fig3\] For the initial states $0_{00}0$, $2_{02}2$ and $6_{06}6$, the distribution of the coefficients with angular momentum $J$, $P_J(t)$, for an electric static field with strength $E_S=\SI{300}{\fieldstrength}$ and an acceleration of the centrifuge $\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$ at $t=20$ ps (a), (c) and (e) and $t=115.5$ ps (b), (d) and (f).](fig3){width=".95\linewidth"}
We now analyze the rotational dynamics of the molecule in terms of the projection of the angular momentum $\vec{\mathbf{J}}$ along the axes of the MFF and the LFF. The optical centrifuge induces a rotation around the $Y$ axis of the LFF, which is measured by ${\left\langle \mathbf{J_Y}^2\right\rangle}$, shown in Fig. \[fig:fig4\](a). ${\left\langle \mathbf{J_Y}^2\right\rangle}$ follows the same behavior as ${\left\langle \mathbf{J}^2\right\rangle}$, increasing during the pulse, until a maximum value which depends on the initial state. During the pulse, we observe oscillations which also appear in the acceleration of linear rotors [@Spanner2001a]. After the pulse, ${\left\langle \mathbf{J_Y}^2\right\rangle}$ remains almost constant, being the weak dc field responsible for the tiny oscillations. In Fig. \[fig:fig4\](b) we show the square of the projection of $\vec{\mathbf{J}}$ along the $x$-axis of the MFF, ${\left\langle \mathbf{J_x}^2\right\rangle}$, for the three initial states. We observe that ${\left\langle \mathbf{J_Y}^2\right\rangle}\approx{\left\langle \mathbf{J_x}^2\right\rangle}$ during and after the optical centrifuge, despite the oscillation of ${\left\langle \mathbf{J_x}^2\right\rangle}$ caused by the asymmetric inertia tensor and the disagreements at low angular momentum. Moreover, these two projections constitute the major contribution to the total angular momentum. For instance, at the end of the pulse, ${\left\langle \mathbf{J}^2\right\rangle}\approx 1662.41\hbar^2$ for the groundstate at $\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$, whereas ${\left\langle \mathbf{J_x}^2\right\rangle}\approx 1235.51\hbar^2$. Therefore, the molecule tends to restrict the motion of the MPA ($y$-axis of the MFF) and the remaining axis with the largest rotational constant ($z$ axis) in the plane of the optical centrifuge ($XZ$ plane), as has been shown experimentally [@Korobenko2015a].
![\[fig:fig4\] For the initial states $0_{00}0$ (red solid), $2_{02}2$ (dashed blue) and $6_{06}6$ (orange dash-dotted) expectation values (a) ${\left\langle \mathbf{J_Y}^2\right\rangle}$ and (b) ${\left\langle \mathbf{J_x}^2\right\rangle}$ as a function of $t$ for an angular acceleration $\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$ and an electric field with strength $E_S=\SI{300}{\fieldstrength}$. The envelope of the centrifuge is also shown (gray).](fig4){width=".95\linewidth"}
Taking all this into account, we are allowed to approximate the superrotor after the pulse as a linear rotor, whose rotational constant is $C$ [@Korobenko2015a]. Then, we can extract the *effective* angular velocity $\omega_{ef}=\sqrt{{\left\langle \omega^2\right\rangle}}$ using ${\left\langle \mathbf{J_x}^2\right\rangle}=I_C^2{\left\langle \omega^2\right\rangle}$, where $I_C$ is the inertia constant around the $x$ axis and $C=\hbar^2/(2I_C)$. We obtain $\omega_{ef}= 221.86,\,195.14,~\text{and}~\SI{174.23}{{\ensuremath{^\circ}}/\pico\second}$ for $0_{00}0$, $2_{02}2$ and $6_{06}6$, respectively. Note that $C$ is the smallest rotational constant, hence, these values are lower bounds of the real $\omega$, which would correspond to $\SI{320}{{\ensuremath{^\circ}}/\pico\second}$ and ${\left\langle \mathbf{J_x}^2\right\rangle}\sim{\left\langle \mathbf{J}^2\right\rangle}=50.51\hbar^2$. Let us remark that the peak of $P_J(t)$ for high $J$’s after the pulse is located around $J\approx 50$ in Fig. \[fig:fig3\] (b), (d) and (f), in agreement with the approximation of SO$_2$ as a linear rotor.
Alignment to the polarization plane {#sec:alignment_polarization_plane}
-----------------------------------
Next, we analyze the alignment induced by the optical centrifuge in the LFF. In the previous section, the comparison between ${\left\langle \mathbf{J_x}^2\right\rangle}~\text{and}~{\left\langle \mathbf{J_Y}^2\right\rangle}$ showed that after the optical centrifuge most of the rotation of the SO$_2$ molecule is restricted around the optical centrifuge propagation axis and the $x$ axis of the MFF. Therefore, the $x$ axis of the MFF tends to align along the $Y$ axis of the LFF, that is to say, the plane defined by the molecule leans towards the $XZ$ plane of the LFF. This effect has been observed experimentally in both linear [@Milner2016a] and asymmetric top molecules [@Korobenko2015a]. To illustrate the alignment of the molecule we show the alignment factors ${\left\langle \cos^2\theta_{Zz}\right\rangle}$, ${\left\langle \cos^2\theta_{Xy}\right\rangle}$ and ${\left\langle \cos^2\theta_{Yx}\right\rangle}$ for the initial states $0_{00}0$, $2_{02}2$ and $6_{06}6$, and the pulse parameters ${\textup{I}_{0}}=\SI{5e12}{\intensity}$, $\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$ and $E_S=\SI{300}{\fieldstrength}$ in Fig. \[fig:fig5\]. During the turning on of the laser, the polarization axis is almost parallel to the $Z$ axis of the LFF. Hence, the MPA aligns along it, and thus ${\left\langle \cos^2\theta_{Xy}\right\rangle}$ decreases to approximately $0.110$ for $0_{00}0$. Similarly, ${\left\langle \cos^2\theta_{Zz}\right\rangle}$ reaches a minimum of approximately $0.168$ at the same times. Due to the optical centrifuge, the projections of these molecular axes onto the lab axes oscillate following the polarization axis rotation. For the three initial states, ${\left\langle \cos^2\theta_{Zz}\right\rangle}$ and ${\left\langle \cos^2\theta_{Xy}\right\rangle}$ follow the same pattern during the pulse, but the amplitude of the oscillations diminishes as the excitation of the state increases. After the pulse, each alignment factor oscillates around approximately the same value for all the states considered, being a bit larger for ${\left\langle \cos^2\theta_{Xy}\right\rangle}$, which involves the MPA. The revivals for both alignment factors during the post pulse propagation are located at the same times. They are more marked for the rotational groundstate as initial state, where the maximum peaks for ${\left\langle \cos^2\theta_{Zz}\right\rangle}$ and ${\left\langle \cos^2\theta_{Xy}\right\rangle}$ are 0.459 and 0.516 and are located at $t\approx 144.5$ ps. The alignment factors are mainly driven by C-type revivals [@Felker1992; @Tenney2016], being the revival time $T_\text{rev}=57$ ps, in accordance to the experimental measurements [@Korobenko2015a] and the rotational dynamics described in Sec. \[sec:centrifugal\].
![\[fig:fig5\] For the initial states $0_{00}0$ (red solid), $2_{02}2$ (blue dashed) and $6_{06}6$ (orange dash-dotted), the expectation values (a) ${\left\langle \cos^2\theta_{Zz}\right\rangle}$, (b) ${\left\langle \cos^2\theta_{Xy}\right\rangle}$ and (c) ${\left\langle \cos^2\theta_{Yx}\right\rangle}$ for an electric field with strength $E_S=\SI{300}{\fieldstrength}$ and a angular acceleration $\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$. The envelope of the centrifuge is also shown (gray).](fig5){width=".95\linewidth"}
In Fig. \[fig:fig5\](c) we illustrate the alignment in the $XZ$ plane of the LFF by ${\left\langle \cos^2\theta_{Yx}\right\rangle}$. Let us remark that $0\le {\left\langle \cos^2\theta_{Yx}\right\rangle}\le 1$, where $0$ means that the molecular plane is perpendicular to the $XZ$ plane of the LFF and, on the contrary, they are coplanar for ${\left\langle \cos^2\theta_{Yx}\right\rangle}=1$. The alignment dynamics is similar for the three states, but, as expected, it is more efficient for the $0_{00}0$. During the turning on, the alignment increases abruptly and the rotation of the polarization axis induces a smooth increasing until the pulse is over. The turning-off of the laser field weakly affects the maximum value reached during the centrifuge, corresponding to approximately $0.528,\,0.463~\text{and }0.424$ for $0_{00}0$, $2_{02}2$ and $6_{06}6$, respectively. These values remain during the post pulse propagation, [i.e.]{}, SO$_2$ remains attached to the $XZ$ plane of the LFF for long times due to angular momentum conservation and the stability of rotations around the $x~(c)$ axis.
Finally, in Fig. \[fig:fig6\], we show the post pulse dynamics of the alignment ${\left\langle \cos^2\theta_{Zz}\right\rangle}$ of the groundstate for the angular accelerations $\gamma=0,\,5~\text{and }\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$. For $\gamma=\SI{0}{{\ensuremath{^\circ}}/\pico\second^2}$, [i.e.]{}, a linearly polarized laser pulse, the alignment presents an irregular behavior without any pattern, ranging from $0.15$ to $0.43$. For $\gamma=\SI{5}{{\ensuremath{^\circ}}/\pico\second^2}$ we observe some revivals separated by approximately $28.4$ ps. There are also weaker revival-like structures between two consecutive main revivals, due to the contribution of other rotation modes associated to SO$_2$. Note that the asymmetry of the SO$_2$ molecule implies that these revivals are not well defined, hence, the interference of different modes causes the damping and vanishing of these structures at long times. For faster rotations, as in the case of $\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$, the main revivals are more pronounced, because the rotation around the $x$ axis of the molecule prevails over the other motions.
![\[fig:fig6\] For the initial state $0_{00}0$, the expectation value ${\left\langle \cos^2\theta_{Zz}\right\rangle}$ for an electric field with strength $E_S=\SI{300}{\fieldstrength}$, a centrifuge with peak intensity ${\textup{I}_{0}}=\SI{5e12}{\intensity}$ and (a) $\gamma=\SI{0}{}$, (b) $\SI{5}{}$ and (c) $\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$.](fig6){width=".95\linewidth"}
Orienting superrotors {#sec:orienting_superrotors}
---------------------
In this section we analyze the orientation induced by the impact of the dc field and the optical centrifuge. Let us recall that the dipole moment of the SO$_2$ molecule defines the $z$ axis of the MFF and coincides with the second MPA. On the contrary, the MPA lies along the $y$ axis of the MFF. Taking into account that even for the largest acceleration considered, $\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$, the polarization axis only rotates $20{\ensuremath{^\circ}}$ during the turning on, we can restrict our study to two limiting cases for linearly polarized lasers ($\gamma=\SI{0}{{\ensuremath{^\circ}}/\pico\second^2}$) to understand the dynamics during the switching on of the centrifuge. First, for parallel fields ($\delta=0{\ensuremath{^\circ}}$), the non-resonant laser pushes the MPA to its polarization axis. However, the dc field acts in the opposite way, forcing the dipole moment $\vec{\mu}$ to orient along the same axis of the LFF. For weak dc fields, the orientation is negligible due to the strong interaction due to the laser. On the other hand, in the perpendicular case ($\delta=90{\ensuremath{^\circ}}$), both fields collaborate and the orientation along the dc field is compatible with the alignment along the polarization axis of the laser. In addition to these considerations, the field configuration determines the number of real and avoided crossings as well as the population transfer among them [@Omiste2013; @Omiste2016_asymmetric_molecules], which can dramatically affect the orientation, even for weak dc fields [@Omiste2011; @Nielsen2012].
![\[fig:fig7\] For the initial state $0_{00}0$, expectation value ${\left\langle \cos\theta_{Zz}\right\rangle}$ during the interaction with the optical centrifuge for an electric static field with strength $E_S=\SI{300}{\fieldstrength}$ (solid red) and $\SI{100}{\kfieldstrength}$ (dashed blue), an angular acceleration (a) $\gamma=\SI{5}{}$ and (b) $\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$ and an initial angle of the centrifuge $\delta=90{\ensuremath{^\circ}}$. The envelope of the centrifuge is also shown (gray).](fig7){width=".95\linewidth"}
We illustrate the orientation of the groundstate in the presence of the optical centrifuge for $\delta=\SI{90}{{\ensuremath{^\circ}}}$ in Fig. \[fig:fig7\]. Before the centrifuge is switched on, the orientation ${\left\langle \cos\theta_{Zz}\right\rangle}=\SI{.234e-2}{}$ and $0.543$ for $E_S=\SI{300}{\fieldstrength}$ and $\SI{100}{\kfieldstrength}$, respectively. The fast switching on of the laser constructs a coherent wavepacket which enables the orientation and antiorientation during the propagation, as it is observed even for $\gamma=\SI{0}{{\ensuremath{^\circ}}/\pico\second^2}$. We see in Fig. \[fig:fig7\](b) that the orientation is fully controlled by the laser field for $\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$, and coincide for both strengths of the dc field. However, for $\gamma=\SI{5}{{\ensuremath{^\circ}}/\pico\second^2}$ the laser is not able to drive the orientation, as we show in Fig \[fig:fig7\](a).
These considerations have important implications in the post pulse propagation, as we illustrate in Figs. \[fig:fig8\]-\[fig:fig10\]. First, let us evaluate the impact of the remaining dc field during the rotation of the molecule after the pulse. We consider that the kinetic energy of the superrotor is mainly due to the rotation around the $x$ axis of the MFF \[see Sec. \[sec:centrifugal\]\], then ${\left\langle \mathbf{H}_{rot}\right\rangle}\sim \hbar^{-2}C{\left\langle \mathbf{J}_x^2\right\rangle}\approx \SI{132.07}{\energy}$ for $\gamma=\SI{5}{{\ensuremath{^\circ}}/\pico\second^2}$, which is much larger than the coupling with $E_S=\SI{100}{\kfieldstrength}$, $\sim E_S{\left\langle \mu_z\right\rangle}\approx {\left\langle \cos\theta_{Zz}\right\rangle}\times \SI{2.72}{\energy}$. Therefore, during the rotation, the superrotor will experience a negligible deceleration (acceleration) impulse when oriented (antioriented) with respect to the electric field. Let us remark that the impact of this kick on the orientation is even much smaller for $\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$, since ${\left\langle \mathbf{H}_{rot}\right\rangle}\sim \hbar^{-2}C{\left\langle \mathbf{J}_x^2\right\rangle}\approx \SI{366.87}{\energy}$.
![\[fig:fig8\] For the initial state $0_{00}0$, expectation value ${\left\langle \cos\theta_{Zz}\right\rangle}$ for an electric field with strength $E_S=\SI{300}{\fieldstrength}$, angular accelerations (a) $\gamma=\SI{0}{}$, (b) $\SI{5}{}$ and (c) $\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$ and an initial angle of the centrifuge $\delta=\SI{90}{{\ensuremath{^\circ}}}$.](fig8){width=".95\linewidth"}
In Fig. \[fig:fig8\] we show the orientation for the most favorable case, [i.e.]{}, $\delta=90{\ensuremath{^\circ}}$. For $\gamma=\SI{0}{{\ensuremath{^\circ}}/\pico\second^2}$ we observe an irregular orientation pattern ranging from approximately $-0.2~\text{to }0.2$, with decreasing amplitude over time. If we increase the acceleration to $\SI{5}{{\ensuremath{^\circ}}/\pico\second^2}$ the peak orientation decreases to 0.08 and the oscillatory pattern is irregular. On the contrary, the orientation for $\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$ shows a clear and well-defined revival structure characterized by revivals located around $85,140$ and $198$ ps, which are separated by approximately $T_\text{rev}=57$ ps, associated to the rotations around the $x$ axis of the MFF. Between these revivals we find other oscillations which are related to other rotational motions.
![\[fig:fig9\] For the initial state $0_{00}0$, expectation value ${\left\langle \cos\theta_{Zz}\right\rangle}$ for an electric field with strength $E_S=\SI{100}{\kfieldstrength}$, angular accelerations (a) $\gamma=\SI{5}{}$ and (b) $\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$ and an initial angle of the centrifuge $\delta=\SI{90}{{\ensuremath{^\circ}}}$.](fig9){width=".95\linewidth"}
If we increase the dc field strength to $E_S=\SI{100}{\kfieldstrength}$ the revivals in the orientation become more regular, as we see in Fig. \[fig:fig9\](b). Specifically, the location of the revivals are separated again by approximately $57$ ps, but, in contrast to the previous case, there are no revivals between these structures. The oscillations of the main revivals in the laser field free region are slightly enhanced with respect to the weak dc scenario.
We illustrate the collinear fields case with a weak dc field, $E_S=\SI{300}{\fieldstrength}$ in Fig. \[fig:fig10\]. As we have discussed above, the dc field and the laser field attempt to orient and align the molecular axes along different directions during the switching on. For $\gamma=\SI{5}{{\ensuremath{^\circ}}/\pico\second^2}$ we find that the orientation is highly oscillatory without a main frequency or revival structure as in $\delta=\SI{90}{{\ensuremath{^\circ}}}$ and the amplitude is smaller than 0.05. However, for $\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$ we do not recover a well-defined revival structure, caused by the mixing during the switching on of the centrifuge.
As we have discussed, the initial configuration of the fields, [i.e.]{}, $\delta$ and the strength of the field, has a strong impact in the orientation for the parameters analyzed in this work. However, for higher values of $\gamma$, the angle covered during the switching on may exceed $\pi$, increasing the population transfer among states with different orientation. This may lead to a vanishing average value of the orientation.
![\[fig:fig10\] For the initial state $0_{00}0$, expectation value ${\left\langle \cos\theta_{Zz}\right\rangle}$ for an electric field with strength $E_S=\SI{300}{\fieldstrength}$, angular accelerations (a) $\gamma=\SI{5}{}$ and (b) $\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$ and an initial angle of the centrifuge $\delta=\SI{0}{{\ensuremath{^\circ}}}$.](fig10){width=".95\linewidth"}
Conclusions {#sec:conclusions}
===========
We have theoretically studied the rotational dynamics of an asymmetric rotor induced by an optical centrifuge and a constant static electric field by solving the TDSE in the rigid-rotor approximation. Specifically, we describe the case of sulfur dioxide, which has been experimentally addressed [@Korobenko2015a]. The accelerated rotation of the polarization axis of the pulse provokes an effective rotation of the molecule paced with the centrifuge, which leads to a strong excitation of states with high angular momentum. We observe that the population of $J$ as the function of time is formed by two well defined parts: the efficiently accelerated and the non accelerated. The latter one is characterized by $J\sim 10$ for $\gamma=\SI{10}{{\ensuremath{^\circ}}/\pico\second^2}$ and remains almost unaltered during the acceleration, whereas the accelerated part moves to higher $J$’s as the centrifuge accelerates. The accelerated population is lower for higher excited initial states.
For the first time, we have shown numerically that a planar asymmetric molecule tends to the plane defined by the polarization axis of the centrifuge as the superrotor accelerates. We confirm that the superrotor remains attached to this plane for long times after the pulse [@Korobenko2015a], being slightly affected by the revivals. On the other hand, the delay between revivals in the alignment ${\left\langle \cos^2\theta_{Zz}\right\rangle}$ and ${\left\langle \cos^2\theta_{Xy}\right\rangle}$ during the post pulse propagation coincide with the experimental measurements [@Korobenko2015a].
The restriction of the rotation to the $XZ$ plane implies that the squared of the projection of the angular momentum along the propagation axis of the laser, ${\left\langle \mathbf{J}_Y^2\right\rangle}$, coincides with the projection along the least polarizable axis (LPA) of the molecule ${\left\langle \mathbf{J}_x^2\right\rangle}$, being the major contribution to the total angular momentum. Therefore, we can extract the *effective angular velocity* $\omega_{ef}$ using the inertia constant of the rotation around the LPA. Finally, we have analyzed the orientation of the superrotor caused by the dc field. We have shown that for the optical centrifuge accelerations under study, the orientation is very sensitive to the initial angle formed by the polarization axis and the dc field. In the case of SO$_2$, the initial perpendicular configuration is the most favorable for the orientation, since the MPA and the dipole moment are perpendicular. In this scenario, we clearly observe a revival structure, which experimentally may allow to locate the high orientation/antiorientation periods during the time evolution. Moreover, for a strong dc field we observe more well defined revivals than in the weak field case, due mainly to the suppression of many rotational modes which do not correspond to the rotation around the $c$ axis. Let us note that faster rotating optical centrifuges may completely frustrate the orientation during the acceleration.
Summing up, we have shown that the optical centrifuge combined with a static electric field contained in the polarization plane allows to control both the alignment and the orientation of a molecular ensemble of asymmetric molecules. This motivates the exploration of other field configurations such as the combination of an optical centrifuge and a perpendicular electric field, which might produce a large orientation perpendicular to the polarization plane.
Derivation of the laser term {#sec:derivation_of_the_laser_term}
============================
The coupling with the laser field in the rotating wave approximation is given by [@stapelfeldt:rev_mod_phys_75_543] $$\label{eq:coupling_laser}
H_L(t)=-\frac{1}{4}\vec{E}^\dagger_L(t) {\ensuremath{\underline{\underline{\alpha}}}}\vec{E}_L(t),$$ where $\vec{E}_L(t)$ is the envelope of the electric field of the laser field and ${\ensuremath{\underline{\underline{\alpha}}}}$ is the polarizability tensor. In the case of SO${}_2$, ${\ensuremath{\underline{\underline{\alpha}}}}$ is diagonal in the MFF, with the elements $\alpha_{xx},~\alpha_{yy}~\text{and}~\alpha_{zz}$. The electric field in the MFF reads as $$\label{eq:el_mff}
\vec{E}_L(t)|_\text{\tiny{MFF}}=R(\phi,\theta,\chi)
E_L(t)\left(
\begin{array}{c}
\sin\beta(t)\\
0 \\
\cos\beta(t)
\end{array}
\right),$$ being $\beta(t)$ the angled formed by the polarization axis of the laser and the $Z$-axis of the LFF. $R(\phi,\theta,\chi)$ is the rotation matrix which links the LFF and the MFF $$\label{eq:rotation_lff_mff}
R(\phi,\theta,\phi)=\left(
\begin{array}{ccc}
\cos\theta_{Xx} & \cos\theta_{Yx} & \cos\theta_{Zx}\\
\cos\theta_{Xy} & \cos\theta_{Yy} & \cos\theta_{Zy}\\
\cos\theta_{Xz} & \cos\theta_{Yz} & \cos\theta_{Zz}
\end{array}
\right),$$ where $\theta_{Pq}$ is the angle formed by the $P$ axis of the LFF and the $q$ axis of the MFF. The analytical expressions of the directional cosines are given in Appendix \[sec:coupling\_wigner\]. The coupling in Eq. may be written
$$\begin{aligned}
\nonumber
H_L(t)&=&-\frac{E_L(t)^2}{4}\left[\sin^2\beta(t)\left(\alpha_{xx}\cos^2\theta_{Xx}+\alpha_{yy}\cos^2\theta_{Xy}+\alpha_{zz}\cos^2\theta_{Xz}\right)+\right.\\
\nonumber
&&\cos^2\beta(t)\left(\alpha_{xx}\cos^2\theta_{Zx}+\alpha_{yy}\cos^2\theta_{Zy}+\alpha_{zz}\cos^2\theta_{Zz}\right)+\\
\label{eq:coupling_laser_expanded}
&&\left.2\sin\beta(t)\cos\beta(t)\left(\alpha_{xx}\cos\theta_{Zx}\cos\theta_{Xx}+\alpha_{yy}\cos\theta_{Zy}\cos\theta_{Xy}+\alpha_{zz}\cos\theta_{Zz}\cos\theta_{Xz}\right)\right].
\end{aligned}$$
Using that $\cos\theta_{Xx}\cos\theta_{Zx}+\cos\theta_{Xy}\cos\theta_{Zy}+\cos\theta_{Xz}\cos\theta_{Zz}=0$ and $\cos^2\theta_{Zx}+\cos^2\theta_{Zy}+\cos^2\theta_{Zz}=1$ in Eq. we obtain $$\begin{aligned}
\nonumber
&&\mathbf{H}_L(t)=-\frac{E_L(t)^2}{4}\left\{\alpha_{xx}+\right.\\
\nonumber
&& \cos^2\beta(t)\left[\left(\alpha_{zz}-\alpha_{xx}\right)\cos^2\theta_{Zz}+\left(\alpha_{yy}-\alpha_{xx}\right)\cos^2\theta_{Zy}\right]+\\
\nonumber
&& \sin^2\beta(t)\left[\left(\alpha_{zz}-\alpha_{xx}\right)\cos^2\theta_{Xz}+\left(\alpha_{yy}-\alpha_{xx}\right)\cos^2\theta_{Xy}\right]+\\
\nonumber
&& \sin 2\beta(t) \left[\left(\alpha_{zz}-\alpha_{xx}\right)\cos\theta_{Xz}\cos\theta_{Zz}+\right.\\
\label{eq:coupling_laser_expanded_def_appendix}
&&\left.\left.\left(\alpha_{yy}-\alpha_{xx}\right)\cos\theta_{Xy}\cos\theta_{Zy}\right]\right\}
\end{aligned}$$
Properties of the Wigner D-matrix elements {#sec:coupling_wigner}
==========================================
We briefly summarize the properties of the Wigner D-matrix elements used throughout this work. First, the basis set functions of the representations of the total Hamiltonian are given by $$\begin{aligned}
&&{\left|JKMs\right\rangle}=\\
&&\left\{
\begin{array}{l}
{\left|J00\right\rangle},~\text{for}~M,K=0 \\
\\
\cfrac{1}{\sqrt{2}}\left( {\left|JKM\right\rangle}+(-1)^{M-K+s}{\left|J-K-M\right\rangle}\right),~\text{otherwise},
\end{array}
\right.
\end{aligned}$$ where the eigenstates of the symmetric top, ${\left|JKM\right\rangle}$, are written in terms of the Wigner D-matrix elements $$\label{eq:symmetric_top_basis}
{\left\langle \Omega|JKM\right\rangle}=\sqrt{\cfrac{2J+1}{8\pi^2}}(-1)^{M-K}D_{-M,-K}^J(\Omega).$$ The trigonometric functions in $\mathbf{H}_{S}$ and $\mathbf{H}_L(t)$ in expressions and , respectively, can also be expressed as linear combinations of $D_{M,K}^J(\Omega)$. The terms involved in $\mathbf{H}_S$ $$\begin{aligned}
\label{eq:czz}
&&\cos\theta_{Zz}= \cos\theta=D_{0,0}^1(\Omega)\\
\label{eq:czx}
&&\cos\theta_{Xz}= \sin\theta\cos\phi=\frac{1}{\sqrt{2}}\left(D_{-1,0}^1(\Omega)-D_{1,0}^1(\Omega)\right)\end{aligned}$$ and in $\mathbf{H}_L(t)$
$$\begin{aligned}
\label{eq:c2zz}
\cos^2\theta_{Zz}&=&\cos^2\theta=\frac{1}{3}\left(2D_{0,0}^2(\Omega)+1\right)\\
\label{eq:c2zy}
\cos^2\theta_{Zy}&=&\sin^2\theta\sin^2\chi=\frac{1}{3}\left(1-D_{0,0}^2(\Omega)\right)-\sqrt{\frac{1}{6}}\left(D^2_{0,2}(\Omega)+D^2_{0,-2}(\Omega)\right)\\
\label{eq:c2xz}
\cos^2\theta_{Xz}&=&\cos^2\phi\sin^2\theta=\frac{1}{3}\left(1-D_{0,0}^2(\Omega)\right)+\sqrt{\frac{1}{6}}\left(D^2_{2,0}(\Omega)+D^2_{2,0}(\Omega)\right)\\
\nonumber
\cos^2\theta_{Xy}&=&(-\cos\phi\cos\theta\sin\chi-\sin\phi\cos\chi)^2=-\frac{1}{4}\left(D_{2,2}^2(\Omega)+D_{2,-2}^2(\Omega)+ D_{-2,2}^2(\Omega)+D_{-2,-2}^2(\Omega)\right)+\\
\label{eq:c2xy}
&&\sqrt{\frac{1}{24}}\left(D_{0,2}^2(\Omega)+D_{0,-2}^2(\Omega)-D_{2,0}^2(\Omega)-D_{-2,0}^2(\Omega)\right)+\frac{1}{3}+\frac{1}{6}D_{0,0}^2(\Omega)\\
\label{eq:cxz_cZz}
\cos\theta_{Zz}\cos\theta_{Xz}&=&\cos\phi\sin\theta\cos\theta = \sqrt{\frac{1}{6}}\left(D_{-1,0}^2(\Omega)-D_{1,0}^2(\Omega)\right)\\
\nonumber
\cos\theta_{Zy}\cos\theta_{Xy}&=&-\sin\theta\sin\chi(\cos\phi\cos\theta\sin\chi+\sin\phi\cos\chi)=-\sqrt{\frac{1}{24}}\left(D_{-1,0}^2(\Omega)-D_{1,0}^2(\Omega)\right)+\\
\label{eq:cxy_czy}
&&\frac{1}{4}\left(-D_{-1,-2}^2(\Omega) -D_{-1,2}^2(\Omega)+D_{1,-2}^2(\Omega)+D_{1,2}^2(\Omega)\right).
\end{aligned}$$
Therefore, the matrix elements of $\mathbf{H}_S$ and $\mathbf{H}_L(t)$ in the basis set of the eigenstates of the symmetric top basis can be computed using $$\begin{aligned}
\nonumber
&&\int\mathrm{d}\Omega D_{M_1,K_1}^{J_1}(\Omega)D_{M_2,K_2}^{J_2}(\Omega)D_{M_3,K_3}^{J_3}(\Omega)=\\
\label{eq:three_wigner_integral}
&&8\pi^2\left(
\begin{array}{ccc}
J_1 & J_2 & J_3\\
K_1 & K_2 & K_3
\end{array}
\right)
\left(
\begin{array}{ccc}
J_1 & J_2 & J_3\\
M_1 & M_2 & M_3
\end{array}
\right),\end{aligned}$$ where $\left(\begin{array}{ccc}
J_1 & J_2 & J_3\\
M_1 & M_2 & M_3
\end{array}\right)$ are the 3J symbols [@Zare1988].
The author acknowledges Dr. Johannes Flo[ß]{}, Dr. Rosario González-Férez and Prof. Dr. Lars Bojer Madsen for fruitful discussions and careful revision of the manuscript. This work was supported by NSERC Canada (via a grant to Prof. P. Brumer). The numerical results presented in this work were obtained at the Centre for Scientific Computing, Aarhus (Denmark).
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---
address: 'Theoretische Physik, ETH-Hönggerberg, CH-8093 Zürich, Switzerland'
author:
- 'D.A. Gorokhov and G. Blatter'
title: Marginal Pinning of Quenched Random Polymers
---
Elastic manifolds subject to a disorder potential [@Halpin] define a rich and challenging problem in modern classical statistical mechanics, with numerous applications in condensed matter physics [@Halpin; @Blatter; @IV; @Gruner]. Fundamental questions arise regarding the relevance of the disorder [@Larkin] and its quantitative effects on the manifold’s quenched fluctuations on short [@Larkin] and large scales [@Halpin]. Such static results then determine, via scaling arguments, the creep type motion of the elastic manifolds under the action of an external force, rendering these studies also relevant in the context of applied physics, e.g., plastic deformations in metals [@IV] or dissipation in superconductors [@Blatter]. Classifying the problems through the dimensionality $d$ of the manifold and the number $n$ of transverse dimensions, the ($1+n$)-dimensional problem describing strings moving in $n$ directions, also known as the random polymer problem, is particularly interesting [@HH; @HHF]. Here, depending on the number of transverse dimensions $n$, the temperature induced fluctuations compete in various ways with the fluctuations due to quenched disorder. In this Letter, we analyze the competition between elasticity, disorder, and temperature for the ($1+n$)-string problem. In particular, we present a functional renormalization group analysis of the ($1+2$)-dimensional problem (a string in 3D space), where the disorder turns marginal at high temperatures; we determine the depinning temperature $T_{\rm dp}$ above which thermal smoothing leads to a collapse of the effective disorder strength and the temperature dependent pinning length $L_c(T)$ beyond which the disorder dominates over the elasticity.
The free energy $F({\bf u},L)= -T\ln Z({\bf u},L)$ of a string starting and ending at the points $({\bf 0}, 0)$ and $({\bf u}, L)$ \[the first (second) coordinate denotes the displacement of (position along) the string)\] is given by the partition function $$Z = \!\!\! \int\limits_{({\bf 0},0)}^{({\bf u},L)} \!\!\!
{\cal D}[{\bf u}^\prime] \,
\exp\bigg\{\!\!-\frac{1}{T} \! \int_0^L \!\!\! dz^\prime
\bigg[ \frac{\epsilon}{2} \bigg(\frac{\partial{\bf u}^\prime}
{\partial{\bf z^\prime}}\bigg)^2
+ U({\bf u}^\prime,z^\prime) \bigg] \bigg\},$$ with $\epsilon$ the elasticity of the string. The disorder potential $U({\bf u},z)$ is assumed to be a random gaussian variable with zero mean and a correlator $$\langle U({\bf u},z) U({\bf u}^\prime, z^\prime)\rangle =
K_0(|{\bf u}-{\bf u}^\prime|)\delta (z-z^\prime),$$ with $K_0(u)$ a smooth function decaying on the scale $\xi$. The partition function $Z$ describes the competition between the elastic energy $(\epsilon/2) (\partial_z{\bf u})^2 > 0$ and the disorder potential $U({\bf u},z)$, from which the string can gain energy by choosing minima with $U({\bf u}, z) <0$.
\[bt\]
The quantity characterizing the behavior of the string is the displacement correlator $\langle u^2 (L)\rangle \equiv \langle[{\bf
u}(L)-{\bf u}(0)]^2\rangle$ $\propto L^{2\zeta}$ describing its wandering with distance $L$. We distinguish between the perturbative (Larkin [@Larkin]) regime at short scales $L < L_c$ and the random manifold regime [@Fisher] at $L > L_c$, as well as temperatures below and above the depinning temperature $T_{\rm dp}\approx (\epsilon
K_0(0) \xi^2)^{1/3}$, see Fig. \[F1\]: For $T < T_{\rm dp}$ the wandering exponent $\zeta_{\rm\scriptscriptstyle L} =3/2$ at small distances $L<L_c$ (see [@Blatter]), while at large distances $L>L_c$, $\zeta_n$ depends on the dimensionality of the space: $\zeta_1 = 2/3$ is an exact result[@HHF] while for $n=2$ numerical simulations [@AV] give a value $\zeta_2 \approx 5/8$. The crossover length $L_c$ separating these two regimes has the form [@Blatter] $L_c\approx [\epsilon^2\xi^4/K_0(0)]^{1/3}$. At high temperatures $T > T_{\rm dp}$ the short scale behavior is subject to thermal smoothing, resulting in a thermal wandering with $\zeta_{\rm
th} = 1/2$, while at large scales $L > L_c(T)$ the disorder induced line wandering characterized by $\zeta_n$ prevails. The crossover length $L_c(T)$ increases algebraically with temperature $L_c(T)\sim
L_c \, (T/T_{\rm dp})^5$ for the ($1+1$)-case [@IV], while for the $(1+2)$-dimensional problem $L_c(T)$ is exponentially sensitive to temperature with [@FV] $L_c(T) \sim A(T)\exp[C(T/T_{\rm dp})^3]$, and $C$ a constant of order unity. It turns out that the usual perturbative methods neither provide the value of $C$ nor that of the prefactor $A(T)$. The exponential sensitivity of the pinning length $L_c(T)$ to the temperature $T$ is related to the appearance of a phase transition in dimensions $n>2$, separating a low temperature disorder dominated phase from a high temperature thermal phase characterized by $\zeta_{\rm th}$ [@Imbrie]. The $(1+2)$-problem then corresponds to the lower critical dimension for this phase transition and thus exhibits [*marginal*]{} behavior.
The main goal of the present work is to develop a consistent scheme for calculating the pinning length $L_c(T)$ in the $(1+2)$-problem specifying the coefficient $C$ in the exponent. While this goal can be achieved within a one-loop calculation including three-replica terms, it turns out that the determination of the prefactor $A(T)$ requires a two-loop analysis (accounting for four-replica terms) which is beyond the scope of this letter. We first discuss perturbative methods for calculating $L_c(T)$ at high temperatures and different dimensions $n$. These techniques fail for the ($1+2$)-problem and we will then turn to the more powerful renormalization group method.
[*Perturbation Theory:*]{} With the weak disorder providing the small parameter $\Delta = \int d^n u\, K_0(u)$ we expand the $k$-fold replicated Green function (see [@GB]) $G({\bf u}_1,\dots, {\bf
u}_k;L)$ $\equiv \langle Z({\bf u}_1,L)\dots Z({\bf u}_k,L)\rangle$ and calculate the mean squared displacement $\langle u^2(L)\rangle =
\lim_{k\rightarrow 0}$ $\int d^n u d^n u_2 \dots d^n u_k \, u^2 G({\bf
u},{\bf u}_2 \dots, {\bf u}_k;L)$. The disorder is relevant when $\langle u^2 (L\rightarrow\infty)\rangle$ grows faster than the thermal wandering $\langle u^2(L)\rangle_{\rm th} = (nT/\epsilon) L$ which is dominant at small lengths; the comparison of the two expressions then defines the characteristic crossover length $L_c(T)$, $(nT/\epsilon)\, L_c (T) \sim \langle u^2(L_c(T)) \rangle$. To lowest order the mean squared displacement takes the form $$\langle u^2(L)\rangle = \frac{nTL}{\epsilon}
\left [1+\frac{(\Delta/T^2)(\epsilon/T)^{n/2}}{(4\pi)^{n/2}(4-n)}
L^{1-n/2} \right];
\label{loword}$$ for $n<2$ the disorder corrections grow more rapidly than the thermal wandering and we obtain [@IV] $L_c(T) \sim L_c \,
[T/(\epsilon\Delta \xi)^{1/3}]^5$ for $n=1$. For $2<n<4$ the contribution from the random potential goes to zero at large scales and the disorder is irrelevant [@Imbrie]. The case $n=2$ is marginal: to lowest order the disorder correction is $L$-independent and the next order term $\propto \Delta^2$ provides a logarithmic correction, $$\langle u^2(L)\rangle= \frac{2TL}{\epsilon}
\left[ 1+\frac{1}{8\pi} \frac{\epsilon\Delta}{T^3} +
\frac{1}{16\pi^2}\frac{\epsilon^{2}\Delta^2}{T^6}\ln (L/\xi)\right],
\label{logcorr}$$ producing an exponential temperature dependence $L_c(T) \propto$ $\exp[CT^3/\epsilon\Delta]$ with $C$ of order unity (we compare terms $\propto \Delta^2$ and $\propto \Delta^3$). The arbitrariness of the criterion prevents us from finding the coefficient in the exponent and the prefactor and we have to develop a more systematic way in order to deal with the problem.
[*Renormalization group:*]{} The basic idea is to construct the renormalized effective correlator $K_l$ of the disorder potential describing the behavior of the manifold on short and large scales. With the disorder becoming irrelevant for (internal) dimensionalities $d>4$ [@Larkin], the effect of disorder is analyzed within an $\epsilon=4-d$-expansion. Starting from short scales, the RG flow goes through a special point where $K_l$ becomes singular. This point is identified with the pinning length $L_c$ of the string [@Bucheli]. Its physical meaning is clear: the fact that the correlator becomes singular at $L_c$ implies that the perturbation theory breaks down and the manifold splits up into independently pinned domains of size $L_c$. By way of introduction we briefly derive the zero-temperature pinning length $L_c$, starting from the one-loop FRG equation [@Fisher; @Balents] $$\begin{aligned}
\partial_l K_l
&=&(4-d-4\zeta -n\zeta)K_l+\zeta\nabla\cdot({\bf u}K_l)\nonumber\\
&+& I [K_l^{\mu\nu}({\bf u}) K_l^{\mu\nu}({\bf u})/2
-K_l^{\mu\nu}({\bf u})K_l^{\mu\nu}({\bf 0})],
\label{frg**} \end{aligned}$$ with $\zeta(l)=(1/2){\partial\ln\langle u^2\rangle}/{\partial\ln L}$ the wandering exponent, $I=A_d/(2\pi)^d \epsilon^2 \Lambda^{4-d}$ ($A_d={2\pi^{d/2}}/{\Gamma (d/2)}$, and $\Lambda^{-1}\sim\xi$ is the short scale cutoff), and the indices $\mu$ and $\nu$ denote derivatives with respect to the Cartesian coordinates $u_{\mu}$ and $u_{\nu}$. The RG variable $l$ is related to the length $L$ via $l=\ln
{\Lambda L}$. We proceed with differentiating (\[frg\*\*\]) four times with respect to $u_\kappa$. Substituting $u=0$, the wandering exponent $\zeta$ drops out and we obtain $$\partial_l \Gamma_l = (4-d) \Gamma_l +
[I(n+8)/3] \Gamma_l^2,
\label{der4}$$ where $\Gamma_l={\partial^4 K_l}/{\partial u_{\kappa}^4}|_{{\bf
u}=0}$. Integrating (\[der4\]) we encounter a divergence at $l_c
\approx \ln[3(4-d)/I(n+8)\Gamma_0]/(4-d)$ (we assume weak disorder with $I \Gamma_0 \ll 1$), allowing us to define the collective pinning or Larkin length $L_c = \Lambda^{-1} \exp(l_c) \approx
[\epsilon^2\xi^4/K(0)]^{1/(4-d)}$ where simple perturbation theory breaks down. The FRG equation (\[frg\*\*\]) allows us to further characterize the Larkin regime: On short scales we can neglect the non-linear term and an expansion of the correlator $K_l(u) \sim
\alpha(l) u^2/2$ produces the wandering exponent $\zeta_{\rm\scriptscriptstyle L}=(4-d)/2$ at the Larkin fixed point [@GiamarchiLeDoussal]. Note that, although Eq. (\[frg\*\*\]) is written in the form of an $\varepsilon$-expansion, the result for $L_c$ is valid for any $\epsilon$ as long as we investigate the short scale behavior, where the functional renormalization group is just a way to sum the perturbation expansion. Indeed, one can show that in the non-marginal situation higher order loop corrections to Eq. (\[frg\*\*\]) do not change the result for $L_c$, while in the marginal case (e.g., for $d=4$ at $T=0$) we have to account for two-loop corrections.
Next, we proceed to study finite temperatures, where besides a simple diffusive term $\propto T \Delta K_l$, new three- and higher replica terms are generated describing non-gaussian fluctuations in the disorder potential $U$. Generalizing the disorder statistics to include non-gaussian terms and replicating $n$ times we arrive at a new Hamiltonian ${\cal H}_n = {\cal H}_{\rm free}+{\cal H}_{\rm int}$ with ${\cal H}_{\rm free} =\int d^d
z\sum_{\alpha}(\epsilon/2)(\partial_{\bf z}{\bf u}_{\alpha})^2$ and $$\frac{{\cal H}_{\rm int}}{T}
=-\int d^d z \biggl[\sum_{\alpha,\beta}
\frac{K({\bf u}_{\alpha\beta})}{2T^2}
+\sum\limits_{\alpha,\beta,\gamma}
\frac{S({\bf u}_{\alpha\beta},{\bf u}_{\alpha\gamma})}{T^3}
+\dots\biggr].$$ Here, ${\bf u}_{\alpha\beta} \equiv {\bf u}_\alpha-{\bf u}_\beta$ and $S({\bf u},\bar{\bf u})$ is the three-replica term describing non-gaussian fluctuations in the disorder potential (the $\dots$ stand for higher-replica terms depending on three or more variables). At finite temperatures, a consistent analysis to one loop order requires to include both two- and three-replica terms $K({\bf u})$ and $S({\bf
u},\bar{\bf u})$ in the FRG flow. After a second order cumulant expansion in the two- and three-body interaction ${\cal H}_{\rm int}$, we proceed with the standard momentum shell RG, expand the ${\bf
u}$-fields into Fourier series and perform the integration over fast modes. The resulting equations take the form $$\begin{aligned}
\partial_l K_l ({\bf u})
&=& (4\!-\!d\!-\!4\zeta\!-\!n\zeta)K_l+\zeta\nabla\cdot({\bf u}K_l)
+ I^{\prime} T K_l^{\mu\mu} \nonumber\\
&+& I [(1/2) K_l^{\mu\nu}({\bf u}) K_l^{\mu\nu}({\bf u})/2
-K_l^{\mu\nu}({\bf u})K_l^{\mu\nu}({\bf 0})]\nonumber\\
&-& \!2I^{\prime}T[S_l^{\mu\mu}({\bf 0},{\bf u})
\!+\!S_l^{{\bar{\mu}}\bar{\mu}}({\bf u},{\bf 0})
\!-\
\nonumber\\
&+& [\partial_{\bf u}\!\cdot\!
\partial_{\bf \bar{u}}S_l]({\bf 0},{\bf u})
+[\partial_{\bf u}\!\cdot\!
\partial_{\bf \bar{u}}S_l]({\bf u},{\bf 0})],
\label{frg}\\
\partial_l S_l({\bf u},{\bf \bar{u}})
&=& (6\!-\!2d\!-\!6\zeta\!-\!2n\zeta)S_l
+\zeta[\nabla\!\cdot\!({\bf u} S_l)
\!+\!{\bar{\nabla}}\!\cdot\!({\bar{\bf u}}S_l)]\nonumber\\
&+& I^{\prime}T[S_l^{\mu\mu}+S_l^{\bar{\mu}\bar{\mu}}
+\partial_{\bf u}\!\cdot\!
\partial_{\bf \bar{u}} S_l]\nonumber\\
&+& ITK_l^{\mu\nu}({\bf u})K_l^{\mu\nu}({\bf \bar{u}})/4,
\label{ffrrgg}\\
\partial_l T_l &=&(2-d-2\zeta)T_l,
\label{frgT}\end{aligned}$$ where $I' = I\epsilon\Lambda^2$ and the superscripts $\mu$ ($\bar\mu$) denote derivatives with respect to the first (second) variable in $S_l({\bf u}, {\bf \bar{u}})$ (implicit summation over double indices is assumed); similarly, the gradients $\nabla$ ($\bar{\nabla}$) denote derivatives with respect to ${\bf u}$ (${\bf{\bar{u}}})$. The system (\[frg\]) – (\[frgT\]) has to be solved with initial conditions $K_{l=0}({\bf u})=K_{0}({\bf u})$, $S_{l=0}({\bf u},{\bf \bar{u}})=0$ (the bare potential is gaussian), and $T_{l=0}=T$. Note that the three-replica term is driven by a source $\propto T K_l^2$ and hence is irrelevant in the $T=0$ analysis. Furthermore, four- and higher replica terms can be omitted within the present one-loop analysis: with $K_l({\bf u})={\cal O}(\Delta)$ the three-replica term $S_l({\bf
u},{\bf{\bar u}})={\cal O}(\Delta^2)$, see (\[ffrrgg\]), while the four-replica term is driven by a term $\propto K_l^3$ and hence $D({\bf u}_1,{\bf u}_2, {\bf u}_3)= {\cal O}(\Delta^3)$. However, going to second-loop order the four-replica term $D_{l}$ has to be included.
The above system (\[frg\]) – (\[frgT\]) can be solved asymptotically exactly for weak disorder and we proceed with the analysis for the elastic string, $d=1$. On short scales the behavior of the string is dominated by thermal fluctuations and the wandering exponent is equal to $\zeta_{\rm th} = 1/2$; as a result, the temperature is not renormalized, see Eq. (\[frgT\]). On the other hand, on large scales $\zeta$ approaches the value $\zeta_2 \approx
5/8$ and the behavior of the string is governed by the zero temperature fixed point. It turns out that the temperature starts renormalizing only close to $L_c(T)$, allowing us to fix $\zeta =1/2$ in Eqs. (\[frg\]) – (\[frgT\]). The non-linear and three-replica terms are of second order in the disorder strength and thus unimportant at small scales $l\ll l_c$, allowing us to neglect the terms $\propto K_l^2$ and $\propto S_l$ in (\[frg\]) in the initial stage of the flow. The Ansatz $$K_l(u)=P_l(u)\exp[(1-n/2)l],
\label{varsubst}$$ then maps (\[frg\]) to the Fokker-Planck equation describing the ‘time’ evolution of the probability distribution function $P_l$ for an overdamped particle in the harmonic potential $V(u)=u^2/4$ at a ‘temperature’ $T^\prime= T/\pi \epsilon\Lambda$, $\partial_l
P_l=\nabla\cdot({\bf u}P_l)/2 +T^\prime\Delta P_l$. At high temperatures $T$, the distribution function $P_l(u)$ rapidly approaches equilibrium and takes the form of a Boltzmann distribution, $$P_{\rm\scriptscriptstyle B}(u)
=\Delta(4\pi T^{\prime})^{-n/2}
\exp(-u^2/4T^\prime).
\label{incondnn}$$ The function $P_{\rm\scriptscriptstyle B}(u)$ depends on the disorder potential only via the integral weight $\Delta =\int d^n u \, K_0(u)$, i.e., the evolution erases the details of the initial condition. Depending on the dimensionality $n$, the following situations arise (c.f., (\[varsubst\])): For $n>2$ the solution of (\[frg\]) decreases with increasing $l$ and the disorder renormalizes to zero, in agreement with the existence of a high temperature phase when $n>2$, see [@Imbrie]. For $n<2$ the correlator $K_l(u)=\exp[(1-n/2)l] P_{\rm\scriptscriptstyle B} (u)$ grows (until reaching the size of the non-linear term) and the disorder is relevant. Comparing terms linear and quadratic in $K_l$ we find the collective pinning length $L_c(T) \simeq
L_c[T/(\epsilon\Delta\xi)^{1/3}]^5$. Most interesting is the marginal case $n=2$ where we find a ‘Boltzmann’ fixed point. In order to obtain $L_c(T)$ we need to take into account the terms ${\cal O}(\Delta^2)$ effectively pumping additional ‘particles’ into the system and leading to an overall growth in the disorder strength $\Delta_l=\int d^n u\,
K_l(u)$. However, for small $\Delta_l$ the strong diffusion will relax this ‘pumping’ towards the ‘Boltzmann’ distribution, i.e., the solution of Eq. (\[frg\]) keeps the form $$K_l({\bf u}) = g_l\, ({T^\prime}^2/I)\,\exp(-{\bf u}^2/4T^{\prime}),
\label{simplef}$$ with the dimensionless coupling $g_l=(\pi/4) \Delta_l\epsilon/T^3$. After diagonalizing the term $[S_l^{\mu\mu}+S_l^{\bar{\mu}\bar{\mu}}
+\partial_{\bf u}\!\cdot\!\partial_{\bf \bar{u}} S_l]$ in (\[ffrrgg\]) we find its Green function $G_S({\bf u},\bar{\bf
u};{\bf u}^\prime,{\bar{\bf u}}^\prime;l) =
1/12[\pi{I^\prime}T(1\!-\!e^{-l})]^2 \exp\{-[({\bf u}\!-\!{\bf
u}^\prime e^{-l/2})^2\!+ \!(\bar{\bf u}\!-\!{\bar{\bf u}}^\prime
e^{-l/2})^2$ $+({\bf u}\!-\!{\bf u}^\prime e^{-l/2})\cdot (\bar{\bf
u}\!-\!{\bar{\bf u}}^\prime e^{-l/2})]/3I^\prime T(1-e^{-l})\}$ and obtain the three-replica term $S_l({\bf u},\bar{\bf u})$ by simple integration. Substituting $S_l$ back into (\[frg\]) and integrating over ${\bf u}$ we find that the three-replica terms mutually cancel and thus do not contribute to the flow of $g_l$. The amplitude $g_l$ then is determined by the remaining non-linear terms alone, $$\partial_l g_l=g_l^2/8,
\label{simplified}$$ with the solution $g_l=g_0/(1-g_0 l/8)$ as derived from the initial condition $g_0=(\pi/4) (\epsilon\Delta/T^3)$. At $l_c=8/g_0$ the amplitude $g_{l_c} \approx 1$ and we find the pinning length $$L_c(T)\sim\exp[(32/\pi)(T^3/T_{\rm dp}^3)],
\qquad T_{\rm dp}^3 = \epsilon\Delta,
\label{mainres}$$ the main result of the paper [@comment]: the one-loop FRG analysis allows us to fix the coefficient $C$ in front of the parameter $1/g_0$ in the exponent. Although the criterion $g_{l_c}\simeq 1$ might seem arbitrary it is sufficient to find the correct asymptotic value of the RG variable $l_c$ as long as the initial value $g_0$ is small. Indeed, the solution $g_l$ remains small in most of the interval $[0,l_c]$ and grows rapidly only very close to $l_c$.
An interesting alternative is offered by mapping the random polymer problem to the Kardar-Parisi-Zhang equation [@KPZ], where the dynamic renormalization group (DRG) [@FNS] provides excellent results for the $n=1$ problem [@MHKZ]. The weight ${\tilde g}$ in the (white noise) Ansatz $K_{{\rm wn}} = {\tilde g} \delta^2 ({\bf
u})$ replacing (\[simplef\]) obeys the equation [@Frey] $\partial_{l^\prime} {\tilde g} = (2-n){\tilde g}+[2(2n-3)/n] {\tilde
g}^2$, with ${\tilde g}(0) =(A_n/(2\pi)^n)\epsilon\Delta/T^3$ (the prime indicates scaling along the transverse dimension). For $n=2$ the solution ‘explodes’ at $l_c^\prime = 2\pi(T/T_{\rm dp})^3$, implying that $L_c(T) \sim \exp[4\pi(T/T_{\rm dp})^3]$ [@FH], different from Eq. (\[mainres\]). Note that, in contrast to the FRG, the DRG scheme does not renormalize the correlator [*function*]{}. Here, we argue in favor of our result (\[mainres\]): While the DRG technique produces accurate results (including a random manifold fixed point) in dimensions $n<3/2$, the results are less convincing in larger dimensions $n>3/2$, where the fixed point gives way to a divergence. On the other hand, the FRG technique features both an apparent divergence at $l_c$ and a random manifold fixed point as $l \rightarrow \infty$ in all dimensions $n \leq 2$, allowing for a consistent physical interpretation of the results in terms of the diagram sketched in Fig. 1.
Above we have assumed that the integral $\int d^n u \, K_0(u)$ converges, however, flux lines in superconductors exhibit a long range interaction with the disorder potential spoiling this assumption: for values $u$ exceeding the coherence length $\xi$, the correlator obeys the asymptotics $K_0(u)\sim K_0(0)(\xi^2/u^2) \ln (u/\xi)$ and the integral $\int_{u<u^\ast}d^n u \,K_0(u)$ diverges as $K_0(0)\xi^2
\ln^2({u^\ast}/\xi)$. We estimate $L_c(T)$ for this case: Neglecting the non-linear term in the first stage of the RG transformation, we approximate the solution of the Fokker-Planck equation in the region $u^2 \alt T\xi/\epsilon$ by $[\Delta_l/4\pi T^\prime]
\exp[-u^2/4T^\prime]$ \[c.f., Eq. (\[incondnn\])\], with an $l$-dependent function $\Delta_l$, a consequence of the divergence of the integral $\int d^n u\,K_0(u)$. The weight $\Delta_l$ collects those particles reaching the central region on scale $l$: a displacement $u^\ast\gg\sqrt{T\xi/\epsilon}$ of the harmonic oscillator takes a ‘time’ $l^\ast\sim\ln(u^\ast
/\sqrt{T\xi/\epsilon})$ to reach the region $u \alt
\sqrt{T\xi/\epsilon}$ and hence $\Delta_{l^\ast} \sim K_0(0)\xi^2
{l^\ast}^2$. The dimensionless coupling $g_l$ then grows already in the linear regime due to the ‘pumping’ from large distances. The pinning length follows from the condition $g_{l_c} =
\epsilon\Delta_{l_c}/T^3 \approx 1$ and we obtain $L_c(T)\sim\exp[C(T^3/\epsilon K(0)\xi^2)^{1/2}]$ with $C$ of order unity. The same result follows from a perturbative treatment.
In order to find the prefactor $A(T)$ in the expression (\[mainres\]) for $L_c(T)$ we have to account for ${\cal O}(\Delta^3)$ corrections, resulting in a flow $\partial_l g_l=g_l^2/8 + \eta g_l^3$ with $\eta>0$ a constant. This equation explodes at $l_c =8/g_0-64 \eta \ln g_0$, producing a prefactor of the form $A(T) \simeq \Lambda^{-1}(T_{\rm
dp}/T)^{192\eta}$. While the two-loop contribution $\eta_{\rm 2-l} =
275/5184$ to $\eta$ can be found from [@Bucheli], the part arising from the four-replica term is more difficult to obtain — whether the four-replica terms mutually cancel (e.g., due to some symmetry), as was the case in the one-loop analysis above, remains an interesting and open problem.
We thank L. Balents, D.S. Fisher, E. Frey, V.B. Geshkenbein, T.Giamarchi, Th. Hwa, O. Narayan, V.M. Vinokur, and J. Zinn-Justin for helpful discussions.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The copper oxides present the highest superconducting temperature and properties at odds with other compounds, suggestive of a fundamentally different superconductivity. In particular, the Abrikosov vortices fail to exhibit localized states expected and observed in all clean superconductors. We have explored the possibility that the elusive vortex-core signatures are actually present but weak. Combining local tunneling measurements with large-scale theoretical modeling, we positively identify the vortex states in YBa$_2$Cu$_3$O$_{7-\delta}$. We explain their spectrum and the observed variations thereof from one vortex to the next by considering the effects of nearby vortices and disorder in the vortex lattice. We argue that the superconductivity of copper oxides is conventional, but the spectroscopic signature does not look so because the superconducting carriers are a minority.'
author:
- Christophe Berthod
- 'Ivan Maggio-Aprile'
- 'Jens Bru[é]{}r'
- Andreas Erb
- Christoph Renner
date: 'April 25, 2017'
title: 'Observation of Caroli–de Gennes–Matricon Vortex States in YBa$_2$Cu$_3$O$_{7-\delta}$'
---
Type-II superconductors immersed in a magnetic field let quantized flux tubes perforate them: the Abrikosov vortices. This remarkable property underlies and often limits many applications of superconductors. In 1964, Caroli, de Gennes, and Matricon used the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity to predict that vortices in type-II superconductors host a collection of localized electrons bound to their core [@Caroli-1964]. The direct observation of these localized states 25 years later by scanning tunneling spectroscopy (STS) is a spectacular verification of the BCS theory [@Hess-1989]. The formation of vortex-core bound states is an immediate consequence of the superconducting condensate being composed of electron pairs, while excitations in the vortex, being unpaired, have a different topology. Core states are, therefore, a robust property of superconductors, like edge states in topological insulators, irrespective of the origin and symmetry of the force that glues the electrons into pairs. In spectroscopy, they appear in the clean limit $\ell\gg\xi$ as a zero-bias peak in the local density of states (LDOS) at the vortex center, where $\ell$ and $\xi$ are the electron mean free path and superconducting coherence length, respectively, or as a structureless LDOS in the dirty limit $\ell\lesssim\xi$ [@Renner-1991]. Next to NbSe$_2$ [@Hess-1989; @Suderow-2014], the core states were seen by STS in several superconducting materials [@deWilde-1997b; @Eskildsen-2002; @Nishimori-2004; @Guillamon-2008a; @Zhou-2013; @Du-2015], including the pnictides, which are believed to host unconventional pairing [@Yin-2009; @Shan-2011; @Song-2011; @Hanaguri-2012; @Fan-2015]. The high-$T_c$ cuprates stand out as the only materials in which the vortex-core states have been looked for but not found. In YBa$_2$Cu$_3$O$_{7-\delta}$ (Y123), discrete finite-energy structures initially believed to be vortex states [@Maggio-Aprile-1995; @Shibata-2003b; @*Shibata-2010] were recently shown to be unrelated to vortices [@Bruer-2016]. In Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212), the vortex cores present no trace of a robust zero-bias peak, but instead very weak finite-energy features apparently related to a charge-density wave order [@Renner-1998b; @Hoogenboom-2000a; @Pan-2000b; @Matsuba-2003a; @*Matsuba-2007; @Levy-2005; @Yoshizawa-2013].
The absence of vortex states in cuprates is challenging the existing theories. Because these states are topological they are robust [@Wang-1995; @Franz-1998b], and they survive modifications of the BCS theory like strong-coupling extensions [@Berthod-2015] that do not change the nature of the condensate. To explain the cuprate vortex phenomenology, one needs either to leave BCS theory [@Arovas-1997; @Himeda-1997; @Andersen-2000; @Kishine-2001; @Berthod-2001b], or to extend it by including additional order parameters that condense inside the vortex cores and gap out the zero-bias peak [@Zhu-2001a; @Maska-2003; @Fogelstrom-2011]. To date, none of these approaches has given a satisfactory account of the phenomenology observed by STS. The discovery that the low-energy structures in Y123 do not belong to vortices [@Bruer-2016] suggests that these theoretical efforts have been misguided.
The electronic structure of the cuprate high-$T_c$ superconductors (HTS) is notoriously complex, as manifested in a rich phase diagram. This complexity reveals a competition of different effective interactions, from which a variety of individual and collective modes emerges progressively as the temperature is lowered towards the ground state in which the system freezes at absolute zero, and which continues to keep the secret of the most stable superconductivity ever observed. It is generally believed that the phenomena taking place close to the Fermi surface—charge, spin, pairing orders, and their fluctuations—all derive from a single band [@Zhang-1988] or a small subset of bands [@Emery-1987] in the CuO$_2$ layer(s). Consistently, the interpretations of STS spectra [@Fischer-2007] have postulated that all electrons contributing to the measured LDOS are excited out of the superconducting condensate, in agreement with Leggett’s theorem [@Leggett-1998]. On the other hand, it is well known that for all cuprates, at any doping, the superfluid density remains much smaller than the electron density [@Uemura-1989; @Bernhard-1995; @Bosovic-2016]. Our recent high-resolution STS experiments on Y123 have also revealed that only a fraction of the signal recorded on the sample surface is of superconducting origin [@Bruer-2016]. Early specific-heat measurements have given a similar hint [@Junod-1999]. We are therefore lead to a new paradigm, in which the low-energy electronic state of the HTS involves a minority superconducting channel in parallel with nonsuperconducting majority charges guilty for the pseudogap and the associated orders. Here we show that the minority carriers are fairly conventional in the superconducting state, showing Caroli–de Gennes–Matricon states in the vortex cores as predicted by the BCS theory for $d$-wave superconductors.
![\[fig:fig1\] (a) The self-consistent LDOS calculated in the BCS theory for an isolated vortex in a $d$-wave superconductor with Y123 band structure is shown along two paths starting at the vortex core (blue) and ending at two points 20 nm away from the core (green) along the nodal (11) and antinodal (10) directions. (b) Same data with the zero-field DOS subtracted from all curves. The vertical scale is arbitrary and the curves are shifted vertically for clarity. The insets show the CuO$_2$ unit cell with the two crystallographic directions, and representations of the vortex core (gray disks and arrows indicating the supercurrent direction) with the color-coded paths of the two spectral traces. ](fig1){width="0.75\columnwidth"}
![\[fig:fig2\] (a) and (b) A $90\times90~\mathrm{nm}^2$ area on the (001) surface of Y123 in a 6 T field, colored by (a) the ratio of the STS tunneling conductance at $+5$ and $+17$ mV, and (b) the conductance at zero bias. The inset in (b) shows the STS spectrum averaged over the small outlined region between vortices. Two series of difference spectra (raw STS data minus average spectrum shown in the inset) along the 20 nm paths indicated in (a) are displayed in (c) and (d). The color encodes distance from the core as in Fig. \[fig:fig1\]. Panels (e) and (f) show the raw $dI/dV$ data along the two paths. ](fig2){width="\columnwidth"}
STS measurements with a normal metal tip do not discriminate the superconducting (SC) and nonsuperconducting (NSC) channels and collect electrons from each. Our working hypothesis is that the tunneling conductances originating from the SC and NSC channels are additive. An earlier report in which vortex-induced changes were tracked in Bi2212 rested on a different scenario, namely, the vortices would suppress locally the condensate and reveal a competing magnetic order [@Hoffman-2002]. As the NSC is not known, a simple subtraction to reveal the SC is not feasible. Yet, inhomogeneities of the SC like those induced by vortices can be singled out by subtracting the tunneling conductance outside vortices from that inside. If the formation of a vortex leaves the NSC unchanged, this procedure eliminates the NSC and permits a comparison of the SC inhomogeneities with the BCS theory. Figure \[fig:fig1\](a) shows the LDOS predicted by the BCS theory at and near the center of an isolated vortex in a two-dimensional superconductor with electronic structure similar to that of Y123 [^1]. The prominent feature is the zero-energy peak localized at the core center. With increasing distance from the core, the peak is suppressed with or without a splitting depending on the direction. Note that this LDOS anisotropy is unrelated to the $d$-wave gap anisotropy. In the quantum regime $k_{\mathrm{F}}\xi\sim1$ relevant for Y123, the vortex size is comparable with the Fermi wavelength and the Fermi-surface anisotropy determines the vortex structure [@Uranga-2016; @Berthod-2016]. In the Supplementary Material, Fig. \[fig:swave\] indeed shows that the zero-bias LDOS is locked to the crystal rather than gap-node directions. For a meaningful comparison with experiment, the LDOS far from the vortex must be subtracted, as done in Fig. \[fig:fig1\](b).
Figure \[fig:fig2\](a) shows a $90\times90~\mathrm{nm}^2$ area on the surface of Y123, where 19 inhomogeneities can be identified as vortex cores. Details about the sample preparation and measurement technique were reported in Ref. . Because of the large NSC, the contrast due to vortices is weak; it is maximized by mapping the ratio of the STS tunneling conductance at 5 and 17 mV bias \[Fig. \[fig:fig2\](a)\]. Local increases of the zero-bias conductance are also seen in the raw data \[Fig. \[fig:fig2\](b)\], and correlate well with the vortex positions determined by the best contrast. In order to remove the NSC, we delineate a small region in-between vortices, calculate the average spectrum in this region \[inset of Fig. \[fig:fig2\](b)\], and subtract this average from all spectra in the map. After subtraction, the spectral traces show the expected vortex signature with a maximum at zero bias in the cores. This is demonstrated in Figs. \[fig:fig2\](c) and \[fig:fig2\](d) with two traces running from one vortex core along the two directions shown in Fig. \[fig:fig2\](a). The peak developing locally at zero bias after subtracting the same background from the spectra of each trace clearly shows there is a larger local density at low energy near the vortex cores. It is not an artifact of the subtraction. Similar results are found in all vortices.
{width="70.00000%"}
A comparison of Figs. \[fig:fig2\](c) and \[fig:fig2\](d) with Fig. \[fig:fig1\](b) reveals evident similarities, but also differences. Similarities include the zero-bias peak, which has its maximum at the core center and is suppressed with increasing distance from the center, the absence of superconducting coherence peaks in the core leading to symmetric dips at $\pm17$ meV in both experiment and theory, a striking spatial anisotropy, with the peak extending farther along path n^o^2 than along path n^o^1, reminiscent of the different decay lengths observed in the theory between directions 10 and 11. Among the differences, one notices the central peak being taller in Fig. \[fig:fig1\](b) than in the measurements, and the spectrum appearing locally reinforced or split at intermediate distances along path n^o^1 and n^o^2, respectively, while Fig. \[fig:fig1\](b) shows a monotonic evolution. We now demonstrate that these differences can be explained by considering that (i) the vortices are not isolated in the experiment, (ii) the relative orientation of the vortex and crystal lattices influences the LDOS, and (iii) the vortex lattice is disordered. Incorporating (i) in the theory reduces drastically the calculated peak height; (ii) means that the LDOS anisotropy depends upon the positions of nearby vortices; finally, (iii) implies that each vortex sits in a specific local environment and presents spectra different from its neighbors.
Figure \[fig:fig3\] displays a series of spectral traces along various paths connecting either nearest-neighbor \[Fig. \[fig:fig3\](a)\] or next-nearest neighbor \[Fig. \[fig:fig3\](b)\] vortices. The notion of nearest- and next-nearest neighbor refers to a local fourfold coordination generally observed among the vortices, despite the long-range disorder in their arrangement. A trend is systematically observed: along a line connecting nearest-neighbor vortices, the zero-bias peak remains visible along the whole path, while it disappears along paths joining next-nearest neighbors. Considering the peak anisotropy as it is predicted by theory (Fig. \[fig:fig1\]), this trend suggests that the locally fourfold-coordinated vortex lattice tends to align along the crystalline axes, as we will confirm by a detailed modeling. In the absence of atomic resolution imaging, we infer the lattice orientation based on optical images of twin boundaries. Another lesson of Fig. \[fig:fig3\] is that all vortices, although similar, are different: some show a single peak, others show a split peak; the height of these peaks is also varying. This variability reflects disorder in the vortex positions, resulting in irregular distributions of supercurrents around each core.
{width="70.00000%"}
We have undertaken large-scale simulations of the LDOS in disordered vortex configurations, in order to study how this modifies the core spectra with respect to the isolated vortex, and thus better understand the observations made by STS. Thanks to a new approach described in Ref. , we are able to compute the LDOS in a disordered vortex lattice with the same accuracy as in the isolated core. All relevant details are given as Supplemental Material [@Note1], and we only briefly review here the key ingredients. The structure of the vortices in a finite field is deduced from the self-consistent solution in the ideal vortex lattice. We find that the LDOS changes dramatically with varying the orientation of the vortex lattice relative to the crystal axes. The system size required in order to compute the LDOS with our target resolution (3 meV) contains half a million unit cells and extends well beyond the area of our STS experiments, which contains roughly 54500 Cu sites. Inside the STS field of view, we locate vortices at the positions indicated in Fig. \[fig:fig4\](a). Outside, we generate vortex positions randomly, however, constraining the intervortex distance to be at least 16 nm. The resulting configurations show no orientational order, but short-range coordination similar to what is seen in the STS image. We do not imply that the actual vortex distribution outside the field of view has no orientational order—in fact, it probably has some [@Maggio-Aprile-1995]—but the available data prevent us from inferring such an order. We also vary the orientation of the crystal lattice with respect to the vortices. For each of 600 generated configurations, we calculate the LDOS along the paths n^o^6 and n^o^8 of Fig. \[fig:fig3\] and compare with the experimental traces. With the configuration giving the smallest difference with experiment on these two traces, we recalculate the LDOS on the whole domain covered by STS with a resolution of 1 nm and deduce the theoretical map and traces shown Fig. \[fig:fig4\](b).
The simulation confirms that quasiparticle scattering off nearby vortices reduces the central peak in each vortex, that the LDOS is different in all cores, and that it depends on the configuration of the vortices outside the STS field of view. We also find that the agreement with experiment is systematically better if the orientation of the microscopic lattice is such that the traces n^o^6 and n^o^8 are close to a nodal direction, in agreement with the twin-boundary directions. Although our search for a good vortex configuration has focused on two traces connecting next-nearest neighbor vortices, the resulting model reproduces the difference between these traces and those connecting nearest-neighbor vortices. It is also striking that the model correctly predicts the contrast of the STS image and the apparent size of the vortex cores without any further adjustments. Figure \[fig:fig4\](b) presents less spectral variations from core to core than Fig. \[fig:fig4\](a), but this appears to be a compromise by which this configuration achieves the best overall agreement. Other vortex configurations that compete closely do show variations comparable with experiment, including split peaks in some of the vortices (Fig. \[fig:statistics\]). We emphasize that our procedure does not deliver the best fit, which would require us to optimize systematically the vortex positions rather than trying random configurations. The vortex positions *within* the field of view should be optimized as well. Figure \[fig:fig4\](b) indeed reveals that the LDOS maxima are in general displaced with respect to the points where the order parameter vanishes: the LDOS gets polarized by asymmetries in the supercurrents [@Berthod-2013a; @Berthod-2016]. Without any doubt, vortex configurations could be found that improve the agreement in Fig. \[fig:fig4\], but we feel that such a costly optimization is unlikely to reveal new physics. Interestingly, the simulation presents vortex cores that appear to be split, e.g., at the beginning of path n^o^6 (see also Fig. \[fig:swave\]). Reference reported a similar observation in Bi2212, which was ascribed to interaction with pinning centers. In Fig. \[fig:fig4\](b) the splitting is merely a LDOS distortion due to an irregular distribution of the supercurrent.
In a clean BCS superconductor, the zero-temperature superfluid density $n_{s0}$ equals the total electron density $n$ [@Leggett-1998]. Estimates based on the penetration depth show instead that $n_{s0}/n=23\%$–$34\%$ in Y123 [^2]. The analysis of STS data yields similar relations between the SC and NSC, with some model dependence, 14% in Ref. , 25% in the present work [@Note1]. In a one-channel picture, the property $n_{s0}\ll n$ requires large pairing fluctuations that would invalidate a mean-field description. Our results show that the mean-field theory works well in vortices, though. Another possibility would be that superconductivity emerges in a population of low-energy quasiparticles carrying only a small fraction of the spectral weight: in the RVB theory, the quasiparticle weight behaves as $2x/(1+x)$ as a function of doping $x$ [@Anderson-2004], as observed in optical data of underdoped cuprates [@Mirzaei-2013], which yields a value close to 25% at optimal doping. In this scenario, the NSC should disappear with overdoping and the vortices should present a clear zero-bias peak, a challenge for future STS experiments. Alternatively, one can imagine two-channel scenarios. It has been recently proposed that the cuprate superconductivity is lead by oxygen $2p$ rather than copper $3d$ holes [@Rybicki-2016]. We speculate that the copper holes remain localized and form the NSC (in the specific case of Y123, where the zero-bias conductance is large, we cannot rule out a parasitic surface channel as another contribution to the NSC), while the oxygen holes are responsible for the recently uncovered universal Fermi-liquid signatures [@Barisic-2013] and enter the SC condensate. While the origin of pairing in this condensate remains mysterious, its spectroscopic properties are well described by the BCS theory, as we have demonstrated by unveiling the Caroli–de Gennes–Matricon-like states in the vortex cores.
We thank T. Giamarchi and D. van der Marel for their comments on the manuscript, N. Hussey and G. Deutscher for discussions, and G. Manfrini and A. Guipet for their technical assistance. This research was supported by the Swiss National Science Foundation under Division II. The calculations were performed in the University of Geneva with the clusters Mafalda and Baobab.
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[**Supplemental Material\
\[0.5em\] [for]{}\
\[0.5em\] Observation of Caroli–de Gennes–Matricon Vortex States in YBa$_2$Cu$_3$O$_{7-\delta}$** ]{}\
Christophe Berthod,$^1$ Ivan Maggio-Aprile,$^1$ Jens Bru[é]{}r,$^1$ Andreas Erb,$^2$ and Christoph Renner$^1$\
*$^1$Department of Quantum Matter Physics, University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva, Switzerland\
$^2$Walther-Meissner-Institut, Bayerische Akademie der Wissenschaften, Walther-Meissner-Strasse 8, D-85748 Garching, Germany*
Microscopic model for the superconducting channel of [YB$_2$C$_3$O$_{7-\delta}$]{} in the mixed state
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Electronic structure
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YBa$_2$Cu$_3$O$_{7-\delta}$ (Y123) has two CuO$_2$ planes in the unit cell, CuO chains running along the $b$ axis, and a small orthorhombic distortion with inequivalent $a$ and $b$ axes ($b>a$) [@Jorgensen-1990-s]. We ignore the CuO chains and the distortion, absent in other cuprates, and therefore irrelevant for high-$T_c$ superconductivity. Some effects of the chains on the vortex-core spectra have been studied in Ref. . We furthermore ignore the bilayer splitting for simplicity and represent the CuO$_2$ layers as a one-band tight-binding model on a perfect square lattice with parameter $a=3.85$ Å. We have also performed calculations for a two-band system including bilayer splitting, and found only inessential quantitative differences in the vortex cores. The one-band model is more convenient for large-scale simulations. We use the tight-binding parameters $t_1=-281$ meV, $t_2=139$ meV, and $t_3=-44$ meV determined by photoemission in Ref. for the first, second, and third neighbor hopping, respectively, ignoring $t_4$ and $t_5$ for simplicity. The chemical potential is set to $\mu=-356$ meV for an electron density $n=0.84$, corresponding to optimally hole-doped Y123 with 0.16 hole per unit cell. The dispersion relation measured from the chemical potential is $\xi_{\vec{k}}=2t_1[\cos(k_xa)+\cos(k_ya)]+4t_2\cos(k_xa)\cos(k_ya)+2t_3[\cos(2k_xa)+\cos(2k_ya)]-\mu$. The average group velocity on the Fermi surface is $\langle v_{\mathrm{F}}\rangle=4.11\times10^7$ cm/s. The Fermi surface is shown in Fig. \[fig:uniform\](a). Due to a van Hove singularity at $-376$ meV, the DOS has a negative slope in the low-energy region, with more weight for the occupied states \[Fig. \[fig:uniform\](b)\]. We set the amplitude of the $d$-wave order parameter to $\Delta_0=19$ meV. In the uniform superconductor, the gap $\Delta_{\vec{k}}=(\Delta_0/2)[\cos(k_xa)-\cos(k_ya)]$ has its maximum at the point $(\pi/a,0.74/a)$ of the Fermi surface, giving coherence peaks at $\pm17$ meV \[Fig. \[fig:uniform\](c)\]. This amplitude and symmetry of the order parameter follow self-consistently from the Bogoliubov-de Gennes equations for an (instantaneous) attractive interaction $V=-247$ meV on nearest-neighbor bonds.
![\[fig:uniform\] (a) Fermi surface and (b) dispersion relation. The green curve in (b) is the normal-state DOS with a van Hove singularity at $-376$ meV. (c) Zero-field tunneling conductance of Y123 at 0.4 Kelvin (solid blue, left scale, from Ref. ) and its decomposition in superconducting (solid green) and non-superconducting (red) channels. The solid green curve (right scale) is the BCS DOS calculated with the dispersion shown in (b). The finite zero-energy DOS in the gap is due to the finite energy resolution of the calculation ($\approx 3$ meV). The dashed green curve is the corresponding normal-state DOS. The value $A=0.25$ eV nS is adjusted such that the non-superconducting channel has no coherence peaks at $\pm17$ meV. The dashed blue curve is the sum of the red and dashed green ones. ](figS1){width="\columnwidth"}
Figure \[fig:uniform\](c) shows the base-temperature zero-field tunneling spectrum of Y123 [@Bruer-2016-s] and a possible decomposition in two channels. The superconducting channel (SC) is calculated with the tight-binding model, and the non-superconducting channel (NSC) is the difference. The relative weight of the two contributions is adjusted in such a way that the NSC has no structure—peak or dip—at the edges of the superconducting gap. The resulting NSC has an overall positive slope. The latter is sensitive to the choice of hopping parameters in the SC, and is therefore not a robust feature of the analysis. This slope is irrelevant for our study of vortex cores focusing on energies $\lesssim50$ meV. In contrast, the dips near $\pm50$ meV, the peaks near $\pm30$ meV, and the subgap peaks near $\pm5$ meV are robust properties of the spectrum measured by STS in regions where superconductivity is suppressed [@Bruer-2016-s]. The dashed blue line shows the spectrum expected in such a region where the superconducting gap is closed, assuming that the relative weight of the two channels remains unchanged. This is fully consistent with the spectra measured in non-superconducting regions [@Bruer-2016-s]. A more elaborate modeling was introduced in Ref. , that involved bilayer splitting as well as an interaction with the spin fluctuations. The main effect of these additional ingredients is to assign (part of) the dips at $\pm50$ meV to the SC rather than the NSC. As these energies are not our main concern and these sophistications are impractical in view of large-scale vortex calculations, here we disregard them.
Isolated vortex, self-consistent solution
-----------------------------------------
We solve the Bogoliubov-de Gennes equations self-consistently with a single vortex at the origin using the method described in Ref. . The reader is referred to Ref. for all practical details, while here we only give the elements specific to the present calculation. As our Hamiltonian extends up to third neighbors, it spreads the wave function on the lattice with a diamond shape. We therefore consider a finite system with diamond shape and linear size $M$, having $1+2M(1+M)$ lattice sites. We use $M=200$ (80401 sites) for calculating the self-consistent order parameter and $M=500$ (501001 sites) for calculating the local density of states (LDOS). The order parameter requires a smaller system because the coherence length imposes a spatial cutoff. With $M=200$, a Chebyshev expansion order $N=6000$, and termination using the Jackson kernel [@Berthod-2016-s], the calculation retrieves the exact order parameter within 0.1%. For the LDOS, the spatial cutoff would be set by the mean free path, which is infinite in our model. With $M=500$ and an expansion order $N=2000$, we reach an energy resolution of roughly $3$ meV without perturbations associated with the system’s boundaries.
The self-consistent order parameter is plotted in Fig. \[fig:vortex\](a). The quantity $|\Delta(\vec{r})|$ is defined as the sum of the order-parameter modulus on the four bonds surrounding the site $\vec{r}$. It is well fitted by the isotropic ansatz $\Delta(r)=\Delta_0/[1+\xi_0/r\exp(-r/\xi_1)]$ with $\xi_0=17a$ and $\xi_1=29.5a$. The difference between the exact and approximate data is negative along the $x$ and $y$ directions and positive along the diagonals. The self-consistent order parameter indeed displays a small in-plane anisotropy unlike the ansatz, and relaxes faster to its asymptotic value along the diagonals than along the lattice axes, as already found in similar calculations [@Ichioka-1996-s; @Berthod-2016-s]. A “core size” $\xi_c$ may be defined by the condition $\Delta(\xi_c)=\Delta_0/2$, yielding $\xi_c=11.5a=4.4$ nm. This agrees very well with the BCS expression of the coherence length $\xi=\hbar v_{\mathrm{F}}/(\pi\Delta_0)=4.5$ nm if the Fermi-surface average of the velocity is substituted for $v_{\mathrm{F}}$.
![\[fig:vortex\] (a) Modulus of the self-consistent order parameter for an isolated vortex (blue) on each site of the tight-binding lattice (black). The difference between the self-consistent and ansatz solutions (see text) is shown in orange. The white disk indicates the core size, defined as the distance at which the order parameter is $\Delta_0/2$. Colored balls mark the sites where the LDOS is plotted along (b) the antinodal direction, (c) the nodal direction, and (d) at a fixed distance as a function of angle. The LDOS curves are shifted vertically in (b), (c), and (d) and the color encodes the distance to the vortex center. Note that the curves show the full LDOS $N(\vec{r},E)$ without subtraction. The dashed curves in (d) are calculated using the isotropic ansatz for the order parameter. The insets in (b) and (c) show the spatial distribution of the LDOS (low, white to high, black) at the indicated energies, with the red circle of radius $\xi_c$ indicating the core size. ](figS2){width="0.96\columnwidth"}
The LDOS plotted in Figs. \[fig:vortex\](b), \[fig:vortex\](c), and \[fig:vortex\](d) displays a zero-energy peak that is maximum at the vortex center and decays differently in all directions. The in-plane anisotropy of the LDOS is not a consequence of the in-plane anisotropy of the order parameter, as illustrated in Fig. \[fig:vortex\](d), where it is seen that the typical angular dependence of the LDOS remains unchanged if an isotropic order parameter is used. It is also not a consequence of the $d$-wave gap symmetry: repeating the calculation for an $s$-wave gap leads to the same anisotropic LDOS with an un-split peak along the antinodal directions and a split peak along the nodal ones. Note that the LDOS peak in Fig. \[fig:vortex\] is a genuine continuum, not the superposition of densely packed discrete core levels as in $s$-wave superconductors [@Franz-1998b-s; @Berthod-2016-s]. In fact, the LDOS anisotropy relates to the dispersion and Fermi-surface anisotropies, which tend to favor low-energy LDOS structures in the directions normal to the Fermi surface. The two traces in Figs. \[fig:vortex\](b) and \[fig:vortex\](c) span different distances from the core; Figure \[fig:fig1\](a) of the main text allows one to compare these traces along the same distance. The specific signature of the core for an isolated vortex \[Fig. \[fig:fig1\](b) of the main text\] is obtained by subtracting the zero-field spectrum (green curve in Fig. \[fig:uniform\]) from the vortex LDOS.
Ideal vortex lattices, self-consistent solutions
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![\[fig:lattice-10\] (a) Modulus of the self-consistent order parameter (blue) and difference between the self-consistent and ansatz solutions (orange) for a square vortex lattice with inter-vortex distance $d$ oriented along the principal directions of the microscopic lattice. Two LDOS traces are shown along (b) the line connecting next-nearest neighbor vortices and (c) the line connecting nearest-neighbor vortices. The colors encode the position with respect to the cores as indicated by the balls in (a). The dashed curves, only half of which are shown for clarity, are obtained using the ansatz order parameter instead of the self-consistent one. (d) Spatial distribution of the LDOS at two energies; the red circles of radius $\xi_c$ show the vortex cores. ](figS3){width="\columnwidth"}
![\[fig:lattice-11\] Same as Fig. \[fig:lattice-10\] for a square vortex lattice with inter-vortex distance $d=38\sqrt{2}a$ oriented along the diagonals of the microscopic lattice. Note that the microscopic lattice is rotated by 45$^{\circ}$ in the three-dimensional plot (a), as compared to Fig. \[fig:lattice-10\](a). In all LDOS maps of (d) and Fig. \[fig:lattice-10\](d), however, the microscopic lattice directions correspond to the horizontal and vertical directions of the maps. ](figS4){width="\columnwidth"}
The self-consistent solution for an ideal square vortex lattice with inter-vortex distance $d=54a$ is displayed in Fig. \[fig:lattice-10\](a). This corresponds to a density of 19 vortices on $90\times 90$ nm$^2$ as observed in the experiment (Fig. \[fig:fig2\] of the main text). The vortex lattice is aligned with the microscopic lattice, the nearest-neighbor vortices being found along the $x$ and $y$ directions. The order parameter is maximum in the center of the squares formed by four nearest-neighbor vortices and has saddle points with $|\Delta(\vec{r})|=12$ meV on the lines joining them, leading to a significant spatial modulation. Note that the solution has been constrained to have a maximum gap of 19 meV like in zero field for simplicity—and also because no measurable reduction of the gap size is observed experimentally at this field—requiring a slight increase of the interaction to $V=-260$ meV. $|\Delta(\vec{r})|$ has a rounded shape in the cores, which is captured by the ansatz (generalized for vortex lattices [@Berthod-2016-s]) with an increased value $\xi_0=115a$ and a reduced value $\xi_1=8a$ with respect to the isolated vortex. The core size defined as $\Delta(\xi_c)=\Delta_0/2$ is slightly increased to $\xi_c=15.9a=6$ nm compared with the isolated vortex, consistently with previous studies in the quantum regime [@Berthod-2016-s].
The corresponding LDOS is shown in Figs. \[fig:lattice-10\](b), \[fig:lattice-10\](c), and \[fig:lattice-10\](d). Although the relative difference between the exact and ansatz solutions is 12% at maximum, the LDOS curves calculated with both order parameters are almost undistinguishable. This is the justification for using the non-self-consistent ansatz when studying the LDOS in disordered vortex configurations, for which a full self-consistent calculation is impractical. The zero-energy LDOS peak is considerably suppressed and broadened in the core with respect to the isolated vortex. We have checked that the vortex-lattice calculation correctly reproduces the isolated-vortex limit as the distance $d$ is increased: both spectra differ by less than 5% for $d\gtrsim 170a$ ($B\lesssim 0.5$ T). At the field considered, however, both the spectral and spatial signatures of the vortex cores differ markedly from those of the isolated vortex seen in Fig. \[fig:vortex\]. We have also verified that the suppression of the zero-energy LDOS peak is not due to the increased value of $\xi_c$ and more rounded order parameter in the core: the spectra shown in Fig. \[fig:lattice-10\] remain qualitatively unchanged if we use the ansatz order parameter with the values of $\xi_0$ and $\xi_1$ corresponding to the isolated vortex. The reason for a suppressed vortex-core peak can be understood by comparing the low-energy LDOS in Figs. \[fig:vortex\](b) and \[fig:lattice-10\](d). Because for the isolated vortex the LDOS extends farther along the (10) direction than along the (11) direction, in the vortex lattice the wavefunctions in different cores strongly overlap and the core states get delocalized. This overlap is suppressed when the vortex lattice is not precisely aligned with the (10) direction and/or the vortex positions are disordered, such that the zero-energy LDOS peak is restored in these situations (see below).
For comparison, we show in Fig. \[fig:lattice-11\] the self-consistent order parameter and LDOS for a square vortex lattice rotated 45$^{\circ}$ with respect to the tight-binding lattice with an inter-vortex distance $d=38\sqrt{2}a$, which corresponds to the same field as in Fig. \[fig:lattice-10\]. There are significant differences between the two vortex-lattice orientations (hereafter I and II), both in the self-consistent order parameter and in the LDOS. While in I the gap has saddle points between nearest-neighbor vortices and maxima between next-nearest-neighbor ones, the situation is reversed in II with the gap maxima between nearest-neighbor vortices. It appears that the order parameter doesn’t move rigidly with the vortex lattice: when rotating the vortex-lattice orientation from I to II, the cores move but the gap maxima and saddle points stay in place. The shape of the core is also quite different in I and II, where a best fit to the ansatz gives $\xi_0=9.9a$, $\xi_1=15a$, and a core size $\xi_c=6.4a=2.5$ nm smaller than in zero field. There is more structure in the case II, because each saddle point is in fact replaced by two saddle points separated by a local minimum. This explains the larger discrepancy between the ansatz and the exact solution, which reaches 32% for II at the local minima. As a result, the differences between the LDOS calculated with the ansatz and self-consistent order parameters are slightly larger in II than in I. These differences remain nevertheless small compared with the qualitative differences between the LDOS in I and II: the zero-energy peak is neither strongly suppressed nor split in II as it is in I; at the position of the local minimum between two saddle points in II, the LDOS has a double peak at zero energy, while at the saddle point in I the LDOS is gapped. We see in Fig. \[fig:lattice-11\](d) that the low-energy LDOS is much more localized in the cores compared with I, which highlights the much weaker wavefunction hybridization along the (10) directions in case II.
The data in Figs. \[fig:vortex\], \[fig:lattice-10\], and \[fig:lattice-11\] demonstrate that the vortex-core LDOS is not only a function of field, but also and more importantly a function of the positions of neighboring vortices. Depending on where the neighbors are, the zero-energy LDOS peak may be sharp or not, split or not, etc. Experimentally, one therefore expects variability in the measured vortex-core spectra when the vortex positions are disordered. These figures also show that, in order to investigate theoretically this variability, it is sufficient to work with the ansatz order parameter, whose only inputs are the vortex positions and the values of $\xi_0$ and $\xi_1$.
LDOS calculations for disordered vortex lattices
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Disordered vortex configurations
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![\[fig:disorder\] (a) Typical disordered vortex configuration. The central square represents the STS field of view of $90\times90~\mathrm{nm}^2$, where the vortices are located as observed in the experiment. The square is rotated by 5.7$^{\circ}$ with respect to the crystal axes. The dotted red square indicates the system size used for calculating the LDOS at the point marked by a cross; as the cross moves, the dotted square moves with it. (b) Isotropic structure factor showing the absence of orientational order in the generated vortex positions. (c) Angular average of the structure factor. The power-law behavior for $k<2\pi/d$ is shown in red. ](figS5){width="\columnwidth"}
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The high sensitivity of the theoretical LDOS to vortex positions prompts us, for a meaningful comparison with the STS experiment, to use in the calculation the vortex positions as they appear under the STM. Three difficulties arise: (i) the LDOS maxima which are accessible experimentally may not sit exactly on the phase singularity points where the order parameter vanishes, due to a polarization of the LDOS by asymmetric supercurrents [@Berthod-2013a-s; @Berthod-2016-s]; (ii) we cannot disregard vortices that are outside the STS field of view although we don’t know their positions; and (iii) due to lack of atomic resolution on the Y123 surface, the orientation of the microscopic lattice is only known approximately via the macroscopic twin boundaries. We ignore (i), expected to be a small effect, and locate the vortices inside the STS field of view at the positions of largest $dI/dV$ contrast \[see Fig. \[fig:fig4\](a) of the main text\]. Outside the field of view, we generate vortex positions with the same density as inside, which corresponds to a field of 4.85 T. The positions are chosen at random, however, with a hard-core repulsion constraining the inter-vortex distance to be at least $d_0=41a$. This value was selected to be as large as possible: for larger values the random generation process would be stuck, not able to fit in the required number of vortices. The resulting vortex configurations show some degree of order, similar to what is seen in the STS field of view. An example is shown in Fig. \[fig:disorder\]. The structure factor $S(\vec{k})=\mathscr{N}^{-1}\left|\sum_ne^{-i\vec{k}\cdot\vec{R}_n}\right|^2$, where $\vec{R}_n$ are the positions of the $\mathscr{N}$ vortices, is isotropic indicating no orientational order. The angular average $\bar{S}(k)=\mathscr{N}^{-1}\sum_{nm}J_0(k|\vec{R}_n-\vec{R}_m|)$, where $J_0$ is the Bessel function, shows oscillations of wavevector $2\pi/d_0$ due to the hard-core repulsion. Furthermore, a power-law suppression of $\bar{S}(k)$ is observed for $k<2\pi/d$, where $d=54a$ is the inter-vortex distance in the equivalent ordered square lattice. Such power law is reminiscent of hyperuniformity, i.e., a type of order characterized by the suppression of density fluctuations at long wavelengths [@Torquato-2003-s], where $\bar{S}(k)\sim k^{2-\eta}$ with $0<\eta\leqslant 2$ in two dimensions. It is likely that in reality the vortices outside the field of view present more order than the configurations generated by our procedure [@Maggio-Aprile-1995-s], but the experimentally available vortex positions are not sufficient for inferring such an order. While certain characteristics of the vortex ordering outside the field of view may influence the LDOS inside, we do not expect this to change any of the conclusions we draw from our analysis.
Search for a good configuration of vortices
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{width="70.00000%"}
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We have generated 600 disordered vortex configurations for various orientations of the microscopic lattice with respect to the STS field of view. For each configuration, we generate the order parameter using the ansatz and the values of $\xi_0$ and $\xi_1$ corresponding to Fig. \[fig:lattice-10\], and we calculate the LDOS along the four paths displayed in Fig. \[fig:optimization\](a), that correspond to paths n^o^6 and n^o^8 in Fig. \[fig:fig4\] of the main text. These four traces share a common point \[the cross in Fig. \[fig:disorder\](a)\]: we use the LDOS at this point as the reference spectrum and subtract it from the calculated LDOS along the four paths. The same procedure applied to the experimental data \[Fig. \[fig:optimization\](b)\], allows us to compute a sum of squared differences as the figure of merit for each vortex configuration. We find that the agreement between measurement and simulation is systematically better when the four paths are close to a nodal direction. The best compromise is reached if the STS field of view is rotated by 5.7$^{\circ}$ relative to the microscopic lattice as shown in Fig. \[fig:disorder\](a). This points to a tendency for the nearest-neighbor vortices to align along the crystal axes, and justifies a posteriori our use of the values of $\xi_0$ and $\xi_1$ corresponding to that orientation rather than that of Fig. \[fig:lattice-11\]. We emphasize again that the precise values of $\xi_0$ and $\xi_1$ play very little role in the LDOS. Figure \[fig:disorder\](a) is the best among the 600 configurations; the four traces are compared with the experimental ones in Fig. \[fig:optimization\]. Using this configuration, we calculate the LDOS in the whole STS field of view with 1 nm resolution. For comparing with Fig. \[fig:fig4\](a) of the main text, we compute $dI/dV$ by adding to our theoretical LDOS the non-superconducting channel using the formula quoted in Fig. \[fig:uniform\], we evaluate the ratio between the calculated $dI/dV$ at 5 and 17 meV, and thus obtain the map and traces shown in Fig. \[fig:optimization\](a) and Fig. \[fig:fig4\](b) of the main text.
Figure \[fig:statistics\] displays three vortex configurations different from the best one shown in Fig. \[fig:disorder\](a) and \[fig:optimization\], and the corresponding theoretical spectral traces. These configurations agree reasonably with experiment as well, with a figure of merit within the best 10% out of the 600 considered. One notices, in particular, four spectroscopically very different cores with configuration (a), a reinforcement of the LDOS at intermediate distance in (b), trace $\alpha'$, similar to what is seen in Fig. \[fig:fig2\](c) of the main text, and split spectra at the center of vortices $\alpha'$ and $\delta'$ for configuration (c), as seen in the experiment \[Fig. \[fig:optimization\](b)\].
Mixed-state LDOS and gap symmetry
---------------------------------
It is tempting to search signatures of the $d_{x^2-y^2}$ symmetry of the order parameter in the LDOS around vortices. In the semiclassical regime $k_{\mathrm{F}}\xi\gg1$, the vortices are slowly varying perturbations of the order parameter compared with atomic distances, and their Fourier components are mostly at low momenta. Therefore, the vortices provide only small momentum transfers and the interaction of the Bogoliubov quasiparticles with the vortex lattice is dominated by forward scattering. In that limit, the details of the Fermi surface are irrelevant and the only source of spatial LDOS anisotropy—apart from the vortex lattice itself—is indeed the order-parameter symmetry. In the quantum regime $k_{\mathrm{F}}\xi\sim1$ relevant for Y123, however, the vortex lattice scatters Bogoliubov quasiparticles with large momentum transfers of the order of $k_{\mathrm{F}}$ and the LDOS is therefore sensitive to the anisotropy of the Fermi surface. This situation generically leads to the development of LDOS structures along the directions normal to the Fermi surface, a fact well known from studies of impurity scattering [@Weismann-2009-s]. In Y123, the Fermi surface segments are mainly oriented along the crystallographic directions \[see Fig. \[fig:uniform\](a)\], such that one expects structure in the vortex-lattice LDOS along the (10) and (01) lattice directions, irrespective of the order-parameter symmetry. This is confirmed by our numerical results shown in Fig. \[fig:swave\]: for both $d$- and $s$-wave symmetries, the LDOS structures are aligned with the microscopic lattice at all energies. No structure is observed along the (11) and equivalent directions, which are the directions of the gap nodes in reciprocal space. Thus the expectation that the LDOS would “leak” out of the vortices along the directions of the gap nodes is not confirmed. There are nevertheless differences between the LDOS calculated for $d_{x^2-y^2}$ and $s$ symmetries. The vortex states are more localized in the $s$-wave case, leading to a better contrast, especially at low energy. However, according to these simulations, an unambiguous determination of the order-parameter symmetry based on experimental LDOS maps around vortices appears to be hopeless.
[12]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty **, [****, ()](\doibase 10.1103/PhysRevB.41.1863) **, [****, ()](\doibase 10.1088/0953-2048/22/1/014005) **, [****, ()](\doibase 10.1103/PhysRevB.57.6090) **, [****, ()](\doibase 10.1103/PhysRevB.57.6107) **, [****, ()](\doibase 10.1038/ncomms11139) **, [****, ()](\doibase 10.1103/PhysRevB.94.184510) **, [****, ()](\doibase 10.1103/PhysRevB.53.15316) **, [****, ()](\doibase 10.1103/PhysRevLett.80.4763) **, [****, ()](\doibase 10.1103/PhysRevB.88.134515) **, [****, ()](\doibase 10.1103/PhysRevE.68.041113) **, [****, ()](\doibase 10.1103/PhysRevLett.75.2754) **, [****, ()](\doibase 10.1126/science.1168738)
[^1]: See Supplemental Material \[url\] for a presentation of the model and calculation methods, which includes Refs. .
[^2]: At an optimal hole doping of 0.16, the nominal carrier density of the bilayer Y123 is $n=2\times0.84/(abc)=9.7~\mathrm{nm}^{-3}$ with the unit-cell parameters $(a,b,c)=(3.82,3.89,11.68)$ Å. With an effective mass of typically 2–3 times the electron mass [@Padilla-2005] and a zero-temperature penetration depth $\lambda=1600$ Å [@Basov-1995], the estimated superfluid density $n_s=m^*/(\mu_0e^2\lambda^2)$ is 2.2–3.3 nm$^{-3}$.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'I define the proton-contributor reference frame in proton nucleus (p–A) collisions as the center of mass of the system formed by the proton and the participant nucleons of the nucleus. Assuming that the rapidity distribution of produced particles is symmetric in the proton-contributor reference frame, several measurements in p–Pb collisions at $\sqrt{s_{\rm NN}} = 5.02~{\rm TeV}$ can be described qualitatively. These include rapidity distributions of charged particles, $J/\psi$ and Z bosons.'
author:
- 'Ginés Martínez-García'
bibliography:
- 'ReferenceInpA.bib'
title: 'Particle rapidity distribution in proton-nucleus collisions using the proton-contributor reference frame'
---
Introduction
============
In proton-proton (pp) and nucleus-nucleus (A–A) collisions, the colliding system defines unambiguously the reference frame in which the rapidity distribution of particles produced in the collisions must be symmetric. In proton-nucleus (p–A) collisions, the situation is more complex since this rapidity distribution is not expected to be necessarily symmetric. Therefore we are used to consider the nucleon-nucleon center-of-mass frame, as for pp and A–A collisions. This is justified for high-energy p–A collisions which can be seen as a multiple interaction of parton pairs, one parton of the pair belonging to the proton and the other belonging to one of the nucleons of the nucleus. I propose hereafter an alternative reference frame, called the [*proton-contributor*]{} reference frame. The main assumption is that particles are produced in p–A collisions with a rapidity distribution identical to that in pp collisions, but shifted by the rapidity gap between the proton-nucleon and the proton-contributor reference frame. Under this simple hypothesis the centrality dependence of charged particle pseudo-rapidity distribution measured by the ATLAS collaboration, the suppression (enhancement) of the $J/\psi$ at forward (backward) rapidity measured by the ALICE collaboration, and the Z boson backward-to-forward ratio measured by the CMS collaboration, are qualitatively understood.
Time scale of the p–A collision
===============================
Many of the numerous theoretical models aiming at describing heavy-ion or proton-nucleus collisions at RHIC and LHC energies assume that, at sufficiently low transferred momentum, the interaction takes place coherently with all the partons of the nuclei or nucleus. Such a coherent interaction will occur when the crossing time of the projectile and the target is smaller than the formation time of a given process. Let us consider the production of a probe involving a momentum transfer $Q$. The formation time of the probe can be estimated as $$\tau_f \approx Q^{-1}.$$ In a proton-nucleus collision, the crossing-time in the reference of the probe (centre of mass frame of the parton-parton interaction in 2$\rightarrow$1 processes) can be estimated as $$\tau_c \approx R/\gamma_R$$ where R is the radius of the nucleus and $\gamma_R$ is the nucleus Lorentz factor in the probe reference frame. $\gamma_R$ can be expressed as $$\gamma_R = \cosh{(y-y_A)}$$ where $y_A$ is the rapidity of the nucleus and $y$ the rapidity of the probe. In p–Pb collisions at 5.02 TeV and for $y=0$, the crossing time $\tau_c$ is smaller than $\tau_f$ for $Q \lesssim70$ GeV. Fig.\[fig:fig1\] shows the formation times for J/$\psi$, $c\bar{c}$ pair and Z particles, compared to the crossing time in p–Pb collisions at 5.02 TeV as a function of the probe rapidity in the LHC reference frame.
The proton-contributor reference frame in p–A collisions
========================================================
We are used to consider the proton-nucleon reference frame to study the production of a probe in p–A collisions, namely when $\tau_c \gg \tau_f$. Indeed, the probe can be viewed as produced in a single collision of the proton with one of the nucleons of the nucleus. However, we have seen in the previous section that most of the time $\tau_c \leq \tau_f$ at LHC energies. Therefore, the whole volume of the nucleus crossed by the proton (a cylinder of about $\sqrt{\sigma_{\rm NN}/\pi}\approx$1.5 fm radius, that I am calling *contributor* in this paper) will coherently contribute to the production of the probe. In addition, the Bjorken-$x$ ($x_{Bj}$) values of the partons involved in the hard collision are small ($x_{Bj}\leq 2 \cdot 10 ^{-2}$ for Q$\leq 100$ GeV/$c$ at y=0 and $\sqrt{s}$=5 TeV). One could wonder whether the belonging of a small $x_{Bj}$ parton to a given nucleon is not *blurred* by the presence of other nucleons contributing to the collision. This is the *main* physics motivation[^1] to make the extreme hypothesis that particles in p–A collisions at the LHC are produced with a rapidity differential cross section which is symmetric in the proton-contributor reference frame with a similar shape as in pp collisions: $$\label{Eq:hypo}
\frac{{\rm d}N^{\rm probe}_{\rm pA(Ap)}}{{\rm d}y} \Bigl( y \Bigr) = {\cal N} \frac{{\rm d}N^{\rm probe}_{\rm pp}}{{\rm d}y} \Bigl( y -(+) \Delta y_{\rm pN-pC} \Bigr)$$ where ${\rm d}N/{\rm d}y$ are the probe yields, ${\cal N}$ is a normalisation parameter, and $\Delta y_{\rm pN-pC}$ is the rapidity gap between the proton-nucleon and proton-contributor frames, which is defined below.
The mass and momentum of the contributor can be obtained using the Glauber model: $$m_{\rm C} = N_{\rm coll}(b) \times m_{\rm N}$$ $$P_{\rm C} = N_{\rm coll}(b) \times P_{\rm Pb}$$ where $m_{\rm N}$ is the mass of the nucleon (here 931 MeV/c$^2$), $P_{\rm Pb}$ is the momentum per nucleon of the Pb LHC beam and $b$ the impact parameter.
The total energy of the proton-contributor system is given by: $$E_{\rm pC} = \sqrt{P_p^2 + m_p^2} + \sqrt{P_{\rm C}^2 + m_{\rm C}^2}$$ where $P_{\rm p}$ is the momentum of the LHC proton beam, and $m_{\rm p}$ is its mass (938 MeV/c$^2$). The total momentum (positive value in the direction of the proton beam) is $$P_{\rm pC} = P_{\rm p} - P_{\rm C}.$$ Finally, the rapidity of the proton-contributor in the laboratory frame is given by $$y_{\rm pC} = \tanh^{-1}{(p_{\rm pC} /E_{\rm pC})}$$
Assuming $N_{\rm coll}$=1, the rapidity becomes $y_{\rm pC}$= 0.465, which is equal to the rapidity of the proton-nucleon frame. In minimum bias p–Pb collisions at 5.02 TeV, the average number of collisions in the nucleonic tube is $\approx$6, therefore the rapidity of the proton-contributor system is $y_{pC}$=-0.430. The rapidity gap between the proton-nucleon and proton-contributor is close to one unit of rapidity, $\Delta y _{\rm pN-pC}$=0.896. For the most central p–A collisions ($N_{\rm coll}=17$), the rapidity is $y_{\rm pC}$=-0.951.
Centrality dependence of the charged particle pseudorapidity distribution in p-Pb collisions at 5.02 TeV
========================================================================================================
Let us assume that the charged particle rapidity density ${\rm d}N^{ch}_{pp}/{\rm d}y$ exhibits a Gaussian shape with a width $\sigma$ of 3.2 rapidity units[^2]. The charged particle rapidity density as a function of the centrality in p–A collisions $dN^{ch}_{pA}/dy$ can be obtained applying Eq.\[Eq:hypo\].
The pseudo-rapidity density is then given by: $$\label{eq:eqdNchdeta}
\frac{{\rm d}N_{ch}}{{\rm d}\eta} = \frac{{\rm d}N_{ch}}{{\rm d}y} \times \frac{{\rm d}y}{{\rm d}\eta}$$ where $$\theta = 2 \cdot \arctan{(e^{-\eta})}$$ $$m_{\rm T} = \sqrt{p_{\rm T}^2+m^2}$$ $$p_{\rm z} = \frac{p_{\rm T}}{\tan{\theta}}$$ and $$y = \sinh^{-1} {\Biggl( \frac{p_{\rm z}}{m_{\rm T}} \Biggr)}$$
The Jacobian depends on the particle mass and transverse momentum. For simplicity, I have assumed a mean charged particle mass of 450 MeV/$c^2$ and a mean transverse momentum of 700 MeV/$c$ [^3]. The charged particle pseudo-rapidity densities as obtained from Eq.\[eq:eqdNchdeta\] for a Gaussian rapidity distribution, are plotted in Fig.\[fig:fig2\] top. The ${\rm d}N^{ch}_{\rm pA}/{\rm d}y$ distributions are normalized (${\cal N}$ parameter) to the charged particle pseudo-rapidity density at $\eta = 0$ measured by the ATLAS collaboration [@ATLASQM2014:2014]. We observe that the shape of ${\rm d}N_{ch}/{\rm d}\eta$ becomes progressively more asymmetric in the Pb-going direction, in accordance with the increase of the contributor size and therefore the increase of the rapidity shift between proton-contributor frame in more central collisions and the proton-contributor frame in the peripheral 60%-90% centrality bin. In Fig.\[fig:fig2\] bottom, the ratio of the pseudo-rapidity densities with respect to that in 60%-90% centrality bin is presented. The double peak structure present in the distributions in Fig.\[fig:fig2\] top disappears in the ratios. The ratios are observed to grow nearly linearly with pseudo-rapidity, and the slope increases from peripheral to central collisions.
The ATLAS collaboration presented at the Quark Matter Conference in Darmstadt, the charged particle pseudo-rapidity distribution in p–Pb collisions at 5.02 TeV as a function of the collision centrality [@ATLASQM2014:2014; @DebbeVelasco:2014]. A linear dependence of the charged particle pseudo-rapidity ratios with a slope increasing from peripheral to central p–Pb collisions, is observed (see Fig.8 in [@ATLASQM2014:2014]), qualitatively agreeing the predictions presented in Fig.\[fig:fig2\] bottom, obtained with the hypothesis in Eq.\[Eq:hypo\].
Centrality dependence of the $J/\psi$ production in p–Pb collisions
===================================================================
Recently, the ALICE collaboration has published results on $J/\psi$ production and nuclear effects in p–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV [@Abelev:2013yxa]. Let us assume a J/$\psi$ rapidity distribution according to the phenomenological parameterization introduced in [@Bossu:2011qe]: $$\frac{{\rm d}\sigma}{{\rm d}y} \, \,/ \,\, \frac{{\rm d}\sigma}{{\rm d}y} \Bigr|_{y=0} = e^{ -(y / y_{\rm max})^2/2\sigma_y^2}$$ where $y_{\rm max}=\ln{(\sqrt{s/}m_{{\rm J}/\psi} ) }$ and $\sigma$=0.38 for pp collisions. In p–A collisions, I am assuming that the rapidity distribution is shifted by the rapidity gap between the proton-nucleon and the proton-contributor frames following Eq.\[Eq:hypo\]. The J/$\psi$ nuclear modification factor is then obtained from the ratio of the two Gaussian distributions. The normalization factor ${\cal N}$ is determined assuming binary scaling and an additional *shadowing-like* factor of 0.85[^4]. This is shown in Fig.3 together with the ALICE measurements. As it can be seen, the simple assumption of a $J/\psi$ rapidity distribution in p–A collisions shifted with respected to that in pp collisions allows to describe the observed $J/\psi$ suppression (enhancement) at forward (backward) rapidity. The rapidity gap between the proton-nucleon and proton-contributor frames, would explain the observed $J/\psi$ suppression at forward rapidity and the enhancement at backward rapidities. As shown in Fig.\[fig:fig3\], the previous pattern is enhanced in central p–Pb collisions since the contributor size increases, thus increasing the rapidity gap. This is in qualitative agreement with results from the ALICE collaboration showing that the J/$\psi$ nuclear modification factor decreases with centrality at forward rapidity, while it increases with centrality at forward rapidity [@MartinBlanco:2014; @Lakomov:2014].
Production of Z in p–A collisions at 5.02 TeV
=============================================
Recently, the ATLAS and CMS collaboration have reported the measurement of the Z differential cross section in p–Pb collisions at 5.02 TeV [@ZATLASQM2014:2014; @ZCMSQM2014:2014; @Cole:2014; @Granier:2014]. ATLAS claimed that the rapidity differential cross section shows a significant asymmetry compared to the simple model based on binary scaling with respect to nucleon-nucleon collisions. Indeed, a relative excess in the Z differential cross section is seen in the backward (Pb-going) part of the rapidity distribution [@ZATLASQM2014:2014]. CMS interpreted such an asymmetry as a consequence of the modification of the parton distribution functions (PDF) in the nucleus and claimed that this measurement is providing new data points in a previously unexplored region of phase space for constraining nuclear PDF fits [@ZCMSQM2014:2014]. Furthermore, ATLAS observed a more pronounced asymmetry in central events, while the asymmetry is apparently absent in peripheral events.
Assuming Z production with a rapidity distribution symmetric in the proton-contributor reference frame (see Eq.\[Eq:hypo\]) can provide a phenomenological explanation to the observation made by the ATLAS and CMS collaborations. As it was quoted above, the rapidity of the proton-contributor reference frame for p–Pb ($N_{coll}\approx$6) is -0.430 in the LHC reference frame. It is, indeed, observed (see Fig.4 of [@ZATLASQM2014:2014]) that the measured Z rapidity distribution exhibits a maximum around this value. Hence, the Z rapidity distributions for 0%-10% ($N_{\rm coll}\approx 14$), 10%-40% ($N_{\rm coll}\approx 10$) and 40%-90% ($N_{\rm coll}\approx 4$) should be centred at $y^{\rm Z}$ equal to -0.85, -0.69 and -0.23, thus qualitatively agreeing with the experimental observations (see Fig.9 of [@ZATLASQM2014:2014]). At Quark Matter Conference, the CMS collaboration presented the backward-to-forward ratio of the Z rapidity distribution in the proton-nucleon centre of mass [@Granier:2014; @Granier:2014c]. A Gaussian rapidity distribution with a width equal to 3 (solid line of Fig.\[fig:fig4\] top), can be considered to model the Powheg-Pythia predictions presented in Fig.2 of reference [@Granier:2014b; @ZCMSQM2014:2014]. Assuming that the Z rapidity distribution in p–Pb collisions has the same shape but is centred at the rapidity of the proton-contributor reference frame ($y_{\rm pC}-y_{\rm pN}$ ), the dashed curve plotted in Fig.\[fig:fig4\] top is obtained. The backward-to-forward ratio can then been easily calculated as the ratio of the dashed and solid curve in Fig.\[fig:fig4\] top and it is plotted in Fig.\[fig:fig4\] bottom. This prediction is in agreement with the backward-to-forward ratio measured by the CMS collaboration [@ZCMSQM2014:2014].
Summary and outlooks
====================
I have defined the proton-contributor the proton-contributor system in p–A collisions as the system formed by the proton and the nucleons of the nucleus participating to the collisions, which is determined with the Glauber model. Assuming that the particle rapidity distribution is identical to that in pp collisions but centred at the rapidity of the proton-contributor system,
- the pattern of the pseudo-rapidty distribution ratios of charged particles as a function of the collision centrality in p–Pb at 5.02 TeV, measured by the ATLAS collaboration [@ATLASQM2014:2014; @DebbeVelasco:2014],
- the nuclear modification of J/$\psi$ at forward and backward rapidity in p–Pb collisions at 5.02 TeV measured by the ALICE collaboration [@Abelev:2013yxa; @Lakomov:2014], and
- the backward-to-forward ratio of Z bosons in p–Pb collisions at 5.02 TeV measured by the CMS collaboration [@ZCMSQM2014:2014; @Granier:2014]
can be understood.
This phenomenological observation might trigger new theoretical ideas on the physics underlying the present hypothesis of the proton-contributor frame. Seemingly, this hypothesis could be also applied to RHIC results in d-Au collisions and to other observables in p–Pb collisions, like $\Upsilon$ or W production. Finally, one could imagine new experimental observables taking into account the rapidity gap $\Delta y _{\rm pN-pC}$ between the proton-nucleon and the proton-contributor systems, like the the backward-to-forward ratio in the proton-contributor frame or a proton-contributor nuclear modification factor that would be defined as: $$R^{pC}_{\rm pA}(y)= \frac{Y_{\rm pA} (y)}{\langle N_{\rm coll} \rangle Y_{\rm pp}(y-\Delta y _{\rm pN-pC})}$$
This is a preliminary draft and comments and suggestions are welcome. I plan to make an oral presentation during the [*Rencontres QGP-France* September 15th-18th 2014 in Etretat](http://llr.in2p3.fr/sites/qgp2014/index.php).
acknowledgements
================
I would like to thank Philippe Crochet and Stéphane Peigné for the fruitful and inspiring discussions, Raphael Granier de Cassagnac and Anna Zsigmond for providing the Powheg-Pythia numerical values from [@ZCMSQM2014:2014], and Philippe Crochet and Ombretta Pinazza for the careful reading of the manuscript.
[^1]: I agree that this physics motivation is weak, as my colleague Stéhane Peigné already told me. Indeed, the main motivation to formulate the proton-contributor hypothesis is the successful explanation of several different phenomenological observations in p–A collisions at LHC energy as it is discussed in the present draft.
[^2]: The $\sigma$ parameter of the charged particle rapidity Gaussian distribution has been chosen arbitrarily to reproduce with ATLAS measurement of the charged particle pseudo-rapidity distribution ratios [@ATLASQM2014:2014].
[^3]: These values represent a first approximation, which could be improved considering realistic particle ratios and realistic particle transverse momentum distributions.
[^4]: The only motivation of this shadowing-like factor is to fit the experimental data. Note that shadowing is expected to be almost constant as a function of rapidity and the 0.85 shadowing factor agrees very well with the predictions [@Abelev:2013yxa].
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"pile_set_name": "ArXiv"
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---
abstract: 'This paper is a sequel to arXiv:1309.0785 were we computed the Weyl anomaly $a$ (Euler density or logarithmic divergence on $S^d$) coefficient for higher-derivative conformal higher spin field in $d=4$ and shown that it matches the expression found in arXiv:1306.5242 by a “holographic” method from a ratio of massless higher spin determinants in AdS$_5$. Here we repeat the same computation in on 6-sphere and demonstrate that the result matches again the one following from AdS$_7$. We also discuss explicitly similar matching in the $d=2$ case.'
---
[ A.A. Tseytlin[^1] ]{} 0.10cm
[ *0.08cm 0.08cm Blackett Laboratory, Imperial College, London SW7 2AZ, U.K.* ]{}
0.10cm
Introduction
============
This paper continues the investigation of quantum conformal higher spin (CHS) models with higher-derivative flat-space action $\int d^d x \, \p_s P_s \del^{2s + d-4} \p_s$ ($P_s$ is transverse traceless symmetric rank $s$ tensor projector). Generalising this action to curved background is a highly non-trivial problem, but as was argued in , at least in the case of a conformally flat Einstein background (i.e. (A)dS$_d$ or $S^d$), the corresponding Weyl-covariant $2s + d-4$ derivative kinetic operator should factorize into product of standard 2nd-derivative operators.
Explicitly, the partition function of a conformal higher spin $s$ field on a $d$-dimensional sphere of unit radius can be written as && Z\_[s]{} ([S\^d]{}) =\_[k=0]{}\^[s-1]{} ([ \_[k ]{} \_[s]{}]{})\^[1/2]{}\
&& \^[-1]{}\_[k’=-[12]{}([ d-4 ]{} )]{} ([1 \_[s]{}]{})\^[1/2]{} , or, equivalently, as && Z\_[s]{} ([S\^d]{}) =\_[k=0]{}\^[s-1]{} ( \_[k ]{} )\^[1/2]{}\
&& \^[s-1]{} \_[k’=[-[12]{}([ d-4 ]{} )]{} ]{} ( \_[s]{} )\^[-1/2]{} , where the 2nd-order differential operator $ (-\nabla^2 + M^2)_{k\pe}$ is defined on transverse traceless symmetric rank $k$ tensors. The first line in [(\[0\])]{} is the contribution of the “partially-massless” modes (with residual gauge invariance and thus “ghost” numerators) while the second corresponds to extra “massive” modes present for $d \not= 4$ (see and refs. there).
This representation allows one to compute the CHS partition function on $S^d$ using standard (e.g., $\zeta$-function) techniques, and, in particular, to find the coefficient of the logarithmic UV divergence or the $\aa$-coefficient of the Euler density term in the corresponding Weyl anomaly. Remarkably, the arguments in suggest that $ Z_{s} ({S^d}) $ in [(\[0\])]{} should have also a “holographic” representation in terms of the ratio of determinants of the standard (second-derivative) massless higher spin $s$ operators with alternate boundary conditions in euclidean AdS$_{d+1}$: && [Z\^[(-)]{}\_[s0]{} ([\_[d+1]{}]{}) Z\^[(+)]{}\_[s0]{} ([\_[d+1]{}]{}) ]{} = Z\_s([S\^d]{}) ,\
&& Z\_[s0]{} ([\_[d+1]{}]{}) = ([ \_[s-1 ]{} \_[s]{}]{})\^[1/2]{} . Here AdS$_{d+1}$ and its boundary $S^d$ are assumed to have unit radius. The subscripts $\pm$ indicate the different boundary conditions.
Let us note that while motivated by the AdS/CFT , the relation [(\[02\])]{} is essentially “kinematical” in nature (i.e. it does not rely on any non-renormalization and should be true for any $d$) belonging to a class of bulk-boundary determinant relations like the one discussed in . One should thus be able to prove it by starting from the one-loop path integral in AdS$_{d+1}$ and “integrating out” the values of the fields in the interior points of AdS$_{d+1}$. As in the scalar case one should pay special attention to regularization. Indeed, the AdS$_{d+1}$ side of [(\[02\])]{} is IR divergent while the $S^d$ side is UV divergent. The logarithm of partition function $Z_{s0}$ on AdS$_{d+1}$ is proportional to its volume which for even $d$ has the following regularized value (we shall keep track of logarithmic divergences only): (AdS\_[d+1]{}) = [2 (-1)\^[d2]{} \^[d2]{} ( [ d+1 2]{}) ]{} L + ... . where $L\to \infty$ is IR cutoff. The free energy on $S^d$ of radius $\rr$ has the following structure && F=-Z = (-\^2 + M\^2) = -B\_d (L ) +... ,\
&& B\_d =d\^d x g b\_d = (S\^d) b\_d , (S\^d) = [ 2 \^[d+12]{} ([ d+12]{})]{} , where $B_d$ is the integrated Seeley coefficient (often called also $a_{d/2}$) of the operator $-\nabla^2 + M^2$ and $L\to \infty $ is UV (heat kernel) cutoff. In the case when the classical theory is conformally invariant $B_d$ represents the integrated Weyl anomaly (see and refs. there). The total coefficient of $\ln L$ term in $\ln Z_s$ can be found by summing the $B_d$-coefficients for the operators in [(\[01\])]{}.
Identifying the IR cutoff in the AdS$_{d+1}$ bulk and the UV cutoff at the $S^d$ boundary the first check of [(\[02\])]{} is the matching of the coefficients of the $\ln L$ terms. Following let us call $a_s$ the coefficient of the IR singular term in the AdS$_{d+1}$ free energy in [(\[02\])]{}. Comparing to the $S^d$ expression [(\[05\])]{} we should get B\_d\^[(s)]{} = - a\_s . Equivalently, $a_s$ should be the coefficient of the $\ln \rr$ term in free energy on $S^d$. In the case of $d=4$ the coefficient of the IR divergent term in the l.h.s. of [(\[02\])]{} was found in to be a\_[s]{} = \_s\^2 ( 14 \_s + 3 ) , \_s = s (s+1) . The same expression was also obtained directly from the spin $s$ CHS partition function [(\[0\])]{} on $S^4$ as (minus) the value of the total Weyl anomaly coefficient $B_4^{(s)}$ &&b\_4 = - å\_s R\^\*R\^\* = - 24 å\_s , B\_4\^[(s)]{} = [ 1 (4)\^2]{} [ 8 \^2 3]{} b\_4 = - 4 å\_s = - a\_s ,\
&& B\_4\^[(s)]{} = - a\_s = - [ s\^2 (s+1)\^2 180]{} ( 14 s\^2 + 14 s + 3 ) . Our aim here will be to perform a further non-trivial test of the relation [(\[02\])]{} by considering the $d=6$ case (and also the $d=2$ case, see Appendix). The case of $d=6$ is of interest in view of the AdS$_7$/CFT$_6$ duality and also because the structure of the CHS partition function [(\[0\])]{} changes for $d\not=4$. We shall first consider the r.h.s. of [(\[02\])]{}, i.e. find the coefficient $B_6$ [(\[06\])]{} of logarithmically divergent term in $F=-\ln Z_s$ in [(\[01\])]{} on $S^6$.
In general, the local Weyl anomaly coefficient has the following structure in $d=6$ b\_6 = åE\_6 + \_[i=1]{}\^3 c\_i I\_i + \_m J\^m , E\_6= - \_6 \_6 RRR , where $I_1\sim C ( \nabla^2 + ...) C, \ I_{2,3} \sim CCC $ contain powers of the Weyl tensor $C$. Then for a unit-radius sphere $S^6$ b\_6 (S\^6)= åE\_6 = - [ 8!7]{} å , B\_6(S\^6) = b\_6 = [ 1 60]{} b\_6 = - 96 å- a . For a conformally coupled scalar ($\dn=-\nabla^2 + { d-2 \ov 4 (d-1)} R$) å\_0= - [5 9!]{} , B\^[(0)]{}\_6 = -a\_0 = [ 1 756]{} . As we shall find below, for a conformal higher spin field in $d=6$ the total value of $B_6$ corresponding to [(\[01\])]{} (generalizing [(\[002\])]{} to any $s \geq 0$) is && B\^[(s)]{}\_6 =- a\_s = - (22 s\^6+198 s\^5+671 s\^4+1056 s\^3+733 s\^2+120 s-50) \
&& = - \_s (88 [\^[3/2]{}\_s]{}-110 \_s-4 \^[1/2]{}\_s +1) , \_s = (s+1)\^2 (s+2)\^2 . Like in the $d=4$ expression [(\[07\])]{} here $\nu_s$ stands for the number of dynamical degrees of a spin $s$ CHS field in $d=6$. Specialising the general expression for the coefficient of the IR divergent part of the AdS$_{d+1}$ side of [(\[02\])]{} to the case of $d=6$ we will also show that it indeed matches [(\[111\])]{} according to [(\[uh\])]{}.
We shall start in section 2 with a general discussion of the values of the $\z$-function and the logarithmic UV divergence coefficient $B_d$ for a massive higher spin operator $( - \na^2 + M^2)_{s\perp} $ on $S^d$, and then specialise to the cases $d=4$ and $d=6$. In section 3 we shall apply the resulting expression for $B_6$ to the operators appearing in [(\[01\])]{} to obtain eq. [(\[111\])]{}. In section 4 we shall rederive [(\[111\])]{} as the coefficient of the IR divergence of the ratio of the AdS$_7$ massless spin $s$ partition functions in [(\[02\])]{}. Section 5 will contain concluding remarks. In Appendix we shall consider the $d=2$ case of [(\[02\])]{} and demonstrate explicitly that the AdS$_3$ expression for $a_s$ matches the coefficient $B_2$ of the UV divergence in the $d=2$ conformal higher-spin partition function [(\[02\])]{}, thus providing another check of [(\[02\])]{},[(\[uh\])]{}.
ł
$\z$-function and $B_d$ coefficient for spin $s$ operators on $S^d$
=====================================================================
To compute $B_d$ we shall use the known solution of the spectral problem for the 2nd-order operator $\dn_{s\perp} $ defined on symmetric traceless transverse tensors of rank $s$ on $S^d$ \_[s]{} (M\^2) ( - \^2 + M\^2)\_[s]{} , \_[s]{} (\_s)\_n = ł\_n (\_s)\_n . The eigen-values and their degeneracy are given by && ł\_n = (n+s) (n+s + d-1) - s + M\^2 , n= 0,1,2,....\
&& \_n = g\_s [ (n+1 ) ( n + 2 s + d-2 ) (2 n + 2s + d-1 ) (n+ s + d-3)! (d-1)! (n + s+1 )! ]{} ,\
&& g\_s = [ ( 2 s + d - 3) ( s + d - 4)! ( d - 3)! s!]{} . Here $g_s \equiv g_s^{(d)} $ is the number of components of the symmetric traceless transverse rank $s$ tensor in $d$ dimensions g\_s N\_[s]{}= N\_s - N\_[s-1]{} , N\_s = [ ( 2 s + d-2) ( s + d-3)! ( d-2)! s! ]{} , g\_s\^[(d)]{} = N\_s\^[ (d-1)]{} , where $N_s\equiv N_s^{(d)} $ is the number of symmetric traceless rank $s$ tensor components. The number of dynamical components of a massless spin $s$ field is (cf. [(\[03\])]{}) \_s = N\_[s]{} - N\_[s-1 ]{} = [ ( 2 s + d-4) ( s + d-5)! ( d-4)! s! ]{} , \_s\^[ (d)]{} = g\_s\^[(d-1)]{}= N\_s\^[ (d-2)]{} . Note also that the number of dynamical degrees of freedom of a conformal higher spin $s$ field is (cf. [(\[01\])]{}) &&\_s = \[ s + (d-4) \] N\_[s]{} - \^[s-1]{}\_[k=0]{} N\_[k]{} = ,\
&& \_s = \_s . The $\z$-function corresponding to the operator [(\[1\])]{} is defined by \_[\_[s]{} ]{} (z) = \^\_[n=0]{} [ \_n( ł\_n )\^z]{} . In general, it is $B_d$ and not $\z_{\dn } (0)$ that governs the scale dependence of $\log \det\, \D$ in [(\[05\])]{}. Note that the definition of $\z$ we use here requires summation over all modes, including the zero ones. Then while for the operator $\dn_{s}$ defined on differentially unconstrained tensors one has $\z_{\dn_{s} } (0) = B_d[\dn_{s}]$, this is not so in general for $\dn_{s\perp}$: $\z_{\D_{s\perp} }(0)$ turns out to be equal to $B_d[ \dn_{s\pe}]$ in [(\[06\])]{} for the operator [(\[1\])]{} only up to the contribution of the zero modes of the operator related to the change of variables from an unconstrained tensor $\p_s$ to its transverse part. In the case of $d=4$ and $s \leq 2$ the reason for this was explained in : to define the operators acting on constrained (transverse) tensors one decomposes the field into its transverse and gradient parts but that introduces $\N $ additional zero modes of the Jacobian of the change of variables. Since these modes were not present for the original unconstrained operator one finds $B_d [ \dn_{s\pe}]= \zeta_{\dn_{s\pe}} (0) - \N$. In more detail, starting with path integral over symmetric traceless tensor $\p_s$ we may change the variables to transverse symmetric traceless rank $s$ tensor $\p_{s\perp} $ and symmetric traceless rank $s-1$ tensor $\vp_{s-1}$ && \_s = \_[s]{} + \_[s-1]{} , \_[s]{}=0 ,\
&&(\_[s-1]{} )\_[m\_1 ...m\_s]{} = \_[(m\_s]{} \_[m\_1 ...m\_[s-1]{})]{} - [ s-1 2(s-2) + d ]{} g\_[(m\_s m\_[s-1]{}]{} \^n \_[m\_1 ...m\_[s-2]{}) n ]{} . Then $\det\,\, \K$ will appear as the Jacobian. The zero modes of $\K$ are rank $s-1$ conformal Killing tensors and their number is dimension of $(s-1,2, 0, ..., 0)$ representation of $SO(d+1,1)$ &&\_[s-1, d]{} = ( 2 s + d - 4) ( 2 s + d- 3) ( 2 s + d- 2) [ (s + d- 4)! ( s+ d - 3)!s! (s - 1)! d! (d - 2)! ]{} . Thus B\_d\[ \_[s]{}\] = \_[ \_[s]{}]{}(0) - , = [dim ker]{} = \_[s-1, d]{} . It should be noted that this subtlety is absent if one considers instead of $S^d$ the non-compact euclidean $H^d=$AdS$_d$ background: then the corresponding $\z_{ \dn_{s\pe}}(0)$-function defined according to matches $B_d[ \dn_{s\pe}]$.
In what follows we shall be interested in the two special cases: the familiar $d=4$ case (to compare to the results of which were found directly from the general expression for $B_4$, i.e. without using the spectrum on $S^4$) and the new $d=6$ one. One finds from [(\[2\])]{}–[(\[4\])]{},[(\[19\])]{} d=4: && ł\_n = n\^2 + (2s+3) n + s(s+2) + M\^2 , g\_4(s) = 2s +1 , \
&& \_n = g\_s (n+1 )(n+ 2s +2 ) (2 n + 2s+3) ,\
&& \_[s-1, 4]{} = (2s+1) s\^2 ( s+1)\^2 , d=6: && ł\_n = n\^2 + (2s+5) n + s(s+4) + M\^2 , g\_s = (s+1) (s+2) (2s +3) , \
&& \_n = g\_s (n+1 ) (n+s + 2 ) (n + s +3) (n +2s +4 ) (2 n + 2s+5) ,\
&& \_[s-1, 6]{} = (2s+3) s (s+1)\^3 (s+2)\^3 (s+3) . Note that in $d=6$ the number of symmetric traceless tensor components is (see [(\[499\])]{})\
$N_s= {1 \ov 12} ( s+1) (s+2)^2 (s+3)$; the number of transverse components is $N_{s\perp}= g_s={\te{ 1 \ov 6}} (s+1) (s+2) (2s +3)$; the number of dynamical degrees of freedom of a massless spin $s$ field [(\[389\])]{} is $\mu_s=(s+1)^2$; the number of dynamical degrees of freedom of a conformal spin $s$ field [(\[479\])]{} is $\nu_s= \frac{1}{4} (s+1)^2 (s+2)^2$.
Let us now consider the computation of the corresponding values of $ \z_{ \dn_{s\pe}}(0) $ in $d=4$ and $d=6$.
$d=4$ case
-----------
The computation of $\z_{ \dn_{s\pe}}(z) $ in $d=4$ was discussed in . First, we write [(\[7\])]{} as \_[ \_[s]{}]{} (z) =( 2s + 1) \^\_[k=s + [32]{} ]{} [ k \[ k\^2 - ( s + )\^2\] k\^[2z]{} (1 - [h\^2k\^2]{})\^z]{} , h\^2 = s + [ ]{}- M\^2 . Then using that (1 - [h\^2k\^2]{})\^[-z]{} = \^\_[m=0]{} c\_m(z) [h\^[2m]{}k\^[2m]{}]{} , c\_m(z) = [ (z + m-1)! m! (z-1)!]{} , we get &&\_[ \_[s]{}]{} (z) =( 2s + 1) \^\_[m=0 ]{} c\_m(z) h\^[2m]{} , where $ \zr(z , b) \equiv \sum_{n=0}^\infty ( n + b)^{-z} $. To find the limit $z\to 0$ we need to use that the terms with $m=1,2$ may have a pole as $\zr(x,b) = { 1 \ov x-1} - \psi (b) + ...$. Then we end up with &&\_[ \_[s]{}]{} (0) = , where $ \zr(-1 , b) =- \ha b^2 + \ha b - { 1 \ov 12}$ and $ \zr(-3 , b) =-\fo b^4 + \ha b^3 - \fo b^2 + { 1 \ov 120}$. Finally, \_[ \_[s]{}]{} (0) = (2s+1) . Then using [(\[vv\])]{},[(\[119\])]{} we get &&B\_4\[ \_[s]{} (M\^2) \] = \_[ \_[s]{}]{} (0) - \_[s-1,4]{}\
&&= (2s+1) . Taking into account [(\[08\])]{} this matches the expression for $\aa[ \dn_{s\pe}(M^2) ] $ which was found directly from the standard algorithm for $B_4$ and using that && \_[s]{} (M\^2) = [ \_[s]{} (M\^2) \_[s-1]{} (M\^2- 2s - d + 3 )]{} ,\
&&B\_d \[ \_[s]{} (M\^2)\] = B\_d \[ \_[s]{} (M\^2)\] - B\_d \[ \_[s-1]{} (M\^2- 2s - d + 3 )\] . Applying [(\[93\])]{} to find the total $B_4$ or $\aa$ coefficient [(\[08\])]{} corresponding to the $d=4$ CHS partition [(\[0\])]{} one ends up with [(\[088\])]{} .
Let us note that the same expression [(\[93\])]{} can be found also by considering instead of $S^4$ the non-compact $H^4$ (euclidean AdS$_4$) background. Indeed, the local expressions for the coefficient $b_4$ in [(\[06\])]{} should match since it depends on the square of the curvature while $R(S^4) = - R(H^4)$ (one should also change the sign of the $M^2$ term as it enters as $M^2 \ep,\ R= d(d-1)\ep, \ \ \ep=\pm 1$). Computing the corresponding value of $\zeta_{ \dn_{s\pe}} (0)$ as in (where its “un-integrated” value was found) and taking into account that the regularized volume of $H^4$ is $\Omega(H^4) = { 4 \pi^2 \ov 3}$ while $ \Omega(S^4) = {8 \pi^2 \ov 3}$ we conclude that $B_4$ and $\zeta^{(H^4)}_{ \dn_{s\pe}} (0)$ should be equal up to the factor of 2 coming from the ratio of the two volumes. Explicitly, given the operator $\dn_{s\pe} (M^2) = (-\nabla^2 + M^2)_{s\pe}$ one finds && B\_4 \[ \_[s]{} (M\^2)\] = \^[(S\^4)]{}\_[ \_[s]{}(M\^2)]{} (0) - \_[s-1,4]{} = 2 \^[(H\^4)]{}\_[ \_[s]{}(-M\^2)]{} (0) ,\
&& \^[(H\^4)]{}\_[ \_[s]{}(-M\^2)]{} (0) = ( 2s + 1) , h\^2 = s + [ 94]{} - M\^2 . The expression [(\[933\])]{} was also used in in the computation of the UV divergent term of the massless higher spin theory in AdS$_4$.
In general, the partition function [(\[03\])]{} of a massless spin $s$ field in AdS$_d$ ($\ep=-1$) or dS$_d$ or $S^d$ space ($\ep=+1$) can be written as Z\_[s0]{}= ([ \_[s-1 ]{}\[M\^2\_[s-1,s]{}\] \_[s]{}\[M\^2\_[s,s-1]{}\]]{} )\^[1/2]{} , M\^2\_[n,k]{} n - (k-1) (k + d-2) . Then we may use [(\[93\])]{} to find the coefficient [(\[05\])]{} of the divergent term in $F=-\ln Z_{s0}$ in $d=4$: && d=4: M\^2\_[s,s-1]{} =- s\^2 + 2s + 2 , M\^2\_[s-1,s]{} = -s\^2 + 1 ,\
&& B\_4\^[(s0)]{}B\_4\[\_[s]{} (M\^2\_[s,s-1]{})\] -B\_4\[\_[s-1 ]{} (M\^2\_[s-1,s]{})\] =- (75 s\^4-15 s\^2 +2) . This is equivalent to the expression obtained in using [(\[933\])]{}. It was found there that the $\z$-function regularized sum of the values of $ \zeta^{(H^4)}(0)$ over all massless spins $s >0$ plus the $s=0$ (scalar) contribution vanishes. Let us note that the same conclusion applies also for the corresponding values of the massless higher spin $\z$-function computed on $S^4$ or dS$_4$: the sum over the zero-mode terms in [(\[23\])]{},[(\[119\])]{} given by (cf. [(\[03\])]{}) $ \sum_{s=1}^\infty ( \kk_{s-1, 4} - \kk_{s-2, 4}) = \sum_{s=1}^\infty \, \te { 1 \ov 6} (s^2 + 5 s^4) $ vanishes separately when $\z$-function regularized.
$d=6$ case
-----------
According to [(\[vv\])]{} we should have the following relation between $B_6$ in [(\[05\])]{} and the corresponding $\zeta$-function on $S^6$ B\_6 \[\_[s]{}\] = \_[\_[s]{} ]{}(0) - \_[s-1, 6]{} , where $ \kk_{s-1, 6}$ is given in [(\[219\])]{}. The computation of the $\z_{\dn_{s\pe} }(0) $ in $d=6$ uses [(\[aa5\])]{},[(\[6\])]{} and follows the same lines as in $d=4$. The counterpart of [(\[99\])]{} is \_[\_[s]{} ]{} (z) = g\_s \_[k=s + [52]{} ]{}\^ , h\^2= s + [ 254]{} - M\^2 , and using [(\[exp\])]{} we get && \_[\_[s]{}]{}(z) = g\_s \^\_[m=0 ]{} c\_m(z) h\^[2m]{} Taking the limit $z\to 0$ gives (cf. [(\[8\])]{}) &&\_[\_[s]{} ]{}(0) = g\_s As a result, && \_[\_[s]{} ]{}(0) = Then eq.[(\[31\])]{} implies that (cf. [(\[93\])]{}) && B\_6\[\_[s]{} (M\^2)\] = . In particular, in the case of the conformal scalar $s=0, \ M^2 = {d-2\ov 4 (d-1)} R = \fo d (d-2) =6$ we get $B_6= { 1 \ov 756}$, i.e. the standard value [(\[002\])]{}.
It should be possible of course to find [(\[36\])]{} directly from the general expression for the $b_6$ heat kernel coefficient of a 2nd-order differential operator in curved space, but in the arbitrary spin $s$ case in $d=6$ this computation appears to be more involved than the one based on the $\z$-function on $S^6$ presented here.
$B^{(s)}_6$ coefficient in conformal spin $s$ partition function on $S^6$
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Let us now apply the general expression [(\[36\])]{} to find the $B_6^{(s)}$ coefficient corresponding to the CHS partition function [(\[01\])]{} on $S^6$. Explicitly, in $d=6$ we get Z\_[s]{}([S\^6]{}) =\_[k=0]{}\^[s-1]{} \^[1/2]{} \^[s-1]{}\_[k’= -1]{} \^[-1/2]{} , M\^2\_[k,m]{} = k - (m-1) ( m + 4). Using [(\[36\])]{} we find for the total anomaly coefficient (cf. [(\[088\])]{}) && B\^[(s)]{}\_6 = \_[k’=-1]{}\^[s-1]{} B\_6\[\_[s]{} ( s - (k’-1) ( k’+ 4) )\] - \_[k=0]{}\^[s-1]{} B\_6\[\_[k]{} ( k - (s-1) ( s+ 4) )\]\
&& = - (22 s\^6+198 s\^5+671 s\^4+1056 s\^3+733 s\^2+120 s-50) . Let us note that the $k=s-1,\ k'=s-1$ terms in [(\[zz\])]{} represent the partition function of massless spin $s$ field on $S^6$ (or dS$_6$) which is the same as the AdS$_6$ one in [(\[03\])]{} up to the sign of the dimensionless mass parameters: on $S^6$ we have M\^2\_[s,s-1]{} = - s\^2 + 6 , M\^2\_[s-1,s]{} = -s\^2 - 2s + 3 . We find for the contribution of this massless spin $s$ factor (cf. [(\[yyy\])]{}) && B\^[(s0)]{}\_6 = B\_6\[\_[s]{} (M\^2\_[s,s-1]{})\] -B\_6\[\_[s-1 ]{} (M\^2\_[s-1,s]{})\]\
&& = - (63 s\^6 + 378 s\^5 + 847 s\^4 + 868 s\^3 + 378 s\^2 + 28s - 20 ) . For $s=0$ this equals to the conformal scalar value [(\[002\])]{} as in this case $B_6[\dn_{s-1\, \pe} (M^2_{s-1,s})]$ vanishes.
$a_s$ coefficient in ratio of massless spin $s$ partition functions in AdS$_{7}$
================================================================================
Let us now show that exactly the same expression [(\[40\])]{} appears as a coefficient of the IR divergent term in the ratio of the massless spin $s$ partition functions in AdS$_7$ in the l.h.s. of eq.[(\[02\])]{}. We shall first review the general expression for this coefficient found in and then apply it to the case of $d=6$.
Starting with a mass $m$ spin $s$ operator in AdS$_{d+1}$ of unit radius ($ \ep=-1$) (M\^2)\_[s]{} = (- \^2 + M\^2 )\_[s]{} , M\^2 = - m\^2 + s - (s-2) (s + d-2) , one finds that the powers of near-boundary asymptotics of the corresponding solutions are $\g_\pm = \Delta_\pm -s$ where && \_(m) = [d ]{} = [d ]{} , \
&& \_+ \_+(0) = s + d -2 , \_- \_-(0) = 2-s , \_- = d-\_+ . These $\De_{\pm}$ apply to the physical (spin $s$) part of [(\[03\])]{} while for the “ghost” (spin $s-1$) part of [(\[03\])]{} $\Delta'_+ = s + d-1, \ \Delta'_- = 1-s$ . As discussed in the Introduction, the partition function of a constant-mass operator on AdS$_{d+1}$ is proportional to its volume which for even $d$ is IR divergent (see [(\[04\])]{}). Calling the coefficient of the $\ln L$ term in the corresponding free energy $F =\ha \ln \det\, \dn_{s\perp}(M^2) $ as $a_s(\Delta)$ where $\Delta= \Delta_+$ in [(\[44\])]{} one finds that && a\_[s]{} () a\_s() - a\_s( d -)\
&&= - [ 2 g\_s\^[(d+1)]{} d!]{} \^\_[[12]{} d]{} dx (x- d) ( x+ s-1) (x- s-d +1) ( x-1) ( d-1-x) ( x ) , where $g_s^{(d+1)}$ is the same as $g_s$ in [(\[4\])]{} with $d \to d+1$. Then the coefficient $a_s$ corresponding to the ratio of the partition functions appearing in the l.h.s. of [(\[02\])]{} can be found as a\_[s]{}= a\_[s]{} (\_+ ) - a\_[s-1]{} (\_+’ ) = a\_[s]{} ( s + d-2 ) - a\_[s-1]{} ( s + d-1 ) . The special cases of $d=2$ and $d=4$ were already discussed in . Doing the integral in [(\[46\])]{} gives && d=2: a\_[s]{} () = (-1) , \
&& d=4: a\_[s]{} () = (-2)\^3 . Using these expressions in [(\[47\])]{} leads to (here for $d=2$ $s \geq 2$ and $a_0= {1\ov 3}, \ a_1= {1\ov 3}$) && d=2: a\_[s]{} = + 4 s (s-1) , \
&& d=4: a\_[s]{} = ( 14 s\^2 + 14 s + 3 ) . Thus in $d=4$ one finds $a_s $ in [(\[07\])]{} that matches $B_4^{(s)}$ [(\[088\])]{} derived in directly from [(\[0\])]{} (see also section 2.1).
The $d=2$ coefficient [(\[494\])]{} (rescaled by $ - { 3}$) was interpreted in as the central charge $c_s= -2 [ 1 + 6 s (s-1)]$ ($s\geq 2$) of the first-order bc-ghost system with weights $s$ and $1-s$ corresponding to spin $s$ W-gravity field . In Appendix we shall demonstrate that the AdS$_3$ prediction [(\[494\])]{} matches the $B_2$ anomaly coefficient for the $d=2$ case of the conformal higher spin partition function [(\[01\])]{}.
Let us now consider the $d=6$ case. Computing the integral in [(\[46\])]{} we get (cf. [(\[493\])]{},[(\[49\])]{}) a\_[s]{} () = (-3)\^3 Let us recall again that the normalization of $a_{s}$ in [(\[46\])]{} is such that it is the coefficient of the logarithm of the radius of $S^d$, i.e. it is equal to minus the corresponding value of $B_d$: in the case of $d=6$ for $s=0,\ \De={d\ov 2} + 1 = 4$ eq.[(\[48\])]{} gives $- {1 \ov 756}=-B^{(0)}_6$ (cf. [(\[002\])]{}).
Applying [(\[48\])]{} to the case of [(\[47\])]{} we find d=6: a\_[s]{}&=& a\_[s]{} ( s + 4 ) - a\_[s-1]{} ( s + 5 )\
&=& (22 s\^6+198 s\^5+671 s\^4+1056 s\^3+733 s\^2+120 s-50) This is the same expression as in [(\[111\])]{}, i.e. it matches the expression [(\[40\])]{} for $-B^{(s)}_6$ found above directly from the CHS partition function on $S^6$.
Concluding remarks
==================
To summarize, in this paper we have shown the agreement [(\[uh\])]{} between the UV divergence coefficient $B^{(s)}_6$ [(\[40\])]{} of the conformal higher spin partition function on $S^6$ and the IR divergence coefficient $a_s$ in the ratio of massless higher spin partition functions with alternate boundary conditions on AdS$_7$. Together with the corresponding $d=4$ results of this provides a non-trivial test of the relation [(\[02\])]{}. We also demonstrate a similar matching in the $d=2$ case in Appendix below.
In $d=4$ the sum of the anomaly coefficients $a_s$ in [(\[07\])]{} over all spins $s$ vanishes when computed using the standard $\z$-function prescription. The same is true also for the sum of the $s \geq 1$ massless spin $s$ divergence coefficients in [(\[yyy\])]{} plus the $s=0$ conformal scalar contribution . In the $d=6$ case we discussed here the corresponding sums of the coefficients in [(\[40\])]{} and in [(\[38\])]{} do not appear to vanish. This may not be surprising since in $d=6$ there is no a priori reason to sum over all spins with weight one and, moreover, to consider only totally symmetric traceless tensor representation.
In general, it would be interesting also to study the $d=6$ conformal higher spin partition function on other backgrounds, e.g., on Ricci-flat one as in $d=4$ case in . The corresponding covariant and Weyl-invariant CHS action should have the structure $\int d^6 x \sqrt g\, \p_s ( \nabla^{2s +2} + ...) \p_s = \int d^6 x \sqrt g\, C_{2s} ( \nabla^2 + ...) C_{2s}$ where the rank $2s$ tensor $C_{2s}$ is a gauge-covariant CHS field strength $C_{2s} \sim P_s \nabla^s \p_s+...$. This action is known explicitly only for lowest values of the spin. For $s=1$ the field strength $C_2$ is the antisymmetric tensor and the 2nd order Weyl-covariant operator $( \nabla^2 + ...) $ acting on it can be found, e.g., in . For $s=2$ the field strength $C_4$ is the same as the Weyl tensor and the corresponding Weyl-covariant operator $( \nabla^2 + ...) $ is the same that appears in the $I_1\sim C ( \nabla^2+...) C $ term in the trace anomaly [(\[09\])]{} (see also ). The “minimal” $d=6$ Weyl gravity action $\int d^6 x \sqrt g\, I_1 $ (which can be expressed in terms of Ricci tensor $I_1 \sim R_{ab} ( \nabla^2+...) R_{mn}$ ) admits an equivalent representation in terms of a collection of fields with ordinary (2nd-derivative) kinetic terms. Such an ordinary-derivative description of the CHS field with any spin $s$ and in any even dimension $d$ is known in flat space , and, following the $s=2$ example , it may serve as a starting point for constructing a covariant CHS $s \geq 2$ actions in generic curved backgrounds.
Let us now see what happens if we formally sum over all spins with equal weight in the $d=6$ case. From [(\[40\])]{} we find \_[s=1]{}\^B\^[(s)]{}\_6 =+-- - - = - . Adding the $d=6$ conformal scalar $s=0$ contribution, or, equivalently, considering the sum in [(\[41\])]{} starting from $s=0$ we also get a non-zero result \_[s=0]{}\^B\^[(s)]{}\_6 = - = . Since this number is smaller than any other standard field contributions it is not clear how it can be cancelled out.
If we consider just the massless spin $s$ contribution to [(\[40\])]{} (which corresponds to the $k=k'=s-1$ terms in the sums in [(\[40\])]{}) given in [(\[38\])]{} we get \_[s=1]{}\^B\^[(s0)]{}\_6 =[ + - - - = -[1918900]{}]{} , \_[s=0]{}\^B\^[(s0)]{}\_6 = . This sum is thus also non-zero for massless spins in AdS$_6$, in contrast to what was found in AdS$_4$ case .
Comparing [(\[41\])]{},[(\[42\])]{} and [(\[39\])]{} we conclude that the non-vanishing of the sum of the conformal higher spin anomaly coefficients is solely due to the massless spin $s$ contribution to the CHS partition function [(\[0\])]{}. It remains to understand if this is just a coincidence or has some useful interpretation.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to A. Barvinsky, R. Metsaev, R. Roiban, E. Skvortsov and M. Vasiliev for useful discussions. We thank I. Klebanov for useful comments on the draft and pointing out that the AdS$_7$ expression [(\[50\])]{} was independently found also in . This work was supported by the ERC Advanced grant No.290456 and also by the STFC grant ST/J000353/1.
Appendix:\
Partition function and $B_2$ coefficient of conformal higher spins on $S^2$ {#appendix-partition-function-and-b_2-coefficient-of-conformal-higher-spins-on-s2 .unnumbered}
===========================================================================
[5.]{}[A.]{}
Here we shall show that the AdS$_3$ prediction for $a_s$ [(\[493\])]{} is indeed the same [(\[uh\])]{} as the logarithmic UV divergence coefficient $B_2$ in the conformal higher-spin partition function [(\[0\])]{} specialised to the $d=2$ case.
Naively, the $d=2$ limit of the conformal higher spin action should start with a $\del^{2s+d-4} = \del^{2s-2}$ term ($s\geq 2$). However, the $d=2$ case of the CHS theory is special – here the number of components $N_s$ [(\[499\])]{} of a symmetric traceless rank $s$ tensor is $s$-independent: $N_s= 2$ for $s \geq 1$ ($N_s=1$ for $s=0$). Then the number of the corresponding transverse components $g_s=N_{s\perp}$ [(\[499\])]{} vanishes for $s\geq 2$: $N_{s\perp}= N_s - N_{s-1} =0$ ($N_{1\perp}=1$). Equivalently, a symmetric rank tensor CHS field $\p_s$ can be completely gauged away by a combination of the gradient gauge symmetry (generalized reparametrizations) and the algebraic gauge symmetry (generalized Weyl symmetry), i.e. there is no non-trivial gauge-invariant field strength $C_{2s}\sim P_s \del^s \p_s $ (this is an $s \geq 3$ generalization of the fact of the absence of Weyl tensor in $d=2$).
Thus the classical $d=2$ CHS action is trivial (a familiar fact for $s=2$ or gravity in $d=2$). Still, non-zero contributions to the corresponding partition function may come from the gauge-fixing or ghost sector. Indeed, the number of dynamical degrees of freedom of a CHS field in $d=2$ as following from the general expression in [(\[477\])]{} is $\nu_s=-2$ (again, a well-known result for $d=2$ gravity with trivial Einstein term action). More precisely, the CHS action in the path integral for the partition function in a background covariant harmonic gauge $(\nabla\cdot \p_s=0$) will have actually a non-trivial $\p_s \del^{2s-2} \p_s + ...$ kinetic term but it will be coming solely from the gauge-fixing term. Thus, despite the triviality of the classical gauge-invariant CHS action, the corresponding partition function will still contain “physical” determinants of spin $s$ operators coming from the gauge-fixing term. Indeed, the $d=2$ limit of the CHS partition function [(\[01\])]{} is found to be Z\_[s]{} ([S\^2]{}) =\_[k=0]{}\^[s-1]{} \^[1/2]{} \^[s-1]{} \_[k’=1 ]{} \^[-1/2]{} . Using [(\[2222\])]{} this may be written explicitly in terms of unconstrained operators as ($\dn_{-1}\equiv 1$) Z\_[s]{} ([S\^2]{}) =\_[k=0]{}\^[s-1]{} \^[1/2]{} \^[s-1]{} \_[k’=1 ]{} \^[ 1/2]{} . Given an operator $\dn_k(M^2)= - \nabla^2 + M^2$ defined on unconstrained symmetric traceless rank $k$ tensor the corresponding Seeley coefficient [(\[06\])]{} in the free energy [(\[05\])]{} on unit-radius $S^2$ (with curvature $R= d(d-1) = 2$) is && B\_2\[ \_[k]{} (M\^2)\] = (S\^2) b\_2 = b\_2 , b\_2 = N\_k ( [ 16]{} R - M\^2 ) , \
&& k1: b\_2= 2 ( [1 3]{} - M\^2 ) , k=0: b\_2= [1 3]{} - M\^2 . Applying [(\[22\])]{} we find && k 2: B\_2 \[ \_[k]{} (M\^2)\] = B\_2 \[ \_[k]{} (M\^2)\] - B\_2 \[ \_[k-1]{} (M\^2- 2k +1 )\] = - 4k + 2 ,\
&& B\_2 \[ \_[1]{} (M\^2)\] = - [ 2 3]{} - M\^2 , B\_2 \[ \_[0]{} (M\^2)\] =B\_2 \[ \_[0 ]{} (M\^2)\]= - M\^2 . Then the total $B_2$ coefficient in free energy [(\[05\])]{} corresponding to [(\[012\])]{} is ($s\geq 2$) B\_2\^[(s)]{}&=& \_[k’=1]{}\^[s-1]{} B\_2 \[ \_[s]{} ( s - k’(k’-1) )\] - \_[k=0]{}\^[s-1]{} B\_2 \[ \_[k]{} ( k - s(s-1) )\]\
&=& B\_2 \[ \_[s]{} ( s )\] - B\_2 \[ \_[1]{} ( 1- s(s-1) )\] - B\_2 \[ \_[0]{} (- s(s-1) )\] - 4\_[k=2]{}\^[s-1]{} (s-k)\
&=& - - 4 s (s-1) . In the conformal 2d vector $s=1$ case (corresponding to the Schwinger $\int F \del^{-2} F = \int A_{m\perp}^2$ action) we get from [(\[012\])]{} $Z_1= \big[\det \dn_{0} ( 0)\big]^{1/2}$ and thus $B_2^{(1)} = - {1 \ov 3}$. This matches the expression for $a_s$ [(\[494\])]{} found from AdS$_3$, in line with the $d=4$ and $d=6$ tests of [(\[02\])]{},[(\[uh\])]{} discussed above.
The $d=2$ CHS model discussed here is, of course, closely related to spin $s$ W-gravity model : both have the same linearized symmetries – generalized reparametrizations and Weyl transformations for spin $s$ field. The resulting conformal anomaly coefficient [(\[007\])]{} is indeed equivalent to the quantum W-gravity anomaly given solely by the corresponding bc ghost contribution to the central charge $c_{gh} = - 2( 1 + 6s^2 - 6s)$ . What is remarkable about the above derivation of this result from the CHS partition function [(\[012\])]{} is that it illustrates that the $d=2$ case, while somewhat degenerate (having trivial classical action), can still be viewed as a limit of $d $-dimensional conformal higher spin theory (which itself may then be interpreted as a natural $d > 2$ generalization of W-gravity).
[30]{}
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[^1]: Also at Lebedev Institute, Moscow. e-mail: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Liouville-Lanczos approach to linear-response time-dependent density-functional theory is generalized so as to encompass electron energy-loss and inelastic X-ray scattering spectroscopies in periodic solids. The computation of virtual orbitals and the manipulation of large matrices are avoided by adopting a representation of response orbitals borrowed from (time-independent) density-functional perturbation theory and a suitable Lanczos recursion scheme. The latter allows the bulk of the numerical work to be performed at any given transferred momentum only once, for a whole extended frequency range. The numerical complexity of the method is thus greatly reduced, making the computation of the loss function over a wide frequency range at any given transferred momentum only slightly more expensive than a single standard ground-state calculation, and opening the way to computations for systems of unprecedented size and complexity. Our method is validated on the paradigmatic examples of bulk silicon and aluminum, for which both experimental and theoretical results already exist in the literature.'
author:
- Iurii Timrov and Nathalie Vast
- Ralph Gebauer
- Stefano Baroni
title: |
Electron energy-loss and inelastic X-ray scattering cross sections\
from time-dependent density-functional perturbation theory
---
\[sec:Introduction\]Introduction
================================
Plasma oscillations in solids are possibly the simplest manifestation of collective effects in condensed matter, and their understanding in terms of *plasmon* modes one of the earliest triumphs of quantum many-body theory. [@Pines:1956; @Nozieres:1959; @Pines:1964] On the experimental side, collective charge-density fluctuations can be probed through electron energy-loss (EEL) or inelastic X-Ray scattering (IXS) spectroscopies, two techniques that have been steadily producing a wealth of data since the early 60s and 70s, respectively.[@Egerton:1996; @Schulke:2007] In the present day the engineering of novel materials down to the nanometer scale makes it possible to design devices where electromagnetic fields interact with collective oscillations of structures of sub-wavelength size. The strong dependence of plasmon dynamics on the size and shape of these nanostructures holds the promise of an extraordinary control over the optical response of the resulting devices, with applications to such diverse fields as photovoltaics,[@Atwater:2011] proton beam acceleration,[@Bartal:2011] or biosensing,[@Anker:2008] to name but a few. This is plasmonics, *i.e* photonics based on collective electronic excitations in strongly heterogeneous systems, where surface effects play a fundamental role. Plasma oscillations at surfaces have recently aroused a renewed attention by themselves, since it was shown that some metal surfaces unexpectedly exhibit *acoustic* plasmons.[@Silkin:2005; @Diaconescu:2007; @Pohl:2010; @Vattuone:2012; @Yan:2012; @Vattuone:2013] These are collective charge excitations localized at the surface, whose frequency vanishes linearly with the wavevector, and are not damped by the bulk electron-hole continuum.[@Silkin:2004; @Pitarke:2004] It is thought that these modes may offer the possibility of light confinement at designated locations on the surface, with possible applications in photonics and nano-optics.[@Pitarke:2007]
Most of the theoretical understanding of the optical response in nano-plasmonic systems relies on a classical approach: the nanostructure is usually described as an assembly of components, each characterized by an effective macroscopic dielectric function, and separated from the others by abrupt interfaces. The overall optical response is then computed by solving Maxwell’s equation for the resulting heterogeneous system.[@Esteban:2012] When distances between the nanoscale components are themselves nanometric, however, quantum effects must be accounted for, and a fully quantum-mechanical description is called for.
Early quantum-mechanical approaches to the dynamics of charge-density fluctuations[@Pines:1956; @Nozieres:1959; @Pines:1964] were based on the random-phase approximation as applied to the jellium model that, albeit exceedingly successful in simple metals and semiconductors, is not suitable for more complex materials, nor can it capture the fine, system-specific, features of even simple ones. The effects of crystal inhomogeneities on plasmon resonances in semiconductors (the so called *local-field* effects) were first addressed in the late 70s,[@Sturm:1978] using the empirical pseudopotential method,[@Cohen:1966] along similar lines as previously followed for the optical spectra.[@Louie:1975] In the present day the method of choice for describing charge dynamics in real materials (as opposed to simplified models, such as the jellium one) is time-dependent (TD) density-functional theory (DFT).[@Runge:1984; @Gross:1996] Although some attempts to investigate EEL and IXS spectra using many-body perturbation theory have been made,[@Caliebe:2000; @Olevano:2001; @Takada:2002; @Arnaud:2005] the vast majority of the studies existing to date relies on TDDFT, which in fact has been successfully used to study plasmons in a number of bulk[@Daling:1992; @Engel:1992; @Sturm:1992; @Quong:1993; @Fleszar:1995; @Ehrnsperger:1997; @Waidmann:2000; @Vast:2002; @Marinopoulos:2002; @Marinopoulos:2003; @Schone:2003; @Dash:2004; @Gurtubay:2004; @Gurtubay:2005; @Weissker:2006; @Kramberger:2008; @Alkauskas:2010; @Huotari:2009; @Weissker:2010; @Cazzaniga:2011; @Yan:2011] and surface[@Silkin:2005; @Diaconescu:2007; @Pohl:2010; @Vattuone:2012; @Yan:2012; @Vattuone:2013] systems.
The conventional TDDFT approach to plasmon dynamics relies on the calculation of the charge-density susceptibility, $\chi$ (or, equivalently, inverse dielectric matrix, $\epsilon^{-1}$), starting from the independent-electron susceptibility, $\chi_0$, via a Dyson-like equation.[@Onida:2002] Although successful in (relatively) simple systems that can be described by unit cells of moderate size, this methodology can hardly be applied to more complex systems, such as low-index or nano-structured surfaces, because of its intrinsic numerical limitations. In particular: *i*) the calculation of $\chi_0$ requires the knowledge of a large number of empty states, which is usually avoided in modern electronic-structure methods; *ii*) the solution of the Dyson-like equation requires the manipulation (multiplication and inversion) of (very) large matrices, and *iii*) all the above calculations have to be repeated independently for each value of the frequency to be sampled.
In this paper we introduce a new method, based on TD density-functional perturbation theory (DFPT),[@Walker:2006; @Rocca:2008; @Malcioglu:2011; @Baroni:2012] that allows to calculate EEL and IXS cross sections avoiding all the above drawbacks, and thus lending itself to numerical simulations in complex systems, potentially as large as several hundred independent atoms. Although the new methodology is general in principle, our implementation relies on the pseudopotential approximation, which limits its applicability to valence (or shallow-core) loss spectra. Inner-core loss spectra are currently addressed using different methods, as explained *e.g.* in Refs. . The salient features of our method are: *i*) the adoption of a representation from time-independent DFPT[@Baroni:2001] allows to avoid the calculation of Kohn-Sham (KS) virtual orbitals and of any large susceptibility matrices ($\chi$ or $\chi_0$) altogether; and *ii*) thanks to the use of a Lanczos recursion scheme, the bulk of the calculations can be performed only once for all the frequencies simultaneously. The numerical complexity of the resulting algorithm is comparable, for the *whole* spectrum in a wide frequency range, to that of a *single* standard ground-state (or static response) calculation.
The paper is organized as follows. In Sec. \[sec:theory\] we describe our basic theoretical and algorithmic frameworks, including the implementation of the newly proposed methodology for the response of a periodic system to a monochromatic perturbation, relevant to the calculation of EEL and IXS cross sections; in Sec. \[sec:applications\] we benchmark our technique on the prototypical examples of bulk silicon and aluminum, for which many experimental and well established theoretical results already exist; finally, our conclusions are presented in Sec. \[sec:conclusions\].
\[sec:theory\] Theory and algorithms
=====================================
Electron energy-loss spectroscopy probes the diffusion of a beam of fast electrons through a solid. According to Van Hove,[@VanHove:1954] the corresponding double-differential cross section for inelastic scattering reads:[@Egerton:1996] $$\left( \frac{d^2\sigma}{d\Omega d\omega} \right)_\mathrm{EEL} =
\left( \frac{4 \pi e^2}{Q^2} \right)^2 \frac{m^2}{4 \pi^2 \hbar^4}
\frac{k_f}{k_i} \, S(\mathbf{Q},\omega) ,
\label{eq:cross_section}$$ where $-e$ and $m$ are the electron charge and mass, $\mathbf{k}_i$, $\mathbf{k}_f$, and $\mathbf{Q} = \mathbf{k}_i - \mathbf{k}_f$ are the incoming, outgoing, and transferred momenta, respectively, and $S(\mathbf{Q},\omega)$ is the dynamic structure factor per unit volume.
While EEL spectroscopy is not suitable for samples enclosed in high-pressure cells, plasmon dynamics under pressure can be probed by IXS spectroscopy.[@Mao:2001; @Loa:2011] The double-differential cross-section reads in this case: $$\left( \frac{d^2\sigma}{d\Omega d\omega} \right)_\mathrm{IXS} =
\left( \frac{e^2}{m c^2} \right)^2 (\mathbf{e}_i \cdot
\mathbf{e}_f)^2 \, \frac{\omega_f}{\omega_i} \, S(\mathbf{Q},\omega)
,
\label{eq:cross_section_IXS}$$ where $\mathbf{e}_i$ and $\mathbf{e}_f$ are the incoming and scattered photon polarization directions, and $\omega_i$ and $\omega_f$ are the corresponding frequencies. According to the fluctuation-dissipation theorem[@Pines:1966] $S(\mathbf{Q},\omega)$ is proportional to the imaginary part of the charge-density susceptibility, $\chi(\mathbf{Q},\mathbf{Q}; \omega)$: $$S(\mathbf{Q},\omega) = - \frac{\hbar}{\pi} \,
\mathrm{Im} \, \chi(\mathbf{Q},\mathbf{Q}; \omega) .
\label{eq:fluct_dissip_theorem}$$ In periodic solids the transferred momentum can be split into a component in the first Brillouin zone, $\mathbf{q} $, and a reciprocal-lattice vector, $\mathbf{G}$, as $\mathbf{Q} =
\mathbf{q}+\mathbf{G}$, and $\chi$ is often expressed in terms of the inverse dielectric matrix, defined as:[@Martin:2004; @Car:1981] $$\epsilon^{-1}_{\mathbf{G}\mathbf{G}'}(\mathbf{q},\omega)
=
\delta_{\mathbf{G}, \mathbf{G}'} + \frac{4\pi
e^2}{|\mathbf{q}+\mathbf{G}|^2} \,
\chi(\mathbf{q}+\mathbf{G}, \mathbf{q}+\mathbf{G}';\omega) ,
\label{eq:def_inv_microscopic_diel_tensor}$$ where $\epsilon^{-1}_{\mathbf{G}\mathbf{G}'}(\mathbf{q},\omega) =
\epsilon^{-1}(\mathbf{Q},\mathbf{Q'};\omega)$. The function $-\mathrm{Im}[\epsilon^{-1}(\mathbf{Q},\mathbf{Q};\omega)]$ is usually referred to as the *loss function*.
Time-dependent density-functional perturbation theory {#sec:TDDFT}
-------------------------------------------------------
In TDDFT electron dynamics is described by TD one-electron equations for the occupied molecular orbitals. These TD KS equations read:[@Martin:2004] $$i \, \frac{\partial \varphi_v(\mathbf{r},t)}{\partial t} =
\hat{H}_{\mathrm{KS}}(t) \, \varphi_v(\mathbf{r},t),
\label{eq:TD-KS_equation}$$ where $\varphi_v(\mathbf{r},t)$ and $\hat{H}_{\mathrm{KS}}(t)$ are the TD KS orbitals and Hamiltonian (quantum mechanical operators are indicated with a caret), respectively, the index $v$ spans the $N_v$ occupied (*valence*) states, and atomic units ($e=m=\hbar=1)$ are used henceforth. The KS Hamiltonian reads: $$\hat{H}_{\mathrm{KS}}(t) = -\frac{1}{2} \nabla^2 +
V_{ext}(\mathbf{r},t) + V_{\mathrm{HXC}}(\mathbf{r},t) ,
\label{eq:KS_Hamiltonian}$$ where $V_{ext}(\mathbf{r},t)$ and $V_{\mathrm{HXC}}(\mathbf{r},t)$ are the external and Hartree-plus-exchange-correlation (HXC) potentials, respectively. Let us assume that the external potential can be split into a static term, plus a small TD perturbation: $$V_{ext}(\mathbf{r},t) = V_{ext}^\circ(\mathbf{r}) +
\lambda(t)V'_{ext}(\mathbf{r}) ,
\label{eq:Vext}$$ where $\lambda(t)$ is the TD strength of the perturbation. The total KS potential is perturbed accordingly: $V'(\mathbf{r},t) = \lambda(t) V'_{ext}(\mathbf{r}) +
V'_{\mathrm{HXC}}(\mathbf{r},t)$, $V'_{\mathrm{HXC}}$ being the response HXC potential. The response of the KS orbitals is defined as $$\varphi_v(\mathbf{r},t) = e^{-i \varepsilon_v t} \, \left (
\varphi_v^\circ(\mathbf{r}) + \varphi'_v(\mathbf{r},t) \right) ,
\label{eq:orbital_response_funct}$$ $\varphi_v^\circ(\mathbf{r})$ and $\varepsilon_v$ being the unperturbed ground-state KS orbitals and energies, respectively. The charge-density susceptibility is the response of the electron charge density, which only depends on the projection of the response of the valence KS orbitals onto the empty-state (conduction) manifold. The Fourier transforms (indicated by tilde $``\tilde{\phantom{a}}"$ hereafter) of such projected response orbitals are obtained from standard first-order perturbation theory via the linear systems: $$\left (\hat{H}^\circ - \varepsilon_v - \omega \right )
\tilde{\varphi}'_v(\mathbf{r},\omega) = - \hat{P}_c
\tilde{V}'(\mathbf{r},\omega) \varphi_v^\circ(\mathbf{r}) ,
\label{eq:lin-resp_w_eq1}$$ where $\hat P_c$ is the projector over the unperturbed conduction-state manifold. Expressing the latter in terms of valence orbitals ($\hat P_c+\hat P_v=1$) allows one to compute response KS orbitals without making any reference to unoccupied states, much in the same way as it is done in time-independent DFPT.[@Baroni:2001] The solution of Eq. (\[eq:lin-resp\_w\_eq1\]) requires one to express the total response potential, $\tilde
V'(\mathbf{r},\omega)$, in terms of its own solutions, through the response charge density, which is the diagonal of the response density matrix, $n'(\mathbf{r},t) =
\rho'(\mathbf{r},\mathbf{r};t)$, whose Fourier transform is defined as: $$\begin{gathered}
\tilde{\rho}'(\mathbf{r},\mathbf{r}';\omega) = 2 \sum_{v=1}^{N_v} \bigl (
\tilde{\varphi}^\prime_v(\mathbf{r},\omega) \,
\varphi^{\circ\,*}_v(\mathbf{r}') \\
+ \, \varphi^\circ_v(\mathbf{r})
\tilde{\varphi}^{\prime\,*}_v(\mathbf{r}',-\omega) \,
\bigr ) ,
\label{eq:charge-dens-response_2}\end{gathered}$$ where the factor two accounts for spin degeneracy in a non-polarized system. Note that $ \tilde{n}'(\mathbf{r},\omega) =
\tilde{n}'^*(\mathbf{r},-\omega) $, as a consequence of the reality of $n'(\mathbf{r},t)$. The equation for the complex conjugate of $\tilde{\varphi}^\prime_v(\mathbf{r},\omega)$ reads: $$\left (\hat{H}^\circ - \varepsilon_v + \omega \right)
\tilde{\varphi}^{\prime\,*}_v(\mathbf{r},-\omega) = - \hat{P}_c
\tilde{V}'(\mathbf{r},\omega) \varphi_v^{\circ\,*}(\mathbf{r}),
\label{eq:lin-resp_w_eq2}$$ where use has been made of the reality of the perturbing potential ($\tilde V'(\omega) = \tilde V'^*(-\omega)$). Equations and describe the *resonant* and *anti-resonant* contributions to charge-density response, respectively. Their left-hand sides just differ by the sign of the frequency, while, by using time-reversal symmetry of the unperturbed system ($\varphi_v^{\circ\,*}=\varphi^\circ_v$) their right-hand side can be made look alike. The equations for the resonant and anti-resonant components of the charge-density response are coupled by the HXC potential, which is determined self-consistently by the density response itself, through the relation: $$\tilde V'_{\mathrm{HXC}}(\mathbf{r},\omega) = \int
\kappa(\mathbf{r},\mathbf{r}') \, \tilde{n}'(\mathbf{r}',\omega) \, d\mathbf{r}',
\label{eq:V_HXC_2}$$ where $$\kappa(\mathbf{r},\mathbf{r}') = \frac{1}{|\mathbf{r-r'}|}+
\frac{\delta V_{\mathrm{XC}}(\mathbf{r})}{\delta n(\mathbf{r}')}
\label{eq:HXC_kernel}$$ is the HXC kernel, which we assume to be independent of frequency, consistently with the adiabatic DFT approximation.[@Gross:1985]
The TD KS equations (\[eq:TD-KS\_equation\]) can be equivalently expressed in terms of a quantum Liouville equation for the one-particle density matrix, $\hat{\rho}(t)$:[@Rocca:2008; @Baroni:2012] $$i \, \frac{d\hat{\rho}(t)}{dt} = \left[ \hat{H}_{\mathrm{KS}}(t),
\hat{\rho}(t) \right].
\label{eq:Liouville_eq_1}$$ Upon linearization and Fourier transformation, Eq. (\[eq:Liouville\_eq\_1\]) takes the form: $$(\omega - \hat{\mathcal{L}}) \cdot \hat{\rho}'(\omega) =
\tilde\lambda(\omega) [\hat{V}'_{ext}, \hat{\rho}^\circ ] ,
\label{eq:Liouville_eq_FT_1_general}$$ where $\hat{\rho}^\circ$ is the unperturbed density matrix and $\hat{\mathcal{L}}$ is the Liouvillian super-operator, defined by the relation:[@Rocca:2008; @Baroni:2012] $$\hat{\mathcal{L}} \cdot \hat{\rho}' = [ \hat{H}^\circ, \hat{\rho}' ] +
\left[ \hat V'_{\mathrm{HXC}}[\hat{\rho}'], \hat{\rho}^\circ \right] .
\label{eq:Liouvillian_def}$$ The response of an arbitrary one-electron Hermitian operator, $\hat{A}$, to an external perturbation, $\hat V_{ext}$, is described by the generalized susceptibility: $$\begin{aligned}
\chi_{AV}(\omega) & \equiv
\frac{1}{\tilde\lambda(\omega)} \mathrm{Tr} \bigl
( \hat{A} \hat{\rho}'(\omega) \bigr )
\label{eq:susceptibility_def_1} \\
&=\bigl ( \hat{A}, (\omega - \hat{\mathcal{L}})^{-1} \cdot
[\hat{V}'_{ext}, \hat{\rho}^0 ] \bigl ) ,
\label{eq:susceptibility_def_2}\end{aligned}$$ where $(\cdot,\cdot)$ indicates a scalar product in an abstract operator manifold.[@Malcioglu:2011] Equation states that, within TDDFT, the most general susceptibility can be expressed as an off-diagonal element of the resolvent of the Liouvillian.
The Liouville-Lanczos algorithm
-------------------------------
The calculation of susceptibilities from Eq. requires the explicit representation of the response density matrix and of the Liouvillian super-operator acting on it. The minimum dimension of such a representation is $2\times N_v\times N_c$, where $N_c=N-N_v$ is the number of virtual (conduction) orbitals and $N$ the dimension of one-electron basis set.[@Timrov:Note:2013:2NcxNv] The inversion of the Liouvillian appearing in Eq. is a formidable task in typical large-scale plane-wave calculations, where the number of occupied states can be as large as several hundreds to a few thousands, and the number of virtual orbitals a hundred times as large. The recursion method by Haydock, Heine, and Kelly [@Bullet:1980] offers an elegant solution to a similar problem, namely the calculation of a diagonal element of the resolvent of a Hermitian matrix, in terms of a continued fraction, whose coefficients are frequency-independent. The [*Lanczos bi-orthogonalization algorithm*]{},[@Saad:2003; @Rocca:2008; @Baroni:2012] allows one to generalize this procedure to the calculation of *off-diagonal* elements of the resolvent of a *non-Hermitian* matrix. The resulting numerical workload for calculating the full spectrum in a whole wide frequency range is comparable to that of a *single* ground-state (or static response) calculation. Other flavours of the Lanczos-type algorithm can be found in Refs. .
### The Lanczos bi-orthogonalization algorithm {#sec:Lanczos_method}
We want to calculate matrix elements such as: $$g(\omega) = \left( u, (\omega - L)^{-1} v \right) ,
\label{eq:off_diag_matrix_element_general}$$ where $L$ is a $P\times P$ non-Hermitian matrix, and $u$ and $v$ are generic $P$-dimensional arrays. To this end we define two sets of *Lanczos vectors*, $\{v_j\}$ and $\{u_j\}$, through the recursive relations:[@Saad:2003] $$\begin{aligned}
\beta_{j+1} \, v_{j+1} & = L \, v_j - \alpha_j \, v_j
- \gamma_j \, v_{j-1} , \label{eq:Lanczos_chain_1} \\
\gamma_{j+1} \, u_{j+1} & = L^\top \,
u_j - \alpha_j \, u_j - \beta_j \, u_{j-1} ,
\label{eq:Lanczos_chain_2}\end{aligned}$$ where one defines $u_0=v_0=0$, $u_1=v_1=v$, and the $\alpha_j$, $\beta_j$, and $\gamma_j$ *Lanczos coefficients* are determined by the *bi-orthogonality* conditions $(u_j,v_j)=1$, and $(u_{j-1},v_j)=(u_{j},v_{j-1})=0$. The set of vectors and coefficients generated through the recursion relations (\[eq:Lanczos\_chain\_1\]-\[eq:Lanczos\_chain\_2\]) is often referred to as a *Lanczos chain*. The details of this algorithm are reviewed *e.g.* in Ref. \[\], and its specialization to TDDFT is presented in Refs. \[\]. For the purposes of the present paper, we limit ourselves to observe that the Lanczos vectors thus generated have the property that they provide a tridiagonal representation of the $L$ matrix. More specifically, if we define the $P\times M$ matrices $^{M\!}U = \{ u_1, u_2, \ldots, u_M \}$ and $^{M\!}V = \{ v_1, v_2, \ldots, v_M \}$ ($M$ being the number of Lanczos iterations), one has: $$\left(^{M\!}U\right)^\top L \,\, ^{M\!}V = \,^{M\!}T ,
\label{eq:L_to_T_tridiag}$$ where $^{M\!}T$ is the tridiagonal matrix $$^{M\!}T = \left(\begin{array}{ccccc}
\alpha_1 & \gamma_2 & 0 & \ldots & 0 \\
\beta_2 & \alpha_2 & \gamma_3 & 0 & \vdots \\
0 & \beta_3 & \alpha_3 & \ddots & 0 \\
\vdots & 0 & \ddots & \ddots & \gamma_M \\
0 & \ldots & 0 & \beta_M & \alpha_M
\end{array}\right) .
\label{eq:tridiagonal_matrix}$$ In this Lanczos representation, the matrix element of Eq. can be expressed as:[@Rocca:2008] $$g(\omega) \simeq \left ( ^{M\!}z , \left ( \omega \, ^{M\!}I - \,
^{M\!}T \right )^{-1} \cdot \, ^{M\!}e_1 \right ) ,
\label{eq:resolvent_g}$$ where $^{M\!}e_1 = \{1,0,\ldots,0\}$ and $^{M\!} z$ is the $M$-dimensional vector defined as:[@Rocca:2008; @Baroni:2012] $$^{M\!} z = \, \left( ^{M\!}V \right )^\top u .
\label{eq:zeta_coef_for_g}$$ The right-hand side of Eq. (\[eq:resolvent\_g\]) can be conveniently computed by solving, for any given value of $\omega$, the equation: $$\left( \omega \, ^{M\!}I - \, ^{M\!}T \right) {^{M\!}x} = \, ^{M\!}e_1 ,
\label{eq:eta_post_processing_for_g}$$ and calculating the scalar product: $$g(\omega) = \left ( {^{M\!} z} , {^{M\!} x} \right ) .
\label{eq:resolvent_Liouvillian_Lanczos_2_for_g}$$ The vector $^{M\!} z$, Eq. (\[eq:zeta\_coef\_for\_g\]), can be computed on the fly during the Lanczos recursion, through the relation $z_j=\left
(u,v_j\right )$. In practice, the procedure outlined above is performed in two steps. In the first step, which is by far the most time consuming, one generates the tridiagonal matrix $^{M\!}T$, Eq. (\[eq:tridiagonal\_matrix\]), and the vector $^{M\!}z$, Eq. (\[eq:zeta\_coef\_for\_g\]). In the second step $g(\omega)$ is calculated from Eq. (\[eq:resolvent\_Liouvillian\_Lanczos\_2\_for\_g\]) upon the solution of Eq. (\[eq:eta\_post\_processing\_for\_g\]), for different frequencies $\omega$. In practice, a small imaginary part $\eta$ is added to the frequency argument, $\omega \rightarrow
\omega + i\eta$, so as to regularize the function $g(\omega)$.[@Rocca:2008; @Baroni:2012] Setting $\eta$ to a non-zero value amounts to broadening each individual spectral line or, alternatively, to convoluting the function $g(\omega)$ with a Lorentzian. Because of the tridiagonal form and the small dimension of the matrix $^{M\!}T$ (a few hundreds to a few thousands), the second step is essentially gratis. Different responses to a same perturbation can be computed simultaneously from a same Lanczos recursion, by computing different $z$ vectors on the fly.
### The batch representation {#sec:Batch_repr_general}
Equation shows that the response density matrix is uniquely determined by the two sets of functions $\{
\tilde{\varphi}^\prime_v(\mathbf{r},\omega) \}$ and $\{
\tilde{\varphi}^{\prime\,*}_v(\mathbf{r},-\omega) \}$. It is convenient to consider a linear combination of these functions, defined as: $$\begin{aligned}
q_v(\mathbf{r}) &= \frac{1}{2} \, \bigl (
\tilde{\varphi}^\prime_v(\mathbf{r},\omega) +
\tilde{\varphi}^{\prime\,*}_v(\mathbf{r},-\omega) \bigr )
, \label{eq:batch_q} \\
p_v(\mathbf{r}) &= \frac{1}{2} \, \bigl (
\tilde{\varphi}^\prime_v(\mathbf{r},\omega) -
\tilde{\varphi}^{\prime\,*}_v(\mathbf{r},-\omega) \bigr ) .
\label{eq:batch_p}\end{aligned}$$ The two sets $\{q_v\}$ and $\{p_v\}$ are called respectively the [ *upper*]{} and [*lower*]{} component of the *standard batch representation* (SBR)[@Rocca:2008; @Baroni:2012] of the response density matrix *super-vector*: $\hat\rho'\SBR \bigl\{ \{q_v\},\{p_v\}
\bigr\}$.[@Malcioglu:2011] The SBR of a Hermitian operator, $\hat
A$, has vanishing lower component, $\hat A \SBR \bigl\{ \{ \hat{P}_c \,
\hat A \, \varphi_v^\circ(\mathbf{r}) \}, 0 \bigr\}$, while that of its commutator with the unperturbed density matrix \[see Eq. (\[eq:Liouville\_eq\_FT\_1\_general\])\] has vanishing upper component, $[\hat A, \hat{\rho}^\circ ] \SBR \bigl\{ 0, \{ \hat{P}_c \,
\hat A \, \varphi_v^\circ(\mathbf{r}) \} \bigr\}$. The SBR of the Liouvillian super-operator has the block form:[@Rocca:2008; @Baroni:2012] $$\hat{\mathcal{L}} =
\left(
\begin{array}{cc}
0 & \hat{\mathcal{D}} \\
\hat{\mathcal{D}} + \hat{\mathcal{K}} & 0
\end{array}
\right) ,
\label{eq:Liouvillian_SBR}$$ where the $\hat{\mathcal{D}}$ and $\hat{\mathcal{K}}$ super-operators are defined by their action on response batches, $$\begin{aligned}
\hat{\mathcal{D}} \{ q_v(\mathbf{r}) \} & = \bigl\{ (\hat{H}^\circ -
\varepsilon_v) q_v(\mathbf{r}) \bigr\},
\label{eq:D_super-operator} \\
\hat{\mathcal{K}} \{ q_v(\mathbf{r}) \} & = \Bigl\{
4 \hat{P}_c \sum_{v'}
\int \kappa(\mathbf{r}, \mathbf{r}')
\varphi^{\circ\,*}_{v'}(\mathbf{r}') q_{v'}(\mathbf{r}')
d\mathbf{r'} \, \varphi^\circ_v(\mathbf{r}) \Bigr\} \nonumber \\
%\label{eq:K_super-operator}
%
& = \left \{ \hat{P}_c
V'_\mathrm{HXC}(\mathbf{r}) \varphi^\circ_v(\mathbf{r}) \right \},
\label{eq:K_super-operator'} \end{aligned}$$ $\kappa(\mathbf{r}, \mathbf{r}')$ is the HXC kernel of Eq. (\[eq:HXC\_kernel\]), and $V'_\mathrm{HXC}$ is the HXC potential (see Eq. ) generated by the response charge density distribution whose SBR is (see Eq. ): $$n'(\mathbf{r}) = 4 \sum_v
\varphi^{\circ\,*}_v(\mathbf{r}) \, q_v(\mathbf{r}) .
\label{eq:SBR_charge-density}$$ According to the above equations, operating with the Liouvillian on a test super-vector essentially requires the calculation of the HXC potential response, its application to each valence KS orbital, as well as the operation of the unperturbed Hamiltonian onto twice the number of valence KS states.
The starting super-vector of the Lanczos recursion is the right-hand side of Eq. whose SBR is: $$v_1 = u_1 \SBR
\left(\begin{array}{c}0 \\
\{ \hat{P}_c \, \tilde{V}'_{ext}(\mathbf{r}) \, \varphi_v^\circ(\mathbf{r}) \}
\end{array}\right) .
\label{eq:Lanczos_starting_vect}$$ Because of the special block structure of the Liouvillian, Eq. (\[eq:Liouvillian\_SBR\]), the SBR of odd Lanczos iterates have vanishing upper components, whereas the even ones have vanishing lower components. As a consequence, the number of response wavefunctions onto which the unperturbed Hamiltonian must operate per Lanczos iteration is halved. Also, the diagonal elements of the resulting tridiagonal matrix (the $\alpha$ coefficients) are all vanishing.
### Lanczos-chain extrapolation {#sec:extrapolation}
It was previously noted that the components of the vector $^{M\!}z$, Eq. (\[eq:zeta\_coef\_for\_g\]), decrease rather rapidly to zero, whereas the $\beta_j$ (and $\gamma_j$) coefficients oscillate around two distinct values for odd and even iterations, whose average is approximatively equal to one half of the kinetic-energy cutoff (in a plane-wave implementation), and whose difference is approximately twice as large as the excitation gap in insulating or semiconducting materials.[@Rocca:2008; @Baroni:2012] This finding can be used to speed up considerably the calculation by adopting a suitable extrapolation technique. In practice, the Lanczos recursion is stoped after $M_0$ iterations, such that the components of the $z$ array are small enough. The dimension $M$ of the linear system, Eq. , is then set to a very large (and to a large extent arbitrary) value. The $z$ components from $M_0+1$ to $M$ are set to zero, whereas the corresponding $\beta$ and $\gamma$ coefficients are set to the average of the values that have been actually computed. The accuracy of the calculated spectrum is then checked [*a posteriori*]{} with respect to the value of $M_0$. In many applications it turns out that $M_0$ may vary from a few hundreds up to a few thousands (depending on the plane-wave kinetic energy cutoff), and $M$ is a (to a large extent arbitrary) number reaching up to several thousands. As the solution of tridiagonal systems can be performed very efficiently via standard factorization techniques, the numerical overhead of this procedure is negligible. More on Lanczos extrapolation can be found in Refs. .
A Liouville-Lanczos approach to EEL and IXS spectroscopies in crystals {#sec:LL_approach_for_EELS_IXS}
----------------------------------------------------------------------
In a periodic solid the unperturbed KS orbitals are $\varphi^\circ_v(\mathbf{r}) = \varphi^\circ_{n,\mathbf{k}}(\mathbf{r})$, where $\{v\} = \{n,\mathbf{k}\}$, $n$ is a band index and $\mathbf{k}$ a point in the Brillouin zone. These KS orbitals can be cast into the Bloch form: $$\varphi^\circ_{n,\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}
\, u^\circ_{n,\mathbf{k}}(\mathbf{r}) , \label{eq:Bloch_function}$$ where $u^\circ_{n,\mathbf{k}}(\mathbf{r})$ is the lattice-periodic function. Similarly, the total perturbing potential can be conveniently decomposed into *Bloch components*: $$\tilde{V}'(\mathbf{r},\omega) =
\sum_{\mathbf{q} } \mathrm{e}^{i\mathbf{q}\cdot\mathbf{r}}
\, \tilde{v}'_\mathbf{q}(\mathbf{r},\omega) ,
\label{eq:Vext_q_decomposition}$$ where $\tilde{v}'_\mathbf{q}(\mathbf{r})$ is also lattice-periodic, and the sum extends over the first Brillouin zone. A similar decomposition can be applied to the external and HXC response potentials. The response of each KS orbital can be correspondingly expressed as a linear combination of the responses to each Bloch component of the perturbing potential: $$\tilde\varphi'_{n\mathbf{k}}(\mathbf{r},\omega) = \sum_\mathbf{q}
\mathrm{e}^{i(\mathbf{k}+\mathbf{q}) \cdot \mathbf{r}}
\tilde u'_{n,\mathbf{k+q}}(\mathbf{r},\omega),$$ where $\tilde u'_{n,\mathbf{k+q}}(\mathbf{r},\omega)$ is a lattice-periodic response orbital that satisfies the equation: $$\begin{gathered}
(\hat{H}^\circ_\mathbf{k+q} - \varepsilon_{n,\mathbf{k}} - \omega) \,
\tilde{u}'_{n,\mathbf{k+q}}(\mathbf{r},\omega) = \\
- \hat{P}_c^\mathbf{k+q} \, \tilde{v}'_\mathbf{q}(\mathbf{r},\omega)
\, u_{n,\mathbf{k}}^\circ(\mathbf{r}).
\label{eq:lin-resp_w_eq6}\end{gathered}$$ In Eq. , as well as in the rest of this paper, quantum-mechanical operators bearing a wave-vector subscript (such as $\hat{H}^\circ_\mathbf{k+q}$) or superscript (such as $\hat{P}_c^\mathbf{k+q}$) are thought to operate on lattice-periodic functions, and are defined in terms of their coordinate representations as: $$\begin{aligned}
H^\circ(\mathbf{r},\mathbf{r}') &= \sum_{\mathbf{k}} \mathrm{e}^{i
\mathbf{k}\cdot(\mathbf{r}-\mathbf{r}')}\, {H}^\circ_\mathbf{k}
(\mathbf{r},\mathbf{r}')\,, \\
P_c(\mathbf{r},\mathbf{r}') &= \sum_{\mathbf{k}} \mathrm{e}^{i
\mathbf{k}\cdot(\mathbf{r}-\mathbf{r}')}
P_c^{\mathbf{k}}(\mathbf{r},\mathbf{r}').
\label{eq:Projector_on_empty_states_2}\end{aligned}$$ The projector onto the conduction manifold in Eq. can be expressed in terms of the periodic parts of the unperturbed Bloch functions as: $${P}_c^{\mathbf{k}}(\mathbf{r,r'}) = \delta (\mathbf{r-r'}) - \sum_n
u^\circ_{n,\mathbf{k}}(\mathbf{r}) \,
u^{\circ\,*}_{n,\mathbf{k}}(\mathbf{r}') ,$$ where the sum extends over all the occupied bands. A similar decomposition into Bloch components holds for the response density matrix, which reads in this case: $$\tilde{\rho}'(\mathbf{r},\mathbf{r}';\omega) =
\sum_\mathbf{q}
\mathrm{e}^{i\mathbf{q\cdot(r-r')}}
\tilde\rho'_\mathbf{q}(\mathbf{r},\mathbf{r}',\omega),$$ where $$\begin{gathered}
\tilde\rho'_\mathbf{q}(\mathbf{r},\mathbf{r}';\omega) =
2 \sum_{n,\mathbf{k}} \bigl (
\tilde{u}^\prime_{n,\mathbf{k+q}}(\mathbf{r},\omega) \,
u^{\circ\,*}_{n,\mathbf{k}}(\mathbf{r}') + \\ u^{\circ\,*}_{n,\mathbf{k}}(\mathbf{r}) \,
\tilde{u}^{\prime\,*}_{n,\mathbf{-k-q}}(\mathbf{r}',-\omega) \,
\bigr ) .
\label{eq:rho_prime-periodic}\end{gathered}$$ The anti-resonant contribution to the density-matrix response in Eq. satisfies the equation: $$\begin{gathered}
(\hat{H}^\circ_\mathbf{k+q} - \varepsilon_{n,\mathbf{k}} + \omega) \,
\tilde{u}^{\prime\,*}_{n,\mathbf{-k-q}}(\mathbf{r},-\omega) = \\
- \hat{P}_c^\mathbf{k+q} \, \tilde{v}'_\mathbf{q}(\mathbf{r},\omega)
\, u_{n,\mathbf{k}}^\circ(\mathbf{r}) ,
\label{eq:lin-resp_w_eq7}\end{gathered}$$ which can be obtained from Eq. by complex conjugation and simple manipulations deriving from time-reversal invariance of the unperturbed system ($u^\circ_{n,\mathbf{k}}=u^{\circ\,*}_{n,\mathbf{-k}}$) and the reality of the perturbing potential ($ \tilde v'_\mathbf{q}(\mathbf{r},\omega)=
\tilde v^{\prime \, *}_\mathbf{-q}(\mathbf{r},-\omega)$).
### Batch representation for periodic solids
In analogy with Eq. , Eq. shows that the response density matrix of a periodic solid to a perturbation of wave-vector $\mathbf{q}$ is uniquely determined by the two sets of response orbitals $\{ \tilde{u}'_{n,\mathbf{k+q}} (\mathbf{r}, \omega) \}$ and $\{ \tilde{u}^{\prime\,*}_{n,\mathbf{-k-q}} (\mathbf{r}, - \omega)
\}$. Note that $n$ and $\mathbf{k}$ are running indices, whereas $\mathbf{q}$ is fixed. The SBR can in this case be defined as: $$\begin{aligned}
q_{n,\mathbf{k+q}}(\mathbf{r}) &= \frac{1}{2} \, \bigl (
\tilde{u}'_{n,\mathbf{k+q}}(\mathbf{r},\omega) +
\tilde{u}^{\prime\,*}_{n,\mathbf{-k-q}}(\mathbf{r},-\omega) \bigr ) ,
\\
p_{n,\mathbf{k+q}}(\mathbf{r}) &= \frac{1}{2} \, \bigl (
\tilde{u}'_{n,\mathbf{k+q}}(\mathbf{r},\omega) -
\tilde{u}^{\prime\,*}_{n,\mathbf{-k-q}}(\mathbf{r},-\omega) \bigl ) . \end{aligned}$$ The two sets of response orbitals, $q_\mathbf{q}=\{q_{n,\mathbf{k+q}}
\} $ and $p_\mathbf{q}=\{p_{n,\mathbf{k+q}}\}$ satisfy the coupled set of equations: $$\left(
\begin{array}{cc}
\omega & -\hat{\mathcal{D}}_\mathbf{q} \\
-\hat{\mathcal{D}}_\mathbf{q}-\hat{\mathcal{K}}_\mathbf{q} &
\omega
\end{array}
\right)
%
\left(
\begin{array}{c}
q_\mathbf{q} \\
p_\mathbf{q}
\end{array}
\right) =
%
\left(
\begin{array}{c}
0 \\
y_\mathbf{q}
\end{array}
\right) ,
\label{eq:Liouvillian_eq_SBR_2}$$ where $ y_\mathbf{q} = \{ \hat{P}_c^\mathbf{k+q} \tilde
v'_{ext,\mathbf{q}}(\mathbf{r}) u_{n,\mathbf{k}}^\circ(\mathbf{r}) \} $, and $\hat{\mathcal{D}}_\mathbf{q}$ and $\hat{\mathcal{K}}_\mathbf{q}$ are the super-operators defined by the relations: $$\begin{aligned}
\hat{\mathcal{D}}_\mathbf{q} q_\mathbf{q} & = \left \{
(\hat{H}^\circ_\mathbf{k+q} - \varepsilon_{n,\mathbf{k}}) \,
q_{n,\mathbf{k+q}}(\mathbf{r}) \right \} \label{eq:D_super-operator_2} \\
\hat{\mathcal{K}}_\mathbf{q} q_\mathbf{q} & = \left \{
\hat{P}_c^\mathbf{k+q} \tilde
v'_{\mathrm{HXC},\mathbf{q}}(\mathbf{r})
u^\circ_{n,\mathbf{k}}(\mathbf{r}) \right \},
\label{eq:K_super-operator_2}\end{aligned}$$ and $$\tilde v'_{\mathrm{HXC},\mathbf{q}}(\mathbf{r}) =
\int\kappa(\mathbf{r},\mathbf{r}')
n'_\mathbf{q}(\mathbf{r}') d\mathbf{r}',
\label{eq:v'_{HXC,q}}$$ is the HXC potential generated by the response charge density: $$n'_\mathbf{q}(\mathbf{r}) = 4 \sum_{n,\mathbf{k}}
u^{\circ\,*}_{n,\mathbf{k}}(\mathbf{r}) \, q_{n,\mathbf{k+q}}(\mathbf{r})
. \label{eq:SBR_charge-density_2}$$ Equations , , and are closely parallel to Eqs. , , and of Sec. \[sec:Batch\_repr\_general\].
In practice, the sum over $\mathbf{k}$ points is limited to the portion of the Brillouin zone that is irreducible with respect to the small group of $\mathbf{q}$ and the resulting function symmetrized accordingly, in close analogy with time-independent DFPT for lattice-dynamical calculations.[@Baroni:2001] More about the exploitation of crystal symmetry in the calculation of dynamical charge-density susceptibilities can be found in Ref. .
The $\chi(\mathbf{Q,Q};\omega)$ component of the charge-density susceptibility is obtained from Eq. as the response of the $\mathbf{Q=q+G}$ Fourier component of the charge-density operator, whose coordinate representation reads $\hat{n}(\mathbf{q+G}) \to \mathrm{e}^{i(\mathbf{q+G})\cdot \mathbf{r}}$, to a monochromatic perturbation, $V'_{ext}(\mathbf{r}) =
\mathrm{e}^{i(\mathbf{q+G})\cdot \mathbf{r}}$. The SBR of the periodic part of $\hat{n}(\mathbf{q+G})$ is $\bigl\{ \{ \hat
P_c^\mathbf{k+q} \mathrm{e}^{i\mathbf{G\cdot r}} u^\circ_{n,\mathbf{k}}
\}, 0 \bigr \}$. The final expression for the susceptibility is: $$\chi(\mathbf{Q,Q};\omega) = \Bigl ( \bigl\{ \{ \hat
P_c^\mathbf{k+q} \mathrm{e}^{i\mathbf{G\cdot r}} u^0_{n,\mathbf{k}}
\}, 0 \bigr \}, \bigl \{q_\mathbf{q},p_\mathbf{q} \bigr \} \Bigl ),
\label{eq:susceptibility_Q}$$ where $ \bigl \{q_\mathbf{q},p_\mathbf{q} \bigr \} $ is the solution of Eq. , obtained when the periodic part of the external perturbing potential is $\tilde
v'_{ext,\mathbf{q}}(\mathbf{r})= \mathrm{e}^{i\mathbf{G\cdot r}}$.
In practice the susceptibility in Eq. is computed following the procedure outlined in Sec. \[sec:Lanczos\_method\] (see Eq. ): $$\chi(\mathbf{Q},\mathbf{Q};\omega) \simeq \left(
^{M\!}z_\mathbf{q} , ( \omega \, ^{M\!}I - \, ^{M\!}T_\mathbf{q}
)^{-1} \cdot \, ^{M\!}e_1 \right) ,
\label{eq:resolvent_Liouvillian_Lanczos_q}$$ where $^{M\!}T_\mathbf{q}$ is a tridiagonal matrix of dimension $M$ of the form (\[eq:tridiagonal\_matrix\]), and $^{M\!}z_\mathbf{q} = (
z_{1,\mathbf{q}}, z_{2,\mathbf{q}}, \ldots, z_{M,\mathbf{q}})$ is an $M$-dimensional array whose coefficients $z_{j,\mathbf{q}}$ are defined as: $$z_{j,\mathbf{q}} = \left( \{ \{ \hat{P}_c^\mathbf{k+q}
e^{i\mathbf{G}\cdot\mathbf{r}} u_{n,\mathbf{k}}^\circ(\mathbf{r}) \} ,
0 \} , v_j \right) .
\label{eq:zeta_coef_q}$$
### Metals
The Liouville-Lanczos approach for EEL and IXS spectroscopies can be extended to metals by a suitable generalization of the smearing technique introduced by de Gironcoli in the static case for lattice-dynamical calculations.[@deGironcoli:1995; @Baroni:2001] In the smearing approach, each KS energy level is broadened by a smearing function $(1/\sigma) \, \tilde{\delta}(\varepsilon/\sigma)$, which is an approximation to the Dirac $\delta$-function in the limit of vanishing smearing width $\sigma$. The monochromatic $\mathbf{q}$ component of the charge-density response Eq. (\[eq:SBR\_charge-density\_2\]) can then be cast into the form: $$\begin{gathered}
n'_\mathbf{q}(\mathbf{r}) = 2 \sum_{n,\mathbf{k}}
u^{\circ\,*}_{n,\mathbf{k}}(\mathbf{r}) \bigl(
\overline{u}^\prime_{n,\mathbf{k+q}}(\mathbf{r},\omega)
\bigr. \\
\bigl. + \,
\overline{u}^{\prime\,*}_{n,\mathbf{-k-q}}(\mathbf{r},-\omega) \,
\bigr) , \label{eq:appendix_charge-dens-response_4}\end{gathered}$$ where the functions $\overline{u}^\prime_{n,\mathbf{k+q}}(\mathbf{r},\omega)$ and $\overline{u}^{\prime\,*}_{n,\mathbf{-k-q}}(\mathbf{r},-\omega)$ satisfy the equations: $$\begin{aligned}
(\hat{H}^\circ_{\mathbf{k+q}} -\varepsilon_{n,\mathbf{k}} & - \omega ) \,
\overline{u}'_{n,\mathbf{k+q}}(\mathbf{r}, \omega) = \nonumber\\
& \quad - \bigl( \tilde{\theta}_{F;n,\mathbf{k}} -
\hat{P}_{n,\mathbf{k}}^{\mathbf{k+q}} \bigr) \,
\tilde{v}'_\mathbf{q}(\mathbf{r},\omega) \,
u^\circ_{n,\mathbf{k}}(\mathbf{r}) , \\
(\hat{H}^\circ_{\mathbf{k+q}} -\varepsilon_{n,\mathbf{k}} & + \omega ) \,
\overline{u}^{\prime\,*}_{n,\mathbf{-k-q}}(\mathbf{r}, -\omega) =
\nonumber \\
& \quad - \bigl( \tilde{\theta}_{F;n,\mathbf{k}} -
\hat{P}_{n,\mathbf{k}}^{\mathbf{k+q}} \bigr) \,
\tilde{v}'_\mathbf{q}(\mathbf{r},\omega) \,
u^\circ_{n,\mathbf{k}}(\mathbf{r}) \end{aligned}$$ (cf. with Eqs. and ), where $$\begin{aligned}
\hat{P}_{n,\mathbf{k}}^{\mathbf{k+q}} & = \sum_m^{occ}
\beta_{n,\mathbf{k};m,\mathbf{k+q}} | u^\circ_{m,\mathbf{k+q}} \rangle
\langle u^\circ_{m,\mathbf{k+q}} | , \\
\beta_{n,\mathbf{k};m,\mathbf{k+q}} & =
\tilde{\theta}_{F;n,\mathbf{k}}
\tilde{\theta}_{n,\mathbf{k};m,\mathbf{k+q}} \nonumber \\
& \qquad \qquad \qquad + \, \tilde{\theta}_{F;m,\mathbf{k+q}}
\tilde{\theta}_{m,\mathbf{k+q};n,\mathbf{k}} , \end{aligned}$$ $\tilde{\theta}_{F;n,\mathbf{k}} \equiv \tilde{\theta} [(\varepsilon_F
- \varepsilon_{n,\mathbf{k}})/\sigma]$ and $\tilde{\theta}_{m,\mathbf{k+q};n,\mathbf{k}} \equiv
\tilde{\theta}[(\varepsilon_{m,\mathbf{k+q}} -
\varepsilon_{n,\mathbf{k}})/\sigma]$ being smooth approximations to the step-function, and $\varepsilon_F$ is the Fermi energy. It can be easily verified that the coefficients $\beta_{n,\mathbf{k};m,\mathbf{k+q}}$ vanish when any of its indices refers to an unoccupied state. Therefore, the operator $\hat{P}_{n,\mathbf{k}}^{\mathbf{k+q}}$ involves only a small number of partially occupied bands, and the first-order variation of the wavefunctions and of the charge density can be computed avoiding any explicit reference to unoccupied states, much in the same way as for insulating materials. More details about the Liouville-Lanczos approach for metals can be found in Ref. .
\[sec:applications\] Application to bulk Si and Al
==================================================
The technique described above has been implemented in the <span style="font-variant:small-caps;">Quantum ESPRESSO</span> suite of computer codes,[@Giannozzi:2009] and is scheduled to be distributed in one of its future releases. We now proceed to validate it by calculating the loss function in bulk silicon and aluminum, for which several TDDFT studies exist, and whose spectra are known to be accurately described within the adiabatic local density (LDA) and generalized gradient (GGA) approximations (see, *e.g.*, Refs. for Si, and for Al).
All the calculations have been performed within the LDA approximation, using the Perdew-Zunger parameterization of the electron-gas data,[@Perdew:1981] norm-conserving pseudopotentials from the <span style="font-variant:small-caps;">Quantum ESPRESSO</span> database[@Timrov:Note:2013:PP] and plane-wave basis sets up to a kinetic-energy cutoff of 16 Ry. The first Brillouin zone has been sampled with a Monkhorst-Pack (MP) $\mathbf{k}$ point mesh, supplemented, in the case of Al, by the Methfessel-Paxton smearing technique[@Methfessel:1989] with a broadening parameter $\sigma=0.02$ Ry. The frequency argument of the susceptibility has been assumed to have a small imaginary part, $\eta$, thus resulting in a Lorentzian smearing of the spectra (see Sec. \[sec:Lanczos\_method\]). For both Si and Al we have used the experimental lattice parameters (10.26 a.u.[@Neuberger:1971] and 7.60 a.u.,[@Wyckoff:1963] respectively), which is very close to the theoretical one and resulting in no appreciable difference in the computed spectra.
Bulk silicon
------------
Figure \[fig:Si\_conv\_iter\] shows the convergence of the loss spectrum of Si, as calculated for a transferred momentum $Q=0.53$ a.u. along the \[100\] direction, as a function of the number of Lanczos iterations. After 400 iterations the spectrum displays spurious wiggles, which disappear by increasing the number of iterations up to 1500. Also displayed are results obtained by the extrapolation procedure outlined at the end of Sec. \[sec:Batch\_repr\_general\], performed with $M_0=400$ Lanczos iterations and extrapolating the results up to a linear system of dimension $M=5000$. We see that the numerical workload can be considerably reduced without any appreciable loss of accuracy. In Fig. \[fig:Si\_conv\_k\] we show the convergence of the loss function with respect to the $\mathbf{k}$ point sampling of the Brillouin zone. The $4 \times 4 \times 4$ MP $\mathbf{k}$ point mesh is not dense enough to obtain a well-converged spectrum, due to the presence of spurious wiggles, which disappear by increasing the size of the MP mesh up to $10 \times 10 \times 10$.
In Fig. \[fig:Si\_theor\_vs\_exp\] we compare our present results with those obtained from the conventional approach based on the Dyson-like equation for the susceptibility[@Onida:2002; @Weissker:2010] and with experiment.[@Weissker:2010] The agreement is excellent in both cases. All the salient features observed in the experiments at small transferred momentum (panel (a)) are correctly predicted: the main plasmon peak around 20 eV, a shoulder around 15 eV, and a weak peak around 6.5 eV. We attribute the slight differences between the two theoretical spectra to the slightly different technical details used in the two works. In particular, the authors of Ref. mimicked electron- and hole-lifetime effects with an energy-dependent broadening, in contrast to the constant Lorentzian broadening, $\eta =
0.035$ Ry, used in our calculations. At larger momentum transfer (panel (b)) the interaction of the plasmon with the electron-hole continuum broadens the spectrum.[@Mahan:1990] The agreement with experiment,[@Timrov:Note:2013:IXS] remarkable also in this case, is enhanced by increasing the Lorentzian broadening up to $\eta =0.080$ Ry, which allowed us to reduce the size of the MP mesh down to $6 \times 6
\times 6$ without any appreciable loss of accuracy.
Bulk aluminum
-------------
Figure \[fig:Al\_conv\_iter\] shows the convergence of the loss function of Al, calculated at a transferred momentum $Q=0.513$ a.u. along the \[100\] direction, as a function of the number of Lanczos iterations. Although the qualitative behavior is similar to that observed in Si (wiggles showing up for a small number of iterations disappear by increasing this number), the convergence appears to be faster in the present case. As for the large-iterate behavior of the Lanczos coefficient, we observe that, in contrast to Si, in Al the odd and even coefficients oscillate around a same value, which is also in this case of the order of one half the plane-wave kinetic-energy cutoff. This is due to the vanishing of the gap, as discussed in Ref. . Figure \[fig:Al\_conv\_k\] shows the convergence with respect to the size of the $\mathbf{k}$ point mesh: very satisfactory convergence is achieved with $10 \times 10 \times 10$ MP mesh and a broadening parameter $\eta = 0.056$ Ry.
In Fig. \[fig:Al\_theor\_vs\_exp\] we compare the loss function of Al as calculated by the present method for two different values of the transferred momentum along the \[100\] direction, with IXS experiments and with previous theoretical work. At small transferred momentum (panel (a)) theoretical predictions agree remarkably well with each other (the slight discrepancies being attributable to the usual small differences between the technical details of the calculations) and with experiment. Both theoretical spectra display a small blueshift ($\sim 0.5$ eV) of the plasmon peak with respect to experiments. At larger transferred momentum (panel (b)) the theoretical spectra display a feature at $\sim 24$ eV, which is not observed experimentally. We attribute the remaining discrepancies to the lifetime effects,[@Cazzaniga:2011] which have been treated in our calculations by a constant Lorentzian broadening parameter ($\eta = 0.068$ Ry, requiring a $14 \times 14 \times 14$ MP mesh).
\[sec:conclusions\] Concluding remarks
======================================
We believe that the Liouville-Lanczos approach introduced in this paper will open new perspectives in the calculation of loss spectra in extended systems. Its main features are the adoption of a representation for the charge-density response borrowed from density-functional perturbation theory, and of a Lanczos recursion scheme for computing selected elements of the inverse of (very) large matrices. The combination of these two elements permits to compute the loss spectrum of a given system, for a given transferred momentum, and for an *entire wide frequency range*, with a numerical workload of the same order as needed for a standard ground-state calculation for a same system (the pre-factor being only a few times larger). In principle, the convergence of the computed loss spectra with respect to the length of the Lanczos chains depends on the spectral range: the lower the frequency, the faster the convergence, as it was already observed for optical spectra in finite systems.[@Walker:2006; @Rocca:2008] In practice, however, adoption of the extrapolation techniques explained in Sec. \[sec:extrapolation\] substantially alleviates this dependence. Also, the spectral range accessible to EEL/IXS spectroscopies is limited by the so-called $f$-sum rule:[@Mahan:1990] $$\int\limits_0^\infty \mathrm{Im} \left[
\epsilon^{-1}(\mathbf{Q},\mathbf{Q};\omega) \right] \omega \,
d\omega = - \frac{\pi}{2} \, \omega_p^2 ,
\label{eq:f_sum_rule}$$ where $\omega_p = 4\pi e^2 n_e/m$ is the plasma frequency, $n_e$ being the average electron density, [*i.e.*]{} the number of electrons (*valence* electrons, in a pseudopotential calculation) per unit volume.[@Timrov:Note:2013:f-sum] Of course, the spectral range that needs to be sampled by Lanczos recursion is correspondingly limited.
The Liouville-Lanczos approach introduced in this paper also lends itself to an easy generalization to those methods (such as hybrid functionals or the static Bethe-Salpeter equation – BSE) that require the full density-matrix (rather than just charge-density) response, which is in fact as easily accessible to the batch representation utilized here.[@Rocca:2012] Further generalization to frequency-dependent XC kernels (or to the BSE with dynamical screening) may simply require computing the loss function at shifted frequencies ($\omega'=\omega+\Sigma(\omega)$), as proposed *e.g.* in Ref. , or further methodological developments. Further work is required to clarify this issue.
All in all we believe that the advances presented in this paper will allow for the simulation of complex, possibly nano-structured, surfaces, as well as of systems where valence and shallow-core loss spectra overlap. Examples of the former include low Miller index surfaces or plasmonic materials, while bulk bismuth is an example of the latter. Work is in progress on both lines.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
We thank S. de Gironcoli, A. Dal Corso, and L. Reining for valuable discussions. Support from the ANR (Project PNANO ACCATTONE) and from DGA are gratefully acknowledged. Computer time was granted by GENCI (Project No. 2210). The work of I.T. and N.V. has been performed under the auspices of the *Laboratoire d’excellence en nanosciences et nanotechnologies Labex Nanosaclay*. N.V. thanks Marco Saitta for discussions about TDDFPT at an early stage of the project. S.B. gratefully acknowledges hospitality at the *Laboratoire des Solides Irradiés* of the École Polytechnique, where this paper was written.
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---
abstract: ', the compact radio source, believed to be the counterpart of the massive black hole at the galactic nucleus, was observed to undergo rapid and intense flaring activity in X-rays with Chandra in October 2000. We report here the detection with XMM-Newton EPIC cameras of the early phase of a similar X-ray flare from this source, which occurred on September 4, 2001. The source 2-10 keV luminosity increased by a factor $\approx$ 20 to reach a level of 4 10$^{34}$ erg s$^{-1}$ in a time interval of about 900 s, just before the end of the observation. The data indicate that the source spectrum was hard during the flare. This XMM-Newton observation confirms the results obtained by Chandra and suggests that, in , rapid and intense X-ray flaring is not a rare event. This can constrain the emission mechanism models proposed for this source, and also implies that the crucial multiwavelength observation programs planned to explore the behaviour of the radio/sub-mm and hard X-ray/gamma-ray emissions during the X-ray flares, have a good chance of success.'
author:
- 'A. Goldwurm, E. Brion, P. Goldoni, P. Ferrando, F. Daigne, A. Decourchelle,'
- 'R.S. Warwick,'
- 'P. Predehl'
title: 'A New X-Ray Flare from the Galactic Nucleus Detected with the XMM-Newton Photon Imaging Cameras'
---
Introduction
============
The bright, compact and variable radio source is believed to be the radiative counterpart of the 2.6 10$^{6}$ M$_{\odot}$ black hole which governs the dynamics of the central pc of our Galaxy [@mefa01]. The compelling evidence for the presence of a dark mass concentration at the Galactic Center [@gen97; @ghe00], which implies the presence of a massive black hole, contrasts remarkably with the weak high-energy activity of such an extreme object. In spite of the fact that some amount of material, either provided by stellar winds from a close stellar cluster or by the hot surrounding medium, is probably feeding a moderate/low level of accretion, the total bolometric luminosity of the source amounts to less than 10$^{-6}$ of the estimated accretion power [@mefa01; @gol01].
This has motivated the development of several black hole accretion flow models with low radiative efficiency, some of which have also been applied to binary systems, low luminosity nuclei of external galaxies and low luminosity active galactic nuclei. These models include spherical Bondi accretion in conditions of magnetic field sub-equipartition with a very small Keplerian disk located within the inner 50 Schwarzschild radii (R$_S$), large hot two-temperature accretion disks dominated by advection (ADAF) or non-thermal emission from the base of a jet of relativistic electrons and pairs, and some other variants or combination of the above (see the review by Melia $\&$ Falcke (2001)). However any such model still predicts some level of X-ray emission from and determining the properties of such emission would constrain the theories of accretion and outflows in the massive black holes and in general in compact objects.
The search for high energy emission from with focussing X-ray telescopes dates back to the end of the ’70s [@pre94; @bag01a; @gol01], but has recently come to a turning point with the remarkable observations made with the Chandra X-ray Observatory in 1999 and in 2000. Baganoff et al. (2001a) first reported the detailed 0.5$''$ resolution images obtained with Chandra in the range 0.5-7 keV, which allowed, finally, the detection of weak X-ray emission from the radio source. The derived luminosity in the 2-10 keV band was 2 10$^{33}$ erg s$^{-1}$, for a distance of 8 kpc, and the measured spectrum was steep, with power law photon index of 2.7. Marginal evidence that the source is extended on a 1$''$ scale was also reported, but at low significance level. Then, in October 2000, the same source was seen to flare up by a factor of $\approx$ 45 in a few hours [@bag01b]. The luminosity increased from a quiescent level similar to the one measured in 1999 to a value of 10$^{35}$ erg s$^{-1}$. The flare lasted a total of 10 ks but the shortest variation took place in about 600 s, implying activity on length scales of $\approx$ 20 R$_S$, for the above quoted mass of the galactic center black hole. Evidence of spectral hardening during the flare was also reported by the authors who determined a source power law photon index during the event of 1.3 ($\pm$ 0.55), significantly flatter than observed during the quiescent state. These results constrain models of the accretion flow and radiation mechanism for . A confirmation of the Chandra results and in particular a better determination of the flaring properties of the source are therefore crucial for the modeling of the physics of the Galactic Center and in general for the theories of accretion in black hole systems.
XMM-Newton, the other large X-ray observatory presently in operation, features three large area X-ray telescopes coupled to three CCD photon imaging cameras (EPIC) operating in the 0.1-15 keV range and to two reflection grating spectrometers (RGS) working in the 0.1-2.5 keV band [@jan01]. Although its angular resolution (6$''$ FWHM) is insufficient for properly resolving in quiescence, the high sensitivity and wide spectral range of XMM-Newton allow deep studies of the X-ray emission of such a complex and crowded region like the Galactic Center. Indeed an intense flare such as the one seen by Chandra can be easily detected with XMM-Newton thanks to its large effective area, and its timing and spectral properties can be studied.
The Galactic Center region is one of the priority targets of the XMM-Newton mission and was included in the guaranteed time program. Visibility constraints and solar flare events have however delayed the monitoring of the very center of our Galaxy. A complete pointed observation was finally performed in fall 2001 and in this letter we report the detection with XMM-Newton of another X-ray flare from which occurred at this time.
Observations and Results
========================
XMM-Newton was pointed towards the galactic nucleus for about 26 ks on 4$^{th}$ September 2001. This observation was part of a large survey program of about 10 overlapping XMM-Newton pointings planned to map the Galactic Plane within 1$^{\circ}$ from the Center. Preliminary mosaiced images of the region have been presented by Warwick (2002) who showed that the region is complex and dominated by diffuse emission and some point-like and extended bright sources. We report here results obtained with the EPIC cameras of XMM-Newton during the observation of the survey which was directly pointed towards (observation GC6). The purpose of our analysis was to search for X-ray variability from this source of the type observed during fall 2000 with the Chandra telescope.
The observation with the EPIC MOS cameras [@tur01] started at 01:27:08 (UT) and lasted 26127 s while the PN [@str01] was activated 4109 s after the MOS for a total exposure of 21748 s. The EPIC cameras were used in standard [*Full Frame*]{} imaging mode with the medium filter while the RGS was used in [*spectral+Q*]{} mode and the OM was blocked. Data reduction was performed using the XMM-Newton Science Analysis Software (SAS) package (version 5.3), using standard choices to select events for both MOS and PN cameras, namely the 0 to 12 patterns for the MOS and both the single and double events for the PN. Inspection of the integrated count rate of the CCDs versus time revealed that the observation was slightly perturbed by weak soft proton flares. The CCD light curves show several peaks all along the observation and in particular at the beginning and at the end of it. The average combined count rate in the central CCD of the two MOS cameras was about 4.50 cts s$^{-1}$, while peaks due to the proton flares reached maximum values of 21 cts s$^{-1}$ for short time intervals. The counts arising from the flares are distributed fairly uniformly across the CCDs (i.e. with only a modest vignetting effect) and indeed derived images did not show particular features due to these background events. The image recorded in the central CCD (11$'$ $\times$ 11$'$ for the MOS) is dominated by the diffuse emission of the Sgr A East region which is thermal in origin and rich in emission lines. Proper analysis of this component involves modelling of the variable background and composition of different images of the survey. The work is in progress and results will be reported elsewhere. Here we rather concentrate on the analysis of the central point-source, , and in particular on the search for variability of its X-ray emission.
In order to optimize the signal to noise of the central source in the presence of the strong diffuse emission and considering the typical width of the XMM-Newton point spread function (15$''$ half power diameter), we extracted and analyzed light curves of events collected within a 10$''$ radius region centered on . As shown in Fig. 1, the 2-10 keV count rate from the combined MOS 1 and MOS 2 events selected in this way, is quite stable around an average value of 0.08 cts s$^{-1}$ till the last 900 s of the observation. Then the count rate gradually increases to reach a value of about a factor 3 higher in the last bin. The integrated count rate in the last 900 s reaches 7 $\sigma$ over the average value measured before the flare and the detected variation has a very low probability to be a statistical fluctuation.
A similar light curve, from a 9 times larger region of the central CCD, far from the source but including a bright part of the diffuse emission, is also reported for comparison in Fig. 1 (scaled for clarity), and does not show any evidence of such an increase in the counting rate. The same trend is observed in the counts extracted from the PN (see Fig. 1 right). Since the PN camera stopped observing about 250 s before the MOS, the last part of the flare is not visible. However the increase in the last (PN) 600 s is also highly significant (4.3 $\sigma$) and again it is not detected in counts extracted from a region away from the source. Some proton flares occurred in the last part of the observation, and they increase the total CCD 2-10 keV count rate of about a factor 1.4 in the last 1000 s. However a detailed light curve of these events show that they occur with a different time behavior than the source flare and, because of their uniform spatial distribution, they give rise to only a $\approx$ 1.5 $\%$ increase in the measured count rate within the 10$''$ radius region during the last 900 s of the observation.
In order to check that the flare actually originated in , we constructed images using MOS events selected in different periods of the observation. In Fig. 2 we report an image of the region around the nucleus integrated during the 1000 s before the flare and a similar image integrated during the last 1000 s and fully including the source flare. The basic image pixels were rebinned by a factor 5, giving image pixels sizes of 5.5$''$ width. The brightening we detected in the light curves is clearly due to the brightening of a central source with the counts in the central pixel of the image which increase from 10 before the flare to 40 during the event. The general level of the image during the flare is a factor 1.4 times higher than before the flare (for pixels $>$ 20$''$ from ) due to the proton flaring events. This difference corresponds to the uniform increase in counts due to the presence of the proton flares in the last part of the observation.
We also used the SAS procedure ([*eboxdetect*]{}) for source detection to determine the location of the excess. On the 2-10 keV MOS 1 and MOS 2 image of the last 1000 s, rebinned to have pixel size of 4$''$, we obtained the centroid of the source at R.A. (2000) = 17$^h$ 45$^m$ 39.99$^s$ Dec (2000) = -29$^{\circ}$ 00$'$ 26.7$''$, with a pure statistical error of 0.4$''$ (90$\%$ confidence level in one parameter) in each direction. To check that the attitude reconstruction does not suffer from large systematic errors, we checked the full low energy (0.5 - 4 keV) MOS images for X-ray sources with stellar counterparts. In the full field of view of MOS 1 we identified 6 point-like sources with stars of known position, we determined the offsets between our derived positions and optical positions, and computed the average and rms values. We obtained an average offset of -0.03$''$ in right ascension and 0.20$''$ in declination, with rms of 0.29$''$ (R.A.) and 1.50$''$ (in Dec.). This implies that the absolute accuracy of location for the central CCD is not worse than the residual systematic uncertainties in the XMM-Newton focal plane, estimated to be 1.5$''$ [@kir02]. The derived flare position is therefore compatible with radio location [@zad99], since it is offset from the latter by only 1.5$''$ (of which 1.4$''$ in Dec.), i.e. within uncertainties.
The detailed study of Chandra data carried out by Baganoff et al. (2001a) showed that a number of other X-ray point sources are present in the vicinity of the galactic nucleus, the nearest of which is associated with the infrared and radio object called IRS 13 at an angular distance of 4$''$. The error box we derived with XMM-Newton excludes the possibility that the observed flare is due to IRS 13 (offset $>$ 4$''$) and also is not consistent with another region of excess counts in the Chandra images (offset $\approx$ 3$''$), not fully recognized as a source but tentatively identified with IRS 16SW by Baganoff et al. (2001a). Of course this does not completely rule out the possibility that another X-ray source in this crowded region of the sky may be the origin of the event. However due to the close resemblance of the event with the Chandra flare, which was more precisely located on , and considering that generally X-ray binaries do not show such large variations on such short timescales, it seems unlikely that another X-ray source in the central star cluster is responsible for this flare. We conclude that the flare detected by XMM-Newton is associated with .
A first spectral analysis of the flaring event was performed by computing a simple hardness ratio (H/S), defined as the ratio between the measured counts (including background) in the hard band 4.5-10 keV, and those in the soft band 2-4.5 keV. Using events collected by both MOS and PN cameras within 10$''$ from we computed the variation of the hardness ratio during the flare compared to the average value measured before the flare. The measure was performed separately for MOS (flare during last 900 s) and PN (last 650 s) events and then we computed the weighted average. The hardness ratio increased by 0.32 $\pm$ 0.127 during the event with respect to the value before the flare. Though the count rate hardening has a modest statistical significance of 2.5 $\sigma$, it is fairly consistent with the trend observed with Chandra for the flare. Unlike Baganoff et al. (2001b) who observed the flare at low energy to follow of few hundred seconds the flare at high energy, we have not revealed any significant lag between the light curve of the soft energy band and the one of the high energy band.
To derive a spectrum of the source during the flare state we have to model the emission which is not due to . Detailed analysis of Chandra data showed that only $\approx$ 10$\%$ of the 2-10 keV emission measured within 10$''$ from the galactic nucleus is due to in quiescence, the rest is mainly due to the local diffuse component (60$\%$), to the contribution of the six other point sources seen by Chandra (20$\%$) and to more diffuse Galactic emission together with the instrumental background (10$\%$). To model the thermal component Baganoff et al. (2001a) used the Raymond $\&$ Smith hot gas spectral model with twice solar abundance [@ray77]. The contribution from six point sources in this region was modeled with a power law of photon index $\approx$ 2.5, while the contribution from was best fitted by a power law of index 2.7 and N$_H$ = 9.8 10$^{22}$ cm$^{-2}$. The Chandra derived best fit parameters (see Table 3 of Baganoff et al. 2001a) were used to evaluate, by convolving the model with XMM-Newton responses, the expected count rates in the 2-10 keV band before the flare. We obtained values very close to the measured ones (demonstrating that the contribution of the instrumental background is not a major influence for this region of high X-ray surface brightness) and therefore we adopted a similar model to fit XMM-Newton data.
We extracted MOS and PN count spectra from the 10$''$ radius circular region centered on before the flare and during the flare (last 900 s for MOS and last 700 s for PN). For the spectra before the flare, rebinned to have 30 counts per bin, we used the above model for which we fitted all parameters simultaneously (tied) on MOS and PN data, apart from the parameters which were kept frozen to Chandra best fit values (with the norm reduced by factor 0.6 to account for the encircled energy loss). Normalizations were left to vary untied between MOS and PN data to allow adjustment to different instrument background and residual normalization differences not accounted for by the responses, and we obtained the best fit parameters reported in Table 1 (left column). The MOS spectrum, compared to the best fit model, is reported in Fig. 3. Allowing the normalization of the power-law to vary freely in the fitting (with fixed slope at 2.7 and fixed N$_H$ at 9.8 10$^{22}$ cm$^{-2}$) we obtained a very low normalization value, indicating that this component, if present, cannot be disentangled from the first power-law supposed to describe point sources and other residual background components. With a reduced $\chi^{2}$ of 1.35 for 121 dof, the fit is acceptable. The best fit parameters match well the values obtained by Chandra, in particular the gas temperature kT of 1.3 keV and its column density are fully compatible with Chandra results, while the power law spectrum for the point sources is slightly flatter and needs a rather higher N$_H$. In the procedure we adopted, this power-law describes also the background and we do not expect it to fully represent the point sources contribution.
We then fixed these parameters, let free the power-law slope and normalization and fit the spectrum extracted during the flare. The fit was performed simultaneously on MOS and PN data of the flare, rebinned to have 10 counts per bin, and fixing the column density to the value measured by Chandra. The power-law normalizations for MOS and PN data were left untied since a different portion of the flare was seen by MOS and PN cameras and we expect different average intensity. We obtained the parameters reported in Table 1 (right column). The MOS count spectrum during the flare, and its best fit model, is shown in Fig. 3, compared to the MOS spectrum before the flare. The power-law photon index of the flaring source is 0.9 $\pm$ 0.5 (error at 1 $\sigma$ for one interesting parameter) which is significantly harder than the spectrum measured with Chandra during the quiescent state (2.7 $\pm$ 1.0), and rather compatible, within uncertainties, to the index measured during the 2000 October flare. By allowing the column density to vary freely during the fit we obtained a lower value for (N$_H$ = 6.4 $^{+4.0}_{-3.3}$ 10$^{22}$ cm$^{-2}$) and an even harder slope. This value is consistent with the value determined by Chandra and which we use to fix the column density in the fits.
Similar parameters were obtained by simply using the spectra extracted before the flare as background components for the flare spectra. After subtraction of the non-flaring count spectrum, the flaring spectrum was fitted with a simple absorbed power law with N$_H$ fixed to the Chandra measured value and leaving untied the MOS and PN normalizations. Results are reported in Table 2, left column. We obtained a photon index of 0.68 $^{+0.53}_{-0.60}$ compatible with the estimate reported in Table 1. This procedure subtracts from the flare spectrum the non flaring component of and therefore assumes that the quiescent emission from is negligible. This is an acceptable approximation since, if the emission level of is comparable to the one observed by Chandra in 1999, it is expected to contribute by only $\approx$ 5$\%$ to the counts of the flare spectrum. On the other hand this procedure allows to subtract the diffuse emission present in the region of the spectral extraction and the instrumental background in a model-independent way. In any case the above results and derived fluxes, are equivalent, within errors, to those obtained adding a power-law component to the non-variable emission model. To directly compare our results with the spectra of the flare observed with Chandra, we also fitted the above spectra with an absorbed power law model modified by the effect of dust scattering [@pre95]. We fixed the parameter of the dust scattering model, the visual extinction A$_V$, to the Galactic Center canonical value of 30 magnitudes and the column density for absorption to N$_H$ = 5.3 10$^{22}$ cm$^{-2}$ as found by Baganoff et al. (2001b). We then fitted the model on the MOS and PN spectra during the flare using the spectra before the flare as background component. The best fit is found for an even harder power law slope of 0.31 (see Table 2, right column). Letting the N$_H$ to vary freely, again we find a best fit value for the column density around 5 – 6 10$^{22}$ cm$^{-2}$.
To compute the observed flux and luminosity we used the normalization value derived from the MOS data, since the MOS observed a bigger fraction of the flare, and we corrected for the fraction of encircled energy at a distance of 10$''$ (60$\%$). We note that we did not corrected for the energy dependence of the encircled energy, which will tend to slightly harden the spectrum, however the statistical errors is by far larger than the systematic bias induced by this procedure. The measured absorbed source flux in the 2-10 keV band is then of (3.3 $\pm$ 0.6) 10$^{-12}$ ph cm$^{-2}$ s$^{-1}$, equivalent to a 2-10 keV luminosity at 8 kpc of (3.8 $\pm$ 0.7) 10$^{34}$ erg s$^{-1}$ (1 $\sigma$ errors computed by fixing all other parameters but the power-law normalization at the best fit values listed in the right column of Table 1). This is the average value in the last 900 s but the last light curve 180 s bin was about a factor 1.4 higher, thus the luminosity reached a value of 5.4 10$^{34}$ erg s$^{-1}$.
These numbers are subject to large errors due to the low statistics available. But the general result which emerges is that the flare we detected presents a harder spectrum than the one measured with Chandra for during the quiescent period. The measured slope is even harder than typically found in X-ray spectra of AGN, however, considering the large uncertainties, this result is not compelling.
Discussion
==========
The XMM-Newton discovery of a new X-ray flare of in September 2001 confirms the results obtained in the earlier Chandra observations. XMM-Newton observed only the first part of the flare, but the recorded event is fully compatible in intensity and time scale with the early phase of the flare seen by Chandra.
The count rate within 10$''$ from increased in 900 s by a factor 3, but if attributed to it implies that this source brightened by a factor about 20-30, which is compatible with the increase in the first 1000 s of the flare observed by Chandra. We have not detected the maximum in the flare rise. Therefore we cannot strictly apply the travel light argument to estimate the size of the emitting region. If we assume that the flare duration (900 s) we observed is the shortest time scale of variation of the present event, it corresponds to a size of about 30 R$_S$. This limit is a factor 1.5 larger than the shortest scale estimated with the Chandra data and does not constrain further the geometry of the region.
On the other hand the detection of another such a flare indicates that the event is not rare. The total reported observation time with Chandra amounts to $\approx$ 75 ks. Considering the XMM-Newton 26 ks exposure, the duty cycle of such event is 0.11 (= 11 ks / 101 ks), but it would increase to 0.18 (= 20 ks / 110 ks) if we assume that the flare we detected for only 1000 s would last for 10 ks. Though not much different than the value determined with Chandra, this estimate of the active time fraction of the source is now based on 2 events and it is therefore more significant.
The radio source on the other hand has been observed many times and the detected flux variability has never exceeded a factor 2 [@zha01]. This implies that it is unlikely that radio or sub-mm emission present a comparable increase in flux. If this is confirmed the flare may not be due to a change in the accretion rate, since this variation would lead, at least in models which attribute the bulk of X-ray emission to self synchrotron Compton emission, to a comparable increase of radio and sub-mm radiation [@mar01].
The X-ray flare from cannot be explained by pure ADAF models (Narayan et al. 1998) as in these models the emission is due to thermal bremsstrahlung from the whole accretion flow and arises from an extended region (between 10$^3$ - 10$^5$ R$_S$) which cannot account for such rapid variability. Models which predict emission from the innermost regions near the black hole involve a mechanism acting either at the base of a jet of relativistic particles [@mar01] or in the hot Keplerian flow present within the circularization radius of a spherical flow [@mel01; @liu02]. In both cases a magnetic field is present in the flow and the sub-mm radiation is attributed to optically thin synchrotron emission from the inner region, while the X-rays are produced by the synchrotron self-Compton (SSC) mechanism whereby radio to mm photons are boosted to X-ray energies by the same relativistic or subrelativistic electrons that are producing the synchrotron radiation. Large flux variations can be produced by a change in accretion rate or, in the jet model, by additional heating of the electrons caused for example by magnetic reconnection. The second mechanism would increase (and harden) the X-ray flux without significantly increasing the radio and sub-mm part of the spectrum and therefore it could be more compatible with the lower amplitude of radio changes compared to X-rays [@mar01].
However even emission from a circularized flow can provide low or anti correlation of the radio emission with the X-rays if the radiation mechanism for the X-rays is bremsstrahlung rather than SSC [@liu02]. The sub-mm and far IR domain on the other hand would in this case show a large correlated increase, but at these frequencies the measurements have not been frequent enough to settle the issue. Though the exact modelling of radiation process depends on viscosity behavior and other uncertain details, the observed hardening of the spectrum during the flare indeed favours the bremsstrahlung emission mechanism in this model rather than the SSC one [@liu02]. More compelling constraints on the models will be set when simultaneous observations in radio/sub-mm and X-ray wavelengths of such a flare are obtained.
Correlated radio and X-ray observations are indeed crucial because, althought the amplitude of the radio variability is low compared to the event recorded in X-rays, an intriguing correlation seems to be present between the X-ray flares and the rise of the radio emission. Indeed Zhao et al. (2001), using Very Large Array (VLA) data collected over two decades, detected a periodicity in the radio variability, with a 106 days cycle and a characteristic timescale of 25 days. Baganoff et al. (2001b) already remarked that the October 2000 X-ray flare occured at a radio-cycle phase corresponding to the beginning of the radio peak. We have computed the 106 days radio cycle phase of the X-ray flare that we detected with XMM-Newton and found that it differs by only 6 days from the phase of the flare detected with Chandra. The flare occurred at the day 64 in the light curve of Fig. 3 of Zhao et al. (2001), while the Chandra flare took place at phase 70 day and the 1.3 cm radio peak rise extends roughly from day 55 to day 75. Even though the light curve radio peak is wide and several other structures are present, both X-ray flares detected till now are very close in phase and take place during the rising part of the main radio flare. We have also compared the time of the flare to a recent radio light curve of obtained at 1.3 cm and 2 cm with the VLA between March and November 2001 [@yua02]. The X-Ray flare occurred 1-2 days after a local maximum of the curve, but no radio data points are reported for the day when our XMM-Newton observation took place.
It will be also important to study the shape of the flare spectrum at energies higher than 10 keV to fully understand the radiation mechanism producing the high energy tail. In particular by measuring the high energy cut-off of the spectrum one could determine the electron temperature for a thermal emission or the Lorentz factor for non-thermal processes. We estimated that such a flare should be marginally visible in the range 10-60 keV with the low energy instruments onboard the new gamma-ray mission INTEGRAL, to be launched in October 2002, if the spectrum extends to these energies with the slope observed with Chandra and XMM-Newton.
Our more secure estimation of the duty cycle of the flares shows that multiwavelength observations of which involve XMM-Newton or Chandra will have good chance of observing an X-ray flare provided the simultaneous coverage is of the order of 100 ks.
Acknowledgments {#acknowledgments .unnumbered}
===============
Based on observations with *XMM-Newton*, an ESA science mission with instruments and contributions funded by ESA member states and the USA (NASA). This observation was performed as part of the *XMM-Newton* guaranteed time program of the XMM-EPIC team. We wish to thank all *XMM-Newton* staff involved in the realization and operation of the mission. F.D. acknowledges financial support from a postdoctoral fellowship from the French Spatial Agency (CNES). We thank the referee Frederick Baganoff for his stimulating comments and suggestions. We also thank Ruby Krishnaswamy for help with installation of data analysis software.
[\*]{} Baganoff, F., et al., 2001a, ApJ, submitted (astro-ph/0102151) Baganoff, F., et al., 2001b, Nature, 413, 45 Ghez, A. M., Morris, M., Becklin, E. E., Tanner, A., Kremenek, T., 2000, Nature, 407, 349 Genzel, R., Eckart, A., Ott, T., Eisenhauer, F., 1997, MNRAS, 291, 219 Goldwurm, A., 2001, Proc. of the 4$^{th}$ INTEGRAL Workshop, ESA-SP 459, 455 Jansen, F., et al., 2001, A$\&$A, 365, L1 Kirsch, M., 2002, XMM-EPIC Status of Calibrations and Data Analysis, XMM-SOC Documentation, ESA-SOC, Vilspa, Spain Liu, S., $\&$ Melia, F., 2002, ApJ, 566, L77 Markoff, S., Falcke, H., Yuan, F., Biermann, L., 2001, A&A, 379, L13 Melia, F. $\&$ Falcke, H., 2001, ARAA, 39, 309 Melia, F., Liu, S., Coker, R. F., 2001, ApJ, 553, 146 Narayan, R., Mahadevan, R., Grindlay, J. E., Popham, R. G., Gammie, C., 1998, ApJ, 492, 554 Predehl, P., $\&$ Truemper, J., 1994, A&A, 290, L29 Predehl, P., $\&$ Schmitt, J.H.M.M., 1995, A&A, 293, 889 Raymond, J. C., $\&$ Smith, B. W., 1977, ApJSS, 35, 419 Strueder, L., et al., 2001, A$\&$A, 365, L18 Turner, M. J. L., et al., 2001, A$\&$A, 365, L27 Warwick, R. S., 2002, Proc. of [*New Visions of the X-ray Universe in the XMM-Newton and Chandra Era*]{}, 26-30 November 2001, ESTEC, The Netherlands, ESA-SP, in press (astro-ph/0203333) Yuan, F. $\&$ Zhao, J., 2002, Chin. J. Astron. Astrophys., in press (astro-ph/0203050) Yusef-Zadeh, F., Choate, D., Cotton, W., 1999, ApJ, 518, L33 Zhao, J., Bower, G. C., Goss, W. M., 2001, ApJ, 547, L29
[llcc]{}\
& $\rm N_H$ \[$10^{22}$ cm$^{-2}$\] & 11.1 $^{+0.6}_{-0.6}$ & 11.1\
& kT \[keV\] & 1.31 $^{+0.08} _{-0.07}$ & 1.31\
& Norm MOS \[$ 10^{-3}$ cm$^{-5}$\] & 4.6 $^{+1.0} _{-1.0}$ & 4.6\
& Norm PN \[$ 10^{-3}$ cm$^{-5}$\] & 3.6 $^{+0.9} _{-0.7}$ & 3.6\
\
& $\rm N_H$ \[$10^{22}$ cm$^{-2}$\] & 21.2 $^{+6.2} _{-4.7}$ & 21.2\
& Photon Index & 2.1 $^{+0.3} _{-0.2}$ & 2.1\
& Norm MOS \[$10^{-4}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$\] & 6.8 $^{+6.0} _{-3.8}$ & 6.8\
& Norm PN \[$10^{-4}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$\] & 6.3 $^{+5.5} _{-3.5}$ & 6.3\
\
& $\rm N_H$ \[$10^{22}$ cm$^{-2}$\] & 9.8 & 9.8\
& Photon Index & 2.7 & 0.9 $^{+0.5} _{-0.5}$\
& Norm MOS \[$10^{-4}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$\] & 2.1 & 2.0 $^{+2.3} _{-1.3}$\
& Norm PN \[$10^{-4}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$\] & 2.1 & 0.6 $^{+0.9} _{-0.4}$\
\
& $\rm \chi^2_{\nu}~(d.o.f.)$ & 1.35 (121) & 0.90 (27)\
[lcc]{} $\rm N_H$ \[$10^{22}$ cm$^{-2}$\] & 9.8 & 5.3\
Photon Index & 0.7 $^{+0.5} _{-0.6}$ & 0.3 $^{+0.6} _{-0.4}$\
Norm MOS \[$10^{-4}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$\] & 1.3 $^{+2.0} _{-1.3}$ & 0.7 $^{+1.0} _{-0.4}$\
Norm PN \[$10^{-4}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$\] & 0.3 $^{+0.6} _{-0.3}$ & 0.2 $^{+0.3} _{-0.1}$\
$\rm \chi^2_{\nu}~(d.o.f.)$ & 0.98 (20) & 0.95 (20)\
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Multi-object tracking has been recently approached with the min-cost network flow optimization techniques. Such methods simultaneously resolve multiple object tracks in a video and enable modeling of dependencies among tracks. Min-cost network flow methods also fit well within the “tracking-by-detection” paradigm where object trajectories are obtained by connecting per-frame outputs of an object detector. Object detectors, however, often fail due to occlusions and clutter in the video. To cope with such situations, we propose to add pairwise costs to the min-cost network flow framework. While integer solutions to such a problem become NP-hard, we design a convex relaxation solution with an efficient rounding heuristic which empirically gives certificates of small suboptimality. We evaluate two particular types of pairwise costs and demonstrate improvements over recent tracking methods in real-world video sequences.'
author:
- |
Visesh Chari[^1] Simon Lacoste-Julien[^2] Ivan LaptevJosef Sivic\
INRIA and Ecole Normale Supérieure, Paris, France\
bibliography:
- 'eccvbib.bib'
nocite:
- '[@andriluka08cvpr]'
- '[@jiang07cvpr]'
- '[@wu06cvpr]'
title: |
On Pairwise Costs for\
Network Flow Multi-Object Tracking
---
Introduction
============
The task of visual multi-object tracking is to recover spatio-temporal trajectories for a number of objects in a video sequence. Tracking multiple objects, like people or vehicles, has a wide range of applications from Robotics to video surveillance [@yilmaz06objecttracking]. Despite recent progress in the field [@andriluka08cvpr; @berclaz11pami; @butt13cvpr; @milan13cvpr; @pellegrini09cvpr; @pirsiavash11cvpr; @yang12cvpr], tracking remains a challenging problem especially in crowded and cluttered scenes.
\
\
With the advances in object detection, “tracking-by-detection” have recently become a popular paradigm for object tracking [@berclaz11pami; @butt13cvpr; @jiang07cvpr; @li09cvpr]. Given object detections in every frame of a video sequence, the tracking is formulated as selection and clustering of corresponding object detections over time. Such selection and clustering problems can be solved in an optimization framework using carefully designed cost functions. Given an appropriate cost function, tracking-by-detection is typically setup as a MAP estimation problem [@zhang08cvpr]. Among different formulations of this problem, min-cost network flow [@ahuja93book] is particularly attractive as it allows for optimal and efficient solutions [@pirsiavash11cvpr]. The energy minimization approach to tracking enables global solutions to track selection and avoids early and error-prone local decisions. Moreover, it also enables for a principled modeling of interactions among different tracks. In the past, models of track interactions have been shown to improve human tracking in crowds [@pellegrini09cvpr], to identify unusual behavior [@kratz09cvpr] as well as to resolve ambiguous tracks [@milan13cvpr; @pirsiavash11cvpr]. Such previous methods, however, either resort to local [*non-convex*]{} optimization [@pellegrini09cvpr; @kratz09cvpr; @milan13cvpr], or use greedy methods to enforce interactions [@pirsiavash11cvpr].
Unlike previous work, we here propose to model track interactions within the min-cost network flow tracking approach. We introduce pairwise costs to the objective function and design a [*convex*]{} relaxation solution with an efficient rounding heuristic. Although our final integer solution can be suboptimal, our method is generic and empirically provides certificates of small suboptimality. Tracking results using two particular examples of pairwise costs discussed in this paper are illustrated in Figure \[fig:opening\].
In summary, this paper makes the following contributions:
- We propose a new *non-greedy* approach to optimize pairwise terms within a min-cost network flow framework. Our solution is generic and allows the simultaneous optimization of any type of pairwise costs.
- We propose a global optimization strategy with a convex relaxation that allows us to minimize pairwise costs using linear optimization, and a principled Frank-Wolfe style rounding procedure to obtain integer solutions with a certificate of suboptimality. The optimization procedure is empirically stable, allowing the practitioner to focus on modeling.
- To illustrate our method, we propose two particular examples of pairwise costs: the first discourages significant overlaps between distinct tracks; the second models the spatial co-occurrence of different types of detections. This allows us to better model complex dynamic scenes with substantial clutter and partial occlusions.
- Using our method, we show improved tracking results on several real-world videos. In addition, we propose a new strategy to evaluate tracking results that better measures the longevity of overlap between output tracks and ground truth.
This paper is organized as follows. Section \[sec:relatedwork\] presents related work and the overview of our approach. Section \[subsec:traditionaltracking\] summarizes min-cost flow tracking. Section \[sec:quadraticcost\] describes our optimization framework with pairwise costs presented in Sections \[subsec:overlap\] and \[subsec:coocurrence\]. The optimization strategy is described in Section \[subsec:optimastrategies\], with initial quadratic optimization formulation in Section \[sec:quadOpt\] and subsequent linear relaxation in Section \[subsubsec:gradient\_based\_search\]. Finally we present results of our method and compare them to the state of the art on challenging datasets in Section \[sec:experiments\], and conclude with a discussion in Section \[sec:discussion\].
Related work {#sec:relatedwork}
============
Recent approaches have formulated multi-frame, multi-object tracking as a min-cost network flow optimization problem [@zhang08cvpr; @pirsiavash11cvpr; @berclaz11pami], where the optimal flow in a connected graph of detections encodes the selected tracks. While earlier min-cost network flow optimization methods have used linear programming, recently proposed solutions to the min-cost flow optimization include push-relabel methods [@zhang08cvpr], successive shortest paths [@pirsiavash11cvpr; @berclaz11pami], and dynamic programming [@pirsiavash11cvpr]. To ensure globally optimal and efficient solutions, previous methods have often restricted the cost to unary terms over all edges. While non-unary terms break the optimality of solutions in general, dependencies between detections have been enforced by greedy approaches, such as greedily eliminating the overlapping detections after each step of a sequential selection of distinct tracks in [@pirsiavash11cvpr]. This non-global optimization approach, however, cannot recover from early suboptimal decisions. Additional dependencies among detections can also be incorporated into the min-cost network flow tracking by modifying the underlying graph structure. Butt and Collins [@butt13cvpr] follows this approach and minimizes the modified objective using Lagrangian methods. While the method works well for the particular type of introduced cost, generalizing this method to the new types of pairwise costs would require appropriate modifications of the graph structure which is non-trivial in general. Moreover, combining multiple costs within such a framework would be difficult. In contrast, our framework allows addition of terms without any modification to the underlying optimization framework.
Brendel et al. [@brendel11cvpr] and Milan et al. [@milan13cvpr; @milan13pami] formulate the problem in a framework that first selects *tracklets* and then connects them using a learned distance measure [@brendel11cvpr] or a CRF [@milan13cvpr; @milan13pami]. Long term occlusions are handled in [@brendel11cvpr] by merging appearance and motion similarity. While [@milan13cvpr; @milan13pami] propose to alternate between discrete and continuous optimizations in order to minimize several cost functions, the presence of two levels of optimization makes theoretical or empirical guarantees of optimality hard to give. Unlike this work, we use a convex relaxation in our approach that allows us to give an empirical guarantee of optimality to our solutions. Other methods [@li09cvpr; @yang11cvpr; @yang12cvpr] use offline or online training to learn a similarity measure between tracklets. These methods do not provide any optimality guarantee, though. In addition, training might be difficult in some conditions. For example, online training to discriminate appearances might be erroneous when objects move very close to each other (Figure \[fig:opening\]). We avoid such problems by using pairwise terms to robustify the tracker to detection errors.
Incorporation of pairwise terms into the min-cost network flow formulation has been previously attempted by Choi and Savarese [@choi12eccv]. Their work, however, is focused on jointly optimizing tracking and activity recognition. In contrast, we focus on tracking in particular, and propose a generic framework enabling inclusion of multiple types of pairwise costs and providing empirical measures of small suboptimality.
Overview of our approach {#subsec:overview}
------------------------
We propose an algorithm that incorporates quadratic pairwise costs into the traditional min-cost flow network. Unlike previous methods [@berclaz11pami; @li09cvpr], which either build on top of min-cost flow solutions [@milan13cvpr] or change the network structure [@butt13cvpr], we propose a modification to the standard optimization algorithm. Such quadratic costs can represent several useful properties like similar motion of people in a rally, co-occurrence of tracks for different parts of the same object instance and others.
While in such a case obtaining the global optimum is NP-hard [@loiola07qap], we outline an approach to obtain near optimum solutions, while we empirically verify its optimality. We present a linear relaxation to the quadratic term that is fast to optimize, followed by a Frank-Wolfe based rounding heuristic to obtain an integer solution.
Background: Min-cost flow tracking {#subsec:traditionaltracking}
==================================
In this section, we describe the traditional formulation of multi-object tracking as a min-cost flow optimization problem [@zhang08cvpr]. We extend this framework in Section \[sec:quadraticcost\].
Given a video with objects in motion, the goal is to simultaneously track $K$ moving objects in a “detect-and-track” framework [@zhang08cvpr]. The input to the approach is two-fold. First a set of candidate object locations is assumed to be given, provided, for example, as output of an object detector. Henceforth we refer to these locations as [*detections*]{}. The approach also requires a measure of correspondence between detections across video frames. This could be obtained for example from optical flow, or using some other form of correspondence. Based on these inputs, the tracking problem is setup as a joint optimization problem of simultaneously selecting detections of objects and connections between them across video frames. Such a problem can be modeled through a MAP objective [@zhang08cvpr] with specific constraints encoding the structure of the tracks. The MAP optimization problem can be cast as the following integer linear program (ILP):
$$\begin{aligned}
\label{eq:mainopt}
\min_{{\mathbf{x}}} & \hspace{-2cm}\sum_i c_i x_i + \sum_{ij \in {E}} c_{ij} x_{ij} \\
\mathrm{s.t.} & \left.
\begin{gathered} 0 \leq x_i \leq 1 \, , \, 0 \leq x_{ij} \leq 1 \\
\sum_{i \, : \, ij \in {E}} x_{ij} = x_j = \sum_{i \, : \, ji \in {E}} x_{ji} \\
\sum_{i} x_{it} = K = \sum_{i} x_{si} \\
\end{gathered} \right\} {\mathbf{x}}\in \operatorname{FLOW}_K \notag \\
& \qquad \quad x_i \, , \, x_{ij} \quad \mathrm{are\quad integer .} \notag \end{aligned}$$
![ Illustration of a graph used in traditional min-cost flow. The detection (red) and connection (cyan) variables are marked as edges and have unit capacity. Every track is a unit flow that starts at a source S and ends at a sink t. S and t are connected to all detections. []{data-label="fig:illustration"}](figures/illustration3.pdf){width="2.5in" height="1.8in"}
The above formulation encodes the joint selection of $K$ tracks using the following selection variables: $x_i \in \{0,1\}$ is a binary indicator variable taking the value $1$ when the *detection* $i$ is selected in some track; $x_{ij} \in \{0,1\}$ is a binary indicator variable taking the value $1$ when detection $i$ and detection $j$ are *connected* through the *same* track in nearby time frames. The index $i$ ranges over possible detections across the whole video. $c_i$ denotes the cost of selecting detection $i$ in a specific frame (and represents the negative detection confidence) while $c_{ij}$ represents the negative of the correspondence strength between detections $i$ and $j$. The set of possible connections between detections is represented by ${E}$ and could be a subset of all pairs of detections in nearby frames by using choice heuristics (such as spatial proximity). The quality of track selection is quantified by the objective in . The constraint $\sum_{i \, : \, ij \in {E}} x_{ij} = x_j = \sum_{i \, : \, ji \in {E}} x_{ji}$, which has the structure of a *flow conservation constraint* [@ahuja93book], encodes the correct claimed semantic that $x_{ij}$ can take the value $1$ if and only if both $x_i$ and $x_j$ take the value $1$, and moreover, that each detection belongs to *at most one track*, enforcing the fact that two objects cannot occupy the same space. Finally, the constraint $\sum_{i} x_{it} = K = \sum_{i} x_{si}$ ensures that exactly $K$ tracks are selected (dummy “source” and “sink” variables with the fixed value $x_s = x_t = K$ are added; the connection variables $x_{si}$ and $x_{it}$ represent the start and end of tracks respectively).
We have grouped the linear constraints in under the name $\operatorname{FLOW}_K$ as they actually correspond to constraints in a min-cost network flow problem where one would like to push $K$ units of flow with minimum cost in a network with unit capacity edges. In fact, these linear constraints have the property of being *totally unimodular* [@ahuja93book]. This implies that the polytope they determine has only vertices with *integer* coordinates, and so relaxing the integer constraints in and solving it as a linear program is still guaranteed to produce *integer solutions*, making it a tight relaxation. Figure \[fig:illustration\] illustrates the correspondence between a network flow structure and the formulation (\[eq:mainopt\]).
To summarize, the above optimization problem with relaxed integer constraint can be solved efficiently using existing network flow or linear algebra packages [@ahuja93book], and provides a convenient framework to transform the tracking problem into a *track selection* problem. We use this conversion as a starting point to add additional constraints and costs on the selection process to influence it in desirable ways to address challenging scenarios that are shown in later sections.
Modeling pairwise costs with an IQP {#sec:quadraticcost}
===================================
The above formulation in represents a linear objective with linear equality constraints (where the integer constraint is not needed). While linear terms are both simple and easy to minimize, higher order models can represent more useful properties [@pellegrini09cvpr]. We suggest to add a quadratic cost between pairs of selection variables. To simplify the notation for the optimization sections, we collect the $x_i$ and $x_{ij}$ variables in a long vector ${\mathbf{z}}$. The product $z_i z_j$ then encodes *joint* selection of $z_i$ and $z_j$ – these choices could correspond to a pair of connections, a pair of detections, or even a connection and a detection. A term of the form $Q_{ij} z_i z_j$ can then either encourage (or discourage) the joint selection of $z_i$ and $z_j$ by having $Q_{ij}$ negative (or positive), respectively. Our approach is to consider a small set ${\mathcal{Q}}$ of such joint selections, and add the term $\sum_{ij
\in {\mathcal{Q}}} z_i z_j Q_{ij}$ to the objective. Our new optimization problem can thus be expressed as the integer quadratic program (IQP): $$\begin{aligned}
\label{eq:quadratic}
\min_{{\mathbf{z}}} & \quad {\mathbf{c}}^\top {\mathbf{z}}+ {\mathbf{z}}^\top {\mathbf{Q}}{\mathbf{z}}\notag \\
\mathrm{s.t.} & \quad {\mathbf{z}}\in \operatorname{FLOW}_K \notag \\
& \quad {\mathbf{z}}\quad \text{integer ,}\end{aligned}$$ where the ${\mathbf{Q}}$ matrix is *sparse* with $Q_{ij}\ne 0$ for $ij \in {\mathcal{Q}}$.
Unfortunately, the above formulation can encode the quadratic assignment problem which is NP-hard to optimize in general [@loiola07qap]. Nevertheless, we propose an efficient (convex) linear relaxation in Section \[subsec:optimastrategies\] as well as a powerful rounding heuristic that provides empirical certificates of suboptimality. Our main modeling strategy is thus twofold: first, we encode our prior knowledge about the joint selection of variables using the sparse cost matrix ${\mathbf{Q}}$ (which can be arbitrary); second we add additional constraints to the IQP as long as they can be encoded as network flow constraints (this is a requirement of our rounding heuristic presented in Section \[subsubsec:gradient\_based\_search\]). In the rest of this section, we provide two examples of pairwise costs used in our experiments. We then focus on the optimization aspects in Section \[subsec:optimastrategies\].
Designing pairwise costs {#sec:differentcosts}
------------------------
In the following subsections, we show how some traditional constraints [@kratz09cvpr; @pellegrini09cvpr] could be incorporated in our quadratic min-cost network flow framework. We focus on elements that cannot be simply encoded with traditional linear terms in .
### Overlap penalty {#subsec:overlap}
Object detectors often produce multiple responses per object. This issue is typically addressed by the *Non Maxima Suppression* (NMS) step, which retains most confident detections within spatial neighborhoods. While NMS works well for tracking isolated objects, independent decisions produced by NMS for each object and frame often become suboptimal in crowded scenes where multiple objects may occupy the same spatial neighborhood. To address this problem, we avoid taking independent decisions and propose to discourage overlapping detections within the network flow tracking framework. For this purpose we extend the cost function with the following [*pairwise overlap cost*]{}: $$\begin{gathered}
\label{eq:overlapterm}
q_{ij}^{\mathrm{ov}} x_i x_j \\
\textrm{for} \,\,\, (i,j) \,\,\, \textrm{s.t.} \,\,\, \mathrm{ov}( \mathrm{box}(x_i), \mathrm{box}(x_j) ) \geq o_{\mathrm{thres}} \nonumber
$$ where $x_i$ and $x_j$ represent two selection variables associated with sufficiently overlapping[^3] detections and $q_{ij}^{\mathrm{ov}} > 0$.
In previous approaches like [@pirsiavash11cvpr], NMS was implemented in a *greedy* fashion. Greedy approaches, however, have the disadvantage of making non-reversible decisions in the early stages of optimization. In contrast, our approach of incorporating the cost (\[eq:overlapterm\]) into the overall cost function ensures that NMS is optimized *simultaneously* with other tracking objectives. As a result, overlapping detections may become tolerated, for example, in situations when two tracks intersect. On the other hand, continuously overlapping tracks resulting from multiple outputs of detectors will be discouraged.
### Enforcing consistency between two signals {#subsec:coocurrence}
In many tracking scenarios, multiple signals are available for use. For example, we might have a body detector as well as a head detector. In case they give complementary information about the presence of the object, we can be more robust to detection noise by ensuring that the two tracks are consistent using a pairwise cost.
For example, let $z^h_i$ and $z^b_i$ denote the selection variables (detection or connection) for the head and body respectively. Each set can be associated with its own flow feasible set $\operatorname{FLOW}^h_K$ and $\operatorname{FLOW}^b_K$. We can *encourage* the consistent “co-occurrence” of the two flows by adding the following negative cost: $$\begin{gathered}
\label{eq:coocurrence}
-q_{ij}^{\mathrm{co}} z_i^h z_j^b \\
\textrm{for $(i,j)$ s.t. $z_i^h$ and $z_j^b$ are \emph{consistent}}. \nonumber
$$ In our experiment, we say that $z_i^h$ and $z_j^b$ are *consistent* in two scenarios. Either $z_i^h$ and $z_j^b$ are detection variables such that their corresponding boxes[^4] overlap more than $o_{\mathrm{thres}}$. Or we have a head detection $z_i^h$ with a box that intersects the edge $z_j^b$ connecting its respective body detection boxes (and similarly for a body detection and head edge). The idea behind the latter possibility is to be more robust to missing detections on some frames: it corresponds to a situation where a head and body detection would have overlapped if we were interpolating detections along an edge that skips frames. Note that the cost is difficult to minimize greedily, since both head and body tracks need to be optimized *simultaneously*.
Optimization {#subsec:optimastrategies}
============
In the previous section, we presented examples of quadratic cost functions that we could include in our extension to the min-cost flow network formulation to encourage co-occurrence preferences for individual variables in the minimization. Finding a global minimum is NP-hard [@loiola07qap] if we keep the integer constraints on the variables (which is necessary to ensure the correct track encoding). Our suggested strategy is to instead find a global solution to the *relaxed* version of the problem with the integer constraints removed, and then use a powerful heuristic to search for nearby integer solution that satisfies the flow constraints (see Section \[subsubsec:gradient\_based\_search\]). By comparing the objective value between the “rounded” integer solution and the global solution to the relaxed problem, which provides a lower bound, we obtain a *certificate of optimality*. In our experiments, we observed that suboptimality upper bounds were quite small, thus indicating that our optimization framework is stable and we can instead focus on designing good cost functions. We now describe several approaches to optimize .
Quadratic optimization {#sec:quadOpt}
----------------------
If ${\mathbf{Q}}$ is positive definite, then the quadratic program (QP) in with relaxed integer constraint is convex and can be robustly optimized using interior point methods implemented in commercial solvers such as MOSEK/CPLEX. These methods can scale to medium-size problems[^5] by exploiting the sparseness of ${\mathbf{Q}}$ suggested in Section \[sec:differentcosts\]. In our general formulation, ${\mathbf{Q}}$ is not necessarily positive definite. We can nevertheless use a standard trick to make it positive definite by defining its diagonal entries to be $Q_{ii}^{\mathrm{new}} = \sum_{j\neq i} |Q_{ij}^{\mathrm{old}}|$, while using $c_i-Q_{ii}^{\mathrm{new}}+Q_{ii}^{\mathrm{old}}$ as the linear coefficient for $z_i$ in the objective. As $z_i^2 = z_i$ for binary variables, this transformation sill yields an (equivalent) IQP. ${\mathbf{Q}}^{\mathrm{new}}$ is now positive semidefinite [@hornAndJohnson Thm. 6.1.10], and so the relaxation gives a convex problem.
In order to scale to very large scale datasets (billions of variables), one could use the Frank-Wolfe algorithm [@jaggi13icml] which is a first order gradient based method that iteratively minimizes a linearization of the quadratic objective. An advantage of this approach is that each step of the Frank-Wolfe algorithm reduces in our case to the minimization of a min-cost network flow problem, which can scale to much larger sizes than a generic linear program solver. Moreover, each step of this algorithm yields an integer solution. Thus, while optimizing the relaxed objective (which will provide a lower bound certificate), we can keep track of which integer iterate had the best objective thus far. This perspective also motivates a powerful rounding heuristic that we describe in Section \[subsubsec:gradient\_based\_search\]. Building on a preliminary version of our paper, [@joulin14FW] used this approach successfully for performing efficient co-localization in videos, where the constraint set also had a network flow structure.
Equivalent integer linear program {#subsec:linearconstraints}
---------------------------------
Another way to make the approach more scalable is to transform the integer QP into an equivalent integer linear program (ILP) by introducing well-chosen additional variables and constraints. We present such an approach in this section, which generalizes the line of reasoning from [@lacoste06qap].
We introduce a new set of variables $u_{ij}$ that encode the joint selection of the edge $z_i$ and $z_j$, and thus we would like to enforce $u_{ij} = z_i z_j$. The quadratic cost component $Q_{ij}
z_i z_j$ could then be replaced with a linear cost $Q_{ij} u_{ij}$. An equivalent integer linear program is thus the following: $$\begin{aligned}
\label{eq:quadlinear}
\min_{{\mathbf{z}}, {\mathbf{u}}} & \qquad \quad {\mathbf{c}}^\top {\mathbf{z}}+ {\mathbf{q}}^\top {\mathbf{u}}\notag \\
& \qquad \quad {\mathbf{z}}\in \operatorname{FLOW}_K \notag \\
\mathrm{s.t.} & \quad \left.
\begin{gathered} 0 \leq u_{ij} \leq 1, \, \forall ij \in {\mathcal{Q}}\notag \\
u_{ij} \leq z_i \, , \, u_{ij} \leq z_j \notag \\
z_i + z_j \leq 1 + u_{ij} \, \\
\end{gathered} \right\}
\begin{gathered}
({\mathbf{z}}, {\mathbf{u}}) \in \\
\operatorname{LOCAL}({\mathcal{Q}})\\
\end{gathered} \\
& \qquad \quad {\mathbf{z}}, {\mathbf{u}}\quad \text{integer .} \end{aligned}$$
Here ${\mathbf{u}}$ and ${\mathbf{q}}$ represents the vector whose elements are $u_{ij}$ and $Q_{ij}$ respectively. The new constraint $ z_i + z_j \leq 1 + u_{ij}$ enforces that $u_{ij}$ should be $1$ if $z_i$ and $z_j$ are both $1$; while the pair of constraints $u_{ij} \leq z_i$ and $u_{ij} \leq z_i$ enforce $u_{ij} = 0$ if either $z_i$ or $z_j$ is zero. We call these constraints ‘$\operatorname{LOCAL}({\mathcal{Q}})$’ as it turns out that they define a polytope which can be obtained by a projection of the *local marginal consistency* polytope for the over-complete representation of a discrete Markov random field (MRF) [@wainwright08 (4.6)] with edges defined by the non-zero entries of ${\mathbf{Q}}$[^6]. Removing the integer constraint in thus yields a LP relaxation that is similar to one for MAP inference in MRFs, but with additional $\operatorname{FLOW}_K$ constraints, yielding a crucial structural difference with the previous works.
An advantage of this formulation is that its relaxed form is a LP, which can usually be optimized by MOSEK or CPLEX to larger scale than the QP formulation, even though there is an increase in the number of variables and constraints. Note though that the number of new variables $u_{ij}$ created is the same as the number of non-zero coefficients in the sparse ${\mathbf{Q}}$, which was indicated by the set ${\mathcal{Q}}$ in to stress that we do not need to look at all pairs of edges. In exploratory experiments, we observed that the LP relaxation yielded similar quality solutions as the QP relaxation, but was faster to optimize; we have thus focused on the LP relaxation in our experiments. Another advantage of is that we can easily generalize it to handle higher order terms in the objective. For a clique $C$ of decision variables that we want to encourage or discourage jointly, we introduce a new variable $u_C :=
\prod_{i \in C} z_i$. This semantic can be readily enforced with the constraints $u_C \leq z_i$ for all $i \in C$, and $\sum_{i \in C} (z_i-1) + 1 \leq u_C$, which generalizes $\operatorname{LOCAL}({\mathcal{Q}})$ for higher order terms and yields another ILP that can be relaxed to a LP.
Frank-Wolfe rounding heuristic {#subsubsec:gradient_based_search}
------------------------------
The solution of the LP relaxation of can have fractional components because the additional linear constraints from $\operatorname{LOCAL}({\mathcal{Q}})$ essentially violate the *total unimodularity* property, in contrast to $\operatorname{FLOW}_K$ which yields a polytope with only integer vertices. Since naively rounding the obtained fractional variables to the nearest integer might not result in a feasible point (in other words a valid flow), we need a strategy to obtain an integer solution with cost similar to the minimum. Given the relaxed global solution ${\mathbf{z}}^*$, the simplest approach would be to look for the point closest in Euclidean norm in $\operatorname{FLOW}_k$ which is an integer. As $z_i^2 = z_i$ for binary variables, we have $||{\mathbf{z}}-{\mathbf{z}}^*||^2 = (\bm{1}-2{\mathbf{z}}^*)^\top {\mathbf{z}}+ ||{\mathbf{z}}^*||^2$ which is a linear function of ${\mathbf{z}}$. We can thus obtain the closest integer point by solving a LP over $\operatorname{FLOW}_k$, as all its vertices are integers. We call this approach *Hamming rounding* as $d_H({\mathbf{z}},{\mathbf{z}}'):=||{\mathbf{z}}-{\mathbf{z}}'||^2$ reduces to the Hamming distance when evaluated on pair of binary vectors. On the other hand, the closest point in Euclidean norm does not necessarily yield a good objective value (as the search was agnostic to the objective). Inspired by the Frank-Wolfe algorithm, our suggested heuristic is to minimize instead the first-order linear under-estimator of the quadratic objective constructed with the gradient at the relaxed global solution ${\mathbf{z}}^*$. Specifically, we obtain the following LP, which has the usual network flow constraint structure and thus can be solved very efficiently: $$\begin{aligned}
\label{eq:gradsearch}
\min_{{\mathbf{z}}} & \quad \left( {\mathbf{c}}+ ({\mathbf{Q}}+ {\mathbf{Q}}^{\top}) {\mathbf{z}}^*\right)^\top {\mathbf{z}}\notag \\
\mathrm{s.t.} & \quad {\mathbf{z}}\in \operatorname{FLOW}_K .\end{aligned}$$ The objective here can be interpreted as modifying the distance function on binary vectors to take the cost function in consideration. As previously mentioned, the relaxed LP solution provides a lower bound on the true ILP (which is equivalent to the IQP) solution. The difference between the objective evaluated on *any* feasible integer solution and the lower bound is thus an [upper bound certificate]{} on its suboptimality. In our experiments, we obtained small suboptimality certificates ($\approx 10^{-3}$) for our returned integer solutions, indicating that our rounding heuristic was effective at returning near-global optimal solutions (we note that we define ${\mathbf{c}}$ and ${\mathbf{Q}}$ so that the objective is normalized between $0$ and $1$). We also observed that Hamming rounding generally produced a suboptimality that was around $3$ to $4$ times *worse* than the solution produced by Frank-Wolfe rounding. These worse objective values also translated in worse tracking accuracy (see Appendix \[sec:FWbetter\] in the supplementary material[^7]). We finally note that in contrast to the previous work [@butt13cvpr] which could not guarantee that their algorithm would converge to an integer solution, our approach will always give *some* integer solution (by solving a simple min-cost network flow problem), and can provide a certificate of suboptimality a-posteriori.
Experiments {#sec:experiments}
===========
In this section, we evaluate our approach on several real world videos and compare results to the state-of-the-art methods [@benfold2011stable; @milan13cvpr; @pirsiavash11cvpr]. First we illustrate the effect of the two pairwise costs proposed in Section \[sec:differentcosts\] and evaluate improvement over the basic min-cost network flow tracking. We also argue that the standard MOTA score is often insufficient to capture the quality of tracking results and propose a new measure for tracking evaluation, termed *re-detection measure* (Section \[para:redetection\]).
Second, we evaluate our method on six videos from the two standard datasets PETS and TUD. For both of these datasets, we obtain part of the input (person detections) from Milan et al. [@milan13cvpr], and show improvements over their approach using the standard MOTA metrics.
Tracking datasets
-----------------
We test our algorithm on several publicly available videos. The first video MarchingRally corresponds to a crowd walking in a rally along a street (see Figure \[fig:opening\], top row). The video consists of 120 frames recorded at 25 fps, and has about 50 people. This video is challenging due to the high number of people moving close to each other. We have manually annotated ground truth tracks for all people in this video for the purpose of tracking evaluation[^8].
The second video illustrated in Figure \[fig:opening\] (bottom row) is called TownCenter [@benfold2011stable] and consists of 4500 frames recorded at 25 fps. The video shows approximately 230 people walking across the street. Finally, we use videos from the well-known PETS and TUD datasets. These videos depict frequently occluded people moving in multiple directions.
#### Preprocessing.
We run a “head” detector [@mikel11iccv] to detect heads of people in every frame of the MarchingRally and PETS videos. While we use only head detections for the MarchingRally sequence, for PETS we use our head detections in combination with readily-available body detections from [@milan13cvpr]. Head detections complement frequently overlapping body detections and help resolving partial occlusions as well as ID-switches. For each of these videos, we run a KLT tracker after initializing features within detection bounding boxes. Finally, for every pair of nearby frames ($<$ 10 frames apart), we connect pairs of detections with high correspondence strength. The strength of correspondence between two detections is the ratio of their common KLT tracks and the total number of KLT tracks passing through both detections.
Tracking in video experiment {#sec:trackingExp}
----------------------------
### Evaluation strategy {#subsec:strategy}
Evaluating results of multi-object tracking is non-trivial because errors might be present in various forms including ID switches, broken tracks, imprecisely localized tracks and false tracks. Measures such as MOTA [@benfold2011stable; @milan13cvpr] combine different errors into a single score and enable the global ranking of tracking methods. Such measures, however, lack interpretability. On the other hand, independent assessment of different errors can also be misleading. For example, in dense crowd videos such as in Figure \[fig:opening\](a), tracks may have relatively low localization error while being incorrect due to switches between neighboring people. Similarly, low error of ID switches can be a consequence of many broken tracks.
We argue that a meaningful evaluation of tracking methods should be related to a task. One task with particular relevance to crowd videos is to detect the location of a given person after $\Delta t$ frames. To evaluate the performance of tracking methods on such a task we propose the re-detection measure as described below.
#### Re-detection measure. {#para:redetection}
The proposed re-detection measure evaluates the ability of a tracker to find the correct location of a given object after $\Delta t$ frames. The measure is inspired by the common evaluation procedure for object detection in still images [@everingham10] and extends it to tracking. For each pair of detections $A_t$ and $B_{t+\Delta t}$ associated to the same track by a tracker, we check if there exists a ground truth track that overlaps with $A_t$ and $B_{t+\Delta t}$ on frames $t$ and $t+\Delta t$ respectively.[^9] If the answer is negative, the subtrack $(A_t,B_{t+\Delta t})$ is labeled as false positive. Otherwise, it is labeled as true positive unless there exist multiple subtracks overlapping with the same ground truth. To avoid multiple responses, in the latter case only one subtrack is labeled as true positive while others are declared as false positives.
For the given $\Delta t$ we collect subtracks from all video intervals $(t, t+\Delta t)$ and sort them according to their confidence.[^10] Given the subtrack labels defined above, we evaluate Precision-Recall and Average Precision (AP). High AP values indicate the good performance of the tracker in the re-detection task. On the other hand, common errors such as ID switches and imprecise localization reduce AP values. Note that in the case of $\Delta t = 0$, our measure reduces to the standard measure for object detection. Larger values of $\Delta t$ enable evaluation of re-detection for longer time intervals. To compare different methods, we plot the re-detection AP for different values of $\Delta t$ as illustrated in Figure \[fig:aplength\].
\
----------------------------------------------------------------------------------------------------
MOTA Prec Recall
------------------- ------------------------------------------------------------- ------ -------- --
NF + Ov. + Co-oc. **[55.9]{}% & **[93.1%]{} & 60.6%\
Ben[@benfold2011stable] Head & 45.4% & 73.8% & **[71.0%]{}\
******
----------------------------------------------------------------------------------------------------
\[tab:simi\]
[|c|c|c|c|c|c|c|c|c|c|c|c|c|c|]{} & & Rcll & Prcn & GT & MT & PT & ML& FP & FN & IDs & FM & MOTA & MOTP\
& NF & 67.9 & 72.0 & 10 & 4 & 6 & 0 & 305 & 371& 26 & 26 & 39.3 & 59.5\
& NF+pairwise & 59.6 & **[89.9]{} & 10 & 2 & 8 & 0 & **[77]{} & 467& 15 & 22 & 51.6 & **[61.6]{}\
& Milan [@milan13cvpr] & **[69.1]{} & 85.6 & 10 & **[4]{} & **[6]{} & 0 & 134 & **[457]{}& 15 & **[13]{} & **[56.2]{} & **[61.6]{}\
********************
& NF & 93.7 & 83.4 & 19 & 17 & 2 & 0 & 870 & 293& 64 & 66 & 73.6 & 72.9\
& NF+pairwise & 92.4 & **[94.3]{} & [19]{} & **[18]{} & **[1]{} & 0 & **[262]{} & 354& 56 & 74 & 85.5 & **[76.2]{}\
& Milan [@milan13cvpr] & **[96.8]{} & 94.1 & [19]{} & 18 & **[1]{} & 0 & 282 & **[148]{} & **[22]{} & **[15]{} & **[90.3]{} & 74.3\
**********************
& NF & 47.7 & 87.6 & 43 & 1 & 37 & 5 & 693 & 5383& 291 & 531 & 38.1 & 60.7\
& NF+pairwise & 60.6 & 88.6 & 43 & 6 & **[34]{} & 3 & 807 & 4050& 244 & 379 & 50.4 & **[60.6]{}\
& Milan [@milan13cvpr] & **[65.1]{} & **[92.4]{} & 43 & **[11]{} & 31 & **[1]{} & **[549]{} & **[3592]{} & **[167]{} & **[153]{} & **[58.1]{} & 59.8\
**********************
& NF & 44.5 &92.2 &44& 9 & 15 & 20 &164& 2428 &121 &189 & 38.0 & 69.3\
& NF+pairwise & **[45.5]{} &91.2 &44& **[12]{} & 15 & **[17]{} &155& **[2125]{} & 44 &50 & **[40.3]{} & 61.2\
& Milan [@milan13cvpr] & 43.0 & **[94.2]{} &44& 8 & **[17]{} & 19 &**[115]{}& 2493 & **[27]{} & **[22]{} & 39.8 & **[65.0]{}\
**********************
& NF & 62.9 & 89.1 & 44 & 18 & 15 & 11 & 295 & 1425 & 289 & 140 & 47.8 & 65.2\
& NF+pairwise & **[68.9]{} & 92.0 & 44 & 20 & **[16]{} & **[8]{} & 230 & **[1198]{} & 35 & 74 & **[62.0]{} & **[62.1]{}\
& Milan [@milan13cvpr] & 64.9 & **[92.4]{} & 44 & **[21]{} & 12 & 11 & **[169]{} & 1349 & **[22]{} & **[19]{} & 60.0 & 61.9\
**********************
& NF & 31.3 & 87.4 & 42 & 4 & 15 & 23 &208 &3501 &101 &243 & 23.7 & 57.9\
& NF+pairwise & **[37.9]{} & 89.6 & 42 & **[4]{} & **[20]{} & **[18]{} & 223 & **[3141]{} & 67 & 122 & **[32.2]{} & 55.0\
& Milan [@milan13cvpr] & 30.9 & **[98.3]{} & 42 & 2 & 19 & 21 & **[27]{} &3494 & **[42]{} & **[34]{} & 29.6 & **[58.8]{}\
**********************
### Experimental results
We compare our algorithm with the state-of-the-art approaches on several video sequences. For the MarchingRally and TownCenter sequences, the baseline approaches for comparison are a greedy implementation of the basic min-cost network flow algorithm with the greedy NMS heuristic from [@pirsiavash11cvpr], and a network flow (NF) implementation as a linear program. In all graphs in Figure \[fig:aplength\], the corresponding results are represented by black (“Greedy + NMS”) and blue (“NF Basic”) curves. We note that we perform a careful grid search over the parameter space for all three algorithms and show the results corresponding to the best parameters, to make sure the differences observed are not arising from different parameter choices, but rather from limitations of the framework. On the other hand, we have used only one fixed set of parameter values to produce the results on the different sequences in the PETS and TUD datasets given in Table \[tab:restab\]. See [@projectwebpage] for the parameters used and information about the runtime. In the MarchingRally video sequence, several people are moving in a crowd in a similar direction. The angle of viewing and the number of people alleviate the issue of clutter, which leads to failure of tracking algorithms that tend to confuse tracking identities. Our algorithm with overlap constraints (red curve) outperforms the state of the art by a large margin. Figure \[fig:aplength\](a) shows the re-detection accuracy results with/without the overlap constraints. Note that the difference in performance between our algorithm and [@pirsiavash11cvpr] grows together with the re-detection time interval. In fact, for the intervals of $40$ frames or more, our algorithm outperforms the baseline by over $20\%$ AP.
The TownCenter sequence is a video with two complementary sets of detections corresponding to heads and upper bodies. While head detections are noisy but have high recall, body detections are more precise but are also prone to more clutter. In such a case, as shown in Figure \[fig:aplength\](b) we leverage body detections to improve noisy head tracks. Again in this case, there is more than $20\%$ improvement in AP over the head baseline. Finally, the table in Figure (\[fig:aplength\]) compares our method with a state-of-the-art [@benfold2011stable] algorithm in terms of traditional MOTA evaluation measure. Note that while we compare with a “greedy” version of the overlap term [@pirsiavash11cvpr], designing a greedy version of the co-occurrence term is not obvious. For the PETS and TUD sequence, we compare the results of our method based on MOTA metrics with those presented in Milan et al. [@milan13cvpr]. These sequences are challenging for a variety of reasons. First, there is a crowd of people walking in different directions and criss-crossing each other, which makes sustained tracking difficult. Second, few full body detections are available per frame in each video, which makes adding new terms to the objective function difficult. Third, since people walk side-by-side there is a lot of overlap between detections that belong to two different persons, hence enforcing the overlap criterion is difficult. However, as can be seen in Table \[tab:restab\], our method generally has comparable MOTA, MOTP and recall scores with [@milan13cvpr]. This shows that our method is able to address complex scenarios effectively and our cost function is easy to adapt to general scenarios. Note also that the camera angle in PETS and TUDS are very different from each other, which means that our algorithm is sufficiently robust to these changes. Thus, we estimate trajectories better (sum of MT and PT of our method is usually high). This also results from the use of both overlap and co-occurrence terms in our approach, which can take into account head detections as additional information.
Discussion and conclusion {#sec:discussion}
=========================
We have presented a generic optimization procedure enabling addition of quadratic costs to the min-cost network flow tracking methods. Our method enables modeling of track interactions in a principled way and provides empirical certificates of small suboptimality. We have shown practical benefits of our method for two particular examples of pairwise costs on challenging video sequences.
Combining different types of pairwise costs into a single (linear) cost opens up the possibility of tracking complicated motions. Moreover, while complex cost functions have more tunable parameters, they could be learnt from labeled data using structured output learning [@lacoste06qap]. This opens up the possibility of learning quadratic costs for specific *crowd actions* such as panic, street crossing or stampede.
Acknowledgements
================
This research was supported in part by the projects FluidTracks, EIT ICT Labs, Google research award, ERC grant Activia (no. 307574) and ERC grant LEAP (no. 336845). We thank Patrick Perez for discussions on the multi-target tracking evaluation.
{width="1.8in"} {width="1.8in"} {width="1.8in"}\
{width="1.8in"} {width="1.8in"} {width="1.8in"}
Superiority of Frank-Wolfe rounding heuristic vs. Hamming rounding {#sec:FWbetter}
==================================================================
In Section \[subsubsec:gradient\_based\_search\], we described two approaches to round the fractional solution ${\mathbf{z}}^*$ obtained after optimizing the LP relaxation . “Rounding” here meant finding a valid track encoding for prediction, i.e. a ${\mathbf{z}}\in \operatorname{FLOW}_k$ with integer coordinates. The first approach was to find the vertex (binary vector) in $\operatorname{FLOW}_k$ with minimal Euclidean distance to ${\mathbf{z}}^*$. We called this approach *Hamming rounding* and is standard for problems operating on binary vectors. We also proposed a novel alternative rounding heuristic called *Frank-Wolfe rounding* which instead minimizes the linear approximation of the quadratic objective , and is given by problem . In our experiments, we observed that Frank-Wolfe rounding yielded solutions with better objective values, as well as better tracking accuracy, than Hamming rounding. We illustrate these observations in this section.
For the MarchingRally experiment (where we only have head detections), we parameterized the objective with two parameters: a multiplicative constant in front of the detection confidences, and the value of the overlap penalty $q_{ij}^{\mathrm{ov}}$ mentioned in (set to a constant).[^11] In Table \[table:suboptim\], we compare the suboptimality certificate values for Frank-Wolfe rounding vs. Hamming rounding for 6 different parameter settings on the MarchingRally dataset. More specifically, for each parameter setting, we first obtain the global relaxed solution ${\mathbf{z}}^*$ to the LP relaxation , then we either round by Frank-Wolfe rounding or by Hamming rounding and compare their suboptimality certificates. We also compare their re-detection accuracy in Figure \[fig:hamming\], which shows that Frank-Wolfe rounding systematically yields better results than Hamming rounding.
Case 1 in Table \[table:suboptim\] is the reference case where we use the best parameter values found by grid search, which were used to produce the results in Figure \[fig:aplength\](a) in the paper. For Case 2 and 3, we vary the overlap penalty weight. Case 2 is a very low value for the overlap term encouraging tracks to criss-cross each other, while Case 3 has a very high overlap weight which means even small amount of overlap is unacceptable. Results for these cases are shown in the first row of Figure \[fig:hamming\]. The next three cases vary the weight for detection confidence. In particular in Case 6, the presence of negative weight actually “discourages” any detections from being picked unless they are connected to edges with extremely high connection strength. This results in poor performance as shown in Figure \[fig:hamming\] but note that even here, Hamming rounding results are worse than the Frank-Wolfe rounding ones. Also note that worse suboptimality certificates usually result in worse tracking.
Detection Overlap FW Ham.
------- ----------- --------- --------- ---------
Case1 0.1 0.0223 4.7e-03 1.4e-02
Case2 0.1 0.0007 8.7e-06 9.3e-03
Case3 0.1 2.23 4.3e-03 1.0e-01
Case4 3.0 0.0223 9.3e-06 8.9e-03
Case5 0.074 0.0223 3.1e-02 1.0e-01
Case6 -1.0 0.0223 1.0e-01 1.3e-01
: Suboptimality certificates for Frank-Wolfe rounding vs. Hamming rounding on the MarchingRally sequence for different parameter value settings of the objective. The first two columns give the parameter value for the detection confidences and the overlap penalty respectively for each case. The last two columns give the suboptimality certificate for Frank-Wolfe rounding and Hamming rounding (lower is better).[]{data-label="table:suboptim"}
Video Results {#sec:MOTAvsUS}
=============
The following images in Figure \[fig:tracks\] shows the tracks overlaid on top of the first frame of the MarchingRally sequence. Each track is shown in a separate color. The output on the top illustrates our result (NF+Overlap) and the one on the bottom illustrates the results of [@pirsiavash11cvpr] (Greedy + NMS). Note how in our case one gets non-overlapping tracks while in the case of [@pirsiavash11cvpr] there are places where tracks overlap and criss-cross. We highlight this in videos available from [@projectwebpage] by drawing cyan colored boxes at places where such ID swaps happen. See Figure \[fig:aplength\](a) for the corresponding re-detection curves. For the more classical metrics, the (MOTA, MOTP, IDswap) numbers for NF+Overlap are (27.7%, 66.5%, 11) vs. (22.5%, 66.0%, 24) for Greedy+NMS.
{width="5in"}\
{width="5in"}
Runtime and Constraints
=======================
For the PETS and TownCenter dataset, typically we have approximately 10–40 detections per frame. For PETS data, each detection is connected to a detection in another frame (with a pairwise term) if they are less than 6 frames apart. On average, each detection is connected to about 10 other detections for pairwise terms (overlap+CO), which means the number of pairwise terms is linear in the number of unary terms. For TownCenter data, we connect detections over 30 frames to account for slower motion of people and missing detections, resulting in about 15 pairwise terms (overlap+CO) per detection on average. While our algorithm runs in about 5–10 seconds on the PETS dataset, it takes about 30–45 minutes on the TownCenter dataset. This difference is due to the larger number of frames in the TownCenter dataset (one order of magnitude greater than for the PETS videos), and also the larger number of pairwise terms per detection on average, resulting in a LP with about 5 million variables in comparison to about 50 thousand for the PETS sequences.
[^1]: WILLOW project-team, Départment d’Informatique de l’Ecole Normale Supérieure, ENS/INRIA/CNRS UMR 8548, Paris, France
[^2]: SIERRA project-team, Départment d’Informatique de l’Ecole Normale Supérieure, ENS/INRIA/CNRS UMR 8548, Paris, France
[^3]: The overlap threshold $o_{\mathrm{thres}}$ is set to 0.5 in our experiments.
[^4]: For the body detection box, we only consider its top 25% region when computing overlap or looking at intersection.
[^5]: A few millions variables, which translates to several hundreds frames with a high number of detections for our datasets.
[^6]: More specifically, this representation defines one indicator variable per possible joint assignment of values on the cliques of the MRF. If we do Fourier-Motzkin elimination [@bertsimas; @wainwright08] on the local consistency polytope to eliminate the extra variables and to only keep the three variables $z_i, z_j, u_{ij}$ for each edge, then we obtain back the constraints for $\operatorname{LOCAL}({\mathcal{Q}})$.
[^7]: The supplementary material (with videos and code) is available at [@projectwebpage].
[^8]: The original MarchingRally video and the corresponding ground truth tracks are available from [@projectwebpage].
[^9]: The overlap between ground truth and detections is measured by the standard Jaccard similarity of corresponding bounding boxes.
[^10]: The confidence for a subtract in this paper is given by the sum of its constituent detection confidences and correspondence strengths.
[^11]: We suppose a multiplicative constant of one in front of the correspondence strengths; changing it as well would just amount to multiply the whole objective by a constant, which would not change the solution.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The k Nearest Neighbors (kNN) method has received much attention in the past decades, where some theoretical bounds on its performance were identified and where practical optimizations were proposed for making it work fairly well in high dimensional spaces and on large datasets. From countless experiments of the past it became widely accepted that the value of $k$ has a significant impact on the performance of this method. However, the efficient optimization of this parameter has not received so much attention in literature. Today, the most common approach is to cross-validate or bootstrap this value for all values in question. This approach forces distances to be recomputed many times, even if efficient methods are used. Hence, estimating the optimal $k$ can become expensive even on modern systems. Frequently, this circumstance leads to a sparse manual search of $k$. In this paper we want to point out that a systematic and thorough estimation of the parameter $k$ can be performed efficiently. The discussed approach relies on large matrices, but we want to argue, that in practice a higher space complexity is often much less of a problem than repetitive distance computations.'
author:
- 'Aleksander Lodwich, Faisal Shafait'
- Thomas Breuel
title: |
Efficient Estimation of $k$ for the Nearest Neighbors Class of Methods\
- unpublished work from 2011 -
---
Introduction
============
Since the introduction of the k Nearest Neighbor (kNN) method by Fix and Hodges in 1951 [@Fix51] a lot of different variants of it have appeared in order to make it suitable to different scenarios. The most notable improvements were done in terms of adaptive distance metrics [@adaptive1][@adaptive2][@adaptive3], fast access via space partitioning (packed R\* trees [@Kame93], kd-trees[@kdtree], X-trees[@xtree], SPY-TEC [@spytec]), knowledge base (prototype) pruning ([@protoopt1],[@protoopt2]) or classification based on sensitive distributed data ([@distributed1],[@distributed2],[@distributed3]). An overview over the state-of-the-art in nearest neighbor techniques is given in [@SurveyKNN].
The continuing richness of investigation work into nearest neighbor can be explained with the omnipresence of CBR (Case Based Reasoning) type of problems [@Aha98theomnipresence] or just from the practical point of view of its massive parallelizability or simply populartiy. After all nearest neighbor is conceptually easy to understand - many students get to learn the nearest neighbor as the first classifier.
All of the beforehand mentioned advances evolve around the question of how to optimize the kNN’s distance measure for retrieving. The method’s apparent laziness might be the reason why a fast preoptimization of $k$ has not been paid a lot attention to.
The proper choice of $k$ is an important factor for achieving maximum performance of a kNN. However, as we will show, conventional $k$ optimization via cross-validation or bootstrapping is slow and promises potential for being sped up. Therefore, we will devote this paper to the concept of fast $k$ optimization.
There is work addressing this issue in an alternative way by introducing *incremental* kNN classifiers based on different types of trees. This class of nearest neighbors attempts to eliminate the influence of k by choosing the right $k$ ad hoc. The rationale behind this method is that for classification tasks the exact majority of a specific class label is not necessarily interesting. It is only interesting when nearest neighbor is used as a density estimator. In other cases it is generally enough to feel safe about which class label rules the nearest set. This class of nearest neighbor starts by polling a minimal amount of nearest neighbors. Then it analyzes the labels and if the retrieved collection of labels is indecisive it will poll more nearest neighbors until it considers the collection decisive enough.
Naturally, the method does not scale to small values of $k$, because e.g. a $k=1$ will never be indecisive. This is a problem because we know from experiments that small $k$ are often optimal. Incremental nearest neighbors have their strength in very large databases where typical queries do not need to compute all relative distances. During a lifetime of a large database some distances might not be computed at all. The lazy distance computation make incremental nearest neighbor methods ideal candidates for real time tasks operating on large volatile data. However, in a cross validation setup which needs to compute all distances incremental kNNs cannot play out their strengths. Because of the restrictions and intended use we exempted incremental methods from investigation in this paper.
We organize this paper in three parts. In the first part we will study the options to make the estimation of $k$ as fast as possible, in the second part we will experimentally compare the result with the conventional approach and in the last part we will draw a conclusion.
Stating the Problem
===================
The kNN is commonly considered a classifier from the area of supervised learning theory. In this theory there exists a training function $T$ that delivers a model $m$ based on a set of options $o$, a matrix of example values $V$ and a vector of labels $l$ of respective size ($m=T(o,V,l)$). In turn, the model $m$ is used in a classification function $C$ that is presented with a matrix of new examples $V'$ and its task is to deliver a vector of new labels $l'$ ($l'=C(m,V')$). $T$ and $C$ must be related but there is no restriction on what the model can be. It can be a set of complex items, a matrix, a vector or just a single value, indeed.
From the perspective of a human the purpose of a model is to make predictions about the future. In order to be able to do this the human brain requires a simplification of the world. Only with the simplification of the world to a reduced number of variables and rules it can compute future states faster than they occur. In case of kNN the model $m$ consists of the data and of the smoothing parameter $k$. This means, that the function $T$ is an identity function between model and parameters. This conflicts with the common notion of a model because the production of models commonly implies reduction. However, the collected data already are a reduction of the world! From a practical point of view they can be considered as representative model states of the world and the data in the model is used to predict the class variable of a new vector before it is actually recorded. This fits perfectly well with the original notion of models. However, the model is only fixed, when $k$ is fixed.
According to the framework, fixing the model (getting the right value for $k$) is the job of the function $T$. Many training techniques in machine learning use optimization strategies developed for real numbers and open parameter spaces. This is not suitable for $k$ as it is an integer value and has known left and right limits: an ideal candidate for full search.
Most frameworks for pattern recognition offer macro optimization for the remaing model parameters that express themself in the initial training options $o$. The structural compatibility of the kNN with the pattern recognition frameworks’ macro optimization functions seduces the users very often to macro optimize $k$. The consequence of this is that kNN must compute distances repetitively as it cannot assume that specific vectors will simply exchange their role between training and testing in the future. In case of kNN this is exceptionally regrettable.
What does macro optimization mean for the computational complexity? Here we assume - but without loose of generality - that the dataset $V$ is of size $n$ ($n$-rows in matrix $V$) and can be exactly divided into $f$ equally sized partitions ready to be rearranged into $f$ different train and test setups. Since everything is being recomputed the computational complexity for this kind of cross-validation of $k$ for a brute kNN is $O\left(\hat k\cdot\frac{f-1}{f}\cdot n^2\right)$. $\hat k$ is the size of the tested range of $k$ and since we are considering full search we accept that $\hat k$ depends on the size $n$ of the dataset and the number of folds $f$. This means that the $\hat k = n\cdot\frac{f-1}{f}$. The scan of nearest sets yields a partial complexity of $O(\frac{f-1}{f}\cdot n^2)$. Hence, for full search the total complexity is $O\left(\left( \frac{f-1}{f} \right)^2\cdot n^3 + \frac{f-1}{f}\cdot n^2 \right)$.
In order to reduce this high complexity it is necessary to optimize kNN within the train function $T$ as it has all necessary information about the relationships among the examples and the labels. This means that $T(o=\{k\},V,l)$ should become $T(o=\{i\},V,l)$ where $i$ is a vector of partition indices of the kind $(0,0,0,...,1,1,1,...,f,f,f...)^T$.
Now, the train function $T$ can utilize the fact that no new data will arrive during the training and all possible distance requests can be computed in advance. Distances within the same partition need not be computed as they will be never requested. These fields can be set to infinity (alternatively they can be filled up with the largest value found in the matrix + 1). The lower triangle is symmetrical to the upper triangle of the matrix because vectors between two points have the same norm. The distance matrix $D$ has a structure as shown in figure \[fig:initial\_matrix\]. The size of $D$ is $n \times n \times 2$. By $D(column,row)_1$ we mean the component *distance* and by $D(column,row)_2$ we mean the associated *label*.
Alone this redesign causes the complexity of the distance computations to get reduced to $O\left( \frac{f-1}{f}\cdot\frac{n^2}{2} \right)$. However this benefit is achieved at the expense of higher memory use. The brute kNN has a space complexity of $O(n)$, now the space complexity has risen to $O(n^2)$.
The next step is to sort the vectors horizontally according to their distance. Although collecting the $k$ best solutions would be faster for a single run it means for a range of $k$ that you effectively obtain the insertion sort. Since there exist faster sorting algorithms we choose to sort but by using a different algorithm.
The fastest algorithm for doing so is the quick sort. Its average complexity is $O\left(n\log(n)\right)$. In the worst case scenario the sorting complexity of this method is $O\left(n^2\right)$. In that case $n$ rows will be sorted with $O\left(n^2\right)$. This means that the worst time complexity so far is $O\left( \frac{f-1}{f}\cdot\frac{n^2}{2} + n^3\right)$ and average case is $O\left( \frac{f-1}{f}\cdot\frac{n^2}{2} + n^2\log(n)\right)$.
In order to obtain nearest neighbors for each vector indexed by the row a counting matrix $M:= {\ensuremath{\mathbb{N}}}^{n\times s}$ is initialized with zeros. $s$ is the number of symbols or classes. For each row in $D$ and $M$ and for the columns $k = 1..\frac{f-1}{f}n$ in $D$ the counters for the specific class label is increased. More precisely, for every row $r = 1..n$ and for every tested $k$ the counters $M\left(D(k,r)_2,r\right)$ are updated by $1$. The complexity of this operation is $O\left(\frac{f-1}{f}\cdot n^2 \right)$. For the overall method this adds up to $O\left(\frac{3}{2}\frac{f-1}{f}\cdot n^2 \right)$. In parallel the level of correct classification must be computed because after every modification of $M$ the state for the smaller $k$ is lost. Therefore a matrix $A:=\mathbb{N}^{\frac{f-1}{f}n \times f}$ for recording the number of correct classfications is required.
How is this number computed? At every round $k$ of the nearest neighbor candidate computations $M_k$ contains in each of its rows a vector that tells how many labels of specific kind are in the nearest neighbors set. The classification label is $l_{rk}' = {\ensuremath{\underset{s}{\operatorname{argmax}}\hspace{0.1cm}}} M_{kr}$. The complexity for this operation is $O( n^2s )$.
This simple method is ambiguous by nature, as there can be many labels that are represented by the same amount of vectors in a nearby set. Computer implementations prefer to return the symbol with the smalest coding. However, it is possible to have a shadow matrix $S := {\ensuremath{\mathbb{R}}}^{n\times s}$ that is the sum of the distances observed for each class label in the set so far. The rule for computing $S$ is the same as for $M$ with the difference that instead of adding ones to the matrix you add distances. When symbol frequency is ambiguous (argmax returns more than one value) it is possible to use $S$ to find which samples are closer overall. Because of the specific interest into fast $k$ optimization the simple argmax processing is used.
Now, every $l_{rk}'$ is compared for equality with $l_r$ (ground truth) and the binary result is added to $A(k, i_r)$. The ${\ensuremath{\underset{k}{\operatorname{argmax}}\hspace{0.1cm}}} A_f$ will return $f$ best $k$. $k^*$ is obtained by averaging.
Considering all parts of the algorithm together the overall complexity is $O\left( \frac{3}{2}\frac{f-1}{f}\cdot n^2 + n^2\log(n) + n^2s \right) \approx O\left( n^2\log(n) \right)$
Experiments
===========
The method for fast $k$ computation (AutokNN) was tested against three other algorithms from the ANN library 1.1[@cit:ann-lib]: brute, kd-tree and bd-tree kNN with default settings. The AutokNN and its competitors performed a complete cross validation run on the ad, diabetes, gene, glass, heart, heartc, horse, ionosphere, iris, mushrooms, soybean, STATLOG australian, STATLOG german, STATLOG heart, STATLOG SAT, STATLOG segment STATLOG shuttle, STATLOG vehicle, thyroid, waveform and wine datasets with 3, 5, 10 and 20 folds. The goal of the experiment was the measurement of the time required to complete the full course of testing different $k$.
The data was separated into stratified partitions which were used in different configurations in order to obtain a training and a testing set. The AutoKNN computes the classification results for all $k$ while the other algorithms are bound to use a logarithmic search. By logarithmic search the following schema is meant: $k \in \{1,2,3,4,5,6,7,8,10,100,200,...,1000,...,\frac{f-1}{f}n\}$. This schema is practically motivated and rational under the assumption that the influence of additional labels on the result diminishes with higher values of $k$. Practical consideration is primarily test time. Example: while AutokNN required 15,35s for a complete scan based on the *ad* dataset, on same data exact brute kNN needed 353,7s in logarithmic mode and 19595,7s in full mode. The use of the logarithmic mode makes results with exactly the same values impossible. However, the differences in resulting $k$ and thus in accuracy were absolutely negligible so that the results are directly comparable nonetheless.
We added the experiment times for all databases up to a total for each cross validation size. The results are shown in the figure and the table under \[fig:result\].
Time measurements were performed on a AMD Phenom II 965 with 8GB of RAM with a Linux 2.6.35 kernel. The algorithms are implemented in C/C++ and were compiled with gcc 4.4.5 with O3 option. Only core algorithm operation was measured and all time for additional I/O was ignored. For best comparability, ANN library sources were statically included.
Discussion and Conclusion
=========================
The Nearest Neighbor approach is considered user friendly and is frequently used for data mining, classification and regression tasks. It is embedded into many automatic environments that make use of kNN’s flexibility. Although kNN has been used, analyzed and advanced for almost six decades a repeating question can not be answered by current literature: What is the fastest way to estimate the right value for $k$ and what are the expenses for doing so.
The approach chosen here is to move the $k$ esimation away from the meta framework right into the training function $T$. The advantage of this is that additional information about the data can be made. This additional information allows to precompute the distances among all vectors without waste and to reuse them numerous times. From this design change which is known to practitioners but not discussed in literature a reduction in time complexity can be observed from $O\left(\left( \frac{f-1}{f} \right)^2\cdot n^3 + \frac{f-1}{f}\cdot n^2 \right)$ to $O\left( \frac{3}{2}\frac{f-1}{f}\cdot n^2 + n^2\log(n) + n^2s \right)$ in average case. The experiments show, that this has significant impact on the speed of the $k$ estimation task. The comparison between kd-tree kNN and the proposed approach proves moreover that having a better time complexity saves practically more time than an efficient distance measure for this task.
The cost of this improvement is a higher space complexity (now $O(n^2)$). In order to esimtate the practical impact of this complexity exchange we studied the contents of the UCI repository [@UCI]. The UCI should be a reasonable crossover of the problems people face in real life.
Out of 162 datasets we found that 90% of them have less than 50K examples, 80% of them have less than 10K examples and half of the UCI’s datasets has less than 1000 examples (for exact distribution see Fig. \[fig:uci\]). These sizes can be easily handled on higher class commodity computers.
This leads to the conclusion that turning in space complexity for time complexity is a good choice most of the time. Future implementations should offer an integrated $k$ searching. The results also show that the so found values for $k$ can be transfered not only to other exact kNN but also to approximate kNN working on kd-tree and bd-tree models.
Acknowledgments
===============
This work has been made possible through the funding of the PaREn (Pattern Recognition and Engineering [@paren]) project by the BMBF (Federal Ministry of Education and Research, Germany).
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APPENDIX {#appendix .unnumbered}
========
Influence of $k$ on the kNN Performance {#apx:perf}
=======================================
The following diagrams are results from kNN based on natural and synthetic datasets. Synthetic datasets were obtained using WGKS [@wgks] The standard deviation was estimated based on a 10x cross-validation. The diagrams are non linear. Sections of little change are compressed, hence x-axis are discontinuous.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We present X-ray data of obtained by . Although it shows a clear orbital intensity modulation with an amplitude of nearly 100% below 0.5 keV in data, the light curves are nearly flat except for a possible dip lasting about one-tenth of the orbital period. We discuss this within the model assumption of a stream-eclipsing geometry as derived from the ROSAT observations.
The X-ray spectrum can be represented by a two temperature optically thin thermal plasma emission model with temperatures of $\sim$1keV and $>$7keV, suggesting postshock cooling as observed in . A remarkable feature of the spectrum is the strong iron emission line whose equivalent width is $\sim$4keV. To account for this, an iron abundance of greater than at least 1.3 times Solar is required. A combined spectral analysis of the PSPC and data indicates that the $N_{\rm H}$-corrected flux ratio of the soft blackbody (0.1–2.4 keV) to the hard optically thin thermal plasma emission (2–10 keV) is as large as $\sim 10^4$.
author:
- 'M. Ishida'
- 'J. Greiner [^1]'
- 'R.A. Remillard'
- 'C. Motch'
date: 'Received 22 December 1997; accepted 14 May 1998'
title: 'ASCA Observation of the polar RXJ1802.1+1804'
---
Introduction
============
A polar (or type object; Cropper 1990) is an accreting binary composed of a mass-donating low mass secondary star and a magnetized white dwarf with a field strength of the order of 10-100MG. Matter from the secondary accretes along the field lines onto a small region of the white dwarf close to the magnetic pole. Since the flow is highly supersonic, a standing shock is formed close to the white dwarf, and a hot plasma with a temperature of $10^8$ K is formed. From the postshock plasma, optical cyclotron emission and optically thin thermal plasma emission in X-rays have been observed. In addition to this, blackbody emission with a temperature of 10–40 eV is observed. Although the blackbody component is considered to be radiated from the surface of the white dwarf around the postshock plasma via reprocessing of the cyclotron and the hard X-ray radiation, its intensity is usually much larger than that of the cyclotron and hard X-ray radiation. This has become known as the so-called ‘soft excess problem’ (Rothschild 1981). Beuermann and Burwitz (1995) recently suggested that the amount of the soft excess is correlated with the strength of the magnetic field of the white dwarf.
It is known that the hard X-ray continuum spectrum of magnetic cataclysmic variables (mCVs) can, to a first approximation, be fitted by an optically thin thermal plasma emission spectrum with a single temperature undergoing photoelectric absorption represented by a single hydrogen column density. To represent the spectra of intermediate polars, Norton and Watson (1989) introduced the so-called ‘partial-covering absorber model’ in which photoelectric absorption was represented by two column densities. Complex absorption was found also in polars with observations (Ishida and Fujimoto 1995).
It has also been expected that the hard X-ray emitting hot plasma is gradually cooled by cyclotron emission and bremsstrahlung (Aizu 1973, Frank, King and Lasota 1983, Imamura and Durisen 1983), and that the hard X-ray spectrum consists of multi-temperature emission components. Such a multi-temperature emission spectrum was first suggested by an observation of (Singh & Swank 1993), and later was established by an observation (Ishida, Mukai and Osborne 1994, Fujimoto and Ishida 1997). Note, however, that it is only for that the existence of multi-temperature emission is confirmed observationally.
was discovered during the search for supersoft X-ray sources in the all-sky survey data (Greiner, Remillard and Motch 1995). Greiner, Remillard and Motch (1995, 1998) have analyzed all the data taken between 1990 September and 1993 September, and found a coherent period of 0.07847977(11) d ( = 1.8835145$\pm$0.000003 hr). The pulse profile in the band $<0.5$ keV is characterized by a deep intensity minimum, with basically no X-ray flux, lasting 0.1 orbital phase. The X-ray spectrum is characterized by strong blackbody emission with a temperature of $20\pm 15$ eV, with a clear excess emission above 1 keV, which has been approximated by thermal bremsstrahlung with a temperature of 20 keV. The absorption-corrected 0.1–2.4 keV fluxes of the blackbody and the thermal bremsstrahlung components are $7 \times 10^{-11}$ erg cm$^{-2}$s$^{-1}$ and $8 \times 10^{-13}$ erg cm$^{-2}$s$^{-1}$, respectively, suggesting a huge soft excess of nearly 90 in the 0.1–2.4 keV band. Szkody (1995) carried out photometry, spectroscopy and polarimetry in the optical band, and found several characteristics of polars such as He[II]{} emission line stronger than H$\beta$ and circular polarization of 4%. All these properties strongly indicate that is a polar.
In this paper, data of taken in 1996 Sep-Oct are presented. In § 2 we describe how the observation was carried out. In § 3 light curve and spectral analysis are presented. We discuss these properties in § 4 in combination with the and data. In § 5 we summarize our results.
Observation
===========
The observation of was carried out between 1996 September 30.75 and October 2.96 (UT). is equipped with four equivalent X-ray Telescopes (XRT: Serlemitsos 1995). In the common focal plane, two Solid-state Imaging Spectrometers (SIS: Burke 1994, Yamashita 1997) and two Gas Imaging Spectrometers (GIS: Makishima 1996, Ohashi 1996) are mounted. The SIS has high sensitivity in the lower energy bandpass and high energy resolution of $\Delta E/E \simeq 0.02$ (at the time of launch), whereas the GIS has high throughput in the higher energy bandpass and high time resolution.
Throughout the observation, the GIS operated in Pulse Height normal mode in which the band 0.7-10 keV is covered by 1024 pulse height channels. The SIS mode was switched between the 1-CCD FAINT mode in high bit rate and the 1-CCD BRIGHT mode in medium bit rate, which cover 0.4-10 keV with 4096 and 2048 pulse height channels, respectively. The observation was performed normally except for loss of data during October 1.57–1.83 (UT) because of a sudden cancelation of the Deep Space Network service.
Analysis and Results
====================
Data Selection
--------------
We have screened the data with the following criteria. The data taken while the spacecraft passes the South Atlantic Anomaly are discarded. In order to avoid the Earth-limb effect, we have only chosen data when the Earth elevation angle of exceeds 5$^\circ$. In addition to this, we have also discarded the SIS data while the elevation angle from the sunny Earth limb is less than 10$^\circ$. For the SIS, we have skipped day-night transition periods of the spacecraft which occur during every satellite orbit. With these selection criteria, some 77 ksec exposure time is retained for both the SIS and the GIS.
For the integration of the X-ray source photons, we have adopted an aperture of 37 and 40 in radius centered on for the SIS and GIS, respectively. For the background, the entire CCD chip outside the aperture is used for the SIS (there are no other X-ray sources within the field of view), whereas an annular region which has the same distance from the boresight of the XRT as the source-integration region is adopted for the GIS.
In Fig. \[LC\], we show the light curve of from all the four detectors in the 0.5–10 keV band with 256 sec binning after the data screening as described above. Because is a low-Earth orbit satellite, the source is usually occulted by the Earth every 96 min. (satellite orbital period). As mentioned in § 2, approximately 6 hrs of data are lost in the middle of the observation due to a failure of data retrieval on a ground station. The average background-subtracted counting rate is 0.052 c s$^{-1}$ with all the four detectors (0.035 c s$^{-1}$ for the two SIS, and 0.017 c s$^{-1}$ for the two GIS).
Energy-resolved Light Curves
----------------------------
Fig. \[FLC\] shows the folded light curves from all the four detectors in the 0.5–10 keV band and in three separate energy bands. In folding the light curves, we have adopted the ephemeris determined from and optical observations (Greiner, Remillard and Motch 1995, 1998), which is
$$T(HJD) = 2449242.3124(21) + 0.07847977(11) \times E.
\label{ROSephem}$$
This ephemeris is accurate enough to predict the time of the dip with an error of only several minutes (phase uncertainty of $\pm 0.02$) at the time of the observation. However, unlike the light curves which show a modulation amplitude of 100% below 0.5 keV, the light curves are extremely flat except for a possible dip during the phase 1.0–1.1. From the average counting rates of the two phase bins around phase 1.08, we have obtained the depth of the dip to be $33\pm 14$%, $35\pm 23$% and $<54$% of the phase-averaged intensity for the bands 0.5–10 keV, 0.5–2 keV and 2–10 keV, respectively, at the 90% confidence level. With the dip duration and depth, it is possible for this dip to correspond to that found in the light curve, although this is not conclusive. If we assume that the dip seen in Fig. \[FLC\] is the same as the one in the observation, the best-fit period would become slightly longer (0.07848022(16) days) but inconsistent with the ROSAT folding.
ASCA Spectra
------------
Since no remarkable intensity variation is found during the observation, we sum up all the data. According to the data selection criteria described in § 2, we have extracted the source and the background spectra separately for the four detectors. Then we have summed the two SIS spectra and the two GIS spectra, and have created background subtracted SIS and GIS spectra, respectively. Hereafter, we derive spectral parameters of by a combined fit of the SIS and GIS spectra using XSPEC version 9.01 (Arnaud 1996).
-------------------- ------------------------------------- --------------- ----------------------- ----------------------- ----------------------- ----------------------- -- -- --
Model 1Abs$\ast$1RS 2Abs$\ast$1RS 2Abs$\ast$1RS 1Abs$\ast$2RS 1Abs$\ast$2RS
+GA +GA
$kT_1$ \[keV\] $\sim$ 0.9 0.97 $^{1.03}_{0.84}$ 0.94 $^{1.02}_{0.82}$ 0.86 $^{0.89}_{0.82}$ 0.87 $^{0.89}_{0.84}$
$kT_2$ \[keV\] — — — 10.3 $^{18.9}_{6.7}$ 30 $^{\infty}_{7.1}$
Abundance \[Solar\] $\sim$ 0.1 0.14 $^{0.23}_{0.09}$ 0.13 $^{0.21}_{0.08}$ 5.8 $^{16.9}_{2.5}$ 0.82 $^{8.3}_{0.26}$
$N_{\rm H1}$ \[$10^{21}$ cm$^{-2}$\] $\sim$ 0.1 $<1.6$ $<$1.9 $<$ 0.6 $<$ 0.8
$N_{\rm H2}$ \[$10^{21}$ cm$^{-2}$\] — 130 $^{170}_{100}$ 130 $^{170}_{100}$ — —
Cover. Frac. \[%\] — 96 $^{97}_{95}$ 97 $^{98}_{96}$ — —
Line Center \[keV\] — — 6.55 $^{6.64}_{6.47}$ — 6.55 $^{6.67}_{6.36}$
Intensity \[$10^{-5}$ phs s$^{-1}$cm$^{-2}$\] — — 1.4 $^{1.9}_{1.0}$ — 0.79 $^{1.16}_{0.42}$
Equiv. Width \[keV\] — — 12 $^{15}_{8}$ — 1.8 $^{2.7}_{1.0}$
$\chi^2_\nu$ (dof) 3.72(52) 1.21(50) 0.68(48) 0.74(50) 0.69(48)
-------------------- ------------------------------------- --------------- ----------------------- ----------------------- ----------------------- ----------------------- -- -- --
We have first tried to fit the spectra with a single temperature optically thin thermal plasma model (Raymond and Smith 1977, hereafter referred to as R&S model) undergoing photoelectric absorption. We note that a thermal bremsstrahlung model has conventionally been used to fit X-ray spectra of mCVs. However, optically thin thermal plasma also produces plenty of emission lines from abundant heavy elements. Among them, the iron $L$-lines appearing between 0.7–2 keV cause a significant excess above the continuum especially in the case that the plasma temperature is lower than $\sim 3$ keV. It is very difficult to resolve them from the continuum spectrum even with the high spectral resolution of the SIS. In addition, there are a few more processes in the optically thin thermal plasma that produce continuum emission, such as the free-bound transition and the two-photon decay. Under these circumstances, we cannot estimate the continuum parameters if we use the thermal bremsstrahlung model. Hence we will substitute the thermal bremsstrahlung model by a R&S model throughout this paper, except when fitting the spectrum in the band 4–10 keV, because the continuum in this band is always dominated by the thermal bremsstrahlung.
The result of the R&S model fit is shown in Fig. \[WARA\], and the best fit parameters are listed in the third column of Table \[ASCApara\]. Although the fit seems good below $\sim$2 keV, it obviously shows excess emission above $\sim$2 keV. The most remarkable structure in the residual is the iron emission line appearing in the 6–7 keV band. If we evaluate this with a thermal bremsstrahlung model plus a Gaussian line in the 4–10 keV band, the equivalent width becomes $4.0\pm1.7$ keV. Note that observations indicated that the equivalent width was in the range 0.2–0.8 keV for a dozen of mCVs (Ishida and Fujimoto 1995). Hence the equivalent width of is roughly an order of magnitude larger than that of mCVs observed by , indicating a greater abundance by the same order.
In order to explain the excess emission above the single component model, we have tried the two possibilities described in § 1, i.e. applying a partial-covering absorption model and introducing a second R&S component.
We first have attempted to apply the partial-covering absorption model to the observed spectra. The result is shown in Fig. \[WARAWARA\] and the best fit parameters are summarized in the 4th column of Table \[ASCApara\]. The reduced $\chi^2$ value of 1.21 means that despite the improvement over the single R&S model the partial-covering absorption model is only marginally acceptable. The best fit values of the two hydrogen column densities are $<2\times 10^{21}$ cm$^{-2}$ and $1.3\times 10^{23}$ cm$^{-2}$ and the covering fraction of the latter over the emission region is $97\pm1$%. From spectra a ratio of soft blackbody flux to hard bremsstrahlung flux of nearly 90 in the 0.1–2.4 keV band was deduced (Greiner, Remillard and Motch 1995, 1998). This extreme soft excess now vanishes because of the high covering fraction of the heavily absorbed component. But the model still does not reproduce the prominent iron emission line which is seen between 6-7 keV. We have thus added a Gaussian, and have fitted the spectra again. The result is summarized in the 5th column of Table \[ASCApara\] and shows that the $\chi^2$ value decreases by nearly 30 after adding two free parameters into the model. Hence, the introduction of the Gaussian is statistically justified. Note, however, that the resulting line equivalent width becomes $\sim$ 12 keV which is unacceptably large. Since the temperature of the emission component is lower than 1 keV, it seems unlikely that this emission line comes from the hot plasma itself. The fluorescent iron emission line is, on the other hand, expected to emanate from the white dwarf surface illuminated by the hard X-ray emission. However, its equivalent width is estimated to be $\sim$ 140 eV (George and Fabian 1991, Done 1994, Beardmore 1995) if the white dwarf surface has solar composition of heavy elements. Therefore, the equivalent width determined from the fit indicates an abundance of the order of $\sim$100 times Solar, which is in strong contradiction to the abundance from the R&S model, $\sim$ 0.1 (Table \[ASCApara\]). We conclude that the partial-covering absorption model cannot reproduce the observed spectrum in a physically consistent manner.
As the next step, we have tried to fit the hard excess component shown in Fig. \[LC\] by introducing a second R&S component. The result of the fit is shown in Fig. \[WARARAGA\], and the best fit parameters are shown in the 6th column of Table \[ASCApara\]. The fit is acceptable with a reduced $\chi^2$ value of 0.74, suggesting that the X-ray spectrum of consists of multi-temperature optically thin thermal plasma emission components. The obtained flux is 4.8$\times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$ in the band 2–10 keV.
Note that this fit still suggests a very high abundance of 6 times Solar with a lower limit of 2.5 times Solar, which seems too high for cataclysmic variables, because they are generally considered to be old systems.
Recently, Hellier (1998) compiled spectra of 15 mCVs from archival data. A total of 14 spectra out of the 15 show a significant fluorescent iron emission line at 6.41 keV, as well as the two thermal plasma components at 6.68 keV and 6.97 keV. Among them, the fluorescent component probably originates from the white dwarf surface (Done, Osborne and Beardmore 1995). Although the statistics of our data is not good enough to resolve these three components, it is necessary to include the fluorescent iron line into the model in evaluating the abundance correctly.
We thus have introduced a Gaussian line as representing the iron line of fluorescence origin. The result is summarized in the last column of Table \[ASCApara\]. Although the best fit abundance is reduced to $\sim$0.9, the equivalent width of the fluorescent iron line becomes around 2 keV. This value again indicates an abundance of more than 10 times Solar. Clearly, the abundances estimated from the intensities of iron lines of the hot plasma origin and of the fluorescence origin should be consistent. This point will be discussed in § 4. Note also that the high abundance can affect the estimation of the bolometric luminosity of the hard component, since the line emission predominates among all the cooling processes in the plasma the temperature of which is less than 2 keV (McCray 1987). We therefore calculate the bolometric luminosity of the hard optically thin thermal component later in relation with the abundance.
Combined Spectral Fit of ROSAT and ASCA
---------------------------------------
Greiner, Remillard and Motch (1995, 1998) reported that the flux of the soft blackbody component is greater than that of the hard thin thermal plasma emission by two orders of magnitude in the band 0.1–2.4 keV. We have attempted to re-examine this extreme soft excess in combination with the hard X-ray data.
pointed four times between 1992 October and 1993 September. We have extracted a mean PSPC spectrum from the observation on 1993 September 11/12 (the exposure time of which was $\sim$ 13 ksec, the longest of all the pointing observations). Details of the observations are presented in Greiner, Remillard and Motch (1998) (see also Greiner, Remillard and Motch 1995).
Since the observation is not simultaneous with the observation, we have first checked if the intensity levels of the two observations are consistent. To do this, we have used the PSPC and the SIS and GIS spectral channels below 2 keV, and have fitted a model consisting of a soft blackbody and a hard thin thermal plasma spectrum undergoing photoelectric absorption represented by a common hydrogen column density. Although the temperatures of both components, the normalization of the blackbody as well as the abundance of the thin thermal plasma are constrained to be the same among the three spectra, the normalization of the thin thermal component is set free to vary independently (note that the blackbody parameters are determined solely by the PSPC spectrum). The resulting normalizations of the hard thin thermal emission of the SIS/GIS relative to that of the PSPC are 1.03 and 1.09, respectively, with a typical statistical error of $\sim \pm 0.3$. We thus regard the intensity level of the hard component as the same between the and the observations.
Next, we have performed a combined spectral fit in the entire 0.1–10 keV band with a model composed of a blackbody and a two temperature thin thermal plasma emission component. The result is shown in Fig. \[WABBRARA\].
The fit is marginally acceptable at the 90% confidence level, with $\chi^{2}_{\nu}$ of 1.15 for 83 degrees of freedom. The confidence contours for the hydrogen column density and the temperature of the blackbody component are also shown in Fig. \[WABBRARA\]. The best fit temperature of the blackbody is obtained to be $15^{+7}_{-5}$ eV for two parameters of interest. The observed flux of the blackbody is $5.6\times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$ in the band 0.1–2.4 keV. After $N_{\rm H}$ correction, the flux becomes $5.1\times 10^{-9}$ erg cm$^{-2}$ s$^{-1}$, which is $\sim 10^4$ times as large as that of the hard component in the 2–10 keV band (§§ 3.3). The major difference to Greiner, Remillard and Motch (1998) is that the blackbody temperature now is even lower than in the fit of the data alone, because a part of the emission around $\approx$0.5 keV is now attributed to the low-kT R&S component.
Because the blackbody temperature is very low, it is very difficult to calculate the bolometric luminosity of the blackbody, because even the PSPC can observe only the high energy end of the Wien tail. Assuming a disc-shape emission region and a distance of 100 pc, we obtain a bolometric luminosity of the blackbody component $L_{\rm BB}$ of $1.6\times 10^{34}$ erg s$^{-1}$/$<\cos \theta>$ for the best fit temperature of 15 eV, where $<\cos \theta>$ implies the cosine of the angle between the normal of the disk and the line of sight. However, the 90% confidence range of $L_{\rm BB}$ becomes $2\times 10^{32} - 1\times 10^{37}$ erg s$^{-1}$ for the temperature of 22–10 eV.
Constraint on the Soft Component from IUE Data
----------------------------------------------
To further constrain the spectral parameters of the soft (blackbody) component, we have utilized the observation performed on 1995 Aug. 31 (Shrader 1997). Among the several exposures of , we collected the data taken out of the X-ray eclipse, which are SWP55775/6 and LWP31382 using the ephemeris of Greiner, Remillard and Motch (1998). For SWP, we took the time average of the two exposures. In Fig. \[AllCombine\], we have plotted the spectra thus obtained together with $N_{\rm H}$-corrected spectra of ($kT=20$ eV) and in a $\nu F(\nu)$ diagram. Note that we have not corrected the color of the spectrum. The correction factor at 1000 [Å]{} is, however, less than 2 from the hydrogen column density $3.2\times 10^{20}$ cm$^{-2}$ (Greiner, Remillard and Motch 1998), and even smaller at longer wavelengths.
We have also shown the $N_{\rm H}$-corrected best fit blackbody spectra obtained from the simultaneous and spectral fitting (§§ 3.4). It is obvious that the shape of the spectrum does not correspond (at least in the long-wavelength region) to that of the blackbody extrapolation, and therefore represents a separate emission component. Thus, the allowed $N_H$-corrected blackbody curves have to be below the spectrum, constraining the blackbody temperature to the range 20–22 eV. The corresponding bolometric luminosity of the blackbody is obtained to be 2–5$\times 10^{32}$ erg s$^{-1}$/$<\cos \theta>$ for a distance of 100 pc. We note that using a white dwarf atmosphere model will predominantly reduce $L_{\rm BB}$ but result in a rather similar effective temperature.
Discussion
==========
Possible Accretion Geometry
---------------------------
As shown in §§ 3.2, the folded light curve shows little evidence of orbital modulation, while the low energy light curves show a deep X-ray intensity modulation with an amplitude of 100%. This probably implies that the accretion pole moves around in the hemisphere of the white dwarf which is visible from the observer, and the X-ray modulation is caused by photoelectric absorption in the accretion column, as is also the case for the recently discovered new polar (Misaki 1996, Thomas and Reinsch 1996).
From the and light curves, we can estimate the hydrogen column density of the accretion column passing over the line of sight. Assuming the absorption by the column can be represented by a single hydrogen column density and the absorbing matter is cold, the PSPC response function predicts $N_{\rm H}>8\times 10^{20}$ cm$^{-2}$ for the counting rate in the band 0.1–0.5 keV to be reduced by 95%, if the blackbody parameters are the same as those out of the eclipse. On the other hand, for the counting rate (SIS+GIS) to be reduced less than 20% in the band 0.5–10 keV, the SIS and the GIS responses require $N_{\rm H}<2\times 10^{21}$ cm$^{-2}$. Therefore, the hydrogen column density of the accretion column passing over the line of sight at the time of eclipse is $\sim 10^{21}$ cm$^{-2}$ on condition that the absorber is cold and can be characterized by a single hydrogen column density. As noted in Greiner, Remillard, Motch (1998), however, a detailed analysis of the energy resolved light curve of indicates that the absorption by the column can hardly be reconciled with a single hydrogen column density. It is possible that the pre-shock column is ionized in part or has a distribution in $N_{\rm H}$ in the range 10$^{21}$ cm$^{-2}$.
Evidence of Postshock Cooling Flow
----------------------------------
As explained in § 1, the hard X-ray spectrum of mCVs can usually be modelled by thermal bremsstrahlung with a single temperature in the range 10–40 keV (Ishida and Fujimoto 1995). Although theories of the postshock accretion flow predict that the postshock plasma is cooled via thermal bremsstrahlung (Aizu 1973) and also cyclotron radiation (Wu 1994, Woelk and Beuermann 1996) evidence of this cooling has been difficult to find observationally, because a thick ($N_{\rm H}\sim 10^{23}$ cm$^{-2}$), partial-covering absorption caused by the accretion column prevents us from measuring the shape of the intrinsic spectrum.
The only exception is in which the hard X-ray continuum emission can be represented by a two temperature R&S model ($kT=$ 0.8 keV and 8 keV) at first order approximation, and the ionization temperatures of heavy elements distribute in the range 0.9–8 keV (Ishida, Mukai and Osborne 1994). Fujimoto and Ishida (1997) showed that the distribution of the ionization temperatures is consistent with the postshock cooling flow predicted by Aizu (1973), and successfully determined the shock temperature and the mass of the white dwarf. The X-ray spectrum of also requires two temperature R&S components, and is similar to that of . We believe that observed the postshock cooling flow in .
This finding probably indicates that the temperature distribution due to the postshock cooling is a common feature among mCVs, and one always finds this as long as the low energy absorption is weak enough ($\leq 10^{21}$ cm$^{-2}$) as in and .
Abundance
---------
As displayed in § 3, the hard part of the X-ray spectrum of has a strong iron emission line with an equivalent width of $\sim$ 4 keV. The line originates from the hot plasma, and also, probably from the white dwarf surface via fluorescence (Hellier, Mukai and Osborne 1998). However, these two components cannot be resolved because of statistical limitations. In obtaining the elemental abundance of the plasma, we have to mix these two components so that the abundances they give are consistent.
As shown in Table \[ASCApara\], the temperature of the hard excess component is uncertain, with a lower limit of $\sim$ 7 keV. Therefore, we have fixed the temperature of the plasma at several trial values between 7 and 30 keV, and have made the following analysis. First, we have adopted thermal bremsstrahlung as the continuum spectrum. Then we have added three Gaussian lines which represent the fluorescent component at $\sim$ 6.4 keV, and He-like and hydrogenic components at 6.68 keV and 6.97 keV, respectively, and have performed the spectral fitting in the band 4–10 keV. In doing this, we have assumed that all the lines are narrow. Also the line central energies of the plasma components are fixed at 6.68 keV and 6.97 keV. The intensities of all the lines are constrained so that they give the same abundance at each fixed temperature. As mentioned in §§ 3.3, the equivalent width of the fluorescent component should be 140 eV, almost irrespective of the plasma temperature, if the white dwarf surface has Solar composition. On the other hand, the equivalent widths of the plasma components can be obtained from the atomic data table in Raymond and Smith (1977) or Mewe (1985) as a function of the plasma temperature in the case of Solar composition plasma. Therefore, the free parameters of the lines are only two — the central energy and the normalization of the fluorescent line. The fitting result thus obtained is shown in Fig. \[AbLumi\].
The resulting abundance is larger for higher trial temperature. This is a result of the fact that $K-$shell electrons are increasingly stripped off for higher temperatures, and hence a higher iron abundance is necessary to account for the observed equivalent width. The smallest abundance is obtained to be 2.4$\pm 1.1\odot$ at a temperature of 7 keV, which implies the lower limit of the abundance to be 1.3$\odot$. Note, however, that this is a very conservative lower limit, and the abundance based on the iron emission line is probably several times as large as that of Solar composition. This is in contrast to the abundances of CVs which have recently been measured to be sub-Solar, such as $0.63\pm 0.08\odot$ for (Fujimoto and Ishida 1997), $0.4^{+0.2}_{-0.1}\odot$ for (Ishida 1997), and $\sim 0.4\odot$ for SS Cyg (Done and Osborne 1997). A hint for a larger abundance than Solar is obtained only for (Misaki 1996).
Luminosity of the Hard X-ray Component and the Soft Excess
----------------------------------------------------------
Since we have obtained the abundance in the previous section, we have next calculated the bolometric luminosity of the hard X-ray component. To do this, we have adopted the volume emissivity formulas of the optically thin plasma approximated by McCray (1987), but modified to take into account the abundance effects. $$\Lambda(T, Z/Z_\odot)\; =\, 1.0\times 10^{-22}
\left( \frac{Z}{Z_\odot}
\right) T_6^{-0.7}\; +$$ $$\hspace{2.4cm}
2.3\times 10^{-24}T_6^{0.5}\hspace{1em} \mbox{[erg cm$^3$ s$^{-1}$]},$$ where $T_6$ is the plasma temperature in $10^6$ K. The first term on the right hand side is the volume emissivity for the line emission which is proportional to the abundance. The second term represents that of the free-free emission. Note that the first term is greater than the second term in the range $T < 2$ keV. The bolometric luminosity of the hard component $L_{\rm H}$ is obtained by $\Lambda \cdot EM$, where $EM$ is the emission measure obtained from the spectral fitting for the 0.8 keV component and the hard excess component separately by assuming a distance to the source. The luminosity thus calculated for the trial temperatures is plotted in the lower panel of Fig. \[AbLumi\] showing a rather flat dependence with temperature in the 7–30 keV range: $L_{\rm H}=0.6-1.4\times 10^{30}$ erg s$^{-1}$. Note that we have not corrected for reflection from the white dwarf surface. One can do this by dividing the above value by $1+a_X$ where $a_X$ is the hard X-ray albedo.
In §§ 3.4, we have obtained the lower limit of the bolometric luminosity of the blackbody component $L_{BB}$ to be $2\times 10^{32}$ erg s$^{-1}$ from and simultaneous spectral fitting. This means $L_{\rm S}/L_{\rm H} > 140/<\cos \theta> $. If we also take data into account (§§ 3.5), $L_{BB}$ is constrained in the range $2-5\times 10^{32}$ erg s$^{-1}$, and hence $L_{\rm S}/L_{\rm H} = (140-830)/<\cos \theta>$ is obtained. Note that white dwarf atmosphere models could possibly reduce the luminosity of the soft component, and thus also $L_{\rm S}/L_{\rm H}$.
Note on Determining Parameters of the Blackbody Spectrum
--------------------------------------------------------
In §§ 3.4, we have derived the temperature of the soft blackbody component to be $15^{+7}_{-5}$ eV. The best fit value is outside the ‘usual’ range derived by Szkody (1995), namely 20–45 eV. In estimating the blackbody temperature, Szkody (1995) assumed a thermal bremsstrahlung component with a temperature of 10 keV for the hard X-ray component. However, based on our data we have found a spectral component which can be represented by a R&S spectrum with $kT \sim$ 0.8 keV. The R&S component with such low temperature has a forest of iron emission lines in the 0.8–1 keV band caused by the iron L-shell transitions (Raymond and Smith 1977). Hence, a significant amount of the flux in the 0.8–2 keV band is attributed to the low temperature R&S component in our modelling. Note that this cannot happen if we assume a thermal bremsstrahlung component with a temperature of 10 keV. As a result, the blackbody temperature becomes lower than the estimates in Szkody (1995).
In analyzing data, one usually assumes the temperature of the hard X-ray component to be around 20 keV (Ramsay 1994, for example). As shown here, however, this may cause a huge systematic error in evaluating the luminosity and the temperature of the soft blackbody component.
Conclusion
==========
We presented X-ray data of obtained by . From the light curves we find only marginal evidence for orbital intensity modulation which is seen in the light curve below 0.5 keV characterized by the sharp and deep minima. From this energy dependence, we conclude that the intensity modulation is caused mostly by photoelectric absorption in the pre-shock accretion column, and the accreting pole moves around on the hemisphere visible from the observer, consistent with the conclusions from Greiner, Remillard & Motch (1998). It is possible that the line of sight absorber is partly ionized or has a distribution in $N_H$ in the range 10$^{21}$ cm$^{-2}$.
The X-ray spectrum can be represented by a two temperature optically thin thermal plasma emission model with temperatures of $\sim$ 1 keV and $>$ 7 keV. In analogy with , we deduce that observed the cooling of the postshock plasma, as indicated by the theory of the postshock accretion flow. A remarkable feature of the X-ray spectrum of is the strong iron emission line whose equivalent width is $\sim 4$ keV. To account for this, an iron abundance greater than Solar by at least 1.3 times is required. From the combined analysis of the PSPC and spectra, the ratio of the bolometric luminosity of the soft component to the hard is revealed to be greater than 140.
We are grateful for Dr. C.R. Shrader for supplying us with his spectra. MI greatly appreciates financial support from JSPS along the Japan-Germany collaboration programme. JG is supported by the Deutsche Agentur für Raumfahrtangelegenheiten (DARA) GmbH under contract numbers FKZ 50 OR 9201 and 50 QQ 96023. RR acknowledges partial support from NASA grant NAG5–1784.
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[^1]: Present address: Astrophysical Institute Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany; [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We define a “nit” as a radix $n$ measure of quantum information which is based on state partitions associated with the outcomes of $n$-ary observables and which, for $n>2$, is fundamentally irreducible to a binary coding. Properties of this measure for entangled many-particle states are discussed. $k$ particles specify $k$ nits in such a way that $k$ mutually commuting measurements of observables with $n$ possible outcomes are sufficient to determine the information.'
author:
- Karl Svozil
title: Quantum information in base $n$ defined by state partitions
---
The formal concept of information is tied to physics, at least as far as applicability is a concern. There seems to be one issue, which, despite notable exceptions (e.g., [@zeil-99 Footnote 6] and [@Muthukrishnan]), has not yet been acknowledged widely: the principal irreducibility of quantum information in base $n$. Define a “nit” as a unit of information equal to the amount of information obtained by learning which of $n$ equally likely events occurred. An $n$-state particle can be prepared in a single one of $n$ possible states. Then, this particle carries one nit of information, namely to “be in a single one from $n$ different states.” Subsequent measurements may confirm this statement. The most natural code basis for such a configuration is $n$, and not a binary one.
Classically, there is no preferred code basis whatsoever. Every classical state is postulated to be determined by a point in phase space. Formally, this amounts to an infinite amount of information in whatever base, since with probability one, all points are random; i.e., algorithmically incompressible [@calude:94]. Operationally, only a finite amount of classical information is accessible. Yet, the particular base in which this finite amount of classical information is represented is purely conventional. The same holds true for discrete classical systems, such as $n$ modes of vibration on a string, where the restriction to these particular states is rather arbitrary.
The fundamental difference between classical and quantum information with respect to code bases could be illustrated by the following example. Physically, each nit could be represented by an $n$-level system. A single measurement collapses an $n$-state superposition and yields only one output, not $\log_2 n$ outputs. In the nonentangled $k$ particle case, the $k$ mutually commuting observables could be some physical quantity (e.g., energy levels) associated with each particle. This sets the stage for the more general observables associated with “entangled” states. References [@zeil-99] and [@DonSvo01] discuss examples with Bell states and Greenberger-Horne-Zeilinger states for the binary case, respectively.
In what follows, let us always consider a complete system of base states ${\cal B}$ associated with a unique “context” or “communication frame” ${\cal F}=\{F_1,F_2, \ldots ,F_{k} \}$, which corresponds to co-measurable observables with $n$ outcomes. For $n=2$, their explicit form has been enumerated in [@DonSvo01]. In this particular case, the $F$’s can be identified with certain projection operators from the set of all possible mutually orthogonal ones, whose two eigenvalues can be identified with the two states. For three or more particles, this is no longer possible.
It should be emphasized that only the case of an entanglement between different particles but not within each particle is considered. If more than one observable could be associated with each particle, then these can become entangled as well, and then $k$ $n$-ary observables will no longer be sufficient to describe $k$ particles.
For a single $n$-state particle, the nit can be formalized as a state partition which is fine grained into $n$ elements, one state per element. That is, if the set of states is represented by $\{1,\ldots ,n\}$, then the nit is defined by $
%\begin{equation}
\{\{1\} ,\ldots ,\{n\}\}.
%\label{2002-statepart-snit}
%\end{equation}
$ Of course, any labeling would suffice, as long as the structure is preserved. It does not matter whether one calls the states, for instance, “+,” “0” and “-”, or “1,” “2” and “3”, resulting in a trit represented by $\{\{+\} ,\{0\} ,\{-\}\}$ or $\{\{1\} ,\{2\} ,\{3\}\}$ (here, the term “trit” stands for a nit with $n=3$). Thus, nits are defined modulo isomorphisms (i.e., one-to-one translations) of the state labels. To complete the setup of the single particle case, let us recall that any such state set would correspond to an orthonormal basis of $n$-dimensional Hilbert space.
Before proceeding to the most general case, we shall consider the case of two particles with three states per particle in all details. We shall adopt an $n$-ary search strategy. Assume that the first and the second particle has three orthogonal states labeled by $a_1,b_1,c_1$ and $a_2,b_2,c_2$, respectively. Then nine product states can be formed and labeled from $1$ to $9$ in lexicographic order; i.e., $
%\begin{equation}
a_1a_2 \equiv 1, \cdots , c_1c_2 \equiv 9
%\label{2002-statepart-ps3}
%\end{equation}
$. Consider a set of two comeasurable three-valued observables inducing two state partitions of the set of states $S=\{1,2,\ldots , 9\}$ with three partition elements with the properties that (i) the set theoretic intersection of any two elements of the two partitions is a single state, and (ii) the union of all these nine intersections is just the set of state $S$. As can be easily checked, an example for such state partitions are $$\begin{array}{llllll}
F_1&=&\{\{1,2,3\},\{4,5,6\},\{7,8,9\}\}&\equiv& \{\{a_1\},\{b_1\},\{c_1\}\},\\
F_2&=&\{\{1,4,7\},\{2,5,8\},\{3,6,9\}\}&\equiv& \{\{a_2\},\{b_2\},\{c_2\}\}.\\
\end{array}
\label{2002-statepart-ps3e}$$ Operationally, the trit $F_1$ can be obtained by measuring the first particle state: $\{1,2,3\}$ is associated with state $a_1$, $\{4,5,6\}$ is associated with $b_1$, and $\{7,8,9\}$ is associated with $c_1$. The trit $F_2$ can be obtained by measuring the state of the second particle: $\{1,4,7\}$ is associated with state $a_2$, $\{2,5,8\}$ is associated with $b_2$, and $\{3,6,9\}$ is associated with $c_2$. This amounts to the operationalization of the trits (\[2002-statepart-ps3e\]) as state filters. In the above case, the filters are “local” and can be realized on single particles, one trit per particle. In the more general case of rotated “entangled” states (cf. below), the trits (more generally, nits) become inevitably associated with joint properties of ensembles of particles. Measurement of the propositions, [*“the particle is in state $\{1,2,3\}$”*]{} and, [*“the particle is in state $\{3,6,9\}$”*]{} can be evaluated by taking the set theoretic intersection of the respective sets; i.e., by the proposition, [*“the particle is in state $\{1,2,3\}\cap \{3,6,9\} = 3$.”*]{} In figure \[2002-statepart1\], the state partitions are drawn as cells of a two-dimensional square spanned by the single cells of the two three-state particles.
[ccccccc]{} 0.37mm
(110.00,125.00) (9.67,25.00)[(90.33,90.00)\[cc\]]{} (40.00,115.00)[(0,-1)[90.00]{}]{} (70.00,115.00)[(0,-1)[90.00]{}]{} (9.67,55.00)[(1,0)[90.33]{}]{} (100.00,85.00)[(-1,0)[90.33]{}]{} (0.00,40.00)[(0,0)\[cc\][$c_2$]{}]{} (0.00,70.00)[(0,0)\[cc\][$b_2$]{}]{} (0.00,100.00)[(0,0)\[cc\][$a_2$]{}]{} (25.00,125.00)[(0,0)\[cc\][$a_1$]{}]{} (55.00,125.00)[(0,0)\[cc\][$b_1$]{}]{} (85.00,125.00)[(0,0)\[cc\][$c_1$]{}]{} (25.33,15.00)[(0,0)\[cc\][$a_1$]{}]{} (55.33,15.00)[(0,0)\[cc\][$b_1$]{}]{} (85.33,15.00)[(0,0)\[cc\][$c_1$]{}]{} (110.00,40.00)[(0,0)\[cc\][$c_2$]{}]{} (110.00,70.00)[(0,0)\[cc\][$b_2$]{}]{} (110.00,100.00)[(0,0)\[cc\][$a_2$]{}]{} (55.00,100.00)[(86.00,26.00)\[\]]{} (25.00,100.00)[(0,0)\[cc\][$1$]{}]{} (55.00,100.00)[(0,0)\[cc\][$2$]{}]{} (85.00,100.00)[(0,0)\[cc\][$3$]{}]{} (55.00,70.00)[(86.00,26.00)\[\]]{} (55.00,40.00)[(86.00,26.00)\[\]]{} (25.00,70.00)[(0,0)\[cc\][$4$]{}]{} (55.00,70.00)[(0,0)\[cc\][$5$]{}]{} (85.00,70.00)[(0,0)\[cc\][$6$]{}]{} (25.00,40.00)[(0,0)\[cc\][$7$]{}]{} (55.00,40.00)[(0,0)\[cc\][$8$]{}]{} (85.00,40.00)[(0,0)\[cc\][$9$]{}]{} (55.00,0.00)[(0,0)\[cc\][$F_1$]{}]{}
& 0.50mm
(0.00,105.00) (0.00,52.00)[(0,0)\[cc\][$\;$]{}]{}
& 0.37mm
(110.00,120.00) (10.00,25.00)[(90.33,90.00)\[cc\]]{} (40.33,115.00)[(0,-1)[90.00]{}]{} (70.33,115.00)[(0,-1)[90.00]{}]{} (10.00,55.00)[(1,0)[90.33]{}]{} (100.33,85.00)[(-1,0)[90.33]{}]{} (0.00,40.00)[(0,0)\[cc\][$c_2$]{}]{} (0.00,70.00)[(0,0)\[cc\][$b_2$]{}]{} (0.00,100.00)[(0,0)\[cc\][$a_2$]{}]{} (25.00,125.00)[(0,0)\[cc\][$a_1$]{}]{} (55.00,125.00)[(0,0)\[cc\][$b_1$]{}]{} (85.00,125.00)[(0,0)\[cc\][$c_1$]{}]{} (25.33,15.00)[(0,0)\[cc\][$a_1$]{}]{} (55.33,15.00)[(0,0)\[cc\][$b_1$]{}]{} (85.33,15.00)[(0,0)\[cc\][$c_1$]{}]{} (110.00,40.00)[(0,0)\[cc\][$c_2$]{}]{} (110.00,70.00)[(0,0)\[cc\][$b_2$]{}]{} (110.00,100.00)[(0,0)\[cc\][$a_2$]{}]{} (25.33,70.00)[(26.00,86.00)\[\]]{} (55.33,70.00)[(26.00,86.00)\[\]]{} (85.33,70.00)[(26.00,86.00)\[\]]{} (27.33,100.00)[(0,0)\[cc\][$1$]{}]{} (27.33,70.00)[(0,0)\[cc\][$4$]{}]{} (27.33,40.00)[(0,0)\[cc\][$7$]{}]{} (57.33,100.00)[(0,0)\[cc\][$2$]{}]{} (57.33,70.00)[(0,0)\[cc\][$5$]{}]{} (57.33,40.00)[(0,0)\[cc\][$8$]{}]{} (87.33,100.00)[(0,0)\[cc\][$3$]{}]{} (87.33,70.00)[(0,0)\[cc\][$6$]{}]{} (87.33,40.00)[(0,0)\[cc\][$9$]{}]{} (55.00,0.00)[(0,0)\[cc\][$F_2$]{}]{}
& 0.50mm
(0.00,105.00) (0.00,52.00)[(0,0)\[cc\][$\;$]{}]{}
& 0.37mm
(110.0,120.00) (10.00,25.00)[(90.33,90.00)\[cc\]]{} (40.33,115.00)[(0,-1)[90.00]{}]{} (70.33,115.00)[(0,-1)[90.00]{}]{} (10.00,55.00)[(1,0)[90.33]{}]{} (100.33,85.00)[(-1,0)[90.33]{}]{} (0.00,40.00)[(0,0)\[cc\][$c_2$]{}]{} (0.00,70.00)[(0,0)\[cc\][$b_2$]{}]{} (0.00,100.00)[(0,0)\[cc\][$a_2$]{}]{} (25.00,125.00)[(0,0)\[cc\][$a_1$]{}]{} (55.00,125.00)[(0,0)\[cc\][$b_1$]{}]{} (85.00,125.00)[(0,0)\[cc\][$c_1$]{}]{} (25.33,15.00)[(0,0)\[cc\][$a_1$]{}]{} (55.33,15.00)[(0,0)\[cc\][$b_1$]{}]{} (85.33,15.00)[(0,0)\[cc\][$c_1$]{}]{} (110.00,40.00)[(0,0)\[cc\][$c_2$]{}]{} (110.00,70.00)[(0,0)\[cc\][$b_2$]{}]{} (110.00,100.00)[(0,0)\[cc\][$a_2$]{}]{} (25.00,100.00)[(26.00,26.00)\[\]]{} (25.00,100.00)[(0,0)\[cc\][1]{}]{} (55.00,100.00)[(26.00,26.00)\[\]]{} (85.00,100.00)[(26.00,26.00)\[\]]{} (55.00,100.00)[(0,0)\[cc\][2]{}]{} (85.00,100.00)[(0,0)\[cc\][3]{}]{} (25.00,70.00)[(26.00,26.00)\[\]]{} (25.00,40.00)[(26.00,26.00)\[\]]{} (25.00,70.00)[(0,0)\[cc\][4]{}]{} (25.00,40.00)[(0,0)\[cc\][7]{}]{} (55.00,70.00)[(26.00,26.00)\[\]]{} (55.00,40.00)[(26.00,26.00)\[\]]{} (85.00,70.00)[(26.00,26.00)\[\]]{} (85.00,40.00)[(26.00,26.00)\[\]]{} (55.00,70.00)[(0,0)\[cc\][5]{}]{} (55.00,40.00)[(0,0)\[cc\][8]{}]{} (85.00,70.00)[(0,0)\[cc\][6]{}]{} (85.00,40.00)[(0,0)\[cc\][9]{}]{} (55.00,0.00)[(0,0)\[cc\][$F_1\wedge F_2$]{}]{}
A Hilbert space representation of this setting can be obtained as follows. Define the states in $S$ to be one-dimensional linear subspaces of nine-dimensional Hilbert space; e.g., $
%\begin{equation}
1 \equiv (1,0,0,0,0,0,0,0,0),\cdots ,
9 \equiv (0,0,0,0,0,0,0,0,1).
%\label{2002-statepart-ps3a}
%\end{equation}
$ The trit operators are given by (the terms “trit operator,” “observable,” and the corresponding state partition will be used synonymously) $$\begin{array}{llll}
F_1&=& \diag (a,a,a,b,b,b,c,c,c),\\
F_2&=& \diag (a,b,c,a,b,c,a,b,c),\\
\end{array}
\label{2002-statepart-ps3top}$$ for $a,b,c \in {\Bbb R}$, $a\neq b\neq c\neq a$.
If $F_2= \diag (d,e,f,d,e,f,d,e,f)$ and $a,b,c,d,e,f,$ are six different prime numbers, then, due to the uniqueness of prime decompositions, the two trit operators can be combined to a single “context” operator $$C=F_1\cdot F_2=F_2\cdot F_1=
\diag (ad,ae,af,bd,be,bf,cd,ce,cf)
\label{2002-statepart-ps3pd}$$ which acts on both particles and has nine different eigenvalues. Just as for the two-particle case [@DonSvo01], there exist $3^2!=9!=362880$ permutations of operators which are all able to separate the nine states. They are obtained by forming a $(2\times 9)$-matrix whose rows are the diagonal components of $F_1$ and $F_2$ from Eq. (\[2002-statepart-ps3top\]) and permuting all the columns. The resulting new operators $F_1'$ and $F_2'$ are also trit operators.
A generalization to $k$ particles in $n$ states per particle is straightforward. We obtain $k$ partitions of the product states with $n$ elements per partition in such a way that every single product state is obtained by the set theoretic intersection of $k$ elements of all the different partitions.
Every single such partition can be interpreted as a nit. All such sets are generated by permuting the set of states, which amounts to $n^k!$ equivalent sets of state partitions. However, since they are mere one-to-one translations, they represent the same nits. This equivalence, however, does not concern the property of (non)entanglement, since the permutations take entangled states into nonentangled ones. We shall give an example below.
Again, the standard orthonormal basis of $n^k$-dimensional Hilbert space is identified with the set of states $S=\{1,2,\ldots ,n^k\}$; i.e., (superscript “$T$” indicates transposition) $$\begin{array}{llll}
1 &\equiv& (1,\ldots,0)^T\equiv \mid 1,\ldots ,1\rangle = \mid 1\rangle \otimes \cdots \otimes \mid 1\rangle ,\\
&\vdots&\\
n^k &\equiv& (0,\ldots,1)^T\equiv \mid n,\ldots ,n\rangle = \mid n\rangle \otimes \cdots \otimes \mid n\rangle .\\
\end{array}
\label{2002-statepart-psma}$$ The single-particle states are also labeled by $1$ through $n$, and the tensor product states are formed and ordered lexicographically ($0<1$).
The nit operators are defined via diagonal matrices which contain equal amounts $n^{k-1}$ of mutually $n$ different numbers such as different primes $q_1,\ldots ,q_n$; i.e., $$\begin{array}{llll}
F_1&=& \diag (\underbrace{\underbrace{q_1,\ldots ,q_1}_{n^{k-1}\;{\rm times}},\ldots ,\underbrace{q_n,\ldots ,q_n}_{n^{k-1}\;{\rm times}}}_{n^0\;{\rm times}}),\\
F_2&=& \diag (\underbrace{\underbrace{q_1,\ldots ,q_1}_{n^{k-2}\;{\rm times}},\ldots ,\underbrace{q_n,\ldots ,q_n}_{n^{k-2}\;{\rm times}}}_{n^1\;{\rm times}}),\\
&\vdots&\\
F_k&=& \diag (\underbrace{q_1,\ldots ,q_n}_{n^{k-1}\;{\rm times}}).
\end{array}
\label{2002-statepart-nitopgen}$$ The operators implement an $n$-ary search strategy, filtering the search space into $n$ equal partitions of states, such that a successive applications of all such filters renders a single state.
There exist $n^k!$ sets of nit operators, which are are obtained by forming a $(k \times n^k)$-matrix whose rows are the diagonal components of $F_1,\ldots,F_k$ from Eq. (\[2002-statepart-nitopgen\]) and permuting all the columns. The resulting new operators $F_1',\ldots,F_k'$ are also nit operators.
All partitions discussed so far are equally weighted and well balanced, as all elements of them contain an equal number of states. In principle, one could also consider nonbalanced partitions. For example, one could take the partition $\overline{F}_1=\{\{1\},\{2,3\},\{4,5,6,7,8,9\}\}$ instead of $F_1$ in (\[2002-statepart-ps3e\]), represented the by trit diagonal operator $\diag (a,b,b,c,c,c,c,c,c)$. Yet any such attempt would result in a deviation from the optimal $n$-ary search strategy, and in an nonoptimal measurement procedures. Another, more principal, disadvantage would be the fact that such a state separation could not reflect the inevitable $n$-arity of the quantum choice.
In terms of partitions, entanglement occurs for diagonal or antidiagonal arrangements of states which do not add up to completed blocks. Take, for example, the state partition scheme of Fig. \[2002-statepart1\], which results in nonentangled states and state measurements. A modified, entangled scheme can be established by just grouping the states into diagonal and counterdiagonal groups as drawn in Fig. \[2002-statepart2\]. The corresponding trits are $$\begin{array}{llll}
F_1&=&\{\{1,5,9\},\{2,6,7\},\{3,4,8\}\},\\
F_2&=&\{\{1,6,8\},\{2,4,9\},\{3,5,7\}\}.\\
\end{array}
\label{2002-statepart-ps3eentan}$$
[ccccccc]{} 0.37mm
(110.00,125.00) (9.67,25.00)[(90.33,90.00)\[cc\]]{} (40.00,115.00)[(0,-1)[90.00]{}]{} (70.00,115.00)[(0,-1)[90.00]{}]{} (9.67,55.00)[(1,0)[90.33]{}]{} (100.00,85.00)[(-1,0)[90.33]{}]{} (0.00,40.00)[(0,0)\[cc\][$c_2$]{}]{} (0.00,70.00)[(0,0)\[cc\][$b_2$]{}]{} (0.00,100.00)[(0,0)\[cc\][$a_2$]{}]{} (25.00,125.00)[(0,0)\[cc\][$a_1$]{}]{} (55.00,125.00)[(0,0)\[cc\][$b_1$]{}]{} (85.00,125.00)[(0,0)\[cc\][$c_1$]{}]{} (25.33,15.00)[(0,0)\[cc\][$a_1$]{}]{} (55.33,15.00)[(0,0)\[cc\][$b_1$]{}]{} (85.33,15.00)[(0,0)\[cc\][$c_1$]{}]{} (110.00,40.00)[(0,0)\[cc\][$c_2$]{}]{} (110.00,70.00)[(0,0)\[cc\][$b_2$]{}]{} (110.00,100.00)[(0,0)\[cc\][$a_2$]{}]{} (12.00,113.00)[(0,-1)[13.00]{}]{} (12.00,100.00)[(1,-1)[73.00]{}]{} (85.00,27.00)[(1,0)[13.00]{}]{} (98.00,27.00)[(0,1)[13.00]{}]{} (98.00,40.00)[(-1,1)[73.00]{}]{} (25.00,113.00)[(-1,0)[13.00]{}]{} (12.00,83.00)[(0,-1)[13.00]{}]{} (12.00,53.00)[(0,-1)[13.00]{}]{} (25.00,83.00)[(-1,0)[13.00]{}]{} (25.00,53.00)[(-1,0)[13.00]{}]{} (55.00,27.00)[(1,0)[13.00]{}]{} (25.00,27.00)[(1,0)[13.00]{}]{} (68.00,27.00)[(0,1)[13.00]{}]{} (38.00,27.00)[(0,1)[13.00]{}]{} (38.00,40.00)[(-1,1)[13.00]{}]{} (55.00,27.00)[(-1,1)[43.00]{}]{} (68.00,40.00)[(-1,1)[43.00]{}]{} (25.00,27.00)[(-1,1)[13.00]{}]{} (42.00,113.00)[(1,0)[13.00]{}]{} (72.00,113.00)[(1,0)[13.00]{}]{} (42.00,100.00)[(0,1)[13.00]{}]{} (72.00,100.00)[(0,1)[13.00]{}]{} (98.00,70.00)[(0,-1)[13.00]{}]{} (98.00,100.00)[(0,-1)[13.00]{}]{} (98.00,57.00)[(-1,0)[13.00]{}]{} (98.00,87.00)[(-1,0)[13.00]{}]{} (85.00,87.00)[(-1,1)[13.00]{}]{} (98.00,70.00)[(-1,1)[43.00]{}]{} (85.00,57.00)[(-1,1)[43.00]{}]{} (98.00,100.00)[(-1,1)[13.00]{}]{} (25.00,100.00)[(0,0)\[cc\][1]{}]{} (55.00,100.00)[(0,0)\[cc\][2]{}]{} (85.00,100.00)[(0,0)\[cc\][3]{}]{} (25.00,70.00)[(0,0)\[cc\][4]{}]{} (25.00,40.00)[(0,0)\[cc\][7]{}]{} (55.00,70.00)[(0,0)\[cc\][5]{}]{} (55.00,40.00)[(0,0)\[cc\][8]{}]{} (85.00,70.00)[(0,0)\[cc\][6]{}]{} (85.00,40.00)[(0,0)\[cc\][9]{}]{} (55.00,0.00)[(0,0)\[cc\][$F_1$]{}]{}
& 0.50mm
(0.00,105.00) (0.00,52.00)[(0,0)\[cc\][$\;$]{}]{}
& 0.37mm
(110.00,125.00) (9.67,25.00)[(90.33,90.00)\[cc\]]{} (40.00,115.00)[(0,-1)[90.00]{}]{} (70.00,115.00)[(0,-1)[90.00]{}]{} (9.67,55.00)[(1,0)[90.33]{}]{} (100.00,85.00)[(-1,0)[90.33]{}]{} (0.00,40.00)[(0,0)\[cc\][$c_2$]{}]{} (0.00,70.00)[(0,0)\[cc\][$b_2$]{}]{} (0.00,100.00)[(0,0)\[cc\][$a_2$]{}]{} (25.00,125.00)[(0,0)\[cc\][$a_1$]{}]{} (55.00,125.00)[(0,0)\[cc\][$b_1$]{}]{} (85.00,125.00)[(0,0)\[cc\][$c_1$]{}]{} (25.33,15.00)[(0,0)\[cc\][$a_1$]{}]{} (55.33,15.00)[(0,0)\[cc\][$b_1$]{}]{} (85.33,15.00)[(0,0)\[cc\][$c_1$]{}]{} (110.00,40.00)[(0,0)\[cc\][$c_2$]{}]{} (110.00,70.00)[(0,0)\[cc\][$b_2$]{}]{} (110.00,100.00)[(0,0)\[cc\][$a_2$]{}]{} (98.00,113.00)[(-1,0)[13.00]{}]{} (85.00,113.00)[(-1,-1)[73.00]{}]{} (12.00,40.00)[(0,-1)[13.00]{}]{} (12.00,27.00)[(1,0)[13.00]{}]{} (25.00,27.00)[(1,1)[73.00]{}]{} (98.00,100.00)[(0,1)[13.00]{}]{} (68.00,113.00)[(-1,0)[13.00]{}]{} (38.00,113.00)[(-1,0)[13.00]{}]{} (68.00,100.00)[(0,1)[13.00]{}]{} (38.00,100.00)[(0,1)[13.00]{}]{} (12.00,70.00)[(0,-1)[13.00]{}]{} (12.00,100.00)[(0,-1)[13.00]{}]{} (12.00,57.00)[(1,0)[13.00]{}]{} (12.00,87.00)[(1,0)[13.00]{}]{} (25.00,87.00)[(1,1)[13.00]{}]{} (12.00,70.00)[(1,1)[43.00]{}]{} (25.00,57.00)[(1,1)[43.00]{}]{} (12.00,100.00)[(1,1)[13.00]{}]{} (98.00,83.00)[(0,-1)[13.00]{}]{} (98.00,53.00)[(0,-1)[13.00]{}]{} (85.00,83.00)[(1,0)[13.00]{}]{} (85.00,53.00)[(1,0)[13.00]{}]{} (55.00,27.00)[(-1,0)[13.00]{}]{} (85.00,27.00)[(-1,0)[13.00]{}]{} (42.00,27.00)[(0,1)[13.00]{}]{} (72.00,27.00)[(0,1)[13.00]{}]{} (72.00,40.00)[(1,1)[13.00]{}]{} (55.00,27.00)[(1,1)[43.00]{}]{} (42.00,40.00)[(1,1)[43.00]{}]{} (85.00,27.00)[(1,1)[13.00]{}]{} (25.00,100.00)[(0,0)\[cc\][1]{}]{} (55.00,100.00)[(0,0)\[cc\][2]{}]{} (85.00,100.00)[(0,0)\[cc\][3]{}]{} (25.00,70.00)[(0,0)\[cc\][4]{}]{} (25.00,40.00)[(0,0)\[cc\][7]{}]{} (55.00,70.00)[(0,0)\[cc\][5]{}]{} (55.00,40.00)[(0,0)\[cc\][8]{}]{} (85.00,70.00)[(0,0)\[cc\][6]{}]{} (85.00,40.00)[(0,0)\[cc\][9]{}]{} (55.00,0.00)[(0,0)\[cc\][$F_2$]{}]{}
& 0.50mm
(0.00,105.00) (0.00,52.00)[(0,0)\[cc\][$\;$]{}]{}
& 0.37mm
(110.0,120.00) (10.00,25.00)[(90.33,90.00)\[cc\]]{} (40.33,115.00)[(0,-1)[90.00]{}]{} (70.33,115.00)[(0,-1)[90.00]{}]{} (10.00,55.00)[(1,0)[90.33]{}]{} (100.33,85.00)[(-1,0)[90.33]{}]{} (0.00,40.00)[(0,0)\[cc\][$c_2$]{}]{} (0.00,70.00)[(0,0)\[cc\][$b_2$]{}]{} (0.00,100.00)[(0,0)\[cc\][$a_2$]{}]{} (25.00,125.00)[(0,0)\[cc\][$a_1$]{}]{} (55.00,125.00)[(0,0)\[cc\][$b_1$]{}]{} (85.00,125.00)[(0,0)\[cc\][$c_1$]{}]{} (25.33,15.00)[(0,0)\[cc\][$a_1$]{}]{} (55.33,15.00)[(0,0)\[cc\][$b_1$]{}]{} (85.33,15.00)[(0,0)\[cc\][$c_1$]{}]{} (110.00,40.00)[(0,0)\[cc\][$c_2$]{}]{} (110.00,70.00)[(0,0)\[cc\][$b_2$]{}]{} (110.00,100.00)[(0,0)\[cc\][$a_2$]{}]{} (25.00,100.00)[(26.00,26.00)\[\]]{} (25.00,100.00)[(0,0)\[cc\][1]{}]{} (55.00,100.00)[(26.00,26.00)\[\]]{} (85.00,100.00)[(26.00,26.00)\[\]]{} (55.00,100.00)[(0,0)\[cc\][2]{}]{} (85.00,100.00)[(0,0)\[cc\][3]{}]{} (25.00,70.00)[(26.00,26.00)\[\]]{} (25.00,40.00)[(26.00,26.00)\[\]]{} (25.00,70.00)[(0,0)\[cc\][4]{}]{} (25.00,40.00)[(0,0)\[cc\][7]{}]{} (55.00,70.00)[(26.00,26.00)\[\]]{} (55.00,40.00)[(26.00,26.00)\[\]]{} (85.00,70.00)[(26.00,26.00)\[\]]{} (85.00,40.00)[(26.00,26.00)\[\]]{} (55.00,70.00)[(0,0)\[cc\][5]{}]{} (55.00,40.00)[(0,0)\[cc\][8]{}]{} (85.00,70.00)[(0,0)\[cc\][6]{}]{} (85.00,40.00)[(0,0)\[cc\][9]{}]{} (55.00,0.00)[(0,0)\[cc\][$F_1\wedge F_2$]{}]{}
We can now introduce new $2\times 3$ basis vectors grouped into the two bases $\{a_1',b_1',c_1'\}$ and $\{a_2',b_2',c_2'\}$ by $$\begin{array}{llll}
\vert a_1' \rangle &=&{1\over \sqrt{3}}
(\vert a_1a_2\rangle + \vert b_1b_2\rangle + \vert c_1c_2\rangle )
% \equiv {1\over \sqrt{3}} (1,0,0,0,1,0,0,0,1)
,\\
\vert b_1' \rangle &=&{1\over \sqrt{3}}
(\vert a_1b_2\rangle + \vert b_1c_2\rangle + \vert c_1a_2\rangle )
%\equiv {1\over \sqrt{3}} (0,1,0,0,0,1,1,0,0)
,\\
\vert c_1' \rangle &=&{1\over \sqrt{3}}
(\vert a_1c_2\rangle + \vert b_1a_2\rangle + \vert c_1b_2\rangle )
%\equiv {1\over \sqrt{3}} (0,0,1,1,0,0,0,1,0)
,\\
\vert a_2' \rangle &=&{1\over \sqrt{3}}
(\vert a_1a_2\rangle + \vert b_1c_2\rangle + \vert c_1b_2\rangle )
%\equiv {1\over \sqrt{3}} (1,0,0,0,0,1,0,1,0)
,\\
\vert b_2' \rangle &=&{1\over \sqrt{3}}
(\vert a_1b_2\rangle + \vert b_1a_2\rangle + \vert c_1c_2\rangle )
%\equiv {1\over \sqrt{3}} (0,1,0,1,0,0,0,0,1)
,\\
\vert c_2' \rangle &=&{1\over \sqrt{3}}
(\vert a_1c_2\rangle + \vert b_1b_2\rangle + \vert c_1a_2\rangle )
%\equiv {1\over \sqrt{3}} (0,0,1,0,1,0,1,0,0)
.
\end{array}
\label{2002-statepart-notb}$$ The new orthonormal basis states are “entangled” with respect to the old bases and [*vice versa*]{}. Their tensor products generate a complete set of basis states in a new nine-dimensional Hilbert space. In terms of the new basis states, the trits can be written as $F_1\equiv \{\{a_1'\},\{b_1'\},\{c_1'\}\}$ and $F_2\equiv \{\{a_2'\},\{b_2'\},\{c_2'\}\}$. The associated bases will be called [*diagonal bases*]{}. Note that the permutation which produces the entangled case (\[2002-statepart-ps3eentan\]) the nonentangled (\[2002-statepart-ps3e\]) one is $1\rightarrow 1$, $2\rightarrow 9$, $3\rightarrow 5$, $4\rightarrow 6$, $5\rightarrow 2$, $6\rightarrow 7$, $7\rightarrow 8$, $8\rightarrow 4$, $9\rightarrow 3$, or $(1)(2,9,3,5)(4,6,7,8)$ in cycle form. A generalization to diagonal bases associated with an arbitrary number of nits is straightforward.
In summary we have shown that, by adopting an $n$-ary search strategy, $k$ particles (entangled or not) specify $k$ nits in such a way that $k$ mutually commuting measurements of independent observables with $n$ outcomes are necessary and sufficient to determine the information. This finding is compatible to Zeilinger’s foundational principle for quantum mechanics [@zeil-99]. In general, the main emphasis in the area of quantum computation has been in the area of binary decision problems. It is suggested that these investigations should be extended to decision problems with $n$ alternatives (e.g., [@kleene-52 pp. 332-340]), for which quantum information theory seems to be extraordinarily well equipped.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The rotational dynamics of CO single molecules solvated in small He clusters (CO@He$_N$) has been studied using Reptation Quantum Monte Carlo for cluster sizes up to $N=30$. Our results are in good agreement with the roto-vibrational features of the infrared spectrum recently determined for this system, and provide a deep insight into the relation between the structure of the cluster and its dynamics. Simulations for large $N$ also provide a prediction of the effective moment of inertia of CO in the He nano-droplet regime, which has not been measured so far.'
author:
- Paolo Cazzato
- Stefano Paolini
- Saverio Moroni
- Stefano Baroni
date:
title: ' Rotational dynamics of CO solvated in small He clusters: a quantum Monte Carlo study '
---
Introduction
============
Thanks to recent progresses in Helium nano-droplet isolation (HENDI) spectroscopy,[@Scoles2001] the infrared and microwave spectra of small molecules solvated in He$_N$ clusters are now becoming accessible in the small- and intermediate-size regimes.[@Jaeger2002; @Jaeger2003; @McKellar2003-OCS; @McKellar2003-CO] In the particular case of carbon monoxide, the roto-vibrational spectrum of the molecule solvated in small He clusters (CO@He$_N$) has been recently studied in the size range $N=2\--
20$.[@McKellar2003-CO] The infrared spectrum in the 2145 cm$^{-1}$ region of the C–O stretch consists of two $R(0)$ transitions which smoothly correlate with the $a$-type ($K=0\leftarrow 0$) and $b$-type ($K=1\leftarrow 0$) $R(0)$ lines of the binary complex, CO@He$_1$. The series of $b$-type transitions—which starts off about 7 times stronger for $N=1$—progressively looses intensity as $N$ increases, until it disappears around $N=7\-- 8$. Around this size, just before it disappears, the $b$-type line seems to split in two. Analogously, around $N=15$ the $a$-type line also seems to split, and the assignment of experimental lines becomes uncertain for larger clusters. Elucidating the relation existing between the position, number, and intensity of the rotational lines and the size and structure of the cluster is the principal goal of the present paper.
Computer simulations of quantum many-body systems have also considerably progressed in recent years, allowing in some cases to determine the low-lying spectrum of excited states. The rotational dynamics of small molecules solvated in He clusters and nanodroplets is one of these favorable instances. The scarcity of low-lying excited states typical of superfluid systems makes it possible in this case to extract information on the location and intensity of the spectral lines from an analysis of the time series generated by quantum Monte Carlo random walks.[@NoAntri-Cornell; @NoAntri-PRL; @POITSEH] The rotational spectrum of OCS@He$_N$ has been studied along these lines in Refs. \[\]. Among the many different flavors of quantum Monte Carlo available in the literature, we adopt [*Reptation Quantum Monte Carlo*]{}[@NoAntri-Cornell; @NoAntri-PRL] which we believe presents distinctive advantages in the present case and which will be briefly introduced in Sec. \[sec:RQMC\]. In Sec. \[sec:results\] we present and discuss our results, whereas Sec. \[sec:conclusions\] contains our conclusions.
Theory, algorithms, and technical details {#sec:RQMC}
=========================================
Virtually all the ground-state quantum simulation methods are based on the prior knowledge of some approximate wave-function, $\Phi_0$, for the system under study. In the Variational Monte Carlo Method (VMC) one contents oneself of this knowledge and the simulation simply aims at calculating the complicated multi-dimensional integrals which are needed to estimate ground-state [*approximate*]{} expectation values, $ \langle \Phi_0 | \widehat A | \Phi_0 \rangle $ (here and in the following quantum-mechanical operators are indicated with a hat, $\widehat{~~}$). To this end, a random walk in configuration space is generated according to the Langevin equation: $$d{\textbf x} = \epsilon ~ {\textbf f}_0({\textbf x}) + d\bm{\xi},
\label{eq:langevin}$$ where ${\textbf x} \equiv \{x_\alpha\} $ indicates the coordinates of the system, $\epsilon$ is the step of time discretization, $${\textbf f}_0({\textbf x}) = 2 {\partial \log\left ( \Phi_0({\textbf
x}) \right ) \over \partial {\textbf x} },$$ and $d\bm{\xi}$ is a Gaussian random variable of variance $2\epsilon$: $\langle d\xi_\alpha d\xi_\beta \rangle = 2\epsilon \delta_{\alpha
\beta}$. [*Approximate*]{} ground-state quantum expectation values are then estimated as time averages over the random walk, Eq. (\[eq:langevin\]).
Within Reptation Quantum Monte Carlo (RQMC), [*exact*]{} ground-state expectation values and imaginary-time correlation functions are calculated as appropriate derivatives of the pseudo partition function, in the low temperature (large $T$) limit: $${\cal Z}_0 = \langle\Phi_0|e^{-T \widehat H}|\Phi_0\rangle ,
\label{eq:Z0}$$ where $\widehat H$ is the Hamiltonian of the system. By breaking the time $T$ into $P$ intervals of length $\epsilon = T/P$, Eq. (\[eq:Z0\]) can be given a path-integral representation: $${\cal Z}_0 \approx \int \Phi_0({\textbf x}_0)
\Pi_{i=0}^{P-1} \langle {\textbf x}_i|
e^{-\epsilon\widehat H} |{\textbf x}_{i+1} \rangle
\Phi_0({\textbf x}_P) d^{P+1}{\textbf x}~. \label{eq:PI}$$ For the relatively small systems considered here, it is sufficient to use the [*primitive approximation*]{} to the imaginary-time propagator: $$\langle {\textbf x}|e^{-\epsilon\widehat H}|{\textbf y}
\rangle \propto
\ealla{
-({\textbf x}-{\textbf y})^2/2\epsilon
-\epsilon[V({\textbf x})+V({\textbf y})]/2
}
+{\cal O}(\epsilon^3),$$ where $V({\textbf x})$ is the potential energy at point ${\textbf x}$. The dynamical variables of the statistical-mechanical system whose partition function is given by Eq. (\[eq:PI\]) are segments of the VMC random walk generated from Eq. (\[eq:langevin\]), ${\textbf
x}(\tau)$ of length $T$, which we call [*reptiles*]{}. As the random walk proceeds, the reptile is allowed to creep back and forth: new configurations of the reptile are accepted or rejected according to a Metropolis test made on the integrand of the path-integral representation of ${\cal Z}_0$, Eq. (\[eq:PI\]). It can be demonstrated[@NoAntri-Cornell; @NoAntri-PRL] that in the large-$T$ limit the sample of reptile configurations thus generated is such that the sample average of quantities like $${\cal A}[{\textbf x}(\tau)] = {1\over T} \int_0^T A({\textbf
x}(\tau)) d\tau
\label{eq:A_estimator}$$ converges without any systematic bias (but those due to the finite values of $\epsilon$ and $T$) to $\langle \widehat A \rangle = \langle
\Psi_0 | \widehat A | \Psi_0 \rangle$, $\Psi_0$ being the [*exact*]{} ground-state wavefunction of the system. Even more interesting is the fact that sample averages of reptile time correlations, $${\cal C}_A(\tau) = {1\over T-\tau} \int_0^{T-\tau} A({\textbf x}(\tau'))
A({\textbf x}(\tau'+\tau)) d\tau', \label{eq:CA_estimator}$$ provide equally unbiased estimates of the corresponding quantum correlation functions in imaginary time: $$\begin{aligned}
C_{\widehat A}(it) &= &
\langle \Psi_0 | \widehat A(it)
\widehat A(0) | \Psi_0 \rangle \nonumber \\
&\equiv&
\langle \Psi_0 | \ealla{\widehat H t} \widehat A
\ealla{-\widehat H t} \widehat A | \Psi_0 \rangle,\end{aligned}$$ ${\cal C}_A(\tau)\approx C_{\widehat A}(i\tau)$. The absorption spectrum of a molecule solvated in a non polar environment is given by the Fourier transform of the autocorrelation function of its electric dipole, $\textbf d$: $$\begin{aligned}
I(\omega) &\propto& 2\pi \sum_n | \langle \Psi_0|\widehat {\textbf
d}|\Psi_n \rangle |^2 \delta(E_n-E_0-\omega) \nonumber \\ &=&
\int_{-\infty}^\infty \ealla{i\omega t} \langle \widehat {\textbf d}(t)
\cdot \widehat {\textbf d} (0) \rangle dt,\end{aligned}$$ where $\Psi_0$ and $\Psi_n$ are ground- and excited-state wavefunctions of the system respectively, and $E_0$ and $E_n$ the corresponding energies. The dipole of a linear molecule—such as CO—is oriented along its axis, so that the optical activity is essentially determined by the autocorrelation function of the molecular orientation versor: $c(t)\equiv C_{\widehat {\textbf n}}
(t)=\langle \Psi_0| \ealla{i\widehat Ht} \widehat {\textbf n}
\ealla{-i\widehat Ht} \widehat {\textbf n} |\Psi_0 \rangle$. We have seen that RQMC gives easy access to the analytic continuation to imaginary time of correlation functions of this kind. From now on, when referring to [*time correlation functions*]{}, we will mean [ *reptile time correlations*]{}, [*i.e.*]{} quantum correlation functions in imaginary time. Continuation to imaginary time transforms the oscillatory behavior of the real-time correlation function—which is responsible for the $\delta$-like peaks in its Fourier transform—into a sum of decaying exponentials whose decay constants are the excitation energies, $E_n-E_0$, and whose spectral weights are proportional to the absorption oscillator strengths, $|\langle
\Psi_0|{\textbf d}|\Psi_n \rangle |^2$. Dipole selection rules imply that only states with $J=1$ can be optically excited from the ground state which has $J=0$. Information on excited states with different angular momenta, $J$, can be easily obtained from the multipole correlation functions, $c_J(\tau)$, defined as the reptile time correlations of the Legendre polynomials: $$\begin{aligned}
c_J(\tau)
&=&
\left \langle
P_J ( {\textbf n}(\tau) \cdot {\textbf n}(0) )
\right \rangle
\nonumber \\
&\equiv&
\left \langle
{4\pi\over 2J+1} \sum_{M=-J}^J Y^*_{JM}\bigl
({\textbf n}(\tau) \bigr ) Y_{JM}\bigl ({\textbf n}(0) )
\right \rangle
\quad {.} \quad \end{aligned}$$
Both the He-He and the He-CO interactions used here are derived from accurate quantum-chemical calculations.[@Korona; @Heijmen] The CO molecule is allowed to perform translational and rotational motions, but it is assumed to be rigid. The trial wavefunction is chosen to be of the Jastrow form: $$\Phi_0 = \exp \left [ -\sum\limits_{i=1}^N {\cal U}_1(r_i,\theta_i)
-\sum\limits_{i<j}^N {\cal U}_2(r_{ij}) \right ],$$ where ${\textbf r}_i$ is the position of the i-th atom with respect to the center of mass of the molecule, $r_i=|{\textbf r}_i|$, $\theta_i$ is the angle between the molecular axis and ${\textbf r}_i$, and $r_{ij}$ is the distance between the i-th and the j-th helium atoms. ${\cal U}_1$ is expressed as a sum of five products of radial functions times Legendre polynomials. All radial functions (including ${\cal U}_2$) are optimized independently for each cluster size with respect to a total of 27 variational parameters. The propagation time is set to $T=\rm 1~ K^{-1}$, with a time step of $\epsilon = \rm
10^{-3}~ K^{-1}$. The effects of the length of the time step and of the projection time have been estimated by test simulations performed by halving the former or doubling the latter. These effects were barely detectable on the total energy, and very small on the excitation energies discussed below (we estimate that more converged simulations would actually improve the already excellent agreement with experimentally observed spectra).
to
The estimate of excitation energies and spectral weights from imaginary-time correlations amounts to performing an inverse Laplace transform, a notoriously ill-conditioned operation which is severely hindered by statistical noise.[@gubernatis] For each value of $J$, we extract the value of the two lowest-lying excitation energies, $\epsilon_{a,b}^J$ —[*i.e.*]{} the two smallest decay constants in $c_J(\tau)$—as well as the corresponding spectral weights, $A_{a,b}^J$, from a fit of $c_J(\tau)$ to a linear combination of three decaying exponentials. This fitting procedure does not solve in general the problem of obtaining the spectrum from a noisy imaginary-time correlation function. However, if we know in advance that very few strong peaks, well separated in energy, nearly exhaust the entire spectral weight, their position and strength can be reliably estimated from this multi-exponential fit. In the present study these favorable conditions are usually met, although the limitations of the procedure will show in some cases, as discussed below.
![ (color) Upper left panel: He-CO interaction potential. C (blue) and O (cyan) atoms are pictured by two circles whose radius is the corresponding Van de Waals radius. The other panels picture the differential He density, $\Delta\rho_N = \rho_N-\rho_{N-1}$ for various sizes of the CO@He$_N$ cluster. Color convention is rainbow: red to purple in order of increasing magnitude. \[fig:effetti-speciali\] ](pannello.eps){width="100mm"}
Results and discussion {#sec:results}
======================
RQMC simulations have been performed for CO@He$_N$ clusters in the size range $N=1\-- 30$. In Fig. \[fig:deltaE\] we report the values of the He atomic binding energy, $\Delta E_N=E_{N-1}-E_N$, as a function of the cluster size. $\Delta E_N$ first increases up to $N=4\-- 5$, and it stays roughly constant in the range $N=5\-- 8$; from this size on $\Delta E_N$ starts decreasing, first slowly, then, from $N=10\-- 11$, rapidly down to a minimum at $N=19$. For $N>19$ $\Delta E_N$ increases again and slowly tends to the nanodroplet regime (where it coincides with the bulk chemical potential, $\mu=7.4~
\rm K$ [@NoAntri-PRL]) which is however attained for much larger cluster sizes than explored here.[@paesani] This behavior can be understood by comparing the shape of the CO–He potential energy function, $v({\textbf r})$, with the incremental atomic density distributions, $\Delta\rho_N({\textbf r}) = \rho_N({\textbf r}) -
\rho_{N-1}({\textbf r})$, where $\rho_N$ is the expectation value of the He density operator: $$\widehat \rho({\textbf r}) = \sum_{i=1}^N \delta({\textbf
r}-{\textbf r}_i)$$ (see Fig. \[fig:effetti-speciali\]). For very small $N$ the atomic binding energy is dominated by the He–CO attraction which is strongest in a well located atop the oxygen atom. As He atoms fill this well, $\Delta E_N$ first slightly increases, as a consequence of the attractive He–He interaction, then, for larger $N$, the increased He–He interaction is counter-balanced by the spill-out of He atoms off the main attractive well, until for $N\approx 9$ the reduction of the He–CO interaction overcomes the increased attraction and the binding energy starts decreasing steeply. For $N$ in the range 10–14 He density accumulates towards the C pole, while, around $N=15$, the first solvation shell is completed and the differential atomic density, $\Delta\rho_N$, is considerably more diffuse starting from $N=16$. $\Delta E_N$ reaches a minimum at $N=19$. For larger sizes, the trend in the atomic binding energy is dominated by the increase of the He–He attraction related to the increase of the cluster size, until it will converge to the bulk chemical potential.
to
to
In Fig. \[fig:EA\_QMC\] we report the positions and spectral weights of the rotational lines, as functions of the cluster size, $N$. In the size range $N=1\-- 9$, analysis of the dipole time correlations clearly reveals the presence of two peaks, with the weight of the higher–energy ($b$-type) rapidly decreasing by almost a factor 2. Note that the sum of the spectral weights of these two lines nicely sums to one, indicating that they exhaust all the oscillator strength available for optical transitions originating from the ground state. For $N$ between 10 and 12 (shaded area in Fig. \[fig:EA\_QMC\]) the situation is less clear. As the weight of the $b$-type line drops to zero, the statistical noise on its position grows enormously. Furthermore the multi-exponential fit introduces some ambiguity, as the results are somewhat sensitive to the number of terms in the sum. However the important information that one line disappears between $N=10$ and 12 is clear. For larger $N$ only one relevant line remains, and the robustness of the fitting procedure is recovered, with the minor exception of the sizes around $N=16$, where the minimum of the $\chi^2$ appears to be less sharp, possibly correlating with the splitting of the line observed in the infrared spectra for $N=15$ (see below).
In the upper panel of Fig. \[fig:E\_expt\] we compare the rotational structure of the observed infrared (vibrational) spectrum[@McKellar2003-CO] with the rotational excitation energies calculated in this work. Experimental data are referred to the center, $\nu_0$, of the vibrational band for $N=0$ (CO monomer). In order to better compare our predictions with experiments, we have corrected the former with an estimate of the [*vibrational shift*]{}, $\Delta
\nu_0$, i.e. the displacement of the vibrational band origin as a function of the number of He atoms. The vibrational shift can be calculated as the difference in the total energy of the cluster obtained with two slightly different potentials,[@Heijmen] $v_{00}$ and $v_{11}$, representing the interaction of a He atom with the CO molecule in its vibrational ground state and first excited states, respectively. Our estimate of the vibrational shift as a function of the cluster size is reported in the lower panel of Fig. \[fig:E\_expt\]. Since the evaluation of a small difference between two large energies is computationally demanding for large clusters, $\Delta \nu_0$ has been evaluated perturbatively with respect to the difference $v_{00}-v_{11}.$[@paesani] We have used the vibrational shift calculated in Ref. after verifying on small clusters that the perturbative treatment is reliable. The agreement between our results and experiments is remarkable. Some of the features of the observed spectrum, however, call for a deeper understanding and theoretical investigation. Two questions, in particular, naturally arise. Why two peaks are observed in the small-size regime, and what determines the disappearance of one of them at $N=8$? What determines the split of the higher-frequency ($b$-type) line at $N=7$ and of the lower-frequency ($a$-type) one at $N=15$?
to
The existence of two lines for small $N$ is likely due to a larger asymmetry of the cluster in this regime. If the CO@He$_N$ complex is described as a rigid rotor, in fact, one would have one rotational line originating from a $J=0$ ground state if the complex has cylindrical symmetry, while this line would double if some of the atomic density accumulates in a longitudinal protrusion. The inertia of the complex would in this case be larger for a rotation about an axis perpendicular to a plane containing the protrusion ([ *end-over-end rotation*]{}) than about an axis lying on such a plane. Given that the He density in the ground state of CO@He$_N$ is cylindrically symmetric, any departure from this symmetry can only show up in higher correlation functions. The situation is conceptually similar to that of a fluid whose density is homogeneous and whose structure at the atomic scale is reflected in the pair correlation function. Analogously, we define an atomic angular correlation function, $C(\phi)$, as the probability of finding two He atoms which form a dihedral angle $\phi$ with respect to the molecular axis: $$C(\phi) =
\left \langle
{1\over N(N-1)} \sum_{i\ne j} \delta(\phi_i-\phi_j-\phi)
\right \rangle.$$ In the upper panel of Fig. \[fig:montarozzi\] we show $C(\phi)$, for different cluster sizes. The depletion of $C$ for $\phi$ larger than $\pi/2$, clearly visible for $N=3$ (green circles), indicates a tendency of the He atoms to cluster on a same side of the molecular axis. For larger clusters, however, this effect weakens to the extent that it becomes difficult to disentangle from the structural information related to the He–He interaction (the dimple at small $\phi$ and the subsequent maximum around $\phi=0.3\-- 0.4~\pi$). A more sensitive measure of the propensity of He atoms to cluster on a side of the molecule is given by the integral of $C(\phi)$ from 0 to $\frac{\pi}{2}$, $$M = \int_0^{\pi\over 2} C(\phi) d\phi-{1\over 2}.
\label{eq_m}$$ In the lower panel of Fig. \[fig:montarozzi\] we display $M$ as a function of the cluster size, $N$: one sees that $M$ decreases with $N$ and reaches a minimum at $N=14$. This is the size at which the first solvation shell is completed, and the cluster asymmetry increases again when the second shell starts to build. The rotational spectrum of the solvated molecule, however, is insensitive to this asymmetry for clusters of this and larger sizes because the motion of He atoms in the second and outer solvation shells is decoupled from that in the first and from molecular rotation. The existence of a longitudinal asymmetry is a necessary condition for the doubling of the rotational line. Whether or not this condition is also sufficient depends on the dynamics: if quantum fluctuations make the motion of the protrusion around the molecular axis fast with respect to the molecular rotation, then the asymmetry is effectively washed out. The existence of two lines in the rotational spectrum of the molecule implies therefore that an asymmetry in the [*classical*]{} distribution of He atoms around the molecular axis exists; that the molecular inertia is sensitive to this asymmetry (the protrusion can be ‘dragged’ along the molecular rotation); and that the motion of this protrusion around the molecular axis is not adiabatically decoupled from the molecular rotation.
to
In order to better characterize the motion of He atoms around the molecule and the coupling of this motion to molecular rotation, we examine the imaginary-time correlations of the versor, $\mathbf{u}$, of the He center of mass, $\mathbf{r}_{CM}$, relative to the molecular center of mass: $${\cal C}_{\mathbf{u}}(\tau)=
\langle
\mathbf{u}(\tau)\cdot \mathbf{u}(0)
\rangle. \label{eq:cuu}$$ For the binary complex, He–CO, $\mathbf{r}_{CM}$ coincides with the position of the helium atom, and we expect its angular dynamics to be strongly correlated to the molecular rotation, at least in the end-over-end mode. In Fig. \[fig:adiab\] we report the frequency of the slowest mode appearing in the spectral analysis of ${\cal
C}_{\mathbf{u}}(\tau)$, $\epsilon_{\mathbf{u}}$, as a function of $N$, and compare it with the corresponding frequencies of the molecular rotation. We see that for cluster sizes up to $N=9\-- 10$, $\epsilon_{\mathbf{u}}$ is degenerate with the $a$-type frequency in the molecular rotational spectrum, with a spectral weight which passes from $A_{\mathbf{u}} \approx 1$ for $N=1$ to $A_{\mathbf{u}} \approx
0.7$ for $N=10$. These findings are a manifestation of the fact that He atoms are dragged along the slowest, end-over-end, rotation of the solvated molecule, and that the effect of this dragging decreases when more He states with $J=1$ become available and subtract spectral weight to the slowest mode. For $N>10$, $\epsilon_{\mathbf{u}}$ further increases and departs from $\epsilon_a$, indicating an effective decoupling of the two kinds of motion. In this regime, the effective rotational constant $B$ of the solvated molecule is almost independent of the cluster size. Free molecular rotation with an increased moment of inertia with respect to the gas phase is the typical signature of superfluid behavior in He nanodroplets. Extrapolating the result obtained for $N$ up to 30 to the nanodroplet limit, we predict a renormalization factor of the $B$ value of 0.78. The lowest atomic mode, $\epsilon_{\mathbf{u}}$, slows down again for $N=15$. This is due, however to the slow He motion in the second solvation shell which hardly affects the rotation of the solvated molecule. Although the resolution that can be achieved with our simulations is not sufficient to detect the doubling of the $a$ and $b$ lines which is experimentally observed for $N=15$ and $N=7$ respectively, it is interesting to notice that the former occurs in correspondence with the crossing between $\epsilon_{\mathbf{u}}$ and $\epsilon_a$, possibly due to the resonant interaction between the two modes. It is tempting to assume that a similar mechanism may be responsible for the doubling of the $b$ line at $N=7$, involving however higher-energy He states. A deeper study of the He dynamics would clarify this point.
Conclusions {#sec:conclusions}
===========
Computer simulations of quantum many-body systems have reached such a degree of sophistication and reliability that in some cases they can be used to provide information, complementary to that which can be obtained in the laboratory, on the dynamical processes probed spectroscopically.
In the case of small polar molecules solvated in He clusters, for instance, the calculation of the time autocorrelation of the molecular dipole (which is the quantity directly coupled to the experimental probe) allows to reproduce rather accurately the roto-vibrational excitation energies which are now becoming experimentally accessible for small clusters ($N=1\--20$). Even more importantly, computer simulations give direct access to quantitities and features (such as, [*e.g.*]{}, static and dynamic properties of the He matrix) which are not accessible to the experiment, and whose knowledge provides the basis for understanding the relation between structure and dynamics in these confined boson systems.
In the specific case of CO@He$_N$, which is the subject of the present study and of a recent infrared spectroscopy experiment,[@McKellar2003-CO] the presence of two spectral lines—$a$-type and $b$-type, evolving respectively from the end-over-end and from the free-molecule rotations of the binary complex—is related to the propensity of the He atoms to cluster on a same side of the molecular axis, which we measure by an angular pair distribution function: as more He atoms progressively fill the first solvation shell, their clustering propensity weakens; the CO impurity gets more isotropically coated, looses a preferred axis for the free-molecule mode, and the $b$–type line disappears.
The time autocorrelation of the versor of the He center of mass provides dynamical information on the He atoms in excited states with $J=1$. We find a substantial spectral weight on a He mode whose energy, $\epsilon_{\mathbf{u}}$, is degenerate with the $a$-type line for $N$ up to about ten. This indicates that some of the He density is dragged along by the molecular rotation—in other words, part of the angular momentum in the cluster mode involving molecular rotation is carried by the He atoms. We also find that for larger clusters the molecular rotation effectively decouples from this He mode, and its energy $\epsilon_a$ becomes essentially independent of the number of He atoms. Based on the nearly constant value of $\epsilon_a$ in the range of $N$ between 15 and 30, well beyond completion of the first solvation shell, we predict the effective rotational constant in the nanodroplet limit to be smaller by a factor 0.78 than its gas phase value.
to 0pt
We would like to thank A.R.W. McKellar for providing us with a preprint of Ref. \[\] prior to publication. We are grateful to G. Scoles for bringing that work to our attention, for his continuous interest in our work, and for many useful discussions. Last but not least, we wish to thank S. Fantoni for his encouragement and for a critical reading of our manuscript. This work has been partial supported by the Italian [*MIUR*]{} through [*PRIN*]{}.
[99]{} C. Callegari, K.K. Lehmann, R. Schmied and G. Scoles, J. Chem. Phys. [**115**]{}, 10090 (2001). J. Tang, Y. Xu, A.R.W. McKellar, and W. Jäger, Science [**297**]{}, 1945 (2002). Y.J. Xu and W. Jäger, J. Chem. Phys. [**119**]{}, 5457 (2003). J. Tang, A.R.W. McKellar, J. Chem. Phys. [**119**]{}, 5467 (2003). J. Tang, A.R.W. McKellar, J. Chem. Phys. [**119**]{}, 754 (2003). S. Baroni and S. Moroni, in [*Quantum Monte Carlo Methods in Physics and Chemistry*]{}, edited by P. Nightingale and C.J. Umrigar. NATO ASI Series, Series C, Mathematical and Physical Sciences, Vol. 525, (Kluwer Academic Publishers, Boston, 1999), p. 313, also available at [cond-mat/9808213]{}. S. Baroni and S. Moroni, Phys. Rev. Lett. [**82**]{}, 4745 (1999). D. Blume, M. Lewerenz, P. Niyaz, and K. B. Whaley, Phys. Rev. E [**55**]{}, 3664 (1997). F. Paesani, A. Viel, F.A. Gianturco, and K.B. Whaley, Phys. Rev. Lett. [**90**]{}, 73401 (2003). S. Moroni, A. Sarsa, S. Fantoni, K.E. Schmidt, and S. Baroni, Phys. Rev. Lett. [**90**]{}, 143401 (2003). T. Korona, H. L. Williams, R. Bukowski, B. Jeziorski and K. Szalewicz, J. Chem. Phys. [**106**]{}, 5109 (1997). T. G. A. Heijmen, R. Moszynski, P. E. S. Wormer, and A. van der Avoird, J. Chem. Phys. [**107**]{}, 9921 (1997). J. E. Gubernatis and M. Jarrell, Phys. Rep. [**269**]{}, 135 (1996). F. Paesani and F. A. Gianturco, J. Chem. Phys. [**116**]{}, 10170 (2002).
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---
author:
- 'Th. Maier'
- 'M. B. Zölfl'
- 'Th. Pruschke'
- 'J. Keller'
date: 'Received: date / Revised version: date'
title: 'Magnetic properties of the three-band Hubbard model'
---
Introduction and Model {#intro}
======================
One of the few undisputed facts about high-T$_c$ materials is that all undoped high-$T_c$ compounds are insulators with antiferromagnetic ordering in the $\rm{CuO_2}$-planes at low enough temperatures [@Alm]. Doping of the systems leads to a strong suppression of the antiferromagnetic order and eventually
superconductivity sets in. While the explanation of this transition and the proximity of antiferromagnetism and superconductivity surely is the most fascinating aspect in the high-T$_c$ compounds, there are a variety of other peculiarities that call for an explanation. One of these side aspects is the observation that, although the basic scenario is the same, hole and electron doped materials show an apparent qualitative difference in that the magnetic phase in the latter appears to be much more stable for the latter (cf. Fig. \[expphase\]).
The physics in the insulating phase is well described by a Heisenberg model since charge fluctuations are suppressed by strong local correlations and, to a first approximation, one is left with the $\rm Cu$-spin degrees of freedom only. The exchange parameter of the resulting effective Heisenberg model is obtained from standard superexchange processes [@Anderson; @Zhang], induced by virtual hopping of a hole from one $\rm Cu$-ion to the neighbouring one over a nonmagnetic $\rm O$-ion.
At any finite doping one has to consider at least also the charge degrees of freedom on the copper sites and would then be left with the usual one-band Hubbard or $t$-$J$ model to describe the interplay between magnetic exchange and itinerancy. However, this scenario completely neglects the existence of the oxygen sites. That they are indeed important, at least for the magnetic properties, can be seen from the following qualitative argument: Due to the strong local correlations at the $\rm Cu$-sites, additional doped holes mainly occupy $\rm O$-sites. The spin of the hole at the $\rm
O$-site induces an effective ferromagnetic interaction [@Aha] between the neighbouring $\rm Cu$-spins, so that the antiferromagnetic ordering is strongly suppressed with increasing hole-doping. In the case of electron doping, on the other hand, the additional particle has to go to the copper sites due to Pauli’s principle, which effectively means that free spins are removed from the system. This obviously also leads to a suppression of magnetic order, but in a weaker fashion. Thus, in order to obtain a more realistic description of the physics of high-T$_C$ compounds one has to take into account also the oxygen degrees of freedom and therefore one-band models like the standard Hubbard or $t$-$J$ model become inadequate.
The simpliest model which includes both the effect of strong correlations and the influence of the oxygen sites, is the three-band Hubbard model or Emery model [@Emer]. The three-band Hubbard Hamiltonian reads $$\begin{aligned}
H&=&\sum\limits_{i,\sigma}\varepsilon_d d_{i\sigma}^\dagger
d_{i\sigma} + \sum\limits_{j,\sigma}\varepsilon_p p_{j\sigma}^\dagger
p_{j\sigma}\nonumber\\
& +& \sum\limits_{\langle ij \rangle ,\sigma}t_{ij}\left(
d_{i\sigma}^\dagger p_{j\sigma}+h.c.\right) + \sum\limits_{i} U_d\,
n_{i\uparrow}^d n_{i\downarrow}^d\quad\mbox{,}
\label{3bhh}\end{aligned}$$ where the vacuum is defined as all orbitals in (\[3bhh\]) filled with electrons. With this convention $d_{i\sigma}^\dagger(p_{j\sigma}^\dagger)$ creates a hole in a $\rm{Cu}\,3d_{x^2-y^2}$-($\rm{O}\,2p_{x/y}$)-orbital at site $i(j)$ with spin $\sigma$ and $\varepsilon_d(\varepsilon_p)$ are the corresponding on-site energies. $t_{ij}=\pm t$ denotes the nearest-neighbour hopping matrix-element between $\rm Cu$- and $\rm O$-sites and $U_d$ stands for the Coulomb-interaction of two holes, residing at the same $\rm Cu$-site $i$ with number operators $n_{i\sigma}^d$.
In this paper we want to study magnetic properties of the Hamiltonian (\[3bhh\]) in the framework of the DMFT. In this theory the dynamical renormalizations of the one particle properties become purely local [@Metzner; @Mueller], so that they can be obtained from an effective impurity problem coupled to a self-consistent medium. Due to the additional orbital degreee of freedom in (\[3bhh\]) the mapping on the corresponding effective impurity model is not unique. In order to treat local spin and charge fluctuations between the $\rm Cu$-site and the surrounding $\rm
O$-sites better than on a mean-field level we use the approach, developed in ref. [@Schmalian], where a cluster of one $\rm{Cu}\,d$-orbital and a normalized bonding combination of the four surrounding $\rm O$-orbitals is coupled to the effective medium. This method indeed leads to the anticipated physics on the one-particle level, namely the formation of a low-lying singlet state – the Zhang-Rice singlet [@Zhang] – and at half filling to a charge-transfer insulator [@Schmalian; @Zaanen; @Zoelfl] in contrast to the Mott-Hubbard scenario for the one-band model. So far, however, only the paramagnetic state has been studied in ref. [@Schmalian]. In ref. [@Doniach] the metal-insulator (MI) transiton was studied with Quantum Monte Carlo in the context of the DMFT. There the model was also embedded on a bipartite lattice in order to take into account the antiferromagnetic symmetry breaking, but in this article the attention was called to the MI-transition.
In order to obtain the phase diagram or look at the behaviour in the antiferromagnetically ordered state the method has to be extended to allow for the calculation of susceptibilities or solve the DMFT equations for lattices with AB-like structure, respectively. A short review of this generalization together with the technique of calculating the magnetic susceptibility will be given in the next section, followed by the discussion of our results in section 3. The paper will conclude with a summary and outlook in section 4.
Method
======
The DMFT for the three-band model\[sec:DMFT\]
---------------------------------------------
Let us begin by summarizing the basic concepts introduced in [@Schmalian] for the three-band Hubbard model. In order to construct the DMFT for a Cu-O plaquette, it is convenient to introduce the Fourier transform of the kinetic part of the Hamiltonian (\[3bhh\]) after generalization to $d$ dimensions, which then reads [@Schmalian] $$\begin{aligned}
H=\sum\limits_{{\bf k},\sigma}h_\sigma({\bf k})+\sum\limits_i U_d\,n_{i\uparrow}^d n_{i\downarrow}^d+H_{\rm non-bond}\quad\mbox{,}
\label{FT}\end{aligned}$$ with $$\begin{aligned}
h_\sigma({\bf k})&=&\varepsilon_d d_{{\bf
k}\sigma}^\dagger d_{{\bf
k}\sigma} + \varepsilon_p p_{{\bf
k}\sigma}^\dagger p_{{\bf
k}\sigma}\nonumber\\
&+&\sqrt{2d}t\gamma_{\bf k}\left(d_{{\bf
k}\sigma}^\dagger p_{{\bf
k}\sigma}+h.c.\right)\quad\mbox{.}\end{aligned}$$ Here, $d_{{\bf k}\sigma}$ is the Fourier transform of $d_{i\sigma}$ and $p_{{\bf k}\sigma}$ is the orthonormalized Fourier transform of the hybridizing combination of the oxygen orbitals surrounding a given copper site [@Zhang; @Schmalian]. The $d-1$ linear combinations, which are orthogonal to $p_{{\bf k}\sigma}$ were collected into $H_{\rm non-bond}$ and will be dropped in the following, because they are decoupled from the remainder of the system. Finally, $\gamma_{\bf k}$ is given by $\gamma_{\bf k}^2=1-\frac{1}{d}\sum\limits_{\nu=1}^d \cos k_\nu$. In ref. [@Schmalian] it was shown, that the rescaling $$\sqrt{2d}t\gamma_{\bf k}\rightarrow 2t^*\gamma_{\bf
k}^*$$ with $\gamma_{\bf
k}^*=\sqrt{1-\frac{\varepsilon_{\bf k}}{\sqrt{2d}}}$, $t^{*}=\mbox{const.}\equiv 1$ and [$\varepsilon_{\bf k}=\sum_{\nu=1}^d
\cos k_\nu$]{} leads to a nontrivial limit for $d\rightarrow\infty$. The $d$-Green’s function in the DMFT now takes the form $$G^d_{{\bf k}\sigma}(z)=\left[z-\varepsilon_d-\Sigma^d_\sigma(z)
-\frac{4t^{*^2}-\frac{4t^{*^2}}{\sqrt{2d}}\varepsilon_{\bf k}}{z-\varepsilon_p}\right]^{-1}
\label{Gdofk}$$ The new ansatz by Schmalian et al. was to write the local $d$-Green’s function to be of the form [@Schmalian] $$G^d_\sigma(z)\begin{array}[t]{l}\displaystyle
=\frac{1}{N}\sum\limits_{\bf k}G^d_{{\bf k}\sigma}(z)\\[5mm]
\displaystyle\stackrel{!}{=}
\Big[z-\varepsilon_d-\Sigma^d_\sigma(z)
-\frac{4t^{*^2}}{z-\varepsilon_p-\Delta_\sigma(z)}\Big]^{-1}\;\;.\end{array}
\label{Gdloc}$$ In the DMFT the effective Cu-O cluster lives in a so-called [*effective medium*]{}, defined via $${\cal G}_\sigma(z)^{-1}=G^d_\sigma(z)^{-1}+\Sigma^d_\sigma(z)=
z-\varepsilon_d-\frac{4t^{*2}}{z-\varepsilon_p-\Delta_\sigma(z)}\;\;.
\label{effMedium}$$ Note that in this form the coupling to the rest of the system, which is described by $\Delta_\sigma(z)$ happens through the $p$-states only. This representation of the local Green’s function is obviously not unique. One could also choose a representation for the local $d$-Green’s function of the form $G^d_\sigma(z)=\Big[z-\varepsilon_d-\Sigma^d_\sigma(z)-\Delta_\sigma(z)\Big]^{-1}$ where the resonance at $z=\varepsilon_p$ is included in $\Delta_\sigma(z)$. But from a numerical point of view the form (\[Gdloc\]) is more convenient because the singularity at $z=\varepsilon_p$ is not included in the hybridization function which therefore becomes smooth as a function of frequency. The form (\[Gdloc\]) of the local Green’s function is just the Dyson equation of an effective impurity problem consisting of one $d$- and one $p$-orbital, where only the $p$-orbital hybridizes with the conduction electrons (see eq. 13 in ref. [@Schmalian]).
DMFT for the Néel state\[sec:DMFTAB\]
-------------------------------------
In the antiferromagnetic phase the period of the unit cell of the lattice is doubled due to the reduced translational symmetry. Consequently, the volume of the magnetic Brillouin zone (MBZ) is reduced to one-half of the volume in the paramagnetic state and the vector ${\bf Q}=(\pi,\pi,\pi,\cdots)$ becomes a reciprocal lattice vector. These changes in the symmetries of the system can be simply taken into account by introduction of an AB-sublattice structure [@Brand] and reformulating the theory on an enlarged unit cell containing exactly one A- and one B-site. Since this procedure does not affect the local two-particle interaction in the Hamiltonian (\[FT\]) we will concentrate on the kinetic part for the derivation of the resulting Hamilton matrix. We first split the kinetic part of (\[FT\]) in the following way: $$\begin{aligned}
H={\sum\limits_{{\bf k}\in{\rm MBZ},\sigma}}\left\{h_\sigma({\bf k})+h_\sigma({\bf k+Q})\right\}
\label{red}\end{aligned}$$ Note that the $\bf k$-sum runs over ${\bf k}$-points in the reduced Brillouin zone only! Rewriting (\[red\]) in terms of the linear combinations $$\begin{array}{lcl}
\displaystyle
d_{A/B{\bf k}\sigma}&=&\frac{1}{\sqrt{2}}\left(d_{{\bf k}\sigma}\pm
d_{{\bf k+Q}\sigma}\right)\\[5mm]
\displaystyle
p_{A/B{\bf k}\sigma}&=&\frac{1}{\sqrt{2}}\left(p_{{\bf k}\sigma}\pm
p_{{\bf k+Q}\sigma}\right)\quad\mbox{,}
\end{array}$$ acting on the A- or B-sublattice, respectively, one obtains $$H={\sum\limits_{{\bf k}\in{\rm MBZ},\sigma}} \Psi_{{\bf k}\sigma}^\dagger
\underline{\underline{H}}_\sigma ({\bf k}) \Psi_{{\bf k}\sigma}\;\;.$$ For simplicity we introduced a spinor notation for the operators $\Psi_{{\bf k}\sigma}^\dagger=(d_{A{\bf k}\sigma}^\dagger\,p_{A{\bf
k}\sigma}^\dagger\,d_{B{\bf k}\sigma}^\dagger\,p_{B{\bf
k}\sigma}^\dagger)$ and the Hamilton matrix on the sublattices $${\underline{\underline{H}}}_\sigma({\bf k})=\left(
\begin{array}{cccc}
\varepsilon_d+\Sigma_A^\sigma&\Pi^+_{{\bf k}}&0&\Pi^-_{{\bf k}}\\
\Pi^+_{{\bf k}}&\varepsilon_p&\Pi^-_{{\bf k}}&0\\
0&\Pi^-_{{\bf k}}&\varepsilon_d+\Sigma_B^\sigma&\Pi^+_{{\bf k}}\\
\Pi^-_{{\bf k}}&0&\Pi^+_{{\bf k}}&\varepsilon_p
\end{array}\right)\quad\mbox{.}$$ The quantities $\Sigma_{A/B}^\sigma$ denote the local self-energies due to the two particle term in (\[FT\]) on A/B-sublattice sites, which are different in the antiferromagnetic state. Furthermore $\Pi^\pm_{{\bf k}}=\frac{2t^*}{\sqrt{2}}\left(\gamma_{\bf
k}^*\pm\gamma_{\bf k+Q}^*\right)$. For the $d$-components of the Green’s function matrix we finally obtain $$\underline{\underline{G}}_d^\sigma({\bf
k},z)=C_{{\bf k}\sigma}\left(
\begin{array}{cc}
\xi_B^\sigma&-\frac{\displaystyle 4t^{*^2}\varepsilon_{\bf
k}}{\displaystyle\sqrt{2d}\xi_p}\\
-\frac{\displaystyle 4t^{*^2}\varepsilon_{\bf
k}}{\displaystyle\sqrt{2d}\xi_p}&\xi_A^\sigma
\end{array}\right)
\label{d_alone}$$ with $C_{{\bf
k}\sigma}=\left[\xi_A^\sigma\xi_B^\sigma-\left(\frac{4t^{*^2}\varepsilon_{\bf
k}}{\sqrt{2d}\xi_p}\right)^2\right]^{-1}$, $\xi_p=z+\mu-\varepsilon_p$ and $\xi_{A/B}^\sigma=z+\mu-\varepsilon_d-\Sigma_{A/B}^\sigma-\frac{4t^{*^2}}{\xi_p}$. The local Green’s function is obtained by taking one of the diagonal elements and summing over $\bf k$ with the result $$G_{d,A/B}^\sigma (z)=\int\limits_{-\infty}^\infty
d\varepsilon\,\rho_o(\varepsilon)\frac{\xi^\sigma_{B/A}}{\xi_A^\sigma
\xi_B^\sigma-\left(\frac{{\displaystyle
4t^{*^2}\varepsilon}}{{\displaystyle \xi_p}}\right)^2}\quad\mbox{,}
\label{G_d_local}$$ where the density of states $\rho_o(\varepsilon)$ corresponding to the dispersion $\varepsilon_{\bf k}$ was introduced. In the paramagnetic state, where $\xi^\sigma_A=\xi^\sigma_B$, one immediately recovers the result of section \[sec:DMFT\].
In the antiferromagnetic state it is sufficient to perform the calculations for the A-sublattice only due to the additional symmetry [@Brand] $G_{d,A}^\sigma=G_{d,B}^{\bar{\sigma}}$ and use the spin-index for book-keeping. The actual calculation is now a straightforward extension of the method used in ref. [@Schmalian] for the paramagnetic phase. The local nature of the selfenergies allows the mapping of the lattice problem on an effective impurity-problem, consisting of a $d$-orbital and the orthonormalized hybridizing combination of the four surrounding $p$-orbitals, coupled to the effective medium, which is described by the propagator (\[effMedium\]) and has to be determined selfconsistently. Again, the coupling to the surrounding clusters is assumed to happen through the $p$-states only.
The remaining local problem is solved with the resolvent method [@Kei; @Bickers] and an extended version of the so called Non Crossing Approximation (NCA) [@Bickers], where the 16 local eigenstates of the impurity are coupled through the hybridization-function $\Delta^\sigma(z)$ [@Schmalian].
On the calculation of the magnetic susceptibility\[sec:MagSusz \]
-----------------------------------------------------------------
On the one-particle level one can obtain magnetic properties by applying a staggered magnetic field and calculating the sublattice magnetization. This technique is very tedious so that we used another method for calculating the magnetic phase diagram.
In addition to the one-particle properties the DMFT also allows to calculate two-particle correlation functions, e.g. the magnetic susceptibility consistently. In analogy to the one-particle case the two-particle self energy becomes purely local in the limit [$d\rightarrow\infty$]{} [@Brand; @Jarr]. This enables us to extract the two-particle self-energy from the effective local problem [@Jarr; @MJ; @ThP] and use it to determine the two-particle correlation function for the lattice. Since the local two-particle propagator is a function of three frequencies in the most general case the algorithm works best for Matsubara frequencies, because all quantities can be represented as matrices in this case. For details of the method see e.g. ref. [@ThP].
The choice of the cluster as effective impurity and the finite value of $U_d$ results in 16 local eigenstates. This leads to a huge number of diagrams for the local two-particle propagator, which have to be calculated as functions of three frequencies and summed up numerically. Although the problem of generating the correct diagrams for the local two-particle propagator can be automated and handled by the computer, the remaining numerical task is still formidable and restrict our calculations to the evaluation of the static susceptibility for the time being. Nevertheless, the study of the dynamical susceptibility is in principle also possible [@ThPII] and will be the subject of a forthcoming publication.
Results\[sec:RESULTS\]
======================
Susceptibility and phase diagram\[sec:SUS\]
-------------------------------------------
Let us start the discussion of our results with the magnetic susceptibility. We calculated the static magnetic susceptibility of the $\rm Cu$-spins in the paramagnetic phase at the points ${\bf q}=0$ and ${\bf q}={\bf
Q}$, which give the homogeneous and staggered susceptibilities, respectively. For the parameters of the three-band Hubbard model we have chosen $U_d=2\Delta=7t^*$, where the charge transfer gap $\Delta$ is defined by $\Delta=\varepsilon_p-\varepsilon_d$. [Fig. \[sus\]a]{} shows typical results for these two susceptibilities as a function of temperature T for a hole-doping $\delta=n_d+n_p-1=0.1$.
For the above choice of $U_d$ and $\Delta$ we observe a finite and slowly varying ferromagnetic susceptibility $\chi_F^d(T,\delta)$, which does not show any tendency towards an instability in the calculated region of temperatures and dopings. The antiferromagnetic susceptibility $\chi_{AF}^d(T,\delta)$, on the other hand, varies strongly as a function of temperature and diverges at a finite temperature $T=T_N$. Fig. \[sus\]b shows the inverse staggered susceptibility for the same parameters. As expected we find the linear variation of $\chi_{AF}^{-1}$, which is typical for a mean-field theory. By calculating the inverse susceptibility for different dopings $\delta$ we obtain the $\delta$-$T$-phase-diagram, shown in [Fig. \[phase\]]{}.
Note, that half-filling ($\delta=0$) does not coincide with the MI-transition. This is so because for the used parameter values of $U_d$ and $\Delta$ the metal-insulator-transition is shifted towards larger hole-filling values $n>1$. Obviously the highest value for the Neél temperature is achieved at the metal-insulator-transition and not at half-filling. As already observed for the one band model [@MJ; @ThP] the antiferromagnetic phase is strongly suppressed upon doping. However, in contrast to the former case one recognizes a pronounced asymmetry in the Neél temperature with respect to hole- and electron-doping. This stronger sensistivity of the antiferromagnetic ordered state in the case of hole doping compared to electron doping is qualitatively in good agreement with experiments (see Fig. \[expphase\]) and is a direct consequence of the oxygen degrees of freedom. The present large $D$ treatment however neglects short range order phenomena so that the argumentation of ref. [@Aha] concerning the frustration of the AF-order by hole doping does not hold on this level. But nevertheless the oxygen degrees of freedom cause a particle hole asymmetry in the half filled three band model which yields the calculated asymmetry in the phase diagram in this large $D$ approach.
In our calculations the ordered phase is stable up to $\delta\approx
0.18$ for low enough temperatures (cf. Fig. \[phase\]). Experiments and other theoretical calculations show much stronger suppression of the antiferromagnetic ordering with doping [@Alm]. The tendency to overestimate the magnetic phase boundary is typical for mean field theories, since they completely neglect fluctuations, which strongly renormalize transition temperatures.
Spectral functions in the ordered phase\[sec:SPEC\]
---------------------------------------------------
With the generalized equation (\[G\_d\_local\]) it is also possible to perform calculations in the antiferromagnetic phase. To allow for solutions with finite sublattice magnetization we apply a small symmetry-breaking staggered magnetic field $h({\bf r}_i)=h
e^{i{\bf Q}\cdot{\bf r}_i}$ in $z$-direction at the beginning of our iteration procedure, which is turned off after a few iterations.
In the following we concentrate on the $d$-part of the spectrum $A_d^\sigma
(\omega)=-\frac{1}{\pi}{\rm Im}G_{d,A}^\sigma
(\omega+i\delta)$ on the A-sublattice, since the $p$-part shows exactly the same features as the $d$-part, only with different spectral weights for the various bands. Fig. \[af\_T\]a,b,c shows the typical behaviour as the temperature $T$ is lowered for fixed values of the parameters [$U_d=2\Delta=7t^*$]{} at finite doping $\delta=0.015$.
For $\beta=18/t^*$ the system is in the paramagnetic state (cf. Fig. \[af\_T\]a), where the general equation (\[G\_d\_local\]) reduces to the form (\[Gdloc\]). We find the same result for the $d$-part of the spectrum as in ref. [@Schmalian], where a detailed discussion of the various bands concerning their doping dependence, transfer of spectral weight and the evolution of coherent quasiparticles near the Fermi-energy can be found. Let us just briefly mention the important low energy parts, namely the so-called lower Hubbard band at $\omega<0$ in the spectrum of Fig. \[af\_T\], which has mainly $d$-character, and the Zhang-Rice band right above the gap, which is generated by the singlet combination of the $p$-and $d$-states on one plaquette [@Schmalian]. Note that $\delta >0$ although the chemical potential is located in the lower Hubbard band. Therefore the MI-transition occurs at larger filling values as already mentioned in section \[sec:SUS\]. With decreasing temperature the system enters the antiferromagnetic phase and the spectral functions of up and down spin become inequivalent (cf. Fig. \[af\_T\]b), yielding a finite sublattice magnetization $m_d$. Note that the major effect is a transfer of spectral weight from the minority spin to the majority spin. In addition the peaks in the spectra are slightly shifted in energy with respect to each other. This effect can be ascribed to an internal molecular field, generated by the finite sublattice magnetization. Therefore, measuring the energy shift one can calculate the internal molecular field and from this the exchange parameter $J$ of a corresponding tJ-model. For still lower temperatures the sublattice magnetization increases and above a value of $m_d\approx 0.5$ a pronounced multipeak structure is evolving (see Fig. \[af\_T\]c). Fig. \[af\_T\]d shows the spectral function for the same system parameters and sublattice magnetization as in Fig. \[af\_T\]c but with the chemical potential right above the gap. Also in this regime the same multipeak structure occurs and the spectral function shows little difference to Fig. \[af\_T\]c. Only the peaks next to the chemical potential have more spectral weight compared with Fig. \[af\_T\]c.
These regular resonances were previously found in DMFT calculations of the tJ-model [@Obermeier]. There the multiple peaks could be related to bound states of one single hole in the Neél background. In ref. [@Strack] it was shown, that this special problem can be solved exactly for $d=\infty$ and $T=0$ within the tJ-model. The most important physical aspect is, that the moving hole feels a binding potential proportional to $J$, growing linearly with the distance from its starting point due to the breaking of antiferromagnetic bonds during its motion [@Strack; @Obermeier]. This linear potential leads to a sequence of discrete poles at frequencies [@Strack] $$\omega_n=-2\hat{t}-\frac{J^*}{2}-a_n\hat{t}\left(\frac{J^*}{2\hat{t}}\right)^\frac{2}{3}
\label{Ai}$$ as spectrum for the one particle excitations. Here, the $a_n$ denote the zeros of the Airy function $Ai\left(4\hat{t}/J^*\right)$, and the renormalized parameters $\hat{t}$ and $J^*$ are given by $\hat{t}=t\sqrt{2d}$, $J^*=J2d$. These exact results can be compared directly to the resonances, found in the DMFT calculations for the tJ-model [@Obermeier]. Since the model used in our calculations is fundamentally different from the tJ-model and also from the one-band Hubbard model, the relevance of this physically intuitive picture to it and especially the proper choice of the parameters for an effective tJ-model to describe the low energy properties is not clear a priori. The approach chosen here is to fix the hopping to $t^2/\Delta$, which reproduces the free bandwidth. In our case we determine an effective exchange interaction $J^*$ from the energy shift $\Delta
E=J^* m_d$ of the bands. Note that this is just the energy shift of the spin-up and spin-down bands of a corresponding tJ-model, treated on a mean-field level [@Mueller; @Obermeier; @Itz]. Another possibilty to obtain the exchange integral $J$ is to use the result of a Schrieffer-Wolf transformation of the 3-band Hubbard model, see e.g. [@Zaanen]. However this transformation holds only for large values of $U_d$ and $
\Delta$, so that we do not expect this procedure to give a meaningful result for our parameter values.
Fixing the paramters in eq. (\[Ai\]) as discussed above, we can indeed directly compare our results with the discrete spectrum (\[Ai\]). Fig. \[comp\] shows some examples for the fit of the low energy part of the $d$-spectrum $A_d^\sigma(\omega)$ by the discrete spectrum (\[Ai\]) at fixed doping $\delta=0.015$ and sublattice magnetization $m_d=0.60$ for various parameters $U_d=2\Delta$.
Note, that the energy scales of up- and down-spin in Fig. \[comp\] are already shifted by $\pm \Delta E/2$ respectively, so that the resonances of majority- and minority-spin bands coincide. We find quite good agreement with the distance of the peak positions. The broadening is expected to result from finite temperature, sublattice magnetization and doping effects [@Obermeier]. This means that the tJ-model with proper choice of the parameters $t$ and $J$ seems to reproduce the low energy one-particle dynamics of the three-band Hubbard model in $d=\infty$ correctly, even in the antiferromagnetic state. In addition, the basic physical picture for the multipeak structures observed for low temperatures appears to be the same as in the simple one-band models.
In order to gain more insight in the effect of doping on these multipeak structure we investigated the spectrum at larger doping far away from the MI-transition. Fig. \[af\_doped\] shows the results for the $d$-part of the spectrum for the same system parameters and sublattice magnetization as in Fig. \[af\_T\]c,d at $\beta=50/t^*$ but at larger doping $\delta=-0.08$ (a) and $\delta=0.13$ (b).
In the electron (Fig. \[af\_doped\]a) as well as in the hole doped regime (Fig \[af\_doped\]b) only the resonances next to the chemical potential survive. Due to the larger doping there are more electrons/holes in the system whose paths can intersect and restore the antiferromagnetic background. Therefore the electrons/holes become more mobile and the resonances at higher energies are washed out.
In finite dimensions the string picture for one hole in the antiferromagnetic background no longer holds and is correct only up to order $\frac{1}{d^2}$ [@Strack] due to the possibility of paths which intersect and touch themselves [@Trug]. Second, fluctuations become more important which can restore the antiferromagnetic background. Thus in low dimensions we expect that the multipeak structure at finite doping will dissappear.
Summary\[SEC:SUM\]
==================
In this paper we presented results for the magnetic properties of the three-band Hubbard model in the limit of high spacial dimensions. These were obtained in the framework of the Dynamical Mean Field Theory, which enabled us to calculate the one particle spectrum as well as two particle correlation functions, namely the magnetic susceptibility. From this we evaluated the $\delta$-$T$-phase diagram, which shows strong suppression of the antiferromagnetic state upon doping. In contrast to one-band models the ordered state is found to be more sensitive upon doping in the case of hole doping in comparison to electron doping. This asymmetric behaviour is qualitatively in good agreement with experiments. The spectral function for single particle excitations in the antiferromagnetic phase shows pronounced features above a sublattice magnetization $m_d\approx 0.5$. These structures are similar to those found in the tJ-model for the special case of one single hole, moving in the Neél background and can be understood by the binding of one hole in a string potential. A quantitive fit of the spectral functions by the exact results for the special case for the tJ-model shows quite good agreement, so that the tJ-model seems to reproduce the correct physics of the three-band Hubbard model as long as one is only interrested in the low energy one-particle physics.
In low dimensions fluctuations become more important which will destroy the multiple peaks found in the spectral function for $d=\infty$. Thus these peaks have not yet been observed in experiments.
C. Almasan, M. B. Maple, Chemistry of High-Temperature Superconductors, edited by C. M. Rao, World Scientific, Singapore (1991) P. W. Anderson, Phys. Rev. [**124**]{}, 41 (1961) F. C. Zhang, T.M. Rice, Phys. Rev. B [**37**]{}, 3759 (1988) A. Aharony et al., Phys. Rev. Lett. [**60**]{}, 1330 (1988); R. J. Birgeneau, M. A. Kastner, A. Aharony, Z. Phys. B [**71**]{}, 57 (1988) V. J. Emery, Phys. Rev. Lett. [**58**]{}, 2794 (1987) W. Metzner, D. Vollhardt, Phys. Rev. Lett. [**62**]{}, 324 ( 1989) E. Müller-Hartmann, Z. Phys., B [**74**]{}, 507 (1989) J. Schmalian, P. Lombardo, M.Avignon, K. H. Bennemann, Physica B [**222-224**]{}, 602 (1996); P.Lombardo, M. Avignon, J. Schmalian, K. H. Bennemann, Phys. Rev. B [**54**]{}, 5317 (1996) J. Zaánen, A. M. Oleś, Phys. Rev. B [**37**]{}, 9423 (1988) M. B. Zoelfl, Th. Maier, Th. Pruschke, J.Keller, to be published H. Watanabe, S. Doniach, Phys. Rev. B [**57**]{}, 3829 (1998) U. Brandt, C. Mielsch, Z. Phys., B [**79**]{}, 295 (1990) H. Keiter, J. C. Kimball, Phys. Rev. Lett. [**25**]{}, 672 (1970) N. E. Bickers, D. L. Cox, J. W. Wilkins, Phys. Rev. B [**36**]{}, 2036 (1987) Th. Pruschke, D. L. Cox, M. Jarrell, Phys. Rev. B [**47**]{}, 3553 (1993) M. Jarrell, Th. Pruschke, Z. Phys. B [**90**]{}, 187 (1993) Th. Pruschke, Q. Qin, Th. Obermeier, J.Keller, J. Phys. Condens. Matter [**8**]{}, 3161, (1996) Th. Pruschke, Th. Obermeier, J. Keller, Physica B [**230-232**]{}, 895(1997). Th. Obermeier, Th. Pruschke, J. Keller, Ann. Physik [**5**]{}, 137 (1996) R. Strack, D. Vollhardt, Phys. Rev. B [**46**]{}, 13852 (1992) C. Itzykson, J.M. Drouffe, Statistical Field Theory Vol. I & II, Cambridge University Press, Cambridge (1989) S. A. Trugman, Phys. Rev. B [**41**]{} 892 (1990)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretic interpretation of cluster mutation in case of acyclic cluster algebras of finite type.'
address:
- |
Institutt for matematiske fag\
Norges teknisk-naturvitenskapelige universitet\
N-7491 Trondheim\
Norway
- |
Department of Mathematics\
University of Leicester\
University Road\
Leicester LE1 7RH\
England
- |
Institutt for matematiske fag\
Norges teknisk-naturvitenskapelige universitet\
N-7491 Trondheim\
Norway
author:
- Aslak Bakke Buan
- 'Robert J. Marsh'
- Idun Reiten
title: Cluster mutation via quiver representations
---
\[section\] \[lem\][Proposition]{} \[lem\][Corollary]{} \[lem\][Theorem]{}
\[lem\][Remark]{} \[lem\][Definition]{}
[^1]
Introduction {#introduction .unnumbered}
============
This paper was motivated by the interplay between the recent development of the theory of cluster algebras defined by Fomin and Zelevinsky in [@fz1] (see [@z] for an introduction) and the subsequent theory of cluster categories and cluster-tilted algebras [@bmrrt; @bmr]. Our main results can be considered to be interpretations within cluster categories of important concepts in the theory of cluster algebras.
Cluster algebras were introduced in order to explain the connection between the canonical basis of a quantized enveloping algebra as defined by Kashiwara and Lusztig and total positivity for algebraic groups. It was also expected that cluster algebras should model the classical and quantized coordinate rings of varieties associated to algebraic groups — see [@bfz] for an example of this phenomenon (double Bruhat cells). Cluster algebras have been used to define generalizations of the Stasheff polytopes (associahedra) to other Dynkin types [@cfz; @fz3]; consequently there are likely to be interesting links with operad theory. They have been used to provide the solution [@fz3] of a conjecture of Zamolodchikov concerning $Y$-systems, a class of functional relations important in the theory of the thermodynamic Bethe Ansatz, as well as solution [@fz4] of various recurrence problems involving Laurent polynomials, including a conjecture of Gale and Robinson on the integrality of generalized Somos sequences. Here the remarkable Laurent properties of the distinguished generators of a cluster algebra play an important role. Cluster algebras have also been related to Poisson geometry [@gsv1], Teichmüller spaces [@gsv2], positive spaces and stacks [@fg1], dual braid monoids [@bes], ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra [@p] as well as representation theory, see amongst others [@bmrrt; @bmr; @cc; @ccs1; @ccs2; @mrz].
A cluster algebra (without coefficients) is defined via a choice of free generating set $\underline{x} = \{ x_1, \dots, x_n \} $ in the field ${\operatorname{\mathcal F}\nolimits}$ of rational polynomials over $\mathbb{Q}$ and a skew-symmetrizable integer matrix $B$ indexed by the elements of $\underline{x}$. The pair $(\underline{x},B)$, called a seed, determines the cluster algebra as a subring of ${\operatorname{\mathcal F}\nolimits}$. More specifically, for each $i= 1, \dots, n$, a new seed $\mu_i(\underline{x},B) = (\underline{x'},B')$ is obtained by replacing $x_i$ in $\underline{x}$ by ${x_i}' \in {\operatorname{\mathcal F}\nolimits}$, where ${x_i}'$ is obtained by a so-called [*exchange multiplication rule*]{} and $B'$ is obtained from $B$ by applying so-called [*matrix mutation*]{} at row/column $i$. Mutation in any direction is also defined for the new seed, and by iterating this process one obtains a countable (sometimes finite) number of seeds. For a seed $(\underline{x},B)$, the set $\underline{x}$ is called a [*cluster*]{}, and the elements in $\underline{x}$ are called [*cluster variables*]{}. The desired subring of ${\operatorname{\mathcal F}\nolimits}$ is by definition generated by the cluster variables.
It is an interesting problem to try to find a categorical/ module theoretical setting with a nice interpretation of the concepts of clusters and cluster variables, and of the matrix mutation and multiplication exchange rule for cluster variables. For the case of acyclic cluster variables so-called [*cluster categories*]{} were introduced as a candidate for such a model [@bmrrt]. Skew-symmetric matrices are in one-one correspondence with finite quivers with no loops or cycles of length two, and the corresponding cluster algebra is called [*acyclic*]{} if there is a seed $(\underline{x},B)$ such that $B$ corresponds to a quiver $Q$ without oriented cycles. There is then, for a field $k$, an associated finite dimensional path algebra $kQ$. The corresponding cluster category ${\operatorname{\mathcal C}\nolimits}$ is defined in [@bmrrt] as a certain quotient of the bounded derived category of $kQ$, which is shown to be canonically triangulated by [@k]. In [@bmrrt] (cluster-)tilting theory is developed in ${\operatorname{\mathcal C}\nolimits}$, with emphasis on connections to cluster algebras. The analogs of clusters are (cluster-)tilting objects, and the analogs of cluster variables are exceptional objects. In case $Q$ is a Dynkin quiver, it was shown in [@bmrrt] that there is a one-one correspondence between cluster variables and exceptional objects in ${\operatorname{\mathcal C}\nolimits}$ (in this case all indecomposables are exceptional) which takes clusters to tilting objects. This was conjectured to hold more generally.
In this paper we show that also the matrix mutation for cluster algebras has a nice interpretation within cluster categories, in terms of the associated cluster-tilted algebras, investigated in [@bmr]. Cluster-tilted algebras are endomorphism algebras of tilting objects in cluster categories. It follows from our results that the quivers of the cluster-tilted algebras arising from a given cluster category are exactly the quivers corresponding to the exchange matrices of the associated cluster algebra. This has further applications to cluster algebras (see [@br]). Another main result of this paper is an interpretation within cluster categories of the exchange multiplication rule of an (acyclic) cluster algebra. So, together with the results from [@bmrrt], all the major ingredients involved in the construction of acyclic cluster algebras have now been interpreted in the cluster category.
Tilting theory for hereditary algebras have been a central topic within representation theory since the early eighties. This involves the study of tilted algebras, and various generalizations. An important motivation for this theory was to compare the representation theory of hereditary algebras with the representation theory of other homologically more complex algebras. The main result of [@bmr] is also in this spirit, showing a close connection between the representation theory of cluster-tilted algebras and hereditary algebras. It is the hope of the authors that our “dictionary” also can be used to obtain further developments in representation theory of finite dimensional algebras. Also new links between this field and other fields of mathematics can be expected, having in mind the influence of cluster algebras on other areas.
In [@ccs1] an alternative description of the cluster category is given for type $A$. The cluster category was also the motivation for a Hall-algebra type definition of a cluster algebra of finite type [@cc; @ck].
The paper is organized as follows. In section 1 we give some preliminaries, allowing us to state the main result more precisely. Most of the necessary background on cluster algebras is however postponed until later (section 6), since most of the paper does not involve cluster algebras. In section 2 we prove the following: If ${\Gamma}$ is cluster-tilted, then so is ${\Gamma}/ {\Gamma}e {\Gamma}$ for an idempotent $e$ in ${\Gamma}$. This is an essential ingredient in the proof of the main result, and also an interesting fact in itself. In section 3 some consequences of this are given. In section 4 we prepare for the proof of our main result. This involves studying cluster-tilted algebras of rank 3, and a crucial result of Kerner [@ker] on hereditary algebras. The main result is proved in section 5, while section 6 deals with the connection to cluster algebras, including necessary background.
The results of this paper have been presented at conferences in Uppsala (June 2004), Mexico (August 2004) and Northeastern University (October 2004).
The first named author spent most of 2004 at the University of Leicester, and would like to thank the Department of Mathematics, and especially Robert J. Marsh, for their kind hospitality. We would like to thank the referee for pointing out an error in an earlier version of this paper and Bernhard Keller and Otto Kerner for helpful comments and conversations.
Preliminaries {#prelim}
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Finite-dimensional algebras
---------------------------
In this section let ${\Lambda}$ be a finite dimensional $K$-algebra, where $K$ is a field. Then $1_{{\Lambda}} = e_1 + e_2 + \cdots + e_n$, where all $e_i$ are primitive idempotents. We always assume that ${\Lambda}$ is basic, that is, ${\Lambda}e_i \not \simeq {\Lambda}e_j$ when $i \neq j$. There are then (up to isomorphism) $n$ indecomposable projective ${\Lambda}$-modules, given by ${\Lambda}e_i$, and $n$ simple modules, given by ${\Lambda}e_i / {\operatorname{\underline{r}}\nolimits}e_i$, where ${\operatorname{\underline{r}}\nolimits}$ is the Jacobson radical of ${\Lambda}$.
If $K$ is an algebraically closed field, then there is a finite quiver $Q$, such that ${\Lambda}$ is isomorphic to $KQ/I$, where $KQ$ is the path-algebra, and $I$ is an admissible ideal, that is there is some $m$, such that ${\operatorname{\underline{r}}\nolimits}^m \subseteq I \subseteq {\operatorname{\underline{r}}\nolimits}^2$. We call $Q$ the quiver of ${\Lambda}$. In case ${\Lambda}$ is hereditary, the ideal $I$ is $0$.
The category ${\operatorname{mod}\nolimits}{\Lambda}$ of finite dimensional left ${\Lambda}$-modules is an abelian category having almost split sequences. In case ${\Lambda}$ is hereditary there is a translation functor $\tau$, which is defined on all modules with no projective (non-zero) direct summands. Here $D$ denotes the ordinary duality for finite-dimensional algebras.
The bounded derived category of ${\Lambda}$, denoted $D^b({\operatorname{mod}\nolimits}{\Lambda})$, is a triangulated category, with suspension given by the shift-functor $[1]$, which is an autoequivalence. We denote its inverse by $[-1]$. In this paper, we only consider derived categories of hereditary algebras $H$. They have an especially nice structure, since the indecomposable objects are given by shifts of indecomposable modules. In this case we also have a translation functor $\tau \colon D^b({\operatorname{mod}\nolimits}H) \to D^b({\operatorname{mod}\nolimits}H)$, extending the functor mentioned above. We have almost split triangles $A \to B \to C \to $ in $D^b({\operatorname{mod}\nolimits}H)$, where $\tau C =A$, for each indecomposable $C$ in $D^b({\operatorname{mod}\nolimits}H)$. We also have the formula ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X, \tau Y) \simeq D{\operatorname{Ext}\nolimits}^1_{{\operatorname{\mathcal D}\nolimits}}(Y,X)$, see [@hap]. Let $H$ be a hereditary finite-dimensional algebra. Then a module $T$ in ${\operatorname{mod}\nolimits}H$ is called a [*tilting module*]{} if ${\operatorname{Ext}\nolimits}^1_H(T,T) = 0$ and $T$ has, up to isomorphism, $n$ indecomposable direct summands. The endomorphism ring ${\operatorname{End}\nolimits}_H(T)^{{{\operatorname{op}}}}$ is called a [*tilted algebra*]{}.
See [@ars] and [@r] for further information on the representation theory of finite dimensional algebras and almost split sequences.
Approximations
--------------
Let ${\operatorname{\mathcal E}\nolimits}$ be an additive category, and ${\operatorname{\mathcal X}\nolimits}$ a full subcategory. Let $E$ be an object in ${\operatorname{\mathcal E}\nolimits}$. If there is an object $X$ in ${\operatorname{\mathcal X}\nolimits}$, and a map $f \colon X \to E$, such that for every object $X'$ in ${\operatorname{\mathcal X}\nolimits}$ and every map $g \colon X' \to E$, there is a map $h \colon X' \to X$, such that $g = fh$, then $f$ is called a right ${\operatorname{\mathcal X}\nolimits}$-approximation [@as]. The approximation map $f \colon X \to E$ is called [*minimal*]{} if no non-zero direct summand of $X$ is mapped to $0$. The concept of (minimal) left ${\operatorname{\mathcal X}\nolimits}$-approximations is defined dually. If there is a field $K$, such that ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal E}\nolimits}}(X,Y)$ is finite dimensional over $K$, for all $X,Y \in {\operatorname{\mathcal E}\nolimits}$, and if ${\operatorname{\mathcal X}\nolimits}= {\operatorname{add}\nolimits}M$ for an object $M$ in ${\operatorname{\mathcal E}\nolimits}$, then (minimal) left and right ${\operatorname{\mathcal X}\nolimits}$-approximations always exist.
Cluster categories and cluster-tilted algebras
----------------------------------------------
We remind the reader of the basic definitions and results from [@bmrrt]. Let $H$ be a hereditary algebra, and let ${\operatorname{\mathcal D}\nolimits}= D^b({\operatorname{mod}\nolimits}H)$ be the bounded derived category.
The *cluster category* is defined as the orbit category ${\operatorname{\mathcal C}\nolimits}_H = {\operatorname{\mathcal D}\nolimits}/ F$, where $F = \tau^{-1}[1]$. The objects of $C_H$ are the same as the objects of ${\operatorname{\mathcal D}\nolimits}$, but maps are given by $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(X,Y) = \amalg_i {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X,F^i Y).$$ Let $Q \colon {\operatorname{\mathcal D}\nolimits}\to {\operatorname{\mathcal C}\nolimits}$ be the canonical functor. We often denote $Q(X)$ by $\widehat{X}$, and use the same notation for maps. Let $\widehat{X}$ be an indecomposable object in the cluster-category. We call ${\operatorname{mod}\nolimits}H \vee {\operatorname{add}\nolimits}H[1] = {\operatorname{mod}\nolimits}H \vee H[1]$ the *standard domain*. There is (up to isomorphism) a unique object $X$ in ${\operatorname{mod}\nolimits}H \vee H[1] \subseteq {\operatorname{\mathcal D}\nolimits}$ such that $Q(X) = \widehat{X}$.
Assume $X_1, X_2$ are indecomposable in the standard domain, then a map $\widehat{f}\colon \widehat{X_1} \to \widehat{X_2}$, can uniquely be written as a sum of maps $\widehat{f_1}+ \widehat{f_2} + \cdots + \widehat{f_r}$, such that $f_i$ is in ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X_1, F^{d_i}X_2)$, for integers $d_i$. In this case $d_i$ is called the [*degree*]{} of $\widehat{f_i}$.
The following summarizes properties about cluster categories that will be freely used later.
Let $H$ be a hereditary algebra, and ${\operatorname{\mathcal C}\nolimits}_H$ the cluster category of $H$. Then
- [${\operatorname{\mathcal C}\nolimits}_H$ is a Krull-Schmidt category and $Q$ preserves indecomposable objects;]{}
- [${\operatorname{\mathcal C}\nolimits}_H$ is triangulated and $Q$ is exact;]{}
- [${\operatorname{\mathcal C}\nolimits}_H$ has AR-triangles and $Q$ preserves AR-triangles.]{}
\(b) is due to Keller [@k], while (a) and (c) are proved in [@bmrrt].
Let us now fix a hereditary algebra $H$, and assume it has, up to isomorphism, $n$ simple modules. A *cluster tilting object* (or for short; tilting object) in the cluster category is an object $T$ with ${\operatorname{Ext}\nolimits}^1_{{\operatorname{\mathcal C}\nolimits}}(T,T) = 0$, and with $n$ non-isomorphic indecomposable direct summands. For an object $X$ in any additive category, we let ${\operatorname{add}\nolimits}X$ denote the smallest full additive subcategory closed under direct sums and containing $X$. Then two tilting objects $T$ and $T'$ are said to be equivalent if and only if ${\operatorname{add}\nolimits}T = {\operatorname{add}\nolimits}T'$. We only consider tilting objects up to equivalence, and therefore we always assume that if $T = \amalg_i T_i$ is a tilting object, with each $T_i$ indecomposable, then $T_i \not \simeq T_j$ for $i \neq j$.
There is a natural embedding of the module category into the bounded derived category, which extends to an embedding of the module category into ${\operatorname{\mathcal C}\nolimits}$. It was shown in [@bmrrt] that the image of a tilting module in ${\operatorname{mod}\nolimits}H$ is a tilting object in ${\operatorname{\mathcal C}\nolimits}_H$. It was also shown that if we choose a tilting object $T$ in ${\operatorname{\mathcal C}\nolimits}_H$, then there is a hereditary algebra $H'$ and an equivalence $D^b(H') \to D^b(H)$, such that $T$ is the image of a tilting module, under the embedding of ${\operatorname{mod}\nolimits}H'$ into ${\operatorname{\mathcal C}\nolimits}_{H'} \simeq C_H$.
If $\bar{T} \amalg X$ is a tilting object, and $X$ is indecomposable, then $\bar{T}$ is called an almost complete tilting object.
The following was shown in [@bmrrt].
\[triangles\] Let $\bar{T}$ be an almost complete tilting object in ${\operatorname{\mathcal C}\nolimits}_H$. Then there are exactly two complements $M$ and $M^{\ast}$. There are uniquely defined non-split triangles $$M^{\ast} \to B \to M \to ,$$ and $$M \to B' \to M^{\ast} \to .$$ The maps $ B \to M$ and $B' \to M^{\ast}$ are minimal right ${\operatorname{add}\nolimits}\bar{T}$-approximations, and the maps $M^{\ast} \to B$ and $M \to B'$ are minimal left ${\operatorname{add}\nolimits}\bar{T}$-approximations.
The endomorphism ring ${\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T)^{{{\operatorname{op}}}}$ of a tilting object in ${\operatorname{\mathcal C}\nolimits}$ is called a *cluster-tilted algebra*. Using the notation of Theorem \[triangles\], we want to compare the quivers of the endomorphism rings ${\Gamma}= {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(\bar{T} \amalg M)^{{{\operatorname{op}}}}$ and ${\Gamma}' = {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(\bar{T} \amalg M^{\ast})^{{{\operatorname{op}}}}$.
Matrix mutation
---------------
Let $X =(x_{ij})$ be an $n \times n$-skew-symmetric matrix with integer entries. Choose $k \in \{1,2,\dots,n\}$ and define a new matrix $\mu_k(X) = X' = (x'_{ij})$ by $$x'_{ij} = \begin{cases}
-x_{ij} & \text{if $k=i$ or $k=j$,} \\
x_{ij} + \frac{{\lvertx_{ik}\rvert}x_{kj} + x_{ik} {\lvertx_{kj}\rvert}}{2} & \text{otherwise.}
\end{cases}$$ The matrix $\mu_k(X) = X'$ is called the mutation of $X$ in direction $k$, and one can show that
- [$\mu_k(X)$ is skew-symmetric, and]{}
- [$\mu_k(\mu_k(X))= X$.]{}
Matrix mutation appears in the definition of cluster algebras by Fomin and Zelevinsky [@fz1].
Main result
-----------
At this point, we have the necessary notation to state the main result of this paper. There are no loops in the quiver of a cluster-tilted algebra [@bmrrt], and we also later show that there are no (oriented) cycles of length two. It follows that one can assign to ${\Gamma}$ a skew-symmetric integer matrix $X_{{\Gamma}}$. Actually, there is a 1–1 correspondence between the skew-symmetric integer matrices and quivers with no loops and no cycles of length two. Fixing an ordering of the vertices of the quiver, this 1–1 correspondence determines mutations $\mu_k$ also on finite quivers (with no loops and no cycles of length 2). The following will be proved in Section 5. The notation is as earlier in this section, especially $K$ is algebraically closed.
Let $\bar{T}$ be an almost complete tilting object with complements $M$ and $M^{\ast}$ and let ${\Gamma}= {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(\bar{T} \amalg M)^{{{\operatorname{op}}}}$ and ${\Gamma}' = {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(\bar{T} \amalg M^{\ast})^{{{\operatorname{op}}}}$. Let $k$ be the vertex of ${\Gamma}$ corresponding to $M$. Then the quivers $Q_{{\Gamma}}$ and $Q_{{\Gamma}'}$, or equivalently the matrices $X_{{\Gamma}} = (x_{ij})$ and $X_{{\Gamma}'} = (x'_{ij})$, are related by the formulas $$x'_{ij} = \begin{cases}
-x_{ij} & \text{if $k=i$ or $k=j$,} \\
x_{ij} + \frac{{\lvertx_{ik}\rvert}x_{kj} + x_{ik} {\lvertx_{kj}\rvert}}{2} & \text{otherwise.}
\end{cases}$$
This is the central result from which the connections with cluster algebras mentioned in the introduction follow.
Factors of cluster-tilted algebras
==================================
In this section, our main result is that for any cluster-tilted algebra ${\Gamma}$, and any primitive idempotent $e$, the factor-algebra ${\Gamma}/ {\Gamma}e {\Gamma}$ is in a natural way also a cluster-tilted algebra. This will give us a powerful reduction-technique, which is of independent interest, and which we use in the proof of our main result in this paper.
Suppose that ${\Gamma}$ is the endomorphism algebra of a tilting object $T$ in the cluster category corresponding to a hereditary algebra $H$. The main idea of the proof is to show that if we localise ${\operatorname{\mathcal D}\nolimits}^b({\operatorname{mod}\nolimits}H)$ at the smallest thick subcategory containing a fixed indecomposable summand $M$ of $T$, then we obtain a category triangle-equivalent to the derived category of a hereditary algebra $H'$. The factor-algebra ${\Gamma}/ {\Gamma}e {\Gamma}$ (where $e$ is the primitive idempotent of ${\Gamma}$ corresponding to $M$) is then shown to be isomorphic to the endomorphism algebra of a tilting object in the cluster category corresponding to $H'$.
Localisation of triangulated categories {#localisationtri}
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We review the basics of localisation in triangulated categories. Let ${\mathcal T}$ be a triangulated category. A subcategory ${\operatorname{\mathcal M}\nolimits}$ of ${\mathcal T}$ is called a *thick subcategory* of ${\mathcal T}$ if it is a full triangulated subcategory of ${\mathcal T}$ closed under taking direct summands.
When ${\operatorname{\mathcal M}\nolimits}$ is a thick subcategory of ${\mathcal T}$, one can form a new triangulated category ${\mathcal T}_{{\operatorname{\mathcal M}\nolimits}}={\mathcal T}/{\operatorname{\mathcal M}\nolimits}$, and there is a canonical exact functor $L_{{\operatorname{\mathcal M}\nolimits}} \colon {\mathcal T}\to {\mathcal T}_{{\operatorname{\mathcal M}\nolimits}}$. See [@rickard] and [@verdier] for details.
For every $M'$ in ${\operatorname{\mathcal M}\nolimits}$, we have $L_{{\operatorname{\mathcal M}\nolimits}}(M') = 0$, and $L_{{\operatorname{\mathcal M}\nolimits}}$ is universal with respect to this property. We also have the following.
\[maps\] Assume ${\mathcal T}$ is a triangulated category, and ${\operatorname{\mathcal M}\nolimits}$ is a thick subcategory of ${\mathcal T}$. Then, for any map $f$ in ${\mathcal T}$ we have $L_{{\operatorname{\mathcal M}\nolimits}}(f) = 0$ if and only if $f$ factors through an object in ${\operatorname{\mathcal M}\nolimits}$.
We will need the following result of Verdier [@verdier Ch. 2, 5-3]:
\[verdieriso\] Let ${\mathcal T}$ be a triangulated category with thick subcategory ${\operatorname{\mathcal M}\nolimits}$, and let ${\mathcal T}_{{\operatorname{\mathcal M}\nolimits}}$ be the quotient category with quotient functor $L_{{\operatorname{\mathcal M}\nolimits}}:{\mathcal T}\to {\mathcal T}_{{\operatorname{\mathcal M}\nolimits}}$. Fix an object $Y$ of ${\mathcal T}$. Then every morphism from an object of ${\operatorname{\mathcal M}\nolimits}$ to $Y$ is zero if and only if for every object $X$ of ${\mathcal T}$ the canonical map $${\operatorname{Hom}\nolimits}_{{\mathcal T}}(X,Y)\to {\operatorname{Hom}\nolimits}_{{\mathcal T}_{{\operatorname{\mathcal M}\nolimits}}}(L_{{\operatorname{\mathcal M}\nolimits}}(X),L_{{\operatorname{\mathcal M}\nolimits}}(Y))$$ is an isomorphism.
In particular, we note that this implies that $L_{{\operatorname{\mathcal M}\nolimits}}$ is fully faithful on the full subcategory of ${\mathcal T}$ with objects given by those objects of ${\mathcal T}$ which have only zero morphisms from objects of ${\operatorname{\mathcal M}\nolimits}$.
Equivalences of module categories {#BongartzHappel}
---------------------------------
Let $H$ be a hereditary algebra and $M$ an indecomposable $H$-module with ${\operatorname{Ext}\nolimits}^1_H(M,M)= 0$. Then there is (up to isomorphism) a unique module $E$ with the following properties:
- [$E$ is a complement of $M$ (that is, $E \amalg M$ is a tilting module)]{}.
- [For any module $X$ in ${\operatorname{mod}\nolimits}H$, we have that ${\operatorname{Ext}\nolimits}^1_H(M,X) = 0$ implies also ${\operatorname{Ext}\nolimits}^1_H(E,X) = 0$]{}.
This is due to Bongartz [@bon], and the module $E$ is sometimes known as the *Bongartz-complement* of $M$. For a module $X$ in ${\operatorname{mod}\nolimits}H$, we denote by $X^{\perp}$ the full subcategory of ${\operatorname{mod}\nolimits}H$ with objects $Y$ satisfying ${\operatorname{Ext}\nolimits}^1_H(X,Y)=0$. If $T$ is a tilting module, then it is well-known that $T^{\perp} = {\operatorname{Fac}\nolimits}T$, where ${\operatorname{Fac}\nolimits}T$ is the full subcategory of all modules that are factors of objects in ${\operatorname{add}\nolimits}T$. Note that B2) can be reformulated as $M^{\perp} = {(M \amalg E)^{\perp}}$.
The following result can be found in [@hap] and [@hris].
\[happelequivalence\]
- [Assume $M$ is an indecomposable non-projective $H$-module with ${\operatorname{Ext}\nolimits}^1_H(M,M)= 0$, and let $E$ be the complement as above. Then the endomorphism ring $H' = {\operatorname{End}\nolimits}_H(E)^{{{\operatorname{op}}}}$ is hereditary, and ${\operatorname{Hom}\nolimits}_H(M,E) = 0$.]{}
- [ Let ${\operatorname{\mathcal U}\nolimits}$ denote the full subcategory of ${\operatorname{mod}\nolimits}H$ with objects $X$ satisfying $${\operatorname{Hom}\nolimits}_H(M,X)=0= {\operatorname{Ext}\nolimits}^1_H(M,X).$$ Then ${\operatorname{\mathcal U}\nolimits}$ is an exact subcategory of ${\operatorname{mod}\nolimits}H$ and the functor ${\operatorname{Hom}\nolimits}_H(E,-)$ from ${\operatorname{mod}\nolimits}H$ to ${\operatorname{mod}\nolimits}H'$ restricts to an exact equivalence between ${\operatorname{\mathcal U}\nolimits}$ and ${\operatorname{mod}\nolimits}H'$.]{}
We note that the above result does not hold in general in the case when $M$ is projective. For example, consider the quiver of type $A_3$ with vertices $1,2$ and $3$ and arrows from $1$ to $2$ and $2$ to $3$. Let $M=P_2$. Then $E=P_1\oplus P_3$ and ${\operatorname{End}\nolimits}_H(E)^{op}$ has three indecomposable objects while ${\operatorname{\mathcal U}\nolimits}$ has only two. The only other complement of $M$ is $E'=P_1\oplus (P_2/P_3)$. Then ${\operatorname{Hom}\nolimits}_H(E',P_3)=0$ although $P_3$ lies in ${\operatorname{\mathcal U}\nolimits}$ and is non-zero. So also in this case the functor ${\operatorname{Hom}\nolimits}_H(E',\ )$ from ${\operatorname{\mathcal U}\nolimits}$ to ${\operatorname{mod}\nolimits}{\operatorname{End}\nolimits}_H(E')^{{{\operatorname{op}}}}$ is not an equivalence. However, we will need the following result which is along similar lines for the case when $M$ is projective.
\[projectiveequivalence\] Let $M$ be an indecomposable projective $H$-module with corresponding idempotent $e_M \in H$. Let $H'=H/{He_M H}$.
- [We have ${\operatorname{Tor}\nolimits}_1^H(H',U)=0$, for any object $U$ in ${\operatorname{\mathcal U}\nolimits}$, where ${\operatorname{\mathcal U}\nolimits}$ is as defined above.]{}
- [We have that ${\operatorname{\mathcal U}\nolimits}$ is an exact subcategory of ${\operatorname{mod}\nolimits}H$ and the functor $H'\otimes_H -$ from ${\operatorname{mod}\nolimits}H$ to ${\operatorname{mod}\nolimits}H'$ restricts to an exact equivalence between ${\operatorname{\mathcal U}\nolimits}$ and ${\operatorname{mod}\nolimits}H'$.]{}
We have that ${\operatorname{\mathcal U}\nolimits}$ is an exact subcategory of ${\operatorname{mod}\nolimits}H$ as in Proposition \[happelequivalence\]. It is easy to see that the functor $H'\otimes_H -$ is an equivalence between ${\operatorname{\mathcal U}\nolimits}$ and ${\operatorname{mod}\nolimits}H'$. To see that it is exact, we consider the following projective resolution of $H'$ as a right $H$-module: $$0\to He_M H\to H\to H'\to 0.$$ Applying $-\otimes_H U$ to this sequence, where $U$ is an object in ${\operatorname{\mathcal U}\nolimits}$, we obtain (part of) the long exact sequence: $${\operatorname{Tor}\nolimits}_1^H(H,U)\to {\operatorname{Tor}\nolimits}_1^H(H',U)\to He_M H\otimes_H U\to
H\otimes_H U\to H'\otimes_H U\to 0.$$ Since $H$ is projective, ${\operatorname{Tor}\nolimits}_1^H(H,U)=0$. We also have $$He_M H\otimes_H U=H\otimes_H He_MU=0$$ since $e_M U=0$. It follows that ${\operatorname{Tor}\nolimits}^H_1(H',U)=0$ and hence that $H'\otimes_H-$ is an exact functor on ${\operatorname{\mathcal U}\nolimits}$.
Localising with respect to an exceptional module {#loc}
------------------------------------------------
Fix a hereditary algebra $H$, and an indecomposable module $M$ in ${\operatorname{mod}\nolimits}H$, with ${\operatorname{Ext}\nolimits}^1_H(M,M) = 0$.
Let ${\operatorname{\mathcal M}\nolimits}= {\operatorname{add}\nolimits}\{ M[i] \}_{i \in {\operatorname{\mathbb Z}\nolimits}}$. Then ${\operatorname{\mathcal M}\nolimits}$ is a thick subcategory in $D^b({\operatorname{mod}\nolimits}H)$.
Straightforward from the fact that any map between indecomposable objects in ${\operatorname{\mathcal M}\nolimits}$ is either zero or an isomorphism.
Let ${\operatorname{\mathcal D}\nolimits}= D^b({\operatorname{mod}\nolimits}H)$, let ${\operatorname{\mathcal D}\nolimits}_{{\operatorname{\mathcal M}\nolimits}}$ be the category obtained from ${\operatorname{\mathcal D}\nolimits}$ by localising with respect to ${\operatorname{\mathcal M}\nolimits}$, and let $L_{{\operatorname{\mathcal M}\nolimits}} \colon {\operatorname{\mathcal D}\nolimits}\to {\operatorname{\mathcal D}\nolimits}_{{\operatorname{\mathcal M}\nolimits}}$ be the localisation functor. Note that ${\operatorname{\mathcal U}\nolimits}$ is the full subcategory of ${\operatorname{mod}\nolimits}H$ consisting of modules $X$ with ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(M,X[i]) = 0$ for all $i$.
\[both\] Let $H$ be a hereditary algebra with $n$ simple modules up to isomorphism. Let $M$ be an indecomposable $H$-module with ${\operatorname{Ext}\nolimits}_H^1(M,M) = 0$, and let ${\operatorname{\mathcal M}\nolimits}$ denote the thick subcategory generated by $M$. Then ${\operatorname{\mathcal D}\nolimits}_{{\operatorname{\mathcal M}\nolimits}}$ is equivalent to the derived category of a hereditary algebra with $n-1$ simple modules (up to isomorphism).
To prove this we show that ${\operatorname{\mathcal D}\nolimits}_{{\operatorname{\mathcal M}\nolimits}}$ is equivalent to the subcategory ${\operatorname{\mathcal D}\nolimits}_0 = {\operatorname{add}\nolimits}\{X[i] \in {\operatorname{\mathcal D}\nolimits}\mid X \in {\operatorname{\mathcal U}\nolimits}, i \in {\operatorname{\mathbb Z}\nolimits}\}$ of ${\operatorname{\mathcal D}\nolimits}$, and that ${\operatorname{\mathcal D}\nolimits}_0$ is equivalent to the derived category of a hereditary algebra with $n-1$ simple modules. This is the content of the following three propositions. We usually denote the object $L_{{\operatorname{\mathcal M}\nolimits}}(X)$ by $\widetilde{X}$.
\[localequiv\] In the setting of Theorem \[both\], the localisation functor $L_{{\operatorname{\mathcal M}\nolimits}}$ induces an equivalence ${\operatorname{\mathcal D}\nolimits}_0 \to {\operatorname{\mathcal D}\nolimits}_{{\operatorname{\mathcal M}\nolimits}}$.
First note that by Proposition \[verdieriso\] we have that $L_{{\operatorname{\mathcal M}\nolimits}} \colon
{\operatorname{\mathcal D}\nolimits}_0 \to {\operatorname{\mathcal D}\nolimits}_{{\operatorname{\mathcal M}\nolimits}}$ is fully faithful. Any object in ${\operatorname{\mathcal D}\nolimits}_{{\operatorname{\mathcal M}\nolimits}}$ is of the form $L_{{\operatorname{\mathcal M}\nolimits}}(X)$ for some object $X$ in ${\operatorname{\mathcal D}\nolimits}$. Let $\widetilde{X}$ be an arbitrary object in ${\operatorname{\mathcal D}\nolimits}_{{\operatorname{\mathcal M}\nolimits}}$ (where $X$ is in ${\operatorname{\mathcal D}\nolimits}$). Then consider the minimal right ${\operatorname{\mathcal M}\nolimits}$-approximation $M_X \to X$, and the induced triangle $M_X \to X \to X_0 \to$. It is clear that $\widetilde{X} = \widetilde{X_0}$. We claim that $X_0$ is in ${\operatorname{\mathcal D}\nolimits}_0$, that is ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(M, X[i]) = 0$ for all $i$. To see this, consider the long exact sequence obtained by applying ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(M,\ )$ to the triangle $M_X \to X \to X_0 \to $. For any $i$, the map ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(M,M_X[i]) \to {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(M,X[i])$ is an epimorphism, since $M_X \to X$ is a right ${\operatorname{\mathcal M}\nolimits}$-approximation. The map is injective since any element in ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(M,M_X[i])$ is either zero or an isomorphism. Thus, $X_0$ is in ${\operatorname{\mathcal D}\nolimits}_0$. This completes the proof that $L_{{\operatorname{\mathcal M}\nolimits}}$ induces an equivalence ${\operatorname{\mathcal D}\nolimits}_0 \to {\operatorname{\mathcal D}\nolimits}_{{\operatorname{\mathcal M}\nolimits}}$.
The next result is an extension of Proposition \[happelequivalence\] to the setting of derived categories.
In the setting of Theorem \[both\], assume $M$ is non-projective. Let $E$ be the Bongartz-complement of $M$, and let $H' = {\operatorname{End}\nolimits}_H(E)^{op}$. Then ${\operatorname{\mathbb{R}Hom}\nolimits}(E,\ )$ induces an equivalence ${\operatorname{\mathcal D}\nolimits}_0 \to {\operatorname{\mathcal D}\nolimits}' = D^b({\operatorname{mod}\nolimits}H')$.
Recall that ${\operatorname{\mathcal U}\nolimits}\subset M^{\perp} = (M \amalg E)^{\perp}$. This implies that for $X \in {\operatorname{\mathcal U}\nolimits}$, we have that ${\operatorname{\mathbb{R}Hom}\nolimits}(E,X)$ is concentrated in degree zero with zero-term ${\operatorname{Hom}\nolimits}_H(E,X)$. Since ${\operatorname{Hom}\nolimits}_H(E,\ )$ is a dense functor from ${\operatorname{\mathcal U}\nolimits}$ to ${\operatorname{mod}\nolimits}H'$, and ${\operatorname{\mathbb{R}Hom}\nolimits}(E,\ )$ commutes with $[1]$, it follows that ${\operatorname{\mathbb{R}Hom}\nolimits}(E,\ )$ restricted to ${\operatorname{\mathcal D}\nolimits}_0$ is dense.
Assume $X,Y$ are indecomposable objects in the same degree in ${\operatorname{\mathcal D}\nolimits}_0$. By the above it now follows directly from Proposition \[happelequivalence\] that $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X,Y) \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}({\operatorname{\mathbb{R}Hom}\nolimits}(E,X), {\operatorname{\mathbb{R}Hom}\nolimits}(E,Y)).$$
We also need to show that ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X,Y[1]) \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}({\operatorname{\mathbb{R}Hom}\nolimits}(E,X), {\operatorname{\mathbb{R}Hom}\nolimits}(E,Y[1]))$. For this note that by Proposition \[happelequivalence\], the equivalence ${\operatorname{Hom}\nolimits}_H(E,\ ) \colon {\operatorname{\mathcal U}\nolimits}\to {\operatorname{mod}\nolimits}H'$ is exact, and that the embedding ${\operatorname{\mathcal U}\nolimits}\hookrightarrow {\operatorname{mod}\nolimits}H$ is exact. This implies that $$\begin{aligned}
{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X,Y[1]) \simeq & {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal U}\nolimits}}(X,Y[1]) \\
\simeq & {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}({\operatorname{Hom}\nolimits}_H(E,X), {\operatorname{Hom}\nolimits}_H(E,Y)[1]) \\
\simeq & {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}({\operatorname{\mathbb{R}Hom}\nolimits}(E,X),{\operatorname{\mathbb{R}Hom}\nolimits}(E,Y)[1]) \\
\simeq & {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}({\operatorname{\mathbb{R}Hom}\nolimits}(E,X),{\operatorname{\mathbb{R}Hom}\nolimits}(E,Y[1])).\end{aligned}$$ Thus the restriction of ${\operatorname{\mathbb{R}Hom}\nolimits}(E,\ )$ to ${\operatorname{\mathcal D}\nolimits}_0$ is fully faithful. This completes the proof.
In the setting of Theorem \[both\], assume $M$ is projective. Assume $M \simeq He_M$ for the primitive idempotent $e_M$ in $H$ and let $H' =H / He_M H$. Then ${\operatorname{\mathbb L}\nolimits}(H' \otimes_H - )$ induces an equivalence ${\operatorname{\mathcal D}\nolimits}_0 \to {\operatorname{\mathcal D}\nolimits}' = D^b({\operatorname{mod}\nolimits}H')$.
First recall from Lemma \[projectiveequivalence\] that ${\operatorname{Tor}\nolimits}_1^H(H',U) = 0$ for any $U$ in ${\operatorname{\mathcal U}\nolimits}$. This means that the image ${\operatorname{\mathbb L}\nolimits}(H' \otimes_H U)$ is just $H' \otimes_H U$ concentrated in degree $0$.
It now follows that ${\operatorname{\mathbb L}\nolimits}(H' \otimes_H -)$ restricted to ${\operatorname{\mathcal D}\nolimits}_0$ is dense, by using that the functor $H' \otimes_H - \colon {\operatorname{\mathcal U}\nolimits}\to {\operatorname{mod}\nolimits}H'$ is dense and that ${\operatorname{\mathbb L}\nolimits}(H' \otimes_H - )$ commutes with $[1]$.
Assume $X,Y$ are indecomposable objects in the same degree in ${\operatorname{\mathcal D}\nolimits}_0$. It follows from Lemma \[projectiveequivalence\] that $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X,Y) \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}({\operatorname{\mathbb L}\nolimits}(H' \otimes_H X), {\operatorname{\mathbb L}\nolimits}(H'
\otimes_H Y)).$$
We need also to show that $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X,Y[1]) \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}({\operatorname{\mathbb L}\nolimits}(H' \otimes_H X), {\operatorname{\mathbb L}\nolimits}(H'
\otimes_H Y[1])).$$ For this recall that the embedding of ${\operatorname{\mathcal U}\nolimits}$ into ${\operatorname{mod}\nolimits}H$ is exact, and that $H' \otimes_H -$ is exact on ${\operatorname{\mathcal U}\nolimits}$ by Lemma \[projectiveequivalence\]. Thus it follows that:
$$\begin{aligned}
{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X,Y[1]) \simeq &{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal U}\nolimits}}(X,Y[1]) \\
\simeq &{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}(H' \otimes_H X, H' \otimes_H Y [1]) \\
\simeq &{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}({\operatorname{\mathbb L}\nolimits}(H' \otimes_H X),{\operatorname{\mathbb L}\nolimits}(H' \otimes_H Y)[1]) \\
\simeq &{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}({\operatorname{\mathbb L}\nolimits}(H' \otimes_H X),{\operatorname{\mathbb L}\nolimits}(H' \otimes_H Y[1])).\end{aligned}$$
This shows that the functor is fully faithful and finishes the proof.
For the remainder of this section, we view the induced equivalence between ${\operatorname{\mathcal D}\nolimits}_{{\operatorname{\mathcal M}\nolimits}}$ and ${\operatorname{\mathcal D}\nolimits}'$ as an identification.
The factor construction
-----------------------
As before, let $M$ be an indecomposable $H$-module with ${\operatorname{Ext}\nolimits}^1_H(M,M)=0$, where $H$ is hereditary, and let $E$ be the Bongartz complement of $M$. We investigate the image of an arbitrary complement $\overline{T}$ of $M$ under the functor $L_{{\operatorname{\mathcal M}\nolimits}}$. For an object $X$ in ${\operatorname{\mathcal D}\nolimits}$, we use the notation $\widetilde{X} = L_{{\operatorname{\mathcal M}\nolimits}}(X)$, as before. Note that $L_{{\operatorname{\mathcal M}\nolimits}}(\overline{T})=
L_{{\operatorname{\mathcal M}\nolimits}}(T)=\widetilde{T}$.
\[orto\] Let the notation be as above.
- [$L_{{\operatorname{\mathcal M}\nolimits}}(\overline{T})= \widetilde{T}$ is in ${\operatorname{mod}\nolimits}H'\vee H'[1]$. ]{}
- [ ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}(\widetilde{T},\widetilde{T}[1])=0$.]{}
\(a) Let $f \colon M'\to \overline{T}$ be a minimal right ${\operatorname{\mathcal M}\nolimits}$-approximation, and consider the induced triangle: $$\label{approxtri}
M'\overset{f}\to \overline{T}\overset{g}\to U_T\to$$ in ${\operatorname{\mathcal D}\nolimits}$. Since ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(M[-1],\overline{T})=0$, we have that $M'$ is in ${\operatorname{add}\nolimits}M$. It is clear that $\widetilde{T} \simeq \widetilde{U_T}$. Now, as in the proof of Proposition \[localequiv\], we get that $U_T$ is in ${\operatorname{\mathcal D}\nolimits}_0$. Here it is clear that $U_T = U_1 \amalg U_2[1]$, where $U_1 = {\operatorname{Coker}\nolimits}f$ and $U_2 = {\operatorname{Ker}\nolimits}f$ are in ${\operatorname{\mathcal U}\nolimits}$. It is clear that $\widetilde{U_1}$ and $\widetilde{U_2}$ are $H'$-modules. We only need to show that $\widetilde{U_2}$ is projective. For an arbitrary $U$ in ${\operatorname{\mathcal U}\nolimits}$, we have that ${\operatorname{Ext}\nolimits}^1_H(U_2,U) = 0$, since ${\operatorname{Ext}\nolimits}^1_H(M,U) = 0$ and $U_2$ is a submodule of $M$. Using that ${\operatorname{\mathcal U}\nolimits}$ is an exact subcategory of ${\operatorname{mod}\nolimits}H$, and that the equivalence ${\operatorname{\mathcal U}\nolimits}\to {\operatorname{mod}\nolimits}H'$ is also exact, it follows that $\widetilde{U_2}$ is projective in ${\operatorname{mod}\nolimits}H'$. Hence $\widetilde{T} \simeq \widetilde{U_T}$ is in ${\operatorname{mod}\nolimits}H'\vee H'[1]$.
\(b) Using again the triangle (\[approxtri\]) we obtain the long exact sequence $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(\overline{T},\overline{T}[1])\to {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(\overline{T},U_T[1])\to
{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(\overline{T},M'[2]).$$ Hence, ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(\overline{T},U_T[1])= 0$. Now, by Proposition \[verdieriso\], it follows that ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}(\widetilde{T}, \widetilde{U_T[1]})= 0$ since $U_T[1]$ is in ${\operatorname{\mathcal D}\nolimits}_0$, and hence ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}(\widetilde{T}, \widetilde{T}[1])= 0$.
Denote as before by $F$ the functor $\tau^{-1}[1] \colon {\operatorname{\mathcal D}\nolimits}\to {\operatorname{\mathcal D}\nolimits}$. When it is not clear which derived category ${\operatorname{\mathcal D}\nolimits}$ we are dealing with, we will denote this functor by $F_{{\operatorname{\mathcal D}\nolimits}}$ and the functor $\tau^{-1}$ by $\tau_{{\operatorname{\mathcal D}\nolimits}}^{-1}$.
\[orto2\] Let $H$ be a hereditary algebra, and let $X$ be an object in ${\operatorname{\mathcal D}\nolimits}$ such that $X$ is in ${\operatorname{mod}\nolimits}H \vee H[1]$. Then ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X,X[1]) = 0$ if and only if ${\operatorname{Ext}\nolimits}^1_{{\operatorname{\mathcal C}\nolimits}_H}(X,X)=0$.
Assume $X$ is in ${\operatorname{mod}\nolimits}H \vee H[1]$, and let $\widehat{X}$ be the image of $X$ in the cluster category ${\operatorname{\mathcal C}\nolimits}_H$ of $H$. Then ${\operatorname{Ext}\nolimits}^1_{{\operatorname{\mathcal C}\nolimits}}(\widehat{X}, \widehat{X}) \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X,X[1])
\amalg D{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X,X[1])$. This follows from ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X, F^{-1}X[1]) = {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X, \tau X) \simeq D{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X,X[1])$ and the easily checked fact that ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(X, F^{i}X[1]) = 0$, whenever $i \not \in \{-1,0 \}$.
Combining these lemmas, and using that a tilting $H$-module induces a tilting object in the cluster category [@bmrrt 3.3], we obtain the following.
Let $T = M \amalg \bar{T}$ be a tilting $H$-module as before. Then the image $\widehat{T}$ of $\widetilde{T}$ in the cluster category ${\operatorname{\mathcal C}\nolimits}_{H'}$ is a tilting object.
Since $T$ is a tilting $H$-module, the triangulated category generated by $T$ is ${\operatorname{\mathcal D}\nolimits}$. Hence the triangulated category generated by $\widetilde{T}$ is ${\operatorname{\mathcal D}\nolimits}'$. By Lemmas \[orto\] and \[orto2\] we have ${\operatorname{Ext}\nolimits}^1_{{\operatorname{\mathcal C}\nolimits}_{H'}}(\widehat{T}, \widehat{T})=0$. By the proof of [@bmrrt Thm. 3.3], we have that $\widetilde{T}$ can be viewed as a direct summand of a tilting $H''$-module, for some hereditary algebra $H''$ derived equivalent to $H'$. We then have ${\operatorname{Ext}\nolimits}^1_{H''}(\widetilde{T}, \widetilde{T})=0$, and hence ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}(\widetilde{T}, \widetilde{T}[i]) = 0$ for all $i \neq 0$. Since ${\operatorname{\mathcal D}\nolimits}'$ is the triangulated category generated by $\widetilde{T}$, it follows that $\widetilde{T}$ is a tilting complex by definition, and consequently a tilting $H''$-module. Then $\widehat{T}$ is a tilting object in ${\operatorname{\mathcal C}\nolimits}_{H'}$.
We can now complete the main result of this section. Let $e$ be the idempotent in ${\Gamma}$, such that ${\Gamma}e \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T,M)$.
\[factor\] With the above notation, there is a natural isomorphism ${\Gamma}/ {\Gamma}e {\Gamma}\simeq {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}_{H'}}(\widehat{T})^{{{\operatorname{op}}}}$.
The remainder of this section will be devoted to proving this theorem. Since the cluster category is defined using the functor $F=\tau^{-1}[1]$, we need to compare $\widetilde{\tau_{{\operatorname{\mathcal D}\nolimits}}^{-1}(X)}$ and $\tau^{-1}_{{\operatorname{\mathcal D}\nolimits}'} \widetilde{X}$ for an indecomposable object $X$ in ${\operatorname{\mathcal D}\nolimits}$. In general $\widetilde{\tau^{-1}_{{\operatorname{\mathcal D}\nolimits}} X} \not \simeq \tau^{-1}_{{\operatorname{\mathcal D}\nolimits}'} \widetilde{X}$, but with extra conditions on $X$, sufficient for our purposes, everything behaves nicely. The next result has been generalised by Keller, with a simpler proof [@k], but for completeness we include our proof here.
\[commutes\] Let $X$ be an indecomposable object in ${\operatorname{\mathcal D}\nolimits}_0\subset {\operatorname{\mathcal D}\nolimits}$. Then $\widetilde{X}$ is indecomposable and $\widetilde{\tau^{-1}_{{\operatorname{\mathcal D}\nolimits}} X} \simeq \tau^{-1}_{{\operatorname{\mathcal D}\nolimits}'} \widetilde{X}$.
Since $L_{{\operatorname{\mathcal M}\nolimits}} \colon {\operatorname{\mathcal D}\nolimits}_0 \to {\operatorname{\mathcal D}\nolimits}_{{\operatorname{\mathcal M}\nolimits}}$ is an equivalence, we have that $\widetilde{X} = L_{{\operatorname{\mathcal M}\nolimits}}(X)$ is indecomposable. Let $f \colon X \to C$ be a minimal left almost split map in ${\operatorname{\mathcal D}\nolimits}$. We want to show that $\widetilde{f} \colon \widetilde{X} \to \widetilde{C}$ has the same property. We first show that $\widetilde{f} \colon \widetilde{X} \to \widetilde{C}$ is not a split monomorphism. Assume to the contrary that there is a map $g' \colon \widetilde{C}
\to \widetilde{X}$ in ${\operatorname{\mathcal D}\nolimits}'$ such that $g' \widetilde{f} = {\operatorname{id}\nolimits}_{\widetilde{X}}$. Since $X$ is in ${\operatorname{\mathcal D}\nolimits}_0$, there is a map $g \colon C \to X$ in ${\operatorname{\mathcal D}\nolimits}$ such that $\widetilde{g} =
g'$ by Proposition \[verdieriso\]. Then $gf \colon X \to X$ is an isomorphism since $\widetilde{g} \widetilde{f} \colon \widetilde{X} \to \widetilde{X}$ is an isomorphism and $X$ is in ${\operatorname{\mathcal D}\nolimits}_0$. Hence $f \colon X \to C$ is a split monomorphism, and we have a contradiction. Hence $\widetilde{f}:\widetilde{X}\to\widetilde{C}$ is not a split monomorphism.
We next show that $\widetilde{f} \colon \widetilde{X} \to \widetilde{C}$ is left almost split. Let $\widetilde{h} \colon \widetilde{X} \to \widetilde{Y}$ be a map in ${\operatorname{\mathcal D}\nolimits}'$ which is not an isomorphism, with $Y$ indecomposable in ${\operatorname{\mathcal D}\nolimits}_0$ and hence $\widetilde{Y}$ indecomposable in ${\operatorname{\mathcal D}\nolimits}'$. Let $h \colon X \to Y$ be the map in ${\operatorname{\mathcal D}\nolimits}$, inducing the map $\widetilde{h} \colon \widetilde{X} \to \widetilde{Y}$. This map is unique by Lemma \[verdieriso\]. Since $f \colon X \to C$ is left almost split, there is some $s \colon C \to Y$ such that $sf =h$. Hence we have $\widetilde{s}\widetilde{f} = \widetilde{h}$ in ${\operatorname{\mathcal D}\nolimits}'$, showing that $\widetilde{f} \colon \widetilde{X} \to \widetilde{C}$ is left almost split.
We also want to show that $\widetilde{f} \colon \widetilde{X} \to \widetilde{C}$ is a left minimal map. Assume to the contrary that there is an indecomposable direct summand $C_1$ of $C$, such that when $f_1 \colon X \to C_1$ is the map induced by $f \colon X \to C$, then $\widetilde{f_1} \colon \widetilde{X} \to \widetilde{C_1}$ is 0, but $\widetilde{C_1} \neq 0$. Then we have a commutative diagram $$\xy
\xymatrix{
X \ar[dr]_{u_1} \ar[rr]^{f_1} & & C_1 \\
& M' \ar[ur]_{v_1} &
}
\endxy$$ with $M'$ in ${\operatorname{\mathcal M}\nolimits}$. Since $f_1 \colon X \to C_1$ is irreducible, we have that either $u_1 \colon X \to M'$ is a split monomorphism, so that $X$ is in ${\operatorname{\mathcal M}\nolimits}$, and hence $\widetilde{X} = 0$, or $v_1 \colon M' \to C_1$ is a split epimorphism and hence $C_1$ is in ${\operatorname{\mathcal M}\nolimits}'$, so that $\widetilde{C_1} = 0$. In both cases we have a contradiction. Hence we can conclude that $\widetilde{f} \colon \widetilde{X} \to \widetilde{C}$ is a left minimal map.
Let $X \overset{f_1}{\to} C \to \tau^{-1} X\to$ be an almost split triangle in ${\operatorname{\mathcal D}\nolimits}$. Then the induced triangle $\widetilde{X} \overset{\widetilde{f_1}}{\to} \widetilde{C} \to
\widetilde{\tau^{-1} X}$ is almost split since $\widetilde{f} \colon \widetilde{X} \to \widetilde{C}$ is a minimal left almost split map. Hence we get $\tau^{-1}_{{\operatorname{\mathcal D}\nolimits}'} \widetilde{X} \simeq \widetilde{\tau^{-1}_{{\operatorname{\mathcal D}\nolimits}}X}$.
Let $T_x$ be an indecomposable direct summand in $T$, not isomorphic to $M$. Let $M_x \to T_x$ be a minimal right ${\operatorname{add}\nolimits}M$-approximation, and consider as before the induced triangle $$M_x \to T_x \to U_x \to$$ in ${\operatorname{\mathcal D}\nolimits}$, where we know that $U_x$ is in ${\operatorname{\mathcal D}\nolimits}_0$ by the proof of Proposition \[localequiv\]. Thus, by applying the above lemma to each of the indecomposable direct summands of $U_x$, we obtain $\widetilde{\tau^{-1}_{{\operatorname{\mathcal D}\nolimits}} U_x} \simeq \tau^{-1}_{{\operatorname{\mathcal D}\nolimits}'} \widetilde{U_x}$, and thus $\widetilde{F_{{\operatorname{\mathcal D}\nolimits}} U_x} \simeq F_{{\operatorname{\mathcal D}\nolimits}'} \widetilde{U_x}$. It is also clear that $\widetilde{U_x} \simeq \widetilde{T_x}$.
Now, pick two (not necessarily different) indecomposable direct summands $T_a$ and $T_b$ of $\bar{T}$. Construct the triangle $$M_b \to T_b \to U_b \to,$$ as above, and apply $F$ to it, to obtain the triangle $$FM_b \to FT_b \to FU_b \to.$$ Apply ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a,\ )$ to this triangle, to obtain the long exact sequence $$\begin{gathered}
\label{unmod}
{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, FM_b) \to {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, FT_b) \to \\
{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, FU_b) \to
{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, FM_b[1]).\end{gathered}$$ The last term vanishes, since $T_a$ and $M_b$ are modules. Since $M_b \to T_b$ is a minimal right ${\operatorname{add}\nolimits}M$-approximation, it follows that $FM_b \to FT_b$ is a minimal right ${\operatorname{add}\nolimits}FM$-approximation. We have that ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, FU_b) \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, FT_b)/(FM)$, where for an object $Z$ we use the notation ${\operatorname{Hom}\nolimits}(X,Y)/(Z)$ to denote the ${\operatorname{Hom}\nolimits}$-space modulo maps factoring through an object in ${\operatorname{add}\nolimits}Z$.
We claim there is an exact sequence $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, FM_b)/(M) \to {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, FT_b)/(M) \to
{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, FU_b)/(M) \to 0$$ induced from the exact sequence (\[unmod\]). For this it is sufficient to show that the kernel of the second map is contained in the image of the first. So let $\alpha \in {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, FT_b)/(M)$, and assume there is a commutative diagram $$\xy
\xymatrix{
T_a \ar[r]^{\alpha} \ar[dr]^{\beta_1} & FT_b \ar[r] & FU_b \\
& M' \ar[ur] &
}
\endxy$$ for some $M'$ in ${\operatorname{add}\nolimits}M$. Since ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(M,FM_b[1]) = 0$, there is $\beta_2 \colon M' \to FT_b$, such that $M' \overset{\beta_2}{\to} FT_b \to FU_b = M' \to FU_b$. In ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a,FT_b)/(M)$ we have $\alpha = \alpha - \beta_2 \beta_1$. By using the long exact sequence (\[unmod\]), we obtain that $\alpha = \alpha - \beta_2 \beta_1$ factors through $FM_b \to FT_b$, so the sequence is exact. It follows from this that ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, FT_b)/(M \amalg FM) \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, FU_b)/(M)$.
Let $f \colon M_1 \to FU_b$ be a minimal right ${\operatorname{\mathcal M}\nolimits}$-approximation, and complete to a triangle $M_1 \overset{f}{\to} FU_b \to (FU_b)'$. Applying ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, \ )$, we get an exact sequence $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, M_1 ) \to {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}( T_a, FU_b) \to {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, (FU_b)' ) \to {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a,M_1[1]).$$ Since $U_b$ is in degree $0$ or $1$, then $FU_b$ is in degree $1$,$2$ or $3$, so $M_1$ is in degree $0$,$1$,$2$ or $3$. Hence the indecomposable direct summands of $M_1[1]$ are in degree at least 1, so that ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a,M_1[1]) = 0$. Since a map $h \colon T_a \to FU_b$ factors through an object in ${\operatorname{\mathcal M}\nolimits}$ if and only if it factors through the minimal right ${\operatorname{\mathcal M}\nolimits}$-approximation of $FU_b$, we get the isomorphism $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, F_{{\operatorname{\mathcal D}\nolimits}}U_b)/ (M) \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, (F_{{\operatorname{\mathcal D}\nolimits}}U_b)').$$ We get that this is isomorphic to ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}(\widetilde{T_a}, \widetilde{F_{{\operatorname{\mathcal D}\nolimits}}U_b})$, since $(F_{{\operatorname{\mathcal D}\nolimits}}U_b)'$ is in ${\operatorname{\mathcal D}\nolimits}_0$. By Lemma \[commutes\] this is isomorphic to ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}(\widetilde{T_a}, F_{{\operatorname{\mathcal D}\nolimits}'}\widetilde{T_b})$. We thus obtain that $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a,F_{{\operatorname{\mathcal D}\nolimits}}T_b)/{(M\amalg FM)}\simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}(\widetilde{T_a},F_{{\operatorname{\mathcal D}\nolimits}'}\widetilde{T_b}).$$
We have ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, T_b) / (M \amalg FM) \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, T_b)/ (M)$. Consider again the triangle $M_b \overset{f_b}{\to} T_b \to U_b$ in ${\operatorname{\mathcal D}\nolimits}$, where $f_b \colon M_b \to T_b$ is a minimal right ${\operatorname{\mathcal M}\nolimits}$-approximation. Applying ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, \ )$ gives an exact sequence $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, M_b) \to {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}( T_a, T_b) \to {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, U_b ) \to {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a,M_b[1]).$$ Since $M_b$ is a module, we have ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, M_b[1]) = 0$, and hence ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, T_b)/(M) \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a, U_b)$, which is isomorphic to ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}(\widetilde{T_a}, \widetilde{U_b})$ by Proposition \[verdieriso\]. We obtain that: $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_a,T_b)/{(M\amalg FM)}\simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}(\widetilde{T_a},\widetilde{T_b}).$$
Therefore ${\Gamma}/ {\Gamma}e {\Gamma}= {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T,T) \amalg {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T,FT) /
(M \amalg FM) \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}(\widetilde{T},\widetilde{T}) \amalg
{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}'}(\widetilde{T}, F_{{\operatorname{\mathcal D}\nolimits}'} \widetilde{T})$ as vector spaces. It is straightforward to check that the map is also a ring map. Theorem \[factor\] is proved.
Comparison with tilted algebras
-------------------------------
We give an example showing that a result similar to Theorem \[factor\] does not hold for tilted algebras. We would like to thank Dieter Happel for providing us with this example. There is a tilting module for the path algebra of a Dynkin quiver of type $D_5$, such that the corresponding tilted algebra ${\Lambda}$ has the quiver $$\xy
\xymatrix{
& & 3 \ar[dr]^{\gamma} & \\
1 \ar[r]^{\alpha} & 2 \ar[ur]^{\beta} \ar[dr]_{\delta} & & 5 \\
& & 4 \ar[ur]_{\epsilon} &
}
\endxy$$ with relations $\alpha \beta = \beta \gamma - \delta \epsilon =0$. If we let $e_4$ be the primitive idempotent corresponding to vertex $4$, then it easy to see that ${\Lambda}/{\Lambda}e_4 {\Lambda}$ is not tilted, since it has global dimension three.
It is well-known that the endomorphism-ring of a partial tilting module is a tilted algebra. However, a similar result does not hold for cluster-tilted algebras. An example of this is the path algebra of an oriented 4-cycle, modulo the cube of its radical. This is a cluster-tilted algebra of type $D_4$.
Cluster-tilted algebras {#cluster-tilted}
=======================
In this section we apply the main result of the previous section to show that (oriented) cycles in the quiver of a cluster-tilted algebra have length at least three. Let $T_1 \amalg T_2 \amalg \dots \amalg T_n$ be a tilting object in the cluster category ${\operatorname{\mathcal C}\nolimits}$. We denote by $\delta_k(T)$ the tilting object $T'$ obtained by exchanging $T_i$ with the second complement of $T_1 \amalg \cdots \amalg T_{i-1} \amalg T_{i+1} \amalg \cdots \amalg T_n$. Let ${\Gamma}= {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T)^{{{\operatorname{op}}}}$ and ${\Gamma}' = {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T')^{{{\operatorname{op}}}}$ be the corresponding cluster-tilted algebras. Passing from ${\Gamma}$ to ${\Gamma}'$ depends on the choice of tilting object $T$. But we still write $\overline{\delta}_k({\Gamma}) = {\Gamma}'$, when either it is clear from the context which tilting object $T$ gives rise to ${\Gamma}$, or when this it is irrelevant. We also say that ${\Gamma}'$ is obtained from ${\Gamma}$ by mutation at $k$.
From [@bmrrt] we know that all tilting objects in ${\operatorname{\mathcal C}\nolimits}_H$ can be obtained from performing a finite number of operations $\delta_k$ to $H$, where $H$ is the hereditary algebra considered as a tilting object in ${\operatorname{\mathcal C}\nolimits}_H$.
If $k$ is a source or a sink in the quiver of a hereditary algebra, then mutation at $k$ coincides with so-called APR-tilting [@apr] (see [@bmr]), and the quiver of the mutated algebra $\overline{\delta}_k(H)$ is obtained by reversing all arrows ending or starting in $k$.
\[ranktwo\] The cluster-tilted algebras of rank at most 2 are hereditary.
This follows from the fact that any cluster-tilted algebra can be obtained by starting with a hereditary algebra, and performing a finite number of mutations. If we start with a hereditary algebra $H$ of rank at most 2, the algebra obtained by mutating at one of the vertices is isomorphic to $H$.
\[noshortcycles\] The quiver of a cluster-tilted algebra has no loops and no cycles of length 2.
This follows directly from combining Lemma \[ranktwo\] with Theorem \[factor\].
This was first proven by Gordana Todorov in case of finite representation type.
Cluster-tilted algebras of rank 3 {#rankthree}
=================================
In this section we specialize to connected hereditary algebras of rank 3, and the cluster-tilted algebras obtained from them. We describe the possible quivers, and give some information on the relation-spaces. Later, this will be used to show our main result for algebras of rank 3. In the proof of our main theorem, we use Theorem \[factor\] to reduce to the case of rank 3. For hereditary algebras of finite representation type, there is up to derived equivalence only one connected algebra of rank 3, and thus up to equivalence only one cluster category ${\operatorname{\mathcal C}\nolimits}$. In this case the technically involved results of this section reduce to just checking [*one case*]{}: The only cluster-tilted algebra of rank 3 which is not hereditary is given by a quiver which is a cycle of length 3, and with the relations that the composition of any two arrows is zero.
The quivers
-----------
We consider quivers of the form $$\xy
\xymatrix{1 \ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}_{r} \ar@<1ex>[rr]_{.}^{t} \ar@<-1ex>[rr]^{.} & &3 \\
& 2 \ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.}_{s} &
}
\endxy$$ where $r > 0$, $s > 0$ and $t \geq 0$ denote the number of arrows as indicated in the above figure. For short, we denote such a quiver by $Q_{rst}$.
Up to derived equivalence, all connected finite dimensional hereditary algebras of rank 3 have a quiver given as above. We first prove that factors of path-algebras of such quivers by admissible ideals are never cluster-tilted. The following is useful for this.
\[almost\] Let $H$ be a hereditary algebra of rank 3, and assume $X, Y$ are indecomposable $H$-modules. Assume that there exists an irreducible map $X \to Y$, and that $X \amalg Y$ is an almost complete tilting module. Then $X$ and $Y$ are either both preprojective or both preinjective, and there exists a complement $Z$ such that $T = X \amalg Y \amalg Z$ has a hereditary endomorphism ring ${\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T)^{{{\operatorname{op}}}} \simeq {\operatorname{End}\nolimits}_H(T)^{{{\operatorname{op}}}}$.
The first claim follows from the fact that regular modules with no self-extensions are quasi-simple, since $H$ has rank 3, and there does not exist an irreducible map between two quasi-simples. Now, it is well-known that $X \amalg Y$ can be completed to a so-called [*slice*]{} [@r], and the endomorphism ring ${\operatorname{End}\nolimits}_H(T)^{{{\operatorname{op}}}}$ is hereditary. Also, for a slice, ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T,\tau^2 T)= 0$, so ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T,FT) = {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T, \tau^{-1}T[1]) \simeq
D{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T, \tau^2 T) = 0$. That is, ${\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T)^{{{\operatorname{op}}}} \simeq {\operatorname{End}\nolimits}_H(T)^{{{\operatorname{op}}}}$.
\[nofactors\] If ${\Gamma}$ is a cluster-tilted algebra with quiver of type $Q_{rst}$, then ${\Gamma}$ is hereditary.
Assume ${\Gamma}= {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T)^{{{\operatorname{op}}}}$, where $T = T_1 \amalg T_2 \amalg T_3$ is a tilting module in ${\operatorname{mod}\nolimits}H$. For $H$ of finite type this is easily checked, since there is, up to triangle-equivalence, only one cluster-category. Therefore we can assume that $H$, and thus ${\Gamma}$, is not of finite type. Thus, we can choose $T$ and $H$ such that $T$ does not have both projective and injective direct summands. And by, if necessary, repeatedly applying $\tau^{-1}$ or $\tau$ we can assume that $T$, $\tau T$ and $\tau^{-1}T$ have no projective or injective direct summands. There is a unique sink in $Q_{rst}$, and we assume that it corresponds to $T_3$. Thus, we can assume that ${\operatorname{Hom}\nolimits}_H(T_3, T_1 \amalg T_2) = 0$, and ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_3, F(T))=0$. We use this to show that ${\Gamma}$ is hereditary.
We now assume ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_3, \tau^{-1}T_i[1])=0$, and by the Auslander-Reiten formula we have ${\operatorname{Hom}\nolimits}_{H}(T_i, \tau^2 T_3)=0$, for $i = 1,2,3$. We also have ${\operatorname{Hom}\nolimits}_H(T_i, \tau T_3) = 0$, using the same formula. Consider now the almost split sequence $$\label{almostsplit}
0 \to \tau T_3 \to X \to T_3 \to 0.$$ We want to show that $X$ is in ${\operatorname{add}\nolimits}T$. Apply $\tau$ to (\[almostsplit\]) to obtain $$\label{tauofit}
0 \to \tau^2 T_3 \to \tau X \to \tau T_3 \to 0.$$ By applying ${\operatorname{Hom}\nolimits}_H(T,\ )$ to (\[tauofit\]), and using a long exact sequence argument, it follows that ${\operatorname{Hom}\nolimits}_H(T, \tau X) = 0$, and so ${\operatorname{Ext}\nolimits}^1_H(X ,T)= 0$ by the AR-formula.
We next want to show that also ${\operatorname{Ext}\nolimits}^1_H(T,X)=0$. For $i = 1,2$ we show that ${\operatorname{Ext}\nolimits}^1_H(T_i,X) \simeq D {\operatorname{Hom}\nolimits}_H(X, \tau T_i)=0$. For this, apply ${\operatorname{Hom}\nolimits}_H(\ ,\tau T_i)$ to the exact sequence (\[almostsplit\]) to obtain the exact sequence $${\operatorname{Hom}\nolimits}_H(T_3, \tau T_i) \to {\operatorname{Hom}\nolimits}_H(X, \tau T_i) \to {\operatorname{Hom}\nolimits}_H(\tau T_3, \tau T_i).$$ By the assumptions both ${\operatorname{Hom}\nolimits}_H(T_3, \tau T_i) \simeq D{\operatorname{Ext}\nolimits}^1_H(T_i,T_3) = 0$ and ${\operatorname{Hom}\nolimits}_H(\tau T_3, \tau T_i) \simeq {\operatorname{Hom}\nolimits}_H(T_3, T_i) =0$. Therefore also the middle term is zero. By applying ${\operatorname{Hom}\nolimits}_H(T_3,\ )$ to the exact sequence (\[almostsplit\]), we obtain the long exact sequence $$\begin{aligned}
{\operatorname{Hom}\nolimits}_H(T_3,\tau T_3) \to {\operatorname{Hom}\nolimits}_H(T_3,X) \to {\operatorname{Hom}\nolimits}_H(T_3,T_3) \to \\
{\operatorname{Ext}\nolimits}^1_H(T_3,\tau T_3) \to {\operatorname{Ext}\nolimits}^1_H(T_3,X) \to {\operatorname{Ext}\nolimits}^1_H(T_3, T_3).\end{aligned}$$ Since (\[almostsplit\]) is an almost split sequence, the map ${\operatorname{Hom}\nolimits}_H(T_3,T_3) \to {\operatorname{Ext}\nolimits}^1_H(T_3,\tau T_3)$ is an isomorphism. We have $ {\operatorname{Ext}\nolimits}^1_H(T_3, T_3) = 0$, and therefore ${\operatorname{Ext}\nolimits}^1_H(T_3,X) = 0$. It follows from the same long exact sequence that ${\operatorname{Hom}\nolimits}_H(T_3,X)= 0$, since ${\operatorname{Hom}\nolimits}_H(T_3,\tau T_3) \simeq D {\operatorname{Ext}\nolimits}^1_H(T_3,T_3) =0$.
Now apply ${\operatorname{Hom}\nolimits}_H(X,\ )$ to (\[almostsplit\]), to obtain the long exact sequence $${\operatorname{Ext}\nolimits}^1_H(X,\tau T_3) \to {\operatorname{Ext}\nolimits}^1_H(X,X) \to {\operatorname{Ext}\nolimits}^1_H(X, T_3).$$ Since ${\operatorname{Ext}\nolimits}^1_H(X,\tau T_3) \simeq D {\operatorname{Hom}\nolimits}_H(T_3,X) = 0$, we get ${\operatorname{Ext}\nolimits}^1_H(X,X) = 0$. Thus $${\operatorname{Ext}\nolimits}^1_H(T \amalg X, T \amalg X)=0,$$ so $X$ is in ${\operatorname{add}\nolimits}T$. Since $T_1$ or $T_2$ must be a direct summand of $X$, there is an irreducible map in ${\operatorname{mod}\nolimits}H$ from $T_1$ and/or $T_2$ to $T_3$.
If there are irreducible maps from both $T_1$ and $T_2$ to $T_3$, the claim follows directly, since $T$ forms a slice in ${\operatorname{mod}\nolimits}H$. Now assume there is an irreducible map $T_2 \to T_3$. For this, we need to discuss two cases. If $t=0$, that is the quiver of $H$ is $$\xy
\xymatrix{1 \ar@<1ex>[r]_{.}^{r} \ar@<-1ex>[r]^{.} & 2 \ar@<1ex>[r]_{.}^{s} \ar@<-1ex>[r]^{.} & 3 \
}
\endxy$$ it can be easily seen that both complements to $T_2 \amalg T_3$ will form a slice, and this will give a hereditary endomorphism ring.
Assume now $t>0$. By Lemma \[almost\], the module $M \amalg T_2 \amalg T_3$ has a hereditary endomorphism ring, either for $M = T_1$ or for $M= T_1^{\ast}$. Here $T_1^{\ast}$ is the second complement of $T_2 \amalg T_3$, using the notation from Theorem \[triangles\]. If $M = T_1$, then we are done. Assume therefore $M = T_1^{\ast}$.
In case $t>0$, there are two types of irreducible maps $X \to Y$: either there exists an indecomposable module $Z$ with irreducible maps $X \to Z$ and $Z \to Y$, or no such $Z$ exists. In the latter case, it is easily seen that both complements of $X \amalg Y$ give a hereditary endomorphism ring. In the first case, the AR-quiver looks like $$\xy
\xymatrix{
{\begin{matrix} & & \\ \cdots & \bullet & \\ & &
\end{matrix}} \ar@<1ex>[rr]_{.} \ar@<-1ex>[rr]^{.} \ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}
& & {\begin{matrix} & & \\ & Z & \\ & & \end{matrix}} \ar@<1ex>[rr]_{.} \ar@<-1ex>[rr]^{.}
\ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}
& & {\begin{matrix} & & \\ & \bullet & \cdots \\ & & \end{matrix}} \\
{\begin{matrix} & & \\ & \cdots & \\ & & \end{matrix}} &
{\begin{matrix} & & \\ & X & \\ & & \end{matrix}} \ar@<1ex>[rr]_{.} \ar@<-1ex>[rr]^{.}
\ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.} & &
{\begin{matrix} & & \\ & Y & \\ & & \end{matrix}}
\ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.} &
{\begin{matrix} & & \\ & \dots & \\ & & \end{matrix}}
}
\endxy$$ One complement of $X \amalg Y$ is clearly $Z$, which gives a hereditary endomorphism ring. The complement $Z^{\ast}$ is obtained from the triangle $Z \to B' \to Z^{\ast} \to$, where $Z \to B$ is a minimal left ${\operatorname{add}\nolimits}(X \amalg Y)$-approximation in ${\operatorname{\mathcal C}\nolimits}$. It is easy to see that this approximation is just the left almost split map $Z \to \amalg Y$, where $\amalg Y$ is a direct sum of copies of $Y$. This means that there are non-zero maps $Y \to Z^{\ast}$ in ${\operatorname{\mathcal C}\nolimits}$. Consider the triangle $Z^{\ast} \to B \to Z \to$, then it is clear that $B \to Z$ is the right almost split map $\amalg X \to Z$, where $\amalg X$ is a direct sum of copies of $X$, and so there are non-zero maps $Z^{\ast} \to X$ in ${\operatorname{\mathcal C}\nolimits}$. Thus there is a cycle in the quiver of ${\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(X \amalg Y \amalg Z^{\ast})^{{{\operatorname{op}}}}$, which gives a contradiction. This completes the proof for this case, and hence the proof of the lemma.
This has the following consequence.
\[cycles\] The quiver of a non-hereditary connected cluster-tilted algebra of rank 3 is of the form $$\xy
\xymatrix{1 \ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}_{r} & &
3 \ar@<1ex>[ll]_{.} \ar@<-1ex>[ll]^{.}_{t} \\
& 2 \ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.}_{s} &
}
\endxy$$ with $r,s,t >0 $.
Combine Lemma \[nofactors\] with Propostion \[noshortcycles\].
In view of this we refer to the cluster-tilted algebras of rank 3 which are non-hereditary as [*cyclic cluster-tilted algebras*]{}.
The relations
-------------
We first show that relations are homogeneous.
\[homogenous\] Let $\Gamma$ be a cluster-tilted algebra of rank 3 with Jacobson radical $\underline{r}$. Then $\underline{r}^6 = 0$, and the relations are homogeneous.
Without loss of generality we can assume that there is a tilting module $T = X \amalg Y \amalg Z$ for a hereditary algebra $H$, such that ${\Gamma}= {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}_H}(T)^{{{\operatorname{op}}}}$.
Using Corollary \[cycles\] it is clear that we can assume that the quiver of ${\Gamma}$ has the form $$\xy
\xymatrix{1 \ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}_{r} & &
3 \ar@<1ex>[ll]_{.} \ar@<-1ex>[ll]^{.}_{t} \\
& 2 \ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.}_{s} &
}
\endxy$$ with $r,s,t >0 $.
Let ${\Lambda}= {\operatorname{End}\nolimits}_{H}(T)^{{{\operatorname{op}}}}$ be the corresponding tilted algebra. There are no cycles in the quiver of a tilted algebra. We can therefore assume that there is a sink in the quiver of ${\Lambda}$, and we assume that this vertex corresponds to $Z$, that is, ${\operatorname{Hom}\nolimits}_H(Z,X \amalg Y) = 0$. We assume $X,Y,Z$ correspond to the vertices $1,2,3$, respectively. If $\widehat{h}$ is a non-zero map in ${\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T}(Z,X)$, it must be of degree $1$, that is, the lifting $h$ is in ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(Z,FX)$. Since this holds for all maps in ${\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T}(Z,X)$, any composition of $6$ arrows will correspond to a map of degree $\geq 2$ from an indecomposable to itself, and therefore must be the zero-map. This follows from the fact that for any indecomposable module $M$, we have ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(M,F^2 M) = 0$. This gives $\underline{r}^6 = 0$.
We can assume that at least one of the arrows (irreducible maps) $X \to Y$ and at least one of the arrows $Y \to Z$ are of degree 0. Otherwise, the tilted algebra ${\Lambda}$ would not be connected.
Now let $\widehat{g}$ be a map in ${\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T}(Y,Z)$. We want to show that it must be of degree 0. Since $X \amalg Y$ is an almost complete tilting object in ${\operatorname{\mathcal C}\nolimits}_H$, there are exactly two complements. Denote as usual the second one by $Z^{\ast}$. The complement $Z^{\ast}$ is either the image of a module or the image of an object of the form $I[-1]$ for an injective indecomposable module $I$. Furthermore, there is a triangle in ${\operatorname{\mathcal C}\nolimits}$ $$\label{exchange}
Z^{\ast} \to Y^r \to Z \to ,$$ for some $r \geq 0$, which can be lifted to a triangle $$F^i Z^{\ast} \overset{\big( \begin{smallmatrix}
\alpha_1 \\ \alpha_2 \end{smallmatrix} \big) } \to
Y^{r_1} \amalg (F^{-1}Y)^{r_2} \to Z \to$$ in ${\operatorname{\mathcal D}\nolimits}$ for some integer $i$ and with $r = r_1 + r_2$. We need to show that $r_2 = 0$. It is sufficient to show that the map $\alpha_2= 0$. We have $r_1 \neq 0$, and thus by minimality $\alpha_1 \neq 0$. It is clear that if also $\alpha_2 \neq 0$, then $i=0$ or $i= -1$.
Assume first $Z^{\ast} \simeq I[-1]$, then $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(I[-1],F^{-1}Y) = {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(I, \tau Y) = 0,$$ so $i=0$ gives $\alpha_2 = 0$. On the other hand, it is clear that $i = -1$ gives $\alpha_1 = 0$.
Assume now that $Z^{\ast}$ is the image of a module. Then there is an exact sequence of modules $$0 \to Z^{\ast} \to Y^s \to Z \to 0,$$ and since $\dim_k {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(Z,Z^{\ast}[1]) =1$ (by [@bmrrt]), it follows that the triangle (\[exchange\]) is induced by this sequence, and thus $r_1 = s$ and $r_2 = 0$.
Now we show that all the irreducible maps $X \to Y$ in ${\operatorname{\mathcal C}\nolimits}_H$ are of degree 0. For this, consider the almost complete tilting object $X \amalg Z$ in $C_H$, with complements $Y$ and $Y^{\ast}$. Consider the triangle $$Y^{\ast} \to X^t \to Y \to ,$$ and the preimage in ${\operatorname{\mathcal D}\nolimits}$, $$F^i Y^{\ast} \overset{\big( \begin{smallmatrix}
\beta_1 \\ \beta_2 \end{smallmatrix} \big) } \to X^{t_1} \amalg (F^{-1}X)^{t_2} \to Y \to .$$
We need to show that $t_2 = 0$. The case where $Y^{\ast} \simeq I[-1]$ is completely similar as for irreducible maps $Y \to Z$. In case $Y^{\ast}$ is the image of a module, it is now more complicated since we have two possibilities. Either there is an exact sequence in ${\operatorname{mod}\nolimits}H$ of the form $$0 \to Y^{\ast} \to X^u \to Y \to 0,$$ or there is an exact sequence of the form $$0 \to Y \to Z^v \to Y^{\ast} \to 0.$$ If we are in the first case, we can use the same argument as for irreducible maps $Y \to Z$. If we are in the second case, note that ${\operatorname{Hom}\nolimits}_H(Y^{\ast},X) = 0$, since ${\operatorname{Hom}\nolimits}_H(Z,X) = 0$. Thus, either $\beta_1= 0$ or $\beta_2= 0$ in our triangle. This completes the proof that all irreducible maps $X \to Y$ are induced by module maps, and thus are of degree 0.
Given that $\underline{r}^6 = 0$, the only possibility for a non-homogeneous relation must involve maps in ${\operatorname{\underline{r}}\nolimits}^2 \setminus {\operatorname{\underline{r}}\nolimits}^3$ and maps in ${\operatorname{\underline{r}}\nolimits}^5$. But, by our description of irreducible maps, this is not possible, because it would involve a relation between maps of different degrees.
Fix a cyclic cluster-tilted algebra of rank 3, and fix a vertex $k$. Let $\alpha$ be an arrow ending in $k$, and $\beta$ an arrow starting in $k$. If $\beta \alpha = 0$, as an element of the algebra, for any choice of $\alpha$ and $\beta$, then we call $k$ a [*zero vertex*]{}.
\[zero\] Let ${\Gamma}$ be a cyclic cluster-tilted algebra, and fix a vertex $k$. Then $k$ is a zero-vertex if and only if $\overline{\delta}_k({\Gamma})$ is hereditary.
We assume the quiver of ${\Gamma}$ is $$\xy
\xymatrix{1 \ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}_{r} & &
3 \ar@<1ex>[ll]_{.} \ar@<-1ex>[ll]^{.}_{t} \\
& 2 \ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.}_{s} &
}
\endxy$$ Let ${\Gamma}= {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T)^{{{\operatorname{op}}}}$, and let $T_i$ be the direct summand of $T$ corresponding to the vertex $i$. Assume that $2$ is a zero-vertex. Then it is clear that ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T_1, T_3 ) = 0$, so the quiver of $\overline{\delta}_2({\Gamma})$ must be $$\xy
\xymatrix{1 & & 3 \ar@<0.5ex>[dl]^{.} \ar@<2.5ex>[dl]_{.}^{r} \ar@<-1ex>[ll]^{.}_{t'} \ar@<1ex>[ll]_{.} \\
& 2^{\ast} \ar@<0.5ex>[ul]^{.} \ar@<2.5ex>[ul]_{.}^{s} &
}
\endxy$$ with $t' \geq 0$. Now $\overline{\delta}_2({\Gamma})$ is hereditary, by Lemma \[nofactors\].
Conversely, assume $\overline{\delta}_k({\Gamma})$ is hereditary. The quiver of $\overline{\delta}_k({\Gamma})$ must be as above, with $t' \geq 0$. This means ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T_1, T_3 ) = 0$, so $2$ is a zero-vertex.
Kerner’s Theorem
----------------
The following result by Kerner [@ker] turns out to be crucial for the proof of the main theorem of this section. There is a more general version of this theorem in [@ker]. We include a proof, for the convenience of the reader. This proof is also due to Kerner, and we thank him for providing us with it.
Let $X,Y$ be regular indecomposable modules over a wild hereditary algebra $H$ of rank 3. If ${\operatorname{Hom}\nolimits}_H(X, \tau Y) = 0$, then also ${\operatorname{Hom}\nolimits}_H(X, \tau^{-1} Y) = 0$.
We first prove the following.
\[special\] Let $U$ be an indecomposable regular module over a wild hereditary algebra of rank 3. Then ${\operatorname{Hom}\nolimits}_H(U, \tau^2 U) \neq 0$.
Assume first ${\operatorname{Ext}\nolimits}^1_H(U,U) \neq 0$. By the AR-formula, then also ${\operatorname{Hom}\nolimits}_H(U, \tau U) \neq 0$. Assume now ${\operatorname{Hom}\nolimits}_H(U, \tau^2 U) = 0$. Then also ${\operatorname{Ext}\nolimits}^1_H(\tau U, U)= 0$ and, by the Happel-Ringel lemma [@hr], a non-zero map $f \colon U \to \tau U$ is either surjective or injective. In either case, $g = \tau (f) \circ f \colon U \to \tau^2 U$ is non-zero. This contradicts ${\operatorname{Hom}\nolimits}_H(U, \tau^2 U) = 0$.
Now assume ${\operatorname{Ext}\nolimits}^1_H(U,U) = 0$. Then by [@Hos], $U$ is quasi-simple. Thus, there is an almost split sequence $0 \to \tau U \to V \to U \to 0$, where $V$ is indecomposable, and by [@ker2] we have ${\operatorname{End}\nolimits}_H(V) \simeq K$, while ${\operatorname{Ext}\nolimits}^1_H(V,V) \neq 0$. Applying ${\operatorname{Hom}\nolimits}_H(U,\ )$ to the almost split sequence, we obtain the exact sequence $${\operatorname{Hom}\nolimits}_H(U, \tau U) \to {\operatorname{Hom}\nolimits}_H(U,V) \to {\operatorname{Hom}\nolimits}_H(U, U) \to {\operatorname{Ext}\nolimits}^1_H(U,\tau U)$$ Since ${\operatorname{Hom}\nolimits}_H(U,U) \to {\operatorname{Ext}\nolimits}^1_H(U,\tau U)$ is an isomorphism and ${\operatorname{Hom}\nolimits}_H(U, \tau U) = 0$, we have that also ${\operatorname{Hom}\nolimits}_H(U,V) = 0$. The long exact sequence obtained by applying ${\operatorname{Hom}\nolimits}_H(\ , \tau U)$ to the almost split sequence, gives ${\operatorname{Hom}\nolimits}_H(V, \tau U) = 0$. Now, this gives ${\operatorname{Hom}\nolimits}_H(V, \tau^2 U) \neq 0$, since there is an exact sequence $$0 \to {\operatorname{Hom}\nolimits}_H(V, \tau^2 U) \to {\operatorname{Hom}\nolimits}_H(V, \tau V) \to {\operatorname{Hom}\nolimits}_H(V, \tau U)$$ and the last term is zero. There is also the long exact sequence $$0 \to {\operatorname{Hom}\nolimits}_H(U, \tau^2 U) \to {\operatorname{Hom}\nolimits}_H(V, \tau^2 U) \to {\operatorname{Hom}\nolimits}_H(\tau U, \tau^2 U)$$ where the last term is zero. This proves ${\operatorname{Hom}\nolimits}_H(U, \tau^2 U) \neq 0$.
Let us now complete the proof of the theorem. Let $X,Y$ be regular indecomposable modules. It suffices to show that ${\operatorname{Hom}\nolimits}_H(X,Y) \neq 0$ implies ${\operatorname{Hom}\nolimits}_H(X, \tau^2 Y) \neq 0$. Let $z \colon X \to Y$ be a non-zero map. Then we can assume there is an indecomposable regular module $U$, such that $z$ factors as $X \overset{p}{\to} U \overset{i}{\to} Y$, where $p$ is surjective and $i$ is injective. Also $\tau^2 i \colon \tau^2 U \to \tau^2 Y$ is injective. By Lemma \[special\], there is a non-zero map $f \colon U \to \tau^2 U$. The composition $\tau^2 i \circ f \circ p$ is non-zero. This completes the proof of the theorem.
The dimensions of relation-spaces
---------------------------------
Let $H$ be a connected hereditary algebra of rank 3. The following notation is used for the rest of this section. Let $\bar{T}$ be an almost complete tilting object with complements $M$ and $M^{\ast}$, and assume there are triangles as in Theorem \[triangles\]. Let $T = \bar{T} \amalg M$ and $T' = \bar{T} \amalg M^{\ast}$ and let ${\Gamma}= {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T)^{{{\operatorname{op}}}}$ and ${\Gamma}' = {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T')^{{{\operatorname{op}}}}$. By now, we know that the quiver of ${\Gamma}$ is either $$\xy
\xymatrix{1 \ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}_{r} & &
3 \ar@<1ex>[ll]_{.} \ar@<-1ex>[ll]^{.}_{t} \\
& 2 \ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.}_{s} &
}
\endxy$$ with $r,s,t >0$ or the quiver $Q_{rst}$ $$\xy
\xymatrix{1 \ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}_{r} \ar@<1ex>[rr]_{.}^{t} \ar@<-1ex>[rr]^{.} & &3 \\
& 2 \ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.}_{s} &
}
\endxy$$ with $r,s > 0$ and $t \geq 0$. We let $M$ correspond to vertex $2$. Then $\bar{T} = T_{B} \amalg T_{B'}$ where $T_B$ corresponds to the vertex $1$ and $T_{B'}$ to $3$. It is then clear that $B = (T_B)^r$ and $B' = (T_{B'})^s$. We label the vertices with the corresponding modules, then the arrows represent irreducible maps in ${\operatorname{add}\nolimits}T$.
We let $I$ denote the ideal such that ${\Gamma}\simeq KQ/I$. In case ${\Gamma}$ is cyclic, we say that ${\Gamma}$ is [*balanced*]{} at the vertex $2$ if $$\dim(({\operatorname{Irr}\nolimits}(T_B,M) \otimes {\operatorname{Irr}\nolimits}(M,T_{B'}) \cap I) = t.$$ We will show that any vertex of a cyclic cluster-tilted algebra is either balanced or a zero-vertex. We first discuss the algebras obtained by mutating hereditary algebras.
\[mutatehereditary\] Let $H$ be a hereditary algebra with quiver $Q_{rst}$ where $r,s>0$ and $t \geq 0$. Then the following hold.
- [The cluster-tilted algebra ${\Gamma}' = \overline{\delta}_2(H)$ is balanced at at the vertices $1$ and $3$.]{}
- [The new vertex $2^{\ast}$ is a zero-vertex.]{}
- [The quiver of ${\Gamma}'$ is $$\xy
\xymatrix{1 \ar@<1ex>[rr]_{.}^{t + rs} \ar@<-1ex>[rr]^{.} & & 3
\ar@<0.5ex>[dl]^{.} \ar@<2.5ex>[dl]_{.}^{s} \\
& 2^{\ast} \ar@<0.5ex>[ul]^{.} \ar@<2.5ex>[ul]_{.}^{r} &
}
\endxy$$ ]{}
Part (b) and (c) follow directly from Lemma \[nofactors\] and Proposition \[zero\]. Let $P_i$ be the indecomposable projective $H$-module corresponding to vertex $i$, and $S_i$ the simple $H$-module $P_i/{\operatorname{\underline{r}}\nolimits}P_i$. Then $P_3 = S_3$ is simple. Consider $P_1 \amalg P_3$ as an almost complete tilting object. There is an exact sequence $$0 \to P_2 \to (P_1)^s \to P_2^{\ast} \to 0,$$ such that the induced triangle in ${\operatorname{\mathcal C}\nolimits}$ is the exchange-triangle of Theorem \[triangles\]. Let $T' = P_1 \amalg P_2^{\ast}
\amalg P_3$. Using the definition of $\tau$, one can show that $S_2 = \tau P_2^{\ast}$, and thus ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(P_2^{\ast}, P_1)= 0$. Since $2^{\ast}$ is a zero-vertex, ${\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(P_3, P_1) = {\operatorname{Hom}\nolimits}_H(P_3, P_1)$, with dimension $rs +t$. We want to compute ${\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(P_2^{\ast}, P_3) \otimes {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(P_3, P_1)
= {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(P_2^{\ast},P_1) \simeq {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(F^{-1}P_2^{\ast},P_1)$. We have ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(F^{-1}P_2^{\ast},P_1) = {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(\tau P_2^{\ast} [-1], P_1) = {\operatorname{Ext}\nolimits}_H^1(S_2,P_1)$. There is an exact sequence $$0 \to (P_3)^r \to P_2 \to S_2 \to 0.$$ Apply ${\operatorname{Hom}\nolimits}_H(\ ,P_1)$ to it, to obtain the long exact sequence $$0 \to {\operatorname{Hom}\nolimits}_H(S_2, P_1) \to {\operatorname{Hom}\nolimits}_H(P_2, P_1) \to {\operatorname{Hom}\nolimits}_H((P_3)^r,P_1) \to {\operatorname{Ext}\nolimits}^1_H(S_2,P_1) \to 0.$$ Since $\dim {\operatorname{Hom}\nolimits}_H(P_2, P_1) = s$, and $\dim {\operatorname{Hom}\nolimits}_H((P_3)^r,P_1) = (rs +t)r$, we have $\dim ({\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(P_2^{\ast}, P_3) \otimes {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(P_3, P_1)) =
r(rs+t)-s$, thus $\dim ({\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(P_2^{\ast}, P_3) \otimes {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(P_3, P_1) \cap I) =s$, and ${\Gamma}'$ is balanced at $3$.
Now apply ${\operatorname{Hom}\nolimits}_H(P_3, \ )$ to the exact sequence $ 0 \to P_2 \to (P_1)^s \to P_2^{\ast} \to 0$ to obtain the exact sequence $$0 \to {\operatorname{Hom}\nolimits}_H(P_3,P_2) \to {\operatorname{Hom}\nolimits}_H(P_3,P_1^{s}) \to {\operatorname{Hom}\nolimits}_H(P_3,P_2^{\ast}) \to 0.$$ Since $\dim {\operatorname{Hom}\nolimits}_H(P_3,P_2) = r$ and $\dim {\operatorname{Hom}\nolimits}_H(P_3,P_1^{s}) = (rs +t)s$, we have $$\dim ({\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(P_3,P_1) \otimes {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(P_1, P_2^{\ast})) = \dim {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(P_3,P_2^{\ast}) =
(rs + t)s - r.$$ This means $\dim ({\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(P_3,P_1) \otimes {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(P_1, P_2^{\ast}) \cap I) = r$, and ${\Gamma}'$ is balanced also at $1$.
\[mutatecyclic\] Let ${\Gamma}$ be a non-hereditary cluster-tilted algebra with quiver $$\xy
\xymatrix{1 \ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}_{r} & &
3 \ar@<1ex>[ll]_{.} \ar@<-1ex>[ll]^{.}_{t} \\
& 2 \ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.}_{s} &
}
\endxy$$
- [If ${\Gamma}$ is balanced at the vertex $2$, then ${\Gamma}' = \overline{\delta}_2({\Gamma})$ is non-hereditary, and thus cyclic, with quiver $$\xy
\xymatrix{1 \ar@<1ex>[rr]_{.}^{rs-t} \ar@<-1ex>[rr]^{.} & & 3
\ar@<0.5ex>[dl]^{.} \ar@<2.5ex>[dl]_{.}^{s} \\
& 2^{\ast} \ar@<0.5ex>[ul]^{.} \ar@<2.5ex>[ul]_{.}^{r} &
}
\endxy$$ It is balanced at the new vertex $2^{\ast}$. Each of the other vertices of ${\Gamma}'$ is either balanced or a zero-vertex.]{}
- [If ${\Gamma}$ has a zero-vertex at $2$, then $\overline{\delta}_2({\Gamma})$ is hereditary with quiver $$\xy
\xymatrix{1 & & 3 \ar@<0.5ex>[dl]^{.} \ar@<2.5ex>[dl]_{.}^{r} \ar@<-1ex>[ll]^{.}_{t -rs} \ar@<1ex>[ll]_{.} \\
& 2^{\ast} \ar@<0.5ex>[ul]^{.} \ar@<2.5ex>[ul]_{.}^{s} &
}
\endxy$$ ]{}
Part (b) follows from Propositions \[homogenous\] and \[zero\].
To prove part (a), we adopt our earlier notation and conventions. Especially, ${\Gamma}= {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T_B \amalg T_{B'} \amalg M)^{{{\operatorname{op}}}}$, and we have the quiver $$\xy
\xymatrix{T_B \ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}_{r} & &
T_{B'} \ar@<1ex>[ll]_{.} \ar@<-1ex>[ll]^{.}_{t} \\
& M \ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.}_{s} &
}
\endxy$$ The quiver of the mutated algebra $\overline{\delta}_2({\Gamma}) = {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T_B \amalg T_{B'} \amalg M^{\ast})^{{{\operatorname{op}}}}$ is $$\xy
\xymatrix{T_B \ar@<1ex>[rr]_{.}^{t'} \ar@<-1ex>[rr]^{.} & & T_{B'}
\ar@<0.5ex>[dl]^{.} \ar@<2.5ex>[dl]_{.}^{s} \\
& M^{\ast} \ar@<0.5ex>[ul]^{.} \ar@<2.5ex>[ul]_{.}^{r} &
}
\endxy$$ Using that ${\Gamma}$ is balanced at $2$, and Proposition \[homogenous\], it follows that $t' = rs -t$. Also by assumption, $M$ does not correspond to a zero-vertex, so there is at least one non-zero composition $T_B \to M \to T_{B'}$. Therefore $rs -t > 0$.
We have $\dim ({\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(T_{B'},M^{\ast}) \otimes {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(M^{\ast}, T_{B})) = \dim
{\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T}(T_{B'},T_B) + \dim ({\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T}(T_{B'},M) \otimes {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T}(M, T_{B}))= t+ 0$. Therefore $\dim ({\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(T_{B'},M^{\ast}) \otimes {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(M^{\ast}, T_{B}) \cap I) = rs-t$, so ${\Gamma}'$ is balanced at $2^{\ast}$.
We now proceed to show that for each of the vertices 1 and 3, ${\Gamma}'$ is either balanced, or a zero-vertex. We assume $T_{B'}$ is not a zero-vertex in ${\Gamma}'$.
The tilted algebra ${\Lambda}= {\operatorname{End}\nolimits}_H(T_B \amalg T_{B'} \amalg M)^{{{\operatorname{op}}}}$ has a unique sink. There is an induced total ordering on the triple $T_B, T_{B'}, M$, where the last element in the ordering corresponds to the sink. Also, by considering the preimage of $M^{\ast}$ in the standard domain of ${\operatorname{\mathcal D}\nolimits}$, the ordering can be extended to the quadruple $B'$, $M^{\ast}$, $B$, $M$. Note that we get the following four possible orderings
- [$(M, T_{B'}, M^{\ast}, T_B)$]{}
- [$(T_{B'}, M^{\ast}, T_B, M)$]{}
- [$(M^{\ast}, T_B, M, T_{B'})$]{}
- [$(T_B, M, T_{B'}, M^{\ast})$.]{}
First we show the claim for the vertex corresponding to $T_{B'}$.
\[magic\] Assume that $T_{B'}$ does not correspond to a zero-vertex in ${\Gamma}'$ and that $M^{\ast}$ is before $T_B$ in the above ordering. Then ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_B, \tau^{-1}M^{\ast})=0$.
Since $\overline{\delta}_2({\Gamma})$ is not hereditary, we have ${\operatorname{Hom}\nolimits}_H(M^{\ast},T_B) \neq 0$. Assume now that ${\operatorname{Hom}\nolimits}_H(T_B,\tau^{-1}M^{\ast}) \neq 0$. Assume first that $T_B$ is regular, then $M^{\ast}$ is also regular. In case $H$ is tame, then there are at most two exceptional modules which are regular. This follows from the fact that $H$ has three simples. But in case there are two exceptional modules which are regular, there is an extension between them. This gives a contradiction. In case $H$ is wild we can apply Kerner’s Theorem, which says that ${\operatorname{Hom}\nolimits}_H(T_B, \tau M^{\ast}) \neq 0$. We have a contradiction, since ${\operatorname{Hom}\nolimits}_H(T_B, \tau M^{\ast}) \simeq D{\operatorname{Ext}\nolimits}^1_H(M^{\ast},T_B)= 0$.
If $B$ is a preprojective or a preinjective module, then ${\operatorname{Hom}\nolimits}_H(M^{\ast},T_B) \neq 0$ and ${\operatorname{Hom}\nolimits}_H(T_B, \tau^{-1}M^{\ast}) \neq 0$ implies that the map $M^{\ast} \to T_B$ is irreducible in the module-category. Thus $M^{\ast} \amalg T_B$ can be complemented to a tilting module with hereditary endomorphism ring. We have seen that the mutated algebra $\overline{\delta}_2({\Gamma})$ is by assumption not hereditary. This means that $T_{B'}$ must correspond to a zero-vertex, so we have a contradiction to ${\operatorname{Hom}\nolimits}_H(B,\tau^{-1}M^{\ast}) \neq 0$ also for $T_B$ being preprojective or preinjective.
Now, let $M \to B' \to M^{\ast} \to $ be the usual triangle. We recall that $\bar{T}= T_B \amalg T_{B'}$. Let $\widetilde{{\operatorname{Hom}\nolimits}}(T_B,M) = {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T}(T_B,M)$, let $\widetilde{{\operatorname{Hom}\nolimits}}(T_B,B') = {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}\bar{T}}(T_B,B')$ and let $\widetilde{{\operatorname{Hom}\nolimits}}(T_B,M^{\ast}) = {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(T_B,M^{\ast})$. Then we claim that there is an exact sequence $$\label{tildes}
0 \to \widetilde{{\operatorname{Hom}\nolimits}}(T_B,M) \overset{\alpha}\to \widetilde{{\operatorname{Hom}\nolimits}}(T_B,B') \to
\widetilde{{\operatorname{Hom}\nolimits}}(T_B,M^{\ast}) \to 0.$$ It is clear from Proposition \[homogenous\], and the fact that $${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_B,M) \to {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_B,B') \to {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_B,M^{\ast}) \to 0$$ is exact, that we only need to show that the map $\alpha$ is a monomorphism. We first assume $M^{\ast}$ is a module. To prove the claim for this case, we consider the four orderings on the quadruple $\{M,B',M^{\ast},B\}$. For each case we show that a map in ${\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T}(T_B,M)$ cannot factor via $M^{\ast}[-1]$ in ${\operatorname{\mathcal C}\nolimits}$.\
$(M,T_{B'},M^{\ast},T_B)\colon$ In this case, a map in ${\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T}(T_B,M)$ is of degree $1$. Assume the lifting is $f \colon \tau T_B[-1] \to M$. There is a non-split exact sequence $0 \to M \to B' \to M^{\ast} \to 0$. We have $\dim {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(M^{\ast},M[1])= 1$, by [@bmrrt], and therefore ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(FM^{\ast},M[1]) =
{\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(\tau^{-1}M^{\ast},M) = 0$. Therefore, if $f\colon T_B \to M$ factors through $M^{\ast}[-1]$ in ${\operatorname{\mathcal C}\nolimits}$, there must be a map $g \colon \tau T_B[-1] \to M$ in ${\operatorname{\mathcal D}\nolimits}$, such that there is a commutative diagram $$\xy
\xymatrix{
& \tau T_B[-1] \ar[d]_{f} \ar[dl]_{g} \\
M^{\ast}[-1] \ar[r] & M
}
\endxy$$ By Lemma \[magic\], we have that ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(\tau T_B,M^{\ast})= {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_B, \tau^{-1}M^{\ast})= 0$, and thus we obtain $f=0$.\
$(T_{B'}, M^{\ast}, T_B, M)$ or $(M^{\ast}, T_B, M, T_{B'}) \colon$ In these cases, a map in ${\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T}(T_B,M)$ is of degree $0$. Assume the lifting of it is $f \colon T_B \to M$. The preimage of $M^{\ast}$ in ${\operatorname{\mathcal D}\nolimits}$ is a module in these cases, so a factorization of $f$ must be of the form $$\xy
\xymatrix{
& T_B \ar[d]_{f} \ar[dl]_{g} \\
\tau^{-1}M^{\ast} \ar[r] & M
}
\endxy$$ Lemma \[magic\] gives $f=0$.\
$(T_B, M, T_{B'}, M^{\ast})\colon$ In this case the preimage of $M^{\ast}$ in ${\operatorname{\mathcal D}\nolimits}$ is either a module or $P[1]$, for an indecomposable projective $H$-module $P$. In both cases, a map in ${\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T}(T_B,M)$ is a map of degree $0$. Assume the lifting is $f\colon T_B \to M$. In the first case there is a non-split exact sequence $0 \to M \to B' \to M^{\ast} \to 0$. Therefore, since $\dim {\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(M^{\ast}, M[1]) = 1$, we have ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(\tau^{-1}M^{\ast},M) = 0$. Since ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_B, M^{\ast}[-1])=0$, we must have $f=0$, if $f$ factors as below $$\xy
\xymatrix{
& T_B \ar[d]_{f} \ar[dl]_{g} \\
M^{\ast}[-1] \ar[r] & M.
}
\endxy$$ Assume $M^{\ast} \simeq P[1]$, with $P$ projective. If ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_B,P) \neq 0$, then $T_B$ is also projective. Therefore ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_B, P[1]) = 0$, and thus ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T_B, M^{\ast})= 0$, which means that $T_{B'}$ is a zero-vertex in ${\Gamma}'$, a contradiction. Then ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(T_B,P) = 0$, but this means that $f \colon T_B \to M$ factors through $M^{\ast}[-1] \simeq P$ only for $f =0$. Thus, the map $\alpha$ is a monomorphism, and the sequence (\[tildes\]) is exact.
Thus, $\dim \widetilde{{\operatorname{Hom}\nolimits}}(T_B,M^{\ast}) = \dim \widetilde{{\operatorname{Hom}\nolimits}}(T_B,B')- \dim \widetilde{{\operatorname{Hom}\nolimits}}(T_B,M)=
t' s - r$. This means that $\dim ({\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(T_B,T_{B'}) \otimes {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(T_{B'}, M^{\ast}) \cap I ) = r$, so ${\Gamma}'$ is balanced at the vertex $3$, corresponding to $T_{B'}$.
We now show that ${\Gamma}'$ is balanced at the vertex $1$, corresponding to $T_B$, or $1$ is a zero-vertex. Assume it is not a zero-vertex. We have the dual version of Lemma \[magic\].
Assume that $T_{B}$ does not correspond to a zero-vertex and that $M^{\ast}$ is before $T_B$ in the above ordering. Then ${\operatorname{Hom}\nolimits}_{{\operatorname{\mathcal D}\nolimits}}(M^{\ast}, \tau T_{B'})=0$.
Similar to the proof of Lemma \[magic\].
Now, consider the triangle $$M^{\ast} \to B \to M \to.$$ We need to show that there is an exact sequence $$\label{tildes2}
0 \to \widetilde{{\operatorname{Hom}\nolimits}}(M,T_{B'}) \to \widetilde{{\operatorname{Hom}\nolimits}}(B,T_{B'}) \to
\widetilde{{\operatorname{Hom}\nolimits}}(M^{\ast},T_{B'}) \to 0,$$ where $\widetilde{{\operatorname{Hom}\nolimits}}(M,T_{B'})= {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T}(M,T_{B'})$, while $\widetilde{{\operatorname{Hom}\nolimits}}(B,T_{B'})=
{\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}\bar{T}}(B,T_{B'})$ and $\widetilde{{\operatorname{Hom}\nolimits}}(M^{\ast},T_{B'}) = {\operatorname{Irr}\nolimits}_{{\operatorname{add}\nolimits}T'}(M^{\ast},T_{B'})$. The proof of this is parallel to the proof for the sequence (\[tildes\]), and therefore omitted. Using the exact sequence (\[tildes2\]), one obtains that ${\Gamma}'$ is balanced at the vertex $1$.
We summarize the results of this section.
\[sum\] Let ${\Gamma}$ be a cluster-tilted algebra of rank 3.
- [${\Gamma}$ is either hereditary, or it is cyclic.]{}
- [ If ${\Gamma}$ is cyclic, then each vertex of ${\Gamma}$ is either balanced or a zero-vertex.]{}
- Let $k$ be a vertex of ${\Gamma}$, let $\overline{\delta}_k({\Gamma})$ be the mutation in direction $k$, and let $k^{\ast}$ be the new vertex of $\overline{\delta}_k({\Gamma})$. Then there are the following possible cases:
- [Both ${\Gamma}$ and $\overline{\delta}_k({\Gamma})$ are hereditary.]{}
- [${\Gamma}$ is hereditary, while $\overline{\delta}_k({\Gamma})$ is cyclic with a zero-vertex at $k^{\ast}$,]{}
- [${\Gamma}$ is cyclic with a zero-vertex at $k$, and $\overline{\delta}_k({\Gamma})$ is hereditary, or]{}
- [${\Gamma}$ is cyclic and balanced at $k$, and $\overline{\delta}_k({\Gamma})$ is cyclic and balanced at $k^{\ast}$]{}.
This follows directly from the previous results in this section, and the fact that all cluster-tilted algebras can be obtained by starting with a hereditary algebra, and then performing a finite number of mutations [@bmrrt], [@bmr].
The above Theorem is very easily verified for algebras of finite type, as indicated in the introduction of this section.
Mutation
========
As mentioned in the introduction, in view of Proposition \[noshortcycles\] it is possible to assign to a cluster-tilted algebra ${\Gamma}$ a skew-symmetric matrix $X_{{\Gamma}} = (x_{ij})$. More precisely, if there is at least one arrow from $i$ to $j$ in the quiver of the endomorphism-algebra ${\Gamma}$, let $x_{ij}$ be the number of arrows from $i$ to $j$. If there are no arrows between $i$ and $j$, let $x_{ij} = 0$. Let $x_{ij} = - x_{ji}$ otherwise.
Now let $\bar{T}$ be an almost complete tilting object with complements $M$ and $M^{\ast}$. Let $T = \bar{T} \amalg M$, let $T' = \bar{T} \amalg M^{\ast}$, let ${\Gamma}= {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T)^{{{\operatorname{op}}}}$ and let ${\Gamma}' = {\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T')^{{{\operatorname{op}}}}$. Then we want to show that the quivers of ${\Gamma}$ and ${\Gamma}'$ are related by the cluster-mutation formula. We use the results of Section \[rankthree\] to show this for cluster-tilted algebras of rank 3, and Theorem \[factor\] to extend to the general case.
\[mutate\] Let $H$ be a hereditary algebra, and let $\bar{T},M , M^{\ast}, {\Gamma}$ and ${\Gamma}'$ be as above. Then the quivers of ${\Gamma}$ and ${\Gamma}'$, or equivalently the matrices $X_{{\Gamma}}$ and $X_{{\Gamma}'}$, are related by the cluster mutation formula.
First, assume $H$ has rank 3. In case $H$ is not connected, the claim is easily checked. Assume $H$ is connected. Fix $k$, the vertex where we mutate. By Theorem \[triangles\], it is clear that $x_{ik}' = - x_{ik}$ for $i= 1,2,3$, and that $x_{kj}' = - x_{kj}$ for $j= 1,2,3$.
Now assume $i \neq k$ and $j \neq k$. By Theorem \[sum\], there are four possible cases.\
Case I: This happens if and only if $k$ is a source or a sink. In this case it is clear that either $x_{ik}= 0$ or $x_{kj}= 0$. For $i \neq k$ and $j \neq k$, it is clear that $x_{ij} = x_{ij}'$, since in this case mutation at $k$ is the same as so-called APR-tilting at $k$. Thus the formula holds.\
Case II: Since $k$ is now not a source or a sink, we can assume ${\Gamma}$ is the path algebra of $$\xy
\xymatrix{1 \ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}_{r} \ar@<1ex>[rr]_{.}^{t} \ar@<-1ex>[rr]^{.} & &3 \\
& 2 \ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.}_{s} &
}
\endxy$$ where $r>0$ and $s > 0$ and $t \geq 0$ and with $k=2$. Then, by Lemma \[mutatehereditary\], the quiver of $\overline{\delta}_2({\Gamma})$ is $$\xy
\xymatrix{1 \ar@<1ex>[rr]_{.}^{t'} \ar@<-1ex>[rr]^{.} & & 3
\ar@<0.5ex>[dl]^{.} \ar@<2.5ex>[dl]_{.}^{s} \\
& 2 \ar@<0.5ex>[ul]^{.} \ar@<2.5ex>[ul]_{.}^{r} &
}
\endxy$$ with $t' = r s + t$. So $x_{13}' = t' = rs +t$, and $$x_{13} + \frac{{\lvertx_{12}\rvert}x_{23} + x_{12} {\lvertx_{23}\rvert}}{2} = t + \frac{rs + rs}{2} = t + rs.$$
Case III: We assume that the quiver of ${\Gamma}$ is $$\xy
\xymatrix{1 \ar@<1ex>[rr]_{.}^{t} \ar@<-1ex>[rr]^{.} & & 3
\ar@<0.5ex>[dl]^{.} \ar@<2.5ex>[dl]_{.}^{s} \\
& 2 \ar@<0.5ex>[ul]^{.} \ar@<2.5ex>[ul]_{.}^{r} &
}
\endxy$$ By Proposition \[mutatecyclic\], the quiver of ${\Gamma}'$ is $$\xy
\xymatrix{1 \ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}_{r} \ar@<1ex>[rr]_{.}^{t'} \ar@<-1ex>[rr]^{.} & &3 \\
& 2 \ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.}_{s} &
}
\endxy$$ with $t' = t-rs$. That is $x_{13}' = t-rs$, and $$x_{13} + \frac{{\lvertx_{12}\rvert}x_{23} + x_{12} {\lvertx_{23}\rvert}}{2} =
t + \frac{{\lvert-r\rvert} (-s) + (-r) {\lvert-s\rvert}}{2} = t-rs,$$ and the formula holds.
Case IV: We assume the quiver of ${\Gamma}$ is the same as in case III. Now the quiver of ${\Gamma}'$ is $$\xy
\xymatrix{1 \ar@<-0.5ex>[dr]_{.} \ar@<-2.5ex>[dr]^{.}_{r} & &
3 \ar@<1ex>[ll]_{.} \ar@<-1ex>[ll]^{.}_{t'} \\
& 2 \ar@<-0.5ex>[ur]_{.} \ar@<-2.5ex>[ur]^{.}_{s} &
}
\endxy$$ where $t' = rs-t$. That is $x_{13}'= -t' = t-rs$, while $$x_{13} + \frac{{\lvertx_{12}\rvert}x_{23}
+ x_{12} {\lvertx_{23}\rvert}}{2} =
t + \frac{{\lvert-r\rvert} (-s) + (-r) {\lvert-s\rvert}}{2} = t-rs,$$ thus the formula holds true also in this case.
Now, assume that $H$ has arbitrary rank. Fix $k$, the vertex where we mutate. By Theorem \[triangles\], it is clear that $x_{ik}' = - x_{ik}$ for any value of $i$, and that $x_{kj}' = - x_{kj}$ for any value of $j$. Assume now that $k \neq i$ and $k \neq j$. Let $e_i, e_j, e_k$ be the primitive idempotents in ${\Gamma}$ corresponding to the vertices $i,j,k$ of the quiver of ${\Gamma}$. Assume $1_{{\Gamma}}= f +e_i +e_j +e_k$. and let ${\Gamma}_{{\operatorname{red}\nolimits}} = {\Gamma}/ {\Gamma}f {\Gamma}$. Let $e_i, e_j, e_k^{\ast}$ be the primitive idempotents corresponding to the vertices $i,j,k^{\ast}$ of the quiver of ${\Gamma}'$. Assume $1_{{\Gamma}'}= f' +e_i +e_j +e_{k^{\ast}}$. and let ${\Gamma}'_{{\operatorname{red}\nolimits}} = {\Gamma}' / {\Gamma}' f' {\Gamma}'$. It is clear that the number of arrows from $i$ to $j$ in the quiver of ${\Gamma}_{{\operatorname{red}\nolimits}}$ is $x_{ij}$ and the number of arrows from $i$ to $j$ in the quiver of ${\Gamma}'_{{\operatorname{red}\nolimits}}$ is $x_{ij}'$. So, by the first part of the proof, $x_{ij}$ and $x_{ij}'$ are related by the matrix mutation formula.
Connections to cluster algebras
===============================
Our main motivation for studying matrix mutation for quivers/matrices associated with tilting objects in cluster categories is the connection to cluster algebras. In this section we explain how Theorem \[mutate\] gives such a connection. In order to formulate our result we first need to give a short introduction to a special type of cluster algebras [@fz1], relevant to our setting [@bfz]. See also [@fz2] for an overview of the theory of cluster algebras.
Let ${\operatorname{\mathcal F}\nolimits}={\operatorname{\mathbb Q}\nolimits}(u_1, \dots, u_n)$ be the field of rational functions in indeterminates $u_1, \dots , u_n$, let $\underline{x} = \{x_1, \dots, x_n\} \subset F$ be a transcendence basis over ${\operatorname{\mathbb Q}\nolimits}$, and $B= (b_{ij})$ an $n \times n$ skew-symmetric integer matrix. A pair $(\underline{x},B)$ is called a [*seed*]{}. The [*cluster algebra*]{} associated to the seed $(\underline{x},B)$ is by definition a certain subring ${\operatorname{\mathcal A}\nolimits}(\underline{x},B)$ of ${\operatorname{\mathcal F}\nolimits}$, as we shall describe. Given such a seed $(\underline{x},B)$ and some $i$, with $1 \leq i \leq n$, define a new element of $x_i'$ of ${\operatorname{\mathcal F}\nolimits}$ by $$x_i x_i' = \prod_{j; b_{ji}>0} x^{b_{ji}}+ \prod_{j; b_{ji}<0} x^{-b_{ji}}.$$ We say that $x_i, x_i'$ form an [*exchange pair*]{}. We obtain a new transcendence basis $\underline{x'} = \{x_1, \dots, x_n\} \cup \{x_i'\} \setminus \{x_i \}$ of ${\operatorname{\mathcal F}\nolimits}$. Then define a new matrix $B' = (b_{ij}')$ associated with $B$ by $$b'_{ij} = \begin{cases}
-b_{ij} & \text{if $k=i$ or $k=j$,} \\
b_{ij} + \frac{{\lvertb_{ik}\rvert}b_{kj} + b_{ik} {\lvertb_{kj}\rvert}}{2} & \text{otherwise.}
\end{cases}$$ The pair $(\underline{x'},B')$ is called the [*mutation*]{} of the seed $(\underline{x},B)$ in direction $i$, written $\mu_i(\underline{x},B) = (\underline{x}',B')$. Let ${\operatorname{\mathcal S}\nolimits}$ be the set of seeds obtained by iterated mutations of $(\underline{x},B)$ (in all possible directions). The set of [*cluster variables*]{} is by definition the union of all transcendence bases appearing in all the seeds in ${\operatorname{\mathcal S}\nolimits}$, and the cluster algebra ${\operatorname{\mathcal A}\nolimits}(\underline{x},B)$ is the subring of ${\operatorname{\mathcal F}\nolimits}$ generated by the cluster variables. The transcendence bases appearing in the seeds are called [*clusters*]{}.
As mentioned earlier, there is a 1–1 correspondence between finite quivers with no loops and no oriented cycle of length two and skew-symmetric integer matrices (up to reordering the columns). The vertices of the quiver of a matrix $B=(b_{ij})$ are $1, \dots, n$, and there are $b_{ij}$ arrows from $i$ to $j$ if $b_{ij}> 0$. The cluster algebra is said to be [*acyclic*]{} if there is some seed where the quiver associated with the matrix has no oriented cycles [@bfz]. We take the corresponding seed as an initial seed. In this case, let $H= KQ$ be the hereditary path algebra associated with an initial seed $(\underline{x},B)$. Let ${\operatorname{\mathcal C}\nolimits}= {\operatorname{\mathcal C}\nolimits}_H$ be the corresponding cluster category, and let $T$ be a tilting object in ${\operatorname{\mathcal C}\nolimits}$. Similar to the above we can associate with $T$ a [*tilting seed*]{} $(T, Q_T)$, where $Q_T$ is the quiver of the endomorphism algebra ${\operatorname{End}\nolimits}_{{\operatorname{\mathcal C}\nolimits}}(T)^{{{\operatorname{op}}}}$. Let $T_1, \dots, T_n$ be the non-isomorphic indecomposable direct summands of $T$. Fix $i$, and let as before $\delta_i(T) = T'$ be the tilting object of ${\operatorname{\mathcal C}\nolimits}$ obtained by exchanging $T_i$ with $T_i^{\ast}$ (using our earlier notation from Theorem \[triangles\]). Define mutation of $(T, Q_T)$ in direction $i$ to be given by $\delta_i(T,Q_T) = (T', Q_{T'})$.
We now want to associate tilting seeds with seeds for acyclic cluster algebras. We first associate $(H[1], Q_H)$ with a fixed initial seed $(\underline{x},B)$, where $Q$ is the quiver for $B$ and $H= KQ$. Let $ (\underline{x}',B')$ be some seed. We then have $ (\underline{x}',B') = \mu_{i_t}\cdots \mu_{i_1} (\underline{x},B)$ for some ordered sequence $(i_1, \dots i_t)$. There are in general several such sequences, and we choose one of minimal length. Associated with $(\underline{x},B)$ is the sequence of length $0$, that is the empty set $\emptyset$. We define $\alpha((\underline{x},B), \emptyset)= (H[1],Q_H)$, and $\alpha((\underline{x}',B'), (i_1, \dots,i_t))= \delta_{i_t} \cdots \delta_{i_1} (H[1], Q_H) =
(T', Q_{T'})$. Fix an ordering on the cluster variables in the cluster $\underline{x}= \{x_1, \dots x_n\}$ of the chosen initial seed and choose a corresponding indexing for the $H_i$ in $H = H_1 \amalg \cdots \amalg H_n$, so that we have a correspondence between $x_i$ and $H_i$. This induces a correspondence between the cluster variables $x'_i$ in the cluster $\underline{x}'$ and the indecomposable direct summands $T'_i$ in $T'$, which we also denote by $\alpha$. We do not know in general if the definition of $\alpha$ only depends on the seed $(\underline{x}',B')$.
We can now formulate the connection between cluster algebras and tilting in cluster categories implied by our main result.
Let the notation be as above, with $(\underline{x},B)$ an initial seed for an acyclic cluster algebra, and $(T', Q_{T'})$ a tilting seed corresponding to a seed $(\underline{x}', B')$, via the correspondence $\alpha$, inducing a correspondence $x'_i \leftrightarrow T'_i$ for $x'_i \in \underline{x}'$ and $T'_i$ an indecomposable direct summand of $T'$.
- [For any $i \in \{1,\dots,n \}$ we have a commutative diagram $$\xy
\xymatrix{
((\underline{x}',B')(i_1,\dots, i_t)) \ar[d]_{\mu_i} \ar[r]^>>>>>>>{\alpha} & (T', Q_{T'}) \ar[d]_{\delta_i} \\
((\underline{x}'',B'')(i_1,\dots, i_t,i)) \ar[r]^>>>>>{\alpha} & (T'', Q_{T''})
}
\endxy$$ where $\underline{x}''$ is the cluster obtained from $\underline{x}'$ by replacing $x'_i \in \underline{x}'$ by $x''_i$, and $T''$ is the tilting object in ${\operatorname{\mathcal C}\nolimits}$ obtained by exchanging the indecomposable summand $T'_i$ by $T''_i$ where $T'= \bar{T} \amalg T'_i$ and $T'' = \bar{T} \amalg T''_i$ are non-isomorphic tilting objects. ]{}
- [Identifying $x'_i$ with $T'_i$ and $x''_i$ with $T''_i$, the multiplication rule for $x'_i x''_i$ is given by $$T'_i T''_i = \prod (T'_j)^{a_j} + \prod (T'_k)^{c_k}$$ where $a_j$ and $c_k$ are determined by the minimal respectively right and left ${\operatorname{add}\nolimits}\bar{T}$-approximations $\amalg (T'_j)^{a_j} \to T'_i$ and $T'_i \to \amalg (T'_k)^{c_k}$. ]{}
(a): This follows by induction, using Theorem \[mutate\], where $\delta_i$ is interpreted as given by a mutation rule like $\mu_i$.
\(b) Let $T'_i$ be the direct summand of $T'$ corresponding to $x'_i$. By (a), $Q_{T'}$ is the quiver of $B'$, and the monomials $M_1$ and $M_2$ are given by the entries of the matrix $B'$, hence by the arrows in the quiver $Q_{T'}$. In particular, the arrows entering and leaving $i$, are given by the minimal right and minimal left ${\operatorname{add}\nolimits}\bar{T}$-approximations of $T'_i$.
Note that with the appropriate formulation, this solves Conjecture 9.3 in [@bmrrt].
For algebras of finite type we know from [@bmrrt] that the map $\alpha$ gives a one-one correspondence between the seeds and tilting seeds, in particular it does not depend on the the $t$-tuple $(i_1, \dots,i_t)$. In fact, we have in this case a 1–1 correspondence between cluster variables and indecomposable objects of ${\operatorname{\mathcal C}\nolimits}$, inducing a 1–1 correspondence between clusters and tilting objects.
Two cluster variables $x_i$ and $x_i^{\ast}$ are said to form an exchange pair if there are $n-1$ cluster variables $\{y_1, \dots, y_{n-1} \}$ such that $\{x_i, y_1, \dots, y_{n-1} \}$ and $\{x_i^{\ast}, y_1, \dots, y_{n-1} \}$ are clusters. Similarly we have exchange pairs with respect to tilting objects. If $\alpha$ identifies $x_i$ and $x_i^{\ast}$ with $T_i$ and $T_i^{\ast}$, respectively, we then have the following.
For a cluster algebra of finite type, let $\alpha$ be the above correspondence between seeds and tilting seeds, and between cluster variables and indecomposable objects in the cluster category.
- [For any $i \in \{1,\dots,n \}$ we have a commutative diagram $$\xy
\xymatrix{
(\underline{x}',B') \ar[d]_{\mu_i} \ar[r]^{\alpha} & (T', Q_{T'}) \ar[d]_{\delta_i} \\
(\underline{x}'',B'') \ar[r]^{\alpha} & (T'', Q_{T''})
}
\endxy$$ ]{}
- [Identify the cluster variables with the indecomposable objects in ${\operatorname{\mathcal C}\nolimits}$ via $\alpha$. We have $$T_i T_i^{\ast} = \prod (T_j)^{a_j} + \prod (T^{\ast}_k)^{c_k}$$ for an exchange pair $T_i$ and $T_i^{\ast}$ where the $a_j$ and $c_k$ appear in the unique non-split triangles $$T_i^{\ast} \to \amalg T_j^{a_j} \to T_i \to ,$$ and $$T_i \to \amalg T_k^{c_k} \to T_i^{\ast} \to$$ in ${\operatorname{\mathcal C}\nolimits}$. ]{}
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[^1]: Aslak Bakke Buan was supported by a grant from the Norwegian Research Council
|
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"pile_set_name": "ArXiv"
}
|
---
abstract: |
We would like to discuss the language to write an extended Standard Model - using renormalizable quantum field theory as the framework; to start with certain basic units together with a certain gauge group. Specifically we use the left-handed and right-handed spinors to form the basic units together with $SU_c(3) \times SU_L(2) \times U(1) \times SU_f(3)$ as the gauge group. We could write down the extended Standard Model, though the details of the Higgs mechanism remains to be worked out. The same general quest appeared about forty years ago - the so-called “How to build up a model”. It is timely to address the same question again especially since we could now put together “Dirac similarity principle” and “Higgs minimum hypothesis” as two additional working rules.
[ PACS Indices: 12.60.-i (Models beyond the standard model); 98.80.Bp (Origin and formation of the Universe); 12.10.-g (Unified field theories and models).]{}
---
[**Leading Questions in an Extended Standard Model**]{}
.5cm
W-Y. Pauchy Hwang$^{1,}$[^1] and Tung-Mow Yan$^2$\
[*$^1$Asia Pacific Organization/Center for Cosmology and Particle Astrophysics,\
Institute of Astrophysics, Center for Theoretical Sciences,\
and Department of Physics, National Taiwan University, Taipei 106, Taiwan\
$^2$Department of Physics, Cornell University, Ithaca, N.Y. 14850*]{} .2cm
[(December 21, 2012)]{}
Introduction
============
As time goes by, our confidence in what we are doing seems to be dwindling - so to look for “superstring”, etc., as alternatives. Nevertheless, the language which was built up during the last century, namely, renormalizable quantum field theory, may still be the language underlying the final Standard Model. It will be the language of this paper.
Usually in a textbook, the QCD chapter precedes the one on Glashow-Weinberg-Salam (GWS) electroweak theory. Nothing is wrong with it but the basic units (or the building blocks) are further divided into the left-handed and right-handed components. It would be nice (in helping us in thinking) if the framework is formulated all at once - in an extended Standard Model we could see everything consistent with one another. Then, the questions which we pose could have broader meanings and implications. Thus, this is what we wish to do.
We shall work with the Lie group $SU_c(3) \times SU_L(2) \times
U(1) \times SU_f(3)$ as the gauge group. Thus, the basic units are made up from quarks (of six flavors, of three colors, and of the two helicities) and leptons (of three generations and of the two helicities), together with all originally massless gauge bosons and the somewhat hidden induced Higgs fields. In view of the search over the last forty years, we could assume “minimum Higgs hypothesis” as the working rule.
If we look at the basic units as compared to the original particle, i.e. the electron, the starting basic units are all “point-like” Dirac particles. Dirac invented Dirac electrons eighty years ago and surprisingly enough these “point-like” Dirac particles are the basic units of the Standard Model. Thus, we call it “Dirac Similarity Principle” - a salute to Dirac; a triumph to mathematics. Our world could indeed be described by the proper mathematics. The proper mathematics may be the renormalizable quantum field theory, although our confidence in it sort of fluctuates in time.
There is no way to “prove” the above two working rules - “Dirac Similarity Principle” and “minimum Higgs hypothesis”. It might be associated with the peculiar property of our Lorentz-invariant space-time. To use these two working rules, we could simplify tremendously the searches for the new extended Standard Models.
The Statement for the Extended Standard Model
=============================================
So far, we have decided on the basic units - those left-handed and right-handed quarks and leptons; the gauge group is chosen to be $SU_c(3) \times SU_L(2) \times U(1) \times SU_f(3)$.
In the gauge sector, the lagrangian is fixed if the gauge group is given; only for a massive gauge theory, Higgs fields are called for and we postpone its discussions until we have spelled out the fermion sector.
For the fermion sector, the story is again fixed if the so-called “gauge-invariant derivative”, i.e. $D_\mu$ in the kinetic-energy term $-\bar \Psi \gamma_\mu D_\mu \Psi$, is given for a given basic unit [@Books].
Thus, we have, for the up-type right-handed quarks $u_R$, $c_R$, and $t_R$, $$D_\mu = \partial_\mu - i g_c {\lambda^a\over 2} G_\mu^a -
i {2\over 3} g'B_\mu,$$ and, for the rotated down-type right-handed quarks $d'_R$, $s'_R$, and $b'_R$, $$D_\mu = \partial_\mu - i g_c {\lambda^a\over 2} G_\mu^a -
i (-{1\over 3}) g' B_\mu.$$
On the other hand, we have, for the $SU_L(2)$ quark doublets, $$D_\mu = \partial_\mu - i g_c {\lambda^a\over 2} G_\mu^a - i g
{\vec \tau\over 2}\cdot \vec A_\mu - i {1\over 6} g'B_\mu.$$
For the lepton side, we introduce the family triplet, $(\nu_\tau^R,\,\nu_\mu^R,\,,\nu_e^R)$ (column), under $SU_f(3)$. Since the minimal Standard Model does not see the right-handed neutrinos, it would be a natural way to make an extension of the minimal Standard Model. We propose that neutrinos are only species seeing the family gauge sector. Or, we have, for $(\nu_\tau^R,\,
\nu_\mu^R,\,\nu_e^R)$ ($\equiv \Psi_R(3,1)$), $$D_\mu = \partial_\mu - i \kappa {\bar\lambda^a\over 2} F_\mu^a.$$ and, for the left-handed $SU_f(3)$-triplet and $SU_L(2)$-doublet $((\nu_\tau^L,\,\tau^L),\, (\nu_\mu^L,\,\mu^L),\, (\nu_e^L,\,e^L))$ (all columns) ($\equiv \Psi_L(3,2)$), $$D_\mu = \partial_\mu - i \kappa {\bar\lambda^a\over 2} F_\mu^a - i g
{\vec \tau\over 2} \cdot \vec A_\mu + i {1\over 2} g' B_\mu.$$ If the right-handed charged leptons were singlets under $SU_f(3)$, then the mass-generation terms for charged leptons would involve the cross terms, such as $\mu\to e$, which is not acceptable at all. Thus, the right-handed charged leptons have to form another triplet $\Psi_R^C(3,1)$ under $SU_f(3)$.
In other words, the quark masses are given by the Higgs mechanism in the minimal Standard Model while the masses of charged leptons are determined by $\bar\Psi_L(3,2)\Psi_R^C(3,1)\Phi(1,2) + c.c.$, only a universal number in the leading-order sense. To make a reasonable theory [@Family], we have to make certain that all gauge bosons and the residual family Higgs particles are very massive, i.e. $\ge$ a few TeV.
As slightly differing from the previous effort [@Family], we would like to write down the $SU_c(3) \times SU_L(2) \times U(1) \times
SU_f(3)$ Standard Model [*all at once*]{}. We introduce the neutrino mass term as follows: $$i {\eta\over 2} {\bar\Psi}_L(3,2) \times \Psi_R(3,1) \cdot \Phi(3,2) + h.c.,$$ where $\Psi(3,i)$ are the lepton multiplets just mentioned above (with the first number for $SU_f(3)$ and the second for $SU_L(2)$). The cross (curl) product is somewhat new [@Family], referring to the singlet combination of three triplets in $SU(3)$ - an $SU(3)$ operation (and not a matrix product). The Higgs field $\Phi(3,2)$ is new in this effort, because it carries some nontrivial $SU_L(2)$ charge.
Lepton-flavor-violating Interaction
===================================
Neutrinos have masses, the tiny masses far below the range of the masses of the quarks and charged leptons. Neutrinos oscillate among themselves, giving rise to a lepton-flavor violation (LFV). There are other oscillation stories, such as the oscillation in the $K^0-{\bar K}^0$ system, but there is a fundamental difference here - the $K^0-{\bar K}^0$ system is composite while neutrinos are “point-like” Dirac particles. It is true that neutrino masses and neutrino oscillations may be regarded as one of the most important experimental facts over the last thirty years [@PDG].
In fact, certain LFV processes such as $\mu \to e + \gamma$ [@PDG] and $\mu + A \to A* + e$ are closely related to the most cited picture of neutrino oscillations so far [@PDG]. In recent publications by one [@Hwang10] of us, it was pointed out that the cross-generation or off-diagonal neutrino-Higgs interaction may serve as the detailed mechanism of neutrino oscillations, with some vacuum expectation value of the new Higgs field(s).
In the other words, the first term in the last equation \[Eq. (6)\] can be used as the basis to analyze the various lepton-flavor-violating decays and reactions.
To illustrate the point further, we calculate the golden lepton-flavor-violating decay $\mu \to e + \gamma$ as the celebrated example. We show in Figures 1(a), 1(b), and 1(c) three leading basic Feynman diagrams. Here the conversion of $\nu_\mu$ into $\nu_e$ is marked by a cross sign and it is a term from the off-diagonal interaction given above with the Higgs vacuum expectation value $u_0$. Here the Higgs masses are assumed to be very large, i.e., greater than a few $TeV$, as in $SU_f(3)$. The only small number (coupling) is $\eta$, consistent with the tiny masses of neutrinos.
{width="4in"}
Using Feynman rules from Wu and Hwang [@Books], we write, for Fig. 1(a),
$$\begin{aligned}
{1\over (2\pi)^4} \int d^4q \cdot {\bar u}(p',s')\cdot &i \cdot {1\over 2 \sqrt 2}
{e\over sin \theta_W}\cdot i \gamma_\lambda (1+ \gamma_5)\nonumber\\
\cdot {1\over i} {m_2-i\gamma\cdot q\over {m_2^2+q^2-i\epsilon}}\cdot
&i \cdot i \eta (-)u_0 \cdot {1\over i} {m_1-i\gamma\cdot q\over
{m_1^2 + q^2-i\epsilon}} \nonumber\\
\cdot i\cdot {1\over 2 \sqrt 2}{e\over sin\theta_W}\cdot &i \gamma_{\lambda'}
(1+\gamma_5)\cdot u(p,s)\nonumber\\
\cdot {1\over i} {\delta_{\lambda'\mu}\over {M_W^2+(p-q)^2-i\epsilon}}\cdot
{\epsilon_\sigma(k)\over \sqrt{2k_0}}\cdot &\Delta_{\sigma\mu\nu} \cdot
{1\over i} {\delta_{\nu\lambda}\over {M_W^2+(p-q-k)^2-i\epsilon}},\end{aligned}$$
with $\Delta_{\sigma\mu\nu}=(-ie)\{\delta_{\mu\nu}(-k-p-q)_\sigma +
\delta_{\nu\sigma}(p-q+p-q-k)_\mu +\delta_{\sigma\mu} (-p+q+k+k)_\nu \}$.
On the other hand, Feynman rules yield, for Fig. 1(b), $$\begin{aligned}
{1\over (2\pi)^4} \int d^4q \cdot {\bar u}(p',s')\cdot &i \cdot {1\over 2 \sqrt 2}
{e\over sin \theta_W}\cdot i \gamma_\lambda (1+ \gamma_5)\nonumber\\
\cdot {1\over i} {m_2-i\gamma\cdot q\over {m_2^2+q^2-i\epsilon}}\cdot
&i\cdot i \eta (-)u_0 \cdot {1\over i} {m_1-i\gamma\cdot q\over
{m_1^2 + q^2-i\epsilon}} \nonumber\\
\cdot i\cdot {1\over 2 \sqrt 2}{e\over sin\theta_W}\cdot &i \gamma_{\lambda'}
(1+\gamma_5)\cdot \nonumber\\
\cdot {1\over i} {\delta_{\lambda\lambda'}\over {M_W^2+(p'-q)^2-i\epsilon}}
\cdot {1\over i} {m_\mu - i\gamma\cdot p'\over {m_\mu^2+ p^{\prime 2}-i\epsilon}}
\cdot &i (-i)e \cdot \gamma\cdot {\epsilon(k)\over \sqrt {2k_0}}. u(p,s),\end{aligned}$$ and a similar result for Fig. 1(c).
The four-dimensional integrations can be carried out, via the dimensional integration formulae (e.g. Ch. 10, Wu/Hwang [@Books]), especially if we drop the small masses compared to the W-boson mass $M_W$ in the denominator. In this way, we obtain $$\begin{aligned}
i T_a={G_F\over \sqrt 2} &\cdot \eta u_0
\cdot (m_1 + m_2)\cdot (-2i){e\over (4\pi)^2}\nonumber\\
&\cdot {\bar u}(p',s') {\gamma\cdot \epsilon\over \sqrt {2k_0}}
(1+\gamma_5) u(p,s).\end{aligned}$$
It is interesting to note that the wave-function renormalization, as shown by Figs. 1(b) and 1(c), yields $$\begin{aligned}
i T_{b+c}= {G_F\over \sqrt 2} &\cdot \eta u_0 (m_1 + m_2)
\cdot (+2i){e\over (4\pi)^2}
\cdot \{{p'^2\over m_\mu^2 + p'^2} + {p^2\over m_e^2 + p^2}\}\nonumber\\ &\cdot
{\bar u}(p',s') {\gamma \cdot \epsilon \over \sqrt{2k_0}} (1+\gamma_5) u(p,s),\end{aligned}$$ noting that $p^2=-m_\mu^2$ and $p'^2=-m_e^2$ would make the contribution from Figs. 1(b) and 1(c) to be of the opposite sign to that from Fig. 1(a).
It is interesting to note that the leading terms all cancel, a result of gauge invariance. We have computed some next-order terms but a complete result seems to be rather difficult to obtain.
In a normal treatment, one ignores the wave-function renormalization diagrams 1(b) and 1(c) in the treatment of the decays $\mu \to e + \gamma$, $\mu \to 3e$, and $\mu+ A \to e+ A^*$.
Comparing this to the dominant mode $\mu \to e {\bar \nu}_e \nu_\mu$ [@Books], we could obtain the branching ratio. Even though the decay rate for $\mu \to e+ \gamma$ would be of the order $O(m_{neutrino}^4/M_W^4)$, which is extremely small. Note that the cancelation does not exist for $\mu \to e + e^+ + e^-$, nor for the conversion process $\mu^- + p \to e^- + p$. So, the rates would be expected to be much larger.
The off-diagonal mass matrix would be modified by the self-energy diagram since the neutrinos form a triplet under $SU_f(3)$. It is presumed that these self-energy diagrams, after the ultraviolet divergences get subtracted, lead to masses of the right order. If the off-diagonal mass matrix is diagonalized alone, the three roots would be two negative and one positive, adding up to zero. This seems like one ordering in the masses of neutrinos - one up and two downs.
Besides the golden decay $\mu \to e+ \gamma$ (much too small) and neutrino oscillations (already observed), violation of the $\tau-\mu-e$ universality is also expected and might be observed. As the baryon asymmetry is sometime attributed to the lepton-antilepton asymmetry, the current scenario for neutrino oscillations [@PDG] seems to be inadequate. If we take the hints from neutrinos rather seriously, there are so much to discover, even though the minimal Standard Model for the ordinary-matter world remains to be pretty much intact.
The Questions
=============
Let us come back to look at the statement of the extended Standard Model. We choose the basic units at first and then the gauge group. The Higgs mechanism would be in the last step.
If that is the case, we have some difficulty in writing down the left-right model [@Salam]. why? If we need to assign a certain left-handed or right-handed spinor into two basic units simultaneously, then the kinetic term appears twice - our language does not go; we believe that a lagrangian should have only one kinetic term.
So, our first question would be: Could the above rationale rule out the right-handed sector, since the simultaneous presence of the left-handed and right-handed basic units as $SU(2)$ doublets are excluded? Experimentally, we should check this point. As long as we could argue, we note that, as long as the left-handed and right-handed components are split in the basic units, parity has to be violated, either V-A or V+A.
In a slightly different context [@Hwang3], It was proposed that we could work with two working rules: “Dirac similarity principle”, based on eighty years of experience, and “minimum Higgs hypothesis”, from the last forty years of experience. Using these two working rules, the extended model mentioned above becomes rather unique - so, it is so much easier to check it against the experiments.
The Model stated in the paper is yet to be completed, in view of the “minimum Higgs hypothesis”. The Higgs mechanism in the previous $SU_f(3)$ family gauge theory is complete since the theory is treated [*alone*]{}. With $SU_f(3)$ and $SU_L(2)$ (3, 2) Higgs multiplet mentioned above plus one (3, 1) Higgs triplet, is it sufficient to do the Higgs-mechanism job - no “unwanted” massless particles? we would like to list this “mathematical” question as the second important question.
We would be curious about how the dark-matter world looks like, though it is difficult to verify experimentally. The first question would be: The dark-matter world, 25 % of the current Universe (in comparison, only 5 % in the ordinary matter), would clusterize to form the dark-matter galaxies, maybe even before the ordinary-matter galaxies. The dark-matter galaxies would then play the hosts of (visible) ordinary-matter galaxies, like our own galaxy, the Milky Way. Note that a dark-matter galaxy is by our definition a galaxy that does not possess any ordinary strong and electromagnetic interactions (with our visible ordinary-matter world). This fundamental question deserves some thoughts, for the structural formation of our Universe.
Of course, we should remind ourselves that, in our ordinary-matter world, those quarks can aggregate in no time, to hadrons, including nuclei, and the electrons serve to neutralize the charges also in no time. Then atoms, molecules, complex molecules, and so on. These serve as the seeds for the clusters, and then stars, and then galaxies, maybe in a time span of $1\, Gyr$ (i.e., the age of our young Universe). The aggregation caused by strong and electromagnetic forces is fast enough to help giving rise to galaxies in a time span of $1\, Gyr$. On the other hand, the seeded clusterings might proceed with abundance of extra-heavy dark-matter particles such as familons and family Higgs, all greater than a few $TeV$ and with relatively long lifetimes (owing to very limited decay channels). So, further simulations on galactic formation and evolution may yield clues on our problem.
So, we could put forward the third important question of this paper: What are the details of the dark-matter world?
Finally, coming back to the fronts of particle physics, neutrinos, especially the right-handed neutrinos, might couple to the dark-matter particles. Any further investigation along this direction would be of utmost importance. It may shed light on the nature of the dark-matter world.
The minimum Higgs picture is as follows: the Standard-Model Higgs doublet $\Phi(1,2)$, the purely family Higgs triplet $\Phi(3,1)$, and the mixed family Higgs $\Phi(3,2)$. The triplet $\Phi(3,1)$ and the neutral part $\Phi^0(3,2)$ undergoes the spontaneous symmetry breaking as in [@Family]. In the U-gauge, the neutral part $\Phi^0{3,2}$ gets projected out and the mass term becomes negative while there remains to have no SSB in the charged part $\Phi^+(3,2)$. For details, please consult [@Hwang12].
Over the years, Pauchy Hwang would like to thank Jen-Chieh Peng and Tony Zee for numerous interactions, those, plus a lot of (unspoken) personal thoughts, lead to this paper. This work is supported in part by National Science Council project (NSC99-2112-M-002-009-MY3).
[99]{}
Ta-You Wu and W-Y. Pauchy Hwang, “Relatistic Quantum Mechanics and Quantum Fields” (World Scientific 1991); Francis Halzen and Alan D. Martin, “Quarks and Leptons” (John Wiley and Sons, Inc. 1984); E.D. Commins and P.H. Bucksbaum, “Weak Interactions of Leptons and Quarks” (Cambridge University Press 1983). We use the first book for the notations and the metrics.
W-Y. Pauchy Hwang, Nucl. Phys. [**A844**]{}, 40c (2010); W-Y. Pauchy Hwang, International J. Mod. Phys. [**A24**]{}, 3366 (2009); the idea first appeared in hep-ph, arXiv: 0808.2091; talk presented at 2008 CosPA Symposium (Pohang, Korea, October 2008), Intern. J. Mod. Phys. Conf. Series [**1**]{}, 5 (2011); plenary talk at the 3rd International Meeting on Frontiers of Physics, 12-16 January 2009, Kuala Lumpur, Malaysia, published in American Institute of Physics 978-0-7354-0687-2/09, pp. 25-30 (2009).
Particle Data Group, “Review of Particle Physics”, J. Phys. G: Nucl. Part. Phys. [**37**]{}, 1 (2010); and its biennual publications.
W-Y. Pauchy Hwang, arXiv:1207.6443v1 \[hep-ph\] 27 Jul 2012; W-Y. Pauchy Hwang, Hyperfine Interactions [**215**]{}, 105 (2013); arXiv:1207.6837v1 \[hep-ph\] 30 Jul 2012; W-Y. Pauchy Hwang, arXiv:1209.5488v1 \[hep-ph\] 25 Sep 2012.
J.C. Pati and A. Salam, Phys. Rev. [**D10**]{}, 275 (1974); R.N. Mohapatra and J.C. Pati, Phys. Rev. [**D11**]{}, 566 (1975); [**D11**]{}, 2559 (1975).
W-Y. P. Hwang, arXiv:11070156v1 (hep-ph, 1 Jul 2011), Plenary talk given at the 10th International Conference on Low Energy Antiproton Physics (Vancouver, Canada, April 27 - May 1, 2011).
W-Y. Pauchy Hwang, arXiv:1301.6464v3 \[hep-ph\] 10 April 2013.
[^1]: Correspondence Author; Email: [email protected]; arXiv:1212.4944v2 \[hep-ph\] 12 April 2013; to be published in “The Universe”.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper proposes a method to compute finite abstractions that can be used for synthesizing robust hybrid control strategies for nonlinear systems. Most existing methods for computing finite abstractions utilize some global, analytical function to provide bounds on the reachable sets of nonlinear systems, which can be conservative and lead to spurious transitions in the abstract systems. This problem is even more pronounced in the presence of imperfect measurements and modelling uncertainties, where control synthesis can easily become infeasible due to added spurious transitions. To mitigate this problem, we propose to compute finite abstractions with robustness margins by over-approximating the local reachable sets of nonlinear systems. We do so by linearizing the nonlinear dynamics into linear affine systems and keeping track of the linearization error. It is shown that this approach provides tighter approximations and several numerical examples are used to illustrate of effectiveness of the proposed methods.'
author:
- 'Yinan Li, Jun Liu, and Necmiye Ozay[^1] [^2] [^3]'
bibliography:
- 'rbabst.bib'
title: '**Computing finite abstractions with robustness margins via local reachable set over-approximation**'
---
Nonlinear systems, temporal logic, control synthesis, reachable set computation.
Introduction
============
Construction of finite abstractions for nonlinear systems is a critical step when applying abstraction-based approaches to hybrid control synthesis [@Alur00]. Such approaches have gained popularity over the past few years for their ability to handle control problems for complex dynamical systems from high-level, rigorous specifications (see, e.g., piecewise affine systems [@Kloetzer08; @YordanovTCBB12], polynomial and nonlinear switched systems [@OzayLPM13; @LiuOTM13].The underlying principle of such approaches is to search for a controller in a finite abstraction of the original continuous system, leveraging formal synthesis techniques developed in computer science. As a result, the fidelity of finite abstractions has a significant influence on the result of control synthesis.
Symbolic models that are approximately similar or bisimilar to continuous-time nonlinear systems have been proposed and studied extensively [@PolaGT08; @GirardPT10; @ZamaniPMT12; @TabuadaBook09], which provide concrete means for computing finite approximate models often based on state-space discretization. For example, the symbolic models proposed in [@PolaGT08] and [@GirardPT10] are based on approximate bisimulation relations, which require incremental input-to-state stability [@Angeli02] of the original system. The work by [@ZamaniPMT12] later relaxes the stability requirement and constructs symbolic models that are essentially approximately alternatingly similar to the original system. Such symbolic models are nondeterministic and the computation of transitions relies on a global, analytical function provided by the incremental forward completeness of dynamics [@ZamaniPMT12].
When dynamical systems are affected by imperfections such as measurement errors, delays, and disturbances, synthesis of robust control strategies using abstraction-based approaches becomes important. Motivated by this, the work by [@LiuO14] introduces a notion of finite abstractions that are equipped with additional robustness margins to account for imperfections in measurements and/or models. These margins also lead to added nondeterminism in the abstractions.
To increase the fidelity of the nondeterminitic finite abstractions, one needs to reduce the number of spurious transitions in the abstractions. One way to do so is to compute tighter approximations of the local reachable sets for nonlinear systems. While local reachable set computation has been used for nonlinear system analysis and verification (see, e.g., [@Althoff10; @Althoff14]), we use it here to compute finite abstractions for robust control synthesis. More specifically, we linearize the nonlinear dynamics and keep track of the linearization errors. Robustness margins are incorporated in the set of initial conditions used for computing local reachable sets. This allows us to use margins that are are state-dependent and take into account variations in local dynamics. One major advantage of the proposed approach is that it provides much less conservative abstractions, compared with existing approaches.
**Notation**: let $\mathbb{Z}$ be the set of integers and $\mathbb{N}$ be the set of all nonnegative integers; $\mathbb{R}$ represents the set of all real numbers; $\mathbb{R}_{\geq 0}$ and $\mathbb{R}_{>0}$ are the sets of all nonnegative and all positive real numbers, respectively; $\mathbb{R}^n$ denotes the $n$-dimensional Euclidean space; ${\mathbb Z}^n$ denotes the $n$-dimensional integer lattice (the set of vectors in $\mathbb{R}^n$ whose components are all integers); given a vector $x=(x_1,\cdots,x_n)$ in ${\mathbb R}^n$, let ${\left\vertx\right\vert}=({\left\vertx_1\right\vert},\cdots,{\left\vertx_n\right\vert})$, i.e., the vector obtained by taking entrywise absolute value of $x$; given two vectors $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,y_n)$, $x\le y$ means $x_i\le y_i$ for all $i\in {\left\{1,\cdots,n\right\}}$ ($x<y$, $x>y$, and $x\ge y$ are similarly defined) and $x\circ y$ indicates the entrywise product, i.e., $x\circ y:=(x_1y_1,\cdots,x_ny_n)$; a vector $x\in{\mathbb R}^n$ is said to be positive if $x>0\in{\mathbb R}^n$ and nonnegative if $x\ge 0\in{\mathbb R}^n$; let $\mathbb{R}^n_{>0}$ and $\mathbb{R}^n_{\ge 0}$ denote the set of positive and nonnegative vectors in ${\mathbb R}^n$; given vectors $\delta\in \mathbb{R}^n_{\ge 0}$ and $x\in{\mathbb R}^n$, define $B_{\delta}(x):={\left\{x'\in{\mathbb R}^n:\,{\left\vertx'-x\right\vert}\le\delta\right\}}$, a hyper-rectangular box centred at $x$; $\mathcal{B}_\delta(0)$ is written as $\mathcal{B}_\delta$ for short; given $\eta \in \mathbb{R}^n_{\ge 0}$, define $[\mathbb{R}^n]_{\eta}:={\left\{\eta\circ k\in{\mathbb R}^n:\,k\in\mathbb{Z}^n\right\}}$ to be a hyper-rectangular grid with granularity parameter $\eta$; given a set $S \subseteq \mathbb{R}^n$ and a vector $\eta \in \mathbb{R}^n_{\ge 0}$, define $[S]_{\eta}:= S\cap [\mathbb{R}^n]_{\eta}$ to be the set of all grid points in $S$; given two sets $X\subseteq {\mathbb R}^n$ and $Y\subseteq {\mathbb R}^n$, $X \oplus Y$ denotes their Minkowski addition defined as $X \oplus Y:= \{ x+y |\; x \in X,\, y \in Y\}$; given a function $f$, dom$(f)$ denotes its domain.
Problem formulation {#sec:prob}
===================
Continuous-time control system
------------------------------
We consider a continuous-time control system described by a tuple ${\mathcal T}:=(X,X_0,U,f,\Pi,L)$, whose execution is governed by the ordinary differential equation with inputs $$\dot{x}(t)=f(x(t),\,u(t)),\label{eq:1}$$ where $t \in \mathbb{R}_{\geq 0}$, $x(t) \in X \subseteq \mathbb{R}^n$ is the system state, $x(0)\in X_0\subseteq \mathbb{R}^n$ is the initial state, and $u(t) \in U \subseteq \mathbb{R}^m$ is the control input. A measurable locally essentially bounded function defined on $[0,\tau]$ taking values in $U$ is called a *control signal* of duration $\tau$. Let $\mathcal{U}$ be the set of all control signals with arbitrary but finite duration. The vector field $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$ is a continuous function that fulfills the basic conditions (see, e.g., [@KhalilBook02]) for existence and uniqueness of solutions: given $x_0 \in X$, $T \in \mathbb{R}_{\geq 0}$, and a control signal $\mathbf{u}$ of duration $T$, there exists a unique solution, denoted by $\xi(t,\,x_0,\,\mathbf{u})$, that satisfies (\[eq:1\]) for $t\in[0,T]$ and the initial condition $x(0)=x_0$. The labeling function $L:X \to 2^\Pi$ is function that maps a state of ${\mathcal T}$ to a set of propositions in $\Pi$ that hold true at this state.
LTL control synthesis problem
-----------------------------
The desired system behaviors for ${\mathcal T}$ are specified using linear temporal logic (LTL). LTL is able to express a combination of safety, reachability, invariance properties. It is built upon the set of atomic propositions $\Pi$, logical operators $\neg$ (negation), $\wedge$ (conjunction) and temporal operators $\bigcirc$ (next), $\mathbf{U}$ (until). An LTL formula $\varphi$ is formed by connecting a finite set of atomic propositions with these operators. In this paper, we use a stutter-invariant fragment of LTL (denoted by $\text{LTL}_{\setminus \bigcirc}$), which excludes operation $\bigcirc$. The synthex of $\text{LTL}_{\setminus \bigcirc}$ can be found in [@ClarkeBook00]. We also assume that all $\text{LTL}_{\setminus \bigcirc}$ formulas have been transformed into negation normal form [@ClarkeBook00 p. 132], by adding the operator $\mathbf{R}$ (release) and replacing any negations of atomic propositions with new atomic propositions.
*$\text{LTL}_{\setminus \bigcirc}$ semantics for continuous trajectories*: Let $\xi$ be a continuous-time trajectory defined on $ \mathbb{R}_{\geq 0}$ and $\varphi$ be a $\text{LTL}_{\setminus \bigcirc}$ formula. Let $\xi[t]$ denote the state at time $t$, and $\xi[t,\infty)$ denotes the part of the trajectory in $[t,\infty), t \geq 0$. Then the semantics of $\xi$ satisfying $\phi$, denoted by $\xi \models \varphi$, is defined as follows:
- $\xi \models \pi, \pi \in \Pi$, iff $\pi \in L(\xi[t_0])$;
- $\xi \models \varphi_1 \wedge \varphi_2$ iff $\xi \models \varphi_1$ and $\xi \models \varphi_2$;
- $\xi \models \varphi_1 \vee \varphi_2$ iff $\xi \models \varphi_1$ or $\xi \models \varphi_2$;
- $\xi \models \varphi_1 \mathbf{U} \varphi_2$ iff there exists $t'> 0$ such that $\xi[t',\infty) \models \varphi_2$ and $\xi[t'',\infty) \models \varphi_1$ for all $t'' \in [0,t')$;
- $\xi \models \varphi_1 \mathbf{R} \varphi_2$ iff for all $t'> 0$ either $\xi[t',\infty) \models \varphi_2$ or there exists $t''\in [0,t')$ such that $\xi[t'',\infty) \models \varphi_1$.
Assume the system state $x_k$ is measured at time $t_k$ with $t_0=0, 0 \leq t_k < t_{k+1}, k \in \mathbb{N}$. A *continuous control strategy* is defined as a function $\sigma:x_0,\cdots,x_i \to \mathbf{u}_i$ that generates a control signal $\mathbf{u}_i \in \mathcal{U}$ for the horizon $[t_i,t_{i+1})$ according to the history of states $x_0,\cdots,x_i$.
We are now ready to formulate the main control synthesis problem this paper aims to address.
***Continuous Synthesis Problem***: Given a continuous-time control system ${\mathcal T}$ and an $\text{LTL}_{\setminus \bigcirc}$ specification $\varphi$, find a nonempty set of initial states $X_0$ and a control strategy $\sigma$ such that the resulting solutions of ${\mathcal T}$ satisfy $\varphi$. The specification $\varphi$ is said to be *realizable* for ${\mathcal T}$ if such $X_0$ exists.
Finite Abstractions with Robustness Margins
===========================================
This section is devoted to formally defining a notion of abstractions useful for solving robust control synthesis problems and proving their correctness and robustness guarantees when solving the continuous synthesis problem by discrete synthesis using these abstractions.
Finite abstractions with robustness margins
-------------------------------------------
In [@LiuO14], the authors introduced a notion of finite abstractions with additional robustness margins that can effectively handle a range of robustness related issues in control synthesis, including modelling uncertainty, measurement errors, and jitter or delays in control signals.
This paper aims to improve its computational procedure in two aspects. First, we define the finite abstractions with a varying (state-dependent) robustness margins while [@LiuO14] use fixed margins which are often conservatively chosen to cope with the worst case. Second, we construct transitions by way of local reachable set computation while the results in [@LiuO14] rely on a global analytical bound that can lead to spurious transitions being added due to variation in local dynamics.
To this end, we shall formally define the notion of finite abstractions with robustness margins using reachable set.
\[def:reach\] Given a control signal $\mathbf{u}\in{\mathcal U}$ of duration $\tau$ and a set of initial states $X_0$, the *reachable set* for system (\[eq:1\]) at time $\tau$ under this control signal $\mathbf{u}$ is defined by $$\mathcal{R}_{\mathbf{u},\,X_0}(\tau):=\{ \xi(\tau,\,x_0,\,\mathbf{u})|\, x_0 \in X_0\}.$$ The *reachable tube* for system (\[eq:1\]) over the interval $[0,\,\tau]$ is the union of all reachable sets during this time interval, which is $$\mathcal{R}_{\mathbf{u},\,X_0}([0,\,\tau]):=\bigcup_{t \in [0,\,\tau]}\{ \xi(t,\,x_0,\,\mathbf{u})|\, x_0 \in X_0\}.$$ With a fixed $u\in U$ and $\tau\in{\mathbb R}_{>0}$, $\mathcal{R}_{u,\,X_0}(\tau)$ and $\mathcal{R}_{u,\,X_0}([0,\,\tau])$ are interpreted as $u$ being a constant control signal on $[0,\tau]$.
We are now ready to define finite abstractions with robustness margins using reachable set.
\[def:abs\]
Given $\delta \in \mathbb{R}^n_{>0}$ and functions $\Gamma_i:\,X \to \mathbb{R}^n_{\geq 0}, i=1,2$, a finite transition system $$\hat{\mathcal{T}}:=(\hat{\mathcal{Q}},\,\hat{\mathcal{Q}}_0,\,\hat{\mathcal{A}},\,\rightarrow_{\hat{\mathcal{T}}},\,\hat{\Pi},\,\hat{L})$$ is said to be a $(\Gamma_1,\Gamma_2,\delta)$-*abstraction* of the continuous-time control system ${\mathcal T}=(X,X_0,U,f,\Pi,L)$, denoted by $\mathcal{T} \preceq_{(\Gamma_1,\Gamma_2,\delta)} \hat{\mathcal{T}}$, if there exists an abstraction map $\Omega:\, X \to \mathcal{\hat{Q}}$ such that
- $\hat{\mathcal{Q}}$ is a finite subset of $X$;
- $\hat{\mathcal{Q}}_0=\bigcup_{x\in X_0}{\left\{\Omega(x)\right\}}$;
- $\hat{\mathcal{A}}$ is a finite subset of $\mathcal{U}$;
- $(\hat{q},\,\hat{\mathbf{u}},\,\hat{q}') \in \rightarrow_{\hat{\mathcal{T}}}$ if, under $\hat{\mathbf{u}} \in \hat{\mathcal{A}}$ with duration $\tau$, $\hat{q}$ and $\hat{q}'$ satisfy $$\big(\Omega^{-1}(\hat{q}') \oplus \mathcal{B}_{\Gamma_2(\hat{q}')}\big) \cap \mathcal{R}_{\mathbf{\hat{u}},\,\Omega^{-1}(\hat{q}) \oplus \mathcal{B}_{\Gamma_1(\hat{q})}}(\tau) \neq \varnothing;$$
- $\hat{L}:\hat{\mathcal{Q}} \to 2^{\hat{\Pi}}$ is defined by $\hat{L}(\hat{q})=\cap_{x \in \mathcal{B}_{\delta}(\hat{q}) \cap X} L(x)$, $\hat{\Pi}=\Pi$.
The parameter $\delta$ is used to guarantee that specifications are satisfied even if the controller is synthesized using a finite abstraction with approximation errors. The functions $\Gamma_{1,2}$ provide additional robustness margins that varies with respect to local dynamics to account for imperfections such as system delay, measurement or modelling errors, at the price of increasing the nondeterminism in the abstraction.
A common and practical type of imperfections involves delays and measurement errors (e.g., noise or quantization). Consider the system ${\mathcal T}$ with a continuous control strategy $\sigma$ subjects to a measurement delay $h(t)\in [0,\Delta]$, $\Delta \in \mathbb{R}_{\geq 0}$, and an error $e(t)$ with $|e(t)|\leq \varepsilon \in \mathbb{R}^n_{\geq 0}$, the system dynamics becomes $$\label{eq:1b}
\left\{
\begin{aligned}
\dot{x}(t)&=f(x(t),\,\mathbf{u}_i),\, \mathbf{u}_i=\sigma(\hat{x}(t_0),\cdots,\hat{x}(t_i)),\\
\hat{x}(t_i)&=x(t_i-h(t_i))+e(t_i).
\end{aligned}
\right.$$ where $\hat{x}$ denotes the measurement of system states, $t \in [t_i,t_{i+1}), t_0=0, t_i<t_{i+1}, i \in \mathbb{N}$ and $\tau_i=t_{i+1}-t_i$ is the time duration of $\mathbf{u}_i$.
Discrete synthesis problem
--------------------------
An $\text{LTL}_{\setminus \bigcirc}$ formula can be interpreted over paths of $\hat{\mathcal{T}}$. A *path* of $\hat{\mathcal{T}}$ is a sequence of states $\hat{\rho}=\hat{q}_0\hat{q}_1\hat{q}_2\cdots$ under the the corresponding action $ \hat{a}_i \in \hat{\mathcal{A}}$ at each state $\hat{q}_i \in \hat{\mathcal{Q}}$ while satisfying $(\hat{q}_i,\,\hat{a}_i,\,\hat{q}_{i+1}) \in \rightarrow_{\hat{\mathcal{T}}},\, i \in \mathbb{N}$.
*$\text{LTL}_{\setminus \bigcirc}$ semantics for discrete sequences*: Let $\rho=q_0q_1q_2\cdots$ be an infinite discrete sequence and $\varphi$ be an $\text{LTL}_{\setminus \bigcirc}$ formula. Let $\rho[i,\infty)$ denote the subsequence $q_iq_{i+1}\cdots, i \in \mathbb{N}$. Then semantics of $\rho$ satisfying $\varphi$, denoted by $\rho \models \varphi$, is defined as follows:
- $\rho \models \pi$, $\pi \in \Pi$, iff $\pi \in L(q_0)$;
- $\rho \models \varphi_1 \wedge \varphi_2$ iff $\rho \models \varphi_1$ and $\rho \models \varphi_2$;
- $\rho \models \varphi_1 \vee \varphi_2$ iff $\rho \models \varphi_1$ or $\rho \models \varphi_2$;
- $\rho \models \varphi_1 \mathbf{U} \varphi_2$ iff there exists $j\geq 0$ such that $\rho[j,\infty) \models \varphi_2$ and $\rho[k,\infty) \models \varphi_1$ for all $0 \leq k < j$;
- $\rho \models \varphi_1 \mathbf{R} \varphi_2$ iff for all $j\geq 0$ either $\rho[j,\infty) \models \varphi_2$ or there exists some $0\le k<j$ such that $\rho[k,\infty) \models \varphi_1$.
Similar to continuous control strategy, a *discrete control strategy* for $\hat{\mathcal{T}}$ is a function $\hat{\sigma}:\hat{q}_0,\cdots,\hat{q}_i \to \hat{a}_i$ that maps the history path to a control action. Then we formulate the discrete synthesis problem as follows.
***Discrete Synthesis Problem*** Given a finite transition system $\hat{\mathcal{T}}$ and an $\text{LTL}_{\setminus \bigcirc}$ specification $\varphi$, find a nonempty set of initial states $\hat{X}_0$ and a control strategy $\hat{\sigma}$ such that any resulting path satisfies $\varphi$. If such $\hat{X}_0$ exists, then $\varphi$ is said to be realizable for $\hat{\mathcal{T}}$.
Correctness and robustness guarantees
-------------------------------------
In general, the existence of a discrete control strategy $\hat{\sigma}$ that solves the discrete synthesis problem with an $\text{LTL}_{\setminus \bigcirc}$ specification $\varphi$ does not guarantee that a control strategy exists for the continuous synthesis problem with the same specification.
As indicated in Definition \[def:abs\], $\hat{\mathcal{T}}$ requires the same propositions of ${\mathcal T}$ to hold within a neighbourhood of radius $\delta$, which is more restrictive. This is because the discrete strategy only guarantees that a sequence of sampled states satisfy a given specification and the parameter $\delta$ accounts for the possible mismatches of the inter-sample states. In addition, the robustness margin functions $\Gamma_i$ ($i=1,2$) are chosen to account for possible imperfections.
To formally reason about the correctness and robustness guarantees of solving the continuous synthesis problem by discrete synthesis using finite abstractions with robustness margins, the following theorem gives a sufficient condition for the realizability of the continuous synthesis problem by the realizability of the discrete synthesis problem.
\[thm1\] Given a continuous-time control system $\mathcal{T}$, its $(\Gamma_1,\Gamma_2,\delta)$-abstraction $\hat{\mathcal{T}}$, and an $\text{LTL}_{\setminus \bigcirc}$ formula $\varphi$,
1. (correctness) $\varphi$ being realizable for $\hat{\mathcal{T}}$ implies that $\varphi$ is realizable for $\mathcal{T}$, provided that, for all $(\hat{q},\,\hat{\mathbf{u}},\,\hat{q}') \in \rightarrow_{\hat{\mathcal{T}}}$, $$\label{eq:thm1}
\mathcal{R}_{\hat{\mathbf{u}},\Omega^{-1}(\hat{q}) \oplus \mathcal{B}_{\Gamma_1(\hat{q})}}(\text{dom}(\hat{\mathbf{u}}))\subseteq \mathcal{B}_\delta(\hat{q}).$$ In particular, if $\hat{\mathcal{T}}$ satisfies $\varphi$ with $\hat{\sigma}$ and $\hat{\mathcal{Q}}_0$, then $\varphi$ is realizable for $\mathcal{T}$ using $X_0=\cup_{q\in \hat{\mathcal{Q}}_0}\Omega^{-1}(q)$ and $\sigma(x_0,\cdots,x_i)=\hat{\sigma}(\Omega(x_0),\cdots,\Omega(x_i))$ where $x_0,\cdots,x_i$ is the sequence of measured states.
2. (robustness) if the system is subjected to measurement delays and errors defined in (\[eq:1b\]), then the same statement holds true, provided additionally that the robustness margins $\Gamma_{i}$ ($i=1,2$) satisfy that, for all $\hat{\mathbf{v}} \in \hat{\mathcal{A}}$ and $\hat{q}\in\hat{{\mathcal Q}}$, $\Gamma_2(\hat{q}) \geq \varepsilon$ and $$\label{eq:thm2}
\mathcal{R}_{\hat{\mathbf{v}},\Omega^{-1}(\hat{q}) \oplus \mathcal{B}_\varepsilon}([0,\Delta]) \subseteq \Omega^{-1}(\hat{q}) \oplus \mathcal{B}_{\Gamma_1(\hat{q})}.$$
\(i) The realizability of $\varphi$ for $\hat{{\mathcal T}}$ implies that there exists an initial set $\hat{\mathcal{Q}}_0$ and a discrete control strategy $\hat{\sigma}$ for $\hat{\mathcal{T}}$ such that all the possible controlled paths from any initial state in $\hat{\mathcal{Q}}_0$ satisfies $\varphi$ (note that $\hat{\mathcal{T}}$ is nondeterministic). We need to show the realizability of $\varphi$ for ${\mathcal T}$. For this purpose, we define an initial set $X_0=\cup_{q\in \hat{\mathcal{Q}}_0}\Omega^{-1}(q)$ and a continuous control strategy by $$\sigma(x_0,\cdots,x_i)=\hat{{\mathbf u}}_i=\hat{\sigma}(\Omega(x_0),\cdots,\Omega(x_i)),$$ where $x_0,\cdots,x_i$ is a sequence of measured states. We write $\hat{q}_i=\Omega(x_i)$ for all $i\ge 0$ and apparently $\hat{q}_0 \in \hat{\mathcal{Q}}_0$. In addition, we denote by $\tau_i$ the duration of $\hat{{\mathbf u}}_i$ and let $t_0=0,t_i=\sum_{k=0}^{i-1}\tau_k,i=1,2,\cdots$. Denote by $\xi$ the trajectory of ${\mathcal T}$ starting from $x_0$ under the control strategy $\sigma$ and by $\hat{\rho}$ the path $\hat{q}_0\hat{q}_1\hat{q}_2\cdots$. This correspondence is illustrated by the diagram below:
The proof consists of two steps: (A) to show that the path $\hat{\rho}=\hat{q}_0\hat{q}_1\hat{q}_2\cdots$ is a valid path in $\hat{{\mathcal T}}$ and, as a result, $\hat{\rho}\models \varphi$; (B) to show from $\hat{\rho}\models \varphi$ that $\xi \models \varphi$.
To show (A), note that, since $x(t_i) \in \Omega^{-1}(\hat{q}_i)$ for all $i\ge 0$, we have $$x(t_{i+1}) \in \mathcal{R}_{\hat{\mathbf{u}}_i,\Omega^{-1}(\hat{q}_i) \oplus \mathcal{B}_{\Gamma_1(\hat{q}_i)}}(\tau_i).$$ It follows from the definition of the transitions of $\hat{\mathcal{T}}$ that $(\hat{q}_{i},\hat{u}_i,\hat{q}_{i+1})\in\rightarrow_{\hat{\mathcal{T}}}$ for all $i\ge 0$.
To show (B), we prove $\xi \models \varphi$ from $\hat{\rho}\models \varphi$ by induction on the form of $\text{LTL}_{\setminus \bigcirc}$ formulas. In fact, we will prove a stronger statement: for each $k\ge 0$, $\hat{\rho}[k,\infty)$ implies that $\xi[t,\infty) \models \varphi_1$ for all $t\in [t_k,t_{k+1})$.
For $\varphi=\pi \in \Pi$, $\hat{\rho}[k,\infty) \models \pi$ iff $\pi \in \hat{L}(\hat{q}_k)$. Since $$x(t)\in \mathcal{R}_{\hat{\mathbf{u}}_k,\Omega^{-1}(\hat{q}_k) \oplus \mathcal{B}_{\Gamma_1(\hat{q}_k)}}([0,\tau_k]),\quad\forall t\in[t_k,t_{k+1}),$$ we have $\pi \in \hat{L}(\hat{q}_k) \subseteq L(x(t))$, i.e., $\xi[t,\infty) \models \varphi=\pi$, for all $t\in [t_k,t_{k+1})$.
The cases for $\xi \models \varphi$ when $\varphi=\varphi_1 \wedge \varphi_2$ or $\varphi=\varphi_1 \vee \varphi_2$ are straightforward to prove. We focus on the case $\varphi=\varphi_1 \mathbf{U} \varphi_2$. Assume $\hat{\rho}[k,\infty) \models \varphi$, which means that there exists some $j\geq k$ such that $\hat{\rho}_0[j,\infty) \models \varphi_2$ and $\hat{\rho}_0[i,\infty) \models \varphi_1$ for all $i$ such that $k \leq i < j$. By the inductive assumption, we have $\xi[t,\infty) \models \varphi_2$ for all $t\in [t_j,t_{j+1})$ and $\xi[t,\infty) \models \varphi_1$ for all $t\in [t_i,t_{i+1})$ and all $i$ such that $k \leq i < j$. This indeed implies that $\xi[t,\infty) \models \varphi=\varphi_1 \mathbf{U} \varphi_2$, for all $t\in [t_k,t_{k+1})$. The proof for the case $\varphi=\varphi_1 \mathbf{R} \varphi_2$ is similar and therefore omitted.
\(ii) Now consider system (\[eq:1b\]) for robustness. The key difference now is that measured states are delayed versions of the longer true states affected by noise. Denote by $\hat{x}(t_i) \in \mathcal{B}_\varepsilon(x(t_i))$ the measured value of $x(t_i)$ and let $\hat{q}_i=\Omega(\hat{x}(t_i))$ for all $i\ge 0$. The corresponding continuous control strategy becomes $$\sigma(\hat{x}(t_0),\cdots,\hat{x}(t_i))=\hat{{\mathbf u}}_i=\hat{\sigma}(\hat{q}_0,\cdots,\hat{q}_i).$$ Each control action $\hat{{\mathbf u}}_i$ is activated when the true state moves to $x(t_i)'=x(t_i+h(t_i)).$ The correspondence between the evolution of a true trajectory and the sequence of measure states are illustrated in the following diagram:
We still need to show the two steps (A) and (B) as in part (i). We start with (A), i.e., show that the path $\hat{\rho}=\hat{q}_0\hat{q}_1\hat{q}_2\cdots$ is a valid path in $\hat{{\mathcal T}}$. Note that, according to (\[eq:thm2\]), we have $$x(t_i)' \in \mathcal{R}_{u_{i-1},\Omega^{-1}(\hat{q}_i) \oplus \mathcal{B}_\varepsilon}([0,\Delta]) \subseteq \Omega^{-1}(\hat{q}_i) \oplus \mathcal{B}_{\Gamma_1(\hat{q}_i)}.$$ Therefore $$x(t_{i+1}) \in \mathcal{R}_{\hat{\mathbf{u}}_i,\Omega^{-1}(\hat{q}_i) \oplus \mathcal{B}_{\Gamma_1(\hat{q}_i)}}(\tau_i) \subseteq \mathcal{B}_\delta(\hat{q}_{i}).$$ Since $\hat{x}(t_{i+1}) \in \mathcal{B}_\varepsilon(x(t_{i+1}))$ and $\hat{q}_{i+1}=\Omega(\hat{x}(t_{i+1}))$, we have $x(t_{i+1})\in \Omega^{-1}(\hat{q}_{i+1})\oplus\mathcal{B}_\varepsilon$. Considering that the transitions for $\hat{{\mathcal T}}$ are constructed according to Definition \[def:abs\] with $\Gamma_2 \geq \varepsilon$, the transition $(\hat{q}_i,\,\hat{\mathbf{u}}_i,\,\hat{q}_{i+1})$ is indeed included in $\rightarrow_{\hat{\mathcal{T}}}$.
Proving step (B) by induction is similar to that for part (i). We prove the claim: for each $k\ge 0$, $\hat{\rho}[k,\infty)$ implies that $\xi[t,\infty) \models \varphi_1$ for all $t\in [t_k,t_{k+1})$. Note that we have $t_k+h(t_k)\in [t_k,t_{k+1})$ and $t_{k+1}-t_k-h(t_k)=\tau_k$, the duration of $\hat{u}_k$. We only prove the case for atomic propositions and the rest is similar to that for part (i).
For $\varphi=\pi \in \Pi$, $\hat{\rho}[k,\infty) \models \pi$ iff $\pi \in \hat{L}(\hat{q}_k)$. Note first that, by (\[eq:thm2\]), $$x(t)\in \mathcal{R}_{\hat{\mathbf{u}}_{k-1},\Omega^{-1}(\hat{q}_k)\oplus\mathcal{B}_{\varepsilon}}([0,\Delta])\subseteq \Omega^{-1}(\hat{q}_i) \oplus \mathcal{B}_{\Gamma_1(\hat{q}_k)}\subseteq \mathcal{B}_\delta(\hat{q}_{k})$$ for all $t\in[t_k,t_{k}+h(t_k)]$. This and (\[eq:thm1\]) further imply that $$x(t)\in \mathcal{R}_{\hat{\mathbf{u}}_{k},\Omega^{-1}(\hat{q}_k) \oplus \mathcal{B}_{\Gamma_1(\hat{q}_i)}}([0,\tau_k])\subseteq \mathcal{B}_\delta(\hat{q}_{k})$$ for all $t\in[t_k+h(t_k),t_{k+1})$. Consequently, we have $\pi \in \hat{L}(\hat{q}_k) \subseteq L(x(t))$, i.e., $\xi[t,\infty) \models \varphi=\pi$, for all $t\in [t_k,t_{k+1})$.
Reachable Set Over-approximation Based on Linearization and Error Estimation
============================================================================
A key step in constructing finite abstractions with robustness margins defined in the previous section is to compute the reachable sets for nonlinear systems. In practice, exact reachable sets of nonlinear systems are difficult to obtain and thus their approximations are usually computed. For example, reachable set over-approximation is implicitly required by the abstraction procedures in [@PolaGT08; @ZamaniPMT12; @LiuO14], where analytical bounds, usually obtained by Lyapunov-like functions, are used to roughly estimate the evolution of trajectories. A more precise computation of reachable sets has the potential to significantly reduce the spurious transitions in the abstraction.
In this section, we present a linearization-based method for the computation of reachable sets for nonlinear systems. For simplicity, we only consider constant control signals, which suffice for the computation of finite abstractions by discretization-based methods to be discussed in Section \[sec:discretization\].
Reachable set computation for linear systems
--------------------------------------------
Consider a class of affine control systems of the form $$\dot{x}(t)=Ax(t)+b+u(t) \label{eq:linaff}$$ where $b \in \mathbb{R}^n$ is a constant vector, $x(t) \in X$ is the state, $u(t) \in U$ is the control signal, and $U \subseteq \mathbb{R}^m$ is a compact convex set.
Similar to Definition \[def:reach\], given an initial set of states $X_0 \subseteq X$, we denote by $\mathcal{R}_{X_0}^{L}(\tau)$ the set of states that are reachable at time $\tau \in \mathbb{R}_{\geq 0}$ under $U$, which is defined by $$\begin{split}
\mathcal{R}^L_{X_0}(\tau):=\{ x(\tau) \in X|\,&\dot{x}(t)=Ax(t)+b+u(t),\forall t \in [0,\tau],\\
&u(t) \in U, x(0) \in X_0\}.
\end{split}$$ The reachable tube over the interval $[0,\,\tau]$ is defined by $$\mathcal{R}^L_{X_0}([0,\tau]):=\bigcup_{t \in [0,\tau]}\mathcal{R}^L_{X_0}(t).$$
Since the control input $u(t)$ is chosen arbitrarily from the set $U$, both the reachable set and tube are difficult to be computed exactly. For linear control systems, their convex over-approximations are used instead (see, e.g., Lemmas 1 and 2 in [@GuernicG10]). The convex hull of two convex sets, which is defined by $$\text{CH}(\mathcal{X},\mathcal{Y})=\{\lambda x+(1-\lambda)y|\,x \in \mathcal{X}, y \in \mathcal{Y}, \lambda \in [0,1] \},$$ is used to compute the reachable tube. For the linear affine control systems, we give the following proposition to over-approximate the reachable sets and tubes.
\[prop1\] For a linear affine control system (\[eq:linaff\]), given a compact convex set $X_0 \subseteq X$ and a time $\tau \in \mathbb{R}_{\geq 0}$, let $$\label{eq:4a}
\begin{split}
Y(\tau)&=e^{A\tau}X_0 \oplus \{G(A,\tau)b \} \oplus \tau U \oplus \mathcal{B}_{\beta_\tau},\\
Y([0,\tau])&=\text{CH}(X_0, Y(\tau) \oplus \mathcal{B}_{\alpha_\tau+\gamma_\tau}),
\end{split}$$ where $$\label{eq:4b}
\begin{split}
\alpha_\tau&=(e^{\tau \|A\|}-1-\tau \|A\|)\max_{x \in X_0}\|x\| \mathbf{1},\\
\beta_\tau&=(e^{\tau \|A\|}-1-\tau \|A\|)\|A\|^{-1}\max_{u \in U}\|u\| \mathbf{1},\\
\gamma_\tau&=(e^{\tau \|A\|}-1-\tau \|A\|)\|A\|^{-1}\|b\|\mathbf{1},
\end{split}$$ with ${\left\Vert\cdot\right\Vert}$ as the infinity norm, $\mathbf{1} \in \mathbb{R}^n$ representing the vector of ones, i.e., each element of it equals to 1, and $G(A,\tau):=\int_0^{\tau}e^{A(\tau-t)}dt$. Then $$\begin{split}
\mathcal{R}^L_{X_0}(\tau) &\subseteq Y(\tau),\\
\mathcal{R}^L_{X_0}([0,\tau]) &\subseteq Y([0,\tau]).
\end{split}$$
Denote by $x(t), t \in [0,\tau]$, a trajectory of the system from a initial state $x_0 \in X_0$ under an input $u(t) \in U$, and $$\begin{split}
x(t)=&e^{tA}x_0+\int_0^te^{A(t-s)}b\,ds\\
&+\int_0^{t}u(s)ds+\int_0^{t}(e^{A(t-s)}-I)u(s)ds\\
=&e^{tA}x_0+G(A,t)b+tu^*(t)\\
&+\int_0^{t}(e^{A(t-s)}-I)u(s)ds,
\end{split}$$ where $u^*(t)=\frac{1}{t}\int_0^tu(s)ds \in U$ for that $U$ is convex. We estimate $x(t)$ by $\hat{x}(t)$, which is given by $$\hat{x}(t)=x_0+\frac{t}{\tau}(e^{\tau A}-I)x_0+\frac{t}{\tau}G(A,\tau)b+tu^*(t).$$ Then $$\label{eq:5}
\begin{split}
\|x(t)-\hat{x}(t)\| \leq &\|e^{tA}x_0-x_0-\frac{t}{\tau}(e^{\tau A}-I)x_0\|\\
& + \|G(A,t)b-\frac{t}{\tau}G(A,\tau)b\| \\
&+ \|\int_0^{t}(e^{A(t-s)}-I)u(s)ds\|\\
\leq &\frac{t}{\tau}(\alpha_\tau+\gamma_\tau+\beta_\tau).
\end{split}$$ This means there exists a vector $\tilde{x}(t)$ in $\mathcal{B}_{\alpha_\tau+\gamma_\tau+\beta_\tau}$ such that $$\begin{split}
x(t)&=\hat{x}(t)+\frac{t}{\tau}\tilde{x}(t)\\
&=(1-\frac{t}{\tau})x_0+\frac{t}{\tau}(e^{\tau A}+G(A,\tau)b+tu^*(t)+\tilde{x}(t)).
\end{split}$$ Therefore $$\begin{aligned}
\mathcal{R}^L_{X_0}([0,\tau]) & \subseteq \text{CH}(X_0, e^{A\tau}X_0 \oplus \{G(A,\tau)b \} \oplus \tau U \oplus \mathcal{B}_{\alpha_\tau+\gamma_\tau+\beta_\tau})\\
&=Y([0,\tau]).\end{aligned}$$
The state estimation error at time $\tau$ reduces to $\|x(\tau)-\hat{x}(\tau)\|\leq \beta_\tau$ by setting $t=\tau$ in (\[eq:5\]). Thus $\mathcal{R}^L_{X_0}(\tau) \subseteq e^{A\tau}X_0 \oplus \{G(A,\tau)b \} \oplus \tau U \oplus \mathcal{B}_{\beta_\tau}=Y(\tau)$.
Proposition \[prop1\] differs from [@GuernicG10] in considering affine systems. Defining $v(t):=b+u(t), v(t) \in V=\{b\} \oplus U$, the method in [@GuernicG10] can also be applied. Yet when $u(t)$ is small compared to $b$, the size of $Y(\tau)$ computed by proposition \[prop1\] is smaller because of a smaller bloating parameter $\beta_\tau$.
Reachable set computation for nonlinear systems
-----------------------------------------------
Reachable set over-approximation for nonlinear systems obtained by a global analytical function can be conservative. To obtain a relatively tighter over-approximation of the one-step reachable set of nonlinear systems, we can write the nonlinear system dynamics as the sum of its linearization in a local area and an approximation error term.
More specifically, for a nonlinear system (\[eq:1\]) under a constant control input $u\in \mathcal{U}$, the dynamics around a center point $x^*\in X$ can be approximated by its first-order Taylor expansion with a Lagrangian remainder: $$\label{eq:lin}
\dot{x}(t) = A_{x^*}(x(t)-x^*)+f(x^*,u)+d_{x^*}(t),$$ where $A_{x^*}=\partial f/ \partial x|_{x^*}$, and $d_{x^*}(t)=(d_1(t),\cdots,d_n(t)) \in \mathbb{R}^n$ is the approximation error with $$d_i(t)=\frac{1}{2}(x(t)-x^*)^TH_i(z_i(t))(x(t)-x^*),$$ $$H_i(z_i(t))=\frac{\partial^2 f_i}{\partial x^2}\bigg|_{z_i(t)},$$ and $z_i(t) \in \mathcal{B}_{|x(t)-x^*|}(x^*)$.
If the system trajectory does not exceed a predefined linearization area $\mathcal{B}_r(x^*)$, where $r \in \mathbb{R}^n_{>0}$, then $d_{x^*}(t)$ belongs to a convex set $\mathcal{D}_{x^*}(r)$ given by $$\label{eq:d}
\begin{split}
\mathcal{D}_{x^*}(r)=\{d=(d_1,\,\dots,\,d_n)|\,&d_i=\frac{1}{2}x^TH_i(z_i)x,\\
& x \in \mathcal{B}_r,\, z_i \in \mathcal{B}_r(x^*)\}.
\end{split}$$
Defining $\tilde{x}(t):=x(t)-x^*$, (\[eq:lin\]) is in the form of (\[eq:linaff\]). Thus, the reachable set and tube of the nonlinear control system (\[eq:1\]) can be computed using Proposition \[prop1\] locally.
Reachable set computation using zonotopes
-----------------------------------------
Since set operations, such as linear transformation, addition and multiplication, are used extensively in the computation of reachable sets, a proper set representation can help expedite the computational process. To this end, zonotope representation is attractive for its efficiency in the aforementioned set operations (see, e.g., [@Girard05; @GirardGM06; @Althoff14]).
\[def:zonotope\] A *zonotope* is a set represented as $$\mathcal{Z}:=\left\{ x \in \mathbb{R}^n|\,x=c+\sum_{i=1}^l\lambda_ig^{(i)},\,\lambda_i \in[-1,\,1]\right\},$$ where $c,\,g^{(i)}(i=1,\,2,\,\dots,\,l) \in \mathbb{R}^n$ are called the central vector and generators, respectively; $l$ is the number of generators. It is often denoted as $\mathcal{Z}=(c,\,g^{(1)},\,\dots,\,g^{(l)})$.
The addition of two zonotopes $\mathcal{Z}_1=(c_1,\,g_1^{(1)},\,\dots,\,g_1^{(l_1)})$ and $\mathcal{Z}_2=(c_2,\,g_2^{(1)},\,\dots,\,g_2^{(l_2)})$ and the multiplication of a zonotope with a matrix $M \in \mathbb{R}^{n\times n}$ can be easily derived as $$\begin{split}
\mathcal{Z}_1 \oplus \mathcal{Z}_2 &= (c_1+c_2,\,g_1^{(1)},\,\dots,\,g_1^{(l_1)},\,g_2^{(1)},\,\dots,\,g_2^{(l_2)}),\\
M\mathcal{Z}_1 &=(Mc_1,\,Mg_1^{(1)},\,\dots,\,Mg_1^{(l_1)}).
\end{split}$$
For a zonotope with $l$ generators in $\mathbb{R}^n$, $l/n$ is called the *order* of the zonotope.
The set $\mathcal{B}_r$ with $r=(r_1,\cdots,r_n), r_i \in \mathbb{R}_{>0}$ can be written in the form of zonotope as $$\label{eq:br}
\mathcal{Z}_{\mathcal{B}_r}=(0,\,g_r^{(1)},g_r^{(2)},\cdots,,g_r^{(n)}),$$ where $g_r^{(i)} \in \mathbb{R}^n$ is a vector with all the elements being zero except that the $i$th element is $r_i$, $i=1,2,\cdots,n$.
The approximation error $\mathcal{D}_{x^*}(r)$ as in (\[eq:d\]) can be over-approximated using the quadratic map [@Althoff14]. Instead of computing $H_i(z_i)$ for every $z_i \in \mathcal{B}_r(x^*)$, we enclose it by an interval matrix $\overline{H}_i(x^*)$. Denote by $\overline{h}_{ij}$ the element of the $i$th row and $j$th column of $\overline{H}_i(x^*)$, then $\overline{h}_{ij}=[h_{ij}^l,h_{ij}^u]$, where $h_{ij}^l$ and $h_{ij}^u$ is the minimum and maximum values of $\overline{h}_{ij}$ in the linearization area respectively. Using $\mathcal{Z}_{\mathcal{B}_r}$ defined in (\[eq:br\]), we can compute an over-approximation of $\mathcal{D}_{x^*}(r)$ by $$\label{eq:dzono}
\mathcal{D}_{x^*}(r)\subseteq \overline{\mathcal{D}}_{x^*}(r):=\text{quad}(\overline{H}_i(x^*),\mathcal{Z}_{\mathcal{B}_r}),$$ where $\text{quad}(\cdot,\cdot)$ is the quadratic map defined in [@Althoff14].
The convex hull operation of two zonotopes can be over-approximated by (see [@Girard05; @Althoff10] for more details) $$\begin{split}
\overline{\text{CH}}(\mathcal{Z}_1,\mathcal{Z}_2)=\frac{1}{2} ( &c_1+c_2, g_1^{(1)}+g_2^{(1)}, \cdots, g_1^{(l)}+g_2^{(l)},\\
&c_1-c_2, g_1^{(1)}-g_2^{(1)},\cdots, g_1^{(l)}-g_2^{(l)} ).
\end{split}$$
To sum up, we give the following proposition, which aims to over-approximate the local reachable sets of nonlinear systems using zonotopes.
Given a nonlinear control system ${\mathcal T}$, the function $\Gamma_1:X \to \mathbb{R}^n_{\geq 0}$, an abstraction map $\Omega: X \to \hat{\mathcal{Q}}$ and a finite set of constant control actions $\hat{\mathcal{A}}$, for any $\hat{q} \in \hat{\mathcal{Q}}$ and $\hat{\mathbf{u}} \in \hat{\mathcal{A}}$ with $\hat{\mathbf{u}}(t)=\hat{u} \in U, \forall t \in [0,\tau]$, denote $$\label{eq:x}
\begin{split}
X_{\hat{q}}=\Omega^{-1}(\hat{q}) \oplus \mathcal{B}_{\Gamma_1(\hat{q})},\; \tilde{X}_{\hat{q}}=\{-\hat{q}\} \oplus X_{\hat{q}}.
\end{split}$$ The reachable set and tube $\mathcal{R}_{\hat{u},X_{\hat{q}}}(\tau)$ and $\mathcal{R}_{\hat{u},X_{\hat{q}}}([0,\tau])$ can be over-approximated by the sets $\overline{\mathcal{R}}_{\hat{u},X_{\hat{q}}}(\tau)$ and $\overline{\mathcal{R}}_{\hat{u},X_{\hat{q}}}([0,\tau])$, respectively, which are computed by $$\label{eq:reachset}
\overline{\mathcal{R}}_{\hat{u},X_{\hat{q}}}(\tau) = {\left\{\hat{q}\right\}} \oplus \tilde{Y}(\tau),$$ and $$\label{eq:reachtube}
\overline{\mathcal{R}}_{\hat{u},X_{\hat{q}}}([0,\tau]) = {\left\{\hat{q}\right\}} \oplus \overline{\text{CH}}(\tilde{X}_{\hat{q}}, \tilde{Y}(\tau) \oplus \mathcal{B}_{\alpha_\tau+\gamma_\tau}),$$ where $$\tilde{Y}(\tau)=e^{A_{\hat{q}}\tau}\tilde{X}_{\hat{q}} \oplus G(A_{\hat{q}},\tau)f(\hat{q},u) \oplus \tau \mathcal{D}_{\hat{q}}(r) \oplus \mathcal{B}_{\beta_\tau},$$ and $\alpha_\tau$, $\beta_\tau$, $\gamma_\tau$, $G(A_{\hat{q}},\tau)$ are defined as in Proposition \[prop1\].
Computation of Abstraction by Discretization and Zonotope Representation {#sec:discretization}
========================================================================
In this section, we discuss how to construct finite abstractions with robustness margins by grid-based discretization.
Grid-based discretization
-------------------------
Consider uniform parameters $\eta \in \mathbb{R}^n_{>0}$, $\mu \in \mathbb{R}^m_{>0}$ and a fixed sampling time $\tau_s \in \mathbb{R}_{>0}$. Let $\hat{\mathcal{Q}}=[X]_{\eta}$ be the set of states in $\hat{{\mathcal T}}$. In this case, $\Omega^{-1}(\hat{q}) \oplus \mathcal{B}_{\Gamma_{i}(\hat{q})}=\mathcal{B}_{\eta/2+\Gamma_{i}(\hat{q})}(\hat{q})$ ($i=1,2$). Using zonotopes with order 1, $X_{\hat{q}}$, $\tilde{X}_{\hat{q}}$ in (\[eq:x\]) become $$\begin{split}
X_{\hat{q}}&=(\hat{q},\,g_\eta^{(1)},g_\eta^{(2)},\cdots,,g_\eta^{(n)}),\\
\tilde{X}_{\hat{q}}&=(0,\,g_\eta^{(1)},g_\eta^{(2)},\cdots,,g_\eta^{(n)}),
\end{split}$$ where $g_\eta^{(i)} \in \mathbb{R}^n$ is a vector with all the elements are zero except the $i$th element being $\eta/2+\Gamma_1(\hat{q})$, $i=1,2,\cdots,n$.
The set of control actions $\hat{\mathcal{A}}$ only contains the control signals that take values in $[U]_{\mu}$ and the time duration are integral multiples of $\tau_s$. Since the computation of reachable sets and tubes are only valid within the linearization area $\mathcal{B}_r(\hat{q})$, the time duration and the value of the control signals should be determined to make sure that the transitions only take place inside it. Furthermore, in order to satisfy Theorem 1, this area should belong to $\mathcal{B}_\delta(\hat{q})$; in other words, $r \leq \delta$.
Algorithm for computing transitions
-----------------------------------
The algorithm for computing transitions is designed to collect all the valid transitions under a grid-based discretization according to Theorem 1. The main steps are devoted to solving the key problem of determining the valid control signal duration $\tau=k\tau_s,k \in \mathbb{N}$ (if it exists) for each element in $[U]_\mu$ and state in $\mathcal{\hat{Q}}$.
Similar to a lazy control strategy, which means that the control action is kept to be the same for as long as possible, we choose $\tau=\tau_{\text{max}}$, where $\tau_{\text{max}}$ is the maximum time of a control signal under which the system remains within a predefined linearization area. A practical consideration for this is that a short time duration can potentially introduce spurious self-transitions that do not exist in the original continuous system.
Out of simplicity in implementation, we use $\hat{\tau}_{\text{max}}=p^*\tau_s, p^* \in \mathbb{N}$ as an under-approximation of $\tau_{\mathrm{max}}$, and approach it iteratively using a lower bound $a$ and an upper bound $b$ ($a,\,b\in\mathbb{N}$ and $a \leq b$). The initial guess equals to the upper bound $b$. If the reachable set is fully inside the linearization area, which means $p^*\ge b$, the bounds shift to $[b,\,b+(b-a)]$; if the reachable set has already move outside the region, the bounds shrink to $[a,\,\lfloor \frac{a+b}{2} \rfloor]$. Considering the situation that reachable sets shrinks around the equilibriums, i.e., $\tau_{\mathrm{max}}=\infty$, we set an upper limit $N \in \mathbb{N}$ for $p$.
Algorithm 1 sketches the computation of transitions in a $(\Gamma_1,\Gamma_2,\delta)$-abstraction. For system (\[eq:1\]), we can use constant margins satisfying $\Gamma_{1,2} \geq 0$. For system (\[eq:1b\]), $\Gamma_2 \geq \varepsilon$ can be set as a constant, whereas the margin $\Gamma_1$ is not predefined, but chosen adaptively according to (\[eq:thm2\]).
$r,\tau_s,\eta,\hat{\mathcal{A}},\hat{\mathcal{Q}}$ and $\Delta$, $\varepsilon$ ($\Delta=0, \varepsilon=0$ for (\[eq:1\])) $\Gamma_1=\varepsilon$, $\Gamma_2 \equiv \varepsilon$, $\{ \rightarrow_{\hat{\mathcal{T}}}\} \leftarrow \varnothing$ Compute $f_{\hat{q}}$, $A_{\hat{q}}$, and $\overline{\mathcal{D}}_{\hat{q}}(r)$ by (\[eq:lin\]) and (\[eq:dzono\])
$X_0'=\varnothing$ $X_0'=X_0' \cup \overline{\mathcal{R}}_{\hat{v},\,\mathcal{B}_{\eta/2+\varepsilon}(\hat{q})}([0,\Delta])$ Choose $\Gamma_1$ s.t. $X_0' \oplus (-\mathcal{B}_{\eta/2}(\hat{q}))\subseteq \mathcal{B}_{\Gamma_1}$ $X_0 = \mathcal{B}_{\eta/2+\Gamma_1}(\hat{q})$, $X_R=\varnothing$
$p=p_0$, $a=0$, $b=p$ Compute $\overline{\mathcal{R}}_{\hat{u},X_0}(p\tau_s)$, $\overline{\mathcal{R}}_{\hat{u},X_0}([0,p\tau_s])$ $X_R=\overline{\mathcal{R}}_{\hat{u},X_0}(p\tau_s)$ $p=2b-a$, $a=b$, $b=p$ $p=\lfloor \frac{a+b}{2} \rfloor$, $b=p$ $\tau=a\tau_s$ $\{ \rightarrow_{\hat{\mathcal{T}}}\} \leftarrow (\hat{q},\,\hat{u},\,\tau,\,\hat{q}')$ $\{ \rightarrow_{\hat{\mathcal{T}}}\}$
Comparison with Lyapunov-based Approximation
============================================
We analyze the performance of the controllers synthesized using finite abstractions with robustness margins by two examples: the pendulum system [@PolaGT08]) and the automatic cruise control [@LiuO14].
Pendulum
--------
The pendulum model considered here is $$\begin{split}
\begin{bmatrix}
\dot{x}_1\\
\dot{x}_2
\end{bmatrix}
&=
\begin{bmatrix}
x_2\\
-\frac{g}{l}\sin x_1-\frac{k}{m}x_2+u
\end{bmatrix},\\
g&=9.8,\,l=5,\,m=0.5,\,k=3,\\
\end{split}$$ where $u \in U=[-1, 0], x \in X=[-0.5, 0]\times[-0.2, 0.2]$; $u$ is the normalized control torque; $x_1, x_2$ represent the angle (rad) and the angular rate (rad/s), respectively. The angle is measured from the perpendicular line to the current ball position. The positive direction is counter clockwise. The constants $g$, $l$, $m$, $k$ denote the gravity acceleration, rod length, mass, and friction coefficient, respectively.
The specification is given by an $\text{LTL}_{\setminus \bigcirc}$ formula $\varphi=\square \varphi_s \wedge \lozenge \square \varphi_t$ with $\varphi_s=X$ and $\varphi_t=[-0.3,\,-0.2] \times [-0.05,\,0.05]$. In our simulation, the abstraction parameters are $\tau_s=0.01s,\,r=[0.04;\,0.04],\,\eta=[0.02;\,0.02],\,\mu=0.01$. As shown in Fig. \[fig:pdl\] (left), the controlled system trajectory satisfies the given specification.
On the other hand, we fail to generate a controller using the abstraction based on Lyapunov-like method, as a result of its greater conservatism. We compare the number of transitions included by different reachable set computation methods. With the same partition, applying the control torque $u=-0.81$ at the state $x_1=-0.3$, $x_2=0.1$, the number of post states computed by our method is 4 while it is 49 using the Lyapunov-based method. As shown in Fig. \[fig:pdl\] (right), the one-step reachable set computed using our method is smaller than that using the Lyapunov-based method.
![Left: The trajectories of the controlled pendulum system states and the corresponding control signal. Right: Comparison of one-step reachable sets generated by two methods: region I indicates the linearization region $\mathcal{B}_r(\hat{q})$; region II is the initial set of states; region III is an over-approximation of the reachable set obtained by the proposed linearization-based method; region IV is an over-approximation of the reachable set obtained by an analytical bound using Lyapunov-based methods.[]{data-label="fig:pdl"}](pendulum)
Automatic cruise control
------------------------
Consider the longitudinal dynamics of automatic cruise control $$\dot{v}=u-c_0-c_1v^2,$$ where $v \in [20,\,30]$, $u \in [-1.5,\,1]$, $c_0=0.1$, and $c_1=0.00016$.
To design a controller satisfying the specification $\varphi=\square(v \leq 30) \wedge \lozenge \square (v \in [22,\,24])$, we set $\tau_s=0.3 \text{s},\,r=0.6,\,\eta=0.1,\,\mu=0.2$. In the simulations, the system is subjected to a maximum delay $d=0.01$s and a measurement error bound $\varepsilon=0.1$m/s. We construct three different abstractions: i) one without robustness margins; ii) one with uniform robustness margins (as defined in [@LiuO14]); iii) one with varying robustness margins (as defined in this paper). Fig. \[fig:cruise\] presents the simulation results of the cruise control system, under controllers synthesized using the first and the third abstractions, respectively. As observed from Fig. \[fig:cruise\] (left), the speed jumps out of the target range as the time lapses because the first abstraction cannot counteract delays or measurement errors, while the result from the third abstraction shown on the right of Fig. \[fig:cruise\] is satisfactory. To compare the second and the third abstractions, we look at their transitions around the state $v=21.4$m/s under the control input $u=0.15$. The second abstraction has $30$ transitions, whereas the third one has only $20$. In fact, due to its greater conservatism, the second abstraction is not able to generate a controller during control synthesis.
![Controlled state evolution synthesized from an abstraction with (right) and without (left) local robustness margins.[]{data-label="fig:cruise"}](acc)
Conclusion
==========
In this paper, we considered the problem of constructing finite abstractions for nonlinear systems that are suitable for synthesizing robust controllers. A notion of finite abstractions with robustness margins that vary with respect to the local dynamics was formally defined. One main contribution of our work was to apply local reachable sets computation techniques in computing finite transitions, which led to reduced degree of nondeterminism in the abstractions. The local reachable sets are computed by linearization and approximation error estimation. As illustrated by numerical examples, the abstractions generated by the proposed method contain fewer spurious transitions than those obtained from Lyapunov-based methods and therefore are more likely to render the control synthesis problem realizable. Future work will combine the abstraction procedures presented in this paper, which take into account local dynamics, with automated refinement procedures to mitigate potential state explosion problem.
[^1]: This work is supported, in part, by EU FP7 Grant PCIG13-GA-2013-617377 and NSF grant CNS-1446298. This is an extended version of the paper [@li2015computing] to appear in the Proceedings of the 2015 IFAC Conference on Analysis and Design of Hybrid Systems (ADHS), Atlanta, GA, USA, October 14-16, 2015.
[^2]: Yinan Li and Jun Liu are with the Department of Automatic Control and System Engineering, University of Sheffield, Sheffield, S1 3JD, UK (e-mails: [email protected]; [email protected]).
[^3]: Necmiye Ozay is with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109, USA (e-mail: [email protected]).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The 2D virtual try-on task has recently attracted a lot of interest from the research community, for its direct potential applications in online shopping as well as for its inherent and non-addressed scientific challenges. This task requires to fit an in-shop cloth image on the image of a person. It is highly challenging because it requires to warp the cloth on the target person while preserving its patterns and characteristics, and to compose the item with the person in a realistic manner. Current state-of-the-art models generate images with visible artifacts, due either to a pixel-level composition step or to the geometric transformation. In this paper, we propose WUTON: a Warping U-net for a Virtual Try-On system. It is a siamese U-net generator whose skip connections are geometrically transformed by a convolutional geometric matcher. The whole architecture is trained end-to-end with a multi-task loss including an adversarial one. This enables our network to generate and use realistic spatial transformations of the cloth to synthesize images of high visual quality. The proposed architecture can be trained end-to-end and allows us to advance towards a detail-preserving and photo-realistic 2D virtual try-on system. Our method outperforms the current state-of-the-art with visual results as well as with the Learned Perceptual Image Similarity (LPIPS) metric.[^1]'
author:
- |
Thibaut Issenhuth, Jérémie Mary, Clément Calauzènes\
Criteo AI Lab\
Paris, France\
` {t.issenhuth, j.mary, c.calauzenes}@criteo.com`
bibliography:
- 'references.bib'
title: 'End-to-End Learning of Geometric Deformations of Feature Maps for Virtual Try-On'
---
Introduction
============
A photo-realistic virtual try-on system would be a significant improvement for online shopping. Whether used to create catalogs of new products or to propose an immersive environment for shoppers, it could impact e-shop and open the door for new easy image editing possibilities. The training data we consider is made of *paired* images that are made of the picture of one cloth and the same cloth worn by a model. Then providing an *unpaired* tuple of images: one picture of cloth and one picture of a model with a different cloth, we aim to replace the cloth worn by the model.
An early line of work addressed this challenge using 3D measurements and model-based methods [@guan2012drape; @hahn2014subspace; @pons2017clothcap]. However, these are by nature computationally intensive and require expensive material, which would not be acceptable at scale for shoppers. Recent works aim to leverage deep generative models to tackle the virtual try-on problem [@jetchev2017conditional; @han2018viton; @wang2018toward; @dong2019towards]. CAGAN [@jetchev2017conditional] proposes a U-net based Cycle-GAN [@isola2017image] approach. However, this method fails to generate realistic results since these networks cannot handle large spatial deformations. In VITON [@han2018viton], the authors use the shape context matching algorithm [@belongie2002shape] to warp the cloth on a target person and learn an image composition with a U-net generator. To improve this model, CP-VTON [@wang2018toward] incorporates a convolutional geometric matcher [@rocco2017convolutional] which learns the parameters of geometric deformations (i.e thin-plate spline transform [@bookstein1989principal]) to align the cloth with the target person. In MG-VTON [@dong2019towards], the task is extended to a multi-pose try-on system, which requires to modify the pose as well as the upper-body cloth of the person.
In this second line of approach, a common practice is to use what we call a *human parser* which is a pre-trained system able to segment the area to replace on the model pictures: the upper-body cloth as well as neck and arms. In the rest of this work, we also assume this parser to be known.
The recent methods for a virtual try-on struggle to generate realistic spatial deformations, which is necessary to warp and render clothes with complex patterns. Indeed, with solid color tees, unrealistic deformations are not an issue because they are not visible. However, for a tee-shirt with stripes or patterns, it will produce unappealing images with curves, compressions and decompressions. Figure \[fig:comparison\] shows these kinds of unrealistic geometric deformations generated by CP-VTON [@wang2018toward].
To alleviate this issue, we propose an end-to-end model composed of two modules: a convolutional geometric matcher [@rocco2017convolutional] and a siamese U-net generator. We train end-to-end, so the geometric matcher benefits from the losses induced by the final synthetic picture generation. Our architecture removes the need for a final image composition step and generates images of high visual quality with realistic geometric deformations. Main contributions of this work are:
- We propose a simple end-to-end architecture able to generate realistic deformations to preserve complex patterns on clothes such as stripes. This is made by back-propagating the loss computed on the final synthesized images to a learnable geometric matcher.
- We suppress the need for a final composition step present in the best current approaches such as [@wang2018toward] using an adversarially trained generator. This performs better on the borders of the replaced object and provides a more natural look of shadows and contrasts.
- We show that our approach significantly outperforms the state-of-the-art with visual results and with a quantitative metric measuring image similarity, LPIPS [@zhang2018unreasonable].
- We identify the contribution of each part of our net by an ablation study. Moreover, we exhibit a good resistance to low-quality human parser at inference time.
Problem statement and related work
==================================
Given the 2D images $p \in \mathbb{R}^{h\times w \times 3}$ of a person and $c \in \mathbb{R}^{h\times w \times3}$ of a clothing item, we want to generate the image $\Tilde{p} \in \mathbb{R}^{h\times w\times 3}$ where the person from $p$ wears the cloth from $c$. The task can be separated in two parts : the geometric deformation $T$ required to align $c$ with $p$, and the refinement that fits the aligned cloth $\Tilde{c} = T(c)$ on $p$. These two sub-tasks can be modelled with learnable neural networks, i.e spatial transformers networks $STN$ [@jaderberg2015spatial; @rocco2017convolutional] that output parameters $\theta = STN(p,c)$ of geometric deformations, and conditional generative networks $G$ that give $\Tilde{p} = G(p,c,\theta)$.
Because it would be costly to construct a dataset with $(p,c,\Tilde{p})$ triplets, previous works [@han2018viton; @wang2018toward] propose to use an agnostic person representation $ap \in \mathbb{R}^{h\times w\times c}$ where the clothing items in $p$ are hidden but identity and shape of the persons are preserved. $ap$ is built with pre-trained human parsers and pose estimators : $ap = h(p) $. These triplets $(ap,c,p)$ allow to train for reconstrution. $(ap,c)$ are the inputs, $\Tilde{p}$ the output and $p$ the ground-truth. We finally have the conditional generative process : $$\Tilde{p} = G(\underbrace{h(p)}_{\text{agnostic person}},\underbrace{c}_{\text{cloth}},\underbrace{STN(h(p),c)}_{\text{geometric transform}})$$
#### **Conditional image generation.**
Generative models for image synthesis have shown impressive results with the arrival of adversarial training [@goodfellow2014generative]. Combined with deep networks [@radford2015unsupervised], this approach has been extended to conditional image generation in [@mirza2014conditional] and performs increasingly well on a wide range of tasks, from image-to-image translation [@isola2017image; @zhu2017unpaired] to video editing [@shetty2018adversarial]. However, these models cannot handle large spatial deformations and fail to modify the shape of objects [@mejjati2018unsupervised], which is necessary for a virtual try-on system.
#### **Image composition.**
Recent approaches for image composition combine STNs [@jaderberg2015spatial] with GANs to align and merge two images. In [@lin2018st], Lin et al. use a sequence of warp generated by an STN to place a foreground object in a background image. Recently, SF-GAN [@zhan2018spatial] separated the task in two stages: an STN warping the object, and a refinement network adapting the texture and appearance of the object.
#### **Geometric deformations in generative models.**
The problem of handling large spatial deformations in generative models has mainly been studied in the context of pose-guided person image generation. This task consists in generating a person image, given a reference person and a target pose. Some approaches use disentanglement to separate pose and shape information from appearance, which allows reconstructing a reference person in a different pose [@ma2018disentangled; @lorenz2019unsupervised; @esser2018variational]. However, recent state-of-the-art approaches for pose-guided person generation include explicit spatial transformations in their architecture, whether learnt [@jaderberg2015spatial] or not. In [@balakrishnan2018synthesizing], the different body parts of a person are segmented and moved to the target pose by part-specific learnable affine transformations, which are applied at the pixel level. The deformable GAN from [@siarohin2018deformable] is a U-net [@ronneberger2015u] generator whose skip connections are deformed by part-specific affine transformations. These transformations are computed from the source and target pose information. Instead, [@dong2018soft] use the convolutional geometric matcher from [@rocco2017convolutional] to learn a thin-plate-spline (TPS) transform between the source human parsing and a synthesized target parsing, and align the deep feature maps of an encoder-decoder generator.
#### **Appearance transfer.**
Close to the virtual try-on task, there is a body of work on human appearance transfer. Given two images of different persons, the goal is to transfer the appearance of a part of the person A on the person B. Approaches using pose and appearance disentanglement mentioned in the previous section [@lorenz2019unsupervised; @ma2018disentangled] fit this task but others are specifically designed for it. SwapNet [@raj2018swapnet] propose a dual path network to generate a new human parsing of the reference person and region of interest pooling to transfer the texture. In [@wu2018m2e], the method relies on DensePose information [@alp2018densepose], which provides a 3D surface estimation of a human body, to perform a warping and align the two persons. The transfer is then done with segmentation masks and refinement networks.
#### **Virtual try-on.**
Most of the approaches for a virtual try-on system come from computer graphics and rely on 3D measurements or representations. Drape [@guan2012drape] learns a deformation model to render clothes on 3D bodies of different shapes. In [@hahn2014subspace], Hahn et al. use subspace methods to accelerate physics-based simulations and generate realistic wrinkles. ClothCap [@pons2017clothcap] aligns a 3D cloth-template to each frame of a sequence of 3D scans of a person in motion. However, these methods are targetting the dressing of virtual avatars, e.g for the gaming or movie industry.
The task we are interested in is the one introduced in CAGAN [@jetchev2017conditional] and further studied by VITON [@han2018viton] and CP-VTON [@wang2018toward], which we defined in the problem statement. In CAGAN [@jetchev2017conditional], Jetchev et al. propose a cycle-GAN approach that requires three images as input: the reference person, the cloth worn by the person and the target in-shop cloth. Thus, it limits its practical uses. VITON [@han2018viton] proposes to learn a generative composition between the warped cloth and a coarse result. The warping is done with a non-parametric geometric transform [@belongie2002shape]. To improve this model, CP-VTON [@wang2018toward] incorporates a learnable geometric matcher $STN$ [@rocco2017convolutional] which aligns $c$ with $p$.
Our approach
============
![\[fig:architecture\] WUTON : our proposed end-to-end warping U-net architecture. Dotted arrows correspond to the forward pass only performed during training. Green arrows are the human parser. The geometric transforms share the same parameters but do not operate on the same spaces. The different training procedure for paired and unpaired pictures is explained in section \[sec:training\].](warping_unet.pdf){width="\linewidth"}
Our task is to build a virtual try-on system that is able to fit a given in-shop cloth on a reference person. We propose a novel architecture trainable end-to-end and composed of two existing modules, a convolutional geometric matcher $STN$ [@rocco2017convolutional] and a U-net [@ronneberger2015u] generator $G$ whose skip connections are deformed by $STN$. The joint training of $G$ and $STN$ allows us to generate realistic deformations that help to synthesize high-quality images. Also, we use an adversarial loss to make the training procedure closer to the actual use of the system which is to replace clothes in the unpaired situation. In previous works [@han2018viton; @wang2018toward; @dong2019towards], the generator is only trained to reconstruct images with supervised triplets (*ap*, *c*, *p*) extracted from the *paired*. Thus, when generating images in the test-setting, it can struggle to generalize and to warp clothes different from the one worn by the reference person. The adversarial training allows us to train our network in the test-setting, where one wants to fit a cloth on a reference person wearing another cloth.
Warping U-net
-------------
Our warping U-net is composed of two connected modules, as shown in Fig.\[fig:architecture\]. The first one is a convolutional geometric matcher, which has a similar architecture as [@rocco2017convolutional; @wang2018toward]. It outputs the parameters $\theta$ of a geometric transformation, a TPS transform in our case. This geometric transformation aligns the in-shop cloth image with the reference person. However, in contrast to previous work [@han2018viton; @wang2018toward; @dong2019towards], we use the geometric transformation on the features maps of the generator rather than at a pixel-level. Thus, we learn to deform the feature maps that pass through the skip connections of the second module, a U-net [@ronneberger2015u] generator which synthesizes the output image $\Tilde{p}$.
The architecture of the convolutional geometric matcher is taken from CP-VTON [@wang2018toward], which reuses the generic geometric matcher from [@rocco2017convolutional]. It is composed of two feature extractors $F_1$ and $F_2$, which are standard convolutional neural networks. The local vectors of feature maps $F_1(c)$ and $F_2(ap)$ are then L2-normalized and a correlation map $C$ is computed as follows : $$C_{ijk} = F_{1_{i,j}}(c) \cdot F_{2_{m,n}}(c)$$ where k is the index for the position (m, n). This correlation map captures dependencies between distant locations of the two feature maps, which is useful to align the two images. $C$ is the input of a regression network, which outputs the parameters $\theta$ and allows to perform the geometric transformation $T_{\theta}$. We use TPS transformations [@bookstein1989principal], which generate smooth sampling grids given control points. Since we transform deep feature maps of a U-net generator, we generate a sampling grid for each scale of the U-net with the same parameters $\theta$.
The input of the U-net generator is also the tuple of pictures $(ap,c)$. Since these two images are not spatially aligned, we cannot simply concatenate them and feed a standard U-net. To alleviate this, we use two different encoders $E_1$ and $E_2$ processing each image independently and with non-shared parameters. Then, the feature maps of the in-shop cloth $E_1(c)$ are transformed at each scale $i$: $E^i_1(c) = T_{\theta} (E^i_1(c))$. Then, the feature maps of the two encoders are concatenated and feed the decoder. With aligned feature maps, the generator is able to compose them and to produce realistic results. Because we simply concatenate the feature maps and let the U-net decoder compose them instead of enforcing a pixel-level composition, experiments will show that it has more flexibility and can produce more natural results. We use instance normalization in the U-net generator, which is more effective than batch normalization [@ioffe2015batch] for image generation [@ulyanov2017improved].
Training procedure {#sec:training}
------------------
Along with a new architecture for the virtual try-on task (Fig. \[fig:architecture\]), we also propose a new training procedure, i.e. a different data representation and an adversarial loss for unpaired images.
While previous works use a rich person representation with more than 20 channels representing human pose, body shape and the RGB image of the head, we only mask the upper-body of the reference person. Our agnostic person representation $ap$ is thus a 3-channel RGB image with a masked area. We compute the upper-body mask from pose and body parsing information provided by a pre-trained neural network from [@liang2019look]. Precisely, we mask the areas corresponding to the arms, the upper-body cloth and a fixed bounding box around the neck keypoint. However, we show in an ablation study that our method is not sensitive to non-accurate masks at inference time since it can generate satisfying images with simple bounding box masks.
Using the dataset from [@dong2019towards], we have pairs of in-shop cloth image $c_a$ and a person wearing the same cloth $p_a$. Using a human parser and a human pose estimator, we generate $ap$. From the parsing information, we can also isolate the cloth on the image $p_a$ and get $c_{a,p}$, the cloth worn by the reference person. Moreover, we get the image of another in-shop cloth, $c_b$. The inputs of our network are the two tuples $(ap, c_a)$ and $(ap, c_b)$. The outputs are respectively $(\Tilde{p_a}, \theta_a)$ and $(\Tilde{p_b}, \theta_b)$.
The cloth worn by the person $c_{a,p}$ allows us to guide directly the geometric matcher with a $L_1$ loss: $$L_{warp} = \lVert T_{\theta_a} (c) - c_{a,p} \rVert_1$$ The image $p$ of the reference person provides a supervision for the whole pipeline. Similarly to CP-VTON [@wang2018toward], we use two different losses to guide the generation of the final image $\Tilde{p}_a$, the pixel-level $L_1$ loss $\lVert \Tilde{p}_a - p_a \rVert_1$ and the perceptual loss [@johnson2016perceptual]. We focus on $L_1$ losses since they are known to generate less blur than $L_2$ for image generation [@zhao2016loss]. The latter consists of using the features extracted with a pre-trained neural network, VGG [@simonyan2014very] in our case. Specifically, our perceptual loss is: $$L_{perceptual} = \sum_{i = 1}^{5} \lVert \phi_i(\Tilde{p}_a) - \phi_i(p_a) \rVert_1$$ where $\phi_i (I)$ are the feature maps of an image I extracted at the i-th layer of the VGG network. Furthermore, we exploit adversarial training to train the network to fit $c_b$ on the same agnostic person representation $ap$, which is extracted from a person wearing $c_a$. This is only feasible with an adversarial loss, since there is no available ground-truth for this pair $(ap, c_b)$. Thus, we feed the discriminator with the synthesized image $\Tilde{p}_b$ and real images of persons from the dataset. This adversarial loss is also back-propagated to the convolutional geometric matcher, which allows to generate much more realistic spatial transformations. We use the relativistic adversarial loss [@jolicoeur-martineau2018] with gradient-penalty [@gulrajani2017improved; @arjovsky2017wasserstein], which trains the discriminator to predict relative realness of real images compared to synthesized ones. Finally, the objective function of our network is : $$L = \lambda_{w} L_{warp} + \lambda_{p} L_{perceptual} + \lambda_{L_1} L_1 + \lambda_{adv} L_{adv}$$ We use the Adam optimizer [@kingma2014adam] to train our network.
Experiments and analysis
========================
----------- -------- --------- -------
Reference Target CP-VTON WUTON
person cloth
----------- -------- --------- -------
\[fig:comparison\]
------------ -------- ----------- ----------- --------- ------------ -- --
Reference Target Unpaired Paired No adv. Not
person cloth adv. loss adv. loss loss end-to-end
\[-0.8ex\]
\[-0.8ex\]
------------ -------- ----------- ----------- --------- ------------ -- --
-------- -------- ------- ----------- -------- -------
Masked Target WUTON Reference Target WUTON
image cloth person cloth
-------- -------- ------- ----------- -------- -------
\[fig:failure\_cases\]
We first describe the dataset. Then, we compare our approach with CP-VTON [^2] [@wang2018toward], the current state-of-the-art for the virtual try-on task. We present visual and quantitative results proving that WUTON significantly outperforms the current state-of-the-art for a virtual try-on. Finally, we describe the impact of each main component of our approach in an ablation study and show that WUTON can also generate high-quality images with a non-accurate mask at inference time.
Dataset
-------
We use the *Image-based Multi-pose Virtual try-on* dataset[^3] from MG-VTON [@dong2019towards]. This dataset contains 35,687/13,524 person/cloth images at (256,192) resolution. For each in-shop cloth image, there are multiple images of a model wearing the given cloth from different views and in different poses. We remove images tagged as back images since the in-shop cloth image is only from the front. We process the images with a neural human parser and pose estimator, specifically the joint body parsing and pose estimation (JPP) network[^4] [@liang2019look; @gong2017look].
Visual results
--------------
Visual results of our method and CP-VTON are shown in Fig. \[fig:comparison\]. CP-VTON has trouble to realistically deform and render complex patterns like stripes or flowers. Control points of the $T_\theta$ transform are visible and lead to unrealistic curves and deformations on the clothes. Also, the edges of cloth patterns and body contours are blurred.
Our method surpasses the previous state-of-the-art on different challenges. On the two first rows, our method generates spatial transformations of a much higher visual quality, which is specifically visible for stripes (2nd row). It is able to preserve complex visual patterns of clothes and presents less blur than CP-VTON on the edges. Also, it can distinguish the relevant parts of the in-shop cloth image (3rd row). Generally, our method generates results of high visual quality while preserving the characteristics of the target cloth. We also show some failure cases in Fig. \[fig:failure\_cases\]. Problems happen when the human parser does not properly detect the original cloth or when models have uncommon poses.
LPIPS metric
------------
To further evaluate our method, we use the linear perceptual image patch similarity (LPIPS) metric developed in [@zhang2018unreasonable]. This metric is very similar to the perceptual loss we use in training (see Section 3.2) since the idea is to use the feature maps extracted by a pre-trained neural network to quantify the perceptual difference between two images. Different from the basic perceptual loss, they first unit-normalize each layer in the channel dimension and then rescale by learned weights $w_i \in \mathbb{R}^{C_i}$ : $$LPIPS(\Tilde{p}_a,p_a) = \sum_{i = 1}^{5} \frac{1}{H_i W_i} \sum_{h,w} \lVert w_i \cdot ( \phi^i_{h,w}(\Tilde{p}_a) - \phi^i_{h,w}(p_a) ) \rVert_2^2$$ where $\phi^i_{h,w}$ is the unit-normalized vector at i-th layer and (h,w) is the spatial location extracted by a neural network, AlexNet [@krizhevsky2014one] in their case.
We evaluate the LPIPS on the test set. We can only use this method in the paired setting since there is no available ground-truth in the unpaired setting. Thus, it does not exactly evaluate the real task we aim for. Results are shown in Table \[table:lpips\]. Our approach significantly outperforms the state-of-the-art on this metric. Here the best model uses adversarial loss on paired data, but visual investigation suggests that the unpaired adversarial loss is better in the real use case of our work. We evaluate CP-VTON [@wang2018toward] on their agnostic person representation $ap_{viton}$ (20 channels with RGB image of head and shape/pose information) and on our $ap_{wuton}$.
[2]{}
Method LPIPS
------------------------- ------------------- --
CP-VTON on $ap_{viton}$ 0.182 $\pm$ 0.049
CP-VTON on $ap_{wuton}$ 0.131 $\pm$ 0.058
WUTON 0.101 $\pm$ 0.047
**Impact of loss**
**functions on WUTON:**
W/o adv. loss 0.107 $\pm$ 0.049
W. paired adv. loss 0.099 $\pm$ 0.046
Not end-to-end 0.112 $\pm$ 0.053
: LPIPS metric on paired setting. Lower is better, $\pm$ reports std. dev.[]{data-label="table:lpips"}
Method LPIPS
---------------------------- ------------------- --
**Impact of composition**
**on WUTON:**
W. composition 0.105 $\pm$ 0.047
**Impact of mask quality**
**box masked person:**
CP-VTON 0.185 $\pm$ 0.078
WUTON 0.151 $\pm$ 0.069
: LPIPS metric on paired setting. Lower is better, $\pm$ reports std. dev.[]{data-label="table:lpips"}
Ablation studies
----------------
To prove the effectiveness of our approach, we perform several ablation studies. In Fig. \[fig:loss\], we show visual comparisons of different variants of our approach: our WUTON with unpaired adversarial loss; with an adversarial loss on paired data (i.e the adversarial loss is computed with the same synthesized image as the L1 and VGG losses); without the adversarial loss; without back-propagating the loss of the synthesized images ($L_1, L_{perceptual}, L_{adv}$) to the geometric matcher.
The adversarial loss generates sharper images and improves the contrast. This is confirmed by the LPIPS metric in Table \[table:lpips\] and with visual results in Fig. \[fig:loss\]. With the unpaired adversarial setting, the system better handles large variations between the shape of the cloth worn by the person and the shape of the new cloth. The results in Fig. \[fig:loss\] as well as the LPIPS score in Table \[table:lpips\] show the importance of our end-to-end learning of geometric deformations. When the geometric matcher only benefits from $L_{warp}$, it only learns to align $c$ with the masked area in $ap$. However, it does not preserve the inner structure of the cloth. Back-propagating the loss computed on the synthesized images $\Tilde{p}$ alleviates this issue. Finally, our approach removes the need for learning a composition between the warped cloth and a coarse result. To prove it, we re-design our U-net to generate a coarse result and a composition mask. The synthesized image is then the composition between the coarse result and the warped cloth. With this configuration, the LPIPS score slightly decreases.
We also show that our method can generate realistic results if the human parser is not accurate at inference time. Hence, we train and test our method with the upper-body of the person masked by a gray bounding box. It is to be noted that we still require the accurate human parsing during training for the warping loss $L_{warp}$.
We also tried to learn the architecture without using $L_{warp}$, but this lead to major trouble with the convergence of the networks. On one hand, a possible future direction is to try to reduce the dependency to the human parser by learning the segmentation in a self-supervised way (providing several pictures of the same cloth on different models or using videos). On the other hand, the presence of this parser can ease the handling of pictures with complex backgrounds.
Conclusion
==========
In this work, we propose an architecture trainable end-to-end which combines a U-net with a convolutional geometric matcher and significantly outperforms the state-of-the-art for the virtual try-on task. The end-to-end training procedure with an unpaired adversarial loss allows to generate realistic geometric deformations, synthesize sharp images with proper contrast and preserve complex patterns, including stripes and logos.
[^1]: Code and implementation details are available at <span style="font-variant:small-caps;">anonymized</span> and supplementary material
[^2]: We use the public implementation from <https://github.com/sergeywong/cp-vton>.
[^3]: The dataset is available at: <http://sysu-hcp.net/lip/overview.php>
[^4]: This human parser and pose estimator is open-source and available at: <https://github.com/Engineering-Course/LIP_JPPNet>.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'It is known that silicon is an indirect band gap material, reducing its efficiency in photovoltaic applications. Using surface plasmons in metallic nanoparticles embedded in a solar cell has recently been proposed as a way to increase the efficiency of thin film silicon solar cells. The dipole mode that dominates the plasmons in small particles produces an electric field having Fourier components with all wave numbers. In this work, we show that such a field creates electron-hole-pairs without phonon assistance, and discuss the importance of this effect compared to radiation from the particle and losses due to heating.'
author:
- 'M. Kirkengen'
- 'J. Bergli'
- 'Y. M. Galperin'
title: 'Direct generation of charge carriers in c-Si solar cells due to embedded nanoparticles'
---
Introduction
============
Present day solar cell industry is completely dominated by the use of silicon as the active material. It’s main advantages are availability, disposability and several decades of industrial metallurgical development, compared to the poisonous or rare elements of, e.g., GaAs. However, silicon is not an ideal material for solar cells. One disadvantage is that it has an indirect band gap. This means that photons with energy close to the band gap can only be absorbed in phonon-assisted processes. Therefore, the absorption of these photons is weak, and the silicon wafer can not be made too thin if one is to absorb this part of the solar spectrum. The material costs and limited production capacity for solar grade silicon mean that the thickness required in todays first generation solar cells is a significant obstacle to their commercial success. Also, due to the limited lifetime of the electron-hole pairs, thicker cells may suffer from larger recombination rate and reduced efficiency.
The question is thus how to increase the optical path lengths of near band gap photons inside the silicon, without increasing wafer thickness. Several approaches have been tried, including texturing of the wafer front or rear surface in various patterns and on different length scales. For length scales larger than the wave length, the incoming light is refracted into angles more parallel to the wafer.[@Green] For length scales close to or smaller than the wavelength, diffraction may couple light into guided modes.[@Sheng] However, the texturing will often lead to an increase in surface defect states and thereby increase recombination rates.
As an alternative to texturing, it has been proposed to place metallic nanoparticles near the surface of the wafer.[@Catchpole] The nanoparticles scatter the incoming light through a surface plasmon resonance. Surface plasmons, or surface plasmon polaritons, are electron density fluctuations at the interface between a metal and a dielectric material. For a good introduction, see, e.g., Raether[@Raether] for plasmons in general, and Bohren&Huffman[@Bohren] for plasmons on small particles. On the surface of nanoparticles, the plasmons can be excited by an incoming plane wave, and they exhibit a marked, tunable resonance. For frequencies near the resonance, nanoparticles have an optical cross section much larger than their geometrical cross section. If this resonance could be tuned to match the band gap of silicon, near bandgap photons could be absorbed into the plasmon state with high probability, while higher energy photons would be unaffected. Certain progress has already been made in the plasmon tuning, e.g., at the University of New South Wales, [@Pillai1; @Pillai2] but further development is required.
We believe that the energy of the surface plasmons can then be used to create electron-hole pairs in two ways. First, the energy can be emitted as light in directions along the wafer. This gives a longer optical path inside the wafer, and thereby increases the indirect absorption.[@Pillai1]
Second, the near field of the nanoparticles can excite electron-hole pairs without phonon assistance, the momentum being transferred to the nanoparticle. This second process has to our knowledge not been considered in the literature, and is the subject of the present paper. Our results indicate that this mechanism will give an extra contribution to the electron-hole pair generation, compared to estimates that only take into account the re-radiation of power, increasing the relative benefit of introducing the nanoparticles.
Some of the plasmon energy goes into heating of the nanoparticles and is obviously lost. This loss should be compared with the losses due to the limited optical path when not exploiting the plasmons, or, if texturing is used to increase the optical path length, with increased losses due to recombination at interfaces. The plasmons will give an improved efficiency if the losses to heating are smaller than previous losses due to optical path length or recombinations. If the resonance is properly tuned, the only photons significantly affected will be those that would otherwise be lost. Any fraction of this near bandgap light that can be used efficiently contributes to a net gain for the cell.
For particles larger than the wavelength, a large fraction of the light will be reflected rather than excite plasmons.[@Bohren] We therefore consider only particles smaller than the wavelength of the incoming light. For such particles, the plasmons can be approximated by a dipole mode, corresponding to uniform polarization of the nanoparticle.[@Bohren] While the dipole approximation is usually only accepted for particles with diameter less than one tenth of a wavelength, we accept it as a first approximation for our order of magnitude estimates. We are not aware of any studies of how the near field is changed by an interface between the layer embeddig nanoparticles and the active layer of the solar cell. For simplicity, we will therefore restrict the further discussion to the case of an electric dipole located inside an infinite medium consisting of silicon. The dipole is excited by an incoming plane wave.
The far-field energy radiated from the dipole represents the maximum energy that can be absorbed by indirect absorption. In real applications, some of this light will inevitably be lost. The presence of an interface may also increase the total emission,[@Mertz; @Benisty] but for the sake of our order of magnitude estimates, we will ignore this effect.
The goal of this paper is to demonstrate that the direct absorption effect should be considered when modeling the effect of plasmons, and that it may have important implications for the optimal sizing and positioning of the plasmons. The fact that plasmons can lead to an increase in efficiency has been experimentally verified.[@Pillai2; @Derkacs] We therefore focus on the *relative* importance of the two mechanisms that could contribute to the increase, and how this could influence cell design considerations.
Theory {#Theory}
======
We use classical electrodynamics to describe both the nanoparticles and the fields. The interaction with the silicon is described by perturbation theory, and we use the tight binding model and the parabolic approximation of the band gap extrema for the wave function of the silicon.
We consider the incoming light to have a frequency close to the band gap of silicon. This corresponds to $\hbar \omega = 1.1$ eV, or $\omega \approx 10^{15}$ s$^{-1}$. We then get for the wavelength of this radiation $\lambda = 2 \pi c / \omega \approx 1$ $\mu$m , and from $c=\omega/k_p$ we get the photon wavenumber $k_p\approx 6\cdot
10^6$ m$^{-1}$.
The vector potential due to a dipole is given as:[@Lorrain] $$\begin{aligned}
\mathbf{A}
&=& \frac{i \omega \mu_0}{4 \pi r} e^{-i(k_p r-\omega t)}\mathbf{p}_0= i
\mathbf{A}_0 A_r\, , \\
\mathbf{A}_0 &=& \frac{\omega \mu_0 k_p p_0}{4 \pi}\mathbf{e_p},\;\;\;\nonumber
A_r=\frac{1}{k_p r} e^{-i(k_p r-\omega t)} \end{aligned}$$ where $\mathbf{p}_0$ is the dipole moment, and $p_0 =
|\mathbf{p}_0|$. $r$ is the distance from the dipole. Using the previous rough estimates for $\omega$ and $k_p$ we get that $A_0/p_0
\approx 10^{15}$ Js/C$^2$m$^2$, while $A_r$ is a dimensionless function containing all spatial dependencies of $\mathbf{A}$. The magnitude of $p_0$ will be addressed later, but is not necessary for the following comparisons of different terms. The scalar potential can be cast in the form $\Phi_0 \Phi_r$ where $$\Phi_0=\frac{k_p^2 p_0}{4 \pi \epsilon_0}, \;\;\;
\Phi_r=\frac{\cos \theta}{k_p r}(i + \frac{1}{k_p r})e^{-i(k_p r -
\omega t)}\, ,$$ $\theta$ is the angle from the dipole axis. We can estimate $\Phi_0/p_0 \approx 3 \cdot 10^{23}$ J/C$^2$m, while $\Phi_r$ is again dimensionless.
The Hamiltonian of the system is:[@LL] $$\begin{aligned}
H&=&\frac{(-i \hbar \nabla+ e \mathbf{A})^2}{2m}-e\Phi \\ &=&
-\frac{\hbar^2 \nabla^2}{2m}
-\frac{i e\hbar ( \nabla\cdot\mathbf{A}+\mathbf{A}\cdot\nabla)}{2m}
+\frac{e^2 \mathbf{A}^2}{2m}
- e\Phi\end{aligned}$$ where $e$ is the positive elementary charge. The $\mathbf{A}^2$-term can safely be neglected. Using the Lorentz gauge, $\nabla \cdot \mathbf{A} +
c^2 \dot{\Phi} = 0$, where $c =(\epsilon_0 \mu_0)^{1/2}$, we rewrite the interaction Hamiltonian as $$\begin{aligned}
H_{\text{int}}&=& A_r \mathbf{C}_a\cdot\nabla - \Phi_r C_b \, ; \\
\mathbf{C_a}&=&2\mu_{\text{B}} \mathbf{A}_0\, , \quad
C_b = \nonumber
\left(
1 + \frac{\hbar \omega}{2 m c^2}
\right) e \Phi_0 \end{aligned}$$ where $\mu_B$ is the Bohr magneton. Since $mc^2 \gg \hbar \omega$, $C_b\approx e \Phi_0$.
For the wave function, we use the standard tight binding approximation,[@Singh] writing the wave function as $$\Psi_\mathbf{k}(r) =
e^{i{\mathbf{k}\cdot\mathbf{r}}}\frac{1}{\sqrt{N_a}}
\sum_n \sum_l b_{\mathbf{k}l} \Psi_{nl}(r)$$ where $\Psi_{nl}(r) = \Psi_l(r-\mathbf{R}_n)$ is the $l$-th orbital corresponding to the atomic wave function centered at the $n$-th atom located at $\mathbf{R_n}$. The parameters $b_{\mathbf{k}l}$ can in principle be found for each point in $k$-space. For states at the valence band maximum, there seems to be good agreement between theory and experiment. For the conduction band minimum, the fitting parameters are still optimized either for the position in $k$-space or for the effective mass in different directions, depending on what is considered the most important. Based on Klimeck *et al.*[@Klimeck; @Martins] we still assume that the minimum can be described by a combination of single electron $p$, $s$ and $s^*$ states, where $s^*$ is an excited $s$-state.
The transition rate for each $\mathbf{k'},\mathbf{k}$ can then be found using Fermi’s golden rule, $$W_{\mathbf{k'k}} = \frac{2 \pi}{\hbar}\left| \langle
\mathbf{k'}|H_{\text{int}}|\mathbf{k}\rangle \right|^2
\delta(E_\mathbf{k'} - E_\mathbf{k}-\hbar \omega)$$ where $\mathbf{k'}$ denotes the final state, and $\mathbf{k}$ the initial one. In the following calculations, we will assume that the final state is near the conduction band minimum. There are six equivalent such minima, the effect of this will be addressed later. The absorbed energy by direct pair creation is given as $P_d=\sum_{\mathbf{k',k}} \hbar \omega
W_{\mathbf{k',k}}$. We calculate this using the parabolic approximation that is valid close to the band edges. Since we are interested in initial states close to the top of the valence band and final states close to a minimum in the conduction band one can assume that the interaction matrix element is weakly dependent on $\mathbf{k}$ and $\mathbf{k}'$, $W_{\mathbf{k}'\mathbf{k}}\approx W_{\mathbf{k_0}\mathbf{0}}$.
Writing $\mathbf{k''}=\mathbf{k}'-\mathbf{k_0}$ we get $$E_\mathbf{k'} = E_g + \frac{\hbar^2 \mathbf{k}''^2}{2 m_c}, E_\mathbf{k} = - \frac{\hbar^2 \mathbf{k}^2}{2 m_v},$$ and using $$\sum_{\mathbf{k}}= V \int d^3\mathbf{k} \frac{1}{(2 \pi)^3}=\frac{4 \pi V}{(2 \pi)^3}\int k^2 \frac{dk}{dE}dE\\$$ this gives $$\begin{aligned}
&\sum_{k}&W_{\mathbf{k}'\mathbf{k}}\delta(E_f-E_i-\hbar \omega)\\
\nonumber &=&
W_{\mathbf{k_0}\mathbf{0}}\frac{2^5 \pi^2 V^2}{(2 \pi)^6\hbar^6}(m_cm_v)^{\frac{3}{2}}\times\\
&\nonumber&
\int_{E_g}^{\infty} dE_f \int_{0}^{-\infty} -dE_i
\sqrt{(E_f-E_g)(-E_i)}\delta(E_f-E_i-\hbar \omega)\\
&=&
W_{\mathbf{k_0}\mathbf{0}}\frac{V^2(m_cm_v)^{\frac{3}{2}}}{2^4 \pi^3\hbar^6}(\hbar\omega - E_g)^2\end{aligned}$$ where $m_c,m_v$ are the effective masses of the valence and conduction bands, respectively, and $E_g$ is the gap energy.
Note that the energy dependence of the absorption is the same as that of indirect absorption, rather than that for direct absorption in direct band gap semiconductors, for which it is proportional to $(\hbar\omega
- E_g)^{1/2}$. The reason for this is the spread of Fourier components in the dipole field, which take the role of the spread in phonon wave numbers in the case of indirect absorption.
Calculations
============
We term the power emitted as radiation $P_r$, the power lost to heating $P_h$, and the power going into direct electron-hole pair generation $P_d$.
After standard calculations we get the absorption:
$$\begin{aligned}
\label{Pd}
P_d &=& \nonumber
\frac{2(m_c m_v)^\frac{3}{2}\omega (\hbar\omega-E_g)^2}{\hbar^6 k_p^2}
\left|
\sum_{l'l} b_{\mathbf{k}'l'}b_{\mathbf{k}l} \left\{
\frac{\alpha_{\mathbf{k'k}}\left(\langle l'|\mathbf{C_a} \cdot \nabla
|l\rangle + i \mathbf{C_a} \cdot \mathbf{k}\right)}{
(|\mathbf{k}'-\mathbf{k}|^2 - k_p^2)} + \frac{i\beta_{\mathbf{k'k}}C_b
\langle l'|l\rangle }{|\mathbf{k}'-\mathbf{k}|}
\right\} \right|^2\, ;
\\
\alpha_{\mathbf{k'k}}&=&
\cos(|\mathbf{k}'-\mathbf{k}| r_a)
+\frac{i k_p}{|\mathbf{k}'-\mathbf{k}|}
\sin(|\mathbf{k}'-\mathbf{k}| r_a)
\, ,
\nonumber \\
\beta_{\mathbf{k'k}} &=& \left\{
\frac{k_p \cos(|\mathbf{k}'-\mathbf{k}| r_a)+ i |\mathbf{k}'-\mathbf{k}| \sin(|\mathbf{k}'-\mathbf{k}| r_a)}{|\mathbf{k}'-\mathbf{k}|^2-k_p^2}
+\frac{\sin(|\mathbf{k}'-\mathbf{k}| r_a)}{|\mathbf{k}'-\mathbf{k}| k_p r_a}
\right\} \, . \end{aligned}$$
Here $r_a$ is the radius of the grain. $\langle l'|\mathcal{O}|l\rangle = \int \Psi_{l'}^* \mathcal{O} \Psi_l d^3r$ denotes integration over the atomic orbitals for the operator $\mathcal{O}$. It can be assumed that the elements of the sum where $n'\neq n$ will only give small corrections.
Writing $\mathbf{p}_0\cdot\nabla = \sum_{i=x,y,z}\mathbf{e}_i p_i
\nabla_i$, and having an initial state that is a combination of $p$-states, only the matrix elements $\langle s| e_i
p_i\nabla_i|p_i\rangle$ do not vanish. They are expected to be of the order of $1/a$ where $a$ is the lattice constant, $5.4 \cdot 10^{-10}$ m for crystalline silicon. Regarding the contributions of the scalar field and Umklapp processes, only the elements with $l=l'$ do not vanish.
While each of the $x,y,z$ give different contributions depending on the orientation of the dipole, it should be noted that there exist six equivalent minima in the conduction band. While the matrix element due to one minimum will be non-isotropic, the sum over all six minima is expected to be isotropic and equivalent to two minima with $\mathbf{p}_0\parallel \mathbf{k}$. We define $k_0 = |\mathbf{k'}-\mathbf{k}|$. At the minima we have $k_0 \approx 0.85 \cdot 2 \pi /a \approx 10^{10}\mbox{m}^{-1}$.
We see that all terms in Eq. \[Pd\] show oscillations with period $1/|\mathbf{k'}-\mathbf{k}|$ with increasing nanoparticle radius. As the nanoparticle diameter cannot be expected to be well defined on this length scale (atomic radius), we will simply take the average over one period. We believe this to be justified both from considering the limited coherence length of the electrons, and from the fact that any physical measurement would include a dispersion of particle sizes. While the limit of $r_a \rightarrow 0$ is mathematically well defined, it is not physically meaningful, as it describes a nanoparticle with less than one atom.
Interestingly, the scalar potential provides much larger contribution than the vector potential. Keeping only the largest terms, we can make an order of magnitude estimate, $$P_d \approx \frac{(m_c m_v)^\frac{3}{2}\omega e^2 p_0^2}
{32 \pi^2 \hbar^6 k_0^4 \epsilon_0^2}
\Delta E^2 \left(k_p+r_a^{-1}\right)^2$$ $P_d$, $P_r$ and $P_h$ are all proportional to $p_0^2$. We define the damping coefficients $\gamma_d = P_d/p_0^2$, $\gamma_r=P_r/p_0^2$, $\gamma_h=P_h/p_0^2$ and $\gamma = \gamma_d+\gamma_r+\gamma_h$. We use these coefficients when comparing the importance of the different mechanisms. To find the total absorbed power, we also need to estimate the dipole moment, $p_0$, which is determined by the amplitude of the incident wave, $E_0$, and the polarizability of the nanoparticles, $\alpha$, as $p_0=\alpha(\omega) E_0$. While the polarizability is in general dependent on the particle volume and shape, at the plasmon resonance it is determined by the damping only. This can be shown from equating the power absorbed from a plane wave by an oscillating dipole, $E_0p_0\omega/2$, with the total emitted power, $\gamma p_0^2$, giving $$\alpha = \frac{\omega}{2 \gamma(\omega)}.$$ Assuming that the incident light is absorbed by a layer of nanoparticles with a 2D density $n$, we get $$\begin{aligned}
\frac{P_d}{W} &\approx& \frac{n (m_c m_v)^{3/2}\omega(\hbar \omega
-E_g)^2 e^2 \alpha^2 }{16\pi^2\hbar^6 c
k_0^4\epsilon_0^3}\left(k_p+r_a^ {-1}\right)^2 \nonumber \\
&\propto&
\frac{\gamma_d}{(\gamma_d+\gamma_r+\gamma_h)^2} \, . \label{PdW}\end{aligned}$$ Here $W=\epsilon_0 E_0^2 c/2$ is the incident power per area. As long as $\gamma_d$ is small compared to $\gamma_r+\gamma_h$, it will not significantly change the polarizability, but if it becomes of the same order, the decrease in polarizability will be more important than the increase in absorption. This can easily be remedied by a higher density of nanoparticles, but if the density becomes very high, interaction between neighboring particles will change both the polarizability and the plasmon resonance frequency.
Results
=======
We see that the direct absorption is proportional to $(\Delta E)^2 \equiv(\hbar \omega -E_g)^2$, the excess energy after bridging the band gap, squared. This is the same energy dependence as found for indirect absorption if the single phonon processes including emission or absorption of a phonon are considered separately.
The resonance of the nanoparticles has a certain width, so $\Delta E$ also has a spread. To get a feeling for the order of magnitude of $P_d$ we define $x=\Delta E/\hbar \omega$ and express the results through this. We are interested in frequencies where direct absorption of a plane wave would be impossible, so an $x$ of close to one is irrelevant. The direct absorption shows no explicit dependence on temperature, as opposed to indirect absorption. This may indicate a method for differentiating between direct and indirect absorption in solar cells containing nanoparticles. The possibility of changes in the band structure of silicon with temperature should still be considered.
The direct absorption, $P_d$, should be compared with the energy lost to heating, $P_h$ or by radiation, $P_r$. From Eq. (\[Pd\]) we have $$P_d = \gamma_d p_0^2,\quad \gamma_d\approx 3\cdot 10^{44}x^2\left(1+\frac{1}{(k_p
r_a)} \right)^2\,\frac{\mbox{J}}{\mbox{sm}^2\mbox{C}^2}\, ,$$ where $x$ will usually be significantly less than one. With $50$ nm particle radius we get $k_p r_a\approx 0.3$. The total integrated radiation from a dipole with the same dipole moment, $p_0$ is: [@Lorrain] $$P_r = \frac{c}{12 \pi \epsilon}\frac{p_0^2 (2
\pi)^4}{\lambda^4}=\gamma_r p_0^2 \, .$$ Using that $k_p \lambda = 2 \pi$ we obtain $$\gamma_r = \frac{c k_p^4}{12 \pi \epsilon}\approx 10^{44}\,
\frac{\mbox{J}}{\mbox{sm}^2\mbox{C}^2}\,$$ giving $$\frac{\gamma_d}{\gamma_r} \approx 3 x^2 \left(1+\frac{1}{(k_p
r_a)^2} \right).$$ As shown in figure 1, we see that for sufficiently small particles and if $x$ is not too small, $\gamma_d$ can be of the same magnitude as $\gamma_r$, or even larger. However, the gain for small particles requires that the particle is very close to the silicon. Designs where the nanoparticles are located outside the silicon may lose the benefit of the $1/k_p r_a$ term.
In the previously mentioned experiments, the nanoparticles were located close to a thin silicon wafer. For dipoles located at such an interface, the total radiation increases, and a large fraction of the radiation is directed into guided modes in the silicon wafer.[@Mertz; @Benisty; @Catchpole; @Pillai1] For the light in the bound modes, it is assumed that the optical path length is sufficient to allow most of the radiation to be absorbed in indirect electron hole pair creations. At the same time the areas where the near field is strongest, has no silicon to absorb the energy. Under such circumstances, it should be assumed that $\gamma_r$ dominates $\gamma_d$.
However, there is also a possibility that radiated energy can excite a plasmon on a neighboring particle, then again to be reemitted. This would reduce the positive contribution of the plasmons in architectures where the radiation along the wafer is exploited, as more energy would be lost to heat. If we instead place the nanoparticles inside the silicon and exploit the direct absorption, less energy would be reabsorbed by neighboring nanoparticles, and thereby less would be lost to heat.
Using a simple resistivity argument, $\gamma_h$ can be estimated as $$\gamma_h\approx \frac{3\omega^2 \rho}{4 \pi r_a^3}\approx
2\cdot 10^{42}\frac{1}{(k_p r_a)^3} \frac{\mbox{J}}{\mbox{sm}^2\mbox{C}^2}$$ for silver. For small particles, heating will take over as the dominant damping mechanism, as shown in figure 1.
{width="3in"}
Conclusions
===========
Our findings indicate that direct absorption due to surface plasmons on metal nanoparticles does occur, and may give important corrections to the total absorption for realistic parameters.
The direct absorption has been found to be
- proportional $\Delta E^2 = (\hbar \omega - E_g)^2$;
- independent of temperature;
- inversely proportional to the k-space position of the conduction band gap minimum to the fourth power;
- comparable in magnitude to radiated energy in some cases.
The existence of the direct absorption mechanism is an argument for placing the nanoparticles inside the silicon, rather than in front of, or at the rear of the cell. This gives the additional requirement that the problem of recombination centers at the particle surface can be kept to a minimum.
Ideally, the size of the nanoparticles should be so small that $\gamma_d$ dominates $\gamma_r$, but not so small that heating takes over as the dominant mechanism. We assume that diameters from about a tenth to half a wavelength could be suitable, depending on the conductivity of the nanoparticle. The plasmon resonance should be tuned using choice of material and particle shape (flattened for red-shift[@Bohren]), to match the band gap of silicon.
The main questions that remain unanswered in our study concern the effects of interfaces and surface states for the direct absorption. We have not considered how an interface changes the near field, and the presence of surface states in the silicon may significantly change the problem in unpredictable ways. As the main contribution is from very near the dipole, both surface electron states and the alteration of the field due to an interface may be very important. There are also some unaddressed problems related to the averaging over particle radii and the finite coherence length of the electrons in the silicon.
It is possible that higher order modes will give larger contributions to the absorption, these modes have been shown to be significant for nanoparticles of sizes where reradiation is larger than heating.
The authors wish to acknowledge the financial support of the Norwegian Research Council. We wish to thank Alexander Ulyashin for introducing us to the idea of using nanoparticle surface plasmons in solar cells, and the SPREE at UNSW for giving us the necessary clues to get started.
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|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We consider the problem of detecting a small subset of defective items from a large set via [*non-adaptive “random pooling” group tests*]{}. We consider both the case when the measurements are noiseless, and the case when the measurements are noisy (the outcome of each group test may be independently faulty with probability $q$). Order-optimal results for these scenarios are known in the literature. We give information-theoretic lower bounds on the query complexity of these problems, and provide corresponding computationally efficient algorithms that match the lower bounds up to a constant factor. To the best of our knowledge this work is the first to explicitly estimate such a constant that characterizes the gap between the upper and lower bounds for these problems.'
author:
-
-
title: 'Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms'
---
Introduction
============
Background
==========
Main Results
============
Proof of lower bounds {#sec:lwrbnd}
=====================
Proofs of upper bounds {#sec:uprbnd}
======================
Simulation
==========
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper we study the excitation spectrum of single- and multi-layer graphene beyond the Dirac cone approximation. The dynamical polarizability of graphene is computed using a full $\pi$-band tight-binding model, considering the possibility of inter-layer hopping in the calculation. The effect of electron-electron interaction is considered within the random phase approximation. We further discuss the effect of disorder in the spectrum, which leads to a smearing of the absorption peaks. Our results show a redshift of the $\pi$-plasmon dispersion of single-layer graphene with respect to graphite, in agreement with experimental results. The inclusion of inter-layer hopping in the kinetic Hamiltonian of multi-layer graphene is found to be very important to properly capture the low energy region of the excitation spectrum.'
author:
- 'Shengjun Yuan, Rafael Roldán and Mikhail I. Katsnelson'
bibliography:
- 'BibliogrGrafeno.bib'
title: 'Excitation spectrum and high energy plasmons in single- and multi-layer graphene'
---
Introduction
============
One of the main issues in the understanding of the physics of graphene is the role played by electron-electron interaction.[@KG10] Several collective modes as low and high energy plasmons, as well as plasmarons, are a consequence of electronic correlations and have been measured in this material. The high energy $\pi$-plasmons have been observed in electron energy-loss spectroscopy (EELS),[@KP08; @EB08; @GG08] inelastic X-ray scattering (IXS)[@RA10] or optical conductivity.[@MSH11] Recently, a plasmaron mode (which is a result of coupling between electrons and plasmons) has been measured in angle-resoved photoemission spectroscopy (ARPES).[@BR10]
At low energies, long range Coulomb interaction leads, in doped graphene, to a gapless plasmon mode which disperses as $\omega_{pl}\sim \sqrt{q}$,[LS08]{} and which can be described theoretically within the random phase approximation (RPA).[@S86; @WSSG06; @HS07; @PM08; @P09; @G11b] The low energy linear dispersion relation of graphene is at the origin of a new series of collective modes predicted for this material and which do not exist for other two-dimensional electron gases (2DEG), as inter-valley plasmons[TM10]{} or linear magneto-plasmons,[@RFG09] which can be described within the RPA as well. For undoped graphene, the inclusion of ladder diagrams in the polarization can lead to a new class of collective modes[@GFM08] as well as to an excitonic instability.[@K01; @AKT07; @GGG09; @WFM10; @G10]
However, much less is known about the high energy $\pi$-plasmon which, in the long wavelength limit, has an energy of the order of 5-6 eV, and which is due to the presence of Van Hove singularities in the band dispersion. For single-layer graphene (SLG), this mode has been studied by Stauber et al.[@SSP10] and by Hill et al.[@HMZ09] in the RPA. Yang et al. have included excitonic effects and found a redshift of the absorption peak,[YL09]{} leading to a better agreement with the experimental results.[EB08]{} Here we extend those previous works and study the excitation spectrum of SLG and multi-layer graphene (MLG) from a tight-binding model on a honeycomb lattice. By means of the Kubo formula, the non-interacting polarization function $\Pi(\mathbf{q},\omega)$ is obtained from the numerical solution of the time-dependent Schrödinger equation. Coulomb interactions are considered in the RPA, the validity of which is discussed. We also consider the effect of disorder in the system, which lead to a considerable smearing of the Van Hove singularities in the spectrum. Our results show a redshift of the $\pi$-plasmon mode in graphene with respect to graphite, as it has been observed in the experiments.[@KP08; @EB08] Furthermore, the inclusion of inter-layer hopping is found to be very important to capture the low energy region of the spectrum in MLG.
The paper is organized as follows. In Sec. \[Sec:Method\] we describe the method that we use to compute the dynamical polarization function of SLG and MLG. In Sec. \[Sec:SLG\] we give results for the excitation spectrum of SLG, considering the effect of disorder and electron-electron interaction. The spectrum of MLG is described in Sec. \[Sec:MLG\]. In Sec. [Sec:ComparExp]{} we compare our results to recent experimental data. Our main conclusions are summarized in Sec. \[Sec:Conclusions\].
Description of the method {#Sec:Method}
=========================
The tight-binding Hamiltonian of a MLG is given by $$H=\sum_{l=1}^{N_{layer}}H_{l}+\sum_{l=1}^{N_{layer}-1}H_{l}^{\prime },
\label{Hamiltonian}$$where $H_{l}$ is the Hamiltonian of the $l$’th layer of graphene,$$H_{l}=-\sum_{<i,j>}(t_{l,ij}a_{l,i}^{\dagger }b_{l,j}+\mathrm{h.c})+\sum_{i}v_{l,i}c_{l,i}^{\dagger }c_{l,i}, \label{Hamiltonian_SLG}$$where $a_{l,i}^{\dagger }$ ($b_{l,i}$) creates (annihilates) an electron on sublattice A (B) of the $l$’th layer, and $t_{l,ij}$ is the nearest neighbor hopping parameter. The second term of $H_l$ accounts for the effect of an on-site potential $v_{l,i}$, where $n_{l,i}=c^{\dagger}_{l, i}c_{l, i}$ is the occupation number operator. In the second term of the Hamiltonian Eq. (\[Hamiltonian\]), $H_{l}^{\prime }$ describes the hopping of electrons between layers $l$ and $l+1$. In nature there are two known forms of stable stacking sequence in bulk graphite, namely ABA (Bernal) and ABC (rhombohedral) stacking, and they are schematically shown in Fig. [Fig:Stacking]{}. For a MLG with an ABA stacking, $H_{l}^{\prime }$ is given by$$H_{l}^{\prime }=-\gamma _{1}\sum_{j}\left[ a_{l,j}^{\dagger }b_{l+1,j}+\mathrm{h.c.}\right] -\gamma _{3}\sum_{j,j^{\prime }}\left[ b_{l,j}^{\dagger
}a_{l+1,j^{\prime }}+\mathrm{h.c.}\right] , \label{Eq:H-interlayer}$$where the inter-layer hopping terms $\gamma_1$ and $\gamma_3$ are shown in Fig. \[Fig:Stacking\]. Thus, all the even layers ($l+1$) are rotated with respect to the odd layers ($l$) by $+120^{\circ }$. The difference between ABA and ABC stacking is that, the third layer(s) is rotated with respect to the second layer by $-120^{\circ }$ (then it will be exactly under the first layer) in ABA stacking, but by $+120^{\circ }$ in ABC stacking.[@K10] In this paper we use the hopping amplitudes $t=3$ eV, $\gamma _{1}=0.4$ eV and $\gamma _{3}=0.3$ eV.[@CG07] The spin degree of freedom contributes only through a degeneracy factor and is omitted for simplicity in Eq. ([Hamiltonian]{}). In our numerical calculations, we use periodic boundary conditions in the plane ($XY$) of graphene layers, and open boundary conditions in the stacking direction ($Z$).
![2D Brillouin zone of SLG. For undoped graphene, the valence and conduction bands touch each other at the vertices of the hexagon, the so called Dirac points (K and K’). The Van Hove singularity lies at the M point, and we have defined $\protect\theta$ as the angle between the wave-vector $q$ and the $k_x$-axis.[]{data-label="Fig:BZ"}](BZHexag.eps){width="6.5cm"}
The dynamical polarization can be obtained from the Kubo formula [@K57] as$$\Pi \left( \mathbf{q},\omega \right) =\frac{i}{V}\int_{0}^{\infty }d\tau
e^{i\omega \tau}\left\langle \left[ \rho \left( \mathbf{q},\tau\right) ,\rho
\left( -\mathbf{q},0\right) \right] \right\rangle , \label{Eq:Kubo}$$where $V$ denotes the volume (or area in 2D) of the unit cell, $\rho \left(
\mathbf{q}\right) $ is the density operator given by$$\rho \left( \mathbf{q}\right)
=\sum_{l=1}^{N_{layer}}\sum_{i}c_{l,i}^{\dagger}c_{l,i}\exp \left( i\mathbf{q\cdot r}_{l,i}\right) ,$$and the average is taken over the canonical ensemble. For the case of the single-particle Hamiltonian, Eq. (\[Eq:Kubo\]) can be written as[YRK10]{}$$\begin{aligned}
&&\Pi \left( \mathbf{q},\omega \right) =-\frac{2}{V}\int_{0}^{\infty }d\tau
e^{i\omega \tau} \notag \label{Eq:Kubo2} \\
&&\times \text{Im}\left\langle \varphi \right\vert n_{F}\left( H\right)
e^{iH\tau}\rho \left( \mathbf{q}\right) e^{-iH\tau}\left[ 1-n_{F}\left(
H\right) \right] \rho \left( -\mathbf{q}\right) \left\vert \varphi
\right\rangle , \notag \\
&&\end{aligned}$$where $n_{F}\left( H\right) =\frac{1}{e^{\beta \left( H-\mu \right) }+1}$ is the Fermi-Dirac distribution operator, $\beta =1/k_{B}T$ where $T$ is the temperature and $k_{B}$ is the Boltzmann constant, and $\mu $ is the chemical potential. In the numerical simulations, we use units such that $\hbar =1 $, and the average in Eq. (\[Eq:Kubo2\]) is performed over a random phase superposition of all the basis states in the real space, i.e.,[@HR00; @YRK10] $$\left\vert \varphi \right\rangle =\sum_{l,i}a_{l,i}c_{l,i}^{\dagger
}\left\vert 0\right\rangle , \label{Eq:phi0}
$$where $a_{l,i}$ are random complex numbers normalized as $\sum_{l,i}\left\vert a_{l,i}\right\vert ^{2}=1$. By introducing the time evolution of two wave functions $$\begin{aligned}
\left\vert \varphi _{1}\left( \mathbf{q,}\tau\right) \right\rangle
&=&e^{-iH\tau}\left[ 1-n_{F}\left( H\right) \right] \rho \left( -\mathbf{q}\right) \left\vert \varphi \right\rangle , \\
\left\vert \varphi _{2}\left( \tau\right) \right\rangle
&=&e^{-iH\tau}n_{F}\left( H\right) \left\vert \varphi \right\rangle ,\end{aligned}$$we get the real and imaginary part of the dynamical polarization as $$\begin{aligned}
\label{Eq:RePi-ImPi}
\text{Re}\Pi \left( \mathbf{q},\omega \right) &=&-\frac{2}{V}\int_{0}^{\infty }d\tau\cos (\omega \tau)~\text{Im}\left\langle \varphi
_{2}\left( \tau\right) \left\vert \rho \left( \mathbf{q}\right) \right\vert
\varphi _{1}\left( \tau\right) \right\rangle , \notag \\
\text{Im}\Pi \left( \mathbf{q},\omega \right) &=&-\frac{2}{V}\int_{0}^{\infty }d\tau\sin (\omega \tau)~\text{Im}\left\langle \varphi
_{2}\left( \tau\right) \left\vert \rho \left( \mathbf{q}\right) \right\vert
\varphi _{1}\left( \tau\right) \right\rangle , \notag \\
&&\end{aligned}$$The Fermi-Dirac distribution operator $n_{F}\left( H\right) $ and the time evolution operator $e^{-iH\tau}$ can be obtained by the standard Chebyshev polynomial decomposition.[@YRK10]
For the case of SLG, we will further compare the polarization function obtained from the Kubo formula Eq. (\[Eq:Kubo\]), to the one obtained from the usual Lindhard function.[@GV05] Notice that this method can be used to calculate the optical conductivity of graphene beyond the Dirac cone approximation. [@SPG08; @YRK10] For pristine graphene, the dynamical polarization obtained from the Lindhard function using the full $\pi $-band tight-binding model is[@S86; @HMZ09; @SSP10] $$\begin{aligned}
\label{Eq:Lindhard}
&&\Pi \left( \mathbf{q},\omega \right) =-\frac{g_{s}}{\left( 2\pi \right)
^{2}}\int_{BZ}d^{2}\mathbf{k} \notag \label{Eq:PolarizationFull} \\
&&\times \sum_{s,s^{\prime }=\pm }f_{s\cdot s^{\prime }}\left( \mathbf{k},\mathbf{q}\right) \frac{n_{F}\left[ E^{s}\left( \mathbf{k}\right) \right]
-n_{F}\left[ E^{s^{\prime }}\left( \mathbf{k}+\mathbf{q}\right) \right] }{E^{s}\left( \mathbf{k}\right) -E^{s^{\prime }}\left( \mathbf{k}+\mathbf{q}\right) +\omega+i\delta }, \notag \\
&&\end{aligned}$$where the integral is over the Brillouin zone, $g_{s}=2$ is the spin degeneracy, $E^{\pm }\left( \mathbf{k}\right) =\pm t\left\vert \phi _{\mathbf{k}}\right\vert -\mu $ is the energy dispersion with respect to the chemical potential, where $$\phi _{\mathbf{k}}=1+2e^{i3k_{x}a/2}\cos \left( \frac{\sqrt{3}}{2}k_{y}a\right) ,$$$a=1.42$Å being the in-plane carbon-carbon distance, and the overlap between the wave-functions of the electron and the hole is given by $$f_{\pm }\left( \mathbf{k},\mathbf{q}\right) =\frac{1}{2}\left( 1\pm \text{Re}\left[ e^{iq_{x}a}\frac{\phi _{\mathbf{k}}}{\left\vert \phi _{\mathbf{k}}\right\vert }\frac{\phi _{\mathbf{k+q}}^{\ast }}{\left\vert \phi _{\mathbf{k+q}}\right\vert }\right] \right) .$$
In the RPA, the response function of SLG due to electron-electron interactions can be calculated as $$\chi \left( \mathbf{q},\omega \right) =\frac{\Pi \left( \mathbf{q},\omega
\right) }{\mathbf{1}-V\left( q\right) \Pi \left( \mathbf{q},\omega \right) },
\label{Eq:chi}$$where $V\left( q\right) =\frac{2\pi e^{2}}{\kappa q}$ is the Fourier component of the Coulomb interaction in two dimensions, in terms of the background dielectric constant $\kappa $, and $$\mathbf{\varepsilon }\left( \mathbf{q},\omega \right) =\mathbf{1}-V\left(
q\right) \Pi \left( \mathbf{q},\omega \right)$$is the dielectric function of the system. We will be interested on the collective modes of the system, which are defined from the zeroes of the dielectric function \[$\varepsilon (\mathbf{q},\omega )=0$\]. The dispersion relation of the collective modes is defined from $$\mathrm{Re}~\varepsilon (\mathbf{q},\omega _{pl})=1-V(q)\Pi (\mathbf{q},\omega _{pl})=0, \label{Eq:Plasmons}$$which leads to poles in the response function (\[Eq:chi\]). The damping $\gamma $ of the mode is proportional to $\mathrm{Im}~\Pi (\mathbf{q},\omega
_{pl})$, and it is given by $$\gamma =\frac{\mathrm{Im}~\Pi (\mathbf{q},\omega _{pl})}{\left. \frac{\partial }{\partial \omega }\mathrm{Re}~\Pi (\mathbf{q},\omega )\right\vert
_{\omega =\omega _{pl}}}. \label{Eq:Damping}$$
For MLG, the response function is calculated as (we use $q_{z}=0$)[@S86] $$\label{Eq:chi3D}
\chi _{3D}\left( \mathbf{q},\omega \right) =\frac{\Pi _{3D}\left( \mathbf{q},\omega \right) }{\mathbf{1}-V\left( q\right) F\left( q\right) \Pi
_{3D}\left( \mathbf{q},\omega \right) d},$$where $d=3.35$Å is the inter-layer separation. Because we use open boundary conditions in the stacking direction, we define the form factor $F\left( q\right) $ as $$F\left( q\right) =\frac{1}{N_{layer}}\sum_{l,l^{\prime
}=1}^{N_{layer}}e^{-q\left\vert l-l^{\prime }\right\vert d}.
\label{Eq:FormFactor}$$The expression (\[Eq:chi3D\]) assumes that the polarization of each layer is the same, and it is exact in two different limits: bilayer graphene and graphite. Notice that a similar effective form factor has been used to study the loss function of multiwall carbon nanotubes.[@SL00] Eq. ([Eq:FormFactor]{}) coincides with the commonly used form factor for a multi-layer system with an infinite number of layers:[@SQ82] $$F\left( q\right)|_{N_{layer}\rightarrow \infty} =\sum_{l^{\prime
}}e^{-q\left\vert l-l^{\prime }\right\vert d},$$where in this last case the periodicity ensures that $F\left( q\right) $ is independent of layer index $l$, with the asymptotic behavior $F(q)=\sinh(qd)/[\cosh(qd)-1]$.[@SQ82]
A crucial issue is the value of the dielectric constant $\kappa $ for each of the cases considered, because it encodes the screening due to high energy ($\sigma $) bands which are not explicitly considered in our calculation. A good estimation for it can be obtained from the expression[@WB11] $$\kappa \left( \mathbf{q}\right) =\frac{\kappa _{1}+1-\left( \kappa
_{1}-1\right) e^{-qL}}{\kappa _{1}+1+\left( \kappa _{1}-1\right) e^{-qL}}\kappa _{1}, \label{Eq:kappa}$$where $\kappa _{1}\approx 2.4$ is the dielectric constant of graphite, $L=d_{m}+\left( N_{layer}-1\right) d$ is the total height of the multi-layer system in terms of the number of layers $N_{layer}$ and the height of a monolayer graphene $d_{m}\approx 2.8$ Å. As expected, Eq. (\[Eq:kappa\]) gives $\kappa =1$ for SLG at $q\rightarrow 0$ and $\kappa =\kappa _{1}$ for graphite.
We notice that the accuracy of the numerical results for the polarization function Eq. (\[Eq:RePi-ImPi\]) is mainly determined by three factors: the time interval of the propagation, the total number of time steps, and the size of the sample. The maximum time interval of the propagation in the time evolution operator is determined by the Nyquist sampling theorem. This implies that employing a sampling interval $\Delta \tau=\pi
/\max_{i}\left\vert E_{i}\right\vert $, where $E_{i}$ are the eigenenergies, is sufficient to cover the full *range* of energy eigenvalues. On the other hand, the *accuracy* of the energy eigenvalues is determined by the total number of the propagation time steps ($N_{\tau}$) that is the number of the data items used in the fast Fourier transform (FFT). Eigenvalues that differ less than $\Delta E=\pi /N_{\tau}\Delta \tau$ cannot be identified properly. However, since $\Delta E$ is proportional to $N_{\tau}^{-1}$ we only have to double the length of the calculation to increase the accuracy by the same factor. The statistic error of our numerical method is inversely proportional to the dimension of the Hilbert space,[@HR00] and in our case (the single particle representation), it is the number of sites in the sample. A sample with more sites in the real space will have more random coefficients ($a_{l,i}$) in the initial state $\left\vert \varphi \right\rangle $, providing a better statistical representation of the superposition of all energy eigenstates.[@YRK10]
Similar algorithm has been successfully used in the numerical calculation of the electronic structure and transport properties of single- and multi-layer graphene, such as the density of states (DOS), or *dc* and *ac* conductivities.[@WK10; @YRK10; @YRK10b] The main advantage of our algorithm is that different kinds of disorders and boundary conditions can be easily introduced in the Hamiltonian, and the computer memory and CPU time is linearly proportional to the size of the sample, which allows us to do the calculations on a sample containing tens of million sites.
Excitation spectrum of single-layer graphene {#Sec:SLG}
============================================
![(Color online) Density of states for SLG considering different kind of disorder. The left inset shows a zoom of the DOS near the Dirac point ($E=0$), whereas the right hand side inset shows the disorder broadening of the Van Hove singularity at $E=t$. The numerical method used in the calculation of DOS is discribed in Ref. , and the sample size of SLG is $4096\times 4096.$[]{data-label="Fig:DOS-disorder"}](dos_disorder.eps){width="6.5cm"}
The particle-hole excitation spectrum is the region of the energy-momentum space which is available for particle-hole excitations. For non-interacting electrons, it is defined as the region where $\mathrm{Im}~\Pi (\mathbf{q},\omega )$, as given by Eq. (\[Eq:Kubo\]) or (\[Eq:Lindhard\]), is non-zero.[@GV05] The linear low energy dispersion relation of graphene as well as the possibility for inter-band transitions lead to a rather peculiar excitation spectrum for SLG as compared to the one of a two-dimensional electron gas (2DEG) with a parabolic band dispersion.[RGF10]{} Here we focus on undoped graphene ($\mu =0$), for which only inter-band transitions are allowed. In Fig. \[Fig:ImPiSLG\] we plot $\mathrm{Im}~\Pi (\mathbf{q},\omega )$ for different wave-vectors at $T=300K$ (which is the temperature that we will use from here on in our results). The first thing one observes is the good agreement between the results obtained from the Kubo formula Eq. (\[Eq:Kubo\]), as compared to Lindhard function Eq. (\[Eq:Lindhard\]), what proofs the validity of our numerical method. Furthermore, for the small wave-vector used in Fig. \[Fig:ImPiSLG\](a)-(b), the results are well described by the Dirac cone approximation,[WSSG06,HS07]{} but only at low energies, around $\omega \sim v_{\mathrm{F}}q$, where $v_{\mathrm{F}}=3at/2$ is the Fermi velocity near the Dirac points. In particular, the continuum approximation cannot capture the peaks of $\mathrm{Im}\Pi (\mathbf{q},\omega )$ around $\omega \approx 2t$. These peaks are related to particle-hole excitations between states of the valence band with energy $E\approx -t$ and states of the conduction band with energy $E\approx t$, which contribute to the polarization with a strong spectral weight due to the enhanced density of states at the Van Hove singularities of the $\pi $-bands (see Fig. \[Fig:DOS-disorder\]).
Second, for larger wave-vectors \[Figs. \[Fig:ImPiSLG\](c)-(f)\] one observes strong differences in the spectrum depending on the orientation of $\mathbf{q}$, effect which has been discussed previously.[@SSP10; @SG10] If $\mathbf{q}$ is along the $\Gamma $-K direction, there is a splitting of the peak associated to the Van Hove singularity at $\omega \sim 2t$. At low energies, we also observe a finite contribution to the spectral weight to the left of the $\omega \approx v_{\mathrm{F}}q$ peak for momenta along the $\Gamma $-M direction \[plots Figs. \[Fig:ImPiSLG\](d) and (f)\]. Finally, trigonal warping effects are important as we increase the magnitude of $|\mathbf{q}|$, due to the deviation of the band dispersion with respect to the linear cone approximation. As a consequence, the constant energy contours are not any more circles around the Dirac points, but present a triangular shape. The consideration of this effect leads to a redshift of the $\omega \approx v_{\mathrm{F}}q$ peak with respect to the Dirac cone approximation, as seen clearly in Fig. \[Fig:ImPiSLG\](e).
Once we have discussed the clean case, we consider the effect of disorder on the excitation spectrum as explained in Sec. \[Sec:Method\]. Two different kinds of disorder are considered: random local change of on-site potentials and random renormalization of the hopping, which correspond to the diagonal and off-digonal disorders in the single-layer Hamiltonian Eq. ([Hamiltonian\_SLG]{}), respectively. The former acts as a chemical potential shift for the Dirac fermions, i.e., shifts locally the Dirac point, and the later rises from the changes of distance or angles between the $p_{z}$ orbitals. In Fig. \[Fig:DOS-disorder\] we show the DOS of SLG for different kinds and magnitudes of disorder. The DOS for clean graphene has been plotted by using the analytical expression given in Ref. . The DOS of the disordered systems are calculated by Fourier transform of the time-dependent correlation functions [@YRK10] $$\rho \left( \varepsilon \right) =\frac{1}{2\pi }\int_{-\infty }^{\infty
}e^{i\varepsilon t}\left\langle \varphi \right\vert e^{-iHt}\left\vert
\varphi \right\rangle dt, \label{Eq:DOS}$$with the same initial state $\left\vert \varphi \right\rangle $ defined in Eq. (\[Eq:phi0\]). As shown in Ref. , the result calculated from a SLG with $4096\times 4096$ lattice sites matches very well with the analytical expression, and here we use the same sample size in the disordered systems. We consider that the on-site potential $v_{i}$ is random and uniformly distributed (independently on each site $i$ ) between $-v_{r}$ and $+v_{r}$. Similarly, the in-plane nearest-neighbor hopping $t_{ij}$ is random and uniformly distributed (independently on sites $i,j$) between $t-t_{r}$ and $t+t_{r}$. The main effect is a smearing of the Van Hove singularity at $E=t$, as observed in the right hand side inset of Fig. [Fig:DOS-disorder]{}.
The effect of disorder is also appreciable in the non-interacting excitation spectrum of the system, as shown by Fig. \[Fig:ImPiSLG-disorder\]. A broadening of the $\omega\approx v_{\mathrm{F}} q$ and $\omega\approx 2t$ peaks is observed in all the cases. Furthermore, disorder leads to a slight but appreciable redshift of the peaks with respect to the clean limit. This effect is more important for higher wave-vectors, as it can be seen in Fig. \[Fig:ImPiSLG-disorder\](c)-(d). Finally, the disorder broadening of the peaks leads in all the cases to a transfer of spectral weight to low energies (below $\omega=v_{\mathrm{F}} q$), as it is appreciable in Fig. [Fig:ImPiSLG-disorder]{}(a)-(d).
The next step is to consider both, disorder and electron-electron interaction in the system. In the RPA, the response function is calculated as in Eq. (\[Eq:chi\]). The results are shown in Fig. \[Fig:ImchiSLG\], where $-\mathrm{Im}~\chi(\mathbf{q},\omega)$ is plotted for the same wave-vectors and disorder used in Fig. \[Fig:ImPiSLG-disorder\]. We observe that the Dirac cone approximation (black line) captures well the low energy region of the spectrum. However, the large peak at $\omega\sim 2t$ cannot be captured by the continuum approximation. They are due to a plasmon mode associated to transitions between electrons in the saddle points of the $\pi$-bands. Strictly speaking, those modes cannot be considered as *fully coherent* collective modes, as for example, the low energy $\sqrt{q}$-plasmon which is present in doped graphene.[@S86] For doped graphene, the acoustic $\sqrt{q}$-plasmon is undamped above the threshold $\omega=v_{\mathrm{F}} q$ until it enters the inter-band particle-hole continuum, when it starts to be damped and decays into electron-hole pairs. However, the $\pi $-plasmon, although it corresponds to a zero of the dielectric function as it can be seen in Fig. \[Fig:ReepsSLG\], it is a mode which lies *inside* the continuum of particle-hole excitations: $-\mathrm{Im}~\Pi(\mathbf{q},\omega_{pl})>0$ at the $\pi$-plasmon energy $\omega_{pl}$, and the mode will be damped even at $q\rightarrow 0$. In any case, it is a well defined mode which has been measured experimentally for SLG and MLG.[@KP08; @EB08; @GG08; @RA10; @MSH11] Coming back to our results, notice that the height of the peaks is reduced when the effect of disorder is considered, although the position is unaffected by it. For small wave-vectors, this mode is highly damped due to the strong spectral weight of the particle-hole excitation spectrum at this energy, as seen e.g. by the peak of $-\mathrm{Im}~\Pi(\mathbf{q},\omega)$ at $\omega=2t$ in Fig. \[Fig:ImPiSLG\](a)-(b). The position of the collective modes can be alternatively seen by the zeroes of the dielectric function Eq. (\[Eq:Plasmons\]), which is shown in Fig. \[Fig:ReepsSLG\]. Notice that the Dirac cone approximation (solid black lines in Fig. \[Fig:ReepsSLG\]) is completely insufficient to capture this high energy $\pi$-plasmon, which predicts always a finite $\mathrm{Re}~\varepsilon(\mathbf{q},\omega)$. As for the polarization, we see that disorder lead to an important smearing of the singularities of the dielectric function, as seen in Fig. \[Fig:ReepsSLG\]. Finally, we mention that the application of our method to even higher wave-vectors and energies as the ones considered in the present work, should be accompanied by the inclusion of local field effects (LFE) in the dielectric function, which are related to the periodicity of the crystalline lattice.[@A62] In fact, for SLG and for wave-vectors along the zone boundary between the M and the K points of the Brillouin zone (see Fig. \[Fig:BZ\]), the inclusion of LFEs leads to a new optical plasmon mode at an energy of 20-25 eV.[@PAP10]
Excitation spectrum for multi-layer graphene {#Sec:MLG}
============================================
In the following, we study the excitation spectrum and collective modes of MLG. For this, we consider not only the Coulomb interaction between electrons on different layers, but also the possibility for the carriers to tunnel between neighboring layers, as described in Sec. \[Sec:Method\]. The importance of considering inter-layer hopping has been already shown in the study of screening properties of MLG.[@G07] First, we see that the results are sensitive to the relative orientation between layers. In Fig. \[Fig:DOS-MLG\] we show the density of states for ABA- and ABC- stacked MLG (see Fig. \[Fig:Stacking\] for details on the difference between those two orientations). As seen in Fig. \[Fig:DOS-MLG\](c)-(d), all the MLGs present a finite DOS at $E=0$, contrary to SLG which has a vanishing DOS at the Dirac point. The main difference between the two kinds of stacking is that for ABC there is a central peak together with a series of satellite peaks around $E=0$ \[Fig. \[Fig:DOS-MLG\](d)\], whereas for ABA the DOS follows closer the behavior of the SLG \[Fig.\[Fig:DOS-MLG\](c)\]. The different structure in the DOS can be understood by looking at Fig. \[Fig:BandStructure\], where we show the low energy band structure of a trilayer graphene with ABA \[Fig. \[Fig:BandStructure\](a)\] and ABC \[Fig. \[Fig:BandStructure\](b)\] orientations. The different jumps and peaks in the DOS of Fig. \[Fig:DOS-MLG\](c)-(d) are associated to the regions of the band dispersion marked by the horizontal red lines of Fig. \[Fig:BandStructure\], the energy of which depends on the values of the tight-binding parameters associated to inter-layer tunneling ($\gamma_1$ and $\gamma_3$ in our case). In the two cases, we observe a splitting of the Van Hove peak, as seen in Fig. [Fig:DOS-MLG]{}(e)-(f). Notice that when we have a high number of layers (e.g. above 10 layers), there is a weak effect on adding a new graphene sheet to the system, as it can be seen from the similar DOS between the 10- and the 50-layers cases of Fig. \[Fig:DOS-MLG\].
In Fig. \[Fig:Pi-ABA-ABC\] we show the non-interacting (left panels) and the RPA (right panels) polarization function of MLG, for systems made of 3, 5 and 20 layers, and for ABA- and ABC-stacking. For the spectrum in the absence of electron-electron interaction, as shown in Fig. [Fig:Pi-ABA-ABC]{}(a), (c) and (e), one does not observe any specific difference in the two spectra apart from different intensities depending on the kind of stacking and on the number of layers considered for the calculation. On the other hand, the energy of the $\pi$-plasmon of ABC samples is redshifted with respect to the ABA-stacking. This can be see from the relative position of the peaks of $-\mathrm{Im}~\chi(\mathbf{q},\omega)$ in Fig. \[Fig:Pi-ABA-ABC\](b), (d) and (e). Also, notice that the separation between the two peaks grows with the number of layers, and for a 20-layers system, the difference can be of the order of 1 eV, as it can be seen in Fig. \[Fig:Pi-ABA-ABC\](f). In the following and unless we say the opposite, all the results will be calculated for the more commonly found ABA-stacking.
![(Color online) (a) $\mathrm{Im}\Pi (\mathbf{q},\protect\omega )$ for SLG and MLG. The results for SLG obtained from the Kubo formula Eq. (\[Eq:Kubo\]) are compared to those obtained from the Lindhard function Eq. (\[Eq:Lindhard\]), and to the Dirac cone approximation. (b) $\mathrm{Re}~\protect\varepsilon (\mathbf{q},\protect\omega )$ for SLG and MLG, and comparison to the Dirac cone approximation, for the same value of $q$ as in (a).[]{data-label="Fig:SpectrumMLG"}](dpIm0.2_a.eps "fig:"){width="6.5cm"} ![(Color online) (a) $\mathrm{Im}\Pi (\mathbf{q},\protect\omega )$ for SLG and MLG. The results for SLG obtained from the Kubo formula Eq. (\[Eq:Kubo\]) are compared to those obtained from the Lindhard function Eq. (\[Eq:Lindhard\]), and to the Dirac cone approximation. (b) $\mathrm{Re}~\protect\varepsilon (\mathbf{q},\protect\omega )$ for SLG and MLG, and comparison to the Dirac cone approximation, for the same value of $q$ as in (a).[]{data-label="Fig:SpectrumMLG"}](eps0.2.eps "fig:"){width="6.5cm"}
For a more clear understanding about the evolution of the particle-hole excitation spectrum with the number of layers, we plot in Fig. [Fig:SpectrumMLG]{}(a) the imaginary part of $\Pi(\mathbf{q},\omega)$ for SLG and MLG of several number of layers, and compare the results to the polarization obtained using the Dirac cone approximation. It is very important to notice that multi-layer graphene presents some spectral weight at low energies as compared to graphene, which can be seen from the finite contribution of $\mathrm{Im}\Pi(\mathbf{q},\omega)$ that appears to the left of the big peak of the graphene polarizability at $\omega=v_{\mathrm{F}} q$ ($\sim 1$eV for the used parameters), in terms of the Fermi velocity near the Dirac point, $v_{\mathrm{F}}=3at/2$. This is due to the low energy parabolic-like dispersion of bilayer and multilayer graphene, as compared to the linear dispersion of single layer graphene, and it can only be captured by considering the inter-layer hopping contribution to the kinetic Hamiltonian Eq. (\[Eq:H-interlayer\]). Furthermore, the spectrum presents a series of peaks for $\omega\approx v_{\mathrm{F}} q$, the number of which depends on the number of layers. This is due to the fact that as we increase the number of coupled graphene planes, the number of bands available for particle-hole excitations also grows leading to peaks at different energies for a given wave-vector.[@ZM08]
The difference between SLG and MLG is also relevant in the low energy region of the dielectric function, as it can be seen in Fig. \[Fig:SpectrumMLG\](b). In fact, the $\omega\rightarrow 0$ limit of $\mathrm{Re}~\varepsilon(\mathbf{q},\omega)$ calculated within the RPA grows with the number of layers. Moreover, as we have discussed above, the zeroes of $\mathrm{Re}~\varepsilon(\mathbf{q},\omega)$ signal the position of collective excitations in the system (plasmons). In Fig. \[Fig:SpectrumMLG\](b) we see that $\mathrm{Re}~\varepsilon(\mathbf{q},\omega)$, for the small wave-vector used, crosses 0 for MLG, revealing the existence of a solution of Eq. (\[Eq:Plasmons\]), but not so for SLG, as it was pointed out in Ref. . However, we emphasize that the very existence of solutions for the $\mathrm{Re}~\varepsilon(\mathbf{q},\omega)=0$ equation for MLG does not imply the existence of long-lived plasmon modes. In fact, as we have already discussed in Sec. \[Sec:SLG\], these modes disperse within the continuum of particle-hole excitations \[$\mathrm{Im}~\Pi(\mathbf{q},\omega_{pl})\ne 0$, where $\omega_{pl}$ is the solution of Eq. ([Eq:Plasmons]{})\], so they will be Landau damped and will decay into electron-hole pairs with a damping given by Eq. (\[Eq:Damping\]). Furthermore, we remember that for a given wave-vector, the energy of the mode is controlled by the background dielectric constant $\kappa$, as given by Eq. (\[Eq:kappa\]). For the systems under consideration, $\kappa$ changes between 1 (for SLG) and 2.4 (for graphite). The value of $\kappa$, together with the form factor Eq. (\[Eq:FormFactor\]) that takes into account inter-layer Coulomb interaction, fix the position of the modes in each case.
The effect of disorder in MLG is considered in Fig. \[Fig:20layer\], where we show the polarization function of a 20-layer graphene system for different kinds of disorder. As in the SLG, we find that disorder leads to a slight redshift of the peaks of the non-interacting spectrum \[Fig. [Fig:20layer]{}(a) and (b)\], together with a smearing of the peaks at $\omega\sim v_{\mathrm{F}} q$ and $\omega\sim 2t$. On the other hand, the interacting polarization function presents a reduction of the intensity of the plasmon peak due to disorder, as seen in Fig. \[Fig:20layer\](c) and (d), also in analogy with the SLG case.
Discussion and comparison to experimental results {#Sec:ComparExp}
=================================================
![(Color online) Loss function $-\mathrm{Im}~1/\protect\varepsilon (\mathbf{q},\protect\omega )$ for SLG and MLG, which is proportional to the spectrum obtained by EEL experiments.[@EB08] We have used the same $q$ as in Fig. \[Fig:SpectrumMLG\].[]{data-label="Fig:EELS"}](imeps0.2.eps){width="6.5cm"}
In this section we compute quantities which are directly comparable to recent experimental results on SLG and MLG. We start by calculating the loss function $-\mathrm{Im}~1/\varepsilon(\mathbf{q},\omega)$, which is proportional to the spectrum measured by EELS. Our results, shown in Fig. \[Fig:EELS\], are in good agreement with the experimental data of Ref. : as in the experiments, we observe a redshift of the plasmon peak as one decrease the number of layers, as well as an increase of the intensity with the number of layers. Notice that, due to finite size effects, there is an infra-red cutoff for the wave-vectors used in our calculations which prevents to reach the long wavelength limit. In Fig. [Fig:EELS]{} we show the results for the smallest wave-vector available, and we emphasize that the peaks will be further shifted to the left for smaller values of $q$. A further redshift of the peaks would be obtained beyond RPA, as it has been reported for single- and bilayer graphene, where excitonic effects have been included.[@YL09]
We have also used our method to study the IXS experiments of Reed *et al*.[@RA10] In Fig. \[Fig:Abb\](a)-(b) we plot the imaginary part of the non-interacting polarization function for SLG and MLG, for two values of $q$ similar to the ones used in Ref. . As we have discussed in Sec. \[Sec:MLG\], inter-layer hopping leads to a finite contribution to the spectral weight in the low energy region of MLG as compared to the SLG spectrum. Notice that the number of peaks at this energy $\omega \approx v_{\mathrm{F}} q$ scales with the number of accessible bands and therefore, with the number of layers. We emphasize that this effect is not included by the usually employed approximation of considering MLG as a series of single-layers of graphene, only coupled via direct Coulomb interaction.[RA10]{} Without the possibility of inter-layer hopping, the polarization function of graphene and graphite are, apart from some multiplicative factor, the same. As we have seen in Sec. \[Sec:MLG\], this simplification does not capture the low energy part of the spectrum, with some finite spectral weight due to low energy inter-band transitions between parabolic-like bands.
At an energy of the order of $\omega\approx 2t$ one observes the peak due to transitions between electrons from the Van Hove singularity of the occupied band to the singularity of the empty band. For SLG, the peak is split into two peaks if the wave-vector points in the $\Gamma$-K direction (as it is the case here), the separation of which increases with the modulus $q=q_x$. However the amplitude of these peaks is highly suppressed from SLG to MLG. Finally, one observes in Fig. \[Fig:Abb\](b) that for higher values of $q$, as the one used here, there is a redshift of the peak of $\mathrm{Im}~\Pi(\mathbf{q},\omega)$ at the energy $\omega\approx v_{\mathrm{F}} q$ with respect to the Dirac cone approximation. This is due to trigonal warping effects, which are beyond the continuum approximation. Summarizing, we find two effects that lead to a global contribution to the polarization at low energies: one is the contribution to the spectral weight due to inter-layer hopping in MLG, and the other is the redshift of the peaks at $\omega\approx
v_{\mathrm{F}} q$ due to trigonal warping effects. Notice that, because we are studying the *non-interacting* polarization function, no excitonic effects are present in the results of Fig. \[Fig:Abb\](a)-(b).
Once the polarization function $\Pi(\mathbf{q},\omega)$ is known, we compute the response function $\chi(\mathbf{q},\omega)$ at the RPA level, as shown in Fig. \[Fig:Abb\](c)-(d). Again, we find a redshift of the position of the peaks as we decrease the number of layers. The different position of the peaks is due to the different contribution of inter-layer electron-electron interaction for each case, as well as to the different value of $\kappa$ as given by Eq. (\[Eq:kappa\]). Our results agree reasonably well with those of Ref. .
![(Color online) Modulus of the screened fine structure constant $|\protect\alpha^{*}|$, calculated from Eq. (\[Eq:alpha\]), for SLG and MLG of a different number of layers. The inset is a zoom for the more experimentally relevant $\protect\omega \rightarrow 0$ region of the spectrum (see text). $|\protect\alpha^*|\approx 0.6$ for SLG, whereas this value is highly reduced for MLG: $|\protect\alpha^*|\approx 0.3$ for a 20-layers sample in our numerical calculation, which has a behavior very similar to graphite.[]{data-label="Fig:alpha"}](alpha_g_0.2.eps){width="6.5cm"}
Finally, we calculate the renormalization of the fine structure constant $\alpha=e^2/v_{\mathrm{F}}$ due to dynamic screening associated to the inter-band transitions from the valence band. For this, and in analogy with Ref. , we define $$\label{Eq:alpha}
\alpha^*(\mathbf{q},\omega)=\frac{\alpha}{\varepsilon(\mathbf{q},\omega)}$$ The results for the modulus $|\alpha^*|$ for SLG and MLG are shown in Fig. \[Fig:alpha\]. In this plot we have used the value of the Fermi velocity valid near the Dirac point, i.e., $v_{\mathrm{F}}=3at/2$. Therefore, we emphasize that these results should be reliable only at low energies. At $\omega \rightarrow 0$ and for the smallest wave-vector we can access ($q=0.2a^{-1}$), RPA predicts $|\alpha^*|\approx 0.6$ for SLG, which is considerably higher than the value estimated in Ref. : $|\alpha^*|\approx 0.15$. However, the results that we obtain for MLG are much closer to this value: we find that $|\alpha^*|\approx 0.3$ for graphite, only slightly higher (a factor of 2) than the experimental results of Ref. , which are actually obtained from graphite.
Conclusions {#Sec:Conclusions}
===========
In conclusion, we have studied the excitation spectrum of single- and multi-layer graphene using a full $\pi$-band tight-binding model in the random phase approximation. We have found that, for MLG, the consideration of inter-layer hopping is very important to properly capture the low energy region ($\omega\sim v_{\mathrm{F}} q$) of the spectrum. This, together with trigonal warping effects, lead to a finite contribution to the spectral weight at low energies as well as a redshift of the peaks with respect to the Dirac cone approximation. We have also studied the high energy plasmons which are present in the spectrum of SLG and MLG at an energy of the order of $\omega\sim 2t\approx 6$eV and which are associated to the enhanced DOS at the Van Hove singularities of the $\pi$-bands. The energy of the $\pi$-plasmon depend also on the orientation between adjacent layers, and we find that, for a given wave-vector, the energy of the mode for ABC-stacked MLG is redshifted with respect to the corresponding energy of ABA ordering. This difference is higher as we increase the number of graphene layers of the system.
The effect of disorder has been considered by the inclusion of a random on-site potential and by a renormalization of the nearest neighbor hopping. Both kinds of disorder lead to a redshift of the $\omega\approx v_{\mathrm{F}} q$ and $\omega\approx 2t$ peaks of the non-interacting excitation spectrum and to a smearing of the Van Hove singularities. The position of the $\pi$-plasmons is unaffected by disorder, although the height of the absorption peaks is reduced as compared to the clean limit.
Finally, we have compared our results to some recent experiments. Our calculations for the loss function $\mathrm{Im}~1/\varepsilon (\mathbf{q},\omega )$ show a redshift of the SLG mode with respect to graphite, and compare reasonably well with experimental EELS data.[@EB08; @GG08] Furthermore, we also obtain good agreement with the IXS results for the response function obtained in Ref. . We obtain a static dielectric function which grows with the number of layers of the system. In the long wavelength and $\omega \rightarrow 0$ limit, the dynamically screened fine structure constant is found to be highly reduced from graphene to graphite. The value that we find for a MLG in the RPA, without considering any excitonic effects, is about two times larger than the one estimated in Ref. for graphene. More accurate results could be obtained going beyond single-band RPA,[@SK11] which is beyond the scope of this work.
Acknowledgement
===============
The support by the Stichting Fundamenteel Onderzoek der Materie (FOM) and the Netherlands National Computing Facilities foundation (NCF) are acknowledged. We thank the EU-India FP-7 collaboration under MONAMI.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The cryptographic protocol of coin tossing consists of two parties, Alice and Bob, that do not trust each other, but want to generate a random bit. If the parties use a classical communication channel and have unlimited computational resources, one of them can always cheat perfectly. Here we analyze in detail how the performance of a quantum coin tossing experiment should be compared to classical protocols, taking into account the inevitable experimental imperfections. We then report an all-optical fiber experiment in which a single coin is tossed whose randomness is higher than achievable by any classical protocol and present some easily realisable cheating strategies by Alice and Bob.'
address:
- '$^1$ Service OPERA photonique, CP 194/5, Université Libre de Bruxelles (U.L.B.), Avenue F. D. Roosevelt 50, B-1050 Bruxelles, Belgium '
- '$^2$ Laboratoire d’Information Quantique, CP 225, Université Libre de Bruxelles (U.L.B.), Boulevard du Triomphe, B-1050 Bruxelles, Belgium'
- '$^3$ Département d’Optique P.M. Duffieux, Institut FEMTO-ST, Centre National de la Recherche Scientifique UMR 6174, Université de Franche-Comté, 25030 Besan[ç]{}on, France'
author:
- 'A. T. Nguyen$^1$, J. Frison$^2$, K. Phan Huy$^3$ and S. Massar$^2$'
title: Experimental quantum tossing of a single coin
---
Introduction
============
The cryptographic protocol of coin tossing introduced by Blum [@Blum] consists of two parties, Alice and Bob, that do not trust each other, but want to generate a random bit. If the parties use a classical communication channel and have unlimited computational resources, one of them can always cheat perfectly. But what if they use a quantum communication channel? Because of its conceptual importance and potential applications, quantum coin tossing was already envisaged by Bennett and Brassard in their seminal paper on quantum cryptography [@BB84]. Later works showed that perfect quantum coin tossing is impossible [@LC; @MSCK; @Kitaev], but that imperfect protocols exist [@GVW; @MSCK; @ATSV; @A2002; @SR2002; @Mo07] that perform better than any classical protocol.
Work on quantum coin tossing distinguishes between “weak coin tossing" and “strong coin tossing". In weak coin tossing Alice and Bob have antagonistic goals: Alice wants the coin to be heads, say, whereas Bob wants the coin to come out tails. Good quantum protocols for weak coin tossing exist, although they seem very difficult to implement[@Mo07]. In strong coin tossing Alice and Bob both want the coin to be perfectly random. Quantum protocols that perform better at strong coin tossing than any classical protocol exist[@A2002; @SR2002] and come close to the known upper bound (for the original unpublished proof of the upperbound, see[@Kitaev]; published proofs can be found in [@ABDR; @GW]).
Quantum coin tossing itself is just one example of several interesting tasks that two parties which do not trust each other can achieve if they share a quantum communication channel, but cannot achieve if they use a classical communication channel. Other examples include multiparty coin tossing[@ABDR] and weak forms of string committment[@BCHLW; @J]. The no go theorems mentioned above [@LC; @MSCK; @Kitaev] rule out most other applications, except if one adds additional assumptions such as bounding the size of quantum memories[@DFSS].
Recently two works [@Expt2; @Expt] have experimentally studied optical implementations of quantum coin tossing. However the experiment of ref. [@Expt2] suffered from important photon loss which made it difficult to assess how the experiment worked when tossing a single coin. This was circumvented, as in [@Expt], by addressing string flipping, [*i.e.*]{} the problem where the parties try to toss a string of coins rather than a single one. These works were however carried out without realizing that good classical protocols exist for string flipping, see e.g. [@Betal] for a presentation of such protocols.
In the present work we go back to the conceptually simpler problem of tossing a single coin, and report an experiment in which a single coin is tossed whose randomness is higher than achievable by any classical protocol. We begin by discussing in detail how the results of such a coin tossing experiment should be compared with classical protocols in view of the inevitable imperfections that will occur in any experimental realisation. Coin tossing in the presence of noise was already studied in [@BM], but with the emphasiz on applications to string flipping, whereas here we are concerned with tossing a single coin. We then present the experimental implementation, which follows closely the earlier work of [@Expt], and present some easily realisable cheating strategies by Alice and Bob.
Formulation of the problem
==========================
A protocol for coin tossing consists in a series of rounds of (classical or quantum) communication at the end of which the parties decide on an outcome. The outcome can be either a decision that the coin has the value $c=0$ or $c=1$, or it can be that the protocol aborts, in which case we say that $c=\perp$. Note that because the rounds of (quantum or classical) communication are sequential, it is logically possible for Alice to choose one output $x$, and for Bob to chose another output $y$. For the sake of generality it is convenient to take this into account and to denote by $$\begin{aligned}
p_{xy}&=&\mbox{Probability that in an honest execution of the}\nonumber\\
& &\mbox{protocol
Alice outputs $x$ and Bob outputs $y$,} \nonumber\\
& &\mbox{where $x,y\in\{0,1,\perp\}$.}\nonumber\end{aligned}$$
We will say that a protocol is [*correct*]{}, if, when both parties are honest, at the end of the protocol they agree on the outcome, and that the results $c=0$ and $c=1$ occur with equal probability: $p_{00}=p_{11}=(1-p_{\perp \perp})/2$. This formulation takes into account that because of experimental imperfections, the outcome $c=\perp$ may occur even when both parties are honest.
The aim of a cheater is to force the outcome of the coin tossing protocol. We denote by $$\begin{aligned}
p_{*y}&=&\mbox{Probability that a dishonest Alice can force
an}\nonumber\\
&& \mbox{ honest Bob to
output y}\nonumber\\
p_{x*}&=&\mbox{Probability that a dishonest Bob can force an}\nonumber\\
&& \mbox{honest Alice to
output x}\nonumber\end{aligned}$$ An alternative notation often used in the litterature is the bias $\epsilon$ which is related to our notation by $$\begin{aligned}
\epsilon_A &=& \max_y (p_{*y}-\frac{1}{2} )\nonumber\\
\epsilon_B &=& \max_x (p_{x*}-\frac{1}{2} )\end{aligned}$$ The bound due to Kitaev [@Kitaev; @ABDR; @GW] states that either $\epsilon_A$ or $\epsilon_B$ is greater or equal to $1/\sqrt{2}$. The best known protocol for strong coin tossing due to Ambainis has $\epsilon_A = \epsilon_B = 1/4$. In our experimental implementation, as we will see later, we will be concerned by a protocol which in the terminology of [@SR2002] has “$\rho_0$ and $\rho_1$ both pure”. For such protocols it is proven in [@SR2002] that $\epsilon_A^2 + \epsilon_B^2 \geq 1/4$.
In the appendix we prove the following (which generalises a result of [@Kitaev] when $p_{\perp\perp}=0$):\
[*Lemma 1: For any correct [*classical*]{} coin tossing protocol with three outcomes $0,1,\perp$ we have: $$\begin{aligned}
(1-p_{0*})(1-p_{*1})&\leq& p_{\perp\perp} \ ,\label{r3}\\
(1-p_{1*})(1-p_{*0})&\leq& p_{\perp\perp}\ . \label{r4}\end{aligned}$$* ]{}
Note that if $p_{\perp\perp}=0$ these inequalities imply that either $p_{0*}=1$ or $p_{*1}=1$, and that either $p_{1*}=1$ or $p_{*0}=1$, thereby showing that classical coin tossing is impossible. When $p_{\perp\perp}\neq 0$ a cheater can no longer necessarily force the outcome he wants. In the supplementary material we show that there exist classical protocols that saturate either one of equations (\[r3\]) or (\[r4\]), and that there exist classical protocols that come close to saturating both equations (\[r3\]) and (\[r4\]).
In view of Lemma 1, it is natural to quantify the quality of quantum coin tossing experiments by the following merit function: $${\cal M} = \frac{ (1-p_{*0})(1-p_{1*})}{2} +
\frac{(1-p_{*1})(1-p_{0*})}{2} - p_{\perp \perp} \label{M}$$ which has the following properties:
1. Positivity of probabilities implies $-1 \leq {\cal M} \leq
+1$
2. For any classical protocol we have ${\cal M}\leq 0$
3. An ideal protocol would have ${\cal M}=1$.
The interpretation of the merit function is most obvious in the weak coin tossing scheme wherein Alice wins if Bob outputs $1$ while Bob wins if Alice outputs $0$ because then the term $(1-p_{*1})(1-p_{0*})$ is the product of how often a dishonest Alice cannot force a win times how often a dishonest Bob cannot force a win (and similarly for the term $(1-p_{*0})(1-p_{1*})$). The better the protocol, the larger these terms.
As illustration let us compute the value of ${\cal M}$ for different protocols. The bound due to Kitaev states with precision, see [@ABDR], that $p_{*1}p_{1*}\geq 1/2$ and $p_{*0}p_{0*}\geq 1/2$. Inserting this into eq . (\[M\]) shows that for all quantum protocols, ${\cal M}\leq (1 -
1/\sqrt{2})^2\simeq 0.086$. For Ambainis’s protocol [@A2002] for instance we have ${\cal M}= 1/16=0.0625$.
Experimental Implementation
===========================
The Protocol
------------
Our implementation of quantum coin tossing uses the following protocol:
1. Alice chooses $a\in\{0,1\}$ at random. She prepares state $\psi_a$, where the two possible states are non orthogonal: $ |
\langle \psi_1 | \psi_0\rangle | = \cos \theta > 0$. She sends $\psi_a$ to Bob.
The states $\psi_{0,1}$ will be taken to be coherent states of light of amplitude $\alpha$ and opposite phase: $$|\psi_{0}\rangle = |+\alpha\rangle \quad , \quad |\psi_{1}\rangle
= |-\alpha\rangle$$ which implies that $$\cos^2 \theta = |\langle\psi_1|\psi_0\rangle|^2= |\langle -\alpha
| + \alpha \rangle |^2 = e^{- 4 \alpha^2}.
\label{cos}$$ In the notation of [@SR2002] we thus have $\rho_0=|\psi_{0}\rangle\langle\psi_{0}|$ and $\rho_1=|\psi_{1}\rangle\langle\psi_{1}|$ both pure. Also note that $\rho_0\neq\rho_1$ prevents from cheating strategies based on entanglement [@M96; @LC].
2. Bob chooses $b\in\{0,1\}$ at random. He tells the value of $b$ to Alice.
3. Alice tells Bob the value of $a$.
4. Bob carries out a measurement which projects onto $\psi_a$ or onto the orthogonal space. If he finds that the state is not equal to $\psi_a$ he aborts, and the outcome of the protocol is $\perp$. If he finds that the state is equal to $\psi_a$ then the outcome of the protocol is $c=a\oplus b$.
Bob’s measurement is carried out as follows: using a local oscillator (LO), he displaces the quantum state by $+\alpha$ if $a=1$ or by $-\alpha$ if $a=0$. If Alice is honest this results in the state becoming the vacuum state. To check this Bob then sends the resulting state onto a single photon detector. If the detector clicks then Bob assumes that Alice was cheating and he aborts: the outcome of the protocol is $\perp$. If the detector does not click, then Bob assumes that Alice is honest. (Note that Bob’s measurement is similar in spirit to the method proposed in [@WV] for quantum state tomography, but Bob’s task is simpler since he only needs to detect if Alice is cheating, and not carry out the full state tomography).
Analysis in the absence of imperfections
----------------------------------------
We now study how the merit function ${\cal M}$ depends on the details of the experiment. For the sake of comparison we first look at the situation in the absence of imperfections.
First of all, in this case $p_{\perp \perp}=0$.
Second, if Alice is dishonest she will send a fixed state $|\phi\rangle$ at step 1 and at step 3 she will choose the value of $a$ which will make her win the protocol, and then she will hope that Bob will not abort. The probability that Bob will abort is given by the overlap of $|\phi\rangle$ with $| \psi_0\rangle$ and $| \psi_1\rangle$. One easily finds (see [@BM]) that Alice’s optimal choice is $|\phi\rangle = N (| \psi_0\rangle + |
\psi_1\rangle ) $ where $N$ a normalization constant, yielding the optimal values: $$p_{*0} = p_{*1} = \frac{1}{2} + \frac{ |\langle
\psi_1|\psi_0\rangle| }{2} = \frac{1}{2} + \frac{\cos \theta}{2}\
. \label{p*c}$$
Third, if Bob is dishonest, he will measure the state sent by Alice at step 2 so as to try to find out whether it is $\psi_0$ or $\psi_1$, and he will then choose the value of $b$ according to the result of his measurement. For the optimal measurement the probability that Bob wins is $$p_{0*} = p_{1*} = \frac{1}{2} + \frac{ \sqrt{1- |\langle
\psi_1|\psi_0\rangle|^2} }{2} = \frac{1}{2} + \frac{\sin
\theta}{2}\ . \label{pc*}$$
The maximal value of the merit function ${\cal M}_{max}= \frac{ (1
- 1/\sqrt{2})^2}{4}\simeq 0.021$ occurs when $\cos(\theta)=\sin(\theta)=1/\sqrt{2}$, corresponding to $\alpha^2=0.17$. Note that this is the maximum value for protocols which in the terminology of Spekkens and Rudolph [@SR2002] fall in the category “$\rho_0$ and $\rho_1$ both pure”.
Analysis in the presence of imperfections
-----------------------------------------
To obtain estimates on $p_{*c}$, $p_{c*}$ and $p_{\perp\perp}$, and hence to estimate ${\cal M}$, in the presence of imperfections requires that we make assumptions on how the experiment is carried out.
The parameter, $p_{\perp\perp}$, which we also call the Quantum Bit Error Rate (QBER), can easily be measured experimentally by tossing a large number of coins with Alice and Bob both following their honest strategy.
### Bob is dishonest
When Bob is dishonest his cheating strategy is, as before, to estimate before step 2 the state $|\psi_{a} \rangle$ prepared by Alice so as to correctly guess the value of $a$. How do experimental imperfections, and in particular the limited visibility $V$ of interferences affect Bob’s success probability $p_{c*}$? To analyse this note that the state Alice sends to Bob is a short laser pulse of known intensity which is then strongly attenuated. Under strong attenuation all quantum states tend towards mixtures of coherent states (see e.g. [@Expt]). Thus we can assume that the states prepared by Alice are coherent states of known intensity $\alpha^2$. These coherent states are not precisely known to Alice. However it is not difficult to show that if two coherent states have intensity $\alpha^2$, their scalar product is lower bounded by $|\langle \psi_1|\psi_0\rangle|
\geq e^{-2 \alpha^2}$. Bob’s cheating probability can then be bounded, as in equation (\[pc\*\]), by the scalar product of the two states prepared by Alice: $$p_{c*}=\frac{1}{2} +
\frac{ \sqrt{1- |\langle \psi_1|\psi_0\rangle|^2} }{2} \leq \frac{1}{2} +
\frac{
\sqrt{1- e^{-4 \alpha^2} }
}{2}\ .
\label{pc*2}$$
### Alice is dishonest
When Alice is dishonest we suppose that she can prepare an arbitrary state just in front of Bob’s laboratory, and then send it to Bob. How do the imperfections in Bob’s laboratory affect $p_{*c}$? To quantify this Bob could carry out a complete tomography of his measurement apparatus, and based on the results compute what is Alice’s best cheating strategy. Here we will make a simple estimate based on easily accessible parameters.
First of all let us consider the effects of the attenuation $A_T$ during transmission between Alice and Bob’s laboratories, of the attenuation $A_B$ in Bob’s apparatus, and of the efficiency $\eta$ of his detector. We take these parameters into account by analysing a fictitious system in which Bob’s apparatus is replaced by a lossless apparatus, and all the attenuation is under Alice’s control, [*i.e.*]{} $\eta^{fict}=100\%$, $A_B^{fict}=1$, and $A_T^{fict}=A_T A_B \eta$. This replacement can only help a cheating Alice. In the fictitious system the state sent by an honest Alice is $|\pm \alpha_B^{fict}\rangle= |\pm \alpha
\sqrt{A_T A_B \eta}\rangle$.
Second we analyse the effect of finite visibility on the performance of the fictitious system just described. Because of the finite visibility, Bob will not be making a projection onto the state $|\pm \alpha_B^{fict}\rangle$, but onto slightly different states. We make the assumption that Bob’s apparatus acts as a passive linear optical system. This implies that the true states onto which Bob projects are slightly modified coherent states $|\pm \alpha_B^{fict} + \delta_\pm\rangle$. The deviations due to $\delta_\pm$ give rise to the optical contribution to the QBER: $$QBER_{opt}=\left( |\delta_+|^2 + |\delta_-|^2\right)/2 =q
|\alpha_B^{fict}|^2\ ,$$ where $q$, the QBER per photon, can be related to the visibility V of interferences by $q\simeq (1-V)/2$. (Note that in addition to $QBER_{opt}$, there is another contribution to the QBER due to the dark counts of the detectors. The total QBER is the sum of these two contributions: $QBER=QBER_{opt}+QBER_{dk}$.)
The distance between the two states onto which Bob projects is given by $$\begin{aligned}
&&|(+\alpha_B^{fict} + \delta_+)-(-\alpha_B^{fict} + \delta_-)|^2\nonumber\\
&\geq& 4 |\alpha_B^{fict}|^2 - 4 |\alpha_B^{fict}| |\delta_+ -
\delta_-|
= 4 |\alpha_B^{fict}|^2 (1 - 2 \sqrt{q}) \ .
\nonumber\\\end{aligned}$$ Inserting this into equation (\[p\*c\]) gives $$p_{*c} \leq \frac{1}{2} + \frac{1}{2}\exp\left[ -A_B A_T
\eta\left(1-2 \sqrt{q} \right) \alpha^2\right]\ . \label{p*cB}$$ Thus the effect of the imperfections is to replace $\alpha^2$ by and effective attenuated intensity $A_{B} A_T \eta
\left(1-2\sqrt{q} \right) \alpha^2$.
Experimental results
--------------------
{width="1\columnwidth"}
Our experimental setup, depicted in Fig. 1, based on the plug and play system developed for long distance quantum key distribution [@RGGGZ], is very similar to the one described in [@Expt]. It consists of an all-fiber (standard SMF-28) passively balanced interferometer, and is therefore well suited to long distance quantum communication. The protocol begins with Bob producing a short (300 ps) intense laser pulse at $\lambda = 1.55 \mu m$ (id300 from idQuantique). The pulse is split in two by the coupler C1 with equal reflection and transmission coefficients $50\%$. The two pulses are delayed one with respect to the other by 134ns. The pulses are then recombined on a polarizing beam splitter (PBS) and sent to Alice. The pulse that propagated along the long arm of the interferometer is strongly attenuated and will play the role of signal. The pulse that propagated along the short arm will play the role of local oscillator (LO). Upon receiving the pulses, Alice splits off part of them using the coupler C2 and sends this to a photodiode that triggers her electronics. At Alice’s site the pulses are further attenuated by the different optical elements. They are reflected by the Faraday mirror. And Alice randomly chooses which phase $\Phi_A=0,\pi$ to put on the signal pulse using her phase modulator. The signal Alice sends back to Bob is thus the coherent state $|\pm \alpha\rangle$ with average photon number $|\alpha|^2=0.27$.
When the pulses come back to Bob’s site, they are sent along the short and long arm of the interferometer by the PBS and interfere at coupler C1. In front of the PBS is a delay line belonging to Bob which ensures that after the pulses enters Bob’s laboratory he bluehas the time to send to Alice the value of the bit $b$ and then receive from her the value of $a$. In our experiment the fiber pigtails of the PBS are sufficient to realize the delay. Upon receiving the value of $a$, Bob puts the corresponding phase $\Phi_B=a \pi$ on the LO. This ensures that there should be destructive interference at the output port that goes to the circulator and then to detector D1 (id200 from idQuantique). If detector D1 registers a click, Bob aborts. If it does not click, the outcome of the coin toss is $c=a\oplus b$. The other output of coupler C1 is monitored by detector D2, although this is not directly used in the experiment.
There are in fact two security loopholes in this experiment. The first arises because Alice does not know the intensity of the signal pulse she attenuates before sending it back to Bob. Thus in principle Bob could send her a more intense state than expected, which would mean that the scalar product of the states prepared by Alice would be smaller than expected. The second security loophole arises because Bob does not know the intensity of the pulse he uses as LO. Thus in principle Alice could send Bob the vacuum state, both in the signal and LO, and cheat perfectly. Both loopholes could be closed by having Alice (Bob) monitor the intensity of the signal (LO) before she (he) attenuates it. This was not realised in the present setup because the laser pulses used were not intense enough, but would be possible using more intense or longer laser pulses as in [@Expt], or by using an isolator combined with an amplitude modulator as in [@MZG2006].
### Both parties are honest
As mentioned above, we performed the experiment with $|\alpha|^2=0.27$. In a typical series 10000 coins were tossed, and we obtained 5066 occurrences of $c=1$, 2 occurrences of $c=\perp$, the other outcomes being $c=0$ (which is consistent with the statistical uncertainty which should be of order $\sqrt{
5000}= 70$). However we insist that the protocol can be used to toss a single coin.
We estimate the merit function as follows. The abort probability is estimated by tossing a large number ($1.5\ 10^5$) of coins with Alice and Bob both honest: $$p_{\perp\perp}\simeq 1.40\pm0.37\; 10^{-4} \ .\label{ExpPperp}$$ where the error comes from statistical uncertainty.
The transmission losses are assumed to be negligible, $A_T=1$, as both parties are separated by a few meters of optical fiber. Bob’s detector D1 has a $\eta =10\%$ quantum efficiency. It is gated using a $2.5\ ns$ gate leading to a dark count probability of $4.7\ 10^{-5}$. The attenuation of the signal in the optical elements of Bob’s laboratory has been measured to be $A_B\simeq
-6$ dB (which includes the $3$ dB losses at coupler C1 where the signal and the LO interfere). Visibilities, as measured using an intense signal, were at least 99.0% (corresponding to $q=5\
10^{-3}$). By inserting these parameters in equations (\[pc\*2\]) and (\[p\*cB\]) we obtain upper bounds for $p_{*c}$ and $p_{c*}$: $$p_{*c} \leq 0.9971 \label{ExpP*c} \quad \mbox{and}\quad p_{c*}\leq
0.906
$$ leading to the lower bound for the merit function: $${\cal M} \geq 1.33\ 10^{-4}\ .\label{ExpM}$$
This bound may seem very small. Its value is roughly explained by noting that the maximal value in the absence of imperfections is ${\cal M}_{max}=0.021$. The main source of imperfections are the efficiency of the detectors (10 dB) and the losses in Bob’s apparatus (6 dB). Thus we should reduce the attainable value of ${\cal M}$ by a factor $40$, yielding approximately equation (\[ExpM\]). This argument shows that the simplest way to improve the experiment would be to use a more efficient detector. It also shows that the value of ${\cal M}$ is rather robust against small variations of the experimental parameters. We have computed that we could keep ${\cal M}$ positive while increasing losses between Alice and Bob to $A_T\simeq 4.4$ dB (more than 20 km of SMF-28 fiber), all other parameters being kept constant.
### Bob is dishonest
In order to cheat Bob must estimate the state $|\psi_{a} \rangle$ prepared by Alice so as to correctly guess the value of $a$ before sending the value of $b$. We implemented a simple cheating strategy in which Bob always applies $\Phi_B=0$ on the LO. If detector D1 clicks Bob assumes that Alice chose $a=1$, whereas if D1 does not click he assumes $a=0$. Implementing this strategy yielded the value $p_{1*}=0.505$. This very low value is due to the small values of $\eta$ and $A_B$. Note that a much better cheating strategy, but which was impossible to implement in our laboratory, would be for Bob to carry out a homodyne measurement and measure the quadrature that gives him the best estimate of $a$.
### Alice is dishonest
As discussed above, when Alice is dishonest her best strategy is to send a fixed state $|\phi\rangle = N (|+\alpha\rangle +
|-\alpha\rangle)$ to Bob. After receiving $b$ she then sends the value of $a$ that makes her win the coin toss and hopes that Bob will not abort. In practice we implemented a strategy where Alice always sends $|+\alpha\rangle$. Even though this strategy is very basic, it leads to $p_{*c}=0.9956$, which is very close to the theoretical maximum equation (\[ExpP\*c\]).
Conclusion
==========
In conclusion we have studied in detail how the performance of quantum coin tossing protocols in the presence of imperfections should be compared to classical protocols. We then reported on a fiber optics experimental realisation of a quantum coin tossing protocol. Our analysis shows that in this realisation the maximum success cheating probabilities for Alice and Bob are respectively 0.9971 and 0.906 when experimental imperfections are taken into account, which is still better than achievable by any classical protocol. We implemented this protocol using an all-optical fiber scheme and tossed a coin whose randomness is higher than achievable by any classical protocol. Finally we implemented simple realisable cheating strategies for both Alice and Bob.
After the present work was completed, we learned of a recent proposal specially designed for carrying out quantum coin tossing in the presence of losses[@BBBG]. Obviously taking into account losses, in particular those that occur in Bob’s apparatus, was an important consideration when choosing and analysing the protocol reported here. The protocol reported in [@BBBG] seems more tolerant to loss then ours. Once the effect of other imperfections (such as finite visibility of interference fringes) are taken into account, it could be compared to ours using the merit function ${\cal M}$ introduced above.
We acknowledge the support of the Fonds pour la formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA, Belgium); of the Interuniversity Attraction Poles Programme - Belgian State - Belgian Science Policy under grant IAP6-10; and of the EU project QAP contract 015848.
Appendix {#appendix .unnumbered}
========
Here we provide bounds on the performance of classical coin tossing protocols when there is some probability that the protocol aborts when both parties are honest. We also show that there exist classical protocols that attain these bounds. We use the notation and terminology introduced in the main text. The idea of the following result is to analyse the performance of a classical protocol with 3 outcomes ([*i.e.*]{} a classical protocol in which the parties try to toss a trit.).
[*Lemma 1: For any correct [*classical*]{} coin tossing protocol with three outcomes $0,1,\perp$ we have: $$\begin{aligned}
(1-p_{0*})(1-p_{*1})&\leq& p_{\perp\perp} \label{rr3}\ ,\\
(1-p_{1*})(1-p_{*0})&\leq& p_{\perp\perp} \label{rr4}\ .\end{aligned}$$* ]{}
[*Proof of Lemma 1*]{}. We need to introduce some notation.
The protocol consists of $K$ rounds of communication, labeled $j=1,\ldots,K$.
Denote by $u_j$ the possible states of the protocol at round $j$.
Denote by $w(u_j)$ the probability of reaching state $u_j$ at round $j$ in an honest execution of the protocol.
Denote by $w(u_{j+1}|u_j)$ the probability that in an honest execution, the protocol will be in state $u_{j+1}$ at round $j+1$ if it is in state $u_j$ at round $j$.
Denote by $p_{*y}(u_j)$ the maximum probability that if Alice is dishonest and Bob is honest, then Alice can force Bob to output $y$ at the end of the protocol if the state at round $j$ is $u_j$.
Denote by $p_{x*}(u_j)$ the maximum probability that if Bob is dishonest and Alice is honest, then Bob can force Alice to output $x$ at the end of the protocol if the state at round $j$ is $u_j$.
Introduce the quantity $T_j$ defined by $$T_j(x,y)=\sum_{u_j}w(u_j)(1-p_{x*}(u_j))(1-p_{*y}(u_j))$$
Note that if we take $x=0$ and $y=1$ the initial value ($j=1$) of $T$ is $T_1(0,1)=(1-p_{0*})(1-p_{*1})$, ie. the left hand side of eq. (\[rr3\]).
Note also that at round $K$, when the protocol has ended, $T_K$ is equal to the sum over the final states of the protocol in an honest execution of the product of the probabilities that the output of Alice is not $x$ and that the output of Bob is not $y$. Thus if we take $x=0$ and $y=1$, then $T_K(0,1)=p_{\perp\perp}$, ie. the right hand side of eq. (\[rr3\]).
To complete the proof we show that $T$ is an increasing function of $j$, ie. $T_{j+1}\geq T_j$. To this end suppose that at round $j$ Bob will send some communication to Alice.
Then Alice cannot influence what will happen at round $j$, hence we have: $p_{*y}(u_j)=\sum_{u_{j+1}}w(u_{j+1}|u_j)
p_{*y}(u_{j+1})$.
Furthermore we have the trivial identity $w(u_{j+1}) = \sum_{u_j}
w(u_{j+1}|u_j) w(u_j)$.
Finally we note that since it is Bob’s turn to talk at round $j$, we have $1-p_{x*}(u_j)\leq 1-p_{x*}(u_{j+1})$ where $u_{j+1}$ is any state at round $j+1$ that can be obtained from state $u_j$ at round $j$ in an honest execution.
Inserting these identities into the definition of $T_j$, we obtained the desired inequality $T_{j+1}\geq T_{j}$.
The proof of eq. (\[rr4\]) is similar.
[*End of proof of Lemma 1.*]{}
We have also obtained a partial converse of Lemma 1:\
[*Lemma 2: There exists a correct classical protocol such that inequality (\[rr3\]) is saturated, and there exists a correct classical protocol such that inequality (\[rr4\]) is saturated. There also exists a correct classical protocol for which $$\begin{aligned}
(1-p_{0*})(1-p_{*1})= (1-p_{1*})(1-p_{*0})=\frac{ p_{\perp\perp}
}{2} \ . \label{sat}\end{aligned}$$*]{}
[*Proof of Lemma 2.*]{} Let us consider the following protocol:
Round 1: Alice excludes one of the outcomes. That is she chooses that the outcome of the protocol will be either in $\{0,1\}$ (she has excluded $\perp$), $\{0,\perp\}$ (she has excluded $1$) or $\{1,\perp\}$ (she has excluded $0$). She tells her choice to Bob. If she is honest she chooses randomly among these three possibilities with a priori probabilities $q_{01}$, $q_{0\perp}$, $q_{1\perp}$.
Round 2: Bob chooses which of the remaining two outcomes is the result of the protocol. He tells Alice what is his choice. Thus for instance if Alice told him that the outcome was in $\{0,1\}$, Bob can choose that the outcome is either $0$ or $1$, but not $\perp$. If he is honest he chooses randomly among the two remaining possibilities with probabilities $q_{0|01}$, $q_{1|01}$; $q_{0|0\perp}$, $q_{\perp|0\perp}$; $q_{1|1\perp}$, $q_{\perp|1\perp}$.
It is easy to check that, if the parties are honest, the probabilities are: $$\begin{aligned}
p_{00} &=& q_{0|01} q_{01}+ q_{0|0\perp}q_{0\perp}\nonumber\\
p_{11} &=& q_{1|01} q_{01}+ q_{1|1\perp}q_{1\perp}\nonumber\\
p_{\perp\perp} &=& q_{\perp|0\perp}q_{0\perp}+ q_{\perp|1\perp}q_{1\perp}\ ; $$ and that, if they are dishonest, the probabilities are: $$\begin{aligned}
p_{*0}&=&\max \{ q_{0|01},q_{0|0\perp} \}\nonumber\\
p_{*1}&=&\max \{ q_{1|01},q_{1|1\perp} \}\nonumber\\
p_{0*}& =& q_{01}+q_{0\perp}\nonumber\\
p_{1*}&=& q_{01}+q_{1\perp}\ .$$
If we choose the parameters such that $q_{0\perp}=q_{1\perp}$, $q_{0|01}=q_{1|01}=1/2$, $q_{0|0\perp}=q_{1|1\perp}\geq 1/2$, then the protocol is correct and eq. (\[sat\]) is verified.
And if we choose $q_{0\perp}=0$ then we have $p_{\perp\perp}=q_{\perp|1\perp}q_{1\perp}=(1-q_{1|1\perp})(1-q_{01})$ and $p_{0*}=q_{01}$, $p_{*1}=q_{1|1\perp}$ thus saturating eq. (\[rr3\]). Note that by adjusting the remaining free parameter $q_{0|01}$ one can make the protocol correct.
Similarly one can saturate inequality (\[rr4\]).
[*End of proof of Lemma 2.*]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
In light of BICEP2, we reexamine single field inflationary models in which the inflaton is a composite state stemming from various four-dimensional strongly coupled theories. We study in the Einstein frame a set of cosmological parameters, the primordial spectral index $n_{s}$ and tensor-to-scalar ratio $r$, predicted by such models. We confront the predicted results with the joint Planck data, and with the recent BICEP2 data. We constrain the number of e-foldings for composite models of inflation in order to obtain a successful inflation. We find the minimal composite inflationary model is fully consistent with the Planck data. However it is in tension with the recent BICEP2 data. The observables predicted by the glueball inflationary model can be consistent with both Planck and BICEP2 contours if a suitable number of e-foldings are chosen. Surprisingly, the super Yang-Mills inflationary prediction is significantly consistent with the Planck and BICEP2 observations.\
\
\
---
July 14, 2014
[**[ BICEP2 constrains Composite Inflation]{}\
**]{}
Introduction
============
Nowadays, inflationary models gain a lot of interest. The inflationary paradigms [@Alex; @KSa; @KSa1; @DKa; @GUT] tend to solve important issues, e.g. the magnetic monopoles, the flatness, and the horizon problems, and subsequently provide the mechanism for generation of density perturbations as seed for the formation of large scale structure in the universe. Most models of inflation so far were formulated by introducing new scalar fields (called inflaton) with a nearly flat potential appearing in many paradigms, e.g. [@new; @new1; @chaotic; @natural; @natural1], and even in superstring [@Espinosa:1998ks; @Casas:1999xj] and supergravity theories [@Yamaguchi:2011kg; @Farakos:2013cqa; @Linde:2013aya; @Ferrara:2013rsa].
However, the theories featuring elementary scalar fields are unnatural meaning that quantum corrections generate unprotected quadratic divergences which must be fine-tuned away if the models must be true till the Planck energy scale. Hence, this is the main reason why the authors in [@Channuie:2011rq; @Bezrukov:2011mv; @Channuie:2012bv] investigated inflation in which the inflaton need not be an elementary degree of freedom. Recent investigations show that it is possible to construct models in which the inflaton emerges as a composite state of a four-dimensional strongly coupled theory. We called these alternative paradigms composite inflation. There were other models of super or holographic composite inflation [@Cvetic:1989eg; @Thomas:1995dq; @GarciaBellido:1997mq; @Allahverdi:2006iq; @Hamaguchi:2008uy; @Evans:2010tf; @Alberghi:2010sn; @Alberghi:2009kk]. In order to constrain the inflationary theory, recent investigations contain hundreds of different scenarios [@Martin:2013tda].
Most frequently, we added a term in which the inflaton field, $\phi$, non-minimally coupled to gravity to the action, say $\xi\phi^{2}R/M^{2}_{\rm P}$. This term has purely phenomenological origin. The reason resides from the fact that one want to relax the unacceptable large amplitude of primordial power spectrum generated if one takes $\xi=0$ or small. However, it is instructive here to provide some insight on how a non-minimal coupling can be naturally formed. An instructive analysis of the generated coupling of a composite scalar field to gravity has been initiated in the Nambu-Jona-Lasinio (NJL) model [@Hill:1991jc]. In this case, a composite field is a chiral condensate, $H$, from the NJL model which couples to gravity via a coupling term very similar to that we deploy in this work. The presence of the induced non-minimal coupling of the boundstate object to gravity, $\xi H^{\dagger}HR$ with $\xi$ a coupling constant, and $R$ the (Ricci) scalar curvature, can be implemented from the symmetric phase of the theory, i.e. a massless case, or from the broken phase when $\textless H\textgreater=v\neq0$ providing the non-minimal term, $\sqrt{2}\xi v \phi R$ where $H\equiv v+\phi/\sqrt{2}$ with $v$ the vacuum expectation value of $H$. However, the results of $\xi$ are the same. In principle, we can transform such a term into another form by applying the conformal transformation (see detailed discussion in [@FuMa]). Among many frames, the Jordan frame and the Einstein frame are those discussed in the community. Roughly speaking, physics looks different in two different conformal frames. However, physical conclusions remains the same in a weak gravitational field limit [@FuMa].
In this work, we study observables, i.e. the scalar spectral index $n_{s}$ and tensor-to-scalar ratio $r$, in the Einstein frame predicted by the composite inflationary models recently proposed. We will briefly present the concept of conformal transformation. We then compute the power spectrum for the primordial curvature perturbations and the tensor-to-scalar ratio for various composite inflationary models in the Einstein frame. Next we constrain the models by placing our results into the $(r, n_{s})$ plane implemented by using the observational bound for $n_s$ and $r$ from the Planck data and also confront the results with the recent BICEP2 data. Finally, some comments about our results are made in the last section.
Composite Setup and Conformal Transformation {#action}
============================================
We consider a generic strongly-interacting theory non-minimally coupled to gravity. For the model of inflation, we identify the inflaton with one of the lightest composite states of the theory and denote it with $\Phi$. This state has mass dimension $d$. The action for the composite models studied below can be written in the general form $$\begin{aligned}
\mathcal{S}_{\rm J}=\int d^{4}x \sqrt{-g}\left[\frac{1}{2}{\cal F}(\Phi)R-{\cal G}(\Phi)g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-{\cal V}({\Phi})\right]\,, \label{nonminimal}\end{aligned}$$ where ${\cal F}(\Phi)$ and ${\cal G}(\Phi)$ can in general be an arbitrary function of the composite field $\Phi$. ${\cal F}(\Phi)$ gives rise to the non-minimal coupling $\xi\Phi^{2/d} R$. To recover the ordinary Einstein theory of gravity at low energy, we deduce ${\cal F}(v)=M^{2}_{P}$ where $v$ is the vacuum expectation value of the field $\Phi$ at the end of inflation and $M^{2}_{P}=(8\pi G)^{-1/2}\simeq 2.436\times 10^{18}\,{\rm GeV}$ is the reduced Planck energy scale. According to the models of single-field inflation [@Channuie:2011rq; @Bezrukov:2011mv; @Channuie:2012bv] in which the inflaton is a composite state stemming from various strongly-interacting theories well-known in particle physics, in this work, we write ${\cal F}(\Phi)$ and ${\cal G}(\Phi)$ in the form as follows: $${\cal F}(\Phi) = M^{2}_{\rm P}+\xi\Phi^{2/d}\,,\quad {\cal G}(\Phi) = \Phi^{(2-2d)/d}\,,$$ where the non-minimal coupling to gravity is controlled by the dimensionless coupling $\xi$. With the non-minimal coupling term phenomenologically introduced, one need to diagonalize into another form by applying a conformal transformation. The conformal transformation transforms a metric tensor $g_{\mu\nu}$ into another metric $\tilde{g}_{\mu\nu}$ according to the rule $$\begin{aligned}
g_{\mu\nu}\rightarrow\tilde{g}_{\mu\nu}=\Omega^{2}(\Phi)g_{\mu\nu}\quad\quad{\rm with}\quad\quad\Omega^{2}(\Phi)=\frac{{\cal F}(\Phi)}{M^{2}_{P}}\,, \label{scale}\end{aligned}$$ such that $$\begin{aligned}
\tilde{g}^{\mu\nu}=\Omega^{-2}(\Phi)g^{\mu\nu}\quad\quad{\rm and}\quad\quad\sqrt{-\tilde{g}}=\Omega^{-4}(\Phi)\sqrt{-{g}}\,.\label{scale2}\end{aligned}$$ Hereafter we drop tildes to express variables in the Einstein frames. Applying the conformal transformation leads to the Einstein frame and the action reads $$\begin{aligned}
\mathcal{S}_{\rm CI,E} &=&\int d^{4}x \sqrt{-g}\Big[\frac{1}{2} M_{\rm P}^2 R - \left(\frac{M_{\rm P}^2{\cal G}(\Phi)}{{\cal F}(\Phi)} + \frac{3}{2} M^{2}_{\rm P} \left(\frac{{\cal F}(\Phi)'}{{\cal F}(\Phi)}\right)^{2} \right)g^{\mu \nu} \partial_{\mu} \Phi \partial_{\nu} \Phi \nonumber\\&&\quad\quad\quad\quad\quad\,\,- \Omega ^{-4} V({\Phi}) \Big]\,, \label{Einsteinframe}\end{aligned}$$ where prime denotes derivative over the field $\Phi$. We arrived at an involved kinetic term for the inflaton. It is convenient to introduce a canonically normalized field $\chi$ related to €$\Phi$ via $$\begin{aligned}
\frac{d\chi}{d\Phi}=\sqrt{2}M_{\rm P}\sqrt{\frac{{\cal G}(\Phi)}{{\cal F}(\Phi)} + \frac{3}{2}\left(\frac{{\cal F}(\Phi)'}{{\cal F}(\Phi)}\right)^{2}}\,.\label{renor}\end{aligned}$$ In terms of the canonically normalized field we come up with the standard fashion $$\begin{aligned}
\mathcal{S}_{\rm CI,E} =\int d^{4}x \sqrt{-g}\Big[\frac{1}{2} M_{\rm P}^2 R - \frac{1}{2}g^{\mu \nu} \partial_{\mu} \chi \partial_{\nu} \chi- {\cal U}({\chi}) \Big]\quad{\rm with}\quad {\cal U}({\chi})\equiv \Omega ^{-4} V({\Phi})\,. \label{EinsteinRe}\end{aligned}$$ In this work, we will examine the dynamics in the Einstein frame, and therefore express the slow-roll parameters in terms of ${\cal U}$ and $\chi$: $$\begin{aligned}
\epsilon &=& \frac{M_{\rm P}^2}{2}\left(\frac{d\,{\cal U}/d\chi}{{\cal U}}\right)^{2}=\frac{M_{\rm P}^2}{2}\left(\frac{{\cal U}'}{{\cal U}}\frac{1}{\chi'}\right)^{2}\,,\label{epsi}\\
\eta &=& M_{\rm P}^2 \frac{d^{2}\,{\cal U}/d\chi^{2}}{{\cal U}}=M_{\rm P}^2 \frac{{\cal U}''\chi'-{\cal U}'\chi''}{{\cal U}\chi'^{3}}\,, \label{eta}\end{aligned}$$ where $\prime$denotes derivative with respect to the field $\Phi$. Slow-roll inflation ends when $\epsilon=1$, with the corresponding field value $\Phi_{end}$. The horizon exist when the field value equals $\Phi_{ini}$ which is determined by the number of e-foldings ${\cal N}$. This parameter is given by $$\begin{aligned}
{\cal N}= \frac{1}{M_{\rm P}^2}\int^{\chi_{ini}}_{\chi_{end}}\frac{{\cal U}}{d\,{\cal U}/d\chi}d\chi=\frac{1}{M_{\rm P}^2}\int^{\Phi_{ini}}_{\Phi_{end}}\frac{{\cal U}\chi'^{2}}{{\cal U}'}d\Phi\,. \label{efold}\end{aligned}$$ Here we express the number of e-foldings for the change of the field $\chi$ (or equivalently $\Phi$) from $\chi_{end}$ to $\chi_{ini}$ and will use the above mathematical framework to examine the composite-field models in Sec.(\[model\]).
Inflationary Observables {#observable}
========================
In this section, we will briefly review the study of the scalar and tensor perturbations. This leads to obtaining the required inflationary predictions, e.g. the spectral index and tensor-to-scalar ratio. Our starting point here is the action given in Eq.(\[EinsteinRe\]). We will derive the parameters by closely following the work presented in [@Kobayashi:2011nu] for Generalised G-inflationusing the unitary gauge. In the unitary gauge $\chi=\chi(t)$, it is kind of tradition to begin with the perturbed metric as [@Arnowitt:1962hi] $$\begin{aligned}
ds^{2} =-N^{2}dt^{2}+\gamma_{ij}\Big(dx^{i}+N^{i}dt\Big)\Big(dx^{j}+N^{j}dt\Big)\,, \label{pert1}\end{aligned}$$ where $N$ is a lab function, and $N^{i}$ is the shift vector which are respectively defined as: $$\begin{aligned}
N=1+\alpha\,,\,\,\,N_{i}=\partial_{i}\beta\,,\,\,\,\gamma_{ij}=a^{2}(t)e^{2\zeta}\Big(\delta_{ij}+h_{ij}+\frac{1}{2}h_{ik}h_{kj}\Big)\,, \label{pert}\end{aligned}$$ where $\alpha,\,\beta$ and $\zeta$ scalar perturbations and $h_{ij}$ is a tensor perturbation satisfying the condition $h_{ii}=0=h_{ij,j}$. Physically, the lapse function represents the rate of flow of proper time with respect to the coordinate time. Substituting the perturbed line-element given in Eq.(\[pert1\]) into the action (\[EinsteinRe\]) and expanding it to the second order, we obtain [@Kobayashi:2011nu] $$\begin{aligned}
{\cal S}^{(2)}_{T} =\frac{1}{8}\int dt d^{3}{\bf x}\,a^{3}\left[\dot{h}^{2}_{ij}-\frac{1}{a^{2}}\Big(\vec{\nabla} h_{ij}\Big)^{2}\right]\,. \label{pertT}\end{aligned}$$ Note that with the definition in [@Kobayashi:2011nu] in this case the sound speed equals unity, $c_{T}=1$. The tensor perturbations can be canonically renormalised via the following re-definitions: $$\begin{aligned}
d\tau := \frac{1}{a}dt\,,\,\,z := \frac{a}{2}\,,\,\,v_{ij} := zh_{ij}\,. \label{define}\end{aligned}$$ Plugging these new parameters, the above quadratic action becomes $$\begin{aligned}
{\cal S}^{(2)}_{T} =\frac{1}{2}\int dt d^{3}{\bf x}\left[\Big(v'_{ij}\Big)^{2}-\Big(\vec{\nabla} v_{ij}\Big)^{2}+\frac{z''}{z}v_{ij}^{2}\right]\,, \label{pertT1}\end{aligned}$$ where a prime represents derivative with respective to the conformal time $\tau$. It is common to transform the action and re-express the resulting one in terms of the Fourier modes, and one can basically show that the evolution equation from the action in Eq.(\[pertT1\]) is given by $$\begin{aligned}
v''_{ij}-\Big(k^{2}-\frac{z''}{z}\Big)v_{ij}=0\,. \label{evolu}\end{aligned}$$ The normalized solution to this perturbation equation can be conventionally written in terms of the Hankel function so that we obtain the power spectrum of the primordial tensor perturbation at the horizon exits $c_{T}k|\tau|=1$ [@Kobayashi:2011nu] as $$\begin{aligned}
{\cal P}_{T}=8\gamma_{T}\frac{H^{2}}{4\pi}\Big|_{c_{T}k|\tau|=1}\,, \label{pT}\end{aligned}$$ where $$\begin{aligned}
\gamma_{T} :=2^{2\nu_{T}-3}\Big|\Gamma(\nu_{T})/\Gamma(3/2)\Big|^{2}\Big(1-\epsilon\Big)\,,\,\,\nu_{T} \simeq 3/2+\epsilon+{\cal O}(\epsilon^{2})\,, \label{defi}\end{aligned}$$ and the tensor spectral tilt is given by $n_{T}=3-2\nu_{T}=-2\epsilon+{\cal O}(\epsilon^{2})$. Now we turn to considering the scalar fluctuations. This can be achieved by setting $h_{ij}=0$ in the perturbed line-element mentioned in Eq.(\[pert\]). Here we proceed in the same manner as for the tensor perturbations. For the scalar perturbations, we arrive with the second order action [@Kobayashi:2011nu] $$\begin{aligned}
{\cal S}^{(2)}_{S} = \int dt d^{3}{\bf x}\,a^{3}\left[\epsilon\dot{\zeta}^{2}-\frac{\epsilon}{a^{2}}\Big(\vec{\nabla} \zeta\Big)^{2}\right]\,, \label{spertT}\end{aligned}$$ where $\epsilon$ is already given in Eq.(\[epsi\]), and the sound speed is also unity, $c_{S}=1$, in this case. The above action can be canonically renormalised via the following re-definitions: $$\begin{aligned}
d\tau := \frac{1}{a}dt\,,\,\,z := \sqrt{2\epsilon}a\,,\,\,u := z\zeta\,. \label{sdefine}\end{aligned}$$ Plugging these new parameters, the above quadratic action becomes $$\begin{aligned}
{\cal S}^{(2)}_{S} =\frac{1}{2}\int dt d^{3}{\bf x}\left[\Big(u'\Big)^{2}-\Big(\vec{\nabla} u\Big)^{2}+\frac{z''}{z}u^{2}\right]\,, \label{spertT1}\end{aligned}$$ where a prime represents derivative with respective to the conformal time $\tau$. It is common to transform the action and re-express the resulting one in terms of the Fourier modes, and one can basically show that the evolution equation from the action (\[spertT1\]) is given by $$\begin{aligned}
u''-\Big(k^{2}-\frac{z''}{z}\Big)u=0\,. \label{sevolu}\end{aligned}$$ The normalized solution to this perturbation equation can be conventionally written in terms of the Hankel function so that we obtain the power spectrum of the primordial scalar perturbation at the horizon exits $c_{S}k|\tau|=1$ [@Kobayashi:2011nu] as $$\begin{aligned}
{\cal P}_{S}=\frac{\gamma_{S}}{2}\frac{H^{2}}{4\pi^2\epsilon}\Big|_{c_{S}k|\tau|=1}\,, \label{spT}\end{aligned}$$ where $$\begin{aligned}
\gamma_{S} :=2^{2\nu_{S}-3}\Big|\Gamma(\nu_{S})/\Gamma(3/2)\Big|^{2}\Big(1-\epsilon\Big)\,,\,\,\nu_{S} \simeq 3/2+3\epsilon-\eta-\epsilon\,\eta+{\cal O}(\epsilon^2,\eta^2)\,, \label{sdefi}\end{aligned}$$ and the spectral index is given by $n_{s}-1=3-2\nu_{S}=-6\epsilon+2\eta+{\cal O}(\epsilon^2,\eta^2)$. Moreover, the tensor-to-scalar ratio is defined by $$\begin{aligned}
r\equiv \frac{{\cal P}_{T}}{{\cal P}_{S}}=16\epsilon\,. \label{t2s}\end{aligned}$$ Recently, the Planck satellite data showed that the spectral index $n_{s}$ of curvature perturbations is constrained to be $n_{s} = 0.9603 \pm 0.0073$ ($68\%$ CL) and ruled out the exact scale-invariance ($n_{s} =1$) at more than $5\sigma$ confident level (CL), whilst the tensor-to-scalar ratio $r$ is bounded to be $r < 0.11$ ($95\%$ CL). Most recently, the observed $B$-mode power spectrum provides the tensor-to-scalar ratio $r=0.20^{+0.07}_{-0.05}$ with $r=0$ disfavoured at $7.0\sigma$ CL [@Ade:2014xna]. These constraints are used to falsify the most popular and simple inflationary models.
Predictions from Composite Inflation {#model}
====================================
According to the composite paradigms considered in this work, it was potentially shown that the models of composite inflation nicely respect tree-level unitary for the scattering of the inflation field during inflation all the way to the Planck scale. In this section, we consider composite inflationary models that can be described by the primordial power spectrum observables consisting of the spectral index, $n_{s}$, and the tensor-to-scalar ratio, $r$, derived in the Einstein frame. In the following computations, we assume a large non-minimal coupling $\xi\gg 1$ and a large field approximation, i.e. $\varphi^{2}\gg M^{2}_{\rm P}/\xi$, is also implemented. For later convenience, we express the inflationary parameters in terms of the number of e-foldings..
$\bullet$[*Minimal Composite Inflation*]{} (MCI)
The first model of composite inflation we will examine is recently investigated in [@Channuie:2011rq]. According to this paradigm, they engaged the simplest models of technicolor passing precision tests well known as the minimal walking technicolor (MWT) theory [@Sannino:2004qp; @Hong:2004td; @Dietrich:2005wk; @Dietrich:2005jn] with the standard (slow-roll) inflationary paradigm as a template for composite inflation. The inflaton field in this case is the lightest composite state emerging from a bilinear condensate of techni-quarks with $d=1$. With the large field approximation, the action in the Jordan frame reads [@Channuie:2011rq] $$\begin{aligned}
{\cal S}_{\rm MCI,\,J}=\int d^{4}x\sqrt{-g}\left[\frac{1 +\xi\varphi^{2}}{2} R - \frac{1}{2}g^{\mu\nu} \partial_{\mu}\varphi\partial_{\nu}\varphi- \frac{\kappa}{4}\varphi^{4} \right]\,,\end{aligned}$$ where $\kappa$ is a self coupling which is constrained by the underlying theory to be of the order of unity. For this model, we have $$\begin{aligned}
F(\varphi) = 1 + \xi\varphi^{2} \quad {\rm and} \quad G = 1\,.
\label{fg-tc}\end{aligned}$$ Here we can diagonalize the inflaton-gravity sector by performing the conformal transformation already discussed in the previous section. After taking the conformal transformation, the resulting action in the Einstein frame reads $$\begin{aligned}
{\cal S}_{\rm MCI,\,E}=\int d^{4}x\sqrt{-g}\left[-\frac{M^{2}_{P}}{2} R - \frac{1}{2}g^{\mu\nu} \partial_{\mu}\chi\partial_{\nu}\chi- {\cal U}_{\rm MCI}(\chi) \right]\,,\end{aligned}$$ where for a large field approximation $$\begin{aligned}
\frac{d\chi}{d\varphi}\simeq \frac{\sqrt{6}M_{\rm P}}{\varphi}\quad{\longrightarrow}\quad \chi \simeq \sqrt{6}M_{\rm P}\ln(\sqrt{\xi}\varphi/M_{\rm P})\,. \label{chi-tc}\end{aligned}$$ This leads to the potential in the Einstein frame as $$\begin{aligned}
{\cal U}_{\rm MCI}(\chi) \simeq \frac{\kappa M^{4}_{P}}{4\xi^{2}}\Big(1+e^{\frac{-2\chi}{\sqrt{6}M_{P}}}\Big)^{-2}\quad{\rm with}\quad \Omega(\chi)^{2}\simeq \exp\left(\frac{2\chi}{\sqrt{6}M_{\rm P}}\right)\,.
\label{u-tc}\end{aligned}$$ With the large field approximation, we can derive the following slow-roll parameter in terms of the field $\varphi({\rm or}\,\chi)$ as $$\begin{aligned}
\epsilon \simeq \frac{4M^{4}_{\rm P}}{3\xi^{2}\varphi^{4}}\left({\rm or}\,\frac{4}{3}\exp\left(-\frac{4\chi}{\sqrt{6}M_{\rm P}}\right)\right)\,.\label{eptc}\end{aligned}$$ Inflation ends when $\epsilon=1$ such that $$\begin{aligned}
\varphi_{\rm end}\simeq \frac{M_{\rm P}}{\sqrt{\xi}}\left({\rm or}\,\chi_{\rm end}\simeq 0.3 M_{\rm P}\right)\,.\label{eptc1}\end{aligned}$$ In the large field limits, the number of e-foldings reads $$\begin{aligned}
{\cal N} \simeq \frac{6}{8M^{2}_{\rm P}/\xi}\Big[\varphi^{2}_{\rm ini}-\varphi^{2}_{\rm end}\Big]\quad{\rm with}\quad \varphi_{\rm ini}\gg\varphi_{\rm end}\,.\label{foldtc}\end{aligned}$$ Using eqs.(\[epsi\])-(\[efold\]), it is straightforward to express the inflationary predictions in terms of the number of e-foldings, and find the Einstein frame parameters: $$\begin{aligned}
n_{s}=1-6\epsilon+2\eta\simeq1-\frac{2}{{\cal N}}\,,\,\,\,\,r=16\epsilon\simeq\frac{12}{{\cal N}^2}\,. \label{paraTC}\end{aligned}$$ Here we kept to the lowest order in $1/\xi$ to derive the spectral index and the tensor-to-scalar ratio. It is worthy to note here that the higher corrections of $1/\xi$ and $1/{\cal N}$ are suppressed to these expressions. This model provides $n_{s}\subseteq [0.933,\,0.975]$ and $r\subseteq [0.002,\,0.013]$ for ${\cal N}\subseteq [30,\,80]$. Unfortunately, from Eq. (\[paraTC\]), the predictions are in tension with the recent BICEP2 observations since a large value of $r$ cannot be produced in this model, see detailed discussions in the last section.
$\bullet$[*Glueball Inflation*]{} (GI)
In this section, we consider the work presented in [@Bezrukov:2011mv] as the model motivated by a pure Yang-Mills theory. In this case, the inflaton is the lightest glueball field with $d=4$. The reason why we want to continue analysing composite inflation by following such model resides from the fact that this theory features the archetype of any composite paradigm in flat space and consequently of models of composite inflation. The low-energy effective Lagrangian of the lightest glueball state can be found in [@Schechter:1980ak; @Migdal:1982jp; @Cornwall:1983zb].
It is worthy to note here that the theory we are using describes the ground state of pure Yang-Mills theory, and of course is not the simple $\phi^4$ theory. Another difference from the $\phi^4$ theory is that the form of the effective potential, before coupling to gravity, is completely fixed by the underlying gauge theory. The action in the Jordan frame in this case reads [@Bezrukov:2011mv] $$\begin{aligned}
{\cal S}_{\rm GI,\,J} = \int d^{4}x\sqrt{-g}\Big[\frac{1 + \xi\varphi^{2}}{2} R - 16g^{\mu\nu} \partial_{\mu}\varphi\partial_{\nu}\varphi - 2\varphi^{4}\ln\left(\varphi/\Lambda\right) \Big]\,, \label{varaction}\end{aligned}$$ which yields $$\begin{aligned}
F(\varphi) = 1 + \xi\,\varphi^{2} \quad {\rm and} \quad G = 16\,.
\label{fg-gbr}\end{aligned}$$ Imposing the conformal transformation, the resulting action in the Einstein frame reads $$\begin{aligned}
{\cal S}_{\rm GI,\,E}=\int d^{4}x\sqrt{-g}\left[-\frac{M^{2}_{P}}{2} R - \frac{1}{2}g^{\mu\nu} \partial_{\mu}\chi\partial_{\nu}\chi- {\cal U}_{\rm GI}(\varphi(\chi)) \right]\,, \end{aligned}$$ which leads to the potential in the Einstein frame for a large field approximation as $$\begin{aligned}
{\cal U}_{\rm GI}(\varphi) \simeq \frac{2M^{4}_{P}}{\xi^{2}}\ln\Big(\varphi/\Lambda\Big)\quad{\rm with}\quad \frac{d\chi}{d\varphi}\simeq \frac{\sqrt{6}M_{\rm P}}{\varphi}\longrightarrow \chi\simeq \sqrt{6}M_{\rm P}\ln(\varphi/\Lambda)\,.
\label{v-gl}\end{aligned}$$ Here we have left the explicit dependence on the field $\varphi$ possessing unity canonical dimension instead of using the canonically normalized new scalar field $\chi$. However, we can express the resulting potential in terms of the field $\chi$ by substituting $\ln(\varphi/\Lambda)=\chi/\sqrt{6}M_{\rm P}$ into the potential given in Eq. (\[v-gl\]). With the large field approximation, we can derive the following slow-roll parameter $$\begin{aligned}
\epsilon \simeq \frac{1}{12\ln(\varphi/\Lambda)^{2}}\,.\label{epgb}\end{aligned}$$ Inflation ends when $\epsilon=1$ such that $$\begin{aligned}
\ln(\varphi_{\rm end}/\Lambda)\simeq \frac{1}{2\sqrt{3}}\,.\label{ep1gb}\end{aligned}$$ In the large field limits, the number of e-foldings reads $$\begin{aligned}
{\cal N} \simeq 3\Big(\ln{\left(}\varphi_{\rm ini}/\Lambda{\right)}^{2} - \ln{\left(}\varphi_{\rm end}/\Lambda{\right)}^{2}\Big)\,.\label{foldgb}\end{aligned}$$ Using eqs.(\[epsi\])-(\[efold\]), the inflationary predictions in terms of the number of e-foldings in the Einstein frame parameters read $$\begin{aligned}
n_{s}=1-6\epsilon+2\eta\simeq1-\frac{3}{2{\cal N}}\,,\,\,\,\,r=16\epsilon\simeq\frac{4}{{\cal N}}\,, \label{paraBlue}\end{aligned}$$ where we have assumed $\varphi_{\rm ini}\gg \varphi_{\rm end}$. We also have kept to the lowest order in $1/\xi$ to derive the spectral index and the tensor-to-scalar ratio and neglected the higher corrections of $1/\xi$ and $1/{\cal N}$ suppressed to these expressions. The predictions of this model from Eq. (\[paraBlue\]) provide $n_{s}\subseteq [0.950,\,0.981]$ and $r\subseteq [0.050,\,0.13]$ for ${\cal N}\subseteq [30,\,80]$. Concretely, more detailed discussions will be added in the last section.
$\bullet$[*Super Yang-Mills Inflation*]{} (SYMI)
The application of the supersymmetric version of a pure Yang-Mills (SYM) has been investigated in various cornerstones. Here we consider for inflation the bosonic part of the Veneziano-Yankielowicz effective theory. For inflationary model building, the gluino-ball state in the super Yang-Mills theory is assigned as the inflaton with $d=3$. The effective Lagrangian in supersymmetric gluodynamics was constructed in [@Veneziano:1982ah]. Having already stamped as the model of composite inflation, the authors of [@Channuie:2012bv] took the scalar component part of the super-glueball and coupled it non-minimally to gravity. The action of the theory in term of the canonical-dimension field $\varphi$ reads [@Channuie:2012bv] $$\begin{aligned}
{\cal S}_{\rm SYMI,\,J} = \int d^{4}x\sqrt{-g}\Big[\frac{1+N^{2}_{c}\xi\varphi^{2}}{2} R - \frac{9N^2_{c}}{\alpha}g^{\mu\nu} \partial_{\mu}\varphi\partial_{\nu}\varphi - 4\alpha N^{2}_{c}\varphi^{4}(\ln[\varphi/\Lambda])^{2} \Big]\,,\end{aligned}$$ with $N_{c}$ a number of colours and $\alpha$ a $N_{c}$-independent quantity. With the action given above, we find $$\begin{aligned}
F(\varphi) = 1 + N^{2}_{c}\xi\,\varphi^{2} \quad {\rm and} \quad G = \frac{9N^{2}_{c}}{\alpha}\,.
\label{fg-sgb}\end{aligned}$$ However, the gravity-scalar coupled sector can basically be diagonalised by imposing the conformal transformation. We find the resulting action in the Einstein frame as $$\begin{aligned}
{\cal S}_{\rm SYMI,\,E}=\int d^{4}x\sqrt{-g}\left[-\frac{M^{2}_{P}}{2} R - \frac{1}{2}g^{\mu\nu} \partial_{\mu}\chi\partial_{\nu}\chi- {\cal U}_{\rm SYMI}(\varphi(\chi)) \right]\,, \end{aligned}$$ where $$\begin{aligned}
{\cal U}_{\rm SYMI}(\varphi) \simeq \frac{4\alpha}{N^{2}_{c}}\frac{M^{4}_{P}}{\xi^{2}}\ln\Big(\varphi/\Lambda\Big)^{2}\quad{\rm with}\quad \frac{d\chi}{d\varphi}\simeq \frac{\sqrt{6}M_{\rm P}}{\varphi}\longrightarrow \chi\simeq \sqrt{6}M_{\rm P}\ln(\varphi/\Lambda)\,, \label{Ymp}
\label{v-ymi}\end{aligned}$$ where we have also left the explicit dependence on the field $\varphi$ instead of using the canonically normalized new field $\chi$. With the large field approximation, we can derive the following slow-roll parameter $$\begin{aligned}
\epsilon \simeq \frac{1}{3\ln(\varphi/\Lambda)^{2}}\,.\label{epsgb}\end{aligned}$$ Inflation ends when $\epsilon=1$ such that $$\begin{aligned}
\ln(\varphi_{\rm end}/\Lambda)\simeq \frac{1}{\sqrt{3}}\,.\label{epsg1b}\end{aligned}$$ Performing the similar approximations to those of the previous subsection, the number of e-foldings for this model in the large $\xi$ limit reads $$\begin{aligned}
{\cal N} \simeq \frac{3}{2}\Big(\ln{\left(}\varphi_{\rm ini}/\Lambda{\right)}^{2} - \ln{\left(}\varphi_{\rm end}/\Lambda{\right)}^{2}\Big)\,.
\label{efold-sgb-l}\end{aligned}$$ Using eqs.(\[epsi\])-(\[efold\]), the inflationary predictions, i.e. $n_{s},\,r$, in terms of the number of e-foldings in the Einstein frame read $$\begin{aligned}
n_{s}=1-6\epsilon+2\eta\simeq1-\frac{2}{{\cal N}}\,,\,\,\,\,r=16\epsilon\simeq\frac{8}{{\cal N}}\,. \label{parasusyym}\end{aligned}$$ In order to derive these parameters, we have kept to the lowest order in $1/\xi$ and neglect the higher corrections of $1/\xi$ and $1/{\cal N}$. We find for this model from Eq. (\[parasusyym\]) that $n_{s}\subseteq [0.933,\,0.975]$ and $r\subseteq [0.100,\,0.267]$ for ${\cal N}\subseteq [30,\,80]$. In the last section, we will summarize our findings for this model.
$\bullet$[*Orientifold Inflation*]{} (OI)
The authors of [@Channuie:2012bv] examined the supersymmetric low-energy effective action to study inflation driven by the gauge dynamics of SU(N) gauge theories adding one Dirac fermion in either the two-index antisymmetric or symmetric representation of the gauge group. Such theories are known as orientifold theories. Here the gluino field of supersymmetric gluodynamics is replaced by two Weyl fields which can be formed as one Dirac spinor. For investigating the inflationary scenario, the orientifold sector non-minimally coupled to gravity in the Jordan frame action is implemented in [@Channuie:2012bv]. For this investigation, we write the action by using the real part of the field $\varphi$ as [@Channuie:2012bv] $$\begin{aligned}
{\cal S}_{\rm OI,\,J}\supset \int d^{4}x\sqrt{-g}\left[-\frac{M^{2}_{P}+N^{2}_{c}\xi\varphi^{2}}{2} R - \frac{9F(N_{c})}{\alpha}g^{\mu\nu} \partial_{\mu}\varphi\partial_{\nu}\varphi- 4\alpha F(N_{c})\varphi^{4}\left(\ln(\varphi/\Lambda)^{2} -\gamma\right) \right]\,, \end{aligned}$$ where $F(N_{c})=N^{2}_{c}(1+{\cal O}(1/N_{c}))$, $\gamma=1/9N_{c}+{\cal O}(1/N^{2}_{c})$ and hereafter we will keep only leading order in $1/N_{c}$. With the action given above, we find $$\begin{aligned}
F(\varphi) = 1 + N^{2}_{c}\xi\,\varphi^{2} \quad {\rm and} \quad G = \frac{9F(N_{c})}{\alpha}\,.
\label{fg-sgb}\end{aligned}$$ As usually proceeded in the standard fashion, we impose the conformal transformation and then find the resulting action in the Einstein frame $$\begin{aligned}
{\cal S}_{\rm OI,\,E}\supset\int d^{4}x\sqrt{-g}\left[-\frac{M^{2}_{P}}{2} R - \frac{1}{2}g^{\mu\nu} \partial_{\mu}\chi\partial_{\nu}\chi- {\cal U}_{\rm OI}(\varphi(\chi)) \right]\,, \end{aligned}$$ where $$\begin{aligned}
{\cal U}_{\rm OI}(\varphi) \simeq \frac{4\alpha F(N_{c})}{N^{4}_{c}}\frac{M^{4}_{P}}{\xi^{2}}\Big[\ln\Big(\varphi/\Lambda\Big)^{2}-\gamma\Big]\quad{\rm with}\quad \frac{d\chi}{d\varphi}\simeq \frac{\sqrt{6}M_{\rm P}}{\varphi}\longrightarrow \chi\simeq \sqrt{6}M_{\rm P}\ln(\varphi/\Lambda)\,,
\label{v-oii}\end{aligned}$$ with $N_{c}$ being a number of colours. Note that at large-$N_{c}$ the theory features certain super Yang-Mills properties, i.e. $F(N_{c})\rightarrow N^{2}_{c}$. With this limit, the transformed potential reduces to that of (\[Ymp\]). With the large field limit, we can derive the following slow-roll parameter $$\begin{aligned}
\epsilon \simeq \epsilon_{\rm SYMI}\Big[1+\frac{2\gamma}{9}\ln(\varphi/\Lambda)^{2}\Big]\quad{\rm with}\quad\epsilon_{\rm SYMI}\simeq \frac{1}{3\ln(\varphi/\Lambda)^{2}}\,.\label{epoi}\end{aligned}$$ Inflation ends when $\epsilon=1$ such that $$\begin{aligned}
\varphi^{3}_{\rm end}\simeq \varphi^{3}_{\rm SYMI,\,end}\Big[1+\frac{\gamma}{9\sqrt{3}}\Big]\,, \label{epsg1oi}\end{aligned}$$ where $\varphi_{\rm SYMI,\,end}$ can be directly obtained from Eq. (\[epsg1b\]). According to this model, the number of e-foldings in the large $\xi$ limit is approximately given by $$\begin{aligned}
{\cal N} \simeq \left[\frac{9\ln(\varphi/\Lambda)}{2\alpha\xi}\left(1-\frac{2\ln(3\ln(\varphi/\Lambda))}{81\ln(\varphi/\Lambda)^{2}}\right)\right]^{\varphi_{\rm ini}}_{\varphi_{\rm end}}\,.
\label{efold-oib-l}\end{aligned}$$ Using eqs.(\[epsi\])-(\[efold\]), the predictions in the Einstein frame for $n_{s}$ and $r$ of this model can consequently written in terms of the number of e-foldings as $$\begin{aligned}
n_{s}=1-6\epsilon+2\eta\simeq 1-\frac{2}{{\cal N}}\left(1+\frac{9\gamma}{2{\cal N}}\right)\,,\,\,\,\,r=16\epsilon\simeq \frac{8}{{\cal N}}\left(1+\frac{3\gamma}{{\cal N}}\right)\,. \label{paraorient}\end{aligned}$$ Notice that for large $N_{c}$ the observables given above features the Super Yang-Mills inflation since $\gamma\rightarrow 0$.
![The contours show the resulting 68$\%$ and 95$\%$ confidence regions for the tensor-to-scalar ratio $r$ and the scalar spectral index $n_{s}$. The red contours are for the Planckdata combination, which for this model extension gives a 95$\%$ bound $r < 0.26$ (Planck Collaboration XVI 2013 [@Ade:2013uln]). The blue contours represent the BICEP2 constraint on $r$ shown in the centre. The dot-line joints show the results of composite models examined in this work (MCI,GI,SYMI) assuming the number of e-foldings ${\cal N}$ to the end of inflation ranges in the interval ${\cal N} \subseteq [30,80]$.[]{data-label="f1"}](Einstein_BICEP2.eps){width="4.5in"}
Summary {#con}
=======
In this last section we summarise our findings by which we confront the predictions of composite inflationary models with Planck and also very recent BICEP2 data and then plot our results on the constraint contours illustrated in Fig.(\[f1\]). Moreover, recent study in the Jordan frame the primordial spectral index and tensor-to-scalar ratio predicted by composite models has been discussed in [@Channuie:2013lla]. For this work, the following is our remarks.
The first model of composite inflation (MCI) is inspired by models of dynamical electroweak symmetry breaking with the Higgs-like potential. Here the Higgs sector of the SM is replaced by a new underlying four-dimensional gauge dynamics free from fundamental scalars and the Higgs field is therefore composite state. We discover for model that the predictions can satisfy the $1\sigma$ CL of the Planck data for $35 \lesssim {\cal N} \lesssim 60$. Moreover, the model predictions can be consistent with the Planck data up to the $2\sigma$ CL if we choose $33 \lesssim {\cal N} \lesssim 80$. However, we find that the MCI model is in tension with the recent BICEP2 data. This is so since the model predictions yield quite small values of both $r$ and $n_{s}$ if we consider large $\xi$ limit. For a small $\xi$ case, there have been discussed in [@Channuie:2013lla]. In order to satisfy up to the $2\sigma$ CL of Planck data, we discover the number of e-foldings ${\cal N}$ should not be greater than $80$.
The underlying theory of the second paradigm (GI) features the archetype of any composite paradigm in flat space and consequently of models of composite inflation. The observables predicted by GI model lie inside the $1\sigma$ CL of the Planck for $30 \lesssim{\cal N} \lesssim 45$, whilst for $25 \lesssim {\cal N} \lesssim 60$ this model can be consistent with the $2\sigma$ region of the Planck contours. Interestingly, we discover for this model that the predictions can satisfy the $1\sigma$ CL of the BICEP2 data for $30 \lesssim {\cal N} \lesssim 35$, and up to $2\sigma$ CL if we choose $25 \lesssim {\cal N} \lesssim 45$. In order to satisfy upto the $2\sigma$ CL of Planck and BICEP2 data, we discover the number of e-foldings ${\cal N}$ should not be greater than $60$ and $45$, respectively.
Finally, the SYMI model features the supersymmetric version of a pure Yang-Mills properties. The model provide an interesting class of inflationary model. Apparently, the SYMI predictions are significantly consistent with the Planck and BICEP2 constraints. We find that in order to satisfy at the $1\sigma$ CL of the BICEP2 data, the model prefers $40\lesssim {\cal N}\lesssim60$. The model can be consistent with the 68$\%$ CL for both Planck and simultaneously BICEP2 data for $45\lesssim {\cal N}\lesssim60$, and with $95\%$ CL for $37\lesssim {\cal N}\lesssim70$. For the OI model, the present of $\gamma$ parameter yields small change of the predictions. However, the results from the OI model coincide with those of the SYMI for a large number of colours. In order to satisfy the $1\sigma$ and $2\sigma$ CL of the BICEP2 data, we discover the number of e-foldings ${\cal N}$ should not be greater than $60$ and $70$, respectively.
Another crucial consequence for the model of inflation is the (p)reheating mechanism. We anticipate to investigate this mechanism by following closely references [@Tsujikawa:1999iv; @Watanabe:2006ku]. Rather interestingly, for instance, the author of [@Ratra:1991bn] studied the consequent of inflation as seed of the present intergalactic magnetic field. However, the author claimed that the results after making a number of simplifying approximations should be considered to be preliminary. Therefore, it is very interesting to study the mechanism for generating an intergalactic magnetic field based on the composite inflationary manners.
: The author thanks Arunee Shunava and Worakrit Thida for assisted graphic works. Remote conversation with the CP$^3$-Origins’s Team, F. Sannino, J. Joergensen and O. Svendsen, is also acknowledged. This work is fully supported by the Research Fund for DPST Graduate with First Placementunder Grant No. 033/2557.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Mark Edelman
title: 'Fractional Maps as Maps with Power-Law Memory'
---
Introduction {#sec:1}
============
Many natural and social systems are systems with memory. Their mathematical description requires solving integro-differential equations and is quite complicated. Maps with memory are used to model real systems with memory in order to derive their basic properties.
Systems with Memory {#sec:1.1}
-------------------
Writing this text I am recalling the content of my latest papers. It is easy to recall the content of my last paper but it becomes more and more difficult as I try to recall papers that are more and more distant in time. Memory is a significant property of human beings and is the subject of extensive biophysical and psychological research. As it has been demonstrated in experiments, forgetting - the accuracy on a memory tasks decays as a power law, $\sim
t^{-\beta}$, with $0<\beta<1$ [@Kahana; @Rubin; @Wixted1; @Wixted2; @Adaptation1]. It is interesting that fractional maps corresponding to fractional differential equations of the order $0<\alpha<1$ are maps with the power-law decaying memory in which the power is $-\beta=\alpha-1$ and $0<\beta<1$ [@DNC]. Human learning is closely related to memory. It also can be described by a power law: the reduction in reaction times that comes with practice is a power function of the number of training trials [@Anderson]. There are multiple publications where power-law adaptation has been applied in describing the dynamics of biological systems at levels ranging from single ion channels up to human psychophysics [@Adaptation3; @Adaptation4; @Adaptation2; @Adaptation5; @Adaptation1; @Adaptation6].
Power-law memory applies not only to the human being as a whole, but also to the hierarchy of its building blocks, from individual neurons and proteins to the tissue of individual organs. It has been shown recently [@Lund2; @Lund1] that processing of external stimuli by individual neurons can be described by fractional differentiation. The orders of fractional derivatives $\alpha$ derived for different types of neurons fall within the interval $[0,1]$. For neocortical pyramidal neurons it is quite small: $\alpha \approx 0.15$. Fluctuations within single protein molecules demonstrate a power-law memory kernel with the exponent $-0.51 \pm 0.07$ [@Protein].
Viscoelastic properties of human tissues were demonstrated in many examples: the brain and the central nervous system in general [@TissueNerv; @TissueBrain2; @TissueBrain1], the breast [@Coussot], the liver [@TissueLiver2; @TissueLiver1], the spleen [@TissueSpleen], the prostate [@TissueProstate1; @TissueProstate2], the arteries [@TissueArteries1; @TissueArteries2], the muscles [@TissueMuscle] (see also references for some other human and animal organs tissues [@DFCV2005; @Magin; @MGE2010; @TissueBov; @Frog2008]). Viscoelastic materials obey the following stress-strain relationship: $$\sigma(t)=E(\gamma)\frac{d^\alpha\gamma(t)}{dt^\alpha},
\label{VE}$$ where $\sigma$ is the stress, $\gamma$ is the strain, $\alpha$ is the order of the fractional derivative, and $t$ is time. In most of the cases for human tissues $0<\alpha<1$ and is close to zero. In some cases, e.g. for modeling of the accurate placement of the needle tip into the target tissue during needle insertion treatments for liver tumors, nonlinearity of $E(\gamma)$ should be taken into account [@TissueLiver2]. In the last example a simple quadratic nonlinearity and $\alpha=0.1$ were used.
A Fourier transform of a fractional derivative is [@KST; @Podlubny; @SKM] $$F\{D^{\alpha}g(t);\omega \}=(-i \omega)^{\alpha}\hat{g}(\omega),
\label{Fourier}$$ where $\hat{g}(\omega)=F\{g(t);\omega \}$. As a result, whenever the term $(\omega)^{\alpha}\hat{g}(\omega)$ appears in the frequency domain, there is a good chance that function $g(t)$ is a solution of a fractional differential equation with a fractional derivative of the order $\alpha$ and the corresponding system is a system with power-law memory. Well known examples of such systems are dielectrics. Electromagnetic fields in dielectric media are described by equations with time fractional derivatives due to the ’universal’ response - the power-law frequency dependence of the dielectric susceptibility in a wide range of frequencies [@TDia2008a; @TDia2008b; @TDia2009; @TarBook]. Similarly, elastic wave attenuation in biological tissue over a wide range of frequencies follows the power law $\alpha(\omega) \propto \omega^{\eta}$ with $\eta \in [0,2]$ [@TissueWaves1; @TissueWaves2; @TissueWaves3; @TissueWaves4] which implies a fractional wave equation. The establishment of accurate fractional wave-propagation models is important for many medical applications [@TissueWaves3].
Above we concentrated on biological systems with memory in order to emphasize the importance of the study of nonlinear fractional dynamical systems described by fractional differential equations of the order $0<\alpha<2$ and especially $\alpha$ close to zero which is a major subject of the following sections (Sec. \[LT1\] and \[BN12\]). Now we’ll list some (not all) other examples of systems with power-law memory. As has been mentioned above, time fractional derivatives and correspondingly systems with power-law memory in many cases are used to describe viscoelasticity and rheology (for the original papers and reviews see [@Visc1; @Visc2; @Visc3; @Visc4; @Visc5; @Visc6; @Visc7; @Visc8], for nonlinear effects see [@NLVisc1; @NLVisc2; @NLVisc3; @NLVisc4]). Electromagnetic fields in dielectric media were also mentioned above. Hamiltonian systems and billiards are also systems with power-law memory, in which the fractal structure of the phase space and stickiness of trajectories in time imply description of transport by the fractional (factional time and space derivatives) Fokker-Plank-Kolmogorov equation [@ZasBook; @ZE2000; @ZE2004; @ZE1997]. In some cases [@KBT1; @KBT2; @KST] fractional differential equations are equivalent to the Volterra integral equations of the second kind. Systems considered in population biology and epidemiology are systems with memory and Volterra integral equations are frequently used to describe such systems [@PopBioBook2001; @HoppBook1975]. Long-term memory provides more robust control in liner and nonlinear control theory (see books [@Control1; @ControlBook2011]).
Maps with Memory {#sec:1.2}
----------------
As in the study of regular dynamics, in the study of systems with memory use of discrete maps significantly simplifies investigation of the general properties of the corresponding systems. In some cases of kicked systems maps are equivalent to the original differential equations. Historically, maps with memory were first considered as analogues of the integro-differential equations of non-equilibrium statistical physics [@MM1; @MM2; @MM6], with regards to thermodynamic theory of systems with memory [@MM5], and to model non-Markovian processes in general [@MM3; @MM4]. The most general form of a map with memory is $$%\bf{x}_{n+1}=\bf{f}_{n+1}(\bf{x}_n,\bf{x}_{n-1},...,\bf{x}_0,P),
\mathbf{x}_{n+1}=\mathbf{f}_{n+1}(\mathbf{x}_n,\mathbf{x}_{n-1},...,\mathbf{x}_0,P),
\label{MMGeneral}$$ where $\bf{x}_k$ are $N$-dimensional vectors, $k,N \in \mathbb{Z}$, $k \ge 0$, and $P$ is a set of parameters. It is almost impossible to derive the general properties of systems with memory from Eq. (\[MMGeneral\]) and simplified forms of maps with memory are used. The most commonly used form is the one-dimensional map with long-term memory $$x_{n+1}=\sum^{n}_{k=0}V_{\alpha}(n,k)G_K(x_k),
\label{LTM}$$ where $V_{\alpha}(n,k)$ and $\alpha$ characterize memory effects and $K$ is a parameter. In many cases weights are taken as convolutions with $V_{\alpha}(n,k)=V_{\alpha}(n-k)$. The particular form of Eq. (\[LTM\]) with constant weights $$x_{n+1}=c\sum^{n}_{k=0}G_K(x_k)
\label{Full}$$ is called a full-memory map. It is easy to note that Eq. (\[Full\]) is equivalent to $$x_{n+1}=x_n+cG_K(x_n),
\label{F1Step}$$ which means that maps with full memory are maps with one-step memory in which all memory is accumulated in the present state of a system and the next values of map variables are fully defined by their present values. We won’t consider maps with short memory in which the number of terms in the sum in Eq. (\[LTM\]) is bounded (from $k=n-M+1$ to $k=n$).
Initial investigations of long-term memory maps were done mostly on different modifications of the logistic map and exponential memory. The general applicability of their results to systems with memory in general is limited. Recently Stanislavsky [@MM7] considered the maps Eq. (\[LTM\]) with $G_K(x)=Kx(1-x)$ (the logistic map) and the weights $V_{\alpha}(n,k)$ as a combination of power-law functions taken from one of algorithms of numerical fractional integration. He came to the conclusion that increase in long-term memory effects leads to a less chaotic behavior.
First maps with power-law memory equivalent to fractional differential equations were derived in [@MM9; @MM8; @TarBook; @MM10; @MM11] by integrating fractional differential equations describing systems under periodic kicks. The method used is similar to the way in which the universal map is derived in regular dynamics.
Universal Map {#sec:1.3}
-------------
In the following section (Sec. \[UFM\]) we will modify the way presented in Sec. \[sec:1.3\] to derive the universal map in regular dynamics (see [@Chir], and Ch. 5 from [@ZasBook]) in order to derive the universal fractional map.
The universal map can be derived from the differential equation $$\ddot{x}+KG(x) \sum^{\infty}_{n=-\infty} \delta \Bigl(\frac{t}{T}-(n+\varepsilon)
\Bigr)=0,
%, \ \ t \ge 0
\label{UMDE}$$ where $0 < \varepsilon < 1$ and $K$ is a parameter, with the initial conditions: $$x(0)=x_0, \ \ p(0)=\dot{x}(0)=p_0.
\label{SMDEIC}$$ This equation is equivalent to the Volterra integral equation of second kind $$x(t)=x_0 + p_0t - K\int^{t}_0 d \tau G(x( \tau )) \sum^{\infty}_{n=-\infty}
\delta \Bigl(\frac{\tau}{T}-(n+\varepsilon)\Bigr)( t-\tau ).
\label{Volt2D}$$ Eq. (\[Volt2D\]) for $(n+\varepsilon)T<t<(n+1+\varepsilon)T$ has a solution $$\begin{aligned}
%\begin{equation}
&&x(t)=x_0 + p_0t -
KT\sum^{n}_{k=0} G(x( Tk+T\varepsilon)) ( t-Tk-T\varepsilon), \nonumber\\
%\end{equation}
%\begin{equation}
&&p(t)=\dot{x}(t)= p_0 -
KT\sum^{n}_{k=0} G(x( Tk+T\varepsilon)).
\label{Volt2Dxp}\end{aligned}$$ After the introduction of the map variables $$x_{n}=x(Tn), \ \ p_{n}=p(Tn)
\label{xnpn}$$ Eq. (\[Volt2Dxp\]) considered for time instances t=(n+1)T gives $$\begin{aligned}
%\begin{equation}
&&x_{n+1}=x_0 + p_0(n+1)T -
KT^2\sum^{n}_{k=0} G(x( Tk+T\varepsilon)) (n-k+1-\varepsilon), \nonumber\\
%\label{Volt2Dxn}
%\end{equation}
%\begin{equation}
&&p_{n+1}= p_0 -
KT\sum^{n}_{k=0} G(x( Tk+T\varepsilon)).
\label{Volt2Dxpn}\end{aligned}$$ As it follows from Eq. (\[Volt2Dxp\]), $\dot{x}(t)=p(t)$ is a bounded function with the discontinuities at the time instances of the kicks (at $t=Tk+T\varepsilon$) and $x(t)$ is a continuous function. This allows us to calculate $G(x)$ at the time instances of the kicks. In the limit $\varepsilon \rightarrow 0$ Eq. (\[Volt2Dxpn\]) gives $$\begin{aligned}
%\begin{equation}
&&x_{n+1}=x_0 + p_0(n+1)T -
KT^2\sum^{n}_{k=0} G(x_k) ( n-k+1), \nonumber\\
%\label{UMapx}
%\end{equation}
%\begin{equation}
&&p_{n+1}= p_0 -
KT\sum^{n}_{k=0} G(x_k).
\label{Umapxp}\end{aligned}$$ Eq. (\[Umapxp\]) is a form of the universal map which allows further simplifications. It can be written in a symmetric form as a map with full memory (see Sec. \[sec:1.2\]): $$\begin{aligned}
%\begin{equation}
&&x_{n+1}= x_0 +
T\sum^{n+1}_{k=1} p_k, \nonumber\\
%\label{UmappSimx}
%\end{equation}
%\begin{equation}
&&p_{n+1}= p_0 -
KT\sum^{n}_{k=0} G(x_k).
\label{UmappSimxp}\end{aligned}$$ As we saw in Sec. \[sec:1.2\], maps with full memory are equivalent to maps with one-step memory. Map Eq. (\[UmappSimxp\]) can be written as the iterative area preserving ($\partial(p_{n+1},x_{n+1})/\partial(p_{n},x_{n})=1$) process with one-step memory which is called the universal map: $$p_{n+1}= p_{n} - KTG(x_n),
\label{UMp}$$ $$x_{n+1}= x_{n}+ p_{n+1}T.
\label{UMx}$$ This map represents the relationship between the values of the physical variables in Eq. (\[UMDE\]) on the left sides of the consecutive kicks. The standard map may be obtained from the universal map by assuming $G(x)=\sin(x)$: $$\begin{aligned}
&&p_{n+1}= p_{n} - K \sin x, \ \ \ \ ({\rm mod} \ 2\pi ),
\nonumber \\
&&x_{n+1}= x_{n}+ p_{n+1}, \ \ \ \ ({\rm mod} \ 2\pi ).
\label{SM}\end{aligned}$$ Here we assumed $T=1$ and consider this map on a torus (${\rm mod} \ 2\pi$).
Derivation of the fractional universal map in the next section (Sec. \[UFM\]) follows [@DNC] and the analysis of this map for $\alpha \in (0,1)$ and $\alpha \in (1,2)$ in Secs. \[LT1\] and \[BN12\] follows [@ME1; @DNC; @ME3; @ME4].
Fractional Universal Map {#UFM}
========================
The one-dimensional logistic map $$x_{n+1}=K x_{n}(1-x_{n})
\label{LM}$$ may be presented in the 2D form $$\begin{aligned}
%\begin{equation}
&&p_{n+1}= - G_{lK}(x_n), \nonumber\\
&&x_{n+1}= x_{n}+ p_{n+1},
\label{LM2Dn}\end{aligned}$$ where $$G_{lK}(x)=x-Kx(1-x).
\label{GLM}$$ It can’t be written as a particular form of the universal map Eqs. (\[UMp\]) and (\[UMx\]). In order to derive the logistic map from the universal map we’ll introduce the notion of the n-dimensional universal map depending on a single parameter.
Universal Integer-Dimensional Maps {#UnM}
----------------------------------
Solution of the one-dimensional analog of Eq. (\[UMDE\]) would require calculations of the function $G(x)$ at the time instances of the kicks $T(n+\varepsilon)$ at which $x(t)$ is discontinuous. To enable us to introduce the universal fractional map we’ll include a time delay $\Delta T$ into the argument of the function $G(x(t))$ (see Fig. \[kicks\]).
![The universal map is a relationship between values of $x(t)$ considered at the times $kT$ (small circles). The kicks occur at the time instances $(k+\varepsilon)T$ (vertical lines). With the time delay $\Delta T$ (distance between the squares and the circles) the function $G_K(x(t))$ is calculated at the time instances $t=(k+\varepsilon-\Delta)T$ (stars). []{data-label="kicks"}](Fig1ME.jpg)
In order to extend the class of maps which are particular forms of the universal map we’ll also consider $K$ not as a factor but as a parameter. Let’s consider the following generating equation: $$\dot{x}+G_K(x(t- \Delta T)) \sum^{\infty}_{n=-\infty} \delta \Bigl(\frac{t}{T}-(n+\varepsilon)
\Bigr)=0,
%, \ \ t \ge - \Delta T
\label{UM1Ddif}$$ where $0 < \varepsilon<1$ and $0< \Delta <1$ with the initial condition: $$x(0)=x_0.
\label{UM1DIC}$$ 1D analog of Eq. (\[Volt2Dxp\]) (for $(n+\varepsilon)T<t<(n+1+\varepsilon)T$) can be written as $$x(t)=x_0 -
T\sum^{n}_{k=0} G_K(x[ T(k+\varepsilon-\Delta)]).
\label{Volt1Dx}$$ From the fact that $\dot{x}=0$ for $t \in
(T(k+\varepsilon-1), T(k+\varepsilon))$ it follows that $x[ T(k+\varepsilon-\Delta)]= x(Tk) $ and the corresponding 1D map can be written as a map with full memory $$x_{n+1}= x_0 -
T\sum^{n}_{k=0} G_K(x_k).
\label{UM1Dfmem}$$ From Sec. \[sec:1.2\] it follows that this map can be written as the 1D form of the universal map with one-step memory $$x_{n+1}= x_n - T G_K(x_n).
\label{UM1D}$$ It would be impossible to derive the logistic map from Eq. (\[UM1D\]) if $K$ were a factor, but from the present form the logistic map can be obtained by assuming $$G_K(x)=G_{lK}(x)=\frac{1}{T}[x-Kx(1-x)].
\label{LMG}$$
In [@PopBioBook2001; @PB3] Eq. (\[UM1Ddif\]) with no time delay, no delta functions, and $G_K(x)$ defined by Eq. (\[LMG\]) is used as one of the most general models in population biology and epidemiology. Three terms in $G_K(x)$ represent a growth rate proportional to the current population, restrictions due to the limited resources, and the death rate. The logistic map appears and plays an important role not only in population biology but also in economics, condensed matter physics, and other areas of science [@PB1; @PB3]. In population biology and epidemiology time delays can be related to the time of the development of an infection in a body until a person becomes infectious, or to the time of the development of an embryo. For the importance of time delay in many scientific applications of the logistic map see e.g. Ch. 3 from [@PB3] and Ch. 3 from [@PB1]. Changes which occur as periodically following discrete events can be modeled by the delta function.
The n-dimensional universal map can be derived from the following generating equation: $$\frac{d^nx}{dt^n}+G_K(x(t- \Delta)) \sum^{\infty}_{k=-\infty} \delta
\Bigl(t-(k+\varepsilon)
\Bigr)=0,
%, \ \ t \ge - \Delta T
\label{UM1D2Ddif}$$ where $n \ge 0$, $n\in\mathbb{Z}$, and $ \varepsilon > \Delta > 0$ in the limit $\varepsilon \rightarrow 0$. This means that in the general case time delay is not essential. Without losing the generality, in Eq. (\[UM1D2Ddif\]) we assumed $T=1$. The case $T \ne 1$ is considered in [@DNC] and can be reduced to this case by rescaling the time variable and the map generating function $G_K(x)$. In Sec. \[AFM\] $T$ denotes periods of trajectories. The 2D universal map Eqs. (\[UMp\]) and (\[UMx\]) corresponds to $n=2$ and the 1D universal map (\[UM1D\]) corresponds to $n=1$. In the consistent introduction of fractional derivatives integer derivatives appear as the limits when the order of a fractional derivative assumes an integer value. Correspondingly, the general form of the n-dimensional universal map appears if we assume an integer value of $\alpha$ in the general form of the fractional universal map. In the following sections we’ll consider the general forms of the fractional universal map which will be derived from Eq. (\[UM1D2Ddif\]) with integer $n$ replaced by $\alpha \in \mathbb{R}$ ($\alpha \ge 0$). The Riemann-Liouville universal map will be derived in Sec. \[RLUM\] and the Caputo universal map will be derived in Sec. \[CUM\].
Riemann-Liouville Universal Map {#RLUM}
-------------------------------
The generating fractional differential equation for the Riemann-Liouville universal map can be written as $$_0D^{\alpha}_tx(t) +G_K(x(t- \Delta )) \sum^{\infty}_{n=-\infty} \delta \Bigl(t-(n+\varepsilon)
\Bigr)=0,
%, \ \ t \ge - \Delta T
\label{UM1D2DdifRL}$$ where $\varepsilon > \Delta > 0$, $\varepsilon \rightarrow 0$, $0 \le
N-1 < \alpha \le N$, $\alpha \in \mathbb{R}$, $N \in \mathbb{Z}$, and the initial conditions $$(_0D^{\alpha-k}_tx)(0+)=c_k, \ \ k=1,...,N.
\label{UM1D2DdifRLic}$$ The left-sided Riemann-Liouville fractional derivative $_0D^{\alpha}_t
x(t)$ is defined for $t>0$ [@KST; @Podlubny; @SKM] as $$_0D^{\alpha}_t x(t)=D^n_t \ _0I^{n-\alpha}_t x(t)=
\frac{1}{\Gamma(n-\alpha)} \frac{d^n}{dt^n} \int^{t}_0
\frac{x(\tau) d \tau}{(t-\tau)^{\alpha-n+1}},
\label{RL}$$ where $n-1 \le \alpha < n$, $D^n_t=d^n/dt^n$, and $ _0I^{\alpha}_t$ is a fractional integral.
For a wide class of functions $G_K(x)$ Eq. (\[UM1D2DdifRL\]) is equivalent to the Volterra integral equation of the second kind ($t>0$) (see [@KBT1; @KBT2; @KST; @TarBook]) $$\begin{aligned}
&&x(t)= \sum^{N}_{k=1}\frac{c_k}{\Gamma(\alpha-k+1)}t^{\alpha -k} \nonumber\\
&&-\frac{1}{\Gamma(\alpha)} \int^{t}_0 d \tau \frac{G_K(x( \tau - \Delta ))}{( t-\tau )^{1-\alpha}} \sum^{\infty}_{k=-\infty}
\delta \Bigl(\tau-(k+\varepsilon)\Bigr).
\label{VoltRL}\end{aligned}$$ Due to the presence of the delta function the integral on the right side of Eq. (\[VoltRL\]) can be easily calculated [@DNC; @MM9; @MM8; @TarBook] for $t>0$: $$\begin{aligned}
&&x(t)= \sum^{N-1}_{k=1}\frac{c_k}{\Gamma(\alpha-k+1)}t^{\alpha -k} \nonumber\\
&&-\frac{1}{\Gamma(\alpha)}
\sum^{[t-\varepsilon]}_{k=0} \frac{G_K(x(k+\varepsilon-\Delta))}{( t-(k+\varepsilon))^{1-\alpha}}
\Theta(t-(k+\varepsilon)),
\label{VoltRLeq}\end{aligned}$$ where $\Theta(t)$ is the Heaviside step function. In Eq. (\[VoltRLeq\]) we took into account that boundedness of $x(t)$ at $t=0$ requires $c_N=0$ and $x(0)=0$. After the introduction (see [@MM11]) $$p(t)= {_0D^{\alpha-N+1}_t}x(t)$$ and $$p^{(s)}(t)= {D^{s}_t}p(t),$$ where $s=0,1,...,N-2$, Eq. (\[VoltRLeq\]) leads to $$\begin{aligned}
&&p^{(s)}(t)= \sum^{N-s-1}_{k=1}\frac{c_k}{(N-s-1-k)!}t^{N -s-1-k} \nonumber \\
&&\hspace{-0.3cm}-\frac{1}{(N-s-2)!}
\sum^{[t-\varepsilon]}_{k=0} G_K(x(k+\varepsilon-\Delta))( t-k )^{N-s-2},
\label{VoltRLeqp}\end{aligned}$$ where $s=0,1,...,N-2$. Assuming $x_n=x(n)$, for $\varepsilon > \Delta > 0$ Eqs. (\[VoltRLeq\]) and (\[VoltRLeqp\]) in the limit $\varepsilon
\rightarrow 0$ give the equations of the Riemann-Liouville universal map
$$\begin{aligned}
&&x_{n+1}= \sum^{N-1}_{k=1}\frac{c_k}{\Gamma(\alpha-k+1)}(n+1)^{\alpha -k} \nonumber \\
&&-\frac{1}{\Gamma(\alpha)}\sum^{n}_{k=0} G_K(x_k) (n-k+1)^{\alpha-1},
\label{FrRLMapx} \\
&&p^s_{n+1}= \sum^{N-s-1}_{k=1}\frac{c_k}{(N-s-1-k)!} (n+1)^{N-s -1-k}
\nonumber \\
&&-\frac{1}{(N-s-2)!}\sum^{n}_{k=0} G_K(x_k) (n-k+1)^{N-s-2}.
\label{FrRLMapp} \end{aligned}$$
Caputo Universal Map {#CUM}
--------------------
Similar to (\[UM1D2DdifRL\]), the generating fractional differential equation for the Caputo universal map can be written as $$_0^CD^{\alpha}_tx(t) +G_K(x(t- \Delta )) \sum^{\infty}_{n=-\infty} \delta \Bigl(t-(n+\varepsilon)
\Bigr)=0,
\label{UM1D2DdifC}$$ where $\varepsilon > \Delta > 0$, $\varepsilon \rightarrow 0$, $0 \le N-1
< \alpha \le N$, $\alpha \in \mathbb{R}$, $N \in \mathbb{Z}$, and the initial conditions $$(D^{k}_tx)(0+)=b_k, \ \ k=0,...,N-1.
\label{UM1D2DdifCic}$$ The left-sided Caputo fractional derivative $_0^CD^{\alpha}_t x(t)$ is defined for $t>0$ [@KST; @Podlubny; @SKM] as $$_0^CD^{\alpha}_t x(t)=_0I^{n-\alpha}_t \ D^n_t x(t) =
\frac{1}{\Gamma(n-\alpha)} \int^{t}_0
\frac{ D^n_{\tau}x(\tau) d \tau}{(t-\tau)^{\alpha-n+1}},
\label{Cap}$$ where $n-1 <\alpha \le n$.
For a wide class of functions $G_K(x)$ Eq. (\[UM1D2DdifC\]) is equivalent to the Volterra integral equation of the second kind ($t>0$) (see [@KBT1; @KBT2; @KST; @TarBook]) $$x(t)= \sum^{N-1}_{k=0}\frac{b_k}{k!}t^{k}
-\frac{1}{\Gamma(\alpha)} \int^{t}_0 d \tau \frac{G_K(x( \tau - \Delta ))}{( t-\tau )^{1-\alpha}} \sum^{\infty}_{k=-\infty}
\delta \Bigl(\tau-(k+\varepsilon)\Bigr).
\label{VoltC}$$ Integration of this equation gives for $t>0$ $$x(t)= \sum^{N-1}_{k=0}\frac{b_k}{k!}t^{k}
-\frac{1}{\Gamma(\alpha)}
\sum^{[t-\varepsilon]}_{k=0} \frac{G_K(x( k+\varepsilon-\Delta ))}{( t-(k+\varepsilon) )^{1-\alpha}}
\Theta(t-(k+\varepsilon)).
\label{VoltCeq}$$ After the introduction $x^{(s)}(t)=D^s_tx(t)$ the Caputo universal map can be derived in the form (see [@TarBook]) [ $$x^{(s)}_{n+1}= \sum^{N-s-1}_{k=0}\frac{x^{(k+s)}_0}{k!}(n+1)^{k}
-\frac{1}{\Gamma(\alpha-s)}\sum^{n}_{k=0} G_K(x_k) (n-k+1)^{\alpha-s-1},
\label{FrCMapx}$$ ]{} where $s=0,1,...,N-1$.
$\alpha$-Families of Maps {#AFM}
=========================
We’ll call Eqs. (\[UM1D2DdifRL\]) and (\[UM1D2DdifRLic\]) with various map generating functions $G_K(x)$ the Riemann-Liouville universal map generating equations and Eqs. (\[FrRLMapx\]) and (\[FrRLMapp\]) the Riemann-Liouville $\alpha$-families of maps corresponding to the functions $G_K(x)$. We’ll call Eqs. (\[UM1D2DdifC\]) and (\[UM1D2DdifCic\]) with various map generating functions $G_K(x)$ the Caputo universal map generating equations and Eqs. (\[FrCMapx\]) the Caputo $\alpha$-families of maps corresponding to the functions $G_K(x)$.
Fractional maps Eqs. , , and are maps with memory in which the next values of map variables depend on all previous values. An increase in $\alpha$ corresponds to the increase in a map dimension. It also corresponds to the increased power in the power-law dependence of weights of previous states which imply increased memory effects. For $\alpha=1$ and $\alpha=2$ the corresponding maps are given by Eqs. (\[UM1D\]), (\[UMp\]), and (\[UMx\]) with $T=1$ and $G_K(x)$ instead of $G(x)$. Eqs. , , and with $\alpha=3$ and variables $y=p$ and $z=\dot{p}$ produce the full-memory 3D Universal Map $$\begin{aligned}
&&x_{n+1}= \frac{z_0}{2}(n+1)^2 + y_0(n+1)+x_0
-\frac{1}{2}\sum^{n}_{k=0} G_K(x_k) (n-k+1)^{2}, \nonumber \\
&&y_{n+1}= z_0(n+1) + y_0 -
\sum^{n}_{k=0} G_K(x_k) (n-k+1), \label{3DUMf} \\
&&z_{n+1}= z_0 -
\sum^{n}_{k=0} G_K(x_k), \nonumber\end{aligned}$$ which is equivalent to the one-step memory (Sec. \[sec:1.2\]) 3D universal map $$\begin{aligned}
&&x_{n+1}= x_n-\frac{1}{2}G_K(x_n)+y_n+\frac{1}{2}z_n, \nonumber \\
&&y_{n+1}=-G_K(x_n)+y_n+z_n, \label{3DUM} \\
&&z_{n+1}=-G_K(x_n)+z_n, \nonumber \end{aligned}$$ or $$\begin{aligned}
&&x_{n+1}= x_n+y_{n+1}-\frac{1}{2}z_{n+1}, \nonumber \\
&&y_{n+1}=y_n+z_{n+1}, \label{3DUMn} \\
&&z_{n+1}=-G_K(x_n)+z_n, \nonumber \end{aligned}$$ which is a volume preserving map. This map has fixed points $z_0=y_0=G_K(x_0)=0$ and stability of these points can be analyzed by considering the eigenvalues $\lambda$ of the matrix (corresponding to the tangent map) $$\left( \begin{array}{ccc}
1-0.5\dot{G_K}(x_0) & 1 & 0.5 \\
-\dot{G_K}(x_0) & 1 & 1 \\
-\dot{G_K}(x_0) & 0 & 1 \end{array} \right).$$ The only case in which the fixed points could be stable is $\dot{G_K}(x_0)=0$, when $\lambda_1=\lambda_2=\lambda_3=1$. From Eq. (\[3DUMn\]) it follows that the only $T=2$ points are the fixed points.
The investigation of the integer members of the $\alpha$-families of maps is a subject of ongoing research. From the examples of maps with the values of $\alpha$ equal to one, two, and three we see that integer values of $\alpha$ correspond to the degenerate cases in which map equations can be written as maps with full memory. They are equivalent to $\alpha$-dimensional one-step memory maps in which map variables at each step accumulate information about all previous states of the corresponding systems.
Corresponding to the fact that in the $\alpha=2$ case the 2D universal family of maps produces the standard map if $G_K(x)=K\sin(x)$ (see Eqs. (\[SM\])) and in the $\alpha=1$ case the logistic map results from $G_K(x)=x-Kx(1-x)$ (see Eqs. (\[UM1D\]) and (\[LMG\])), we’ll call:
- [the Riemann-Liouville $\alpha$-family of maps Eqs. (\[FrRLMapx\]) and (\[FrRLMapp\]) with $G_K(x)=K\sin(x)$ the **[standard $\alpha$-RL-family of maps]{};**]{}
- [the Caputo $\alpha$-family of maps Eqs. (\[FrCMapx\]) with $G_K(x)=K\sin(x)$ the **[standard $\alpha$-Caputo-family of maps]{};**]{}
- [the Riemann-Liouville $\alpha$-family of maps with $G_K(x)=x-Kx(1-x)$ the **[logistic $\alpha$-RL-family of maps]{};**]{}
- [the Caputo $\alpha$-family of maps with $G_K(x)=x-Kx(1-x)$ the **[logistic $\alpha$-Caputo-family of maps]{}**]{}.
For $\alpha=0$ the solution of Eq. and correspondingly, the universal map is identically zero. For $\alpha <1 $ the Riemann-Liouville $\alpha$-families of maps Eqs. and (\[FrRLMapp\]) corresponding to the functions $G_K(x)$ satisfying the condition $G_K(0)=0$, which is true for the standard and logistic $\alpha$-RL-families of maps, also produces identically zero.
Integer-Dimensional Standard and Logistic Maps {#IntStLog}
----------------------------------------------
In general, properties of fractional maps converge to the corresponding properties of integer maps when $\alpha$ approaches integer values. To better understand properties of fractional maps we’ll start with the consideration of the integer members of the corresponding families of maps.
### One-Dimensional Logistic and Standard Maps {#1DStLog}
The one-dimensional logistic map Eq. (\[LM\]) is one of the best investigated maps. This map has been used as a playground for investigation of the essential property of nonlinear systems - transition from order to chaos through a sequence of period-doubling bifurcations, which is called cascade of bifurcations, and scaling properties of the corresponding systems (see [@LM1; @LM2; @LM3; @LM4; @LM6]). In our investigation of fractional maps we’ll use the well known stability properties of the logistic map (see [@LM5]), which for $0<K<4$ are summarized in the bifurcation diagram in Fig. \[BD1D\](a).
![(a) The bifurcation diagram for the logistic map $x=Kx(1-x)$. (b) The bifurcation diagram for the 1D standard map (circle map) Eq. (\[SM1D\]). []{data-label="BD1D"}](Fig2ME.jpg){width="100.00000%"}
The $x=0$ fixed point (sink) is stable for $K < 1$, the $(K-1)/K$ fixed point (sink) is stable for $1 < K < 3$, the $T=2$ sink is stable for $3 \le K < 1-\sqrt{6} \approx 3.449$, the $T=4$ sink is stable when $3.449 < K < 3.544$, and the onset of chaos as a result of the period-doubling cascade of bifurcations occurs at $K \approx 3.56995$.
The one-dimensional standard map ($\alpha=1$) considered on a circle $$x_{n+1}= x_n - K \sin (x_n), \ \ \ \ ({\rm mod} \ 2\pi )
\label{SM1D}$$ is a particular form of the circle map with zero driving phase. It has attracting fixed points $2\pi n$ for $0<K \le K_{c1}(1)=2$ and $ \pi + 2 \pi n$ when $-2 \le K < 0$ (for the bifurcation diagram of the 1D standard map see Fig. \[BD1D\](b)). The antisymmetric $T=2$ sink $$x_{n+1}= -x_n
\label{T21DSym}$$ is stable for $2 < |K| < \pi$, while $x_{n+1} = x_n+\pi$ two sinks ($T=2$) are stable when $\pi < |K| < \sqrt{\pi^2+2} \approx 3.445$. The stable $T=4$ sinks appear at $|K| \approx 3.445$ and the sequence of bifurcations $T=4$ $\rightarrow$ $T=8$ at $K \approx 3.513$, $T=8$ $\rightarrow$ $T=16$ at $K \approx 3.526$, and so on leads to the transition to chaos at $K \approx 3.532$. Antisymmetric $T=2$ trajectories ($K=2.4$), $T=4$ trajectories ($K=3.49$), and two cases of chaotic trajectories ($K=4.1$ and $K=5.1$) are presented in Fig. \[SM1Dpp\].
![\[SM1Dpp\] Attractors in the one-dimensional standard map; $x_n$ vs. $x_{n+1}$ plots (seven trajectories with different initial conditions in each plot): (a). $K=2.4$; antisymmetric $T=2$ sink. (b). $K=3.49$; $T=4$ trajectories. (c). $K=4.1$; proper attractor (width of the chaotic area is less than $2\pi$). (d). $K=5.1$; improper attractor (width of the chaotic area is $2\pi$). ](Fig3ME.jpg){width="100.00000%"}
In the 1D standard map with $K>0$ the full phase space $x \in [-\pi, \pi]$ becomes involved in chaotic motion (we’ll call this case “improper attractor”) when the maximum of the function $f_K(x)=x-K \sin x$ is equal to $\pi$ which occurs at $K_{max1D}= 4.603339$ when $x_{max1D}=-1.351817$ (see Figs. \[SM1Dpp\] (c) and (d)). Narrow bands with $|K|$ above $2\pi |n| $ (see Fig. \[BD1D\](b) for $K>2\pi$) are accelerator mode bands with zero acceleration within which in the unbounded space (no ${\rm mod} \ 2\pi $) $x$ is increasing/decreasing with the rate equal approximately to $ 2\pi |n|$.
### Two-Dimensional Logistic and Standard Maps {#2DStLog}
The two-dimensional logistic map $$\begin{aligned}
&& p_{n+1}= p_n+Kx_n(1-x_n)-x_n,
\nonumber \\
&& x_{n+1}= x_n + p_{n+1}
\label{LFMalp2}\end{aligned}$$ is a quadratic area preserving map. Its phase space contains stable elliptic islands and chaotic areas (no attractors). Quadratic area preserving maps which have a stable fixed point at the origin were investigated by Hénon [@Henon69] (for a recent review on 2D quadratic maps see [@ZeraS2010]). To investigate the logistic $\alpha$-families of maps we need to know the evolution of the periodic points of the 2D logistic map with the increase of the map parameter $K$. For $K \in (-3,1)$ the map Eq. has the stable fixed point $(0,0)$ which turns into the fixed point $((K-1)/K,0)$ stable for $K \in (1,5)$. The $T=2$ elliptic point [ $$\begin{aligned}
&&x = \frac{K+3 \pm \sqrt{(K+3)(K-5)}}{2K}, \nonumber \\
&&p=\pm \frac{\sqrt{(K+3)(K-5)}}{K}
\label{LFMalp2T2}\end{aligned}$$ ]{}
![ Bifurcations in the 2D Logistic Map: (a) $T=1$ $\rightarrow$ $T=2$ bifurcation at $K = 5$ ($K=5.05$ on the figure). (b) $T=8$ $\rightarrow$ $T=16$ bifurcation at $K \approx 5.5319$ ($K=5.53194$ on the figure). []{data-label="FigLog2D"}](Fig4ME.jpg){width="100.00000%"}
is stable for $-2 \sqrt{5}+1<K<-3$ and $5<K<2 \sqrt{5}+1$. The period doubling cascade of bifurcations (for $K>0$) follows the scenario of the elliptic-hyperbolic point transitions with the births of the double periodicity islands inside the original island which has been investigated in [@Schmidt] and applied to investigate the standard map stochasticity at low values of the map parameter. Further bifurcations in the 2D logistic map, $T=2$ $\rightarrow$ $T=4$ at $K \approx 5.472$, $T=4$ $\rightarrow$ $T=8$ at $K \approx 5.527$, $T=8$ $\rightarrow$ $T=16$ at $K \approx 5.5319$, $T=16$ $\rightarrow$ $T=32$ at $K \approx 5.53253$, etc., and the corresponding decrease in the areas of the islands of stability (see Fig. \[FigLog2D\]) lead to chaos.
The two-dimensional standard map on a torus Eq. (\[SM\]) (Chirikov standard map) is one of the best investigated 2D maps. It demonstrates a universal generic behavior of the area-preserving maps whose phase space is divided into elliptic islands of stability and areas of chaotic motion (see, e.g., [@Chir; @LL]). Elliptic islands of the standard map in the case of the standard $\alpha$-families of maps with $1 < \alpha < 2$ evolve into periodic sinks (see Sec. \[BN12\]). Properties of phase space and appearance of different types of attractors in the fractional case, as in the case of the fractional logistic map, are connected to the evolution (with the increase in parameter $K$) of the 2D standard map’s islands originating from the stable (for $K<4$) fixed point (0,0). At $K=4$ the fixed point becomes unstable (elliptic-hyperbolic point transition [@Schmidt]) and two elliptic islands around the stable for $4 < K <2 \pi$ period 2 antisymmetric point $$\label{SimP2D}
p_{n+1}=-p_n, \ \ x_{n+1}=-x_n$$ appear. At $K=2 \pi$ this point transforms into two $T=2$ points $$p_{n+1}=-p_n, \ \ x_{n+1}=x_n-\pi,
\label{ASimP2D}$$ which are stable when $2 \pi <K<6.59$. These points transform into $T=4$ stable elliptic points at $K \approx 6.59$ and the period doubling cascade of bifurcations leads to the disappearance of islands of stability in the chaotic sea at $K \approx 6.6344$ [@Chir; @LL]. The 2D standard map has a set of bands for $K$ above $2\pi n$ of the accelerator mode sticky islands in which the momentum $p$ increases proportionally to the number of iterations $k$ and the coordinate $x$ increases as $k^2$. The role of accelerator mode islands (for $K$ above $2\pi$) in the anomalous diffusion and the corresponding fractional kinetics is well investigated (see, for example, [@ZasBook; @ZE1997]).
### Three-Dimensional Logistic and Standard Maps {#3DStLog}
Eq. (\[3DUMn\]) with $G_K(x)=x-Kx(1-x)$ (see Eq. (\[LMG\]) produces the 3D logistic map $$\begin{aligned}
&&x_{n+1}= x_n+y_{n+1}-\frac{1}{2}z_{n+1}, \nonumber \\
&&y_{n+1}=y_n+z_{n+1}, \label{3DLMn} \\
&&z_{n+1}=Kx_n(1-x_n)-x_n+z_n. \nonumber \end{aligned}$$ Three-dimensional quadratic volume preserving maps were investigated in [@Moser1994; @LoMeiss1998]. Everything stated in Sec. \[AFM\] for the 3D universal map is still valid for the 3D logistic map.
The three-dimensional standard map with $G_K(x)=K \sin(x)$ $$\begin{aligned}
&&x_{n+1}= x_n+y_{n+1}-\frac{1}{2}z_{n+1},\ \ ({\rm mod} \ 2\pi ),
\nonumber \\
&&y_{n+1}=y_n+z_{n+1}, \ \ ({\rm mod} \ 2\pi ), \label{3DSMn} \\
&&z_{n+1}=-K\sin(x_n)+z_n, ,\ \ ({\rm mod} \ 4\pi ) \nonumber \end{aligned}$$ has unstable fixed points $(2\pi n, 2\pi m, 4 \pi k)$ and $(2\pi n+ \pi , 2\pi m, 4 \pi k)$ , $n \in \mathbb{Z}$, $m \in \mathbb{Z}$, $k \in \mathbb{Z}$. Ballistic points $K\sin(x)=-4\pi n$, $y=2\pi m$, $z=4\pi k$, which appear for $|K| \ge 4\pi$, are also unstable.
Stability of $T=2$ ballistic points is defined by the eigenvalues of the matrix $$\left( \begin{array}{ccc}
1-0.5K \cos x_1 & 1 & 0.5 \\
-K \cos x_1 & 1 & 1 \\
-K \cos x_1 & 0 & 1 \end{array} \right) \times
\left( \begin{array}{ccc}
1-0.5 K \cos x_2 & 1 & 0.5 \\
-K \cos x_2 & 1 & 1 \\
-K \cos x_2 & 0 & 1 \end{array} \right).$$ For the period two on the torus ballistic points $$\begin{aligned}
&&z_1,\ \ y_1=\frac{z_1}{2}-\pi(2n+1),\ \ K \sin x_1=2z_1, \nonumber \\
&& z_2=-z_1, \ \ y_2=-\frac{z_1}{2}-\pi(2n+1),\ \ x_2=x_1-\pi(2n-1),
\label{3DT2StablePoint}\end{aligned}$$ where $n \in \mathbb{Z}$, the eigenvalues are $$\Bigl\{1, \ \ \frac{1}{8}(8-K^2 \cos^2 x_1 \pm K \cos x_1 \sqrt{K^2\cos^2
x_1-16}) \Bigr\}
\label{3DT2StablePointEigenV}$$
![\[3D\] Phase space of the 3D standard map (\[3DSMn\]) with $K=3$: (a). Three dimensional phase space. (b). A projection of the 3D phase space on the $x$-$y$ plane. ](Fig5ME.jpg){width="100.00000%"}
Ballistic $T=2$ points are stable along a line defined by Eqs. (\[3DT2StablePoint\]) for all values of $z$ satisfying the condition $$K^2-16<4z^2<K^2.
\label{3DT2zCond}$$ An example of the phase space for $K=3$ in three dimensions and its projection on the $x$-$y$ plane is given in Fig. \[3D\]. For this value of $K$ ballistic $T=2$ points are stable when $-1.5<z<1.5$ and the space around the line of stability presents a series of islands (invariant curves), islands around islands, and separatrix layers. When $K \rightarrow 0$, the volume of the regular motion shrinks. When $K$ is small, the line of the stable $T=2$ ballistic points exists for $-K/2<z<K/2$. A different form of the 3D volume preserving standard map was introduced and investigated in detail in [@DM].
$\alpha$-Families of Maps ($0<\alpha<1$) {#LT1}
----------------------------------------
As we mentioned at the end of Sec. \[AFM\], members of the logistic and standard $\alpha$-families of maps corresponding to $\alpha=0$ and RL-families’ members with $0<\alpha<1$ are identically zeros. The only fractional logistic and standard maps with $0<\alpha<1$ which are not identically zeros are $\alpha$-Caputo-families of maps. The $\alpha$-Caputo-universal map ($0<\alpha<1$) $$x_{n+1}= x_0-
\frac{1}{\Gamma(\alpha)}\sum^{n}_{k=0} G_K(x_k) (n-k+1)^{\alpha-1}
\label{FrCMapxlt1}$$ in the limit $\alpha \rightarrow 1$ is identical to the one-dimensional universal map Eq. (\[UM1D\]) and in this limit properties of fractional maps are similar to properties of the corresponding 1D maps. Eq. (\[FrCMapxlt1\]) with $G_K(x)=x-Kx(1-x)$ is the logistic $\alpha$-Caputo-family of maps for $0<\alpha<1$ $$x_{n}= x_0+
\frac{1}{\Gamma(\alpha)}\sum^{n-1}_{k=0} \frac{Kx_k(1-x_k)-x_k}{(n-k)^{1-\alpha}}
\label{FrCMapLM}$$ and with $G_K(x)=K\sin(x)$ is the standard $\alpha$-Caputo-family of maps for $0<\alpha<1$ $$x_{n}= x_0-
\frac{K}{\Gamma(\alpha)}\sum^{n-1}_{k=0} \frac{\sin{x_k}}{(n-k)^{1-\alpha}},
\ \ ({\rm mod} \ 2\pi )
\label{FrCMapSM}$$ These maps are one-dimensional maps with power-law decreasing memory [@DNC]. The bifurcation diagrams for these maps are similar to the corresponding diagrams for the $\alpha=1$ case Fig. \[BD1D\].
![Bifurcation diagrams for the logistic and standard $\alpha$-Caputo-families of maps with $0<\alpha<1$. In (a)-(f) the bifurcation diagrams obtained after performing $10^4$ iterations on a single trajectory with $x_0=0.1$ for various values of $K$. (a), (c), and (e) - the logistic $\alpha$-Caputo-family. (b), (d), and (f) - the standard $\alpha$-Caputo-family. In (a) and (b) $\alpha=0.8$. In (c) and (d) $\alpha=0.3$. In (e) and (f) $\alpha=0.1$. []{data-label="LowAlpBif"}](Fig6ME.jpg){width="95.00000%"}
A decrease in $\alpha$ and the corresponding decrease in weights of the earlier states (decrease in memory effects) leads to the stretchiness of the corresponding bifurcation diagrams along the parameter $K$-axis and this stretchiness increases as $\alpha$ gets smaller Fig. \[LowAlpBif\].
Within a band of values of $K$, above the value which corresponds to the appearance of $T=4$ trajectories, map trajectories are attracting cascade of bifurcations type trajectories (CBTT) (see Fig. \[CBTT1D\]). On CBTT an increase in the number of map iterations leads to the change in the map’s stability properties. A trajectory which converges to a $T>4$ periodic point or becomes a chaotic trajectory (depending on the value of $K$) evolves according to a certain scenario: it first converges to a $T=4$ point; then it bifurcates, always at the same place for the given values of the parameter $K$ and the order $\alpha$, and converges to a $T=8$ trajectory; then to a $T=16$ trajectory; and so on. Power-law decaying memory with power $\beta \approx 0.9$ corresponding to small values of $\alpha \approx 0.1$ (see Sec. \[sec:1.1\]) appears in biological applications. Attracting CBTT in, for example, adaptive biological systems may represent not simply a change of a state of a biological system according to a change in a parameter, but rather a change in the evolution of the system according to the change in the parameter. Examples of CBTT in the logistic and standard $\alpha$-Caputo-families of maps with $\alpha=0.1$ are presented in Fig. \[CBTT1D\].
![Cascade of bifurcations type trajectories in the logistic and standard $\alpha$-Caputo-families of maps with $\alpha=0.1$. (a) The fractional logistic map with $\alpha=0.1$ and $K=22.65$. (b) The fractional standard map with $\alpha=0.1$ and $K=26.65$. []{data-label="CBTT1D"}](Fig7ME.jpg){width="98.00000%"}
It also should be noted that bifurcation diagrams of the fractional maps depend on the number of iterations used in their calculations. This is a consequence of the existence of CBTT. Trajectories which after 100 iterations converged to a fixed point in Fig. \[BifOfNI\](b) after 10000 iterations became $T=2$ trajectories in Fig. \[BifOfNI\](a). With an increase in the number of iterations the whole bifurcation diagram shifts to the left.
![Dependence of bifurcation diagrams of the fractional maps on the number of iterations on a single trajectory used in their calculation. Bifurcation diagrams for the fractional logistic map with $\alpha=0.1$. (a) 10000 iterations on each trajectory. (b) 100 iterations on each trajectory. []{data-label="BifOfNI"}](Fig8ME.jpg){width="100.00000%"}
$\alpha$-Families of Maps ($1<\alpha<2$) {#BN12}
----------------------------------------
For $1<\alpha<2$ the logistic and standard $\alpha$-families of maps assume the following forms:
- [ The RL-standard map on a cylinder $$\begin{aligned}
&&p_{n+1} = p_n - K \sin x_n , \label{FSMRLp} \\
&&x_{n+1} = \frac{1}{\Gamma (\alpha )}
\sum_{i=0}^{n} p_{i+1}V^1_{\alpha}(n-i+1)
, \ \ \ \ ({\rm mod} \ 2\pi ), \label{FSMRLx} \end{aligned}$$ where $$\label{V1}
V^k_{\alpha}(m)=m^{\alpha -k}-(m-1)^{\alpha -k}.$$ This map requires the initial condition $x_0=0$ and can’t be considered on a torus.]{}
- [The Caputo-standard map on a torus $$\begin{aligned}
&& p_{n+1} = p_n
-\frac{K}{\Gamma (\alpha -1 )}
\Bigl[ \sum_{i=0}^{n-1} V^2_{\alpha}(n-i+1) \sin x_i
+ \sin x_n \Bigr],\ \ ({\rm mod} \ 2\pi ), \label{FSMCp} \\
&&x_{n+1} = x_n + p_0
-\frac{K}{\Gamma (\alpha)}
\sum_{i=0}^{n} V^1_{\alpha}(n-i+1) \sin x_i,\ \ ({\rm mod} \ 2\pi ).
\label{FSMCx}\end{aligned}$$ ]{}
- [ The RL-logistic map $$\begin{aligned}
&&p_{n+1} = p_n - Kx_n (1-x_n)-x_n, \label{LMRLp} \\
&&x_{n+1} = \frac{1}{\Gamma (\alpha )}
\sum_{i=0}^{n} p_{i+1}V^1_{\alpha}(n-i+1), \label{LMRLx} \end{aligned}$$ which requires the initial condition $x_0=0$.]{}
- [ The Caputo-logistic map $$\begin{aligned}
&&x_{n+1}=x_0+ p(n+1)^{k}
-\frac{1}{\Gamma(\alpha)}\sum^{n}_{k=0} [x_k-Kx_k(1-x_k)] (n-k+1)^{\alpha-1},
\label{LMCx} \\
&&p_{n+1}=p_0
-\frac{1}{\Gamma(\alpha-1)}\sum^{n}_{k=0} [x_k-Kx_k(1-x_k)] (n-k+1)^{\alpha-2}.
\label{LMCp}\end{aligned}$$ Here and in Eqs. (\[FSMCp\]) and (\[FSMCx\]) we assumed $x \equiv
x^0$ and $p \equiv x^1$ in the Caputo universal map Eq. (\[FrCMapx\]). ]{}
The fractional standard maps Eqs. (\[FSMRLp\]), (\[FSMRLx\]), (\[FSMCp\]), and (\[FSMCx\]) are well investigated (see [@ME1; @DNC; @ME3; @ME4]) and the logistic maps are the subject of ongoing research.
Evolution of trajectories in fractional maps depends on two parameters: the map parameter $K$ and the fractional order $\alpha$. Fig. \[figBif\] reflects this dependence in the case of the standard $\alpha$-families of maps with $1<\alpha<2$.
![ Bifurcations in the standard $\alpha$-families of maps with $1<\alpha<2$. Below $K=K_{c1}$ curve the fixed point $(0,0)$ is stable. It becomes unstable at $K=K_{c1}$ and gives birth to the antisymmetric $T=2$ sink which is stable for $K_{c1}<K<K_{c2}$. A pair of the $T=2$ sinks with $x_{n+1}=x_n-\pi$, $p_{n+1}=-p_n$ is stable in the band above $K=K_{c2}$ curve. Cascade of bifurcations type trajectories (CBTT) appear and exist in the narrow band which ends at the cusp at the top right corner of the figure. $(x_c,p_c)$ is the point at which the standard map’s ($\alpha=2$) $T=2$ elliptic points with $x_{n+1}=x_n-\pi$, $p_{n+1}=-p_n$ become unstable and bifurcate. In the area below $K_{c3}$ (above the CBTT band) the chaotic attractor is restricted to a band whose width is less than $2\pi$. On the upper curves and above them the full phase space is chaotic.[]{data-label="figBif"}](Fig9ME.jpg){width="70.00000%"}
### $T=2$ Antisymmetric Sink {#T2}
It is obvious that the fractional standard and logistic maps have the fixed points at the origin $(0,0)$. But we’ll start the fractional maps’ phase space analysis with the consideration of the $T=2$ antisymmetric sinks. We’ll present most of the analysis for the fractional RL-standard map (Fig. \[figBif\]). Results of numerical simulations suggest that the fractional Caputo-standard map has similar properties and the results for the logistic map are submitted for publication.
The 1D standard map has the $T=2$ antisymmetric sink Eq. (\[T21DSym\]) and the 2D standard map has the $T=2$ antisymmetric elliptic point Eq. (\[SimP2D\]). Numerical experiments (Fig. \[T2sink\]) show that the antisymmetric $T=2$ sinks persist in the fractional standard maps with $1<\alpha<2$. In the RL-standard map these sinks in the RL-standard map attract most of the trajectories with small $p_0$. Assuming the existence of an antisymmetric $T=2$ sink $$\label{RLT2sink}
p_n=p_l(-1)^n, \ \ x_n=x_l(-1)^n ,$$ it is possible to calculate the coordinates of its attracting points $(x_l, p_l)$ and $(-x_l, -p_l)$. In the limit $n \rightarrow \infty$ Eqs. (\[FSMRLp\]) and (\[FSMRLx\]) can be written as $$\begin{aligned}
&&p_l = \frac{K}{2} \sin(x_l), \label{T2Limp} \\
&&x_{l} = \lim_{n \rightarrow \infty}x_{2n}=\frac{p_l}{\Gamma(\alpha)}
\lim_{n \rightarrow \infty}\sum^{2n-1}_{k=0}(-1)^{k+1} V_{\alpha}^1(2n-i)
=\frac{p_l}{\Gamma(\alpha)}V_{\alpha l}(k) \label{T2Limx},\end{aligned}$$ where $$\label{Valpl}
V_{\alpha l} = \sum_{k=1}^{\infty} (-1)^{k+1} V_{\alpha}^1(k).$$
![\[T2sink\] The RL-standard map’s period 2 sink: (a). An example of the $T=2$ attractor for $K=4.5$, $\alpha =1.9$. One trajectory with $x_0=0$, $p_0=0.513$. (b). $p_l$ of $x_l$ for the case of $K=4.5$. (c). $p_l$ of $\alpha $ for the case of $K=4.5$. (d). $x_l$ of $\alpha $ for the case of $K=4.5$. (e). $p_n-p_l$ for the trajectory in (a). After 1000 iterations $|p_n-p_l| < 10^{-7}$. (f). $x_n-x_l$ for the trajectory in (a). After 1000 iterations $|x_n-x_l| < 10^{-7}$. ](Fig10ME.jpg){width="95.00000%"}
Finally, the equation for the $x_l$ takes the form $$\label{T2AntiXleq}
x_l = \frac{K}{2 \Gamma(\alpha)} V_{\alpha l} \sin(x_l).$$ The numerical solution of Eqs. (\[T2AntiXleq\]) and (\[T2Limp\]) for $K=4.5$ when $1<\alpha<2$ is presented in Figs. \[T2sink\] (b)-(d). Figs. \[T2sink\] (e) and (f) show how well this solution agrees with the results of numerical simulations of individual trajectories. After 1000 iterations presented in Figs. \[T2sink\] (e) and (f) the values of deviations $|p_n-p_l|$ and $|x_n-x_l|$ are less than $10^{-7}$.
The condition of the existence of a solution for Eq. (\[T2AntiXleq\]) $$\ \label{Kc1}
K > K_{c1}(\alpha) = \frac{2 \Gamma(\alpha)}{V_{\alpha l}}
%\eqno (13)$$ is the condition of the existence of the antisymmetric $T=2$ sink. This sink exists above the curve $K = K_{c1}$ on Fig. \[figBif\]. For $\alpha=2$ Eq. (\[Kc1\]) produces the standard map condition $K>4$ (see Sec \[2DStLog\]) and for $\alpha=1$ it gives $K>2$ (see Sec \[1DStLog\]).
### Fixed Points {#Fixed}
Numerical simulations show that as in the 1D and 2D cases, in the case of fractional maps with $1<\alpha<2$ the condition of the appearance of $T=2$ trajectories coincides with the condition of the disappearance of the stable fixed point. This result for the fractional standard map was demonstrated in [@ME4] and for the fractional logistic map was submitted for publication. Below we present two ways in which stability of the RL-standard map’s $(0,0)$ fixed point can investigated.
In the vicinity of the fixed point $(0,0)$ the equation for the deviation of a trajectory from the fixed point can be written as $$\begin{aligned}
&& \delta p_{n+1} = \delta p_n - K \delta x_n , \label{00fixP} \\
&&\delta x_{n+1} = \frac{1}{\Gamma (\alpha )}
\sum_{i=0}^{n} \delta p_{i+1}V_{\alpha}(n-i+1) .\label{00fixX}\end{aligned}$$
![\[FigFix\] Stability of the fixed point $(0,0)$ in the RL-standard map with $1<\alpha<2$: (a). The fixed point is stable below the curve $K=K_c(\alpha )$. (b). Values of $S_\infty$ and $I_\infty$ obtained after 20000 iterations of Eq. \[SI\]. The values of $S_\infty$ and $I_\infty$ increase rapidly when $\alpha \rightarrow 2$; for example, $S_\infty \approx 276$ and $I_\infty \approx 552$ after 20000 iterations when $\alpha=1.999$. (c). An example of the typical evolution of $S_\infty$ and $I_\infty$ over the first 200 iterations for $1 < \alpha <2$. This particular figure corresponds to $\alpha =1.8$. (d). Deviation of the values $S_n$ and $I_n$ from the values $S_\infty \approx 2.04337$ and $I_\infty \approx 3.37416$ for $\alpha =1.8$ during the first 20000 iterations (this type of behavior remains for $1 < \alpha <2$). (e). Evolution of trajectories with $p_0=1.5+0.0005i$, $0 \le i < 200$ for the case $K=3$, $\alpha=1.9$. The line segments correspond to the $n$th iteration on the set of trajectories with close initial conditions. The evolution of the trajectories with smaller $p_0$ is similar. (f). $10^5$ iterations on both of two trajectories for $K=2$, $\alpha =1.4$. The one at the bottom with $p_0=0.3$ is a fast converging trajectory. The upper trajectory with $p_0=5.3$ is an example of an attracting slow converging trajectory in which $p_{100000}
\approx 0.042$. ](Fig11ME.jpg){width="89.00000%"}
Based on the results of Sec. \[T2\] let’s look for a solution in the form $$\begin{aligned}
&&\delta p_{n} = p_0\sum_{i=0}^{n-1}p_{n,i}\Bigl(\frac{2}{V_{\alpha
l}}\Bigr)^i\Bigl(\frac{K}
{K_{c1}(\alpha)}\Bigr)^i, \quad ( n > 0) , \label{00fixPSol} \\
&&\delta x_{n} = \frac{p_0}{\Gamma (\alpha )}\sum_{i=0}^{n-1}x_{n,i}
\Bigl(\frac{2}{V_{\alpha
l}}\Bigr)^i\Bigl(\frac{K}{K_{c1}(\alpha)}\Bigr)^i, \quad (n > 0),
\label{00fixXSol}\end{aligned}$$ where $p_{n,i}$ and $x_{n,i}$ satisfy the following iterative equations $$\begin{aligned}
&&x_{n+1,i}=-\sum_{m=i}^{n}(n-m+1)^{\alpha-1}x_{m,i-1} , \quad ( 0 <i \le n) ,
\label{00fixXIter} \\
&&p_{n+1,i}=-\sum_{m=i}^{n}x_{m,i-1} , \quad ( 0 <i < n),
\label{00fixPIter} \end{aligned}$$ for which the initial and boundary conditions are $$\label{IC}
p_{n+1,n}=x_{n+1,n}=(-1)^n, \quad p_{n+1,0}=1, \quad x_{n+1,0}=(n+1)^{\alpha-1}.$$ To verify the convergence of the alternating series Eqs. (\[00fixPSol\]) and (\[00fixXSol\]) we apply the Dirichlet’s test by considering the totals $$\label{Dirichlet}
S_n=\sum_{i=0}^{n-1}x_{n,i}\Bigl(\frac{2}{V_{\alpha l}}\Bigr)^i, \quad
I_n=\sum_{i=0}^{n-1}p_{n,i}\Bigl(\frac{2}{V_{\alpha l}}\Bigr)^i.$$ They obey the following iterative rules $$\label{SI}
S_n= n^{\alpha -1}-\frac{2}{V_{\alpha l}} \sum_{i=1}^{n-1}(n-i)^{\alpha -1}S_i,
\quad
I_n=1 -\frac{2}{V_{\alpha l}} \sum_{i=1}^{n-1}S_i,$$ where $S_1=1$. Numerical simulations demonstrate that values of $S_n$ and $I_n$ converge to the values $(-1)^{n+1}S_\infty $ and $(-1)^{n+1}I_\infty $ presented in Fig. \[FigFix\](b). Figs. \[FigFix\](c) and (d) show an example of the typical evolution of $S_n$ and $I_n$ over the first 20000 iterations. There is still no strict mathematical proof of the convergence. From the boundedness of $S_n$ and $I_n$ the convergence of $\delta p_{n}$ and $\delta x_{n}$ requires the following condition $$\label{fixStable}
\frac{K}{K_{c1}(\alpha )} <1,$$ which, as we expected, is exactly opposite to the condition of the existence of the antisymmetric $T=2$ sink Eq. (\[Kc1\]). Hundreds of runs of computer simulations confirmed that the transition from the stable fixed point $(0,0)$ to the stable antisymmetric $T=2$ sink in both the RL-standard map and the Caputo-standard map occurs on the curve $K=K_{c1}$ depicted in Fig. \[FigFix\](a).
The second way to investigate stability of the $(0,0)$ fixed point is by using generating functions [@Fel], which in the case of convolutions allows transformations of sums of products into products of sums. After the introduction $$\tilde{W}_{\alpha}(t)= \frac{K}{\Gamma (\alpha)}
\sum_{i=0}^{\infty }[(i+1)^{\alpha-1}-i^{\alpha-1}]t^i, \quad
\tilde{X}(t)=\sum_{i=0}^{\infty }\delta x_i t^i,
\quad
\tilde{P}(t)=\sum_{i=0}^{\infty }\delta p_i t^i
\label{GF}$$ system Eqs. (\[00fixP\]) and (\[00fixX\]) can be written as $$\begin{aligned}
&&\tilde{X}(t)=\frac{p_0 \tilde{W}_{\alpha}(t)}{K}
\frac{t}{1 - t \Bigl(1- \tilde{W}_{\alpha}(t) \Bigr) }, \label{GFeqx} \\
&&\tilde{P}(t)=p_0 \frac{1+ \tilde{W}_{\alpha}(t) }
{ 1- t \Bigl( 1- \tilde{W}_{\alpha}(t) \Bigr) } . \label{GFeqp}\end{aligned}$$ We see that the original problem can be solved by investigating the asymptotic behavior at $t=0$ of the derivatives of the analytic functions $\tilde{X}(t)$ and $\tilde{P}(t)$. This is still a complex unresolved problem.
When $K<K_{c1}$ and the fixed point is stable, in phase space it is surrounded by a finite basin of attraction, whose width $w$ depends on the values of $K$ and $\alpha$. For example, for $K=3$ and $\alpha=1.9$ the width of the basin of attraction is $1.6<w<1.7$. Numeric simulations of thousands of trajectories with $p_0 < 1.6$ performed by the authors of [@ME4], of which only 200 (with $1.5< p_0 < 1.6$) are presented in Fig. \[FigFix\](e), show only converging trajectories, whereas among 50 trajectories with $1.6<p_0<1.7$ in Fig. \[Fig12\](a) there are trajectories converging to the fixed point as well as some trajectories converging to attracting slow diverging trajectories, whose properties will be discussed in the following section (Sec. \[SecAttractors\]). Fig. \[FigFix\](e) shows fast converging trajectories. In the case $K=2$ and $\alpha=1.4$ in addition to the fast converging trajectories and attracting slow diverging trajectories there exist attracting slow converging trajectories (Fig. \[FigFix\](f)).
### Attractors Below Cascade of Bifurcations Band {#SecAttractors}
In the following most of the statements are conjectures made on the basis of the results of numerical simulations performed for some values of parameters $K$ and $\alpha$ which then were verified for additional parameter values.
![\[Fig12\] The RL- and Caputo-standard maps’ phase spaces for $K<K{c1}$: (a). The RL-standard map with the same values of parameters as in Fig. \[FigFix\](e) but $p_0=1.6+0.002i$, $0 \le i < 50$. (b). The Caputo-standard map with the same values of parameters as in Fig. \[FigFix\](e) but $p_0=1.7+0.002i$, $0 \le i < 50$. (c). 400 iterations on the RL-standard map trajectories with $p_0=4+0.08i$, $0 \le i < 125$ for the case $K=2$, $\alpha=1.9$. Trajectories converge to the fixed point and two types of attracting slow diverging trajectories: with $x_{lim}=0$ ($T=1$) and $T=4$. (d). 100 iterations on the Caputo-standard map trajectories with $p_0=-3.14+0.0314i$, $0 \le i < 200$ for the same case as in (c) ($K=2$, $\alpha=1.9$) but considered on a torus. In this case all trajectories converge to the fixed point or $T=4$ sink. (e). 400 iterations on trajectories with $p_0=2+0.04i$, $0 \le i < 50$ for the RL-standard map case $K=0.6$, $\alpha=1.9$. Trajectories converge to the fixed point and two attracting slow diverging trajectories ($T=2$ and $T=3$). (f). 100 iterations on the Caputo-standard map trajectories with $p_0=-3.14+0.0314i$, $0 \le i < 200$ for the same case as in (e) ($K=0.6$, $\alpha=1.9$) considered on a torus. In this case all trajectories converge to the fixed point, period two and period three sinks. ](Fig12ME.jpg){width="85.00000%"}
The structure of the fractional standard map’s phase space preserves some features which exist in the $\alpha = 2$ case. For example, for $K<K_{c1}$ stable higher period points, which exist in the standard map, still exist in the fractional standard maps Fig. \[Fig12\], but they exist in the asymptotic sense and they transform from elliptic points into sinks and (in the case of the RL-standard map) into attracting slow ($p_n \sim n^{2-\alpha}$) diverging trajectories. In the area preserving standard map stable fixed and periodic points are surrounded by islands of regular motion which in the case of fractional maps turn into basins of attraction associated with sinks or slowly diverging attracting trajectories. In the standard map islands are surrounded by chaotic areas. For $K<K_{c1}$ and $1< \alpha <2$ in the fractional standard maps there are no chaotic or regular trajectories. Chaos exists in the following sense: two initially close trajectories that start in an area between basins of attractions at first diverge, but then converge to the same or different attractors.
![\[Fig13\] Different types of convergence of trajectories to the fixed point in the RL-standard map ((a) and (b)) and the Caputo-standard map (c): (a). Time dependence of the coordinate and momentum for the fast converging trajectory with $K=2$, $\alpha =1.4$ and the initial conditions $x_0=0$ and $p_0=0.3$ from Fig. \[FigFix\](f). (b). The same as in (a) but for the attracting slow converging trajectory with the initial conditions $x_0=0$ and $p_0=5.3$. (c). $x$ and $p$ time dependence for the Caputo-standard map with $K=2$, $\alpha =1.4$, and the initial conditions $x_0=0$ and $p_0=0.3$. ](Fig13ME.jpg){width="100.00000%"}
There are differences not only between properties of the regular and fractional standard maps but also between phase space structures of the RL- and Caputo-standard maps. There is more than one way to approach an attracting periodic or fixed point of the RL-standard map. In Fig. \[Fig13\] the examples of three trajectories, two for the RL-standard map and one for the Caputo-standard map, are used to demonstrate the differences in the rates of convergence. In the RL-standard map trajectories starting from attractors’ basins of attractions demonstrate fast convergence with $$\label{FastConv}
\delta x_n \sim n^{-1-\alpha}, \ \ \delta p_n
\sim n^{-\alpha}$$ and trajectories with the initial conditions from chaotic areas demonstrate slow convergence: $$\label{SlowConv}
\delta x_n \sim n^{-\alpha}, \ \ \delta p_n
\sim n^{1-\alpha}.$$ There is only one type of convergence in the Caputo-standard map: $$\label{CaputoConv}
\delta x_n \sim n^{1-\alpha}, \ \ \delta p_n
\sim n^{1-\alpha}.$$ The same rates of convergence were observed also for antisimmetric (see Sec. \[T2\] and Fig. \[Fig15\]) and $x_{n+1}=x_n-\pi$, $p_{n+1}=-p_n$ period two ($T=2$) points (Fig. \[Fig16\]).
From Figs. \[Fig15\] (a) and (b) one can see that phase portraits on cylinders of the fractional standard maps with $K=3$ and $\alpha=1.9$ contain, in addition to the $(0,0)$ fixed point, attracting slow diverging trajectories (RL-case), or fixed points (Caputo-case) approximately equally spaced along the $p$-axis. This result agrees with the fact that the standard map with $K=3$ has only one central island. More complex structures of the fractional standard maps’ phase spaces, for $K=2$ with $T=4$ sinks (Figs. \[Fig15\] (c) and (d)) and for $K=0.6$ with $T=2$ and $T=3$ sinks (Figs. \[Fig15\] (e) and (f)), can be explained by the presence of the islands with the same periodicity in the standard map with the same $K$.
![\[Fig14\] Evaluation of the behavior of the attracting slow diverging trajectories: (a). Momenta for two trajectories with $x_n \approx 2\pi n$ in unbounded space (in this example $K=2$). The solid line is related to a trajectory with $\alpha = 1.9$ and its slope is 0.1. The dashed line corresponds to a trajectory with $\alpha = 1.5$ and its slope is 0.5. (b). Deviation of momenta from the asymptotic formula for two trajectories with $x_n \approx 2\pi n$ in unbounded space, $\alpha = 1.9$, and $K=2$. The dashed line has $p_0=7$ and the solid one $p_0=6$. (c). Relative deviation of the momenta for the trajectories in (b) from the asymptotic formula. (d). Deviation of the $x$-coordinates for the trajectories in (b) from the asymptotic formula. ](Fig14ME.jpg){width="100.00000%"}
Numerical evaluations (see Fig. \[Fig14\]) lead to the suggestion that attracting slow diverging trajectories which converge to trajectories along the $p$-axis ($x \rightarrow x_{lim}=0$) in the area of parameters of their stability for large $n$ demonstrate the following asymptotic behavior $$\label{5i}
p_n = C n^{2-\alpha }.$$ The constant C can be evaluated for $1.8< \alpha <2$. Consider a trajectory on a cylinder with $ x_{lim}=0$, $T=1$, and constant step in $x$ in the unbounded space $2 \pi M$, where $M$ is an integer. Then from Eq. (\[FSMRLx\]) follows $$\label{5}
x_{n+1}-x_{n} = \frac{1}{\Gamma (\alpha )}
\sum_{k=1}^{n} (p_{k+1}-p_k)V_{\alpha}^1(n-k+1)
+ \frac{p_1}{\Gamma (\alpha)} V_{\alpha}^1(n+1) .$$ For large $n$ the last term is small ($\sim n^{\alpha-2}$) and the following holds $$\label{6}
\sum_{k=1}^{n} (p_{k+1}-p_k)V_{\alpha}^1(n-k+1) = 2 \pi M \Gamma (\alpha).
%\eqno (6)$$
It can be shown, assuming $p_n \sim n^{2-\alpha}$, that for values of $\alpha>1.8$ the terms in the last sum with large $k$ are small and in the series representation of $V_{\alpha}^1(n-k+1)$ only terms of the highest order in $k/n$ can be kept. In this case, Eq. (\[6\]) leads to the approximations $$\label{7}
p_n \approx p_0 + \frac{2 \pi M \Gamma (\alpha) n^{2-\alpha}}{\alpha-1},
\quad
x_n \approx -\frac{2\pi M(2-\alpha ) \Gamma (\alpha) }{ K(\alpha-1) n^{\alpha-1}}.$$ In the case $K=2$, $\alpha=1.9$ Figs. \[Fig14\] (b)-(d) show for two trajectories with $M=1$ (initial momenta $p_0=6$ and $p_0=7$) approaching an attracting slow diverging trajectory the deviation from the asymptotic formula Eq. (\[7\]) and the relative difference with respect to Eq. (\[7\]).
![\[Fig15\]Stable antisymmetric $x_{n+1}=-x_n$, $p_{n+1}=-p_n$ period $T=2$ trajectories for $K=4.5$: (a). 1000 iterations on each of 25 trajectories for the standard map with $K=4.5$. The only feature is a system of two islands associated with the period two elliptic point. (b). RL-standard map stable $T=2$ antisymmetric sink for $\alpha=1.8$. 500 iterations on each of 25 trajectories: $p_0=0.0001+0.08i$, $0 \le i <25$. Slow and fast converging trajectories. (c). Caputo-standard map stable $T=2$ antisymmetric sink for $\alpha=1.8$. 1000 iterations on each of 10 trajectories: $p_0=-3.1415+0.628i$, $0 \le i <10$. ](Fig15ME.jpg){width="100.00000%"}
As for $K<K_{c1}$, in the case $K_{c1}(\alpha)<K<K_{c2}(\alpha)$ asymptotic existence and stability of the antisymmetric sink (Sec. \[T2\]) is a result of the gradual transformation of the standard map’s elliptic point with the decrease in the order of derivative from $\alpha =2$ (see Fig. \[Fig15\]). Convergence of trajectories follows Eqs. (\[FastConv\])-(\[CaputoConv\]).
The standard map’s antisymmetric $T=2$ trajectory becomes unstable when $K=2\pi$ and at the point $(\pi/2,0)$ in phase space a pair of $T=2$ trajectories with $x_{n+1}=x_n-\pi$, $p_{n+1}=-p_n$ appears. Numerical simulations of the fractional standard maps (see Fig. \[Fig16\]) show that they demonstrate similar behavior. With the assumption that the RL-standard map Eqs. (\[FSMRLp\]) and (\[FSMRLx\]) have an asymptotic solution $$\label{T2nonAS}
p_{n} = (-1)^np_l, \ \ x_{n} = x_l-\frac{\pi}{2}[1-(-1)^n]$$ it can be shown from Eq. (\[FSMRLp\]) that the relationship $p_l = K/2 \sin(x_l)$ (Eq. (\[T2Limp\])) is valid in this case too.
Numerical simulations similar to those presented in Fig. \[Fig13\] show that for $K>K_{c2}$ (see Fig. \[figBif\]) the RL-standard map has the asymptotic behavior $$\label{T2nonLimAS}
p_{n} = (-1)^np_l+An^{1-\alpha},$$ where $A$ is the same for both even and odd values of $n$.
![\[Fig16\] Stable $x_{n+1}=x_n-\pi$, $p_{n+1}=-p_n$ period $T=2$ trajectories for $K>K_{c2}$: (a). 500 iterations on each of 50 trajectories for the standard map with $K=6.4$. The main features are two accelerator mode sticky islands around points $(-1.379,0)$ and $(1.379,0)$ which define the dynamics. Additional features - dark spots at the top and the bottom of the figure (which are clear on a zoom) - two systems of $T=2$ tiny islands associated with two $T=2$ elliptic points: $(1.379,\pi)$, $(1.379-\pi,-\pi)$ and $(\pi-1.379,\pi)$, $(-1.379,-\pi)$. (b). Two RL-standard map’s stable $T=2$ sinks for $K=4.5$, $\alpha=1.71$. 500 iterations on each of 25 trajectories: $p_0=0.0001+0.08i$, $0 \le i <25$. (c). Two Caputo-standard map’s stable $T=2$ sinks for $K=4.5$, $\alpha=1.71$. 1000 iterations on each of 10 trajectories: $p_0=-3.1415+0.628i$, $0 \le i <10$. ](Fig16ME.jpg){width="100.00000%"}
After substituting (\[T2nonLimAS\]) in (\[FSMRLx\]) in the limit $n \rightarrow \infty$ one can derive $$\label{T2nonASxsol}
\sin(x_l)= \frac{\pi \Gamma(\alpha)}{K V_{\alpha l}},$$ which has solutions when $$\label{T2nonASKc}
K>K_{c2}= \frac{\pi \Gamma(\alpha)}{V_{\alpha l}}$$ (see Fig. \[figBif\]). The value of $A$ can also be calculated: $$\label{T2nonASA}
A= \frac{2 x_l-\pi}{2 \Gamma(2-\alpha)}.$$ Results of the analytic estimations Eqs. (\[T2nonASxsol\]) -(\[T2nonASA\]) are in good agreement with the direct numerical simulations of the fractional standard maps.
### Cascade of Bifurcations Band {#BisBand}
At $K \approx 6.59$ in the standard map $T = 2$ points become unstable and stable $T = 4$ elliptic points appear. Further increase in $K$ results in the period doubling cascade of bifurcations which leads to the disappearance of the corresponding islands of stability in the chaotic sea at $K \approx 6.6344$ (see Sec. \[2DStLog\]). The cusp in Fig. \[figBif\](a) points to a point $\alpha = 2$ and $6.59<K_*<6.63$. Inside the band leading to the cusp a new type of attractors, cascade of bifurcations type trajectories (CBTT), appears (see Fig. \[Cascades\]). The lower boundary of the band approximately corresponds to the transition from the $T=2$ sink $x_{n+1}=x_n-\pi$, $p_{n+1}=-p_n$ to the $T=4$ sink and the upper boundary corresponds to the transition to chaos. At $\alpha=1$ the lower and upper boundaries correspond to the $T=2$ $\rightarrow$ $T=4$ transition and the transition to chaos in the 1D standard map (see Sec. \[1DStLog\]).
![ Cascade of bifurcations type trajectories in the RL-standard map: (a). $\alpha=1.65, K=4.5$; one intermittent trajectory in phase space. (b). Time dependence of the coordinate $x$ ($x$ of $n$) for the case (a). (c). $\alpha=1.98, K=6.46$; zoom of a small feature for a single intermittent trajectory in phase space. (d). $\alpha=1.1, K=3.5$; a single trajectory enters the cascade after a few iterations and stays there during 500000 iterations. []{data-label="Cascades"}](Fig17ME.jpg){width="1.\textwidth"}
In CBTT period doubling cascade of bifurcations occurs on a single trajectory with a fixed value of the map parameter. A typical CBTT’s behavior is similar to the behavior of trajectories in Hamiltonian dynamics in the presence of sticky islands: occasionally a trajectory enters a CBTT and then leaves it and enters the chaotic sea (Figs. \[Cascades\] (a) and (b)). With the decreases in $\alpha$ the relative time trajectories spend in CBTT increases. CBTT are barely distinguishable near the cusp (Fig. \[Cascades\](c)) and trajectories spend relatively little time in CBTT. A trajectory enters a CBTT after a few iterations and stays there over the longest computational time we were running our codes - 500000 iterations when $\alpha$ is close to one.
The CBTT in Fig. \[Cascades\] were obtained for the RL-standard map. In many cases it is difficult to find CBTT in phase space of the Caputo-standard map but they look almost the same for both fractional maps on the $x$ vs. $n$ plot (see Fig. \[Cascades\](b)).
Results of numerical simulations submitted for publication show that not CBTT but inverse (in time) CBTT, are present within the CBTT band (from the $T=2$ $\rightarrow$ $T=4$ transition to the transition to chaos) of the fractional logistic maps.
### More Fractional Attractors {#MoreAttractors}
In the one-dimensional standard map with $K>0$ the “proper” chaotic attractor exists for $3.532<K< 4.603339$ (see Sec. \[1DStLog\]). This is the interval between the upper boundary of the CBTT band for $\alpha=1$ and $K=K_{c3}(1)$ in Fig. \[figBif\]. In the area between $K=K_{c3}(\alpha)$ curve and the upper border of the CBTT band (in Fig. \[figBif\]) the fractional chaotic attractors are proper (see Fig. \[proper\](a)) and above $K=K_{c3}(\alpha)$ the entire phase space is chaotic (Fig. \[proper\](b)).
![\[proper\] “Proper” and “improper” attractors in the RL-standard map. 3000 iterations on ten trajectories with the initial conditions $x_0=0$, $p_0=0.001+1.65i$, $i=0,1,...9$: (a). A “proper” chaotic attractor for $K=4.2$, $\alpha=1.1$. (b). An “improper” chaotic attractor for $K=4.4$, $\alpha=1.1$. ](Fig18ME.jpg){width="1.\textwidth"}
The standard map has a set of bands for $K$ above $2\pi n$ of the accelerator mode sticky islands in which momentum increases proportionally to the number of iterations $n$ and coordinate increases as $n^2$ (see Sec. \[2DStLog\]). In the one-dimensional standard map the corresponding bands demonstrate cascades of bifurcations (see Fig. \[figBif\](b)) for $|K|$ above $2\pi |n| $. The acceleration in those bands is zero and $x$ increases proportionally to $n$ (see Sec. \[1DStLog\]).
![\[Ballistic\] RL-standard map’s accelerator mode attractors. 25000 iterations on a single trajectory with the initial conditions $x_0=0$, $p_0=0.1$: (a). CBTT-type accelerator mode attractor for $K=5.7$, $\alpha=1.03$. (b). Accelerator mode attractor for $K=7.6$, $\alpha=1.97$. ](Fig19ME.jpg){width="1.\textwidth"}
Accelerator mode attractors in the case $1<\alpha<2$ are not fully investigated. The standard map’s accelerator mode islands evolve into the accelerator mode (ballistic) attracting sticky trajectories when $\alpha$ is reduced from $2$ for the values of $K$ which increase with the decrease in $\alpha$ (Fig. \[Ballistic\](b)). When the value of $\alpha$ increases from 1, the corresponding ballistic attractors evolve into the cascade of bifurcation type ballistic trajectories (see Fig. \[Ballistic\](a)) for the values of $K$ which decrease with the increase in $\alpha$. This could mean that corresponding features in the one- and two-dimensional maps (at least for $K=2\pi$) are not connected by the continuous change in $\alpha$.
$\alpha$-Families of Maps ($2<\alpha<3$) {#BN23}
----------------------------------------
Fractional maps for $\alpha>2$ are not yet investigated. Here we’ll present the first results [@DNC] for the RL-standard map.
![\[RL2\_3D\] RL-standard map for $2< \alpha <3$: (a). 3D phase space for $K=1$, $\alpha=2.01$ obtained on a single trajectory with $x_0=p_0=0$ and $p^1_0=0.01$. (b). Projection of the phase space in (a) on the $x$-$y$ plane. (c). Projection of the phase space for $K=0.2$, $\alpha=2.01$, $x_0=p_0=0$ on the $x$-$y$ plane obtained using 20 trajectories with different initial values of $p^1_0$. (d). The same as in (c), but for $K=4$ and $\alpha=2.9$. ](Fig20ME.jpg){width="1.\textwidth"}
With $G_K(x)=K \sin (x)$ in Eqs. (\[FrRLMapx\]) and (\[FrRLMapp\]), the RL-standard map for $2< \alpha \le 3$ can be written as $$\begin{aligned}
&&p^1_{n+1}= p^1_n-K\sin(x_n), \nonumber
\label{SMRLalp2n3p1} \\
&&p_{n+1}= p^1_n +p_n-K\sin(x_n), \ \ ({\rm mod} \ 2\pi ),
\label{SMRLMalp2n3p} \\
&&x_{n+1}=\frac{p_0}{\Gamma(\alpha-1)}(n+1)^{\alpha-2}+\frac{1}{\Gamma(\alpha)}\sum^{n}_{k=0}
p^1_{k+1}V^1_{\alpha}(n-k+1), \ \ ({\rm mod} \ 2\pi ). \nonumber
\label{SMRLalp2n3x} \end{aligned}$$ In our simulations we did not find a stable fixed point even for small values of $K$ (see Fig. \[RL2\_3D\] (c)). Simulations show that for this map there are attractors in the form of the attracting multi-period lines with constant $x$ (see Fig. \[RL2\_3D\] (a), (b), and (d)). For most of the values of the map parameters the phase space is highly chaotic.
This case and the transition from the 2D standard map to the 3D standard map is not yet fully investigated.
Conclusion {#Conclusion}
==========
The systems with long-term memory that are most frequently encountered in nature are systems with power-law memory. In many applications, including biological applications, the exponent in power law, $\sim t^{-\beta}$, is $0<\beta<1$. This is true, in particular, for adaptive systems and for viscoelastic properties of human tissues. These systems can be described by nonlinear fractional differential equations with fractional derivatives of the order $\alpha=1-\beta$ with $0<\alpha<1$. Fractional differential equations can be modeled by discrete nonlinear maps with power-law memory. We studied maps which model fractional differential equations with $0<\alpha<2$ and, correspondingly, $-1<\beta<1$. Decrease in $\beta$ and, correspondingly, increase in $\alpha$ means an increase in the memory effects - older states have higher weights in the definition of the present state of a system.
In Sec. \[AFM\] we showed that an increase in memory effects leads to more complicated and chaotic behavior. As can be seen in Fig. \[LowAlpBif\], systems with small $\alpha$ are more stable. At the values of system parameters, corresponding to the periodic behavior and transition to chaos, behavior of such systems follows a well defined cascade of bifurcations pattern Fig. \[CBTT1D\]. This type of evolution may mean a slow adaptation when a system changes its state long after a change in a parameter occurred.
Increase in memory effects with the transition from $0<\alpha<1$ to $1<\alpha<2$ leads to increased diversity in systems’ behavior. Systems with $1<\alpha<2$ may demonstrate periodic sinks, attracting slow diverging trajectories, attracting accelerator mode trajectories, chaotic attractors, and cascade of bifurcations and inverse cascade of bifurcations type attracting trajectories. An intermittent cascade of bifurcations type behavior (Figs. \[Cascades\] (a) and (b)) may correspond to a scenario of the evolution of chronic diseases, to some mental disorders, or to the evolution of some social systems.
The way in which systems with power-law memory approach fixed and periodic points (Eqs. (\[FastConv\])-(\[CaputoConv\])) can be used to identify systems with memory in an analysis of experimental data.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author expresses his gratitude to V. E. Tarasov for useful discussions, to E. Hameiri and H. Weitzner for the opportunity to complete this work at the Courant Institute, and to V. Donnelly for technical help.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we provide a much simplified proof of the main result in [@Lin12] concerning the global existence and uniqueness of smooth solutions to the Cauchy problem for a 3D incompressible complex fluid model under the assumption that the initial data are close to some equilibrium states. Beside the classical energy method, the interpolating inequalities and the algebraic structure of the equations coming from the incompressibility of the fluid are crucial in our arguments. We combine the energy estimates with the $L^\infty$ estimates for time slices to deduce the key $L^1$ in time estimates. The latter is responsible for the global in time existence.'
author:
- |
Fanghua Lin$^1$, Ting Zhang$^2$\
1. Courant Institute, New York University, New York, NY 10012, USA\
2. Department of Mathematics, Zhejiang University, Hangzhou 310027, China
title: Global Small Solutions to a Complex Fluid Model in 3D
---
Introduction.
=============
In this paper, we consider the global existence of classical solutions to the following simple model for a complex fluid flows $$\left\{
\begin{array}{l}
\partial_t \phi +v\cdot\nabla \phi=0,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^3,\\
\partial_t v+v\cdot\nabla v-\Delta v+\nabla p=-\mathrm{div}[\nabla \phi\otimes\nabla \phi],\\
\mathrm{div}v=0,\\
(\phi,v)|_{t=0}=(\phi_0,v_0).
\end{array}
\right.\label{m0-E1.1-1111}$$ Here $\phi$, $v=(v_1,v_2,v_3)^\top$ and $p$ denote the scalar potential, velocity field and scalar pressure of the fluid respectively. As in [@Lin12], we shall consider the initial data $(\phi_0,v_0)$ is close to a non-trivial equilibrium, e.g. $(\phi_0,v_0)\simeq(x_3,(0,0,0)^\top)$. Note that instead of $x_3$, any non-constant linear functions would work as well by our method (cf.[@Lin12] for explanations). What would be important is that $\nabla \phi_0$ is close to a constant non-zero vector field in a suitable way. Thus, we may write $(\phi,v)=(x_3+\psi,v)$, and substitute it into (\[m0-E1.1-1111\]) with $x\in \mathbb{R}^3$, to obtain the following equivalent system for $(\psi,v)$, $$\left\{
\begin{array}{l}
\partial_t \psi +v\cdot\nabla \psi+ v_3=0,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^3,\\
\partial_t v_h+v\cdot\nabla v_h -\Delta v_h+\nabla_h p+ \nabla_h\partial_3\psi=
-\mathrm{div}[\nabla_h \psi\otimes\nabla \psi],\\
\partial_t v_3+v\cdot\nabla v_3 -\Delta v_3+\partial_3 p+ (\Delta+\partial_3^2)\psi=
-\mathrm{div}[\partial_3 \psi \nabla \psi],\\
\mathrm{div}v=0,\\
(\psi,v)|_{t=0}=(\psi_0,v_0),
\end{array}
\right.\label{m0-E1.5-N3}$$ where $v_h=(v_1,v_2)^\top$, $\nabla_h=(\partial_1,\partial_2)^\top$. For the rest of the paper we shall work on the equations (\[m0-E1.5-N3\]).
We remark that the nonlinear hyperbolic-parabolic system (\[m0-E1.1-1111\]) has been used for describing many fluid dynamic models, see [@Constantin; @Constantin2; @Lin12-3]. Indeed, when $\phi=(\phi_1,\phi_2)^\top$ is a vector-valued function on $\mathbb{R}^+\times\mathbb{R}^2$ with $\det(\nabla\phi)=1$, the system (\[m0-E1.1-1111\]) is equivalent to the well-known Oldroyd-B model for viscoelastic fluids equations, see [@Saut; @Larson; @Lin12-2; @Renardy]. It is also closely related to the evolution equation of nematic liquid crystal as well as the diffusive sharp interface motion and immersed boundary in flow fields [@Peskin], see also the recent survey article [@Lin12-3]. In [@Lin12], authors used the system (\[m0-E1.1-1111\]) as a toy model for the 3D incompressible viscous and non-resistive MHD system.
In fact, the system (\[m0-E1.1-1111\]) is exactly the incompressible MHD equations with zero magnetic diffusion when the space dimension is two. Recall that the 2D incompressible MHD system reads, $$\left\{
\begin{array}{l}
\partial_t b+v\cdot\nabla b-\eta\Delta b=b\cdot\nabla v,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^2,\\
\partial_t v+v\cdot\nabla v-\nu \Delta v+\nabla p=b\cdot\nabla b,\\
\mathrm{div}v=\mathrm{div}b=0,\\
b|_{t=0}=b_0,\ v|_{t=0}=v_0,
\end{array}
\right.\label{m0-E1.2}$$ where $b=(b_1,b_2)^{\top}$, $v=(v_1,v_2)^{\top}$ and $p$ denote the magnetic field, velocity field and scalar pressure of the fluid respectively. In (\[m0-E1.2\]), the condition $\mathrm{div}b=0$ implies the existence of a scalar function $\phi$ such that $b=(\partial_2\phi,-\partial_1\phi)^\top$, and the corresponding system becomes the following 2D incompressible MHD type system, $$\left\{
\begin{array}{l}
\partial_t \phi +v\cdot\nabla \phi-\eta\Delta\phi=0,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^2,\\
\partial_t v+v\cdot\nabla v-\Delta v+\nabla p=-\mathrm{div}[\nabla \phi\otimes\nabla \phi],\\
\mathrm{div}v=0,\\
(\phi,v)|_{t=0}=(\phi_0,v_0).
\end{array}
\right.$$ There are some global wellposedness results for the system (\[m0-E1.2\]), see [@Duvaut; @Sermange] for the case when $\eta>0$ and $\nu>0$, [@Cao] for the case when $\eta>0$ and $\nu=0$, as well as the case with mixed partial dissipation and additional (artificial) magnetic diffusion. In [@Bardos], authored considered the case when $\eta=\nu=0$ and the initial data $(b_0,v_0)$ close to the equilibrium state $(B_0,0)$. In [@Lin12-2], the case when $\eta=0$, $\nu>0$ was studied. Under the assumption that the initial data $(b_0,v_0)$ is close to the equilibrium state $((1,0)^\top,0)$, the global wellposedness was proven. We should note that authors observed in [@Bardos] that the fluctuations $v+b-B_0$ and $v-b+B_0$ propagate along the $B_0$ magnetic field in opposite directions. Thus, a strong enough magnetic field will reduce the nonlinear interactions and prevent formations of strong gradients [@Bardos; @Frisch; @Kraichnan]. Unfortunately, the method applied in [@Bardos] is purely hyperbolic (characteristic method) and hence could not be applied in our case.
In [@Lin12], using the anisotropic Littlewood-Paley analysis, the first author and Ping Zhang [@Lin12] proved a global wellposedness result of the system (\[m0-E1.1-1111\]). The arguments involved, despite its general interests, were rather complicated. The aim of this note is to give a new and simple proof, which involves only the energy estimate method, interpolating inequalities and couple elementary observations.
\[m0-Thm1.2\] Assume that the initial data $(\psi_0,v_0)$ satisfy $
(\nabla \psi_0,v_0)\in
H^2(\mathbb{R}^3)\times H^2(\mathbb{R}^3),
$ $\mathrm{div}v_0=0$, then there exists a positive constant $c_0$ such that if $$B_0=\|\nabla \psi_0\|_{H^2}+
\|v_0\|_{H^2}\leq c_0,\label{1.6}$$ then the system (\[m0-E1.5-N3\]) has a unique global solution $(\psi,v,\nabla p)\in F^2$ satisfying $$\begin{aligned}
B_T^2=\|v\|^2_{L^\infty([0,T]; H^2 )}+\|\nabla \psi\|^2_{L^\infty([0,T];H^2)}
+\|\nabla v\|^2_{L^2([0,T]; H^2 )}+\|\nabla_h\nabla \psi\|^2_{L^2([0,T]; H^1 )}
\leq CB_{0}^2,\label{1.7}
\end{aligned}$$ and $$\|\nabla p\|_{L^\infty([0,T]; H^1 )}
\leq C B_0,\label{1.8}$$ for all $T>0$, where $C$ is a positive constant independent of $T$, $$F^n= \left\{ (\psi,v,\nabla p)\left| \begin{array}{l}
(\nabla \psi,v,\nabla p)\in C([0,\infty);H^n\times H^n)\times C([0,\infty); H^{n- 1}), \\
(\nabla_h\nabla \psi, \nabla v)
\in L^2([0,\infty);H^{n- 1}\times H^n).
\end{array}
\right.
\right\}$$
We note that under the assumptions of Theorem \[m0-Thm1.2\], if $(\nabla
\psi_0,v_0)\in H^n(\mathbb{R}^3)\times H^n(\mathbb{R}^3)$, $n\geq3$, then we can easily obtain that $(\psi,v,\nabla p)\in F^n$ and omit the details.
There are three key technical points in our proofs:
(1)
: interpolating estimates, see for example, Lemma \[m0-L2.2\];
(2)
: using the algebraic structure: $\mathrm{div} v =0$ to inter-changing the estimates for the vertical ($\partial_3$) and the horizontal ($\nabla_h$) derivatives;
(3)
: using the first equation of (\[m0-E1.5-N3\]) to reduce “$L^1$ in time estimates” ([@Lin12]) which is the key to the global existence result to “energy estimates and $L^\infty$ estimates for time slices” that are relatively easy to obtain.
In fact, the basic strategy for the proofs is rather clear. Using the basic energy laws, one reduces the problems to estimating certain terms of particular forms. For example, one of the difficulties of the proofs would be to control the following type term, $$\int^T_0\int_{\mathbb{R}^3} \partial_3 v_3 (\partial_3^3\psi)^2 dxdt.$$ Since the horizontal derivatives of $\psi$, $\nabla_h\psi$ decay faster than $\partial_3\psi$ (by energy laws), in [@Lin12] authors explored such anisotropic behavior by using the anisotropic Littlewood-Paley theory to conclude the key estimate that $v_3\in L^1(\mathbb{R}^+;Lip(\mathbb{R}^3))$. Here we will show that first by interpolating inequalities $\nabla
\psi\in L^4_T(L^{ \infty})$ in Lemma \[m0-L2.2\]. Then we use the first equation of (\[m0-E1.5-N3\])$_1$ twice and proceed the estimates as follows: $$\begin{aligned}
&&\left| \int^T_0\int_{\mathbb{R}^3} \partial_3 v_3 (\partial_3^3\psi)^2 dxdt\right|\nonumber\\
&=&\left|\int^T_0\int_{\mathbb{R}^3}
\partial_3(\partial_t\psi+v\cdot\nabla\psi )(\partial_3^3\psi)^2 dxdt\right|\nonumber\\
&\leq&\left|\int_{\mathbb{R}^3}
\partial_3 \psi(\partial_3^3\psi)^2 dx\big|^T_0\right|+\left|\int^T_0\int_{\mathbb{R}^3}
\partial_3 v_3\partial_3\psi(\partial_3^3\psi)^2 dxdt\right|+\ldots\nonumber\\
&=&\ldots+\left|\int^T_0\int_{\mathbb{R}^3}
\partial_3 (\partial_t\psi+v\cdot\nabla\psi ) \partial_3\psi(\partial_3^3\psi)^2 dxdt\right|+\ldots\nonumber\\
&\leq&\ldots+ C\|\nabla
\psi\|_{L^4_T(L^\infty(\mathbb{R}^3))}^2\|\nabla v\|_{L^2_T(L^2(\mathbb{R}^3))}\|\partial_3^3 \psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}^2+\ldots.\label{idea}
\end{aligned}$$ We refere the details to Lemma \[m0-L2.5\]. This is a simple idea works well for the issue concerning various anisotropic dissipative system similar to (\[m0-E1.5-N3\]).
We also note that in a recent preprint [@Lin12-2] authores embedded the system (\[m0-E1.2\]) with $\eta=0$ into a 2D viscoelastic fluid system. Then they use equations in Lagrangian coordinates and the anisotropic Littlewood-Paley analysis techniques, to obtain a global wellposedness result. One can apply the methods in this paper to obtain similar results as theirs.
The organization of this paper can be as the following: we shall present some *a priori* estimates in Section \[m0-S2\], and prove Theorem \[m0-Thm1.2\] in Section \[m0-S3\].
Let us complete this section by the notation we shall use in this paper.
**Notation.** We shall denote by $(a|b)$ the $L^2$ inner product of $a$ and $b$, and $(a|b)_{H^s}$ the standard $H^s$ inner product of $a$ and $b$. Finally, we denote $L^p_T(L^q_h(L^r_v))$ the space $L^p([0,T];L^q(\mathbb{R}_{x_1}\times\mathbb{R}_{x_{2}};L^r(\mathbb{R}_{x_3})))$, $C_T(X)$ the space $C([0,T];X)$.
*A priori* estimates {#m0-S2}
====================
In this section, we prove a set of *a priori* estimates which are crucial for the global existence of solutions for the system (\[m0-E1.5-N3\]). We begin with the following Gagliardo-Nirenberg-Sobolev type estimate, see [@Nirenberg].
\[m0-L2.2\] If the function $\psi$ satisfies that $\nabla \psi\in L^\infty_T(H^2(\mathbb{R}^3))$ and $\nabla_h\nabla \psi\in L^2_T(H^1(\mathbb{R}^3))$, then there hold $$\begin{aligned}
\|\nabla \psi\|_{L^4_T(L^4(\mathbb{R}^3))} &\leq& C \| \nabla_h\nabla \psi\|_{L^2_T(L^2(\mathbb{R}^3))}^\frac{1}{2}
\| \nabla \psi\|_{L^\infty_T(H^1(\mathbb{R}^3))}^\frac{1}{2},\label{3D-E2.3-00}
\end{aligned}$$ $$\|\nabla^2 \psi\|_{L^4_T(L^4(\mathbb{R}^3))}\leq C \| \nabla_h\nabla \psi\|_{L^2_T(H^1(\mathbb{R}^3))}^\frac{1}{2}
\| \nabla \psi\|_{L^\infty_T(H^2(\mathbb{R}^3))}^\frac{1}{2},\label{3D-E2.3-0}$$ $$\|\nabla \psi\|_{L^4_T(L^{\infty}(\mathbb{R}^3))}\leq C \| \nabla_h\nabla \psi\|_{L^2_T(H^1(\mathbb{R}^3))}^\frac{1}{2}
\| \nabla \psi\|_{L^\infty_T(H^2(\mathbb{R}^3))}^\frac{1}{2},\label{m0-E2.2}$$ where $C$ is a positive constant independent of $T$.
Using Gagliardo-Nirenberg-Sobolev’s inequality and Minkowski’s inequality, we obtain $$\begin{aligned}
\|\nabla \psi\|_{L^4_T(L^4(\mathbb{R}^3))}&\leq& C
\left\|
\|\nabla \psi\|_{L^2_h}^\frac{1}{2}
\|\nabla \nabla_h \psi\|_{L^2_h}^\frac{1}{2}
\right\|_{L^4_T(L^4_v)}\nonumber\\
&\leq&C
\|\nabla \psi\|_{L^\infty_T(L^\infty_v(L^2_h))}^\frac{1}{2}
\|\nabla \nabla_h \psi\|_{L^2(L^2(\mathbb{R}^3))}^\frac{1}{2}\nonumber\\
&\leq&C
\|\nabla \psi\|_{L^\infty_T(L^2_h(L^\infty_v))}^\frac{1}{2}
\|\nabla \nabla_h \psi\|_{L^2(L^2(\mathbb{R}^3))}^\frac{1}{2}\nonumber\\
&\leq&C
\left\|
\|\nabla \psi\|_{L^2_v}^\frac{1}{2}
\|\nabla \partial_3\psi\|_{L^2_v}^\frac{1}{2}
\right\|_{L^\infty_T(L^2_h)}^\frac{1}{2}
\|\nabla \nabla_h \psi\|_{L^2(L^2(\mathbb{R}^3))}^\frac{1}{2}\nonumber\\
&\leq&C
\|\nabla \psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}^\frac{1}{4}
\|\nabla \partial_3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}^\frac{1}{4}
\|\nabla \nabla_h \psi\|_{L^2_T(L^2(\mathbb{R}^3))}^\frac{1}{2}\nonumber\\
&\leq& C \| \nabla_h\nabla \psi\|_{L^2_T(L^2(\mathbb{R}^3))}^\frac{1}{2}
\| \nabla \psi\|_{L^\infty_T(H^1(\mathbb{R}^3))}^\frac{1}{2}.
\end{aligned}$$ This proves (\[3D-E2.3-00\]). Inequality (\[3D-E2.3-0\]) is a direct consequence of (\[3D-E2.3-00\]). Combining (\[3D-E2.3-00\])-(\[3D-E2.3-0\]) with Gagliardo-Nirenberg-Sobolev’s inequality again, we obtain (\[m0-E2.2\]).
By taking divergence of the $v$ equation of (\[m0-E1.5-N3\]), we can express the pressure function $p$ via $$p=-2 \partial_3 \psi +\sum_{i,j=1}^3(-\Delta)^{-1}
[\partial_iv_j\partial_j v_i+\partial_i\partial_j(\partial_i \psi\partial_j \psi)
].\label{m0-E1.6}$$ As in [@Lin12], we substitute (\[m0-E1.6\]) into (\[m0-E1.5-N3\]) to obtain $$\left\{
\begin{array}{l}
\partial_t \psi +v\cdot\nabla \psi + v_3=0,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^3,\\
\partial_t v_h+v\cdot\nabla v_h -\Delta v_h - \nabla_h\partial_3\psi =f^h\\
\ \ \ \ \ \ \ \:=-\displaystyle{\sum_{i,j=1}^3}\nabla_h(-\Delta)^{-1}
[\partial_iv_j\partial_j v_i+\partial_i\partial_j(\partial_i \psi\partial_j \psi)
]-\displaystyle{\sum_{j=1}^3}\partial_j[\nabla_h \psi\partial_j \psi],\\
\partial_t v_3+v\cdot\nabla v_3 -\Delta v_3 + \Delta_h\psi =f^v\\
\ \ \ \ \ \ \ \:=-\displaystyle{\sum_{i,j=1}^3}\partial_3(-\Delta)^{-1}
[\partial_iv_j\partial_j v_i+\partial_i\partial_j(\partial_i \psi\partial_j \psi)
]-\displaystyle{\sum_{j=1}^3}\partial_j[\partial_3 \psi\partial_j \psi],\\
\mathrm{div}v=0,\\
(\psi,v)|_{t=0}=(\psi_0,v_0).
\end{array}
\right.\label{m0-E1.7}$$ Here, $\Delta_h=\partial_{x_1}^2+\partial_{x_{2}}^2$.
The next Lemma is a standard energy estimate.
\[m0-L2.6\] Let $(\psi,v)$ be sufficiently smooth functions which solve (\[m0-E1.5-N3\]), then there holds $$\begin{aligned}
&&\frac{d}{dt}\left\{
\frac{1}{2}\left(
\|v\|_{H^2}^2+\|\nabla \psi\|_{H^2}^2+\frac{1}{4 }\|\Delta\psi\|_{H^1}^2
\right)+\frac{1}{4}(v_3|\Delta\psi)_{H^1}
\right\} \label{m0-L2.6-111} \\
&&+\|\nabla v \|_{H^2}^2-\frac{1 }{4}\|\nabla v_3\|_{H^1}^2
+\frac{1 }{4}\|\nabla\nabla_h\psi\|_{H^1}^2\nonumber\\
&=&-(v\cdot\nabla v|v)_{H^2}
+(v\cdot\nabla\psi|\Delta\psi)_{H^2}
-(\mathrm{div}(\nabla\psi\otimes\nabla\psi)|v)_{H^2}
-\frac{1}{4}(v\cdot\nabla v_3|\Delta\psi)_{H^1}
\nonumber\\
&&
+\frac{1}{4}(f^v|\Delta\psi)_{H^1}
+\frac{1}{4}(\nabla v_3|\nabla (v\cdot\nabla \psi))_{H^1}
-\frac{1}{4 }(\Delta(v\cdot\nabla\psi)|\Delta\psi)_{H^1} .
\nonumber
\end{aligned}$$
Taking the standard $H^2$ inner product of (\[m0-E1.5-N3\])$_{2,3}$ with $v$ and then using the integration by parts, we have $$\begin{aligned}
&&\frac{1}{2}\frac{d}{dt}\|v\|_{H^2}^2
+(v\cdot\nabla v|v)_{H^2}
+\|\nabla v\|_{H^2}^2\nonumber\\
&=&-( \nabla_h\partial_3\psi|v_h)_{H^2}
-( (\Delta+\partial_3^2)\psi|v_3)_{H^2}
-(\mathrm{div}(\nabla\psi\otimes\nabla \psi)|v)_{H^2}.\label{m0-E2.11}
\end{aligned}$$ Since $\mathrm{div}v=0$, the integration by parts gives $$\begin{aligned}
& &-( \nabla_h\partial_3\psi|v_h)_{H^2}
= ( \partial_3 \psi|\mathrm{div}_h v_h)_{H^2}\nonumber\\
& =& -( \partial_3 \psi|\partial_3v_3)_{H^2}
= ( \partial_3^2 \psi| v_3)_{H^2}.
\end{aligned}$$ From (\[m0-E1.5-N3\])$_1$, we have $$\begin{aligned}
& &-( \Delta\psi|v_3)_{H^2}
= ( \Delta \psi|(\partial_t\psi+v\cdot\nabla \psi ))_{H^2}\nonumber\\
& =&-\frac{1}{2}\frac{d}{dt}\|\nabla\psi\|_{H^2}^2+
( \Delta \psi| v\cdot\nabla \psi )_{H^2} .\label{m0-E2.13}
\end{aligned}$$ Combining (\[m0-E2.11\])-(\[m0-E2.13\]), we deduce that $$\begin{aligned}
&&\frac{1}{2}\frac{d}{dt}\left(\|v\|_{H^2}^2
+\|\nabla\psi\|_{H^2}^2\right)
+\|\nabla v\|_{H^2}^2\nonumber\\
&=& -(v\cdot\nabla v|v)_{H^2}
+
( \Delta \psi| v\cdot\nabla \psi )_{H^2}
-(\mathrm{div}(\nabla\psi\otimes\nabla \psi)|v)_{H^2}.\label{m0-E2.14}
\end{aligned}$$ Next we take the standard $H^1$ inner product of (\[m0-E1.7\])$_{3}$ with $\Delta\psi$, using again the integration by parts, to obtain $$\begin{aligned}
(\partial_tv_3|\Delta\psi)_{H^1}+(v\cdot\nabla v_3|\Delta\psi)_{H^1}
-(\Delta v_3|\Delta\psi)_{H^1}
&=& - \|\nabla_h\nabla \psi\|_{H^1}^2
+(f^v|\Delta\psi)_{H^1} .\label{m0-E2.15}
\end{aligned}$$ From the equation (\[m0-E1.5-N3\])$_1$ and again the integration by parts, we get $$\begin{aligned}
&&(\partial_tv_3|\Delta\psi)_{H^1}\nonumber\\
&=&\frac{d}{dt}(v_3|\Delta\psi)_{H^1}
-(v_3|\Delta\partial_t\psi)_{H^1}\nonumber\\
&=&\frac{d}{dt}(v_3|\Delta\psi)_{H^1}
+(v_3|\Delta(v\cdot\nabla\psi + v_3))_{H^1}\nonumber\\
&=&\frac{d}{dt}(v_3|\Delta\psi)_{H^1}
-(\nabla v_3|\nabla(v\cdot\nabla\psi))_{H^1}
-\|\nabla v_3\|_{H^1}^2.
\end{aligned}$$ We observe, by (\[m0-E1.5-N3\])$_1$, that $$\begin{aligned}
&&-(\Delta v_3|\Delta\psi)_{H^1}\nonumber\\
&=&(\Delta(\partial_t\psi+v\cdot\nabla\psi )
|\Delta\psi)_{H^1}\nonumber\\
&=&\frac{1}{2 }\frac{d}{dt}\|\Delta\psi\|_{H^1}^2
+(\Delta( v\cdot\nabla\psi )
|\Delta\psi)_{H^1}.\label{m0-E2.17}
\end{aligned}$$ Combining (\[m0-E2.15\])-(\[m0-E2.17\]), we hence conclude $$\begin{aligned}
&&\frac{d}{dt}\left\{
\frac{1}{2}\|\Delta\psi\|_{H^1}^2+(v_3|\Delta\psi)_{H^1}
\right\}+ \|\nabla_h\nabla\psi\|_{H^1}^2- \|\nabla v_3\|_{H^1}^2\nonumber\\
&=&-(v\cdot\nabla v_3|\Delta\psi)_{H^1}
+(f^v|\Delta\psi)_{H^1}+(\nabla v_3|\nabla(v\cdot\nabla \psi))_{H^1}
-(\Delta(v\cdot\nabla\psi)|\Delta\psi) .\label{m0-E2.18}
\end{aligned}$$ With (\[m0-E2.14\]) and (\[m0-E2.18\]), one can complete the proof.
The following is the key *a priori* estimate which is essential to the proof of the main result of this paper.
\[m0-L2.3\] Let $(\psi,v)$ be sufficiently smooth functions which solve (\[m0-E1.5-N3\]) and satisfy $\nabla \psi\in L^\infty_T(H^2(\mathbb{R}^3))$, $\nabla_h\nabla \psi\in L^2_T(H^1(\mathbb{R}^3))$, $v\in L^\infty_T(H^2(\mathbb{R}^3))$ and $\nabla v\in L^2_T(H^2(\mathbb{R}^3))$, then there holds $$B_T^2 \leq C
(\|v_0\|_{H^2(\mathbb{R}^3)}^2+\|\nabla\psi_0\|_{H^2(\mathbb{R}^3)}^2)
+ C B_T^3(1+ B_T)^2.\label{m0-E2.19}$$ where $C$ is a positive constant independent of $T$.
By the energy estimate (\[m0-L2.6-111\]) and the definition of $ B_T$, we get for a positive constant $C$ (independent of $T$) that $$\begin{aligned}
B_T^2&\leq&CB_0^2+C\left|\int^T_0(v\cdot\nabla v|v)_{H^2}dt\right|
+C\left|\int^T_0(v\cdot\nabla\psi|\Delta\psi)_{H^2}dt\right|
\nonumber\\
&&
+C\left|\int^T_0(\mathrm{div}(\nabla\psi\otimes\nabla\psi)|v)_{H^2}dt\right| +C\left|\int^T_0(v\cdot\nabla v_3|\Delta\psi)_{H^1}dt\right|
+C\left|\int^T_0(f^v|\Delta\psi)_{H^1}dt\right|\nonumber\\
&&
+C\left|\int^T_0(\nabla v_3|\nabla (v\cdot\nabla \psi))_{H^1}dt\right|
+C\left|\int^T_0(\Delta(v\cdot\nabla\psi)|\Delta\psi)_{H^1} dt\right|\nonumber\\
&:=&CB_0^2+\sum_{j=1}^7I_j.\label{m0-E2.18-00}
\end{aligned}$$
We are going to estimate term by term the right hand side of the inequality. The basic strategies involved in estimating all such quantities are the same. More precisely, we estimate separately terms involving horizontal derivatives and terms with vertical derivatives. For terms with horizontal derivatives $\nabla_h \psi$, one can use the dissipations implied by the energy equality (\[m0-L2.6-111\]). For terms containing vertical derivatives, we use the algebriac relation (deduced from that $\mathrm{div}v=0$) and the transport equations. The latter reduces space-time estimates to bounds on time-slices and terms with either horizontal derivatives or of higher order nonlinearities (hence they are smaller under our smallness assumptions on the initial data). To illustrate the basic idea, we start with the second term $I_2$. Applying the Gagliardo-Nirenberg-Sobolev type estimates in Lemma \[m0-L2.2\], and use the fact that $\mathrm{div}v=0$, Hölder and Sobolev inequalities, we deduce that $$\begin{aligned}
&& I_2=C\left|\int^T_0(v\cdot\nabla\psi|\Delta\psi)_{H^2}dt\right|\nonumber\\
&=&C\left| \sum_{|\alpha|\leq2}\sum_{i=1}^3\int^T_0\int
[\partial^\alpha\partial_i(v\cdot\nabla\psi)-v\cdot\nabla\partial^\alpha\partial_i\psi]\partial^\alpha\partial_i\psi dxdt
\right|
\nonumber\\
&\leq& C\|\nabla v\|_{L^2_T(L^2(\mathbb{R}^3))}\|\nabla \psi\|_{L^4_T(L^4(\mathbb{R}^3))}^2
+C\|\nabla^2\psi\|_{L^4_T(L^4(\mathbb{R}^3))}(\|\nabla^2 v\|_{L^2_T(L^2(\mathbb{R}^3))}\|\nabla\psi\|_{L^4_T(L^4(\mathbb{R}^3))}\nonumber\\
&&
+\|\nabla v\|_{L^2_T(L^2(\mathbb{R}^3))}\|\nabla^2\psi\|_{L^4_T(L^4(\mathbb{R}^3))})
+C\|\nabla_h\nabla^2\psi\|_{L^2_T(L^2(\mathbb{R}^3))}(\|\nabla^3 v\|_{L^2_T(L^2(\mathbb{R}^3))}\|\nabla\psi\|_{L^\infty_T(L^\infty(\mathbb{R}^3))}
\nonumber\\
&& +\|\nabla^2 v\|_{L^2_T(L^4(\mathbb{R}^3))}\|\nabla^2\psi\|_{L^\infty_T(L^4(\mathbb{R}^3))}
+\|\nabla v\|_{L^2_T(L^\infty(\mathbb{R}^3))}\|\nabla^3 \psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}
)\nonumber\\
&&+C\|\nabla^3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}
(\|\nabla^3 v_h\|_{L^2_T(L^2(\mathbb{R}^3))}\|\nabla_h\psi\|_{L^2_T(L^\infty(\mathbb{R}^3))}
+\|\nabla^2 v_h\|_{L^2_T(L^4(\mathbb{R}^3))}\|\nabla\nabla_h\psi\|_{L^2_T(L^4(\mathbb{R}^3))}\nonumber\\
&&
+\|\nabla v_h\|_{L^2_T(L^\infty(\mathbb{R}^3))}\|\nabla^2\nabla_h \psi\|_{L^2_T(L^2(\mathbb{R}^3))}
)\nonumber\\
&&+C\left|\int^T_0\int \partial_3^3\psi(\partial_3^3v_3\partial_3\psi+
3\partial_3^2v_3\partial_3^2\psi
+3\partial_3v_3\partial_3^3\psi)dxdt\right|. \label{m0-E2.19-0}
\end{aligned}$$ To estimate the last term in the above inequality (\[m0-E2.19-0\]), we need the following two technical lemmas. The proofs of these two Lemmas will be given in the Appendix.
\[m0-L2.4\] Under the conditions in Lemma \[m0-L2.3\], then there holds $$\begin{aligned}
&& \left|\int^T_0\int_{\mathbb{R}^3} \partial_3\psi \partial_3^3\psi\partial_3^3 v_3 dxdt
\right|
+\left|\int^T_0\int_{\mathbb{R}^3} \partial_3^2\psi \partial_3^3\psi\partial_3^2 v_3 dxdt
\right|
+\left|\int^T_0\int_{\mathbb{R}^3} \partial_3 \psi \partial_3^3\psi\partial_3^2 v_3\partial_3^2\psi dxdt
\right|
\nonumber \\
&\leq& C\|\nabla\psi\|_{L^\infty_T(H^2(\mathbb{R}^3))}\|\nabla_h\nabla\psi\|_{L^2_T(H^1(\mathbb{R}^3))}
\|\nabla v\|_{L^2_T(H^2(\mathbb{R}^3))}(1+\|\nabla\psi\|_{L^\infty_T(H^2(\mathbb{R}^3))}),\label{m0-E2.7}
\end{aligned}$$ where $C$ is a positive constant independent of $T$.
\[m0-L2.5\] Under the conditions in Lemma \[m0-L2.3\], then there holds $$\begin{aligned}
\left|\int^T_0\int_{\mathbb{R}^3} \partial_3 v_3(\partial_3^3\psi)^2 dxdt
\right|
&\leq& C B_T^3(1+B_T^2),
\end{aligned}$$ where $C$ is a positive constant independent of $T$.
Accepting these two lemmas, we proceed with our proof of the key estimate in Lemma \[m0-L2.3\]. By (\[m0-E2.19-0\]) and Lemmas \[m0-L2.4\]-\[m0-L2.5\], we obtain $$\begin{aligned}
I_2=C\left|\int^T_0(v\cdot\nabla\psi|\Delta\psi)_{H^2}dt\right|
&\leq& C B_T^3(1 +B_T)^2. \label{m0-E2.20}
\end{aligned}$$ Similarly, one can estimate that $$\begin{aligned}
I_7&=&C\left|\int^T_0(\Delta(v\cdot\nabla\psi)|\Delta\psi)_{H^1}dt\right|\nonumber\\
&=& C\left| \sum_{i=1}^3
\int^T_0 (\partial_i(v\cdot\nabla\psi)|
\partial_i\Delta\psi)_{H^1}dt\right| \nonumber\\
&\leq& C B_T^3(1+ B_T)^2.
\end{aligned}$$ Next, we estimate $I_3$ as follows: $$\begin{aligned}
I_3&=&C\left|\int^T_0(\mathrm{div}(\nabla\psi\otimes\nabla\psi)|v)_{H^2}dt\right|\nonumber\\
&=&C\left| \sum_{|\alpha|\leq2}\sum_{i,j=1}^3\int^T_0\int
\partial^\alpha(\partial_i\psi\partial_j\psi)\partial^\alpha\partial_i v_j dxdt\right|
\nonumber\\
&\leq& C\|\nabla v\|_{L^2_T(L^2(\mathbb{R}^3))}\|\nabla\psi\|_{L^4_T(L^4(\mathbb{R}^3))}^2
+C\|\nabla^2 v\|_{L^2_T(L^2(\mathbb{R}^3))}\|\nabla\psi\|_{L^4_T(L^4(\mathbb{R}^3))}\|\nabla^2\psi\|_{L^4_T(L^4(\mathbb{R}^3))}
\nonumber\\
&&+C\|\nabla^3v\|_{L^2_T(L^2(\mathbb{R}^3))}
(\|\nabla^2\psi\|_{L^4_T(L^4(\mathbb{R}^3))}^2
+\|\nabla_h\psi\|_{L^2_T(L^\infty(\mathbb{R}^3))}\|\nabla^3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}
\nonumber\\
&&+\|\nabla\psi\|_{L^\infty_T(L^\infty(\mathbb{R}^3))}\|\nabla^2\nabla_h\psi\|_{L^2_T(L^2(\mathbb{R}^3))}
)+C\left|\int^T_0\int\partial_3^3v_3\partial_3\psi\partial_3^3\psi dxdt\right|
\nonumber\\
&\leq& C B_T^3.\label{m0-E2.23}
\end{aligned}$$ Apply the same line of arguments, one can deduce that $$\begin{aligned}
&&\left|\sum_{i,j=1}^3\int^T_0
(\partial_3(-\Delta)^{-1}(\partial_iv_j\partial_j v_i)|\Delta\psi)_{H^1}dt\right|\nonumber\\
&\leq&C\int^T_0\|(\nabla v)^2\|_{H^1(\mathbb{R}^3)}\|\nabla\psi\|_{H^1(\mathbb{R}^3)} dt\nonumber\\
&\leq& C\|\nabla
v\|_{L^2_T(H^2(\mathbb{R}^3))}^2\|\nabla\psi\|_{L^\infty_T(H^2(\mathbb{R}^3))}. \label{m0-E2.24}
\end{aligned}$$
Similarly, one has $$\begin{aligned}
&&\left|\int^T_0
\left(\left.-\sum_{i,j=1}^3\partial_3(-\Delta)^{-1}(
\partial_i\partial_j(\partial_i\psi\partial_j\psi))-
\sum_{j=1}^3\partial_j(\partial_3\psi\partial_j\psi)\right|\Delta\psi
\right)_{H^1}
dt\right|\nonumber\\
&=&\left|\int^T_0
\left(\left.-\sum_{i,j=1}^2\partial_3(-\Delta)^{-1}(
\partial_i\partial_j(\partial_i\psi\partial_j\psi)) \right|\Delta\psi
\right)_{H^1}
dt\right|\nonumber\\
&&+\left|\int^T_0
\sum_{j=1}^2 \left(\left. 2\partial_3(-\Delta)^{-1} (
\partial_3(\partial_3\psi\partial_j\psi))+
(\partial_3\psi\partial_j\psi)\right|\partial_j\Delta\psi
\right)_{H^1}
dt\right|\nonumber\\
&&+\left|\int^T_0
( - (-\Delta)^{-1}
\partial_3^3 (\partial_3\psi)^2 -
\partial_3(\partial_3\psi )^2 |\Delta\psi
)_{H^1}dt\right|\nonumber\\
&\leq& C\int^T_0
\|\nabla (\nabla_h\psi)^2\|_{H^1(\mathbb{R}^3)}\|\Delta\psi\|_{H^1(\mathbb{R}^3)}dt
+C\int^T_0
\|\nabla (\nabla \psi\nabla_h\psi) \|_{H^1(\mathbb{R}^3)}\|\nabla\nabla_h\psi\|_{H^1(\mathbb{R}^3)}dt
\nonumber\\
&&+\left|\int^T_0
( \sum_{i=1}^2 (-\Delta)^{-1}
\partial_3\partial_i^2 (\partial_3\psi)^2 |\Delta\psi
)_{H^1}dt\right|\nonumber\\
&\leq& C\|\nabla\nabla_h \psi\|_{L^2_T(H^1(\mathbb{R}^3))}^2\|\nabla\psi\|_{L^\infty_T(H^2(\mathbb{R}^3))}
+C\int^T_0\|\nabla_h(\nabla\psi)^2\|_{H^1(\mathbb{R}^3)}\|\nabla\nabla_h\psi\|_{H^1(\mathbb{R}^3)}dt\nonumber\\
&\leq& C\|\nabla\nabla_h \psi\|_{L^2_T(H^1(\mathbb{R}^3))}^2\|\nabla\psi\|_{L^\infty_T(H^2(\mathbb{R}^3))}.
\label{m0-E2.25}
\end{aligned}$$ Combining (\[m0-E2.24\])-(\[m0-E2.25\]), one concludes $$I_5=C\left| \int^T_0
(f^v|\Delta\psi)_{H^1}dt\right|\leq CB_T^3. \label{m0-E2.26}$$
One can obtain the following estimates in the same way, to save the ink, we omit the details. $$I_1=C\left| \int^T_0(v\cdot\nabla v|v)_{H^2}dt\right|\leq C\|\nabla v\|_{L^2_T(H^2(\mathbb{R}^3))}^2\|v\|_{L^\infty_T(H^2(\mathbb{R}^3))},$$ $$I_4=C\left|\int^T_0 (v\cdot\nabla v_3|\Delta\psi)_{H^1}dt\right|
\leq C\|\nabla v\|_{L^2_T(H^2(\mathbb{R}^3))}^2\|\nabla \psi\|_{L^\infty_T(H^2(\mathbb{R}^3))},$$ $$I_6=C\left| \int^T_0 (\nabla v_3|\nabla (v\cdot\nabla \psi))_{H^1}dt\right|
\leq C\|\nabla v\|_{L^2_T(H^2(\mathbb{R}^3))}^2\|\nabla \psi\|_{L^\infty_T(H^2(\mathbb{R}^3))},\label{m0-E2.34}$$ Summing up (\[m0-E2.18-00\]), (\[m0-E2.20\])-(\[m0-E2.23\]) and (\[m0-E2.26\])-(\[m0-E2.34\]), we conclude (\[m0-E2.19\]).
In several places of our proofs, we have used the fact that $$\|\nabla_h\psi\|_{L^2_T(L^\infty(\mathbb{R}^3))}\leq C \|\nabla_h\psi\|_{L^2_T(W^{1,6}(\mathbb{R}^3))}
\leq C \|\nabla\nabla_h\psi\|_{L^2_T(H^{1}(\mathbb{R}^3))},$$ which is a direct consequence of Sobolev embedding Theorem. When the spatial dimension is two, we cannot use $\|\nabla\nabla_h\psi\|_{L^2_T(H^{1}(\mathbb{R}^2))}$ to bound $\|\nabla_h\psi\|_{L^2_T(L^\infty(\mathbb{R}^2))}$. So it is necessary to make various changes in order for the proofs in this article to work in the case that the spatial dimension is two. On the other hand, if one assume that the initial data are in $H^2\times \dot{H}^{-s}(\mathbb{R}^2)$, $s\in (\frac{1}{2},1)$, then use $\|\nabla^{1+s}\nabla_h\psi\|_{L^2_T(H^{1}(\mathbb{R}^2))}$ to bound $\|\nabla_h\psi\|_{L^2_T(L^\infty(\mathbb{R}^2))}$ and obtain the similar result though the arguments are technically more complicated.
Proof of Theorem \[m0-Thm1.2\] {#m0-S3}
==============================
Via the analysis in [@Majda84], one can get the following local existence result following now the standard arguments:
\[m0-Thm3.1\] Assume that the initial data $(\psi_0,v_0)$ satisfy $(\nabla \psi_0,v_0)\in H^2(\mathbb{R}^3)\times H^2(\mathbb{R}^3)$, then there exists $T_0>0$ such that the system (\[m0-E1.5-N3\]) has a unique local solution $(\psi,v,\nabla p)$ on $[0,T_0]$ satisfying $$\nabla \psi, v\in C([0,T_0];H^2(\mathbb{R}^3)),\ \nabla v\in L^2([0,T_0];H^2(\mathbb{R}^3)), \label{m0-E3.1}$$ $$\nabla p\in L^\infty([0,T_0];H^{ 1} (\mathbb{R}^3)) .\label{m0-E3.3}$$
**Proof of Theorem \[m0-Thm1.2\].** Theorem \[m0-Thm3.1\] implies that the system (\[m0-E1.5-N3\]) has a unique local strong solution $(\psi,v,\nabla p)$ on $[0,T^*)$, where $[0,T^{*})$ is the maximal existence time interval for the above solution. Our goal is to prove $T^*=\infty$ provided that the initial data $(\psi_0,v_0)$ satisfy (\[1.6\]).
Assume that $(\psi,v,\nabla p)$ is the unique local strong solution of (\[m0-E1.5-N3\]) on $[0,T^*)$, and satisfies (\[m0-E3.1\])-(\[m0-E3.3\]). From (\[m0-E2.19\]), we have $$B_T^2 \leq C
(\|v_0\|_{H^2(\mathbb{R}^3)}^2+\|\nabla\psi_0\|_{H^2(\mathbb{R}^3)}^2)
+ C B_T^3(1+ B_T)^2,$$ for all $T\in (0,T^*)$. If the initial data $(\psi_0,v_0)$ satisfy (\[1.6\]), where $c_0$ satisfies $$C\sqrt{2C }c_0(1+ \sqrt{2C }c_0)^2\leq \frac{1}{2},$$ then one can easily obtain $$B_T^2\leq 2CB_0^2, \ \textrm{ for all }T\in (0,T^*).$$ As the right hand side of the last inequality above remains to be small, we must have that $T^*=\infty$, and hence (\[1.7\]) holds. From (\[m0-E1.6\]), we see that $\nabla p\in L^\infty([0,\infty);H^1)$ and (\[1.8\]) holds. This finishes the proof of Theorem \[m0-Thm1.2\]. [$\square$]{}
Appendix {#appendix .unnumbered}
========
Here we shall give the proofs of two technical Lemmas \[m0-L2.4\]-\[m0-L2.5\] that are needed in establishing the key *a priori* estimates. As $\mathrm{div}v=0$, we can replace $\mathrm{div}_hv_h=\partial_1 v_1+\partial_2 v_2$ by $\partial_3 v_3$ in various calculations in the proof of Lemma \[m0-L2.4\]. Sometime, it would be useful (and it may be also necessary) to replace $v_3$. In fact, via (\[m0-E1.5-N3\])$_1$, we can re-write $$v_3=-(\partial_t\psi+v\cdot\nabla \psi ).\label{m0-E2.5}$$ The above substitution for $v_3$ has the advantage that it reduces space-time integral estimates to estimates on time slices and space times integral with higher order nonlinearities and fast dissipation. The latter is smaller by the initial smallness assumptions.
**Proof of Lemma \[m0-L2.4\].** Using the integration by parts, the fact that $\mathrm{div} v=0$, the Hölder’s inequality and the Sobolev embedding Theorem, we can estimate the first term in the lemma \[m0-L2.4\] as follows: $$\begin{aligned}
&&\left|\int^T_0\int \partial_3\psi \partial_3^3\psi\partial_3^3 v_3 dxdt
\right|\\
&=&\left|\int^T_0\int \partial_3\psi \partial_3^3\psi\partial_3^2\mathrm{div}_h v_h dxdt
\right|\\
&=&\left| \int^T_0\int \left(-\partial_3^2 v_h\cdot\nabla_h\partial_3\psi \partial_3^3\psi
-\partial_3\psi \partial_3^2 v_h\cdot\nabla_h\partial_3^3\psi
\right) dxdt
\right|\\
&=&\left| \int^T_0\int \left(-\partial_3^2 v_h\cdot\nabla_h\partial_3\psi \partial_3^3\psi
+\partial_3^2\psi \partial_3^2 v_h\cdot\nabla_h\partial_3^2\psi
+\partial_3\psi \partial_3^3 v_h\cdot\nabla_h\partial_3^2\psi
\right) dxdt
\right|\\
&\leq& C\|\partial_3^2 v_h\|_{L^2_T(L^4(\mathbb{R}^3))}\|\nabla_h\partial_3 \psi\|_{L^2_T(L^4(\mathbb{R}^3))}
\|\partial_3^3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}\\
&&+C\|\partial_3^2\psi\|_{L^\infty_T(L^4(\mathbb{R}^3))}
\|\partial_3^2 v_h\|_{L^2_T(L^4(\mathbb{R}^3))}\|\nabla_h\partial_3^2\psi\|_{L^2_T(L^2(\mathbb{R}^3))}\\
&&
+C\|\partial_3\psi\|_{L^\infty_T(L^\infty(\mathbb{R}^3))}
\|\partial_3^3 v_h\|_{L^2_T(L^2(\mathbb{R}^3))}\|\nabla_h\partial_3^2\psi\|_{L^2_T(L^2(\mathbb{R}^3))}\\
&\leq& C\|\nabla\psi\|_{L^\infty_T(H^2(\mathbb{R}^3))}\|\nabla_h\nabla\psi\|_{L^2_T(H^1(\mathbb{R}^3))}
\|\nabla v\|_{L^2_T(H^2(\mathbb{R}^3))}.
\end{aligned}$$ The other terms in (\[m0-E2.7\]) can be treated similarly, and we can conclude Lemma \[m0-L2.4\]. [$\square$]{}
We shall now proceed with the proof lemma \[m0-L2.5\]. The basic strategy has been described earlier, see for example, (\[idea\]). For this purpose, we first prove the following lemma. Here we use the equation (\[m0-E1.5-N3\])$_1$ to bounded the term $\int^T_0\int \partial_3\psi \partial_3 v_3(\partial_3^3\psi)^2 dxdt$.
\[m0-L2.2-2\] Under the conditions of Lemma \[m0-L2.3\], then there holds $$\begin{aligned}
\left|\int^T_0\int \partial_3\psi \partial_3 v_3(\partial_3^3\psi)^2 dxdt
\right|
&\leq& CB_T^4 (1+B_T), \label{m0-E2.5-0}
\end{aligned}$$ where $C$ is a positive constant independent of $T$.
Applying (\[m0-E2.2\]), (\[m0-E2.5\]), the integration by parts, Hölder’s inequality and Sobolev embedding Theorem, we get $$\begin{aligned}
&&\left|\int^T_0\int \partial_3\psi \partial_3 v_3(\partial_3^3\psi)^2 dxdt
\right|\\
&=&\left|\int^T_0\int \partial_3\psi \partial_3
(\partial_t\psi+v\cdot\nabla \psi )(\partial_3^3\psi)^2 dxdt
\right|\\
&=&\left| \left.\int \frac{1}{2 } (\partial_3\psi)^2 (\partial_3^3\psi)^2 dx\right|^T_0
+\int^T_0\int \left[ - (\partial_3\psi)^2 \partial_3^3\psi\partial_3^3\partial_t\psi
+ \partial_3\psi \partial_3
(v\cdot\nabla \psi )(\partial_3^3\psi)^2 \right]dxdt
\right|\\
&\leq& C\|\partial_3\psi\|_{L^\infty_T(L^\infty(\mathbb{R}^3))}^2
\|\partial_3^3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}^2 \\
&& +\left|
\int^T_0\int \left[ (\partial_3\psi)^2 \partial_3^3\psi\partial_3^3(v\cdot\nabla\psi + v_3)
+\partial_3\psi\partial_3(v\cdot\nabla\psi) (\partial_3^3\psi)^2 \right]dxdt
\right|\\
&\leq& C\|\nabla\psi\|^4_{L^\infty_T(H^2(\mathbb{R}^3))} +C\|\partial_3\psi\|_{L^4_T(L^\infty(\mathbb{R}^3))}^2
\|\partial_3^3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}
\|\partial_3^3(v\cdot\nabla\psi)-v\cdot\nabla\partial_3^3\psi\|_{L^2_T(L^2(\mathbb{R}^3))} \\
&& +C\|\partial_3\psi\|_{L^4_T(L^\infty(\mathbb{R}^3))}^2
\|\partial_3^3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}
\|\partial_3^3v_3 \|_{L^2_T(L^2(\mathbb{R}^3))}\\
&&+\left|
\int^T_0\int \left\{\frac{1}{2 }
\left[(\partial_3\psi)^2 v\cdot\nabla(\partial_3^3\psi)^2
+
v\cdot\nabla (\partial_3\psi)^2 (\partial_3^3\psi)^2\right] \right\}dxdt
\right|\\
&&+\left|
\int^T_0\int \left\{ \partial_3\psi\partial_3 v\cdot\nabla \psi (\partial_3^3\psi)^2 \right\}dxdt
\right|\\
&\leq& CB_T^4(1+B_T).
\end{aligned}$$ In deriving the last inequality above, we have used the following calculations: $$\begin{aligned}
&&\|\partial_3^3(v\cdot\nabla\psi)-v\cdot\nabla\partial_3^3\psi\|_{L^2_T(L^2(\mathbb{R}^3))}\\
&\leq&C \|\nabla^3 v\|_{L^2_T(L^2(\mathbb{R}^3))}\|\nabla \psi\|_{L^\infty_T(L^\infty(\mathbb{R}^3))}
+C\|\nabla^2 v\|_{L^2_T(L^4(\mathbb{R}^3))}\|\nabla^2 \psi\|_{L^\infty_T(L^4(\mathbb{R}^3))}
\\
&& +C\|\nabla v\|_{L^2_T(L^\infty(\mathbb{R}^3))}\|\nabla^3 \psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}\\
&\leq&C\|\nabla v\|_{L^2_T(H^2(\mathbb{R}^3))}\|\nabla \psi\|_{L^\infty_T(H^2(\mathbb{R}^3))},
\end{aligned}$$ and also the estimation: $$\begin{aligned}
&&\left|
\int^T_0\int \left\{ \partial_3\psi\partial_3 v\cdot\nabla \psi (\partial_3^3\psi)^2 \right\}dxdt
\right| \\
&\leq& \|\nabla\psi\|_{L^4_T(L^\infty(\mathbb{R}^3))}^2\|\nabla v\|_{L^2_T(L^\infty(\mathbb{R}^3))}\|\nabla^3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}^2.
\end{aligned}$$
**Proof of Lemma \[m0-L2.5\].** We use Lemma \[m0-L2.2\], (\[m0-E2.7\]), (\[m0-E2.5\]), (\[m0-E2.5-0\]), the integration by parts, Hölder’s inequality and Sobolev embedding Theorem to do following derivations: $$\begin{aligned}
&&\left|\int^T_0\int \partial_3 v_3(\partial_3^3\psi)^2 dxdt
\right|\\
&=&\left|\int^T_0\int \partial_3 (\partial_t
\psi+v\cdot\nabla\psi )(\partial_3^3\psi)^2 dxdt
\right|\\
&=&\left|\left.\int \partial_3
\psi (\partial_3^3\psi)^2 dx\right|^T_0
+\int^T_0\int \left\{ -2\partial_3\psi\partial_3^3\psi\partial_3^3\partial_t\psi
+ \partial_3 ( v\cdot\nabla\psi )(\partial_3^3\psi)^2\right\}dxdt
\right|\\
&\leq& C\|\partial_3\psi\|_{L^\infty_T(L^\infty(\mathbb{R}^3))}\|\partial_3^3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}^2 \\
&& +\left|
\int^T_0\int \left\{ 2\partial_3\psi\partial_3^3\psi\partial_3^3(v\cdot\nabla\psi+
v_3) + \partial_3 ( v\cdot\nabla\psi )(\partial_3^3\psi)^2\right\}dxdt
\right|\\
&\leq& C\|\nabla\psi\|_{L^\infty_T(H^2(\mathbb{R}^3))}^3
+C\|\nabla \psi\|_{L^\infty_T(H^2(\mathbb{R}^3))}\|\nabla\nabla_h\psi\|_{L^2_T(H^1(\mathbb{R}^3))}\|\nabla v\|_{L^2_T(H^2(\mathbb{R}^3))}\\
&& +\left|
\int^T_0\int \left\{ 2\partial_3\psi\partial_3^3\psi[\partial_3^3(v\cdot\nabla\psi)-v\cdot\nabla\partial_3^3\psi]
+
\partial_3\psi v \cdot\nabla(\partial_3^3\psi)^2 \right.\right.\\
&&\left.\left. + v\cdot\nabla \partial_3\psi(\partial_3^3\psi)^2 + \partial_3 v\cdot\nabla\psi (\partial_3^3\psi)^2\right\}dxdt
\right|\\
&\leq&
C B_T^3(1+B_T^2).
\end{aligned}$$ In the last step above, we have also applied the following estimate, $$\begin{aligned}
&& \left|
\int^T_0\int \left\{ 2\partial_3\psi\partial_3^3\psi[\partial_3^3(v\cdot\nabla\psi)-v\cdot\nabla\partial_3^3\psi]
+ \partial_3 v\cdot\nabla\psi (\partial_3^3\psi)^2\right\}dxdt
\right|\\
&\leq& \left|
\int^T_0\int 2\partial_3\psi\partial_3^3\psi(\partial_3^3v\cdot\nabla\psi+3\partial_3^2v\cdot\nabla\partial_3\psi
) dxdt
\right|+\left|
\int^T_0\int 6\partial_3\psi\partial_3^3\psi
\partial_3 v_h\cdot\nabla_h\partial_3^2\psi dxdt
\right|\\
&&
+\left|
\int^T_0\int 7\partial_3\psi
\partial_3 v_3 (\partial_3^3\psi)^2 dxdt
\right| +\left|
\int^T_0\int
\partial_3 v_h\cdot\nabla_h \psi (\partial_3^3\psi)^2 dxdt
\right|\\
&\leq&C\|\nabla\psi\|_{L^4_T(L^{ \infty}(\mathbb{R}^3))}^2\|\partial_3^3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}\|\nabla^3 v\|_{L^2_T(L^2(\mathbb{R}^3))}
\\
&&+C\|\nabla\psi\|_{L^4_T(L^{ \infty}(\mathbb{R}^3))}\|\partial_3^3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}\|\nabla^2 v\|_{L^2_T(L^4(\mathbb{R}^3))}
\|\nabla^2\psi\|_{L^4_T(L^4(\mathbb{R}^3))}
\\
&&+C\|\partial_3\psi\|_{L^\infty_T(L^\infty(\mathbb{R}^3))}
\|\partial_3^3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}\|\partial_3 v\|_{L^2_T(L^\infty(\mathbb{R}^3))}
\|\nabla_h\partial_3^2\psi\|_{L^2_T(L^2(\mathbb{R}^3))}\\
&&+ CB_T^4(1+B_T) +C\|\partial_3 v\|_{L^2_T(L^\infty(\mathbb{R}^3))}
\|\nabla_h\psi\|_{L^2_T(L^\infty(\mathbb{R}^3))}\|\partial_3^3\psi\|_{L^\infty_T(L^2(\mathbb{R}^3))}^2 \\
&\leq& C B_T^4(1+B_T).
\end{aligned}$$
The remaining parts have already shown to have the desired estimates. Thus we complete the proof of Lemma. [$\square$]{}
Acknowledgements {#acknowledgements .unnumbered}
================
The research of F.H.Lin is partial supported by the NSF grants, DMS 1065964 and DMS 1159313. The research of T. Zhang is partially supported by NSF of China under Grants 11271322, 11331005 and 11271017, National Program for Special Support of Top-Notch Young Professionals, Program for New Century Excellent Talents in University NCET-11-0462, the Fundamental Research Funds for the Central Universities (2012QNA3001). Part of the work was done while the second author was visiting the Courant Institute Mathematical Sciences. T.Z. wants to thank the Courant Institute for the warm hospitality.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We perform a global fit of the parameters of the Standard Model with a sequential fourth generation (SM4) to LHC and Tevatron Higgs data and electroweak precision data. Using several likelihood ratio tests we compare the performance of the SM4 and SM3 at describing the measured data. Since the SM3 and SM4 are not *nested* (i.e. the SM3 can not be considered as a special case of the SM4 with some parameters fixed) the usual analytical formulae for $p$-values in likelihood ratio tests do not hold. We thus apply a new method to compute these $p$-values. For a Higgs mass of $126.5{\ensuremath{\,\text{GeV}}}$ and fourth-generation quark masses above [600[$\,\text{GeV}$]{}]{} we find that the SM4 is excluded at $3.1\sigma$.'
author:
- |
Otto Eberhardt$^{\,a}$, Alexander Lenz$^{\,b}$, Andreas Menzel$^{\,c}$,\
Ulrich Nierste$^{\,a}$, and Martin Wiebusch$^{\,a}$
bibliography:
- 'sm4higgs2.bib'
title: |
Status of the fourth fermion generation before ICHEP2012:\
Higgs data and electroweak precision observables
---
Method and inputs
=================
In this paper we study the SM4, which differs from the established Standard Model (denoted by SM3) by an additional fermion generation. We treat the masses of the extra fermions as free parameters and allow for arbitrary flavor mixings among the quarks of the four generations in our fits. Large mixings of the fourth-generation lepton doublet with those of the first three generations are ruled out [@Lacker:2010zz] from data on lepton-flavor violating decays and lepton-flavor universality [@NA62:2011aa]. Recent NA62 data constrain these mixing angles even further [@NA48/2:2012gh]. Including lepton mixing within the allowed range has a negligible impact on the electroweak precision observables (EWPOs). In the absence of lepton mixing the decay of the Higgs boson into neutrinos is invisible as long as the fourth-generation charged lepton is heavier than the corresponding neutrino. This invisible Higgs decay mode increases the total Higgs width and potentially counterbalances the effect of the enhanced $gg\to H$ production mechanism [@Gunion:1995tp; @Kribs:2007nz], because the branching fractions into the observed final states are reduced [@Cetin:2011fp; @Eberhardt:2012sb]. Allowing for (even small) mixing of the fourth with the other lepton doublets can render the neutrino decay mode visible. Since we want to quantify the level at which the SM4 is ruled out, we may confine ourselves to the most conservative scenario with an unmixed fourth-generation lepton doublet. Like the SM3, the SM4 can be studied with Dirac or Majorana neutrinos. In the fits presented in this paper we use Dirac neutrinos. In our conclusions we briefly discuss the (marginal) changes in the results expected for Majorana neutrinos. From a model-building point of view, the hierarchy between three almost massless neutrinos and a fourth neutrino with mass of order of the electroweak scale can be motivated by a symmetry enforcing massless neutrinos in the exact symmetry limit: e.g. three right-handed neutrino fields might carry some U(1) charge while the fourth neutrino field and the left-handed lepton doublets are uncharged under this new symmetry. The Yukawa couplings are small spurions breaking this symmetry, leading to three tiny neutrino masses and tiny mixings between the fourth and the other generations.
A sequential fourth generation of fermions decouples neither from the production cross section $\sigma(gg \to H)$ nor from the Higgs decay rate into photons. Consequently, current LHC Higgs data put the SM4 under serious pressure [@Djouadi:2012ae; @Kuflik:2012ai; @Eberhardt:2012sb; @Buchkremer:2012yy]. In a recent publication [@Eberhardt:2012sb] we presented a global fit of the SM4 parameters to EWPOs and Higgs signal strengths measured at Tevatron and the LHC. The signal strength is defined as $$\label{eq:signalstrength}
\hat\mu(X\to H\to Y) = \frac{\sigma(X\to H){\mathcal{B}}(H\to Y)|_\text{SM4}}
{\sigma(X\to H){\mathcal{B}}(H\to Y)|_\text{SM3}}
\quad.$$ Here we update our results with all available data and analyse the status of the SM4 prior to the ICHEP2012 conference. We also compute the statistical significance ($p$-value) at which the SM4 is excluded. As explained in [@Eberhardt:2012sb] the computation of the $p$-value is non-trivial: due to the non-decoupling nature of the fourth-generation fermions the SM3 can not be regarded as a special case of the SM4, i.e. the two models are not *nested*. Analytical formulae for $p$-values only hold for nested models and thus the $p$-value of the SM4 has to be computed numerically. To this end, a new C++ framework for maximum likelihood fits and likelihood ratio tests called [[*my*Fitter]{}]{} [@myFitter-page] was written. The implementation is discussed in [@myFitter].
In total, the following aspects of our previous analysis have been improved:
1. The masses of all four fourth-generation fermions are now consistently treated as free parameters. To avoid non-perturbative Yukawa couplings and constraints from direct searches of fourth-generation quarks we require $600{\ensuremath{\,\text{GeV}}}\leq m_{t'},m_{b'}\leq 800{\ensuremath{\,\text{GeV}}}$. We are aware that for fermion masses of $800{\ensuremath{\,\text{GeV}}}$ the validity of perturbation theory is questionable at best. However, reducing the upper limit for the fermion masses can only lead to larger $\chi^2$ values in the SM4 and thus to smaller $p$-values. In this sense, the upper limit of $800{\ensuremath{\,\text{GeV}}}$ is a *conservative* estimate.
2. The signal strength for $pp\to H\to\tau\tau$ measured at the LHC [@ATLAS-CONF-2012-019] is included in the analysis.
3. In the global fit, the Higgs mass is no longer fixed at $125{\ensuremath{\,\text{GeV}}}$, but is allowed to float in the range where experimental data on the Higgs signal strengths is available, i.e. $115{\ensuremath{\,\text{GeV}}}\leq m_H\leq 150{\ensuremath{\,\text{GeV}}}$.[^1]
4. Since, for a variable Higgs mass, no separate $H\to\gamma\gamma$ signal strengths for the gluon fusion and vector boson fusion production modes are available we only use the combined signal strength for $pp\to
H\to\gamma\gamma$ as input.
5. For the two cases $m_H=126.5{\ensuremath{\,\text{GeV}}}$ (the preferred Higgs mass of the SM3) and $m_H=147{\ensuremath{\,\text{GeV}}}$ (the preferred Higgs mass of the SM4) we perform likelihood ratio tests to compare the performance of the SM3 and SM4 at describing the measured data.
Regarding the last point, a few more comments are in order. In likelihood ratio tests the difference $\Delta\chi^2$ of minimal $\chi^2$ values obtained in the SM3 and the SM4 is used as a test statistic. One then assumes that the measured observables are random variables distributed around the prediction of one model (e.g. the SM4) with a spread determined by their errors and computes the probability ($p$-value) that a random set of “toy-observables” leads to a $\Delta\chi^2$ which is more extreme (e.g. more SM3-like) than the $\Delta\chi^2$-value obtained from the real data. Note that this is different from the goodness-of-fit analysis presented in [@Kuflik:2012ai], which used the $\chi^2$ value of the SM4 as a test statistic and therefore did not compare the performance of the SM3 and the SM4. Also, the $H\to\tau\tau$ signal strengths were not included in their analysis.
Unfortunately, the likelihood ratio tests can not be done (by us) if the Higgs mass is treated as a free parameter. In that case, the signal strengths measured in each invariant mass bin of each Higgs decay mode would have to be treated as separate observables, and we do not have any information on statistical correlations between adjacent bins. Thus we only perform likelihood ratio tests for *specialisations* of the SM3 and SM4, where the Higgs mass is fixed to $m_H=126.5{\ensuremath{\,\text{GeV}}}$ (the value preferred by the global SM3 fit) or $m_H=147{\ensuremath{\,\text{GeV}}}$ (the value preferred by the global SM4 fit). Then only the signal strengths at $m_H=126.5{\ensuremath{\,\text{GeV}}}$ and $m_H=147{\ensuremath{\,\text{GeV}}}$ have to be treated as independent observables and correlations between these observables can safely be neglected.
Note, however, that the information from *all* invariant mass bins is encoded in our $\chi^2$ function. So, for example, the $\chi^2$ value at $m_H=147{\ensuremath{\,\text{GeV}}}$ has a contribution due to the fact that there is a signal at $m_H=126.5{\ensuremath{\,\text{GeV}}}$. If the model under consideration had a Higgs boson outside the discovery reach of LHC (or no Higgs boson at all), the theory prediction for all signal strengths in all invariant mass bins would be zero. This leads to a constant contribution to the $\chi^2$, which we are allowed to drop. Now assume that the model has a Higgs boson with some mass $m_H$ and a predicted signal strength $\hat\mu_\text{th}(m_H)$. Let $\hat\mu_\text{ex}(m_H)$ and $\Delta\hat\mu(m_H)$ be the measured signal strength and experimental error for the corresponding invariant mass bin. After dropping the constant, the $\chi^2$ function is $$\chi^2(m_H)=
\frac{[\hat\mu_\text{th}(m_H)-\hat\mu_\text{ex}(m_H)]^2
- [\hat\mu_\text{ex}(m_H)]^2}{[\Delta\hat\mu(m_H)]^2}
\quad.$$ If there is a clear signal at the Higgs mass $m_H$, the second term gives a large negative contribution to the $\chi^2$ function. This contribution is not present if $m_H$ is in a region without a signal, so the minimum of the $\chi^2$ function will usually be at a Higgs mass close to the signal.
In the present analysis, the following experimental inputs are used:
- $\hat\mu(pp\to H\to WW^*)$ measured by ATLAS [@ATLAS:2012sc],
- $\hat\mu(pp\to H\to\gamma\gamma)$ measured by ATLAS [@ATLAS:2012ad],
- $\hat\mu(p\bar p\to HV\to Vb{\bar{b}})$ measured by CDF and D0 [@Knoepfel:2012xk],
- $\hat\mu(pp\to H\to ZZ^*)$ and $\hat\mu(pp\to H\to \tau {\bar{\tau}})$ measured by ATLAS [@ATLAS-CONF-2012-019],
- the electroweak precision observables (EWPOs) $M_Z$, $\Gamma_Z$, $\sigma_\text{had}$, $A_\text{FB}^l$, $A_\text{FB}^c$, $A_\text{FB}^b$, $A_l$, $A_c$, $A_b$, $R_l=\Gamma_{l^+l^-}/\Gamma_\text{had}$, $R_c$, $R_b$, $\sin^2\theta_l^\text{eff}$ measured at LEP and SLC [@EWWG:2010vi] as well as $m_t$, $M_W$, $\Gamma_W$ and $\Delta\alpha_\text{had}^{(5)}$ [@Nakamura:2010zzi].
- the lower bounds $m_{t^\prime,b^\prime}\gtrsim 600\;$[$\,\text{GeV}$]{} (from the LHC) [@Aad:2012xc; @Aad:2012us; @CMS:2012ye; @CMS-PAS-EXO-11-099] and $m_{l_4}>101\,\text{GeV}$ (from LEP2) [@Nakamura:2010zzi].
Unfortunately, there is no data for signal strengths as a function of the Higgs mass from CMS.
On the theory side, the global fits with a variable Higgs mass were done with the CKMfitter software [@Hocker:2001xe]. The EWPOs in the SM4 were calculated with the method described in [@Gonzalez:2011he], using FeynArts, FormCalc and LoopTools [@Hahn:1998yk; @Hahn:2000kx; @Hahn:2006qw] to compute the SM4 corrections to the EWPOs. The EWPOs in the SM3 were calculated with the ZFitter software [@Bardin:1989tq; @Bardin:1999yd; @Arbuzov:2005ma]. The Higgs width and branching ratios in the SM4 and SM3 were calculated with HDECAY v. 4.45 [@Djouadi:1997yw], which implements results of [@Djouadi:1994gf; @Djouadi:1994ge; @Passarino:2011kv; @Denner:2011vt]. The SM3 Higgs production cross sections were taken from [@Dittmaier:2011ti] (LHC) and [@Brein:2003wg; @Baglio:2010um] (Tevatron). For the numerical integration required to compute the $p$-values we use the Dvegas code [@Dvegas] which was developed in the context of [@Kauer:2001sp; @Kauer:2002sn].
Results
=======
To show the impact of the $H\to\tau\tau$ signal strength we plot the minimal $\chi^2$ value with and without the $H\to\tau\tau$ input as a function of the mass $m_{\nu_4}$ of the fourth-generation neutrino in Fig. \[fig:mnu\]. We see that for $m_{\nu_4}\lesssim 60{\ensuremath{\,\text{GeV}}}$ the minimum $\chi^2$ values are almost the same with and without the $H\to\tau\tau$ input. For $m_{\nu_4}\gtrsim 65{\ensuremath{\,\text{GeV}}}$ the $H\to\tau\tau$ input increases the minimum $\chi^2$ by more than 20. We also see that without the $H\to\tau\tau$ input the SM4 favours large values of $m_{\nu_4}$.With the $H\to\tau\tau$ signal strengths included, the smallest $\chi^2$ values are obtained for $m_{\nu_4}$ between $50$ and $60{\ensuremath{\,\text{GeV}}}$.
This can be understood as follows: the production rate of Higgs bosons in gluon fusion is enhanced by a factor of 9 in the SM4 due to the contributions from additional heavy quark loops. On the other hand, the effective $HWW$, $HZZ$ and $H\gamma\gamma$ couplings are suppressed by the higher order corrections discussed in [@Denner:2011vt]. No such suppression is possible for $H\to\tau\tau$, so we would expect a $H\to\tau\tau$ signal strength of $9$. The only way to reduce this signal strength is to open the invisible $H\to\nu_4\bar\nu_4$ decay mode, which then suppresses all branching ratios by a common factor. Thus, for large values of $m_{\nu_4}$, the fit gets considerably worse if the $H\to\tau\tau$ channel is included.
![Minimum $\chi^2$ values for a fixed neutrino mass as a function of $m_{\nu_4}$. The blue (red) lines show the results from the combined analysis of EWPOs and Higgs signal strengths with (without) the $H\to\tau\tau$ channel. The solid and dashed lines correspond to the SM4 and SM3, respectively.[]{data-label="fig:mnu"}](mnu){width="45.00000%"}
Figs. \[fig:mHSM3\] and \[fig:mHSM4\] show the minimum $\chi^2$ value as a function of the Higgs mass $m_H$ in the SM3 and SM4, respectively. The solid lines show the results of the combined analysis of signal strengths and EWPOs while for the dashed lines only the Higgs signal strengths (including $H\to\tau\tau$) were used as inputs. We see that the SM3 clearly prefers a Higgs mass near $126.5{\ensuremath{\,\text{GeV}}}$. This is in agreement with a similar analysis presented in [@Erler:2012uu]. There is another local minimum at $m_H=145{\ensuremath{\,\text{GeV}}}$, but with a considerably larger $\chi^2$ value. The $\chi^2$ function of the SM4 in the combined analysis of signal strengths and EWPOs also has one minimum at $m_H=126.5{\ensuremath{\,\text{GeV}}}$ and another one at $m_H=147{\ensuremath{\,\text{GeV}}}$. Here, the $\chi^2$ values are almost the same, but still larger than the minimal $\chi^2$ value of the SM3, $\chi^2 =23.3$, by about 8 units. Note that for non-nested models or models with bounded parameters the relation between $\chi^2$ values and p-values is no longer given by Wilks’ theorem. Thus, in the case of the SM4, the number of degrees of freedom is an ill-defined concept and the $p$-values have to be calculated by numerical simulation. For the simulations we used the [[*my*Fitter]{}]{} package [@myFitter-page]. Further details on the statistical issues and the [[*my*Fitter]{}]{} simulation method can be found in [@myFitter]. For $m_H=147{\ensuremath{\,\text{GeV}}}$ the signals at invariant masses near $126.5{\ensuremath{\,\text{GeV}}}$ would be interpreted as statistical fluctuations. Then the data would be better described by the SM4 because it has more mechanisms for suppressing its Higgs signals. These mechanisms were discussed in [@Eberhardt:2012sb].
![The minimum $\chi^2$ value of the SM3 as a function of the Higgs mass $m_H$. The solid line shows the results of the combined analysis of signal strengths and EWPOs. For the dashed line only the signal strengths were included in the fit.[]{data-label="fig:mHSM3"}](mHSM3){width="45.00000%"}
![The minimum $\chi^2$ value of the SM4 as a function of the Higgs mass $m_H$. The solid line shows the results of the combined analysis of signal strengths and EWPOs. For the dashed line only the signal strengths were included in the fit.[]{data-label="fig:mHSM4"}](mHSM4){width="45.00000%"}
Fig. \[fig:pulls\] shows the pulls of the Higgs signal strengths for the SM3 with a Higgs mass of $126.5{\ensuremath{\,\text{GeV}}}$ and the SM4 with a Higgs mass of $126.5{\ensuremath{\,\text{GeV}}}$ or $147{\ensuremath{\,\text{GeV}}}$. We see that in the SM4 with $m_H=126.5{\ensuremath{\,\text{GeV}}}$ the measured $H\to\tau\tau$ signal strength deviates by more than $4\sigma$ from its predicted value. This is due to the effect mentioned in the discussion of Fig. \[fig:mnu\]. For the SM4 with $m_H=147{\ensuremath{\,\text{GeV}}}$ the measured signal strengths for the invariant mass bin at $147{\ensuremath{\,\text{GeV}}}$ are in good agreement with their theory predictions. However, in that case the $\chi^2$ receives a large contribution due to the fact that the measured values of the signal strengths in the invariant mass bin at $126.5{\ensuremath{\,\text{GeV}}}$ deviate from their predicted values of zero.
![Pulls of the Higgs signal strengths for the SM3 with $m_H=126.5{\ensuremath{\,\text{GeV}}}$ and the SM4 with $m_H=126.5{\ensuremath{\,\text{GeV}}}$ and $m_H=147{\ensuremath{\,\text{GeV}}}$.[]{data-label="fig:pulls"}](Pullplot){width="45.00000%"}
Table \[tab:pvalues\] shows the $p$-values obtained from the likelihood ratio tests for the two SM4 Higgs masses. We see that, based on the Higgs signal strengths alone, the SM4 scenario with $m_H=126.5{\ensuremath{\,\text{GeV}}}$ is ruled out at almost $4\sigma$ while the scenario with $m_H=147{\ensuremath{\,\text{GeV}}}$ is only excluded at $3\sigma$. At a fixed Higgs mass of $126.5{\ensuremath{\,\text{GeV}}}$ the electroweak fit is actually better in the SM4 than in the SM3. Thus, if the EWPOs are included in the fit, the $p$-value increases to $2$ permille, which corresponds to $3.1\sigma$. The lower bound $m_{t^\prime,b^\prime}\gtrsim 600\;$[$\,\text{GeV}$]{} is not essential for this result, relaxing this bound to $m_{t^\prime,b^\prime}\gtrsim
400\;$[$\,\text{GeV}$]{} decreases the minimum-$\chi^2$ by $0.6$. For the SM4 scenario with $m_H=147{\ensuremath{\,\text{GeV}}}$ the $p$-value drops to $0.74$ permille ($3.4\sigma$). In any case, the SM4 is excluded at more than $3\sigma$.
SM4 @ 126.5[$\,\text{GeV}$]{} SM4 @ 147[$\,\text{GeV}$]{}
------------- ------------------------------------ -----------------------------------
Higgs only $0.088\cdot 10^{-3}$ ($3.9\sigma$) $2.4\cdot 10^{-3}$ ($3.0\sigma$)
Higgs+EWPOs $2.0\cdot 10^{-3}$ ($3.1\sigma$) $0.74\cdot 10^{-3}$ ($3.4\sigma$)
: $p$-values obtained from the likelihood ratio tests for fixed Higgs mass. Both, the SM4 with $m_H=126.5{\ensuremath{\,\text{GeV}}}$ and $m_H=147{\ensuremath{\,\text{GeV}}}$ are compared to the SM3 with a fixed Higgs mass of $126.5{\ensuremath{\,\text{GeV}}}$. In the first row, only the Higgs signal strengths were used as inputs. The second row contains the results of the combined analysis of signal strengths and EWPOs. The number of standard deviations corresponding to each $p$-value are shown in parentheses.[]{data-label="tab:pvalues"}
Conclusions
===========
We presented a combined analysis of Higgs signal strengths and EWPOs in the context of the Standard Model with three or four fermion generations. The SM3 is in good agreement with the experimental data and the best-fit Higgs mass is $126.5{\ensuremath{\,\text{GeV}}}$. The SM4, on the other hand, struggles to describe the Higgs signal strengths measured at Tevatron and the LHC. The $\chi^2$ function of the SM4 has two minima at $m_H=126.5{\ensuremath{\,\text{GeV}}}$ and $m_H=147{\ensuremath{\,\text{GeV}}}$ with essentially the same $\chi^2$ value, which is larger than the minimal $\chi^2$ value of the SM3 by 8 units. The second minimum of the SM4 $\chi^2$ function occurs because the SM4 cannot reproduce the signal strengths measured at $126.5{\ensuremath{\,\text{GeV}}}$ very well, so that an SM4 with a Higgs mass nowhere near the observed signals describes the data equally well as an SM4 with $m_H=126.5{\ensuremath{\,\text{GeV}}}$. To quantitatively compare the performance of the SM3 and SM4 at describing the data we performed likelihood ratio tests for fixed Higgs masses $m_H$ of $126.5{\ensuremath{\,\text{GeV}}}$ in the SM3 and $m_H=126.5{\ensuremath{\,\text{GeV}}},\ 147{\ensuremath{\,\text{GeV}}}$ in the SM4. The $p$-values were computed with a new numerical method [@myFitter] for likelihood ratio tests of non-nested models. If EWPOs and signal strengths are included in the fit we find $p$-values of $2.0\cdot10^{-3}$ and $0.74\cdot10^{-3}$, respectively, which means that the SM4 is excluded at the $3\sigma$ level. While this result is obtained for Dirac neutrinos, it will change only marginally for the case of Majorana neutrinos with two fourth-generation mass eigenstates $\nu_4$, $\nu_5$: the fit to the signal sthrengths will return the same invisible Higgs width, now corresponding to the sum of the four decay rates $\Gamma(H\to \nu_{4,5}\nu_{4,5})$. A marginal difference occurs once the EWPOs are included: choosing the $\nu_4$–$\nu_5$ mass splitting such that the eigenstate with the larger SU(2) doublet component becomes heavier, one can slightly improve the quality of the electroweak fit. The improvement is negligible, as indicated by the shallowness of the minimum of the SM4 $\chi^2$ function in [Fig. \[fig:mnu\]]{}. While the SM4 is under severe pressure, a sequential fourth generation may still be viable in conjunction with an extended Higgs sector [@Hung:2010xh; @BarShalom:2011zj; @BarShalom:2012ms; @He:2011ti].
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Geoffrey Herbert and Heiko Lacker for important contributions to the SM4 part of the CKMfitter code. We also thank Jérôme Charles for CKMFitter software support and very useful inputs regarding likelihood ratio tests for non-nested models.
We acknowledge support by the DFG through grants NI1105/2-1, LA2541/1-1, LE1246/9-1, and Le1246/10-1.
[^1]: A lattice study has found the lower bound $m_H \gtrsim 500{\ensuremath{\,\text{GeV}}}$ for $m_{t'}=m_{b'}=700{\ensuremath{\,\text{GeV}}}$ [@Gerhold:2010wv]. We interpret this result such that the perturbative vacuum state is metastable for $m_H \approx 125 {\ensuremath{\,\text{GeV}}}$ and the heavy quark masses used by us. Therefore Ref. [@Gerhold:2010wv] per se does not invalidate our analysis.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We prove a nearly optimal bound on the number of stable homotopy types occurring in a $k$-parameter semi-algebraic family of sets in ${\mbox{\rm R}}^\ell$, each defined in terms of $m$ quadratic inequalities. Our bound is exponential in $k$ and $m$, but polynomial in $\ell$. More precisely, we prove the following. Let ${\mbox{\rm R}}$ be a real closed field and let $${\mathcal P} = \{P_1,\ldots,P_m\} \subset {\mbox{\rm R}}[Y_1,\ldots,Y_\ell,X_1,\ldots,X_k],$$ with ${\rm deg}_Y(P_i) \leq 2, {\rm deg}_X(P_i) \leq d, 1 \leq i \leq m$. Let $S \subset {\mbox{\rm R}}^{\ell+k}$ be a semi-algebraic set, defined by a Boolean formula without negations, whose atoms are of the form, $P \geq 0, P\leq 0, P \in {\mathcal P}$. Let $\pi: {\mbox{\rm R}}^{\ell+k} \rightarrow {\mbox{\rm R}}^k$ be the projection on the last $k$ co-ordinates. Then, the number of stable homotopy types amongst the fibers $S_{{\mathbf{x}}} = \pi^{-1}({\mathbf{x}}) \cap S$ is bounded by $
\displaystyle{
(2^m\ell k d)^{O(mk)}.
}
$
address: 'School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.'
author:
- Saugata Basu
- Michael Kettner
title: 'Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities'
---
Introduction {#sec:intro}
============
Let $S \subset {\mbox{\rm R}}^{\ell+k}$ be a semi-algebraic set over a real closed field ${\mbox{\rm R}}$. Let ${\pi:{\mbox{\rm R}}^{\ell + k} \to {\mbox{\rm R}}^k}$ be the projection map on the last $k$ co-ordinates, and for any $S \subset {\mbox{\rm R}}^{\ell+k}$ we will denote by $\pi_S$ the restriction of $\pi$ to $S$. Moreover, when the map $\pi$ is clear from context, for any ${\mathbf{x}}\in {\mbox{\rm R}}^k$ we will denote by $S_{{\mathbf{x}}}$ the fiber $\pi_S^{-1}({\mathbf{x}})$.
A fundamental theorem in semi-algebraic geometry states,
(Hardt’s triviality theorem [@Hardt]) \[the:hardt\] There exists a semei-algebraic partition of ${\mbox{\rm R}}^k$, $\{T_i\}_{i \in I}$, such that the map $\pi_{S}$ is definably trivial over each $T_i$.
Theorem \[the:hardt\] implies that for each $i \in I$ and any point ${\mathbf{y}}\in T_i$, the pre-image $\pi^{-1}(T_i) \cap S$ is semi-algebraically homeomorphic to $S_{{\mathbf{y}}} \times T_i$ by a fiber preserving homeomorphism. In particular, for each $i \in I$, all fibers $S_{{\mathbf{y}}}$, ${\mathbf{y}}\in T_i$, are semi-algebraically homeomorphic.
Hardt’s theorem is a corollary of the existence of [*cylindrical algebraic decompositions*]{} (see [@BPR03]), which implies a double exponential (in $k$ and $\ell$) upper bound on the cardinality of $I$ and hence on the number of homeomorphism types of the fibers of the map $\pi_S$. No better bounds than the double exponential bound are known, even though it seems reasonable to conjecture a single exponential upper bound on the number of homeomorphism types of the fibers of the map $\pi_S$.
In [@BV06], the weaker problem of bounding the number of distinct [*homotopy types*]{}, occurring amongst the set of all fibers of $\pi_S$ was considered, and a single exponential upper bound was proved on the number of homotopy types of such fibers.
Before stating this result more precisely we need to introduce a few notation. Let ${\mbox{\rm R}}$ be a real closed field, ${\mathcal P} \subset {\mbox{\rm R}}[Y_1,\ldots,Y_\ell,X_1,\ldots,X_k]$, and let $\phi$ be a Boolean formula with atoms of the form $P=0$, $P > 0$, or $P< 0$, where $P \in {\mathcal P}$. We call $\phi$ a ${\mathcal P}$-formula, and the semi-algebraic set $S \subset {\mbox{\rm R}}^{\ell+k}$ defined by $\phi$, a ${\mathcal P}$-semi-algebraic set.
If the Boolean formula $\phi$ contains no negations, and its atoms are of the form $P= 0$, $P \geq 0$, or $P \leq 0$, with $P \in {\mathcal P}$, then we call $\phi$ a ${\mathcal P}$-closed formula, and the semi-algebraic set $S \subset {\mbox{\rm R}}^{\ell+k}$ defined by ${\phi}$, a ${\mathcal P}$-closed semi-algebraic set.
The following theorem appears in [@BV06].
[@BV06] \[the:mainBV\] Let ${\mathcal P} \subset {\mbox{\rm R}}[Y_1,\ldots,Y_\ell,X_1,\ldots,X_k]$, with $\deg(P) \leq d$ for each $P \in {\mathcal P}$ and cardinality $\#{\mathcal P} = m$. Then, there exists a finite set $A \subset {\mbox{\rm R}}^k$, with $$\# A \leq (2^\ell mkd)^{O(k\ell)},$$ such that for every ${\mathbf{x}}\in {\mbox{\rm R}}^k$ there exists ${\mathbf{z}}\in A$ such that for every ${\mathcal P}$-semi-algebraic set $S \subset {\mbox{\rm R}}^{\ell+k}$, the set $S_{{\mathbf{x}}}$ is semi-algebraically homotopy equivalent to $S_{{\mathbf{z}}}$. In particular, for any fixed ${\mathcal P}$-semi-algebraic set $S$, the number of different homotopy types of fibers $S_{{\mathbf{x}}}$ for various ${\mathbf{x}}\in \pi(S)$ is also bounded by $$(2^{\ell} mkd)^{O(k\ell)}.$$
A result similar to Theorem \[the:mainBV\] has been proved for semi-Pfaffian sets as well in [@BV06], and has been extended to arbitrary o-minimal structures in [@Basu07b]. The bounds on the number of homotopy types proved in [@BV06; @Basu07b] are all exponential in $\ell$ as well as $k$. The following example, which appears in [@BV06], shows that in this generality the single exponential dependence on $\ell$ is unavoidable.
\[eg:exp\] Let $P \in {\mbox{\rm R}}[Y_1,\ldots,Y_\ell] \hookrightarrow {\mbox{\rm R}}[Y_1,\ldots,Y_\ell,X]$ be the polynomial defined by $$P = \sum_{i=1}^{\ell} \prod_{j=0}^{d-1}(Y_i - j)^2.$$ The algebraic set defined by $P=0$ in ${\mbox{\rm R}}^{\ell+1}$ with co-ordinates $Y_1,\ldots,Y_\ell,X$, consists of $d^\ell$ lines all parallel to the $X$ axis. Consider now the semi-algebraic set $S \subset {\mbox{\rm R}}^{\ell+1}$ defined by $$\displaylines{
(P = 0)\; \wedge \;
(0 \le X \le Y_1+ dY_2 + d^2 Y_3 + \cdots + d^{\ell-1}Y_\ell).
}$$ It is easy to verify that, if $\pi:\> {\mbox{\rm R}}^{\ell+1} \to {\mbox{\rm R}}$ is the projection map on the $X$ coordinate, then the fibers $S_{{\mathbf{x}}}$, for ${\mathbf{x}}\in \{0,1, 2, \ldots ,d^\ell-1 \} \subset {\mbox{\rm R}}$ are 0-dimensional and of different cardinality, and hence have different homotopy types.
Semi-algebraic sets defined by quadratic inequalities
-----------------------------------------------------
One particularly interesting class of semi-algebraic sets is the class of semi-algebraic sets defined by quadratic inequalities. This class of sets has been investigated from an algorithmic standpoint [@Barvinok93; @GP; @Basu05a; @Basu07a; @BZ], as well as from the point of view topological complexity, [@Agrachev; @Barvinok97; @BK].
Semi-algebraic sets defined by quadratic inequalities are distinguished from arbitrary semi-algebraic sets by the fact that, if the number of inequalities is fixed, then the sum of their Betti numbers is bounded polynomially in the dimension. The following bound was proved by Barvinok [@Barvinok97].
\[the:barvinok\] Let $S \subset {\mbox{\rm R}}^\ell$ be a semi-algebraic set defined by the inequalities, $P_1 \geq 0,\ldots,P_m \geq 0$, $\deg(P_i) \leq 2, 1 \leq i \leq m$. Then, $\displaystyle{
\sum_{i=0}^\ell b_i(S) \leq (m \ell)^{O(m)},
}
$ where $b_i(S)$ denotes the $i$-th Betti number of $S$.
An extension of Barvinok’s bound to arbitrary ${\mathcal P}$-closed (not just basic closed) semi-algebraic sets defined in terms of quadratic inequalities has been done recently in [@BP'R07].
Now suppose that we have a parametrized family of sets, each defined in terms of $m$ quadratic inequalities. More precisely, let $${\mathcal P} = \{P_1,\ldots,P_m\} \subset {\mbox{\rm R}}[Y_1,\ldots,Y_\ell,X_1,\ldots,X_k],$$ with $
{\rm deg}_Y(P_i) \leq 2, {\rm deg}_X(P_i) \leq d, 1 \leq i \leq m
$ ($X_1,\ldots,X_k$ are the [*parameters*]{}), and let $S \subset {\mbox{\rm R}}^{\ell+k}$ be a ${\mathcal P}$-closed semi-algebraic set. Let $\pi: {\mbox{\rm R}}^{\ell+k} \rightarrow {\mbox{\rm R}}^k$ denote the projection on the last $k$ co-ordinates. Then, for each ${\mathbf{x}}\in {\mbox{\rm R}}^k$ the semi-algebraic set $S_{{\mathbf{x}}}$ is defined by a Boolean formula involving at most $m$ quadratic polynomials in $Y_1,\ldots,Y_\ell$.
Bounding the number of topological types amongst the fibers, $S_{{\mathbf{x}}}, {\mathbf{x}}\in R^k$, is an interesting special case of the more general problem mentioned in the last section. In view of the topological simplicity of semi-algebraic sets defined by few quadratic inequalities as opposed to general semi-algebraic sets (cf. Theorem \[the:barvinok\]), one might expect a much tighter bound on the number of topological types compared to the general case. However one should be cautious, since a tight bound on the Betti numbers of a class of semi-algebraic sets does not automatically imply a similar bound on the number of topological or even homotopy types occurring in that class.
In this paper we consider the problem of bounding the number of *stable homotopy types* (see Definition \[def:S-equivalence\] below) of fibers $S_{{\mathbf{x}}}$, where $\pi$ and $S$ are as defined above. We prove a bound which for each fixed $m$, is polynomial in $\ell$ (the dimension of the fibers). In some special cases our bound can be extended to the number of homotopy types (see Theorem \[the:union\]).
Our result can be seen as a follow-up to the recent work on bounding the number of homotopy types of fibers of general semi-algebraic maps studied in [@BV06]. However, the bound in [@BV06] applied to the special case of sets defined by quadratic inequalities would yield a bound exponential in both $k$ and $\ell$, as shown by Example \[eg:exp\], where the semi-algebraic set $S$ is defined in terms of three polynomials.
Note that the notions of homeomorphism type, homotopy type and stable homotopy type are each strictly weaker than the previous one, since two semi-algebraic sets might be stable homotopy equivalent, without being homotopy equivalent (see [@Spanier], p. 462), and also homotopy equivalent without being homeomorphic. However, two closed and bounded semi-algebraic sets which are stable homotopy equivalent have isomorphic homology groups.
Prior and Related Work
----------------------
Since sets defined by quadratic equalities and inequalities are the simplest class of topologically non-trivial semi-algebraic sets, the problem of classifying such sets topologically has attracted the attention of many researchers.
Motivated by problems related to stability of maps, Wall [@Wall] considered the special case of real algebraic sets defined by two simultaneously diagonalizable quadratic forms in $\ell$ variables. He obtained a full topological classification of such varieties making use of Gale diagrams (from the theory of convex polytopes). In our notation, letting $$\displaylines{
Q_1=\sum_{i=1}^\ell X_iY_i^2, \cr
Q_2=\sum_{i=1}^\ell X_{i+\ell} Y_i^2,
}$$ and $$S=\{({\mathbf{y}},{\mathbf{x}})\in{{\mbox{${\Bbb R}^{}$}}}^{3\ell} \mid \quad\parallel{\mathbf{y}}\parallel=1,\quad Q_1({\mathbf{y}},{\mathbf{x}})=Q_2({\mathbf{y}},{\mathbf{x}})=0\},$$ Wall obtains as a consequence of his classification theorem, that the number of different topological types of fibers $S_{{\mathbf{x}}}$ is bounded by $2^{\ell-1}$. Notice that in this case the number of parameters ($X_1,\ldots,X_{2\ell}$), as well as the number of variables ($Y_1,\ldots,Y_\ell$), are both $O(\ell)$. Similar results were also obtained by López [@Lopez] using different techniques. Much more recently Briand [@Briand07] has obtained explicit characterization of the isotopy classes of real varieties defined by two general conics in ${{\mbox{${\Bbb R}^{}$}}}{{\Bbb P}}^2$ in terms of the coefficients of the polynomials. His method also gives a decision algorithm for testing whether two such given varieties are isotopic.
In another direction Agrachev [@Agrachev] studied the topology of semi-algebraic sets defined by quadratic inequalities, and he defined a certain spectral sequence converging to the homology groups of such sets. A parametrized version of Agrachev’s construction is in fact a starting point of our proof of the main theorem in this paper.
Main Result
===========
The main result of this paper is the following theorem.
\[the:main\] Let ${\mbox{\rm R}}$ be a real closed field and let $${\mathcal P} = \{P_1,\ldots,P_m\} \subset {\mbox{\rm R}}[Y_1,\ldots,Y_\ell,X_1,\ldots,X_k],$$ with ${\rm deg}_Y(P_i) \leq 2, {\rm deg}_X(P_i) \leq d, 1 \leq i \leq m$. Let $\pi: {\mbox{\rm R}}^{\ell+k} \rightarrow {\mbox{\rm R}}^k$ be the projection on the last $k$ co-ordinates. Then, for any ${\mathcal P}$-closed semi-algebraic set $S \subset {\mbox{\rm R}}^{\ell+k}$, the number of stable homotopy types amongst the fibers, $S_{{\mathbf{x}}}$, is bounded by $\displaystyle{
(2^m\ell k d)^{O(mk)}.
}
$
Note that the bound in Theorem \[the:main\] (unlike that in Theorem \[the:mainBV\]) is polynomial in $\ell$ for fixed $m$ and $k$. The exponential dependence on $m$ is unavoidable, as can be seen from a slight modification of Example \[eg:exp\] above. Consider the semi-algebraic set $S \subset {\mbox{\rm R}}^{\ell+1}$ defined by $$\displaylines{
Y_i(Y_i-1) = 0, \; 1 \leq i \leq m \leq \ell, \cr
0 \leq X \leq Y_1 +2\cdot Y_2 + \ldots + 2^{m-1}\cdot Y_m.
}$$
Let $\pi: {\mbox{\rm R}}^{\ell+1} \rightarrow {\mbox{\rm R}}$ be the projection on the $X$-coordinate. Then, the sets $S_{{\mathbf{x}}}$, ${{\mathbf{x}}\in \{0,1\ldots,2^{m-1}\}}$, have different number of connected components, and hence have distinct (stable) homotopy types.
Note that the technique used to prove Theorem \[the:mainBV\] in [@BV06] does not directly produce better bounds in the quadratic case, and hence we need a new approach to prove a substantially better bound in this case. However, due to technical reasons, we only obtain a bound on the number of stable homotopy types, rather than homotopy types.
Mathematical Preliminaries
==========================
We first need to fix some notation and a few preliminary results needed later in the proof.
Some Notation {#sub:notations}
-------------
Let ${\mbox{\rm R}}$ be a real closed field. For an element $a \in {\mbox{\rm R}}$ introduce $${\mbox{\rm sign}}(a) = \Biggl\{
\begin{tabular}{ccc}
0 & \mbox{ if } $a=0$,\\
1 & \mbox{ if } $a> 0$,\\
$-1$& \mbox{ if } $a< 0$.
\end{tabular}$$ If ${\mathcal P} \subset {{\mbox{\rm R}}} [X_1, \ldots , X_k]$ is finite, we write the [*set of zeros*]{} of ${\mathcal P}$ in ${{\mbox{\rm R}}}^k$ as $${{\rm Z}}({\mathcal P})=\Bigl\{ {\mathbf{x}}\in {{\mbox{\rm R}}}^k\mid\bigwedge_{P\in{\mathcal P}}P({\mathbf{x}})= 0 \Bigr\}.$$ A [*sign condition*]{} $\sigma$ on ${\mathcal P}$ is an element of $\{0,1,- 1\}^{\mathcal P}$. The [*realization of the sign condition $\sigma$*]{} is the basic semi-algebraic set $${{\mathcal R}}(\sigma) = \Bigl\{ {\mathbf{x}}\in {{\mbox{\rm R}}}^k\;\mid\;
\bigwedge_{P\in{\mathcal P}} {\mbox{\rm sign}}({P}({\mathbf{x}}))=\sigma(P) \Bigr\}.$$ A sign condition $\sigma$ is [*realizable*]{} if ${{\mathcal R}}(\sigma) \neq \emptyset$. We denote by ${\rm Sign}({\mathcal P})$ the set of realizable sign conditions on ${\mathcal P}.$ For $\sigma \in {\rm Sign}({\mathcal P})$ we define the [*level of*]{} $\sigma$ as the cardinality $\#\{P \in {\mathcal P}| \sigma(P) = 0 \}$. For each level $p$, $0 \leq p \leq \# {\mathcal P}$, we denote by ${\rm Sign}_{p}({\mathcal P})$ the subset of ${\rm Sign}({\mathcal P})$ of elements of level $p$. Moreover, for a sign condition $\sigma$ let $${\mathcal Z}(\sigma) = \Bigl\{ {\mathbf{x}}\in {{\mbox{\rm R}}}^k\;\mid\;
\bigwedge_{P\in{\mathcal P},\ \sigma (P)=0} P({\mathbf{x}})=0 \Bigr\}.$$
Use of Infinitesimals
---------------------
Later in the paper, we will extend the ground field ${\mbox{\rm R}}$ by infinitesimal elements. We denote by ${\mbox{\rm R}}\langle \zeta\rangle$ the real closed field of algebraic Puiseux series in $\zeta$ with coefficients in ${\mbox{\rm R}}$ (see [@BPR03] for more details). The sign of a Puiseux series in ${\mbox{\rm R}}\langle \zeta\rangle$ agrees with the sign of the coefficient of the lowest degree term in $\zeta$. This induces a unique order on ${\mbox{\rm R}}\langle \zeta\rangle$ which makes $\zeta$ infinitesimal: $\zeta$ is positive and smaller than any positive element of ${\mbox{\rm R}}$. When $a \in {\mbox{\rm R}}{{\langle}}\zeta {{\rangle}}$ is bounded from above and below by some elements of ${\mbox{\rm R}}$, $\lim_\zeta(a)$ is the constant term of $a$, obtained by substituting 0 for $\zeta$ in $a$. Given a semi-algebraic set $S$ in ${{\mbox{\rm R}}}^k$, the [*extension*]{} of $S$ to ${\mbox{\rm R}}'$, denoted ${{\rm Ext}}(S,{\mbox{\rm R}}'),$ is the semi-algebraic subset of ${ {\mbox{\rm R}}'}^k$ defined by the same quantifier free formula that defines $S$. The set ${{\rm Ext}}(S,{\mbox{\rm R}}')$ is well defined (i.e. it only depends on the set $S$ and not on the quantifier free formula chosen to describe it). This is an easy consequence of the Tarski-Seidenberg transfer principle (see for instance [@BPR03]).
We will also need the following remark about extensions which is again a consequence of the Tarski-Seidenberg transfer principle.
\[rem:transfer\] Let $S,T$ be two closed and bounded semi-algebraic subsets of ${\mbox{\rm R}}^k$, and let $R'$ be a real closed extension of ${\mbox{\rm R}}$. Then, $S$ and $T$ are semi-algebraically homotopy equivalent if and only if ${{\rm Ext}}(S,{\mbox{\rm R}}')$ and ${{\rm Ext}}(T,{\mbox{\rm R}}')$ are semi-algebraically homotopy equivalent.
We will need a few results from algebraic topology, which we state here without proof referring the reader to papers where the proofs appear.
The following inequalities are consequences of the Mayer-Vietoris exact sequence.
Betti numbers and Mayer-Vietoris Inequalities
---------------------------------------------
We will use the following notation.
\[not:\[m\]\] For each $m \in {{\mathbb Z}}_{\geq 0}$ we will denote by $[m]$ the set $\{1,\ldots, m\}$.
\[prop:MV\] Let the subsets $W_1, \ldots , W_r \subset {\mbox{\rm R}}^n$ be all open or all closed. Then, for each $i \geq 0$ we have, $$\label{eq:MV1}
{b}_i \left( \bigcup_{1 \le j \le r} W_j \right) \le \sum_{J \subset [r]}
{b}_{i- (\# J) +1} \left( \bigcap_{j \in J} W_j \right)$$ and $$\label{eq:MV2}
{b}_i \left( \bigcap_{1 \le j \le r} W_j \right) \le \sum_{J \subset [r]}
{b}_{i + (\# J) -1} \left( \bigcup_{j \in J} W_j \right).$$
See [@BPR03].
The following proposition gives a bound on the Betti numbers of the projection $\pi(V)$ of a closed and bounded semi-algebraic set $V$ in terms of the number and degrees of polynomials defining $V$.
[@GVZ] \[prop:GVZ\] Let $R$ be a real closed field and let $\psi:{\mbox{\rm R}}^{m + k} \to {\mbox{\rm R}}^k$ be the projection map on to last $k$ co-ordinates. Let $V \subset {{\mbox{\rm R}}}^{m+k}$ be a closed and bounded semi-algebraic set defined by a Boolean formula with $s$ distinct polynomials of degrees not exceeding $d$. Then the $n$-th Betti number of the projection $${\rm b}_n (\psi (V)) \le (nsd)^{O(k+nm)}.$$
See [@GVZ].
Stable homotopy equivalence {#subsec:stable}
---------------------------
For any finite CW-complex $X$ we will denote by ${{\mathbf S}}(X)$ the suspension of $X$.
Recall from [@Spanier-Whitehead] that for two finite CW-complexes $X$ and $Y$, an element of $$\label{eqn:defofS-maps}
\{X;Y\}= \varinjlim_i \; [{{\mathbf S}}^i(X),{{\mathbf S}}^i(Y)]$$ is called an [*S-map*]{} (or map in the [*suspension category*]{}). (When the context is clear we will sometime denote an S-map $f \in \{X;Y\}$ by $f: X \rightarrow Y$).
\[def:S-equivalence\] An S-map $f \in \{X;Y\}$ is an S-equivalence (also called a stable homotopy equivalence) if it admits an inverse $f^{-1} \in \{Y;X\}$. In this case we say that $X$ and $Y$ are stable homotopy equivalent.
If $f \in \{X;Y\}$ is an S-map, then $f$ induces a homomorphism, $$f_* : {{\rm H}}_*(X) \rightarrow {{\rm H}}_*(Y).$$
The following theorem characterizes stable homotopy equivalence in terms of homology.
[@Spanier] \[the:stable\] Let $X$ and $Y$ be two finite CW-complexes. Then $X$ and $Y$ are stable homotopy equivalent if and only if there exists an S-map $f \in \{X;Y\}$ which induces isomorphisms $f_* : {{\rm H}}_i(X) \rightarrow {{\rm H}}_i(Y)$ (see [@Dieudonne], pp. 604).
### Spanier-Whitehead duality
In order to compare the complements of closed and bounded semi-algebraic sets which are homotopy equivalent, we will use the duality theory due to Spanier and Whitehead [@Spanier-Whitehead]. We will need the following facts about Spanier-Whitehead duality (see [@Dieudonne], pp. 603 for more details). Let $X \subset {\mbox{${\bf S}$}}^n$ be a finite CW-complex. Then there exists (up to stable homotopy equivalence) a dual complex, denoted $D_n X \subset {\mbox{${\bf S}$}}^n \setminus X$. The dual complex $D_n X$ is defined only up to S-equivalence. In particular, any deformation retract of ${\mbox{${\bf S}$}}^n \setminus X$ represents $D_n X$. Moreover, the functor $D_n$ has the following property. If $Y \subset {\mbox{${\bf S}$}}^n$ is another finite CW-complex, and the S-map represented by $\phi: X \rightarrow Y$ is a stable homotopy equivalence, then there exists a stable homotopy equivalence $D_n \phi$. Moreover, if the map $\phi: X \rightarrow Y$ is an inclusion, then the dual S-map $D_n \phi$ is also represented by a corresponding inclusion.
\[rem:spanier-whitehead\] Note that, since Spanier-Whitehead duality theory deals only with finite polyhedra over ${{\mbox{${\Bbb R}^{}$}}}$, it extends without difficulty to general real closed fields using the Tarski-Seidenberg transfer principle.
Homotopy colimits
-----------------
Let ${\mathcal A} = \{A_1,\ldots,A_n\}$, where each $A_i$ is a sub-complex of a finite CW-complex.
Let $\Delta_{[n]}$ denote the standard simplex of dimension $n-1$ with vertices in $[n]$. For $I \subset [n]$, we denote by $\Delta_I$ the $(\#I-1)$-dimensional face of $\Delta_{[n]}$ corresponding to $I$, and by $A_I$ the CW-complex $\displaystyle{\bigcap_{i \in I} A_i}$.
The homotopy colimit, ${{\rm hocolim}}({{\mathcal A}})$, is a CW-complex defined as follows.
\[def:hocolimit\] $${{\rm hocolim}}({{\mathcal A}}) = {\mathop{\makebox[0pt]{\hskip 1.4em $\boldsymbol\cdot$}\bigcup}}_{I \subset [n]} \Delta_I \times A_I/\sim$$ where the equivalence relation $\sim$ is defined as follows.
For $I \subset J \subset [n]$, let $s_{I,J}: \Delta_I \hookrightarrow \Delta_J$ denote the inclusion map of the face $\Delta_I$ in $\Delta_J$, and let $i_{I,J}: A_J \hookrightarrow A_I$ denote the inclusion map of $A_J$ in $A_I$.
Given $({\mathbf s},{\mathbf{x}}) \in \Delta_I \times A_I$ and $({\mathbf t},{\mathbf{y}}) \in \Delta_J \times A_J$ with $I \subset J$, then $({\mathbf s},{\mathbf{x}}) \sim
({\mathbf t},{\mathbf{y}})$ if and only if ${\mathbf t} = s_{I,J}({\mathbf s})$ and ${\mathbf{x}}= i_{I,J}({\mathbf{y}})$.
We have a obvious map $$f: {{\rm hocolim}}({{\mathcal A}}) \longrightarrow {{\rm colim}}({{\mathcal A}}) = \bigcup_{i \in [n]} A_i$$ sending $({\mathbf s},{\mathbf{x}}) \mapsto {\mathbf{x}}$. It is a consequence of the Smale-Vietoris theorem [@Smale] that
\[lem:hocolimit1\] The map $$f: {{\rm hocolim}}({{\mathcal A}}) \longrightarrow {{\rm colim}}({{\mathcal A}}) = \bigcup_{i \in [n]} A_i$$ is a homotopy equivalence.
Now let ${{\mathcal A}}= \{A_1,\ldots,A_n\}$ (resp. ${\mathcal B} = \{B_1,\ldots,B_n\}$) be a set of sub-complexes of a finite CW-complex. For each $I \subset [n]$ let $f_I \in \{A_I;B_I\}$ be a stable homotopy equivalence, having the property that for each $I \subset J \subset [n]$, $f_J = f_I|_{A_J}$. Then, we have an induced S-map, $f \in \{{{\rm hocolim}}({{\mathcal A}});{{\rm hocolim}}({\mathcal B})\}$, and we have that
\[lem:hocolimit2\] The induced S-map $f\in \{{{\rm hocolim}}({{\mathcal A}}); {{\rm hocolim}}({\mathcal B})\}$ is a stable homotopy equivalence.
Using the Mayer-Vietoris exact sequence it is easy to see that if the $f_I$’s induce isomorphisms in homology, so does the map $f$. Now apply Theorem \[the:stable\].
Proof of Theorem \[the:main\]
=============================
Proof Strategy
--------------
The strategy underlying our proof of Theorem \[the:main\] is as follows. We first consider the special case of a semi-algebraic subset, $A \subset {\mbox{${\bf S}$}}^{\ell}$, defined by a disjunction of $m$ homogeneous quadratic inequalities restricted to the unit sphere in ${\mbox{\rm R}}^{\ell+1}$. We then show that there exists a closed and bounded semi-algebraic set $C'$ (see (\[eqn:defofC’\]) below for the precise definition of the semi-algebraic set $C'$), consisting of certain sphere bundles, glued along certain sub-sphere bundles, which is homotopy equivalent to $A$. The number of these sphere bundles, as well descriptions of their bases, are bounded polynomially in $\ell$ (for fixed $m$).
In the presence of parameters $X_1,\ldots,X_k$, the set $A$, as well as $C'$, will depend on the values of the parameters. However, using some basic homotopy properties of bundles, we show that the homotopy type of the set $C'$ stays invariant, under continuous deformation of the bases of the different sphere bundles which constitute $C'$. These bases also depend on the parameters, $X_1,\ldots,X_k$, but the degrees of the polynomials defining them have degrees bounded by $O(\ell d)$ in $X_1,\ldots,X_k$. Now, using techniques similar to those used in [@BV06], we are able to control the number of isotopy types of the bases which occur, as the parameters vary over ${\mbox{\rm R}}^k$. The bound on the number of isotopy types, also gives a bound on the number of possible homotopy types of the set $C'$ and hence of $A$, for different values of the parameter.
In order to prove the results for semi-algebraic sets defined by more general formulas than disjunctions of weak inequalities, we first use Spanier-Whitehead duality to obtain a bound in the case of conjunctions, and then use the construction of homotopy colimits to prove the theorem for general ${\mathcal P}$-closed sets. Because of the use of Spanier-Whitehead duality we get bounds on the number of stable homotopy types, rather than homotopy types.
Topology of sets defined by quadratic constraints
-------------------------------------------------
One of the main ideas behind our proof of Theorem \[the:main\] is to parametrize a construction introduced by Agrachev in [@Agrachev] while studying the topology of sets defined by (purely) quadratic inequalities (that is without the parameters $X_1,\ldots,X_k$ in our notation). However, we avoid construction of Leray spectral sequences as done in [@Agrachev]. For the rest of this section, we fix a set of polynomials $${\mathcal Q} = \{Q_1,\ldots,Q_{m}\} \subset {\mbox{\rm R}}[Y_0,\ldots,Y_\ell,X_1,\ldots,X_k]$$ which are homogeneous of degree $2$ in $Y_0,\ldots,Y_\ell$, and of degree at most $d$ in $X_1,\ldots,X_k$.
We will denote by $$Q = (Q_1,\ldots,Q_m): {\mbox{\rm R}}^{\ell+1} \times {\mbox{\rm R}}^k \rightarrow {\mbox{\rm R}}^m,$$ the map defined by the polynomials $Q_1,\ldots,Q_m$, and generally, for $I \subset \{1,\ldots, m\}$, we denote by $Q_I: {\mbox{\rm R}}^{\ell+1} \times {\mbox{\rm R}}^k \rightarrow {\mbox{\rm R}}^I$, the map whose co-ordinates are given by $Q_i$, $i \in I$. We will often drop the subscript $I$ from our notation, when $I = [m]$.
For any subset $I \subset [m]$, let $A_I \subset {\mbox{${\bf S}$}}^{\ell} \times {\mbox{\rm R}}^k$ be the semi-algebraic set defined by $$\label{eqn:defofA_I}
A_I = \bigcup_{i\in I}
\{ ({\mathbf{y}},{\mathbf{x}}) \;\mid\; |{\mathbf{y}}|=1\; \wedge\; Q_i({\mathbf{y}},{\mathbf{x}}) \leq 0\},$$ and let $$\label{eqn:defofOmega_I}
\Omega_I = \{\omega \in {\mbox{\rm R}}^{m} \mid |\omega| = 1,
\omega_i = 0, i \not\in I,
\omega_i \leq 0, i \in I\}.$$
For $\omega \in \Omega_I$ we denote by ${\omega}Q \in {\mbox{\rm R}}[Y_0,\ldots,Y_\ell,X_1,\ldots,X_k]$ the polynomial defined by $$\label{eqn:defofomegaQ}
{\omega} Q = \sum_{i=0}^{m} \omega_i Q_i.$$
For $(\omega,{\mathbf{x}}) \in F_I = \Omega_I \times {\mbox{\rm R}}^k$, we will denote by $\omega Q(\cdot,{\mathbf{x}})$ the quadratic form in $Y_0,\ldots,Y_\ell$ obtained from $\omega Q$ by specializing $X_i = {\mathbf{x}}_i, 1 \leq i \leq k$.
Let $B_I \subset \Omega_I \times {\mbox{${\bf S}$}}^{\ell} \times {\mbox{\rm R}}^k$ be the semi-algebraic set defined by
$$\label{eqn:defofB_I}
B_I = \{ (\omega,{\mathbf{y}},{\mathbf{x}})\mid \omega \in
\Omega_I,
{\mathbf{y}}\in {\mbox{${\bf S}$}}^{\ell},
{\mathbf{x}}\in{\mbox{\rm R}}^k, \; {\omega}Q({\mathbf{y}},{\mathbf{x}}) \geq 0\}.$$
We denote by $\phi_1: B_I \rightarrow F_I$ and $\phi_2: B_I \rightarrow {\mbox{${\bf S}$}}^{\ell} \times{\mbox{\rm R}}^k$ the two projection maps (see diagram below). $$\label{eqn:maindiagram}
\begin{diagram}
\node{}
\node{B_I} \arrow{sw,t}{\phi_{I,1}}\arrow{s,..}\arrow{se,t}{\phi_{I,2}} \\
\node{F_I = \Omega_I \times{\mbox{\rm R}}^k} \arrow{e} \node{{\mbox{\rm R}}^k} \node{{\mbox{${\bf S}$}}^{\ell} \times{\mbox{\rm R}}^k} \arrow{w} \\
\end{diagram}$$ The following key proposition was proved by Agrachev [@Agrachev] in the unparametrized situation, but as we see below it works in the parametrized case as well.
\[prop:homotopy2\] The map $\phi_2$ gives a homotopy equivalence between $B_I$ and $\phi_2(B_I) = A_I$.
In order to simplify notation we prove it in the case $I = [m]$, and the case for any other $I$ would follow immediately. We first prove that $\phi_2(B) = A.$ If $({\mathbf{y}},{\mathbf{x}}) \in A,$ then there exists some $i, 1 \leq i \leq m,$ such that $Q_i({\mathbf{y}},{\mathbf{x}}) \leq 0$. Then for $\omega = (-\delta_{1,i},\ldots,-\delta_{m,i})$ (where $\delta_{i,j} = 1$ if $i=j$, and $0$ otherwise), we see that $(\omega,{\mathbf{y}},{\mathbf{x}}) \in B$. Conversely, if $({\mathbf{y}},{\mathbf{x}}) \in \phi_2(B),$ then there exists $\omega = (\omega_1,\ldots,\omega_m) \in \Omega$ such that, $$\sum_{i=1}^m \omega_i Q_i({\mathbf{y}},{\mathbf{x}}) \geq 0.$$ Since, $\omega_i \leq 0, 1\leq i \leq m,$ and not all $\omega_i = 0$. This implies that $Q_i({\mathbf{y}},{\mathbf{x}}) \leq 0$ for some $i, 1 \leq i \leq m$. This shows that $({\mathbf{y}},{\mathbf{x}}) \in A$.
For $({\mathbf{y}},{\mathbf{x}}) \in \phi_2(B)$, the fiber $$\phi_2^{-1}({\mathbf{y}},{\mathbf{x}}) = \{ (\omega,{\mathbf{y}},{\mathbf{x}}) \mid
\omega \in \Omega \;\mbox{such that} \; {\omega}Q({\mathbf{y}},{\mathbf{x}}) \geq 0\},$$ is a non-empty subset of $\Omega$ defined by a single linear inequality. Thus, each non-empty fiber is an intersection of a convex cone with ${\mbox{${\bf S}$}}^{m-1}$, and hence contractible.
The proposition now follows from the well-known Smale-Vietoris theorem [@Smale].
We will use the following notation.
For any quadratic form $Q \in {\mbox{\rm R}}[Y_0,\ldots,Y_\ell]$, we will denote by ${\rm index}(Q)$, the number of negative eigenvalues of the symmetric matrix of the corresponding bilinear form, that is of the matrix $M_Q$ such that, $Q({\mathbf{y}}) = \langle M_Q {\mathbf{y}}, {\mathbf{y}}\rangle$ for all ${\mathbf{y}}\in {\mbox{\rm R}}^{\ell+1}$ (here $\langle\cdot,\cdot\rangle$ denotes the usual inner product). We will also denote by $\lambda_i(Q), 0 \leq i \leq \ell$, the eigenvalues of $Q$, in non-decreasing order, i.e. $$\lambda_0(Q) \leq \lambda_1(Q) \leq \cdots \leq \lambda_\ell(Q).$$
For $I \subset [m]$, we denote by
$$\label{eqn:defofF_Ij}
F_{I,j} = \{(\omega,{\mathbf{x}}) \in \Omega_I \times {\mbox{\rm R}}^k \;
\mid \; {\rm index}({\omega}Q(\cdot,{\mathbf{x}})) \leq j \}.$$
It is clear that each $F_{I,j}$ is a closed semi-algebraic subset of $F_I$ and that they induce a filtration of the space $F_I$, given by $$F_{I,0} \subset F_{I,1} \subset \cdots \subset F_{I,\ell+1} =F_I.$$
\[lem:sphere\] The fiber of the map $\phi_{I,1}$ over a point $(\omega,{\mathbf{x}})\in F_{I,j}\setminus F_{I,j-1}$ has the homotopy type of a sphere of dimension $\ell-j$.
As before, we prove the lemma only for $I = [m]$. The proof for a general $I$ is identical. First notice that for $(\omega,{\mathbf{x}}) \in
F_{j}\setminus F_{j-1}$, the first $j$ eigenvalues of $\omega Q(\cdot,{\mathbf{x}})$, $$\lambda_0({\omega}Q(\cdot,{\mathbf{x}})),\ldots, \lambda_{j-1}({\omega}Q(\cdot,{\mathbf{x}})) < 0.$$ Moreover, letting $W_0({\omega}Q(\cdot,{\mathbf{x}})),\ldots,W_{\ell}({\omega}Q(\cdot,{\mathbf{x}}))$ be the co-ordinates with respect to an orthonormal basis, $e_0({\omega}Q(\cdot,{\mathbf{x}})),\ldots,e_{\ell}({\omega}Q(\cdot,{\mathbf{x}}))$ , consisting of eigenvectors of ${\omega}Q(\cdot,{\mathbf{x}})$, we have that $\phi_1^{-1}(\omega,{\mathbf{x}})$ is the subset of ${\mbox{${\bf S}$}}^{\ell} = \{\omega\} \times {\mbox{${\bf S}$}}^{\ell} \times \{{\mathbf{x}}\}$ defined by $$\displaylines{
\sum_{i=0}^{\ell} \lambda_i({\omega}Q(\cdot,{\mathbf{x}}))W_i({\omega}Q(\cdot,{\mathbf{x}}))^2 \geq 0, \cr
\sum_{i=0}^{\ell} W_i({\omega}Q(\cdot,{\mathbf{x}}))^2 = 1.
}$$ Since, $\lambda_i({\omega}Q(\cdot,{\mathbf{x}})) < 0, 0 \leq i < j,$ it follows that for $(\omega,{\mathbf{x}}) \in F_{j}\setminus F_{j-1}$, the fiber $\phi_1^{-1}(\omega,{\mathbf{x}})$ is homotopy equivalent to the $(\ell-j)$-dimensional sphere defined by setting $$W_0({\omega}Q(\cdot,{\mathbf{x}})) = \cdots = W_{j-1}({\omega}Q(\cdot,{\mathbf{x}})) = 0$$ on the sphere defined by $\sum_{i=0}^{\ell}W_i({\omega}Q(\cdot,{\mathbf{x}}))^2 = 1$.
For each $(\omega,{\mathbf{x}}) \in F_{I,j} \setminus F_{I,j-1}$, let $L_j^+(\omega,{\mathbf{x}}) \subset {\mbox{\rm R}}^{\ell+1}$ denote the sum of the non-negative eigenspaces of $\omega Q(\cdot,{\mathbf{x}})$ (i.e. $L_j^+(\omega,{\mathbf{x}})$ is the largest linear subspace of ${\mbox{\rm R}}^{\ell+1}$ on which $\omega Q(\cdot,{\mathbf{x}})$ is positive semi-definite). Since ${\rm index}(\omega Q(\cdot,{\mathbf{x}})) = j$ stays invariant as $(\omega,{\mathbf{x}})$ varies over $F_{I,j}\setminus F_{I,j-1}$, $L_j^+(\omega,{\mathbf{x}})$ varies continuously with $(\omega,{\mathbf{x}})$.
We will denote by $C_I$ the semi-algebraic set defined by
$$\label{eqn:definition_of_C}
C_I = \bigcup_{j=0}^{\ell+1} \{(\omega,{\mathbf{y}},{\mathbf{x}}) \;\mid\; (\omega,{\mathbf{x}}) \in
F_{I,j}\setminus F_{I,j-1},
{\mathbf{y}}\in L_j^+(\omega,{\mathbf{x}}), |{\mathbf{y}}| = 1\}.$$
The following proposition relates the homotopy type of $B_I$ to that of $C_I$.
\[prop:homotopy1\] The semi-algebraic set $C_I$ defined above is homotopy equivalent to $B_I$ (see (\[eqn:defofB\_I\]) for the definition of $B_I$).
We give a deformation retraction of $B_I$ to $C_I$ constructed as follows. For each $(\omega,x) \in F_{I,\ell} \setminus
F_{I,\ell-1}$, we can retract the fiber $\phi_1^{-1}(\omega,x)$ to the zero-dimensional sphere, $L_{\ell}^+(\omega,x) \cap {\mbox{${\bf S}$}}^{\ell}$ by the following retraction. Let $$W_0({\omega}Q_I(\cdot,x)),\ldots,W_{\ell}({\omega}Q_I(\cdot,x))$$ be the co-ordinates with respect to an orthonormal basis $e_0({\omega}Q(\cdot,{\mathbf{x}})),\ldots,e_{\ell}({\omega}Q(\cdot,{\mathbf{x}}))$, consisting of eigenvectors of ${\omega}Q_I(\cdot,x)$ corresponding to non-decreasing order of the eigenvalues of ${\omega}Q(\cdot,{\mathbf{x}})$. Then, $\phi_1^{-1}(\omega,x)$ is the subset of ${\mbox{${\bf S}$}}^{\ell}$ defined by $$\displaylines{
\sum_{i=0}^{\ell} \lambda_i({\omega}Q_I(\cdot,x))W_i({\omega}Q_I(\cdot,x))^2 \geq 0, \cr
\sum_{i=0}^{\ell} W_i({\omega}Q_I(\cdot,x))^2 = 1.
}$$ and $L_{\ell}^+(\omega,x)$ is defined by $W_0({\omega}Q_I(\cdot,x)) = \cdots =
W_{\ell-1}({\omega}Q_I(\cdot,x)) = 0$. We retract $\phi_1^{-1}(\omega,x)$ to the zero-dimensional sphere, $L_{\ell}^+(\omega,x) \cap {\mbox{${\bf S}$}}^{\ell}$ by the retraction sending, $(w_0,\ldots,w_\ell) \in \phi_1^{-1}(\omega,x)$, at time $t$ to $((1-t)w_0,\ldots,(1-t)w_{\ell-1},t'w_\ell)$, where $0 \leq t \leq 1$, and $
\displaystyle{
t' = \left(\frac{1 - (1-t)^2 \sum_{i=0}^{\ell-1}w_i^2}{w_\ell^2}\right)^{1/2}.
}
$ Notice that even though the local co-ordinates $(W_0,\ldots,W_\ell)$ in ${\mbox{\rm R}}^{\ell+1}$ with respect to the orthonormal basis $(e_0,\ldots,e_\ell)$ may not be uniquely defined at the point $(\omega,x)$ (for instance, if the quadratic form ${\omega}Q_I(\cdot,x)$ has multiple eigen-values), the retraction is still well-defined since it only depends on the decomposition of $R^{\ell+1}$ into orthogonal complements ${{\rm span}}(e_0,\ldots,e_{\ell-1})$ and ${{\rm span}}(e_\ell)$ which is well defined. We can thus retract simultaneously all fibers over $F_{I\ell} \setminus F_{I,\ell-1}$ continuously, to obtain a semi-algebraic set $B_{I,\ell} \subset B_I$, which is moreover homotopy equivalent to $B_I$.
This retraction is schematically shown in Figure \[fig:figure2\], where $F_{I,\ell}$ is the closed segment, and $F_{I,\ell-1}$ are its end points.
Now starting from $B_{I,\ell}$, retract all fibers over $F_{I,\ell-1} \setminus F_{I,\ell-2}$ to the corresponding one dimensional spheres, by the retraction sending, $(w_0,\ldots,w_\ell) \in \phi_1^{-1}(\omega,x)$, at time $t$ to $((1-t)w_0,\ldots,(1-t)w_{\ell-2},t'w_{\ell-1}, t'w_\ell)$, where $0 \leq t \leq 1$, and $
\displaystyle{
t' = \left(\frac{1 - (1-t)^2 \sum_{i=0}^{\ell-2}w_i^2}{\sum_{i=\ell-1}^{\ell}
w_i^2}\right)^{1/2}
}
$ to obtain $B_{I,\ell-1}$, which is homotopy equivalent to $B_{I,\ell}$. Continuing this process we finally obtain $B_{I,0} = C_I$, which is clearly homotopy equivalent to $B_I$ by construction.
Notice that the semi-algebraic set $\phi_1^{-1}(F_{I,j} \setminus F_{I,j-1})\cap C_I$ is a ${\mbox{${\bf S}$}}^{\ell - j}$-bundle over $F_{I,j} \setminus F_{I,j-1}$ under the map $\phi_1$, and $C_I$ is a union of these sphere bundles. We have good control over the bases, $F_{I,j} \setminus F_{I,j-1}$, of these bundles, that is we have good bounds on the number as well as the degrees of polynomials used to define them. However, these bundles could be possibly glued to each other in complicated ways, and it is not immediate how to control this glueing data, since different types of glueing could give rise to different homotopy types of the underlying space. In order to get around this difficulty, we consider certain closed subsets, $F_{I,j}'$ of $F_I$, where each $F_{I,j}'$ is an infinitesimal deformation of $F_{I,j} \setminus F_{I,j-1}$, and form the base of a ${\mbox{${\bf S}$}}^{\ell - j}$-bundle. Moreover, these new sphere bundles are glued to each other along sphere bundles over $F_{I,j}' \cap F_{I,j-1}'$, and their union, $C'_I$, is homotopy equivalent to $C_I$. Finally, the polynomials defining the sets $F_{I,j}'$ are in general position in a very strong sense, and this property is used later to bound the number of isotopy classes of the sets $F_{I,j}'$ in the parametrized situation.
We now make precise the argument outlined above. Let $\Lambda_I$ be the polynomial in $ {\mbox{\rm R}}[Z_1,\ldots,Z_m,X_1,\ldots,X_k,T]$ defined by $$\begin{aligned}
\Lambda_I &=& \det(M_{Z_I \cdot Q} + T\; {{\rm Id}}_{\ell+1}),\\
&=& T^{\ell+1} + H_{I,\ell} T^\ell + \cdots + H_{I,0},\end{aligned}$$ where $Z_I \cdot Q = \sum_{i \in I} Z_i Q_i$, and each $H_{I,j} \in {\mbox{\rm R}}[Z_1,\ldots,Z_m,X_1,\ldots,X_k]$.
Notice, that $H_{I,j}$ is obtained from $H_j = H_{[m],j}$ by setting for each $i \not\in I$, the variable $Z_i$ to $0$ in the polynomial $H_j$.
Note also that for $({\mathbf{z}},{\mathbf{x}}) \in {\mbox{\rm R}}^m\times{\mbox{\rm R}}^k$, the polynomial $\Lambda_I({\mathbf{z}},{\mathbf{x}},T)$ being the characteristic polynomial of a real symmetric matrix has all its roots real. It then follows from Descartes’ rule of signs (see for instance [@BPR03]), that for each $({\mathbf{z}},{\mathbf{x}}) \in {\mbox{\rm R}}^m \times {\mbox{\rm R}}^k$, where ${\mathbf{z}}_i = 0$ for all $ i \not\in I$, ${\rm index}({\mathbf{z}}Q(\cdot,{\mathbf{x}}))$ is determined by the sign vector $$({\rm sign}(H_{I,\ell}({\mathbf{z}},{\mathbf{x}})),\ldots,{\rm sign}(H_{I,0}({\mathbf{z}},{\mathbf{x}}))).$$ Hence, denoting by $$\label{eqn:defofH_I}
{\mathcal H}_I =
\{H_{I,0},\ldots,H_{I,\ell}\} \subset {\mbox{\rm R}}[Z_1,\ldots,Z_m,X_1,\ldots,X_k],$$ we have
For each $j, 0 \leq j \leq \ell+1$, $F_{I,j}$ is the intersection of $F_I$ with a ${\mathcal H_I}$-closed semi-algebraic set $D_{I,j} \subset {\mbox{\rm R}}^{m+k}$.
Let $D_{I,j}$ be defined by the formula $$\label{eqn:defofD_Ij}
D_{I,j} = \bigcup_{\sigma \in \Sigma_{I,j}} {{\mathcal R}}(\sigma),$$ for some $\Sigma_{I,j} \subset {\rm Sign}({\mathcal H_I})$. Note that, ${\rm Sign}({\mathcal H}_I) \subset {\rm Sign}({\mathcal H})$ and $\Sigma_{I,j} \subset \Sigma_j$ for all $I \subset [m]$.
Now, let $\bar{\delta}=(\delta_\ell,\ldots,\delta_0)$ and $\bar{{{\varepsilon}}}=({{\varepsilon}}_{\ell+1},\ldots,{{\varepsilon}}_0)$ be infinitesimals such that $$0 < \delta_0 \ll \cdots\ll \delta_{\ell} \ll {{\varepsilon}}_{0} \ll \cdots \ll {{\varepsilon}}_{\ell+1} \ll 1,$$ and let $$\label{eqn:defofR'}
{\mbox{\rm R}}' = {\mbox{\rm R}}{{\langle}}\bar{{{\varepsilon}}},\bar{\delta}{{\rangle}}$$
Given $\sigma \in {\rm Sign}({\mathcal H}_I)$, and $0 \leq j \leq \ell+1$, we denote by ${{\mathcal R}}(\sigma^c_j) \subset {\mbox{\rm R}}'^{m+k}$ the set defined by the formula $\sigma^c_j$ obtained by taking the conjunction of $$\begin{array}{l}
-{{\varepsilon}}_j - \delta_i \leq H_{I,i} \leq {{\varepsilon}}_j + \delta_i \mbox{ for each }
H_{I,i} \in {\mathcal H}_I
\mbox{ such that } \sigma(H_{I,i}) = 0, \cr
H_{I,i} \geq - {{\varepsilon}}_j - \delta_i, \mbox{ for each } H_{I,i} \in {\mathcal H}_I
\mbox{ such that } \sigma(H_{I,i}) = 1, \cr
H_{I,i} \leq {{\varepsilon}}_j + \delta_i, \mbox{ for each } H_{I,i} \in {\mathcal H}_I
\mbox{ such that } \sigma(H_{I,i}) = -1.
\end{array}$$
Similarly, we denote by ${{\mathcal R}}(\sigma^o_j) \subset {\mbox{\rm R}}'^{m+k}$ the set defined by the formula $\sigma^o$ obtained by taking the conjunction of $$\begin{array}{l}
- {{\varepsilon}}_j - \delta_i < H_{I,i} < {{\varepsilon}}_j + \delta_i \mbox{ for each }
H_{i,I} \in {\mathcal H}_I
\mbox{ such that } \sigma(H_{I,i}) = 0, \cr
H_{I,i} > - {{\varepsilon}}_j - \delta_i, \mbox{ for each } H_{I,i} \in {\mathcal H}_I
\mbox{ such that } \sigma(H_{I,i}) = 1, \cr
H_{I,i} < {{\varepsilon}}_j + \delta_i, \mbox{ for each } H_{I,i} \in {\mathcal H}_I
\mbox{ such that } \sigma(H_{I,i}) = -1.
\end{array}$$
For each $j, 0 \leq j \leq \ell+1$, let
$$\begin{aligned}
D_{I,j}^o &=& \bigcup_{\sigma \in \Sigma_{I,j}} {{\mathcal R}}(\sigma_j^o),\nonumber \\
D_{I,j}^c &=& \bigcup_{\sigma \in \Sigma_{I,j}} {{\mathcal R}}(\sigma_j^c), \nonumber\\
D_{I,j}' &=& D_{I,j}^c \setminus D_{I,j-1}^o,\nonumber \\
F_{I,j}' &=& {{\rm Ext}}(F_I,{\mbox{\rm R}}') \cap D_{I,j}'. \label{def:Fprime}\end{aligned}$$
where we denote by $D_{I,-1}^o = \emptyset~$. We also denote by $F'_I = {{\rm Ext}}(F_I,{\mbox{\rm R}}')$.
We now note some extra properties of the sets $D'_{I,j}$’s.
For each $j, 0 \leq j \leq \ell+1$, $D_{I,j}'$ is a ${\mathcal H}_I'$-closed semi-algebraic set, where $$\label{eqn:defofH_I'}
{\mathcal H}'_I = \bigcup_{i=0}^{\ell} \bigcup_{j=0}^{\ell+1}\{
H_{I,i} + {{\varepsilon}}_j + \delta_i,
H_{I,i} - {{\varepsilon}}_j -\delta_i\}.$$
Follows from the definition of the sets $D_{I,j}'$.
\[lem:local\] For $0 \leq j+1 < i \leq \ell+1$, $$D_{I,i}' \cap D_{I,j}' = \emptyset.$$
In order to keep notation simple we prove the proposition only for ${I = [m]}$. The proof for a general $I$ is identical. The inclusions, $$\displaylines{
D_{j-1} \subset D_j \subset D_{i-1} \subset D_i,\cr
D_{j-1}^o \subset
D_j^c \subset D_{i-1}^o \subset D_i^c.
}$$ follow directly from the definitions of the sets $$D_i,D_j,D_{j-1},D_i^c,D_j^c,D_{i-1}^o, D_{j-1}^o,$$ and the fact that, $${{\varepsilon}}_{j-1} \ll {{\varepsilon}}_{j} \ll {{\varepsilon}}_{i-1} \ll {{\varepsilon}}_{i}.$$
It follows immediately that, $$D_i' = D_i^c\setminus D_{i-1}^o$$ is disjoint from $D_j^c$, and hence from $D_j'$.
We now associate to each $F'_{I,j}$ a $(\ell - j)$-dimensional sphere bundle as follows. For each $(\omega,{\mathbf{x}}) \in F_{I,j}'' = F_{I,j}\setminus F'_{I,j-1}$, let $L_j^+(\omega,{\mathbf{x}}) \subset {\mbox{\rm R}}^{\ell+1}$ denote the sum of the non-negative eigenspaces of $\omega Q(\cdot,{\mathbf{x}})$ (i.e. $L_j^+(\omega,{\mathbf{x}})$ is the largest linear subspace of ${\mbox{\rm R}}^{\ell+1}$ on which $\omega Q(\cdot,{\mathbf{x}})$ is positive semi-definite). Since ${\rm index}(\omega Q(\cdot,{\mathbf{x}})) = j$ stays invariant as $(\omega,{\mathbf{x}})$ varies over $F''_{I,j}$, $L_j^+(\omega,{\mathbf{x}})$ varies continuously with $(\omega,{\mathbf{x}})$.
Let, $$\lambda_0(\omega,{\mathbf{x}}) \leq \cdots \leq \lambda_{j-1}(\omega,{\mathbf{x}}) < 0 \leq \lambda_j(\omega,{\mathbf{x}}) \leq \cdots \leq \lambda_{\ell}(\omega,{\mathbf{x}}),$$ be the eigenvalues of $\omega Q(\cdot,{\mathbf{x}})$ for $(\omega,{\mathbf{x}}) \in F''_{I,j}$. There is a continuous extension of the map sending $(\omega,{\mathbf{x}}) \mapsto L_j^+(\omega,{\mathbf{x}})$ to $(\omega,{\mathbf{x}}) \in F'_{I,j}$.
To see this observe that for $(\omega,{\mathbf{x}}) \in F''_{I,j}$ the block of the first $j$ (negative) eigenvalues, ${\lambda_0(\omega,{\mathbf{x}}) \leq \cdots \leq \lambda_{j-1}(\omega,{\mathbf{x}})}$, and hence the sum of the eigenspaces corresponding to them can be extended continuously to any infinitesimal neighborhood of $F''_{I,j}$, and in particular to $F'_{I,j}$. Now $L_j^+(\omega,{\mathbf{x}})$ is the orthogonal complement of the sum of the eigenspaces corresponding to the block of negative eigenvalues, $\lambda_0(\omega,{\mathbf{x}}) \leq \cdots \leq \lambda_{j-1}(\omega,{\mathbf{x}})$.
We will denote by $C'_{I,j}\subset F'_{I,j} \times {\mbox{\rm R}}'^{\ell+1}$ the semi-algebraic set defined by
$$\label{eqn:defofC_Ij'}
C'_{I,j} = \{(\omega,{\mathbf{y}},{\mathbf{x}}) \;\mid\; (\omega,{\mathbf{x}}) \in F'_{I,j},
{\mathbf{y}}\in L_j^+(\omega,{\mathbf{x}}), |{\mathbf{y}}| = 1\}.$$
Note that the projection $\pi_{I,j}: C'_{I,j} \rightarrow F'_{I,j}$, makes $C'_{I,j}$ the total space of a $(\ell - j)$-dimensional sphere bundle over $F'_{I,j}$.
Now observe that, $$C'_{I,j-1} \cap C'_{I,j} = \pi_{I,j}^{-1}( F'_{I,j} \cap F'_{I,j-1} ),$$ and $$\pi_{I,j}|_{C'_{I,j-1} \cap C'_{I,j}}:C'_{I,j-1} \cap C'_{I,j} \rightarrow
F'_{I,j} \cap F'_{I,j-1}$$ is also a $(\ell - j)$ dimensional sphere bundle over $F'_{I,j} \cap F'_{I,j-1}$.
Let $$\label{eqn:defofC'}
C'_I = \bigcup_{j=0}^{\ell+1} C'_{I,j}.$$
We have that
\[prop:homotopy3\] $C'_I$ is homotopy equivalent to ${{\rm Ext}}(C_I,{\mbox{\rm R}}')$, where $C_I$ and ${\mbox{\rm R}}'$ are defined in (\[eqn:definition\_of\_C\]) and (\[eqn:defofR’\]) respectively.
Let $\bar{{{\varepsilon}}}=({{\varepsilon}}_{\ell+1},\ldots,{{\varepsilon}}_0)$ and let $$R_i=
\begin{cases}
R{{\langle}}\bar{{{\varepsilon}}},\delta_\ell,\ldots,\delta_i{{\rangle}}\text{, }0\le i\le \ell,\\
R{{\langle}}{{\varepsilon}}_{\ell+1},\ldots,{{\varepsilon}}_{i-\ell-1} {{\rangle}}\text{, }\ell+1\le i\le 2\ell+2,\\
R \text{, }i = 2\ell+3.
\end{cases}$$ First observe that $C_I = \lim_{{{\varepsilon}}_{\ell+1}} C_I'$ where $C_I$ is the semi-algebraic set defined in (\[eqn:definition\_of\_C\]) above.
Now let, $$\begin{aligned}
C_{I,-1} &=& C_I',\\
C_{I,0} &=& \lim_{\delta_0} C_I', \\
C_{I,i} &=& \lim_{\delta_i} C_{I,i-1}, 1 \leq i \leq \ell, \\
C_{I,\ell+1} &=& \lim_{{{\varepsilon}}_0} C_{I,\ell}, \\
C_{I,i} &=& \lim_{{{\varepsilon}}_{i-\ell-2}} C_{I,i-1}, \ell+2\le i\le 2\ell+3.\end{aligned}$$
Notice that each $C_{I,i}$ is a closed and bounded semi-algebraic set. Also, for $i\ge 0$, let $C_{I,i-1,t} \subset {\mbox{\rm R}}_{i}^{m+\ell+k}$ be the semi-algebraic set obtained by replacing $\delta_i$ (resp., ${{\varepsilon}}_i$) in the definition of $C_{I,i-1}$ by the variable $t$. Then, there exists $t_0 > 0$, such that for all $0 < t_1 < t_2 \leq t_0$, $C_{I,i-1,t_1} \subset C_{I,i-1,t_2}$.
It follows (see Lemma 16.17 in [@BPR03]) that for each $i$, $0 \leq i \leq 2\ell+3$, ${{\rm Ext}}(C_{I,i}, {\mbox{\rm R}}_i)$ is homotopy equivalent to $C_{I,i-1}$.
### Partitioning the parameter space {#sec:whitney}
The goal of this section is to prove the following proposition (Proposition \[prop:main\]). The techniques used in the proof are similar to those used in [@BV06] for proving a similar result. We go through the proof in detail in order to extract the right bound in terms of the parameters $d,k,\ell$ and $m$.
\[prop:main\] There exists a finite set of points $T\subset{\mbox{\rm R}}^k$, with $$\# T \leq (2^m\ell k d)^{O(mk)},$$ such that for any ${\mathbf{x}}\in {\mbox{\rm R}}^k$, there exists ${\mathbf{z}}\in T$, with the following property.
There is a semi-algebraic path, $\gamma: [0,1] \rightarrow {\mbox{\rm R}}'^k$ and a continuous semi-algebraic map, $\phi: \Omega \times [0,1] \rightarrow \Omega $ (see (\[eqn:defofOmega\_I\]) and (\[eqn:defofR’\]) for the definition of $\Omega$ and ${\mbox{\rm R}}'$), with $\gamma(0) = {\mathbf{x}}$, $\gamma(1) = {\mathbf{z}}$, and for each $I \subset [m]$, $$\phi(\cdot,t)|_{F'_{I,j,{\mathbf{x}}}}: F'_{I,j,{\mathbf{x}}} \rightarrow F_{I,j,\gamma(t)}'$$ is a homeomorphism for each $0 \leq t \leq 1$.
Before proving Proposition \[prop:main\] we need a few preliminary results. Let $$\label{eqn:defofH''}
{\mathcal H}'' =
{\mathcal H}' \cup \{Z_1,\ldots,Z_m, Z_1^2 + \cdots + Z_m^2 -1\},$$ where ${\mathcal H}' = {\mathcal H}'_{[m]}$ is defined in (\[eqn:defofH\_I’\]) above.
Note that for each $j$, $0 \leq j \leq \ell+1$, $F'_{I,j}$ is a ${\mathcal H}''$-closed semi-algebraic set. Moreover, let $\psi:{\mbox{\rm R}}'^{m+k}\rightarrow{\mbox{\rm R}}'^k$ be the projection onto the last $k$ co-ordinates.
\[not:T\] We fix a finite set of points $T \subset {\mbox{\rm R}}^k$ such that for every ${\mathbf{x}}\in {\mbox{\rm R}}^k$ there exists ${\mathbf{z}}\in T$ such that for every $\mathcal{H}''$-semi-algebraic set $V$, the set $\psi^{-1}({\mathbf{x}})\cap V$ is homeomorphic to $\psi^{-1}({\mathbf{z}})\cap V$.
The existence of a finite set $T$ with this property follows from Hardt’s triviality theorem (Theorem \[the:hardt\]) and the Tarski-Seidenberg transfer principle, as well as the fact that the number of ${\mathcal H}''$-semi-algebraic sets is finite.
Now, we note some extra properties of the family ${\mathcal H}''$.
\[prop:A\] If $\sigma \in {\rm Sign}_{p}({\mathcal H}'')$, then $p \le k+m$ and ${{\mathcal R}}(\sigma) \subset {{\mbox{\rm R}}'}^{m+k}$ is a non-singular $(m+k-p)$-dimensional manifold such that at every point $({\mathbf{z}},{\mathbf{x}}) \in {{\mathcal R}}(\sigma)$, the $(p \times (m+k))$-Jacobi matrix, $$\left( \frac{\partial P}{\partial Z_i} , \frac{\partial P}{\partial Y_j}
\right)_{P \in{\mathcal H}'',\ \sigma(P) = 0,\
1\leq i \leq m,\ 1 \leq j \leq k}$$ has the maximal rank $p$.
Let ${{\rm Ext}}({\mbox{${\bf S}$}}^{m-1},{\mbox{\rm R}}')$ be the unit sphere in $R'^m$. Suppose without loss of generality that $$\{ P \in {\mathcal H}'' |\> \sigma (P)=0 \}= \{
H_{i_1}-{{\varepsilon}}_{j_1}-\delta_{i_1},
\ldots ,H_{i_{p-1}}-{{\varepsilon}}_{j_{p-1}}-\delta_{i_{p-1}}, \sum_{i=1}^m Z_i^2-1 \}$$ since the equation $Z_i=0$ eliminates the variable $Z_i$ from the polynomials. It follows that it suffices to show that the algebraic set $$\label{algset}
V=\bigcap_{r=1}^{p-1}\{ ({\mathbf{z}}, {\mathbf{x}})\in{{\rm Ext}}({\mbox{${\bf S}$}}^{m-1},{\mbox{\rm R}}')\times{\mbox{\rm R}}'^k \mid
H_{i_r}({\mathbf{z}}, {\mathbf{x}})={{\varepsilon}}_{j_r}+\delta_{i_r}
\}$$ is a smooth $((m-1)+k-(p-1))$-dimensional manifold such that at every point on it the $(p \times (m+k))$-Jacobi matrix, $$\left( \frac{\partial P}{\partial Z_i} , \frac{\partial P}{\partial Y_j}
\right)_{P \in{\mathcal H}'',\ \sigma(P) = 0,\
1\leq i \leq m,\ 1 \leq j \leq k}$$ has the maximal rank $p$.
Let $p \le m+k$. Consider the semi-algebraic map $P_{i_1,\ldots,i_{p-1}}: {\mbox{${\bf S}$}}^{m-1}\times{\mbox{\rm R}}^k \rightarrow {{\mbox{\rm R}}}^{p-1}$ defined by $$({\mathbf{z}},{\mathbf{x}}) \mapsto (H_{i_1}({\mathbf{z}},{\mathbf{x}}),\ldots,H_{i_{p-1}}({\mathbf{z}},{\mathbf{x}})).$$ By the semi-algebraic version of Sard’s theorem (see [@BCR]), the set of critical values of $P_{i_1,\ldots,i_{p-1}}$ is a semi-algebraic subset $C$ of ${{\mbox{\rm R}}}^{p-1}$ of dimension strictly less than $p-1$. Since $\bar\delta$ and $\bar{{\varepsilon}}$ are infinitesimals, it follows that $$({{\varepsilon}}_{j_1}+\delta_{i_1},\ldots,{{\varepsilon}}_{j_{p-1}}+\delta_{i_{p-1}})\notin{{\rm Ext}}(C,R').$$ Hence, the algebraic set $V$ defined in (\[algset\]) has the desired properties, and the same is true for the basic semi-algebraic set ${{\mathcal R}}(\sigma)$.
We now prove that $p \le m+k$. Suppose that $p > m+k$. As we have just proved, $$\{ H_{i_1}({\mathbf{z}}, {\mathbf{x}})={{\varepsilon}}_{j_1}+\delta_{i_1},\ldots ,
H_{i_{m+k-1}}({\mathbf{z}}, {\mathbf{x}})={{\varepsilon}}_{j_{m+k-1}}+\delta_{i_{m+k-1}} \}$$ is a finite set of points. But the polynomial $H_{i_{p-1}}-{{\varepsilon}}_{j_{p-1}}-\delta_{i_{p-1}}$ cannot vanish on each of these points as $\bar\delta$ and $\bar{{\varepsilon}}$ are infinitesimals.
\[prop:B\] For every ${\mathbf{x}}\in {\mbox{\rm R}}^k$, and $\sigma \in {\rm Sign}_{p}({\mathcal H}''_{{\mathbf{x}}})$, where $${\mathcal H}''_{{\mathbf{x}}}= \{ P(Z_1, \ldots ,Z_m, {\mathbf{x}})|\> P \in {\mathcal H}'' \},$$ the following holds.
1. $0 \leq p \leq m$, and ${{\mathcal R}}(\sigma) \cap \psi^{-1}({\mathbf{x}})$ is a non-singular $(m-p)$-dimensional manifold such that at every point $({\mathbf{z}},{\mathbf{x}}) \in {{\mathcal R}}(\sigma) \cap \psi^{-1}({\mathbf{x}})$, the [$(p \times m)$-Jacobi matrix]{}, $$\left( \frac{\partial P}{\partial Z_i} \right)_{P \in{\mathcal H}_{{\mathbf{x}}}'',
\sigma(P) = 0, 1 \leq i \leq m}$$ has the maximal rank $p$.
Note that $P_{{\mathbf{x}}}=P(Z_1, \ldots ,Z_m, {\mathbf{x}})\in{\mbox{\rm R}}'[Z_1,\ldots,Z_m]$ for each $P\in\mathcal{H}''$ and ${\mathbf{x}}\in{\mbox{\rm R}}^k$. The proof is now identical to the proof of Lemma \[prop:A\].
\[Whitney\] For any bounded ${\mathcal H}''$-semi-algebraic set $V$ defined by $$V = \bigcup_{\sigma \in \Sigma_V \subset {\rm Sign}({\mathcal H}'')}
{{\mathcal R}}(\sigma),$$ the partitions $$\begin{aligned}
\label{partition}
{\mbox{\rm R}}'^{m+k} &=& \bigcup_{ \sigma \in {\rm Sign}({\mathcal H}'')} {{\mathcal R}}(\sigma),\\
V &=& \bigcup_{ \sigma \in \Sigma_V} {{\mathcal R}}(\sigma),\end{aligned}$$ are compatible Whitney stratifications of ${\mbox{\rm R}}'^{m+k}$ and $V$ respectively.
Follows directly from the definition of Whitney stratification (see [@GM; @CS]), and Lemma \[prop:A\].
Fix some sign condition $\sigma \in {\rm Sign}({\mathcal H}'')$. Recall that $({\mathbf{z}},{\mathbf{x}}) \in {{\mathcal R}}(\sigma)$ is a [*critical point*]{} of the map $\psi_{{{{\mathcal R}}(\sigma)}}$ if the Jacobi matrix, $$\left( \frac{\partial P}{\partial Z_i} \right)_{P \in{\mathcal H}'',
\sigma(P) = 0,\
1 \leq i \leq m}$$ at $({\mathbf{z}}, {\mathbf{x}})$ is not of the maximal possible rank. The projection $\psi ({\mathbf{z}}, {\mathbf{x}})$ of a critical point is a [*critical value*]{} of $\psi_{{{{\mathcal R}}(\sigma)}}$.
Let $C_1\subset {\mbox{\rm R}}'^{m+k}$ be the set of critical points of $\psi_{{{{\mathcal R}}(\sigma)}}$ over all sign conditions $$\sigma \in \bigcup_{p \le m} {\rm Sign}_{p}({\mathcal H}''),$$ (i.e., over all $\sigma \in {\rm Sign}_{p}({\mathcal H}'')$ with $\dim ({{\mathcal R}}(\sigma)) \ge k $). For a bounded ${\mathcal H}''$-semi-algebraic set $V$, let $C_1(V)\subset V$ be the set of critical points of $\psi_{{{{\mathcal R}}(\sigma)}}$ over all sign conditions $$\sigma \in \bigcup_{p \le m} {\rm Sign}_{p}({\mathcal H}'')\cap
\Sigma_V$$ (i.e., over all $\sigma \in \Sigma_V$ with $\dim ({{\mathcal R}}(\sigma)) \ge k$).
Let $C_2 \subset {\mbox{\rm R}}'^{m+k}$ be the union of ${{\mathcal R}}(\sigma)$ over all $$\sigma \in \bigcup_{p > m} {\rm Sign}_{p}({\mathcal H}'')$$ (i.e., over all $\sigma \in {\rm Sign}_{p}({\mathcal H}'')$ with $\dim ({{\mathcal R}}(\sigma)) < k$). For a bounded ${\mathcal H}''$-semi-algebraic set $V$, let $C_2(V) \subset V$ be the union of ${{\mathcal R}}(\sigma)$ over all $$\sigma \in \bigcup_{p > m} {\rm Sign}_{p}({\mathcal H}'') \cap \Sigma_V$$ (i.e., over all $\sigma \in \Sigma_V$ with $\dim ({{\mathcal R}}(\sigma)) < k$).
Denote $C = C_1 \cup C_2$, and $C(V)= C_1(V) \cup C_2(V)$.
\[closed\] For each bounded ${\mathcal H}''$-semi-algebraic $V$, the set $C(V)$ is closed and bounded.
The set $C(V)$ is bounded since $V$ is bounded. The union $C_2(V)$ of strata of dimensions less than $k$ is closed since $V$ is closed.
Let $\sigma_1 \in {\rm Sign}_{p_1}({\mathcal H}'') \cap \Sigma_V$, $\sigma_2 \in {\rm Sign}_{p_2}({\mathcal H}'') \cap \Sigma_V$, where $p_1 \le m$, $p_1 < p_2$, and if $\sigma_1 (P)=0$, then $\sigma_2 (P)=0$ for any $P \in {\mathcal H}''$. It follows that stratum ${{\mathcal R}}(\sigma_2)$ lies in the closure of the stratum ${{\mathcal R}}(\sigma_1)$. Let ${\mathcal J}$ be the finite family of $(p_1 \times p_1)$-minors such that $Z( {\mathcal J}) \cap {{\mathcal R}}(\sigma_1)$ is the set of all critical points of $\pi_{{{\mathcal R}}(\sigma_1)}$. Then $Z( {\mathcal J}) \cap {{\mathcal R}}(\sigma_2)$ is either contained in $C_2(V)$ (when $\dim ({{\mathcal R}}(\sigma_2)) <k$), or is contained in the set of all critical points of $\pi_{{{\mathcal R}}(\sigma_2)}$ (when $\dim ({{\mathcal R}}(\sigma_2)) \ge k$). It follows that the closure of $Z( {\mathcal J}) \cap {{\mathcal R}}(\sigma_1)$ lies in the union of the following sets:
1. $Z( {\mathcal J}) \cap {{\mathcal R}}(\sigma_1)$,
2. \[case2\_a\] sets of critical points of some strata of dimensions less than $m+k- p_1$,
3. some strata of dimension less than $k$.
Using induction on descending dimensions in case (\[case2\_a\]), we conclude that the closure of $Z( {\mathcal J}) \cap {{\mathcal R}}(\sigma_1)$ is contained in $C(V)$. Hence, $C(V)$ is closed.
\[def:criticalvalues\] We denote by $G_i = \psi(C_i), i= 1,2$, and $G = G_1 \cup G_2$. Similarly, for each bounded ${\mathcal H}''$-semi-algebraic set $V$, we denote by $G_i(V) = \psi(C_i(V))$, ${i= 1,2}$, and $G(V) = G_1(V) \cup G_2(V)$.
\[representatives\] We have $T \cap G = \emptyset$. In particular, $T \cap G(V) = \emptyset$ for every bounded [${\mathcal H}''$-semi-algebraic]{} set $V$.
By Lemma \[prop:B\], for all ${\mathbf{x}}\in T$, and $\sigma \in {\rm Sign}_{p}({\mathcal H}_{{\mathbf{x}}}'')$,
1. \[lem:rep:1\] $0 \leq p \leq m$, and
2. \[lem:rep:2\] ${{\mathcal R}}(\sigma) \cap \psi^{-1}({\mathbf{x}})$ is a non-singular $(m-p)$-dimensional manifold such that at every point $({\mathbf{z}},{\mathbf{x}}) \in {{\mathcal R}}(\sigma) \cap \psi^{-1}({\mathbf{x}})$, the $(p \times m)$-Jacobi matrix, $$\left( \frac{\partial P}{\partial Z_i} \right)_{P \in{\mathcal H}_{{\mathbf{x}}}'',
\sigma(P) = 0, 1 \leq i \leq m}$$ has the maximal rank $p$.
If a point ${\mathbf{x}}\in T \cap G_1 = T \cap \psi(C_1)$, then there exists ${\mathbf{z}}\in {\mbox{\rm R}}'^m$ such that $({\mathbf{z}},{\mathbf{x}})$ is a critical point of $\psi_{{{\mathcal R}}(\sigma)}$ for some $\sigma \in \bigcup_{p \le m} {\rm Sign}_{p}({\mathcal H}'')$, and this is impossible by (\[lem:rep:2\]).
Similarly, ${\mathbf{x}}\in T \cap G_2 = T \cap \psi(C_2)$, implies that there exists ${\mathbf{z}}\in {\mbox{\rm R}}'^m$ such that $({\mathbf{z}},{\mathbf{x}}) \in {{\mathcal R}}(\sigma)$ for some $\sigma \in \bigcup_{p > m} {\rm Sign}_{p}({\mathcal H}'')$, and this is impossible by (\[lem:rep:1\]).
Let $D$ be a connected component of ${\mbox{\rm R}}'^{k} \setminus G$, and for a bounded ${\mathcal H}''$-semi-algebraic set $V$, let $D(V)$ be a connected component of $\psi(V) \setminus G(V)$.
\[lem:discriminant\] For every bounded ${\mathcal H}''$-semi-algebraic set $V$, all fibers $\psi^{-1}({\mathbf{x}}) \cap V$, ${\mathbf{x}}\in D$ are homeomorphic.
Lemma \[prop:B\] and Lemma \[Whitney\] imply that $\widehat V=\psi^{-1}(\psi(V) \setminus G(V))\cap V$ is a Whitney stratified set having strata of dimensions at least $k$. Moreover, $\psi|_{\widehat V}$ is a proper stratified submersion. By Thom’s first isotopy lemma (in the semi-algebraic version, over real closed fields [@CS]) the map $\psi|_{\widehat V}$ is a locally trivial fibration. In particular, all fibers $\psi^{-1}({\mathbf{x}})\cap V$, ${\mathbf{x}}\in D(V)$ are homeomorphic for every connected component $D(V)$. The lemma follows, since the inclusion $G(V) \subset G$ implies that either $D \subset D(V)$ for some connected component $D(V)$, or $D \cap \psi(V)= \emptyset$.
\[components\] For each ${\mathbf{x}}\in T$, there exists a connected component $D$ of ${\mbox{\rm R}}'^k \setminus G$, such that $\psi^{-1}({\mathbf{x}}) \cap V$ is homeomorphic to $\psi^{-1}({\mathbf{x}}_1) \cap V$ for every bounded ${\mathcal H}''$-semi-algebraic set $V$ and for every ${\mathbf{x}}_1 \in D$.
Let $V$ be a bounded ${\mathcal H}''$-semi-algebraic set and ${\mathbf{x}}\in T$. By Lemma \[representatives\], ${\mathbf{x}}$ belongs to some connected component $D$ of ${\mbox{\rm R}}'^k \setminus G$. Lemma \[lem:discriminant\] implies that $\psi^{-1}({\mathbf{x}}) \cap V$ is homeomorphic to $\psi^{-1}({\mathbf{x}}_1) \cap V$ for every ${\mathbf{x}}_1 \in D$.
We now are able to proof Proposition \[prop:main\].
Recall that $G=G_1 \cup G_2$, where $G_1$ is the union of sets of critical values of $\psi_{{{\mathcal R}}(\sigma)}$ over all strata ${{\mathcal R}}(\sigma)$ of dimensions at least $k$, and $G_2$ is the union of projections of all strata of dimensions less than $k$.
By Lemma \[components\] it suffices to bound the number of connected components of the set ${\mbox{\rm R}}'^k \setminus G$. Denote by ${\mathcal E}_1$ the family of closed sets of critical points of $\psi_{{\mathcal Z} (\sigma)}$, over all sign conditions $\sigma$ such that strata ${{\mathcal R}}(\sigma)$ have dimensions at least $k$ (the notation ${\mathcal Z} (\sigma)$ was introduced in Section \[sub:notations\]). Let ${\mathcal E}_2$ be the family of closed sets ${\mathcal Z} (\sigma)$, over all sign conditions $\sigma$ such that strata ${{\mathcal R}}(\sigma)$ have dimensions equal to $k-1$. Let ${\mathcal E}= {\mathcal E}_1\cup {\mathcal E}_2$. Denote by $E$ the image under the projection $\psi$ of the union of all sets in the family ${\mathcal E}$.
Because of the transversality condition, every stratum of the stratification of $V$, having the dimension less than $m+k$, lies in the closure of a stratum, having the next higher dimension. In particular, this is true for strata of dimensions less than $k-1$. It follows that $G \subset E$, and thus every connected component of the complement ${\mbox{\rm R}}'^k \setminus E$ is contained in a connected component of ${\mbox{\rm R}}'^k \setminus G$. Since $\dim (E)<k$, every connected component of ${\mbox{\rm R}}'^k \setminus G$ contains a connected component of ${\mbox{\rm R}}'^k \setminus E$. Therefore, it is sufficient to estimate from above the Betti number ${\rm b}_0 ({\mbox{\rm R}}'^k \setminus E)$ which is equal to ${\rm b}_{k-1}(E)$ by the Alexander’s duality.
The total number of sets ${\mathcal Z} (\sigma)$, such that $\sigma \in {\rm Sign}({\mathcal H}'')$ and $\dim ({\mathcal Z} (\sigma)) \ge k-1$, is $O(\ell^{2(m+1)})$ because each ${\mathcal Z} (\sigma)$ is defined by a conjunction of at most $m+1$ of possible $O(\ell^2+m)$ polynomial equations.
Thus, the cardinality $\# {\mathcal E}$, as well as the number of images under the projection $\pi$ of sets in ${\mathcal E}$ is $O(\ell^{2(m+1)})$. According to (\[eq:MV1\]) in Proposition \[prop:MV\], ${\rm b}_{k-1}(E)$ does not exceed the sum of certain Betti numbers of sets of the type $$\Phi =\bigcap_{1 \le i \le p} \pi (U_i),$$ where every $U_i \in {\mathcal E}$ and $1 \leq p \leq k$. More precisely, we have $$\displaylines{
{\rm b}_{k-1}(E) \;\leq \;\sum_{1 \le p \le k}\quad
\sum_{ \{ U_{1}, \ldots ,U_{p} \} \subset\ {\mathcal E}}
{\rm b}_{k-p} \left( \bigcap_{1 \le i \le p} \pi (U_i) \right).
}$$ Obviously, there are $O(\ell^{2(m+1)k})$ sets of the kind $\Phi$.
Using inequality (\[eq:MV2\]) in Proposition \[prop:MV\], we have that for each $\Phi$ as above, the Betti number ${\rm b}_{k-p}(\Phi)$ does not exceed the sum of certain Betti numbers of unions of the kind, $$\Psi = \bigcup_{1 \le j \le q} \pi (U_{i_j}) =
\pi \left( \bigcup_{1 \le j \le q} U_{i_j} \right),$$ with $1 \leq q \leq p$. More precisely, $$\begin{aligned}
{\rm b}_{k-p} (\Phi) &\;\leq\;&
\sum_{1 \le q \le p}\quad \sum_{1 \leq i_1 < \cdots< i_q \leq p}
{\rm b}_{k-p+q-1} \left( \pi \left( \bigcup_{1 \le j \le q} U_{i_j} \right) \right).\end{aligned}$$ It is clear that there are at most $2^{p} \leq 2^k$ sets of the kind $\Psi$.
If a set $U \in {\mathcal E}_1$, then it is defined by $m$ polynomials of degrees at most $O(\ell d)$. If a set $U \in {\mathcal E}_2$, then it is defined by $O(2^m)$ polynomials of degrees $O(m\ell d)$, since the critical points on strata of dimensions at least $k$ are defined by $O(2^m)$ determinantal equations, the corresponding matrices have orders $O(m)$, and the entries of these matrices are polynomials of degrees at most $O(\ell d)$.
It follows that the closed and bounded set $$\bigcup_{1 \le j \le q} U_{i_j}$$ is defined by $O(k2^m))$ polynomials of degrees $O(\ell d)$.
By Proposition \[prop:GVZ\], ${\rm b}_{k-p+q-1}(\Psi) \le (2^mk\ell d)^{O(mk)}$ for all $1 \le p \le k$, $1 \le q \le p$. Then ${\rm b}_{k-p} (\Phi) \le (2^mk\ell d)^{O(mk)}$ for every $1 \le p \le k$. Since there are $O(\ell^{2(m+1)k})$ sets of the kind $\Phi$, we get the claimed bound $${\rm b}_{k-1}(E) \le (2^mk\ell d)^{O(mk)}.$$ The rest of the proof follows from Proposition \[components\].
The Homogeneous Case {#subsec:homogeneous}
--------------------
We first consider the case, where all the polynomials in ${\mathcal Q}$ are homogeneous in variables $Y_0,\ldots,Y_\ell$ and we bound the number of homotopy types among the fibers $S_{{\mathbf{x}}}$, defined by the ${\mathcal Q}$-closed semi-algebraic subsets $S$ of ${\mbox{${\bf S}$}}^{\ell} \times {\mbox{\rm R}}^k$. We first the prove the following theorems for the special cases of unions and intersections.
\[the:union\] Let ${\mbox{\rm R}}$ be a real closed field and let $${\mathcal Q} = \{Q_1,\ldots,Q_m\} \subset {\mbox{\rm R}}[Y_0,\ldots,Y_\ell,X_1,\ldots,X_k],$$ where each $Q_i$ is homogeneous of degree $2$ in the variables $Y_0,\ldots,Y_\ell$, and of degree at most $d$ in $X_1,\ldots,X_k$.
For $i\in [m]$, let $A_i\subset {\mbox{${\bf S}$}}^{\ell} \times {\mbox{\rm R}}^k$ be semi-algebraic sets defined by $$A_i = \{ ({\mathbf{y}},{\mathbf{x}}) \;\mid\; |{\mathbf{y}}|=1\; \wedge\; Q_i({\mathbf{y}},{\mathbf{x}}) \leq 0)\},$$ Let $\pi: {\mbox{${\bf S}$}}^{\ell} \times {\mbox{\rm R}}^{k} \rightarrow {\mbox{\rm R}}^k$ be the projection on the last $k$ co-ordinates.
Then, the number of homotopy types amongst the fibers $\displaystyle{\bigcup_{i=1}^m A_{i,{\mathbf{x}}}}$ is bounded by $$(2^m\ell k d)^{O(mk)}.$$
With the same assumptions as in Theorem \[the:union\] we have
\[the:intersection\] The number of stable homotopy types amongst the fibers $\displaystyle{\bigcap_{i=1}^m A_{i,{\mathbf{x}}}}$ is bounded by $$(2^m\ell k d)^{O(mk)}.$$
Before proving Theorems \[the:union\] and \[the:intersection\] we first prove two preliminary lemmas.
\[lem:prelim\_union\] There exists a finite set $T \subset {\mbox{\rm R}}^k$, with $$\# T \leq (2^m\ell kd)^{O(mk)},$$ such that for every ${\mathbf{x}}\in {\mbox{\rm R}}^k$ there exists ${\mathbf{z}}\in T$, a semi-algebraic set $D_{{\mathbf{x}},{\mathbf{z}}} \subset {\mbox{\rm R}}'^{m+\ell}$, and semi-algebraic maps $f_{\mathbf{x}},f_{\mathbf{z}}$, as shown in the diagram below, such that $f_{\mathbf{x}},f_{\mathbf{z}}$ are both homotopy equivalences. $$\begin{diagram}
\node{}\node{D_{{\mathbf{x}},{\mathbf{z}}}}\arrow{sw,tb}{f_{\mathbf{x}}}{\sim}\arrow{se,tb}{f_{\mathbf{z}}}{\sim}
\node{}\\
\node{{{\rm Ext}}(\bigcup_{i \in [m]}A_{i,{\mathbf{x}}},{\mbox{\rm R}}')} \node{}
\node{{{\rm Ext}}(\bigcup_{i \in [m]}A_{i,{\mathbf{z}}},{\mbox{\rm R}}')}
\end{diagram}$$
Moreover, for each $I \subset [m]$, there exists a subset $D_{I,{\mathbf{x}},{\mathbf{z}}} \subset D_{{\mathbf{x}},{\mathbf{z}}}$, such that the restrictions, $f_{I,{\mathbf{x}}},f_{I,{\mathbf{z}}}$, of $f_{\mathbf{x}},f_{\mathbf{z}}$ to $D_{I,{\mathbf{x}},{\mathbf{z}}}$ give rise to the following diagram in which all maps are again homotopy equivalences.
$$\begin{diagram}
\node{}\node{D_{I,{\mathbf{x}},{\mathbf{z}}}}\arrow{sw,tb}{f_{I,{\mathbf{x}}}}{\sim}\arrow{se,tb}
{f_{I,{\mathbf{z}}}}{\sim}\node{}\\
\node{{{\rm Ext}}(\bigcup_{i \in I}A_{i,{\mathbf{x}}},{\mbox{\rm R}}')} \node{} \node{{{\rm Ext}}(
\bigcup_{i \in I}A_{i,{\mathbf{z}}},{\mbox{\rm R}}')}
\end{diagram}$$
For each $I \subset J \subset [m]$, $D_{I,{\mathbf{x}},{\mathbf{z}}} \subset D_{J,{\mathbf{x}},{\mathbf{z}}}$ and the maps $f_{I,{\mathbf{x}}},f_{I,{\mathbf{z}}}$ are restrictions of $f_{J,{\mathbf{x}}},f_{J,{\mathbf{z}}}$.
By Proposition \[prop:main\], there exists $T \subset {\mbox{\rm R}}^k$ with $$\#T \le (2^m\ell k d)^{O(m k)},$$ such that for every ${\mathbf{x}}\in {\mbox{\rm R}}^k$, there exists ${\mathbf{z}}\in T$, with the following property.
There is a semi-algebraic path, $\gamma: [0,1] \rightarrow {\mbox{\rm R}}'^k$ and a continuous semi-algebraic map, $\phi: \Omega \times [0,1] \rightarrow \Omega $, with $\gamma(0) = {\mathbf{x}}$, $\gamma(1) = {\mathbf{z}}$, and for each $I \subset [m]$, $$\phi(\cdot,t)|_{F'_{I,j,{\mathbf{x}}}}: F'_{I,j,{\mathbf{x}}} \rightarrow F'_{I,j,\gamma(t)},$$ is a homeomorphism for each $0 \leq t \leq 1$ (see (\[eqn:defofOmega\_I\]), (\[eqn:defofR’\]) and (\[def:Fprime\]) for the definition of $\Omega$, ${\mbox{\rm R}}'$ and $F'_{I,j}$). Now, observe that $C_{I,j,{\mathbf{x}}}'$ (resp. $C_{I,j,{\mathbf{z}}}'$) is a sphere bundle over $F_{I,j,{\mathbf{x}}}'$ (resp. $F_{I,j,{\mathbf{z}}}'$). Moreover $$C'_{I,j,{\mathbf{x}}} = \{(\omega,{\mathbf{y}}) \;\mid\; \omega \in F'_{I,j,{\mathbf{x}}},
{\mathbf{y}}\in L_j^+(\omega,{\mathbf{x}}), |{\mathbf{y}}| = 1\},$$ and, for $\omega \in F_{I,j,{\mathbf{x}}}' \cap F_{I,j-1,{\mathbf{x}}}'$, we have $L_j^+(\omega,{\mathbf{x}}) \subset L_{j-1}^+(\omega,{\mathbf{x}})$.
We now prove that the map $\phi$ induces a homeomorphism $\tilde{\phi}:C_{{\mathbf{x}}}' \rightarrow C_{{\mathbf{z}}}'$, which for each $I \subset [m]$ and $0 \leq j \leq \ell$, restricts to a homeomorphism ${\tilde{\phi}_{I,j}: C_{I,j,{\mathbf{x}}}' \rightarrow C_{I,j,{\mathbf{z}}}'}$.
First recall that by a standard result in the theory of bundles (see for instance, [@Fuks], p. 313, Lemma 5), the isomorphism class of the sphere bundle $C_{I,j,{\mathbf{x}}}' \rightarrow F_{I,j,{\mathbf{x}}}'$, is determined by the homotopy class of the map, $$\begin{aligned}
F_{I,j,{\mathbf{x}}}' &\rightarrow & Gr(\ell+1-j,\ell+1) \\
\omega &\mapsto& L_j^+(\omega,{\mathbf{x}}),\end{aligned}$$ where $Gr(m,n)$ denotes the Grassmannian variety of $m$ dimensional subspaces of ${\mbox{\rm R}}'^n$.
The map $\phi$ induces for each $j, 0 \leq j \leq \ell$, a homotopy between the maps $$\begin{aligned}
f_0:F_{I,j,{\mathbf{x}}}'&\rightarrow & Gr(\ell+1-j,\ell+1)\\
\omega &\mapsto & L_j^+(\omega,{\mathbf{x}})\end{aligned}$$ and $$\begin{aligned}
f_1:F_{I,j,{\mathbf{z}}}'&\rightarrow & Gr(\ell+1-j,\ell+1)\\
\omega & \mapsto & L_j^+(\omega,{\mathbf{z}})\end{aligned}$$ (after indentifying the sets $F_{I,j,{\mathbf{x}}}'$ and $F_{I,j,{\mathbf{z}}}'$ since they are homeomorphic) which respects the inclusions $L_j^+(\omega,{\mathbf{x}}) \subset L_{j-1}^+(\omega,{\mathbf{x}})$, and $L_j^+(\omega,{\mathbf{z}}) \subset L_{j-1}^+(\omega,{\mathbf{z}})$.
The above observation in conjunction with Lemma 5 in [@Fuks] is sufficient to prove the equivalence of the sphere bundles $C_{I,j,{\mathbf{x}}}'$ and $C_{I,j,{\mathbf{z}}}'$. But we need to prove a more general equivalence, involving all the sphere bundles $C_{I,j,{\mathbf{x}}}'$ simultaneously, for $0 \leq j \leq \ell$.
However, note that the proof of Lemma 5 in [@Fuks] proceeds by induction on the skeleton of the CW-complex of the base of the bundle. After choosing a sufficiently fine triangulation of the set $F_{I,{\mathbf{x}}}' \cong F_{I,{\mathbf{z}}}'$ compatible with the closed subsets $F_{I,j,{\mathbf{x}}}' \cong F_{I,j,{\mathbf{z}}}'$, the same proof extends without difficulty to this slightly more general situation to give a fiber preserving homeomorphism, $\tilde{\phi}:C_{{\mathbf{x}}}' \rightarrow C_{{\mathbf{z}}}'$, which restricts to an isomorphism of sphere bundles, $
\displaystyle{
\tilde{\phi}_{I,j}: C_{I,j,{\mathbf{x}}}' \rightarrow C_{I,j,{\mathbf{z}}}',
}
$ for each $I \subset [m]$ and $0 \leq j \leq \ell$.
We have the following maps.
$$\label{eqn:diagramofmaps}
\begin{diagram}
\node{{{\rm Ext}}(A_{{\mathbf{x}}},{\mbox{\rm R}}')} \node{{{\rm Ext}}(B_{{\mathbf{x}}},{\mbox{\rm R}}')}\arrow{w,t}{\phi_2}
\node{{{\rm Ext}}(C_{{\mathbf{x}}},{\mbox{\rm R}}')} \arrow{w,t}{i} \node{C_{{\mathbf{x}}}'}\arrow{w,t}{r}
\arrow{s,r}{\tilde{\phi}} \\
\node{{{\rm Ext}}(A_{{\mathbf{z}}},{\mbox{\rm R}}')} \node{{{\rm Ext}}(B_{{\mathbf{z}}},{\mbox{\rm R}}')}\arrow{w,t}{\phi_2}
\node{{{\rm Ext}}(C_{{\mathbf{z}}},{\mbox{\rm R}}')} \arrow{w,t}{i} \node{C_{{\mathbf{z}}}'}\arrow{w,t}{r}
\end{diagram}$$
The map $i$ is the inclusion map, and $r$ is a retraction shown to exist by Proposition \[prop:homotopy3\].
Since all the maps $\phi_2,i,r$ have been shown to be homotopy equivalences, by Propositions \[prop:homotopy2\], \[prop:homotopy1\], and \[prop:homotopy3\] respectively, their composition is also a homotopy equivalence.
Moreover, for each $I \subset [m]$, the maps in the above diagram restrict properly to give a corresponding diagram:
$$\label{eqn:diagramofmapsI}
\begin{diagram}
\node{{{\rm Ext}}(A_{I,{\mathbf{x}}},{\mbox{\rm R}}')} \node{{{\rm Ext}}(B_{I,{\mathbf{x}}},{\mbox{\rm R}}')}\arrow{w,t}{\phi_2}
\node{{{\rm Ext}}(C_{I,{\mathbf{x}}},{\mbox{\rm R}}')} \arrow{w,t}{i} \node{C_{I,{\mathbf{x}}}'}\arrow{w,t}{r}
\arrow{s,r}{\tilde{\phi}} \\
\node{{{\rm Ext}}(A_{I,{\mathbf{z}}},{\mbox{\rm R}}')} \node{{{\rm Ext}}(B_{I,{\mathbf{z}}},{\mbox{\rm R}}')}\arrow{w,t}{\phi_2}
\node{{{\rm Ext}}(C_{I,{\mathbf{z}}},{\mbox{\rm R}}')} \arrow{w,t}{i} \node{C_{I,{\mathbf{z}}}'}\arrow{w,t}{r}
\end{diagram}$$
Now let $D_{{\mathbf{x}},{\mathbf{z}}} = C_{{\mathbf{x}}}'$, and $f_{\mathbf{x}}= \phi_2 \circ i\circ r$ and $f_{\mathbf{z}}= \phi_2 \circ i\circ r \circ \tilde{\phi}$. Finally, for each $I \subset [m]$, let $D_{I,{\mathbf{x}},{\mathbf{z}}} = C_{I,{\mathbf{x}}}'$ and the maps $f_{I,{\mathbf{x}}}, f_{I,{\mathbf{z}}}$ the restrictions of $f_{\mathbf{x}}$ and $f_{\mathbf{z}}$ respectively to $D_{I,{\mathbf{x}},{\mathbf{z}}}$. The collection of sets $D_{I,{\mathbf{x}},{\mathbf{z}}}$ and the maps $f_{I,{\mathbf{x}}}, f_{I,{\mathbf{z}}}$ clearly satisfy the conditions of the lemma. This completes the proof of the lemma.
\[rem:prelim\_union\] Note that if ${\mbox{\rm R}}_1$ is a real closed sub-field of ${\mbox{\rm R}}$, then Lemma \[lem:prelim\_union\] continues to hold after we substitute “$T \subset {\mbox{\rm R}}_1^k$” and “for all ${\mathbf{x}}\in {\mbox{\rm R}}_1^k$” in place of “$T \subset {\mbox{\rm R}}^k$” and “for all ${\mathbf{x}}\in {\mbox{\rm R}}^k$” in the statement of the lemma. This is a consequence of the Tarski-Seidenberg transfer principle.
With the same hypothesis as in Lemma \[lem:prelim\_union\] we also have,
\[lem:prelim\_intersection\] There exists a finite set $T \subset {\mbox{\rm R}}^k$, with $$\# T \leq (2^m\ell kd)^{O(mk)},$$ such that for every ${\mathbf{x}}\in {\mbox{\rm R}}^k$ there exists ${\mathbf{z}}\in T$, for each $I \subset [m]$, a semi-algebraic set $E_{I,{\mathbf{x}},{\mathbf{z}}}$ defined over ${\mbox{\rm R}}''$, where ${\mbox{\rm R}}'' = {\mbox{\rm R}}{{\langle}}{{\varepsilon}},\bar{{{\varepsilon}}},\bar{\delta}{{\rangle}}$ (see (\[eqn:defofR’\] for the definition of $\bar{{{\varepsilon}}}$ and $\bar{\delta}$), and S-maps $g_{I,{\mathbf{x}}},g_{I,{\mathbf{z}}}$ as shown in the diagram below such that $g_{I,{\mathbf{x}}},g_{I,{\mathbf{z}}}$ are both stable homotopy equivalences. $$\begin{diagram}
\node{}\node{E_{I,{\mathbf{x}},{\mathbf{z}}}}
\node{}\\
\node{{{\rm Ext}}(\bigcap_{i \in I}A_{i, {\mathbf{x}}},{\mbox{\rm R}}'')}\arrow{ne,tb}{g_{\mathbf{x}}}{\sim}
\node{}
\node{{{\rm Ext}}(\bigcap_{i \in I}A_{i,{\mathbf{z}}},{\mbox{\rm R}}'')}\arrow{nw,tb}{g_{\mathbf{z}}}{\sim}
\end{diagram}$$
For each $I \subset J \subset [m]$, $E_{J,{\mathbf{x}},{\mathbf{z}}} \subset E_{I,{\mathbf{x}},{\mathbf{z}}}$ and the maps $g_{J,{\mathbf{x}}},g_{J,{\mathbf{z}}}$ are restrictions of of $g_{I,{\mathbf{x}}},g_{I,{\mathbf{z}}}$.
Let $1\gg {{\varepsilon}}> 0$ be an infinitesimal. For $1 \leq i \leq m$, we define $$\label{eqn:tildeQ}
\tilde{Q}_i = Q_i + {{\varepsilon}}(Y_0^2 + \cdots + Y_\ell^2),$$ $$\label{eqn:tildeA}
\tilde{A}_i = \{ ({\mathbf{y}},{\mathbf{x}}) \;\mid\; |{\mathbf{y}}|=1\; \wedge\; \tilde{Q_i}({\mathbf{y}},{\mathbf{x}})
\leq 0)\}.$$
Note that the set $\displaystyle{\bigcap_{i \in I} \tilde{A}_{i,{\mathbf{x}}}}$ is homotopy equivalent to $\displaystyle{{{\rm Ext}}(\bigcap_{i \in I} A_{i,{\mathbf{x}}},{\mbox{\rm R}}{{\langle}}{{\varepsilon}}{{\rangle}})}$ for each $I \subset [m]$ and ${\mathbf{x}}\in {\mbox{\rm R}}^k$. Applying Lemma \[lem:prelim\_union\] (see Remark \[rem:prelim\_union\]) to the family $\tilde{\mathcal Q} = \{-\tilde{Q}_1, \ldots, -\tilde{Q}_m\}$, we have, that there exists a finite set $T \subset {\mbox{\rm R}}^k$, with $$\# T \leq (2^m\ell kd)^{O(mk)},$$ such that for every ${\mathbf{x}}\in {\mbox{\rm R}}^k$ there exists ${\mathbf{z}}\in T$ such that for each $I \subset [m]$, the following diagram $$\begin{diagram}
\label{eqn:diaginlemma}
\node{}\node{\tilde{D}_{I,{\mathbf{x}},{\mathbf{z}}}}
\arrow{sw,tb}{\tilde{f}_{I,{\mathbf{x}}}}{\sim}
\arrow{se,tb}{\tilde{f}_{I,{\mathbf{z}}}}{\sim}
\node{}\\
\node{{{\rm Ext}}(\bigcup_{i \in I}\tilde{A}_{i,{\mathbf{x}}},{\mbox{\rm R}}'')} \node{}
\node{{{\rm Ext}}(\bigcup_{i \in I} \tilde{A}_{i,{\mathbf{z}}},{\mbox{\rm R}}'')}
\end{diagram}$$ where for each ${\mathbf{x}}\in {\mbox{\rm R}}^k$ we denote $$\tilde{A}_{i,{\mathbf{x}}} = \{ ({\mathbf{y}},{\mathbf{x}}) \;\mid\; |{\mathbf{y}}|=1\; \wedge\;
-\tilde{Q}_i({\mathbf{y}},{\mathbf{x}}) \leq 0)\},$$ $\tilde{f}_{I,{\mathbf{x}}},\tilde{f}_{I,{\mathbf{z}}}$ are homotopy equivalences.
Note that for each ${\mathbf{x}}\in {\mbox{\rm R}}^k$, the set $
\displaystyle{
{{\rm Ext}}(\bigcap_{i \in I} A_{i,{\mathbf{x}}},{\mbox{\rm R}}'')
}
$ is a deformation retract of the complement of $\displaystyle{{{\rm Ext}}(\bigcup_{i \in I}\tilde{A}_{i,{\mathbf{x}}},{\mbox{\rm R}}'')}$ and hence is Spanier-Whitehead dual to $\displaystyle{{{\rm Ext}}(\bigcup_{i \in I}\tilde{A}_{i,{\mathbf{x}}},{\mbox{\rm R}}'')}$. The lemma now follows by taking the Spanier-Whitehead dual of diagram (\[eqn:diaginlemma\]) above for each $I \subset [m]$.
Follows directly from Lemma \[lem:prelim\_union\].
Follows directly from Lemma \[lem:prelim\_intersection\].
We now prove a homogenous version of Theorem \[the:main\]
\[the:homogeneous\] Let ${\mbox{\rm R}}$ be a real closed field and let $${\mathcal Q} = \{Q_1,\ldots,Q_m\} \subset {\mbox{\rm R}}[Y_0,\ldots,Y_\ell,X_1,\ldots,X_k],$$ where each $Q_i$ is homogeneous of degree $2$ in the variables $Y_0,\ldots,Y_\ell$, and of degree at most $d$ in $X_1,\ldots,X_k$.
Let $\pi: {\mbox{${\bf S}$}}^{\ell} \times {\mbox{\rm R}}^{k} \rightarrow {\mbox{\rm R}}^k$ be the projection on the last $k$ co-ordinates. Then, for any ${\mathcal Q}$-closed semi-algebraic set $S \subset {\mbox{${\bf S}$}}^\ell\times{\mbox{\rm R}}^k$, the number of stable homotopy types amongst the fibers $S_{{\mathbf{x}}}$ is bounded by $$(2^m\ell k d)^{O(mk)}.$$
We first replace the family ${\mathcal Q}$ by the family, $${\mathcal Q}' = \{Q_1,\ldots,Q_{2m}\} =
\{Q,-Q \;\mid\; Q \in {\mathcal Q}\}.$$ Note that the cardinality of ${\mathcal Q}'$ is $2m$. Let $$A_i = \{ ({\mathbf{y}},{\mathbf{x}}) \;\mid\; |{\mathbf{y}}|=1\; \wedge\; Q_i({\mathbf{y}},{\mathbf{x}}) \leq 0)\}.$$ It follows from Lemma \[lem:prelim\_intersection\] that, there exists a set $T\subset{\mbox{\rm R}}^k$ and with $$\# T\le (2^m\ell k d)^{O(mk)}$$ such that for every $I \subset [2m]$ and ${\mathbf{x}}\in {\mbox{\rm R}}^k$, there exists ${\mathbf{z}}\in T$ and a semi-algebraic set $E_{I,{\mathbf{x}},{\mathbf{z}}}$ defined over ${\mbox{\rm R}}''={\mbox{\rm R}}{{\langle}}{{\varepsilon}},\bar{{{\varepsilon}}},\bar{\delta}{{\rangle}}$ and S-maps $g_{I,{\mathbf{x}}},g_{I,{\mathbf{z}}}$ as shown in the diagram below such that $g_{I,{\mathbf{x}}},g_{I,{\mathbf{z}}}$ are both stable homotopy equivalences. $$\begin{diagram}
\node{}\node{E_{I,{\mathbf{x}},{\mathbf{z}}}}\node{}\\
\node{{{\rm Ext}}(\bigcap_{i \in I} A_{i,{\mathbf{x}}},{\mbox{\rm R}}'')}\arrow{ne,tb}{g_{I,{\mathbf{x}}}}{\sim}
\node{}
\node{{{\rm Ext}}(\bigcap_{i \in I}A_{i,{\mathbf{z}}},{\mbox{\rm R}}'')}
\arrow{nw,tb}{g_{I,{\mathbf{z}}}}{\sim}
\end{diagram}$$
Now notice that each $\mathcal{Q}$-closed set $S$ is a union of sets of the form $\displaystyle{\bigcap_{i \in I} A_{i}}$ with $I \subset [2m]$. Let $$S = \bigcup_{I \in \Sigma \subset 2^{[2m]}} \bigcap_{i \in I} A_{i}.$$ Moreover, the intersection of any sub-collection of sets of the kind, $\bigcap_{i \in I} A_{i}$ with $I \subset [2m]$, is also a set of the same kind. More precisely, for any $\Sigma' \subset \Sigma$ there exists $I_{\Sigma'} \in 2^{[2m]}$ such that $$\bigcap_{I \in \Sigma'} \bigcap_{i \in I} A_{i} =
\bigcap_{i \in I_{\Sigma'}} A_{i}.$$
We are not able to show directly a stable homotopy equivalence between $S_{\mathbf{x}}$ and $S_{\mathbf{z}}$. Instead, we note that the S-maps $g_{I,{\mathbf{x}}}$ and $g_{I,{\mathbf{z}}}$ induce S-maps (cf. Definition \[def:hocolimit\]) $$\displaylines{
\tilde{g}_{\mathbf{x}}:
{{\rm hocolim}}(\{{{\rm Ext}}(\bigcap_{i \in I} A_{i,{\mathbf{x}}},{\mbox{\rm R}}'') \mid I \in \Sigma\}) \longrightarrow
{{\rm hocolim}}(\{ E_{I,{\mathbf{x}},{\mathbf{z}}}\mid I \in \Sigma \} ) \cr
\tilde{g}_{\mathbf{z}}:
{{\rm hocolim}}(\{{{\rm Ext}}(\bigcap_{i \in I} A_{i,{\mathbf{z}}},{\mbox{\rm R}}'') \mid I \in \Sigma\} ) \longrightarrow
{{\rm hocolim}}(\{E_{I,{\mathbf{x}},{\mathbf{z}}} \mid I \in \Sigma\} )
}$$
which are stable homotopy equivalences by Lemma \[lem:hocolimit2\] since each $g_{I,{\mathbf{x}}}$ and $g_{I,{\mathbf{z}}}$ is a stable homotopy equivalence.
Since $
\displaystyle{
{{\rm hocolim}}(\{ \bigcap_{i \in I} A_{i,{\mathbf{x}}} \mid I \in \Sigma\})
}$ (resp. $
\displaystyle{
{{\rm hocolim}}(\{ \bigcap_{i \in I} A_{i,{\mathbf{z}}} \mid I \in \Sigma\})
}
$) is homotopy equivalent by Lemma \[lem:hocolimit1\] to $
\displaystyle{
\bigcup_{I \in \Sigma} \bigcap_{i \in I} A_{i,{\mathbf{x}}}
}
$ (resp. $
\displaystyle{
\bigcup_{I \in \Sigma} \bigcap_{i \in I} A_{i,{\mathbf{z}}}
}
$), it follows (see Remark \[rem:transfer\]) that $
\displaystyle{
S_{\mathbf{x}}= \bigcup_{I \in \Sigma}\bigcap_{i \in I} A_{i,{\mathbf{x}}}
}
$ is stable homotopy equivalent to $
\displaystyle{
S_{\mathbf{z}}= \bigcup_{I \in \Sigma}\bigcap_{i \in I} A_{i,{\mathbf{z}}}
}
$. This proves the theorem.
Inhomogeneous case
------------------
We are now in a position to prove Theorem \[the:main\].
Let $\phi$ be a ${\mathcal P}$-closed formula defining the ${\mathcal P}$-closed semi-algebraic set $S \subset {\mbox{\rm R}}^{\ell+k}$. Let $1 \gg {{\varepsilon}}> 0$ be an infinitesimal, and let $$P_0={{\varepsilon}}^2\left(
\sum_{i=1}^\ell Y_i^2 + \sum_{i=1}^k X_i^2
\right) - 1.$$ Let $\tilde{\mathcal{P}}=\mathcal{P}\cup\{P_0\}$, and let $\tilde{\phi}$ be the $\tilde{\mathcal{P}}$-closed formula defined by $$\tilde{\phi}=\phi\wedge \{P_0\leq 0\},$$ defining the $\tilde{\mathcal{P}}$-closed semi-algebraic set $S_b\subset {\mbox{\rm R}}{{\langle}}{{\varepsilon}}{{\rangle}}^{\ell+k}$. Note that the set $S_b$ is bounded.
It follows from the local conical structure of semi-algebraic sets at infinity [@BCR] that the semi-algebraic set $S_b$ has the same homotopy type as ${{\rm Ext}}(S,{\mbox{\rm R}}{{\langle}}{{\varepsilon}}{{\rangle}})$.
Considering each $P_i$ as a polynomial in the variables $Y_1,\ldots,Y_\ell$ with coefficients in ${\mbox{\rm R}}[X_1,\ldots,X_k]$, and let $P_i^h$ denote the homogenization of $P_i$. Thus, the polynomials $P_i^h \in {\mbox{\rm R}}[Y_0,\ldots,Y_\ell,X_1,\ldots,X_k]$ and are homogeneous of degree $2$ in the variables $Y_0,\ldots,Y_\ell$.
Let $S_b^h\subset{\mbox{${\bf S}$}}^\ell\times{\mbox{\rm R}}{{\langle}}{{\varepsilon}}{{\rangle}}^{k}$ be the semi-algebraic set defined by the $\tilde{\mathcal{P}}^h$-closed formula $\tilde{\phi}^h$ (replacing $P_i$ by $P_i^h$ in $\tilde{\phi}$). It is clear that $S_b^h$ is a union of two disjoint, closed and bounded semi-algebraic sets each homeomorphic to $S_b$, which has the same homotopy type as ${{\rm Ext}}(S,{\mbox{\rm R}}{{\langle}}{{\varepsilon}}{{\rangle}})$.
The theorem is now proved by applying Theorem \[the:homogeneous\] to the family $\tilde{\mathcal{P}}^h$ and the semi-algebraic set $S_b^h$. Note that two fibers $S_{{\mathbf{x}}}$ and $S_{{\mathbf{y}}}$ are stable homotopy equivalent if and only if ${{\rm Ext}}(S_{{\mathbf{x}}},{\mbox{\rm R}}{{\langle}}{{\varepsilon}}{{\rangle}})$ and ${{\rm Ext}}(S_{{\mathbf{y}}},{\mbox{\rm R}}{{\langle}}{{\varepsilon}}{{\rangle}})$ are stable homotopy equivalent (see Remark \[rem:transfer\]).
Metric upper bounds {#sec:metric}
===================
In [@BV06] certain metric upper bounds related to homotopy types were proved as applications of the main result. Similar results hold in the quadratic case, except now the bounds have a better dependence on $\ell$. We state these results without proofs.
We first recall the following results from [@BV06]. Let $V \subset {{\mbox{\rm R}}}^\ell$ be a ${\mathcal P}$-semi-algebraic set, where ${\mathcal P} \subset {{\mathbb Z}}[Y_1, \ldots , Y_\ell]$. Let for each $P \in {\mathcal P}$, $\deg (P) < d$, and the maximum of the absolute values of coefficients in $P$ be less than some constant $M$, $0 < M \in {{\mathbb Z}}$. For $a > 0$ we denote by $B_\ell(0,a)$ the open ball of radius $a$ in ${\mbox{\rm R}}^\ell$ centered at the origin.
\[the:ball\] There exists a constant $c > 0$, such that for any $r_1 > r_2 > M^{d^{c\ell}}$ we have,
1. $V \cap B_\ell(0,r_1)$ and $V \cap B_\ell(0,r_2)$ are homotopy equivalent, and
2. $V \setminus B_\ell(0,r_1)$ and $V \setminus B_\ell(0,r_2)$ are homotopy equivalent.
In the special case of quadratic polynomials we get the following improvement of Theorem \[the:ball\].
\[the:ballquad\] Let ${\mbox{\rm R}}$ be a real closed field. Let $V \subset {{\mbox{\rm R}}}^\ell$ be a ${\mathcal P}$-semi-algebraic set, where $${\mathcal P} = \{P_1,\ldots,P_m\} \subset {\mbox{\rm R}}[Y_1,\ldots,Y_\ell],$$ with ${\rm deg}(P_i) \leq 2$, $1 \leq i \leq m$ and the maximum of the absolute values of coefficients in ${\mathcal P}$ is less than some constant $M$, $0 < M \in {{\mathbb Z}}$.
There exists a constant $c > 0$, such that for any $r_1 > r_2 > M^{\ell^{cm}}$ we have,
1. \[the:ballquad:1\] $V \cap B_\ell(0,r_1)$ and $V \cap B_\ell(0,r_2)$ are stable homotopy equivalent, and
2. \[the:ballquad:2\] $V \setminus B_\ell(0,r_1)$ and $V \setminus B_\ell(0,r_2)$ are stable homotopy equivalent.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The total number of globular clusters (GCs) in a galaxy rises continuously with the galaxy luminosity $L$, while the relative number of galaxies decreases with $L$ following the Schechter function. The product of these two very nonlinear functions gives the relative number of GCs contained by all galaxies at a given $L$. It is shown that GCs, in this universal sense, are most commonly found in galaxies within a narrow range around $L_{\star}$. In addition, blue (metal-poor) GCs outnumber the red (metal-richer) ones globally by 4 to 1 when all galaxies are added, pointing to the conclusion that the earliest stages of galaxy formation were especially favorable to forming massive, dense star clusters.'
author:
- 'William E. Harris'
title: 'Where Are Most of the Globular Clusters in Today’s Universe?'
---
Introduction
============
Galaxies with luminosities higher than $L \sim 10^7 L_{\odot}$ – in essence, all but the very smallest dwarfs – have measurable numbers of globular clusters (GCs), the massive compact star clusters that were preferentially formed during the earliest stages of star formation during galaxy evolution. The number of GCs present in a given galaxy increases dramatically with host galaxy mass or luminosity, but not in a simple linear way [@harris_etal13 hereafter HHA13]. At the same time, the relative number of galaxies decreases continuously with $L$ following the empirically based Schechter function [@schechter1976].
Combining these two opposing trends leads to a rather simple question: which galaxies contribute the most to the total number of globular clusters in the universe? Dwarf galaxies have very few GCs individually, but there are huge numbers of such galaxies. Contrarily, the biggest GC populations are to be found in central supergiant ellipticals like M87, but these are very rare galaxies. Which ones are the most important when added up over the entire galaxy population?
At the same time, we can address the question of the two classic subpopulations of GCs, the blue (metal-poor) and red (metal-rich) ones that are consistently seen to form a bimodal distribution in GC luminosity versus color [e.g. @brodie_strader06]. Color index increases monotonically with GC metallicity and thus is a useful proxy for \[Fe/H\], with the dividing line between blue and red near \[Fe/H\] $\simeq -1$. Whereas the blue GCs are consistently found in all galaxies from dwarfs to giants, the red ones reside preferentially in massive galaxies; quantitative discussions of this trend are given by, e.g., @peng_etal06 [@peng_etal08] and @harris_etal2015 (hereafter HHH15). Because the metal-poor blue GCs are found in all galaxies, it could therefore be expected that they would outnumber the metal-richer ones in total, but it is not immediately clear by how much.
In this paper, some simple GC demographics are calculated to gain a first answer to these questions. As will be seen below, the discussion draws heavily on recent observational gains that establish the numbers of blue, red, and all GCs within galaxies covering their entire luminosity range (see HHA13, HHA15). In what follows I have adopted a distance scale of $H_0 = 70$ km s$^{-1}$ Mpc$^{-1}$ wherever necessary.
Analysis
========
In Figure \[fig:ngctot\], the total number of GCs ($N_{GC}$) is plotted versus host galaxy luminosity ($L$), from data for 418 galaxies of all types as listed in the recent catalog of HHA13. Here, no discrimination is made by galaxy type (spiral, S0, E), but as shown in HHA13 and HHH15, differences by type appear to have only second-order effects. For about half the observed sample (n=216), the original observations are of sufficient photometric precision and depth to resolve the standard bimodal distribution in GC colors and thus to obtain the red and blue fractions as well.
{width="50.00000%"}
\[fig:ngctot\]
For our purposes here, a useful interpolation curve giving the trend of $N_{GC}$ versus $L(gal)$, is $${\rm log} N_{GC} =
\begin{cases}
-3.71 + 0.548 x & (x < 9.35), \\
-0.30 + 0.66 x - 0.1815 x^2 + 0.014 x^3 & (x \geq 9.35)
\end{cases}$$ where $x$ = log $(L_V/L_{\odot})$. A marked change in the slope of this relation happens near $L \sim 2 \times 10^9 L_{\odot}$; for the dwarfs fainter than that transition we find $N_{GC} \sim L^{0.5}$, while for large galaxies above it we find roughly $N_{GC} \sim L^{1.4}$. The ratio of these two quantities is the classic specific frequency $S_N = const (N_{GC}/L_V)$ [@harris_vandenbergh1981], which has a well known characteristic U-shaped dependence on $L$.
In Figure \[fig:redblue\], the same relation is shown but now divided into the blue and red subpopulations. The approximate interpolation curves shown in the Figure are, for the blue GCs, $${\rm log} N_{GC} =
\begin{cases}
-3.39 + 0.51 x & (x < 9.35), \\
15.418 - 3.84 x + 0.25 x^2 & (x \geq 9.35)
\end{cases}$$ and for the red GCs, $${\rm log} N_{GC} =
\begin{cases}
-3.75 + 0.48 x & (x < 9.35), \\
-13.31 + 1.50 x & (x \geq 9.35) \, .
\end{cases}$$
The Schechter function giving the relative number of galaxies per unit luminosity is $$\phi(L) dL \, = \, \phi_o ({L \over L_{\star}})^{\alpha} e^{-(L/L_{\star})} dL \, .$$ The parameters ($\alpha, L_{\star}$) may empirically depend somewhat on environment, but in this case the goal is simply to track the first-order behavior of GC populations averaged over all environments. The values adopted here are $\alpha = -1.26$ and $L_{\star} = 2.77 \times 10^{10} L_{\odot}$ from the SDSS DR6 database discussed by @montero-dorta2009; many other versions close to this pair of values can be found in the recent literature, but the precise numbers do not affect the results of the following discussion in any significant way.
{width="50.00000%"}
\[fig:redblue\]
Calculating the total number of GCs in all galaxies at a given $L$ is then a matter of multiplying Equation (4) numerically with either (1), (2), or (3) depending on which GC subpopulation we want to track. The results in smoothed histogram form are shown in Figure \[fig:nbin\], which gives the total number of GCs in all galaxies within a constant *logarithmic* bin size $\Delta {\rm log} L = 0.01$. In rough terms, this graph gives the relative probability that a globular cluster anywhere in the universe will be sitting in a host galaxy of luminosity (log $L$), or equivalently at a given absolute magnitude.
The shapes of all three curves in Fig. \[fig:nbin\] peak strongly at intermediate luminosities very near $L_{\star}$, with a long gradual ramp down towards the dwarf galaxies at lower $L$. Dwarf galaxies are very common but they do not have enough GCs per galaxy to dominate the totals; and contrarily, the highest$-L$ supergiant ellipticals have tens of thousands of GCs each but they are too rare to dominate. What is perhaps surprising is the height and relative sharpness of the population peak. Specifically, we find the following features:
- For all GCs combined, the peak is at log $(L/L_{\odot}) \simeq 10.53$ and 50% of the population lies in the range log $(L/L_{\odot}) = 9.86 - 10.95$ (a factor of 12 in $L$).
- For the blue GCs, the peak is at log $(L/L_{\odot}) \simeq 10.5$ and 50% of the population lies between log $(L/L_{\odot}) = 9.78 - 10.93$ (a factor of 14).
- For the red GCs, the peak is at log $(L/L_{\odot}) \simeq 10.6$ and 50% of them lie between log $(L/L_{\odot}) = 10.03 - 10.99$ (a factor of 9).
The reason why these peaks are rather high and narrow can be seen from Figs. \[fig:ngctot\] and \[fig:redblue\]. $N_{GC}$ begins rising steeply near log $(L/L_{\odot}) \sim 9.5$, which is still a decade below $L_{\star}$. Thus the Schechter function is still on the flat part of its curve ($L < L_{\star}$) and the number of galaxies at a given $L$ is declining only slowly. Once $L$ passes $L_{\star}$, however, the number of host galaxies declines so steeply that it forces all the curves in Fig. \[fig:nbin\] rapidly downward. In short, the galaxies near $L_{\star}$ provide the “best compromise” situation for GC populations in a universal sense: they have typically several hundred GCs per galaxy, and are still numerous enough cosmologically to dominate the GC totals.
The general appearance of Fig. \[fig:nbin\] to some extent resembles the @li_white2009 model calculation of the total amount of stellar mass contributed by galaxies of a given luminosity or baryonic mass (see their Fig. 5). In both cases the distribution is sharply peaked near $\L_{\star}$. However, the long tail towards low $L$ for the GC numbers is noticeably more prominent than for all stellar mass, reflecting the empirical fact that dwarf galaxies have higher average specific frequencies than $L_{\star}$-type ones.
For comparison, Figure \[fig:ntot\] presents the cumulative distribution. Half the population of blue GCs resides in galaxies with $L < 1.3 \times 10^{10} L_{\odot}$, whereas half the red GC population falls within galaxies with $L < 2.8 \times 10^{10} L_{\odot}$, a crossing point more than twice as high.
It is also noteworthy from Fig. \[fig:ntot\] that the blue, metal-poor GCs make up almost 80% of all globular clusters in the universe. A major reason for this predominance is that in the dwarf-galaxy regime (log $(L/L_{\odot} \lesssim 9.5$) there are almost no metal-rich GCs present, and it is only in the very luminous (and rare) supergiants that they make up comparable numbers to the metal-poor ones [see, e.g., @peng_etal06; @harris09a; @harris_etal2014 for recent examples].
{width="50.00000%"}
\[fig:nbin\]
{width="50.00000%"}
\[fig:ntot\]
In Figure \[fig:nbin\_mh\], another version of the probability distribution is shown, but now plotted versus galaxy halo mass $M_h$ rather than luminosity; $M_h$ is dominated by dark matter. The Figure shows the total *mass* in GCs within all galaxies at a given $M_h$ rather than the total *number*, but these are nearly equivalent given the very shallow increase of mean GC mass with galaxy mass (HHA13). This graph was generated through the combination of (a) a double-Schechter-function form of the number of galaxies per unit stellar mass $M_{\star}$, from [@kelvin_etal2014], (b) conversion of $M_{\star}$ to $M_h$ via the stellar-to-halo mass ratio SHMR with the convenient parametrization of @guo_etal2010, and finally (c) the total mass in GCs within a galaxy of a given $M_h$, which has a simple linear form (HHH15).
The graph indicates that GCs are most likely to be found within galaxy halos near $\sim 10^{13} M_{\odot}$, but the peak is much broader than in Fig. \[fig:nbin\], a result of the very nonlinear conversion of $M_{\star}$ (baryonic mass) to $M_h$. The slight upturn of the curve for $\lesssim 10^{11} M_{\odot}$ is quite uncertain (see HHH15 for a discussion of the data), but is partly determined by the steeper slope of the double Schechter function for the smallest dwarfs [@kelvin_etal2014]. However, this calculation is more or less arbitrarily cut off below $M_h = 10^{10} M_{\odot}$, since dwarfs below this limit have $< 1$ GC each according to the empirical evidence (see HHH15).
{width="50.00000%"}
\[fig:nbin\_mh\]
Summary and Discussion
======================
In this paper some broad-brush demographics of globular cluster populations are discussed; for the first time, it is possible to estimate quantitatively (though admittedly only to first order) which galaxies are responsible for contributing most of the GCs in the present-day universe. The combination of the nonlinear shapes of both the $N_{GC}$ versus $L$ function, and the Schechter function for galaxy numbers, demonstrates that galaxies in a narrow range around the Schechter $L_{\star}$ contribute the most.
Expressed in terms of galaxy halo mass (i.e. total mass) rather than luminosity, GCs are predominantly found within halos in the broad range $\sim 10^{12-14} M_{\odot}$, with the peak near $10^{13} M_{\odot}$.
The primary result of this discussion is shown in Fig. \[fig:nbin\]. It should be seen essentially as a snapshot in time, valid only for the present day: as the universe evolves and the continual process of galaxy merging continues, the biggest galaxies grow by absorbing their small neighbors. Thus over time, the peak in Fig. \[fig:nbin\] will shift to higher $L$ and the tail at lower $L$ will shrink. In the past at much higher redshift, the galaxy population was much more dominated by dwarfs and the GC population peak was correspondingly at lower $L$.
The result that the metal-poor GCs outnumber the metal-richer ones by a global ratio of roughly 4 to 1 is striking. The implication for galaxy evolution is that the very earliest stages of hierarchical merging, when baryonic matter was predominantly in quite low-metallicity gas, was exceptionally favorable for the formation of dense massive star clusters [see also @kruijssen2015]. The bulk of low-metallicity GC formation appears to have happened near $z \sim 5$ [e.g. @vandenberg_etal13; @forbes_etal2015 and references cited there], while the metal-richer “red” population predominated later near $z \sim 2-3$, much nearer the peak of the cosmic star formation rate [e.g. @madau_dickinson2014]. At that later time, the remaining gas was much more enriched, but it was not as successful at producing the extremely dense $\sim 10^5-10^7 M_{\odot}$ protocluster clouds within which GCs could form [@harris_pudritz1994; @kravtsov_gnedin05; @elmegreen2012; @li_gnedin14; @kruijssen2015].
In the discussion above, the term “globular cluster” is taken implicitly to mean classically old, massive star clusters. If we broaden that definition to include massive star clusters formed at any time, then it would be appropriate to include YMCs (young massive star clusters, also sometimes referred to as super star clusters in the literature) formed in low-redshift mergers between galaxies, as is seen in nearby active merger remnants [e.g. @trancho_etal2014; @trancho_etal2007; @whitmore_etal2014; @goudfrooij2012; @zepf_etal1999; @carlson_etal1998 among others]. These young GCs will add to the metal-rich GC population and to some extent increase their fraction of the universal population. However, even in the most prominent mergers (see the citations above for examples) only some dozens of clusters are added that are $\gtrsim 10^5 M_{\odot}$ and thus likely to survive for many Gyr. These will not add significantly to the old clusters already present when added up over all galaxies. In essence, the GC production in any merger old or young will depend critically on the amount of cold gas available, so most GC formation happened in the early universe [see @li_gnedin14 for discussion].
Lastly, the present discussion does not account for GCs either in the Intergalactic Medium (IGM) or Intracluster Medium (ICM), i.e. ones not definitely associated with any individual galaxy. True IGM clusters far from any galaxy are extremely hard to find and their numbers are generally presumed to be be very small, lacking any evidence to the contrary. ICM populations of GCs are also not well studied as yet, but are known to exist in a few rich clusters of galaxies such as Virgo or Coma [@peng_etal11; @durrell_etal2014; @west_etal2011; @alamo-martinez_etal2013]. The cases studied so far indicate that these ICM GCs add up to roughly the same numbers as are associated with the central Brightest Cluster Galaxy in their local environment, and would therefore not change the totals estimated here significantly.
Acknowledgements {#acknowledgements .unnumbered}
================
The author acknowledges financial support from NSERC (Natural Sciences and Engineering Research Council of Canada).
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Pedro Osuna, Jesus Salgado'
title: Dimensional Analysis applied to Spectrum Handling in Virtual Observatory Context
---
Introduction
============
Handling of units in spectra can be cumbersome when dealing with different Flux energy densities. For instance, the conversion between $F_{\lambda}$ and $F_{\nu}$ is normally done by parsing the unit name strings. Despite several efforts to homogenize unit name strings (See [@GREISEN2004], [@TAYLOR1995], [@GEORGE1995]), several different standards do exist, forcing the parsing-string mechanism to stick to one or several of those standards. On top of this, the adaptation of already existing legacy data to one or other standard might be a cumbersome -and even sometimes impossible- task.
The aforementioned problems led us to try and figure out a way to automatically handle units without the necessity to parse strings. Using dimensional analysis, we have devised an algorithmic way to convert between dimensionally homogeneous quantities. As a simple example, the string-parsing algorithm needed to convert between $W/cm^{2}/\mu m$ and $erg/cm^{2}/s/\AA$ (both $F_{\lambda}$) would be substituted for an algorithm to go from $10^{10} M L^{-1}
T^{-3}$ to $10^{7} M L^{-1} T^{-3}$, i.e., dividing by a factor of $10^{3}$.
In section 2 we give a general mathematical formalism of the relevant parts of Dimensional Analysis techniques that will be needed for the handling of this problem. In this section we will follow with very slight modifications the excellent book by [@SZIRTES1997]. We also describe the handling of the dimensional matrix to unveil how to extract dimensional relations between different units on the same problem. In section 3 we apply the theory to the case of $F_{\lambda}$ and $F_{\nu}$ conversions. And in section 4, we give a general algorithmic method for the conversion between different unit systems for the case of spectra together with details about the use of this technique in the VOSpec, a tool to handle SSAP (Simple Spectrum Access Protocol) compatible spectra developed at the European Space Astronomy Centre (ESAC) of the European Space Agency.
Dimensional Analysis overview
=============================
Dimensional analysis helps in the understanding of certain problems for which no analytic mathematical formulation exists, or for which the mathematical formulation is too complex. In these cases, the dimensional analysis allows to extract certain conclusions about the behavior of the system without the need of specific mathematical formulae relating the different variables of the problem at hand.
In physics we deal with quantities which have certain dimensions. These are combinations of a reduced number of basic or fundamental dimensions. These fundamental dimensions form a dimensional system. Dimensional systems can range from mono-dimensional (only a fundamental dimension is used to represent any physical quantity) to omni-dimensional (all dimensions are fundamental dimensions)(See ref [@SZIRTES1997] for examples). The intermediate, and mostly used, is the multidimensional system, in which a reduced set of fundamental dimensions is used. In the examples that will follow, we will adhere to the most widely used system of MASS-LENGTH-TIME system with the addition of Temperature, Electric current and luminosity to deal with other more complex problems. Following a long tradition in dimensional analysis (see [@MAXWELL1890]) will call M-L-T these fundamental dimensions. Following this convention, a physical quantity, e.g., a Force, would be represented as follows:
$[F] = [m] [a] = M L T^{-2 }$
where square brackets should be read as “dimensions of” and we will call the rightmost part the “dimensional equation” of the quantity under consideration.
Dimensional Matrix
------------------
Let $F(V_{1}, V_{2}, ..., V_{n})=0$ be a physical relation among a set of variable quantities $V_{i}$. The dimensional equations of the different variables will be:
$$\begin{aligned}
[V_{1}]&=&d_{1}^{\alpha_{11}}d_{2}^{\alpha_{12}}...d_{n}^{\alpha_{1n}}\\
\left [V_{2}\right ]&=&d_{1}^{\alpha_{21}}d_{2}^{\alpha_{22}}...d_{n}^{\alpha_{2n}}\\
...\\
\left [V_{n}\right
]&=&d_{1}^{\alpha_{m1}}d_{2}^{\alpha_{m2}}...d_{n}^{\alpha_{mn}}\end{aligned}$$
The matrix formed with the exponents of the dimensions is called the Dimensional Matrix:
$$\label{dimensional_matrix}
\mathbf{D} = \left [ \begin{array}{cccc}
\alpha_{11}&\alpha_{12}&\ldots&\alpha_{1n}\\
\alpha_{21}&\alpha_{22}&\ldots&\alpha_{2n}\\
\vdots&\vdots&\ldots&\vdots\\
\alpha_{m1}&\alpha_{m2}&\ldots&\alpha_{mn} \end{array} \right ]$$
From the dimensional matrix, we can now tackle the following problem: how can we find dimensional (or dimensionless) products of the variables at hand?. In other words, how do we find the $\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{n}$ exponents that solve the following equation:
$$\label{}
[V_{1}^{\varepsilon_{1}}V_{2}^{\varepsilon_{2}}...V_{n}^{\varepsilon_{n}}]=
d_{1}^{q_{1}}d_{2}^{q_{2}}...d_{p}^{q_{p}}$$
where: $V_{1},...,V_{n}$ are the variables of the problem, $d_{1},...,d_{p}$ are the fundamental dimensions of the problem, $q_{1},...,q_{p}$ are the *given* exponents (sought combinations of fundamental dimensions; to be set equal to zero for dimensionless products).
The problem is therefore reduced to solving for $\varepsilon_{j}$ the following system of linear equations:
$$\label{exp_eqns}
\sum_{j=1}^n \: D_{ij} \: \varepsilon_{j}\: =\: q_{i}$$
It can be shown that (see [@SZIRTES1997]) the solution to the above is:
$$\label{solution_for_epsilon}
\varepsilon_{j} \: = \:\sum_{k=1}^n \: E_{jk} \: \eta_{k}$$
where:
$$\mathbf{E} = \left [ \begin{array}{cccc} \mathbf{1} & \vdots&\mathbf{0} \\
\ldots&&\ldots\\
-\mathbf{A^{-1}}\mathbf{B} &\vdots& \mathbf{A^{-1}}
\end{array} \right ]$$
and $\mathbf{A}$ is a nonsingular square matrix of size the rank of the dimensional matrix, and $\mathbf{B}$ is formed by the rest of columns included in $\mathbf{D}$ and not included in $\mathbf{A}$. The sizes of the different matrices are:
$$\label{matrices}
\begin{array}{rlr}
\mathbf{E}:&(n)\times(n)&square \: matrix\nonumber\\
\mathbf{I}:&(n-p)\times(n-p)&identity \: matrix\nonumber\\
\mathbf{0}:&(n-p)\times(p) &null \: matrix\nonumber\\
\mathbf{A}:&(p)\times(p)&square \: matrix\nonumber\\
\mathbf{B}:&(p)\times(n-p)& \: matrix\\
\mathbf{A^{-1}B}:&(p)\times(n-p)&\: matrix\nonumber
\end{array}$$
the column matrix $\eta_{k}$ is formed by $\varepsilon_{n-p}$ arbitrary exponents followed by $q_{p}$ sought dimensional exponents. The first $\varepsilon_{n-p}$ are arbitrary due to the fact that $R_{D}\equiv rank(\mathbf{D})=p$ and therefore out of the n exponents we can only determine p, leaving n-p arbitrary. Therefore, $\eta_{k}$ is:
$$\eta_{k}=\left [ \begin{array}{l} \varepsilon_{1}\\
\varepsilon_{2}\\
\vdots\\ \varepsilon_{n-p}\ \\ q_{1} \\ q_{2}\\
\vdots \\ q_{p}\end{array} \right ]$$
The problem is thus reduced to finding the matrix $\mathbf{P}$ result of the product:
$$\label{exp_eqns}
P_{ij}\: = \: \sum_{k=1}^n \: E_{ik} \: Z_{kj}\:$$
where $E_{}ik$ is as defined earlier and $Z_{kj}$ is the result of superimposing the $\eta_{k}$ columns for the different variables:
$$\label{zeta_matrix}
\mathbf{Z} = \left [ \begin{array}{cccc}
\varepsilon_{11}&\varepsilon_{12}&\ldots&\varepsilon_{1p}\\
\varepsilon_{21}&\varepsilon_{22}&\ldots&\varepsilon_{2p}\\
\vdots&\vdots&\ldots&\vdots\\
\varepsilon_{(n-p)1}&\varepsilon_{(n-p)2}&\ldots&\varepsilon_{(n-p)p}\\
q_{1}&q_{1}&\ldots&q_{1}\\
q_{2}&q_{2}&\ldots&q_{2}\\
\vdots&\vdots&\ldots&\vdots\\
q_{p}&q_{p}&\ldots&q_{p}
\end{array} \right ]$$
both $\mathbf{P}$ and $\mathbf{Z}$ are of size nxp.
In the particular case where we are seeking for *dimensionless* products, all the $q_{i}$ will be set to zero (dimensional exponents) as a *dimensionless* quantity is considered to have dimension=1 (for a dissertation on the convenience to call dimensionless these type of quantities, check [@SZIRTES1997]).
Complete set of dimensional products: Buckingham theorem
--------------------------------------------------------
The so called Buckingham’s theorem reads as follows:
*The number of independent dimensional products $n_{\pi}$ which can be composed for a given number of variables and dimensions is:* $$\begin{array}{rlll}
n_{\pi}\:&=&n\:-\:R_{D}&for \:dimensionless\:products\\
n_{\pi}\:&=&n\:-\:R_{D}\:+\:1&for \:dimensional\:products
\end{array}$$
where $n$ is the number of variables and $R_{DM}$ is the rank of the dimensional matrix.
In this case, the sizes of the matrices at \[matrices\] would be:
$$\label{matrices}
\begin{array}{rlr}
\mathbf{I}:&(n-R_{D})\times(n-R_{D})&identity \: matrix\nonumber\\
\mathbf{0}:&(n-R_{D})\times(R_{D})&null \: matrix\nonumber\\
\mathbf{A}:&(R_{D})\times(R_{D})&square \: matrix\nonumber\\
\mathbf{B}:&(R_{D})\times(n-R_{D})& \: matrix\\
\mathbf{E}:&(n)\times(n)&square \: matrix\nonumber\\
\mathbf{A^{-1}B}:&(R_{D})\times(n-R_{D})& \: matrix\nonumber
\end{array}$$
An obvious advantage of finding the dimensionless products of the problem under investigation is that it reduces the number of parameters relevant to the system. In the case that the precise mathematical formula governing the behavior of the system is known, e.g., the Navier-Stokes equation for a fluid system, then the dimensionless products give information as well on the relative importance of each of the terms in the equation, helping in the simplification of the equations for certain combinations of the dimensionless products (for a beautiful example regarding the Navier-Stokes equations, see [@ROCHE1980]).
Application to Spectral Fluxes
==============================
Using the dimensional analysis principles, we will extract the dimensionless products for flux densities. These products will be used later, in section 4, to achieve the unit conversion between flux densities in different units.
$F_{\lambda}$ versus $\lambda$
------------------------------
To construct the dimensional matrix for $F_{\lambda}$ we will consider a spectral energy distribution of the form:
$$\label{f_l_relation}
F_{\lambda}(c,\lambda,h)=0$$
The dimensional equations of the different members of the previous relation are:
$$\label{fl_vs_l_dim}
[F_{\lambda}]=M L^{-1}T^{-3}, \;
[c]=LT^{-1},\;
[\lambda]=L,\;
[h]=ML^2T^{-1}$$
we now construct the following table:
$$\label{f_l_vs_l}
\begin{tabular}{ l | c c c c }
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
& $F_{\lambda}$ & $c$ & $\lambda$ & $h$\\
\hline
M & 1 & 0 & 0 & 1 \\
L & -1 & 1 & 1 & 2 \\
T & -3 & -1 & 0 & -1 \\
\end{tabular}$$
according to previous sections, we will have the following set of matrices:
$$\begin{aligned}
\mathbf{A} =\left[%
\begin{array}{rrr}
0 & 0 & 1 \\
1 & 1 & 2 \\
-1 & 0 & -1 \\
\end{array}%
\right] \hspace{2cm}
\mathbf{B} =\left[%
\begin{array}{r}
1 \\
-1 \\
-3 \\
\end{array}%
\right]\end{aligned}$$
$$\begin{aligned}
\mathbf{-A^{-1}B} =\left[%
\begin{array}{r}
-2 \\
5 \\
-1 \\
\end{array}%
\right] \hspace{2cm} \mathbf{E} =\left[%
\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
-2 & -1 & 0 & -1 \\
5 & -1 & 1 & 1 \\
-1 & 1 & 0 & 0 \\
\end{array}%
\right]\end{aligned}$$
$$\begin{aligned}
\mathbf{Z} =\left[%
\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}%
\right]\hspace{0.5cm} \mathbf{P} = \mathbf{E}*\mathbf{Z}=\left[%
\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
-2 & -2 & -2 & -2 \\
5 & 5 & 5 & 5 \\
-1 & -1 & -1 & -1 \nonumber\\
\end{array}%
\right]\end{aligned}$$
where following the general practice (see [@SZIRTES1997]) we have chosen the matrix A to be composed of the independent variables in relation (\[f\_l\_relation\]), and he matrix $\mathbf{P}$ gives the exponents(columns) on the dimensions (rows) that give rise to dimensionless products. According to the theory before, the number of dimensionless products in this case would be $n-R_{DM}$, i.e., 4-3=1, hence the fact that three of the columns are linear combinations of the remaining one (identical, in this case) in the P matrix only giving result to one dimensionless product combination:
$$\begin{aligned}
\begin{array}{lr}
%
\mathbf{P} =\left[%
\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
-2 & -2 & -2 & -2 \\
5 & 5 & 5 & 5 \\
-1 & -1 & -1 & -1 \\
\end{array}%
\right]
%
&
%
\end{array}\end{aligned}$$
where rows identify the variables in (\[fl\_vs\_l\_dim\]) and columns correspond to the exponents of those variables.
Therefore, the unique dimensional product thus for this case would be:
$$\label{dimless_fl_vs_l}
\pi = F_{\lambda}\frac{\lambda^{5}}{c^2 h}$$
$F_{\nu}$ versus $\lambda$
--------------------------
Following exactly the same procedure as before, we would have:
$$\label{f_n_vs_l}
\begin{tabular}{ l | c c c c }
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
& $F_{\nu}$ & $c$ & $\lambda$ & $h$\\
\hline
M & 1 & 0 & 0 & 1 \\
L & 0 & 1 & 1 & 2 \\
T & -2 & -1 & 0 & -1 \\
\end{tabular}$$
and therefore:
$$\begin{aligned}
\mathbf{A} =\left[%
\begin{array}{rrr}
0 & 0 & 1 \\
1 & 1 & 2 \\
-1 & 0 & -1 \\
\end{array}%
\right] \hspace{2cm}
\mathbf{B} =\left[%
\begin{array}{r}
1 \\
0 \\
-2 \\
\end{array}%
\right]\end{aligned}$$
$$\begin{aligned}
\mathbf{-A^{-1}B} =\left[%
\begin{array}{r}
-1 \\
3 \\
-1 \\
\end{array}%
\right] \hspace{2cm} \mathbf{E} =\left[%
\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
-1 & -1 & 0 & -1 \\
3 & -1 & 1 & 1 \\
-1 & 1 & 0 & 0 \\
\end{array}%
\right]\end{aligned}$$
$$\begin{aligned}
\mathbf{Z} =\left[%
\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}%
\right]\hspace{0.5cm} \mathbf{P} = \mathbf{E}*\mathbf{Z}=\left[%
\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
-1 & -1 & -1 & -1 \\
3 & 3 & 3 & 3 \\
-1 & -1 & -1 & -1 \nonumber\\
\end{array}%
\right]\end{aligned}$$
And therefore the only dimensionless product would be:
$$\label{dimless_fn_vs_l}
\pi' = F_{\nu}\frac{\lambda^{3}}{c h}$$
Both dimensionless quantities in (\[dimless\_fl\_vs\_l\]) and (\[dimless\_fn\_vs\_l\]) are descriptions of the same physical problem. As they are lineal in the dependent variable ($F_{\lambda}$ and $F_{\nu}$ respectively) we are able to conclude that they will be equivalent, i.e., that we can write in this case:
$\pi=\pi'$
and therefore:
$$\label{relation_fl_fn}
F_{\lambda}\frac{\lambda^{5}}{c^2 h} = F_{\nu}\frac{\lambda^{3}}{c
h} \:\:\Longrightarrow \:\: F_{\lambda}=\frac{c}{\lambda^2} F_{\nu}$$
which is the physical result expected when transforming flux densities.
The dimensionless product obtained when repeating the above procedure for $F_{\nu}$ and $F_{\lambda}$ with $\nu$ as the independent variable give:
$$\begin{aligned}
\pi=F_{\nu}\frac{c^{2}}{\nu^{3} h} \:\:\:\:\mbox{and}\:\:
\:\:\:\pi=F_{\lambda}\frac{c^{3}}{\nu^{5} h}
\:\:\:\mbox{respectively}\end{aligned}$$
This working example shows how the dimensional analysis can be used in the handling of spectrum fluxes and unit transformation. Much more complex problems can be tackled using this approach. For a nice compilation of literally hundreds of examples, consult the book by [@SZIRTES1997].
In what follows, we give an algorithm to bring the aforementioned ideas to practice when designing a client tool to handle Virtual Observatory spectra.
Comment on apparent velocity as X-axis spectral coordinate
----------------------------------------------------------
In the paper by [@GREISEN2004], the apparent radial velocity is considered as one of the possible spectral coordinates. A whole set of possible transformations between the different spectral coordinates (“x-axes”) is given together with their derivatives. The paper deals spectral coordinate transformations, rather than with flux transformations (on the “y-axis”). In this work, we deal with energy densities which depend, generally, on $\lambda$ or $\nu$.
Conversions in the x-axis to velocity space do need a central reference wavelength (frequency) which will have to accompany the metadata for the data. Once this value is known, the transformation of the x-axis will just consist of a translation plus a dilation, and the Y-axis will be kept as is. In the case that the data are coming with velocity in the x-axis, they will have to contain the reference lambda giving rise to those velocity values, and the process to convert the data to wavelength values would be simply inverted.
Algorithmic approach and use in VOSpec
======================================
In the Virtual Observatory context, there is a need to have a standard protocol to make spectra accessible from different projects in a simple way. The idea is to create a protocol for spectra in the same line as the already standard Simple Image Access Protocol (SIAP), that would allow for the creation of on-the-fly Spectral Energy Distributions (SEDs) from heterogeneous data sets.
This protocol has been called Simple Spectrum Access Protocol (SSAP) and basically implements a two-step process:
- In the first step, a cone search is done on available services and the match results are sent, together with metadata, in a VO standard VOTable.
- In the second step, the pointers to the real data files (spectra) are called and data are retrieved.
The main problem faced when trying to create an SED using data from different projects/formats is to compare data different units. We should be able to transform spectral coordinate units and fluxes to a common unit system.
Our proposed solution is to specify in the metadata (first SSAP step) for every spectrum, the dimensional equation for the spectrum axes, and use dimensional analysis to extract the conversion formulae needed to go from one to the other.
To prove that this on-the-fly conversion was possible, we developed a tool called VOSpec, able to request different SSAP servers and produce a common SED from different spectra in different formats from different projects.
The application is already available to the general public at: **http://esavo.esa.int/vospec/** and it has been used for the AVO demo which took place on Jan 25 & 26, 2005 at ESAC. See [@AVODEMO2005] for reference.
{width="9cm"}
At the time of writing this paper, spectra from **ISO**, **IUE**, **HST (FOS)**, **SDSS**, **HyperLeda** and **FUSE** projects are already providing SSAP access to their data. All the spectra from these services can be superimposed in the same display, and the user can generate on-the-fly SEDs as can be seen in Figure \[fig:VOSpec\]. Spectra from projects that don’t have SSAP services, can be loaded locally using the SSAP Wrapper Creator integrated in VOSpec.
Algorithmic approach
--------------------
There are several ways to describe a spectrum flux and the spectral coordinate inside a 1-D spectrum. A conversion table could be used to make the transformations accordingly, but it is not easy to use in an automatic algorithm, and this would limit the number of possible transformations allowed by the system.
In this section, we will describe an algorithmic way to approach the units problem from the dimensional point of view and show how this is used in the VOSpec application.\
A unit can be described in the following way:
$$[UNIT] = Scaling * M^{a}L^{b}T^{c}$$
where the scaling factor is defined with respect to a certain common system of units and the exponents a,b,c define the unit dimensionally. We will choose the SI as our base dimensional system of reference.
In order to understand how the algorithm works, suppose we are dealing with a spectrum in Jansky (y-axis) and Hertz (x-axis). The units dimensions (and scale factors) turn out to be:
$$\begin{aligned}
\nu (Hz) &=& T^{-1}\\
F_{\nu }(Jy) &=& 10^{-26} MT^{-2}\end{aligned}$$
where $1$ and $10^{-26}$ are the reference scaling factors to the SI units system for $Hz$ and $Jy$ respectively.
Suppose now, we want to convert one spectrum point defined by the pair $\left[\nu (Hz),F_{\nu }(Jy)\right]$ to other ones, e.g. $\left[\lambda (\mu m), F_{\lambda }(\frac{W}{cm^{2} \mu
m})\right]$, i.e.: $$\begin{aligned}
\lambda (\mu m) &=& 10^{-6} L \\
F_{\lambda }(\frac{W}{cm^{2} \mu m}) &=& 10^{10} ML^{-1}T^{-3}\end{aligned}$$
First we need to generate the matrix $E$ for the original system, as we saw in section 3:
$$\label{f_l_vs_l}
\begin{tabular}{ l | c c c c }
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
& $F_{\nu }$ & $c$ & $\nu $ & $h$\\
\hline
M & 1 & 0 & 0 & 1 \\
L & 0 & 1 & 0 & 2 \\
T & -2 & -1 & -1 & -1 \\
\end{tabular}$$
so, in this case: $$\begin{aligned}
\mathbf{A} =
\left[%
\begin{array}{rrr}
0&0&1\\
1&0&2\\
-1&-1&-1\\
\end{array}%
\right] \hspace{0.5cm} \mathbf{A^{-1}} =
\left[%
\begin{array}{rrr}
-2&1&0\\
1&-1&-1\\
1&0&0\\
\end{array}%
\right]\end{aligned}$$
and constructing $\mathbf{B}$ from the Flux density units:
$$\begin{aligned}
\mathbf{B} =
\left[%
\begin{array}{r}
1\\
0\\
-2\\
\end{array}%
\right] \hspace{0.5cm} \mathbf{-A^{-1} B} =
\left[%
\begin{array}{r}
2\\
-3\\
-1\\
\end{array}%
\right]\end{aligned}$$
so finally, constructing the matrix $\mathbf{E}$ using the rules described in section 3:
$$\begin{aligned}
\mathbf{E} =
\left[%
\begin{array}{rrrr}
1&0&0&0\\
2&-2&1&0\\
-3&1&-1&-1\\
-1&1&0&0\\
\end{array}%
\right]\end{aligned}$$
Now, to go from one system to the other, we have to generate two different Z matrices, one for the spectrum coordinate transformation and the second one for the flux transformation. As we are looking for certain target dimensions, the Z matrix will have to include the sought dimensional exponents, and therefore, as the final spectrum coordinates are $\mu m$ ($L$) we will have:
$$\begin{aligned}
\begin{array}{lr}
%
\mathbf{Z_{1}} =
\left[%
\begin{array}{r}
0 \\
0 \\
1 \\
0 \\
\end{array}%
\right]
%
&
%
\begin{array}{rrrr}
F_{\nu }\\
M \\
L\\
T\\
\end{array}%
\end{array}\end{aligned}$$
where the first zero is imposing no dependence in the flux, and the rest are the exponents in $M$, $L$, $T$ respectively.
For the final system of units, the $F_{\lambda }$ is $\frac{W}{cm^{2} \mu m}$ ($ML^{-1}T^{-3}$):
$$\begin{aligned}
\begin{array}{lr}
%
\mathbf{Z_{2}} =
\left[%
\begin{array}{r}
1 \\
1 \\
-1 \\
-3 \\
\end{array}%
\right]
%
&
%
\begin{array}{rrrr}
F_{\nu }\\
M \\
L\\
T\\
\end{array}%
\end{array}\end{aligned}$$
where, as it was defined in previous section, the first element is the dependence in the flux and the rest of the elements in the $Z$ are the dimensional equation exponents for $M$,$L$ and $T$ respectively.
If we multiply then the original E matrix with these Z vectors we obtain for the spectral coordinate:
$$\begin{aligned}
\begin{array}{lr}
%
\mathbf{P_1} = \mathbf{E * Z_1} =
\left[%
\begin{array}{r}
0 \\
1 \\
-1 \\
0 \\
\end{array}%
\right]
%
&
%
\begin{array}{rrrr}
F_{\nu }\\
c \\
\nu\\
h\\
\end{array}%
\end{array}\end{aligned}$$
and for the flux:
$$\begin{aligned}
\begin{array}{lr}
%
\mathbf{P_2} = \mathbf{E * Z_2} =
\left[%
\begin{array}{r}
1 \\
-1 \\
2 \\
0 \\
\end{array}%
\right]
%
&
%
\begin{array}{rrrr}
F_{\nu }\\
c \\
\nu\\
h\\
\end{array}%
\end{array}\end{aligned}$$
That means, respectively:
$$\begin{aligned}
\left[ \lambda \right] &=& \frac{\left[c\right]}{\left[ \nu \right] } \\
\left[ \frac{W}{cm^{2} \mu m} \right] &=& \left[ Jy\right] .
\frac{\left[\nu \right] ^2}{\left[c\right]}\end{aligned}$$
To finalize the transformation, we must include the scaling factors of the different units. To achieve this goal, we note that every time a magnitude is used, the scaling must appear, i.e.,
$$\begin{aligned}
S_{(\mu m)} \lambda (\mu m)&=& \frac{c}{S_{(Hz)} \nu (Hz)} \\
S_{(\frac{W}{cm^{2} \mu m})} F_{\lambda }(\frac{W}{cm^{2} \mu m})
&=& S_{(Jy)} F_{\nu }(Jy) . \frac{(S_{(Hz)} \nu (Hz))^2}{c}\end{aligned}$$
Where the S elements correspond to the scalings with respect to a common system of reference units (in this case SI).\
Finally we obtain:
$$\begin{aligned}
\lambda (\mu m) &=& \frac{1}{S_{(\mu m)}.S_{(Hz)}}
.\frac{c(m/s)}{\nu (Hz)} \\
&=& 10^6 . \frac{c(m/s)}{\nu (Hz)} \\
F_{\lambda }(\frac{W}{cm^{2} \mu m}) &=& \frac{S_{(Jy)} .
S^2_{(Hz)}}{S_{(\frac{W}{cm^{2} \mu m})}} F_{\nu }(Jy) \frac{\nu
^2
(Hz)}{c(m/s)} \\
&=& \frac{10^{-26}}{10^{10}} F_{\nu }(Jy) \frac{\nu ^2
(Hz)}{c(m/s)} \\
&=& 10^{-36} F_{\nu }(Jy) \frac{\nu ^2 (Hz)}{c(m/s)}\end{aligned}$$
These final formulae tell us how to express the values of a $\left[x,y\right]$ point in the final units as a function of the point in the original ones.
The algorithm can be summarized as follows:
1. Construct the matrix A, using the spectral coordinate, c and h dimensional equations.
2. Construct the vector B using the flux density dimensional equation.
3. Invert the matrix A, and construct the matrix E as it was described in point 1.
4. Construct two Z vectors using spectral coordinate and flux density dimensional equations of the final units.
5. Multiply the matrix E with the two Z vectors to obtain the conversion factors.
6. Finally, use the scaling factors to finish the conversion.
Conclusions
===========
We have shown how to make use of Dimensional Analysis techniques to handle unit conversion in an automated way for the case of spectral flux densities.
We have proposed the IVOA to include the SCALEQ and DIMEQ parameters as part of the Simple Spectrum Access Protocol so that clients can use Dimensional Analysis algorithms to handle units automatically.
The approach shown is not only relevant to the Spectral access within the VO, and can be extended -as mentioned in the introduction- to any physical problem by including the relevant dimensions.
In this sense, we imagine any unit as composed of three main attributes:
1. A name (and possibly a symbol)
2. A SCALEQ (giving the Scaling of the unit with respect to the International System of Units)
3. A DIMEQ (giving the dimensions of the unit)
An example of a serialization of this units model could be the FITS representation, in which a unit like the “Jansky” could be represented in the following way:
$$\begin{aligned}
\begin{array}{lll}
CUNIT='Jansky' \\
CSCALEQ='1E-26' \\
CDIMEQ='MT**-2'
\end{array}\end{aligned}$$
Certainly, to cover all possible physical units, more dimensions than M-L-T would be needed, but the set will always be far smaller than the units they can represent.
We acknowledge Jose Tomas Diez-Roche, Matteo Guainazzi and Andy Pollock for their useful comments and discussions.
[99]{} \[1\] E.W. Greisen, M.R. Calabretta, F.G. Valdes, S.L.Allen: “Representations of Spectral Coordinates in FITS” \[2\] Barry N. Taylor: “Guide for the Use of the International System of Units (SI)”, NIST Special Publication 811 \[3\] I.M. George, L. Angelini: “Specification of Physical Units within OGIP FITS files”, OGIP Memo OGIP/93-001 \[4\] Thomas Szirtes: “Applied Dimensional Analysis and Modelling” , McGraw Hill 1997 \[5\] Jose Tomas Diez-Roche: “Fluid Mechanics”, ETSIN publications \[6\] James Clerck Maxwell: “Electricity and Magnetism” \[7\] AVO Demo 2005 European Space Astronomy Centre (ESAC) (Madrid, Spain) ESA **http://euro-vo.org/twiki.bin/view/Avo/AvoDemo2005Star** \[8\] International Virtual Observatory Alliance (IVOA) **http://www.ivoa.net**
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Dimer algebras arise from a particular type of quiver gauge theory. However, part of the input to such a theory is the gauge group, and this choice may impose additional constraints on the algebra. If the gauge group of a dimer theory is abelian, then the algebra that arises is not actually the dimer algebra itself, but a particular quotient we introduce called the ‘homotopy algebra’. We show that a homotopy algebra $\Lambda$ on a torus is a dimer algebra if and only if it is noetherian, and otherwise $\Lambda$ is the quotient of a dimer algebra by homotopy relations. Stated in physics terms, a dimer theory is superconformal if and only if the corresponding dimer and homotopy algebras coincide. We also give an explicit description of the center of a homotopy algebra in terms of a special subset of its perfect matchings. In our proofs we introduce formalized notions of Higgsing and the mesonic chiral ring from quiver gauge theory.'
address: 'Institut für Mathematik und Wissenschaftliches Rechnen, Universität Graz, Heinrichstrasse 36, 8010 Graz, Austria.'
author:
- Charlie Beil
title: Homotopy dimer algebras and cyclic contractions
---
=1
Introduction
============
Let $k$ be an algebraically closed field. A dimer quiver $Q$ is a quiver that embeds into a compact surface $M$ such that each connected component of $M \setminus Q$ is simply connected and bounded by an oriented cycle, called a unit cycle. A perfect matching $D$ of $Q$ is a subset of arrows such that each unit cycle contains precisely one arrow in $D$. We will often assume that $Q$ is *nondegenerate*, that is, each arrow is contained in at least one perfect matching. One can imagine associating a different color to each perfect matching, and coloring each arrow $a$ by all the perfect matchings that contain $a$. In this article we introduce the *homotopy algebra* of $Q$, which is a quotient of the path algebra $kQ$ where paths with coincident heads and coincident tails are deemed equal if they have the same coloring.
To be more precise, denote by $\mathcal{P}$ the set of perfect matchings of $Q$, and by $n := |Q_0|$ the number of vertices of $Q$. Consider the algebra homomorphism from $kQ$ to the $n \times n$ matrix ring over the polynomial ring generated by $\mathcal{P}$, $$\label{eta}
\eta: kQ \to M_n\left(k[x_D \ | \ D \in \mathcal{P}] \right),$$ defined on the vertices $e_i \in Q_0$ and arrows $a \in Q_1$ by $$\label{eta def}
e_i \mapsto e_{ii} \ \ \ \text{ and } \ \ \ a \mapsto \prod_{a \in D \in \mathcal{P}} x_D \cdot e_{\operatorname{h}(a),\operatorname{t}(a)},$$ and extended multiplicatively and $k$-linearly to $kQ$. The homotopy algebra of $Q$ is the quotient $$\Lambda := kQ/\operatorname{ker} \eta.$$ $\Lambda$ may be viewed as a tiled matrix ring by identifying it with its induced image under $\eta$. In companion articles we will find that homotopy algebras, although often nonnoetherian, have rich geometric, algebraic, and homological properties [@B2; @B3; @B5; @B6].
Homotopy algebras generalize cancellative dimer algebras. The *dimer algebra* of a dimer quiver $Q$ is the superpotential algebra $A = kQ/I$, where $I$ is the ideal $$\label{I def}
I := \left\langle p-q \ | \ \exists a \in Q_1 \text{ s.t.\ } ap,aq \text{ are unit cycles} \right\rangle \subset kQ.$$ Dimer algebras originated in string theory, and have found wide application to many areas of mathematics (e.g., [@BKM; @Br; @IU; @FHKV; @H; @IN]). Although dimer algebras describe a class of abelian superconformal quiver gauge theories, it is really homotopy algebras–and not dimer algebras–that describe abelian quiver gauge theories which are not superconformal. As we will show, homotopy algebras are quotients of superpotential algebras, and the additional relations arise from the assumption that the gauge group is abelian.
We consider the following questions for dimer quivers $Q$ on a torus:
- *What is a minimal subset of perfect matchings $\mathcal{P}_0 \subseteq \mathcal{P}$ with the property that the algebra homomorphism $$\label{P'}
\tau: kQ \to M_n\left(k[ x_D \ | \ D \in \mathcal{P}_0] \right),$$ defined by (\[eta def\]) with $\mathcal{P}_0$ in place of $\mathcal{P}$, satisfies $kQ/\ker \eta \cong kQ/ \ker \tau$?*
- *How is the homotopy algebra $\Lambda$ related to the dimer algebra $A$?*
- *How is the center and representation theory of $\Lambda$ related to the perfect matchings in $\mathcal{P}_0$?*
Typically $Q$ contains *non-cancellative pairs* of paths, which are distinct paths $p,q$ in $A$ with the property that there is some path $r$ satisfying $$pr = qr \not = 0 \ \ \ \text{ or } \ \ \ rp = rq \not = 0.$$ If $A$ has a non-cancellative pair, then $A$ and $Q$ are said to be *non-cancellative*, and otherwise $A$ and $Q$ are *cancellative*. A fundamental characterization of this property is that $A$ is cancellative if and only if $A$ is noetherian [@B3 Theorem 1.1].
To address our questions, we introduce the notion of a *cyclic contraction*. A cyclic contraction is a $k$-linear map of dimer algebras $$\psi: A = kQ/I \to A' = kQ'/I',$$ where $Q'$ is cancellative and is obtained by contracting a set of arrows in $Q$ such that the cycles in $Q$ are suitably preserved (Definition \[cyclic contraction\]). Cyclic contractions are useful because they allow non-cancellative dimer algebras to be studied by relating them to well-understood cancellative dimer algebras that share similar structure. Moreover, cyclic contractions exist for all nondegenerate dimer algebras [@B1 Theorem 1.1].
A perfect matching $D \in \mathcal{P}$ is called *simple* if there is an oriented path between any two vertices in $Q \setminus D$; in this case, $Q \setminus D$ supports a simple $\Lambda$-module (and simple $A$-module) of dimension vector $1^{Q_0}$. Let $\mathcal{S}$ be the set of simple matchings of $Q$. We will show that if $Q$ is cancellative, then we may take $\mathcal{P}_0$ to be the simple matchings $\mathcal{S}$. However, in general $\mathcal{P}_0$ cannot equal $\mathcal{S}$ since there are (nondegenerate) dimer quivers for which $\mathcal{S} = \emptyset$; see Figure \[deformation figure\].i. One may ask if $\mathcal{P}_0$ is simply a minimal set of perfect matchings such that each arrow of $Q$ is contained in some $D \in \mathcal{P}_0$. To the contrary: for our choice of $\mathcal{P}_0$, there will always be arrows which are not contained in any $D \in \mathcal{P}_0$ whenever $Q$ is non-cancellative.
Our main theorem is the following.
\[main\] (Theorems \[first main\] and \[impression prop\].) Let $Q$ be a nondegenerate dimer quiver on a torus, and fix a cyclic contraction $\psi: A \to A'$. Let $\mathcal{P}_0 \subseteq \mathcal{P}$ be the perfect matchings $D$ of $Q$ for which $\psi(D)$ is a simple matching of $Q'$. Set $B := k[x_D \ | \ D \in \mathcal{P}_0]$.
1. Let $\tau_{\psi}: kQ \to M_n(B)$ be the algebra homomorphism defined in (\[eta def\]) with $\mathcal{P}_0$ in place of $\mathcal{P}$. There are algebra isomorphisms $$\begin{array}{rcl}
\Lambda := kQ/\ker \eta & \cong & kQ/\ker \tau_{\psi} \\
& = & A/\left\langle p-q \ | \ p,q \text{ is a non-cancellative pair} \right\rangle.
\end{array}$$
2. Suppose $k$ is uncountable. The induced homomorphism $\tau_{\psi}: \Lambda \to M_n(B)$ classifies all simple $\Lambda$-module isoclasses of maximal $k$-dimension: for each such module $V$, there is a maximal ideal $\mathfrak{b} \in \operatorname{Max}B$ such that $$V \cong (B/\mathfrak{b})^n,$$ where $av := \tau_{\psi}(a)v$ for each $a \in \Lambda$, $v \in (B/\mathfrak{b})^n$.
3. The centers of $\Lambda$ and $\Lambda' \cong A'$ are given by the intersection and union of the vertex corner rings of $\Lambda$, $$Z(\Lambda) \cong k\left[ \cap_{i \in Q_0} \bar{\tau}_{\psi}\left( e_i \Lambda e_i \right) \right] \subseteq k\left[ \cup_{i \in Q_0} \bar{\tau}_{\psi}\left( e_i\Lambda e_i \right) \right] \cong Z(\Lambda').$$
To define the homotopy algebra $\Lambda$ it thus suffices to keep only those perfect matchings $D \in \mathcal{P}$ for which $\psi(D)$ is a simple matching of $Q'$.
$$\begin{array}{ccccc}
(i) & \ \ & \xy
(-12,6)*+{\text{\scriptsize{$2$}}}="1";(0,6)*+{\text{\scriptsize{$1$}}}="2";(12,6)*+{\text{\scriptsize{$2$}}}="3";
(-12,-6)*+{\text{\scriptsize{$1$}}}="4";(0,-6)*+{\text{\scriptsize{$2$}}}="5";(12,-6)*+{\text{\scriptsize{$1$}}}="6";
(-12,0)*{\cdot}="7";(0,0)*{\cdot}="8";(12,0)*{\cdot}="9";
(-6,6)*{\cdot}="10";(6,6)*{\cdot}="11";(-6,-6)*{\cdot}="12";(6,-6)*{\cdot}="13";
{\ar@[green]"2";"10"};{\ar"10";"1"};{\ar^{}"7";"4"};{\ar@[green]"4";"12"};{\ar"12";"5"};{\ar@[green]"5";"8"};{\ar@[green]"2";"11"};{\ar"11";"3"};{\ar^{}"9";"6"};
{\ar@[green]"6";"13"};{\ar"13";"5"};
{\ar@[green]"3";"9"};{\ar@[green]"1";"7"};{\ar"8";"2"};
\endxy
& \stackrel{\psi}{\longrightarrow} &
\xy
(-12,6)*+{\text{\scriptsize{$2$}}}="1";(0,6)*+{\text{\scriptsize{$1$}}}="2";(12,6)*+{\text{\scriptsize{$2$}}}="3";
(-12,-6)*+{\text{\scriptsize{$1$}}}="4";(0,-6)*+{\text{\scriptsize{$2$}}}="5";(12,-6)*+{\text{\scriptsize{$1$}}}="6";
{\ar^{}"2";"1"};{\ar^{}"1";"4"};{\ar^{}"4";"5"};{\ar^{}"5";"2"};{\ar^{}"2";"3"};{\ar^{}"3";"6"};{\ar^{}"6";"5"}; \endxy \\
&&Q &\ \ \ \ & Q' \\
\\
(ii) & \ \ & \xy
(-12,6)*+{\text{\scriptsize{$2$}}}="1";(0,6)*+{\text{\scriptsize{$1$}}}="2";(12,6)*+{\text{\scriptsize{$2$}}}="3";
(-12,-6)*+{\text{\scriptsize{$1$}}}="4";(0,-6)*+{\text{\scriptsize{$2$}}}="5";(12,-6)*+{\text{\scriptsize{$1$}}}="6";
(-12,0)*{\cdot}="7";(0,0)*{\cdot}="8";(12,0)*{\cdot}="9";
{\ar^{}"2";"1"};{\ar^{}"7";"4"};{\ar^{}"4";"5"};{\ar^{}"5";"8"};{\ar^{}"2";"3"};{\ar^{}"9";"6"};
{\ar^{}"6";"5"};
{\ar@[green]"3";"9"};{\ar@[green]"1";"7"};{\ar@[green]"8";"2"};
\endxy
& \stackrel{\psi}{\longrightarrow} &
\xy
(-12,6)*+{\text{\scriptsize{$2$}}}="1";(0,6)*+{\text{\scriptsize{$1$}}}="2";(12,6)*+{\text{\scriptsize{$2$}}}="3";
(-12,-6)*+{\text{\scriptsize{$1$}}}="4";(0,-6)*+{\text{\scriptsize{$2$}}}="5";(12,-6)*+{\text{\scriptsize{$1$}}}="6";
{\ar^{}"2";"1"};{\ar^{}"1";"4"};{\ar^{}"4";"5"};{\ar^{}"5";"2"};{\ar^{}"2";"3"};{\ar^{}"3";"6"};{\ar^{}"6";"5"};
\endxy
\\
&&Q & \ \ \ \ & Q' \\
\end{array}$$ $$\begin{array}{ccccccc}
(iii) & \ & \xy
(-12,6)*+{\text{\scriptsize{$2$}}}="1";(0,6)*+{\text{\scriptsize{$1$}}}="2";(12,6)*+{\text{\scriptsize{$2$}}}="3";
(-12,-6)*+{\text{\scriptsize{$1$}}}="4";(0,-6)*+{\text{\scriptsize{$2$}}}="5";(12,-6)*+{\text{\scriptsize{$1$}}}="6";
(0,0)*{\cdot}="7";
{\ar^{}"1";"2"};{\ar^{}"2";"7"};{\ar^{}"7";"1"};{\ar@[red]"1";"5"};{\ar^{}"5";"4"};{\ar^{}"4";"1"};{\ar^{}"5";"6"};{\ar"6";"3"};{\ar@[blue]"3";"5"};{\ar^{}"7";"3"};{\ar^{}"3";"2"};
{\ar@[green]"5";"7"};
\endxy
& \stackrel{\psi}{\longrightarrow} &
\xy
(-12,6)*+{\text{\scriptsize{$2$}}}="1";(0,6)*+{\text{\scriptsize{$1$}}}="2";(12,6)*+{\text{\scriptsize{$2$}}}="3";
(-12,-6)*+{\text{\scriptsize{$1$}}}="4";(0,-6)*+{\text{\scriptsize{$2$}}}="5";(12,-6)*+{\text{\scriptsize{$1$}}}="6";
{\ar^{}"1";"2"};{\ar^{}"2";"5"};{\ar^{}"5";"4"};{\ar^{}"4";"1"};{\ar^{}"5";"6"};{\ar^{}"6";"3"};{\ar^{}"3";"2"};{\ar@/_.3pc/"1";"5"};{\ar@/_.3pc/"5";"1"};{\ar@/^.3pc/"5";"3"};{\ar@/^.3pc/"3";"5"};
\endxy
& &
\xy
(-12,6)*+{\text{\scriptsize{$2$}}}="1";(0,6)*+{\text{\scriptsize{$1$}}}="2";(12,6)*+{\text{\scriptsize{$2$}}}="3";
(-12,-6)*+{\text{\scriptsize{$1$}}}="4";(0,-6)*+{\text{\scriptsize{$2$}}}="5";(12,-6)*+{\text{\scriptsize{$1$}}}="6";
{\ar^{}"1";"2"};{\ar^{}"2";"5"};{\ar^{}"5";"4"};{\ar^{}"4";"1"};{\ar^{}"5";"6"};{\ar^{}"6";"3"};{\ar^{}"3";"2"};
\endxy
\\
&& Q & \ \ \ & Q' & \ \ \ &
\\ \\
(iv) & \ & \xy
(-12,12)*+{\text{\scriptsize{$3$}}}="1";(0,12)*+{\text{\scriptsize{$2$}}}="2";(-12,0)*+{\text{\scriptsize{$2$}}}="3";(0,0)*{\cdot}="4";(12,0)*+{\text{\scriptsize{$1$}}}="5";(0,-12)*+{\text{\scriptsize{$1$}}}="6";(12,-12)*+{\text{\scriptsize{$3$}}}="7";
(-18,6)*{}="9";(-6,18)*{}="10";(6,-18)*{}="11";(18,-6)*{}="12";
(6,0)*{\cdot}="8";
{\ar^{}"1";"2"};{\ar^{}"2";"8"};{\ar^{}"4";"3"};{\ar^{}"3";"1"};{\ar^{}"6";"4"};{\ar^{}"4";"7"};{\ar^{}"7";"8"};{\ar^{}"8";"5"};{\ar^{}"5";"7"};{\ar^{}"7";"6"};
{\ar@{..}^{}"9";"10"};{\ar@{..}^{}"10";"12"};{\ar@{..}^{}"12";"11"};{\ar@{..}^{}"11";"9"};
{\ar@[green]"8";"4"};
\endxy
& \stackrel{\psi}{\longrightarrow} &
\xy
(-12,12)*+{\text{\scriptsize{$3$}}}="1";(0,12)*+{\text{\scriptsize{$2$}}}="2";(-12,0)*+{\text{\scriptsize{$2$}}}="3";(0,0)*{\cdot}="4";(12,0)*+{\text{\scriptsize{$1$}}}="5";(0,-12)*+{\text{\scriptsize{$1$}}}="6";(12,-12)*+{\text{\scriptsize{$3$}}}="7";
(-18,6)*{}="9";(-6,18)*{}="10";(6,-18)*{}="11";(18,-6)*{}="12";
{\ar^{}"1";"2"};{\ar^{}"2";"4"};{\ar^{}"4";"3"};{\ar^{}"3";"1"};{\ar^{}"4";"5"};{\ar^{}"5";"7"};{\ar^{}"7";"6"};{\ar^{}"6";"4"};
{\ar@{..}^{}"9";"10"};{\ar@{..}^{}"10";"12"};{\ar@{..}^{}"12";"11"};{\ar@{..}^{}"11";"9"};
{\ar@/_.3pc/"7";"4"};{\ar@/_.3pc/"4";"7"};
\endxy
& &
\xy
(-12,12)*+{\text{\scriptsize{$3$}}}="1";(0,12)*+{\text{\scriptsize{$2$}}}="2";(-12,0)*+{\text{\scriptsize{$2$}}}="3";(0,0)*{\cdot}="4";(12,0)*+{\text{\scriptsize{$1$}}}="5";(0,-12)*+{\text{\scriptsize{$1$}}}="6";(12,-12)*+{\text{\scriptsize{$3$}}}="7";
(-18,6)*{}="9";(-6,18)*{}="10";(6,-18)*{}="11";(18,-6)*{}="12";
{\ar^{}"1";"2"};{\ar^{}"2";"4"};{\ar^{}"4";"3"};{\ar^{}"3";"1"};{\ar^{}"4";"5"};{\ar^{}"5";"7"};{\ar^{}"7";"6"};{\ar^{}"6";"4"};
{\ar@{..}^{}"9";"10"};{\ar@{..}^{}"10";"12"};{\ar@{..}^{}"12";"11"};{\ar@{..}^{}"11";"9"};
\endxy
\\
&&Q & \ \ \ & Q' & \ \ \ &
\end{array}$$
[ Four cyclic contractions are given in Figure \[deformation figure\]. The non-cancellative quivers $Q$ in (ii) and (iv) have appeared in the physics literature (e.g., [@FHPR Section 4], [@FKR]; and [@DHP Table 6, 2.6]). The unit 2-cycles in (iii) and (iv) consist of arrows that are redundant generators for $A' = kQ'/I'$, and so may be removed from $Q'$. In (iii), let $a,b,c$ be the respective red, blue, and green arrows in $Q$. Observe that in $A = kQ/I$, we have $$ab \not = ba \ \ \ \text{ and } \ \ \ cab = cba.$$ Thus the pair $ab$, $ba$ is non-cancellative (in fact, $a$ and $b$ generate a free subalgebra of $A$). In (i), $Q$ has no simple matchings. These examples are considered in more detail in Example \[four examples\]. ]{}
Preliminaries
=============
Let $R$ be an integral domain and a $k$-algebra. We will denote by $\operatorname{Max}R$ the set of maximal ideals of $R$, and by $\mathcal{Z}(\mathfrak{a})$ the closed set $\left\{ \mathfrak{m} \in \operatorname{Max}R \ | \ \mathfrak{m} \supseteq \mathfrak{a} \right\}$ of $\operatorname{Max}R$ defined by the subset $\mathfrak{a} \subset R$.
We will denote by $Q = \left( Q_0,Q_1,\operatorname{t}, \operatorname{h} \right)$ the quiver with vertex set $Q_0$, arrow set $Q_1$, and head and tail maps $\operatorname{h},\operatorname{t}: Q_1 \to Q_0$. We will denote by $kQ$ the path algebra of $Q$, and by $e_i$ the idempotent corresponding to vertex $i \in Q_0$. Multiplication of paths is read right to left, following the composition of maps. By module we mean left module. Finally, we will denote by $e_{ij} \in M_d(k)$ the $d \times d$ matrix with a 1 in the $ij$-th slot and zeros elsewhere.
Algebra homomorphisms from perfect matchings
--------------------------------------------
Let $A = kQ/I$ be a dimer algebra on a torus. Denote by $\mathcal{P}$ and $\mathcal{S}$ the sets of perfect and simple matchings of $Q$ respectively.
Consider the algebra homomorphisms $$\label{te}
\tau: kQ \to M_{|Q_0|}\left(k\left[ x_D \ | \ D \in \mathcal{S} \right]\right) \ \ \ \ \text{ and } \ \ \ \ \eta: kQ \to M_{|Q_0|}\left(k\left[ x_D \ | \ D \in \mathcal{P} \right] \right)$$ defined on $i \in Q_0$ and $a \in Q_1$ by $$\label{taua}
\begin{array}{ccc}
\tau(e_i) = e_{ii}, & \ \ \ & \tau(a) = \prod_{a \in D \in \mathcal{S}} x_D \cdot e_{\operatorname{h}(a),\operatorname{t}(a)},\\
\eta(e_i) = e_{ii}, & \ \ \ & \eta(a) = \prod_{a \in D \in \mathcal{P}} x_D \cdot e_{\operatorname{h}(a),\operatorname{t}(a)},
\end{array}$$ and extended multiplicatively and $k$-linearly to $kQ$.
\[tau’A’\] The ideal $I \subset kQ$ given in (\[I def\]) is contained in the kernels of $\tau$ and $\eta$. Therefore $\tau$ and $\eta$ induce algebra homomorphisms on the dimer algebra $A$, $$\label{tau eta}
\tau: A \to M_{|Q_0|}\left(k\left[ x_D \ | \ D \in \mathcal{S} \right]\right) \ \ \ \ \text{ and } \ \ \ \ \eta: A \to M_{|Q_0|}\left(k\left[ x_D \ | \ D \in \mathcal{P} \right] \right).$$
Let $p - q$ be a generator for $I$ as given in (\[I def\]); that is, there is an arrow $a \in Q_1$ such that $pa$ and $qa$ are unit cycles. Then $$\tau(p) = \prod_{a \not \in D \in \mathcal{S}} x_D \cdot e_{\operatorname{t}(a),\operatorname{h}(a)} = \tau(q).$$ Similarly, $\eta(p) = \eta(q)$. Therefor $p - q$ is in the kernels of $\tau$ and $\eta$.
The following lemma is clear.
\[sigma\] If $\sigma_i, \sigma'_i$ are two unit cycles at $i \in Q_0$, then $\sigma_i = \sigma'_i$ in $A$. Furthermore, the sum $\sum_{i \in Q_0}\sigma_i$ is in the center of $A$.
We will denote by $\sigma_i \in A$ the unique unit cycle at vertex $i$.
Each unit cycle $\sigma_i \in e_iAe_i$ satisfies $$\label{prod D}
\tau\left( \sigma_i \right) = \prod_{D \in \mathcal{S}} x_D \cdot e_{ii} \ \ \ \ \text{ and } \ \ \ \ \eta\left( \sigma_i \right) = \prod_{D \in \mathcal{P}} x_D \cdot e_{ii}.$$
Each perfect matching contains precisely one arrow in each unit cycle.
\[tau bar notation\] [ For each $i,j \in Q_0$, denote by $$\bar{\tau}: e_jAe_i \to B := k\left[ x_D \ | \ D \in \mathcal{S} \right] \ \ \ \ \text{ and } \ \ \ \ \bar{\eta}: e_jAe_i \to k\left[ x_D \ | \ D \in \mathcal{P} \right]$$ the respective $k$-linear maps defined on $p \in e_jAe_i$ by $$\tau(p) = \bar{\tau}(p)e_{ji} \ \ \ \ \text{ and } \ \ \ \ \eta(p) = \bar{\eta}(p)e_{ji}.$$ In particular, $\bar{\tau}(p)$ and $\bar{\eta}(p)$ are the single nonzero matrix entries of $\tau(p)$ and $\eta(p)$. In Section \[Cycle structure\], we will set $$\label{A notation}
{\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} := \bar{\tau}(p) \ \ \ \text{ and } \ \ \ \sigma := \prod_{D \in \mathcal{S}} x_D \ \ \ \text{(or occasionally, $\sigma := \prod_{D \in \mathcal{P}} x_D$).}$$ In Section \[Cancellative dimer algebras\], given a cyclic contraction $\psi: A \to A'$ and elements $p \in e_jAe_i$, $q \in e_{\ell}A'e_k$, we will set $$\label{contraction notation}
{\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} := \bar{\tau}_{\psi}(p) := \bar{\tau}\psi(p), \ \ \ \ {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} := \bar{\tau}(q), \ \ \ \text{ and } \ \ \ \sigma := \prod_{D \in \mathcal{S}'} x_D,$$ where $\mathcal{S}'$ is the set of simple matchings of $A'$. ]{}
Impressions
-----------
The following definition, introduced in [@B7], captures a useful matrix ring embedding.
\[impression definition\]
[@B7 Definition 2.1] Let $A$ be a finitely generated $k$-algebra and let $Z$ be its center. An *impression* of $A$ is an algebra monomorphism $\tau: A \hookrightarrow M_d(B)$ to a matrix ring over a commutative finitely generated $k$-algebra $B$, such that
- for generic $\mathfrak{b} \in \operatorname{Max}B$, the composition $$\label{composition}
A \stackrel{\tau}{\longrightarrow} M_d(B) \stackrel{1}{\longrightarrow} M_d\left(B/\mathfrak{b} \right) \cong M_d(k)$$ is surjective; and
- the morphism $\operatorname{Max}B \rightarrow \operatorname{Max}\tau(Z)$, $\mathfrak{b} \mapsto \mathfrak{b} \cap \tau(Z)$, is surjective.
Surjectivity of (\[composition\]) implies that the center $Z$ is given by $$\label{Ziso}
Z \cong \left\{ f \in B \ | \ f1_d \in \operatorname{im}\tau \right\} \subseteq B.$$ If in addition $A$ is a finitely generated module over its center, then $\tau$ classifies all simple $A$-module isoclasses of maximal $k$-dimension [@B7 Proposition 2.5]. Specifically, for each such module $V$, there is some $\mathfrak{b} \in \operatorname{Max}B$ such that $$\label{Vcong}
V \cong (B/\mathfrak{b})^d,$$ where $av := \tau(a)v$ for each $a \in A$, $v \in (B/\mathfrak{b})^d$. If $A$ is nonnoetherian, then $\tau$ may characterize the central geometry of $A$ using the framework of depictions [@B4 Section 3]; this relationship is used to study the central geometry of nonnoetherian homotopy and dimer algebras in [@B6].
We will show that for a homotopy algebra $\Lambda$, the homomorphism $\tau_{\psi}: \Lambda \to M_n(B)$ is an impression (Theorem \[impression prop\].1). Furthermore, a dimer algebra admits an impression if and only if it equals its homotopy algebra (Corollary \[cr\]).
Cyclic contractions {#cyclicconractions}
===================
In this section, we introduce a new method for studying non-cancellative dimer algebras that is based on the notion of Higgsing, or more generally symmetry breaking, in physics. Using this strategy, we gain information about non-cancellative dimer algebras by relating them to cancellative dimer algebras with similar structure. Throughout, $A = kQ/I$ is a dimer algebra, typically non-cancellative.
In the following, we introduce a $k$-linear map induced from the operation of edge contraction in graph theory.
\[contraction\] [ Let $Q$ be a dimer quiver, let $Q_1^* \subset Q_1$ be a subset of arrows, and let $Q'$ be the quiver obtained by contracting each arrow in $Q_1^*$. Specifically, $Q'$ is formed by simultaneously removing each arrow $a$ in $Q_1^*$, and merging together the head and tail vertices of $a$. This operation defines a $k$-linear map of path algebras $$\psi: kQ \rightarrow kQ',$$ where $$\psi(a) = \left\{ \begin{array}{cl} a & \text{ if } \ a \in Q_0 \cup Q_1 \setminus Q_1^* \\ e_{\operatorname{t}(a)} & \text{ if } \ a \ \in Q_1^* \end{array} \right.$$ and extended multiplicatively to paths and $k$-linearly to $kQ$. If $\psi$ induces a $k$-linear map of dimer algebras $$\psi: A = kQ/I \to A' = kQ'/I',$$ that is, $\psi(I) \subseteq I'$, then we call $\psi$ a *contraction of dimer algebras*. ]{}
[ The containment $\psi(I) \subseteq I'$ may be proper. Indeed, $\psi(I) \not = I'$ whenever $\psi$ contracts a unit cycle to a removable 2-cycle (Definition \[removable2\]). ]{}
We now describe the structure we wish to preserve under a contraction. To specify this structure, we introduce the following commutative algebras.
\[cyclic contraction\] [ Let $\psi: A \to A'$ be a contraction to a cancellative dimer algebra. If $$S := k\left[ \cup_{i \in Q_0} \bar{\tau}\psi\left(e_iAe_i\right) \right] = k\left[ \cup_{i \in Q'_0} \bar{\tau}\left( e_iA'e_i \right) \right],$$ then we say $\psi$ is *cyclic*, and call $S$ the *cycle algebra* of $A$. ]{}
The cycle algebra is independent of the choice of $\psi$ by [@B2 Theorem 3.13].
[ For $g,h \in B$, by $g \mid h$ we mean $g$ divides $h$ in $B$, even if $g$ or $h$ is assumed to be in $S$. ]{}
\[cannot contract\] Suppose $\psi: A \to A'$ is a contraction of dimer algebras, and $A'$ has a perfect matching. Then $\psi$ cannot contract a unit cycle of $A$ to a vertex.
Assume to the contrary that $\psi$ contracts the unit cycle $\sigma_j \in A$ to the vertex $e_{\psi(j)} \in A'$. Fix a unit cycle $\sigma_{i'} \in A'$. Since $\psi$ is surjective on $Q'_0$, there is a vertex $i \in Q_0$ such that $\psi(i) = i'$. Let $p \in A$ be a path from $i$ to $j$, and set $p' := \psi(p)$. Then $$\label{p' sigma}
p' \sigma_{i'} \stackrel{\textsc{(i)}}{=} \psi(p \sigma_i) \stackrel{\textsc{(ii)}}{=} \psi(\sigma_j p) \stackrel{\textsc{(iii)}}{=} \psi(\sigma_j) p' = e_{\psi(j)}p' = p'.$$ Indeed, (<span style="font-variant:small-caps;">i</span>) and (<span style="font-variant:small-caps;">iii</span>) hold by Definition \[contraction\], and (<span style="font-variant:small-caps;">ii</span>) holds by Lemma \[sigma\].
Denote by $\mathcal{P}'$ is the set of perfect matchings of $A'$. Set $\sigma := \prod_{D \in \mathcal{P}'}x_D$. Then (\[p’ sigma\]) implies $$\bar{\eta}(p') \sigma = \bar{\eta}(p' \sigma_{i'}) = \bar{\eta}(p') \in k\left[x_D \ | \ D \in \mathcal{P}' \right],$$ by Lemma \[tau’A’\]. Whence $\sigma = 1$. But this contradicts our assumption that $\mathcal{P}' \not = \emptyset$.
An example where a unit cycle is contracted to a vertex is given in Figure \[non-example unit\].
$$\xy 0;/r.4pc/:
(-5,5)*{}="1";(5,5)*{}="2";(7.07,0)*{}="3";(5,-5)*{}="4";(-5,-5)*{}="5";(-7.07,0)*{}="6";
(0,2)*{\cdot}="7";(2,-2)*{\cdot}="8";(-2,-2)*{\cdot}="9";
{\ar"7";"1"};{\ar"2";"7"};{\ar"4";"8"};{\ar"6";"9"};{\ar"9";"5"};{\ar"8";"3"};
{\ar@[green]"7";"8"};{\ar@[green]"8";"9"};{\ar@[green]"9";"7"};\endxy
\ \ \ \ \stackrel{\psi}{\longrightarrow} \ \ \ \
\xy 0;/r.4pc/:
(-5,5)*{}="1";(5,5)*{}="2";(7.07,0)*{}="3";(5,-5)*{}="4";(-5,-5)*{}="5";(-7.07,0)*{}="6";(0,0)*{\cdot}="7";
{\ar"7";"1"};{\ar"2";"7"};{\ar"4";"7"};{\ar"7";"5"};{\ar"6";"7"};{\ar"7";"3"};\endxy$$
\[positive length cycle\] Suppose $\psi: A \to A'$ is a contraction of dimer algebras, and $A'$ has a perfect matching. Then $\psi$ cannot contract a cycle in the underlying graph ${\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}$ of $Q$ to a vertex. In particular,
1. $\psi$ cannot contract a cycle in $Q$ to a vertex;
2. if $p \in \mathcal{C} \setminus \mathcal{C}^0$, then $\psi(p) \in \mathcal{C}' \setminus \mathcal{C}'^0$; and
3. $A$ does not have a non-cancellative pair where one of the paths is a vertex.
The number of vertices, edges, and faces in the underlying graphs ${\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}$ and ${\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}'$ of $Q$ and $Q'$ are given by $$\begin{array}{lclcl}
V = \left| Q_0 \right|, & \ \ \ & E = \left| Q_1 \right|, & \ \ \ & F = \# \text{ of connected components of } T^2 \setminus {\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}, \\
V' = \left| Q'_0 \right|, & & E' = \left| Q'_1 \right|, & & F' = \# \text{ of connected components of } T^2 \setminus {\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}'.
\end{array}$$ Since ${\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}$ and ${\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}'$ each embed into a two-torus, their respective Euler characteristics vanish: $$\label{VEF}
V - E + F = 0, \ \ \ \ V' - E' + F' = 0.$$
Assume to the contrary that $\psi$ contracts the cycles $p_1,\ldots, p_{\ell}$ in ${\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}$ to vertices in ${\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}'$. Denote by $n_0$ and $n_1$ the respective number of vertices and arrows in $Q$ which are subpaths of some $p_i$, $1 \leq i \leq \ell$. Denote by $m$ the number of vertices in $Q'_0$ of the form $\psi(p_i)$ for some $1 \leq i \leq \ell$. By assumption, $m \geq 1$.
In any cycle, the number of trivial subpaths equals the number of arrow subpaths. Furthermore, if two cycles share a common edge, then they also share a common vertex. Therefore $$n_1 \geq n_0.$$ Whence $$\begin{array}{rcl}
0 & = & F' - E' + V'\\
& = & F' - (E - n_1) + (V - n_0 + m)\\
& = & F' + (-E + V) + (n_1 - n_0) + m\\
& \geq & F' - F + m.
\end{array}$$ Thus $F' < F$ since $m \geq 1$. Therefore $\psi$ contracts a face of $Q$ to a vertex. In particular, some unit cycle in $Q$ is contracted to a vertex. But this is not possible by Lemma \[cannot contract\].
[ Suppose $\psi: A \to A'$ is a cyclic contraction. We will show in Lemma \[at least one\] below that $A'$, being cancellative, necessarily has a perfect matching. Therefore Lemmas \[cannot contract\] and \[positive length cycle\] hold in the case $\psi$ is cyclic. ]{}
An immediate question is whether all non-cancellative dimer algebras admit cyclic contractions. If $Q$ is nondegenerate, then $A$ admits a cyclic contraction [@B1 Theorem 1.1]. However, there are degenerate dimer algebras that do not admit contractions (cyclic or not) to cancellative dimer algebras. For example, dimer algebras that contain permanent 2-cycles (Definition \[removable2\]) cannot contract to cancellative dimer algebras, by Lemma \[positive length cycle\].1.
A similar question is whether cyclic contractions can exist between two different cancellative dimer algebras. The answer to this question is negative: if $\psi: A \to A'$ is a cyclic contraction and $A$ is cancellative, then $\psi$ must be the identity map [@B3 Theorem 1.1].
[ Suppose $\psi: A \to A'$ is a contraction. Consider a path $p = a_n \cdots a_2a_1 \in kQ$ with $a_1, \ldots, a_n \in Q_0 \cup Q_1$. If $p \not = 0$, then by definition $$\psi(p) = \psi(a_n) \cdots \psi(a_1) \in kQ'.$$ However, we claim that if $\psi$ is nontrivial, then it is not an algebra homomorphism. Indeed, let $\delta \in Q_1^*$. Consider a path $a_2 \delta a_1 \not = 0$ in $A$. By Lemma \[positive length cycle\].1, $\delta$ is not a cycle. In particular, $\operatorname{h}(a_1) \not = \operatorname{t}(a_2)$. Whence $$\psi(a_2a_1) = \psi(0) = 0 \not = \psi(a_2)\psi(a_1).$$ We note, however, that the restriction $$\psi: \epsilon_0A\epsilon_0 \to A' \ \ \ \text{ where } \ \epsilon_0 := 1_A - \sum_{\delta \in Q_1^*}e_{\operatorname{h}(\delta)},$$ is an algebra homomorphism [@B2 Proposition 2.12.1]. ]{}
Cycle structure {#Cycle structure}
===============
Let $A = kQ/I$ be a dimer algebra, possibly with no perfect matchings. Unless stated otherwise, by path or cycle we mean path or cycle modulo $I$. Throughout, we use the notation (\[A notation\]).
Let $\pi: \mathbb{R}^2 \rightarrow T^2$ be a covering map such that for some $i \in Q_0$, $$\pi\left(\mathbb{Z}^2 \right) = i.$$ Denote by $Q^+ := \pi^{-1}(Q) \subset \mathbb{R}^2$ the covering quiver of $Q$. For each path $p$ in $Q$, denote by $p^+$ the unique path in $Q^+$ with tail in the unit square $[0,1) \times [0,1) \subset \mathbb{R}^2$ satisfying $\pi(p^+) = p$.
Furthermore, for paths $p$, $q$ satisfying $$\label{exception}
\operatorname{t}(p^+) = \operatorname{t}(q^+) \ \ \text{ and } \ \ \operatorname{h}(p^+) = \operatorname{h}(q^+),$$ denote by $\mathcal{R}_{\tilde{p},\tilde{q}}$ the compact region in $\mathbb{R}^2 \supset Q^+$ bounded by representatives $\tilde{p}^+$, $\tilde{q}^+$ of $p^+$, $q^+$.[^1] If the representatives are fixed (or arbitrary), we will write $\mathcal{R}_{p,q}$ for $\mathcal{R}_{\tilde{p},\tilde{q}}$.
\[non-cancellative pair def\]
- We say say two paths $p,q \in A$ are a *non-cancellative pair* if $p \not = q$, and there is a path $r \in A$ such that $$rp = rq \not = 0 \ \ \text{ or } \ \ pr = qr \not = 0.$$
- We say a non-cancellative pair $p,q$ is *minimal* if for each non-cancellative pair $s,t$, $$\mathcal{R}_{s,t} \subseteq \mathcal{R}_{p,q} \ \ \text{ implies } \ \ \{s,t\} = \{ p,q \}.$$
\[here2\] Let $p,q \in e_jAe_i$ be distinct paths such that $$\label{p+}
\operatorname{t}(p^+) = \operatorname{t}(q^+) \ \ \text{ and } \ \ \operatorname{h}(p^+) = \operatorname{h}(q^+).$$ The following hold.
1. $p \sigma_i^m = q \sigma_i^n$ for some $m, n \geq 0$.
2. $\bar{\tau}(p) = \bar{\tau}(q) \sigma^m$ for some $m \in \mathbb{Z}$.
3. If $p,q$ is a non-cancellative pair, then $\bar{\tau}(p) = \bar{\tau}(q)$ and $\bar{\eta}(p) = \bar{\eta}(q)$.
4. If $\mathcal{P} \not = \emptyset$ and $\bar{\eta}(p) = \bar{\eta}(q)$, then $p,q$ is a non-cancellative pair.
\(1) We proceed by induction on the region $\mathcal{R}_{p,q} \subset \mathbb{R}^2$ bounded by $p^+$ and $q^+$, with respect to inclusion.
If there are unit cycles $sa$ and $ta$ with $a \in Q_1$, and $\mathcal{R}_{p,q} = \mathcal{R}_{s,t}$, then the claim is clear. So suppose the claim holds for all pairs of paths $s,t$ such that $$\operatorname{t}(s^+) = \operatorname{t}(t^+), \ \ \ \operatorname{h}(s^+) = \operatorname{h}(t^+), \ \ \text{ and } \ \ \mathcal{R}_{s,t} \subset \mathcal{R}_{p,q}.$$
Factor $p$ and $q$ into subpaths, $$p = p_m p_{m-1} \cdots p_1 \ \ \text{ and } \ \ q = q_n q_{n-1} \cdots q_1,$$ such that for each $1 \leq \alpha \leq m$ and $1 \leq \beta \leq n$, there are paths $p'_{\alpha}$ and $q'_{\beta}$ for which $$p'_{\alpha}p_{\alpha} \ \ \text{ and } \ \ q'_{\beta}q_{\beta}$$ are unit cycles, and ${p'_{\alpha}}^+$ and ${q'_{\beta}}^+$ lie in the region $\mathcal{R}_{p,q}$. See Figure \[purple\]. Note that if $p_{\alpha}$ is itself a unit cycle, then $p'_{\alpha}$ is the vertex $\operatorname{t}(p_{\alpha})$, and similarly for $q_{\beta}$.
Consider the paths $$p' := p'_1p'_2 \cdots p'_m \ \ \text{ and } \ \ q' := q'_1q'_2 \cdots q'_n.$$ Then by Lemma \[sigma\] there is some $c,d \geq 0$ such that $$\label{p'p =}
p'p = \sigma_i^c \ \ \text{ and } \ \ q'q = \sigma_i^d.$$
Now $$\operatorname{t}(p'^+) = \operatorname{t}(q'^+), \ \ \ \operatorname{h}(p'^+) = \operatorname{h}(q'^+), \ \ \text{ and } \ \ \mathcal{R}_{p',q'} \subset \mathcal{R}_{p,q}.$$ Thus, by induction, there is some $c',d' \geq 0$ such that $$p' \sigma_i^{c'} = q' \sigma_i^{d'}.$$ Therefore by Lemma \[sigma\], $$p \sigma_i^{d+d'} = p q'q \sigma_i^{d'} = pq'\sigma_j^{d'}q = pp'\sigma_j^{c'}q = p p' q \sigma_i^{c'} = q \sigma_i^{c+c'},$$ proving our claim.
\(2) By Claim (1) there is some $m, n \geq 0$ such that $$\label{sigmaj}
p \sigma^m_i = q \sigma_i^n.$$ Thus by Lemma \[tau’A’\], $$\label{sigmaj2}
\bar{\tau}(p)\sigma^m = \bar{\tau}\left( p\sigma^m_i \right) = \bar{\tau}\left( q\sigma_i^n \right) = \bar{\tau}(q)\sigma^n \in B.$$ Claim (2) then follows since $B$ is an integral domain.
\(3) Suppose $p,q$ is a non-cancellative pair. Then there is a path $r$ such that $$rp = rq \not = 0 \ \ \ \text{ or } \ \ \ pr = qr \not = 0;$$ say $rp = rq$. Whence $$\bar{\eta}(r)\bar{\eta}(p) = \bar{\eta}(rp) = \bar{\eta}(rq) = \bar{\eta}(r)\bar{\eta}(q),$$ by Lemma \[tau’A’\]. Therefore $\bar{\eta}(p) = \bar{\eta}(q)$ since $B$ is an integral domain. Similarly, $\bar{\tau}(p) = \bar{\tau}(q)$.
\(4) Finally, suppose $\mathcal{P} \not = \emptyset$ and $\bar{\eta}(p) = \bar{\eta}(q)$. Set $\sigma := \prod_{D \in \mathcal{P}} x_D$. Then (\[sigmaj2\]) implies $$\sigma^m = \sigma^n$$ since $B$ is an integral domain (with $\bar{\eta}$ in place of $\bar{\tau}$). By assumption, $\mathcal{P} \not = \emptyset$. Whence $\sigma \not = 1$. Therefore $m = n$. Consequently, the path $$r = \sigma_i^m$$ satisfies $pr = qr \not = 0$ by (\[sigmaj\]).
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\[r+\]
1. Suppose paths $p,q$ are either equal modulo $I$, or form a non-cancellative pair. Then their lifts $p^+$ and $q^+$ bound a compact region $\mathcal{R}_{p,q}$ in $\mathbb{R}^2$.
2. Suppose paths $p,q$ are equal modulo $I$. If $i^+$ is a vertex in $\mathcal{R}_{p,q}$, then there is a path $r^+$ from $\operatorname{t}(p^+)$ to $\operatorname{h}(p^+)$ that is contained in $\mathcal{R}_{p,q}$, passes through $i^+$, and satisfies $$\label{r++}
p = r = q \ \ \text{ (modulo $I$)}.$$
(1.i) First suppose $p,q$ are equal modulo $I$. The relations generated by $I$ in (\[I def\]) lift to homotopy relations on the paths of $Q^+$. Thus $\operatorname{t}(p^+) = \operatorname{t}(q^+)$ and $\operatorname{h}(p^+) = \operatorname{h}(q^+)$. Therefore $p^+$ and $q^+$ bound a compact region in $\mathbb{R}^2$.
(1.ii) Now suppose $p,q$ is a non-cancellative pair. Then there is a path $r$ such that $rp = rq \not = 0$, say. In particular, $\operatorname{t}((rp)^+) = \operatorname{t}((rq)^+)$ and $\operatorname{h}((rp)^+) = \operatorname{h}((rq)^+)$ by Claim (1.i.). Therefore $\operatorname{t}(p^+) = \operatorname{t}(q^+)$ and $\operatorname{h}(p^+) = \operatorname{h}(q^+)$ as well.
\(2) The ideal $I$ is generated by relations of the form $s - t$, where there is an arrow $a$ such that $sa$ and $ta$ are unit cycles. The claim then follows since each trivial subpath of the unit cycle $sa$ (resp. $ta$) is a trivial subpath of $s$ (resp. $t$).
\[removable2\] [ A unit cycle $\sigma_i \in A$ of length 2 is a *removable 2-cycle* if the two arrows it is composed of are redundant generators for $A$, and otherwise $\sigma_i$ is a *permanent 2-cycle*. ]{}
\[permanent 2-cycles\] There are precisely two types of permanent 2-cycles, given in Figures \[2-cycle\].ii and \[2-cycle\].iii. Consequently, if a dimer algebra $A$ has a permanent 2-cycle, then $A$ is degenerate.
Let $ab$ be a permanent 2-cycle, with $a,b \in Q_1$. Let $$\sigma_{\operatorname{t}(a)} = sa \ \ \text{ and } \ \ \sigma'_{\operatorname{t}(b)} = tb$$ be the complementary unit cycles to $ab$ containing $a$ and $b$ respectively. Since $ab$ is permanent, $b$ is a subpath of $s$, or $a$ is a subpath of $t$.
Suppose $b$ is a subpath of $s$. Then there are (possibly trivial) paths $p,q$ such that $s = qbp$. We therefore have either case (ii) or case (iii) shown in Figure \[2-cycle\].
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(i) & (ii) & (iii)
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By a *cyclic subpath* of a path $p$, we mean a proper subpath of $p$ that is a nontrivial cycle. Consider the following sets of cycles in $A$:
- Let $\mathcal{C}$ be the set of cycles in $A$ (i.e., cycles in $Q$ modulo $I$).
- For $u \in \mathbb{Z}^2$, let $\mathcal{C}^u$ be the set of cycles $p \in \mathcal{C}$ such that $$\operatorname{h}(p^+) = \operatorname{t}(p^+) + u \in Q_0^+.$$
- For $i \in Q_0$, let $\mathcal{C}_i$ be the set of cycles in the vertex corner ring $e_iAe_i$.
- Let $\hat{\mathcal{C}}$ be the set of cycles $p \in \mathcal{C}$ with a representative $\tilde{p}$ such that $(\tilde{p}^2)^+$ does not have a cyclic subpath; equivalently, the lift of each cyclic permutation of $\tilde{p}$ does not have a cyclic subpath.
We denote the intersection $\hat{\mathcal{C}} \cap \mathcal{C}^u \cap \mathcal{C}_i$, for example, by $\hat{\mathcal{C}}^u_i$. Note that $\mathcal{C}^0$ is the set of cycles whose lifts are cycles in $Q^+$. In particular, $\hat{\mathcal{C}}^0 = \emptyset$. Furthermore, the lift of any cycle $p$ in $\hat{\mathcal{C}}$ has no cyclic subpaths, although $p$ itself may have cyclic subpaths.
\[here3’\] Suppose $A$ does not have a non-cancellative pair where one of the paths is a vertex. Let $p$ be a nontrivial cycle.
1. If $p \in \mathcal{C}^0$, then ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = \sigma^m$ for some $m \geq 1$.
2. If $p \in \mathcal{C}^0$ and $A$ is cancellative, then $p = \sigma_i^m$ for some $m \geq 1$.
3. If $p \in \mathcal{C} \setminus \hat{\mathcal{C}}$, then $\sigma \mid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$.
4. If $p$ is a path for which $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$, then $p$ is a subpath of a cycle in $\hat{\mathcal{C}}$.
\(1) Suppose $p \in \mathcal{C}^0$, that is, $p^+$ is a cycle in $Q^+$. Set $i := \operatorname{t}(p)$. By Lemma \[here2\].1, there is some $m,n \geq 0$ such that $$\label{n > m}
p\sigma_i^m = \sigma_i^n.$$ If $m \geq n$, then the paths $p \sigma_i^{m-n}$ and $e_{\operatorname{t}(p)}$ form a non-cancellative pair. But this is contrary to assumption. Thus $n - m \geq 1$.
Furthermore, $\bar{\tau}$ is an algebra homomorphism on $e_iAe_i$ by Lemma \[tau’A’\]. In particular, (\[n > m\]) implies $${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}\sigma^m = \sigma^n.$$ Therefore ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = \sigma^{n - m}$ since $B$ is an integral domain.
\(2) If $A$ is cancellative, then (\[n > m\]) implies $p = \sigma_i^{n-m}$.
\(3) Suppose $p \in \mathcal{C} \setminus \hat{\mathcal{C}}$. Then there is a cyclic permutation of $p^+$ which contains a cyclic subpath $q^+$. In particular, ${\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} \mid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$. Furthermore, since $q \in \mathcal{C}^0$, Claim (1) implies ${\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} = \sigma^m$ for some $m \geq 1$. Therefore $\sigma \mid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$.
\(4) Let $p$ be a path for which $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$. Then there is a simple matching $D \in \mathcal{S}$ such that $x_D \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$. In particular, $p$ is supported on $Q \setminus D$. Since $D$ is simple, $p$ is a subpath of a cycle $q$ supported on $Q \setminus D$. Whence $x_D \nmid {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}$. Thus $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}$. Therefore $q$ is in $\hat{\mathcal{C}}$ by the contrapositive of Claim (3).
\[weird\]
In Lemma \[here3’\] we assumed that $A$ does not have a non-cancellative pair where one of the paths is a vertex. Such non-cancellative pairs exist: if $A$ contains a permanent 2-cycle as in Figure \[2-cycle\].ii, then $p\sigma_{\operatorname{t}(p)} = \sigma_{\operatorname{t}(p)}$. In particular, $p, e_{\operatorname{t}(p)}$ is a non-cancellative pair.
Furthermore, it is possible for $m > n$ in (\[n > m\]). Consider a dimer algebra with the subquiver given in Figure \[weird fig\]. Here $a,b,c,d$ are arrows, $q$ is a nontrivial path, and $ad$, $bc$, $qdcba$ are unit cycles. Let $m \geq 1$, and set $p := baq^{m+1}dc$. Then $$p \sigma_i^m = baq^{m+1}dc \sigma_i^m \stackrel{\textsc{(i)}}{=} ba(q \sigma_j)^m qdc \stackrel{\textsc{(ii)}}{=} b(aqd)^{m+1}c \stackrel{\textsc{(iii)}}{=} bc = \sigma_i.$$ Indeed, (<span style="font-variant:small-caps;">i</span>) holds by Lemma \[sigma\]; (<span style="font-variant:small-caps;">ii</span>)holds since $\sigma_j = da$; and (<span style="font-variant:small-caps;">iii</span>) holds since $b = baqd$ in $A$. Note, however, that such a dimer quiver does not admit a perfect matching.
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For the remainder of this section, we assume that $Q$ has at least one perfect matching, $\mathcal{P} \not = \emptyset$, unless stated otherwise.
By the definition of $\hat{\mathcal{C}}$, each cycle $p \in \mathcal{C}^u_i \setminus \hat{\mathcal{C}}$ has a representative $\tilde{p}$ that factors into subpaths $\tilde{p} = \tilde{p}_3 \tilde{p}_2 \tilde{p}_1$, where $$\label{C0}
p_1p_3 = \tilde{p}_1 \tilde{p}_3 + I \in \mathcal{C}^0 \ \ \text{ and } \ \ p_2 = \tilde{p}_2 + I \in \hat{\mathcal{C}}^u.$$
\[here\] If $\hat{\mathcal{C}}^u_i = \emptyset$ for some $u \in \mathbb{Z}^2 \setminus 0$ and $i \in Q_0$, then $A$ is non-cancellative.
In the following, we will use the notation (\[C0\]). Set $\sigma := \prod_{D \in \mathcal{P}} x_D$.
Suppose $\hat{\mathcal{C}}^u_i = \emptyset$. Let $p,q \in \mathcal{C}^u_i$ be cycles such that the region $$\mathcal{R}_{\tilde{p}_3\tilde{p}_2^2\tilde{p}_1,\tilde{q}_3\tilde{q}_2^2\tilde{q}_1}$$ contains the vertex $i^+ + u \in Q_0^+$. Furthermore, suppose $p$ and $q$ admit representatives $\tilde{p}'$ and $\tilde{q}'$ (possibly distinct from $\tilde{p}$ and $\tilde{q}$) such that the region $\mathcal{R}_{\tilde{p}',\tilde{q}'}$ has minimal area among all such pairs of cycles $p,q$. See Figure \[6figure\].
By Lemmas \[tau’A’\] and \[here2\].1, there is some $m \in \mathbb{Z}$ such that $$\bar{\eta}(p_3p_2^2p_1) = \bar{\eta}(q_3q_2^2q_1) \sigma^m.$$ Suppose $m \geq 0$. Set $$s := p_3p_2^2p_1 \ \ \text{ and } \ \ t := q_3q_2^2q_1 \sigma_i^m.$$ Then $$\label{s == t}
\bar{\eta}(s) = \bar{\eta}(t).$$
Assume to the contrary that $A$ is cancellative. If $s \not = t$, then $s,t$ would be a non-cancellative pair by Lemma \[here2\].4. Therefore $s = t$. Furthermore, there is a path $r^+$ in $\mathcal{R}_{\tilde{s},\tilde{t}}$ which passes through the vertex $$i^+ + u \in Q_0^+$$ and is homotopic to $s^+$ (by the relations $I$), by Lemma \[r+\].2. In particular, $r$ factors into paths $r = r_2e_ir_1 = r_2r_1$, where $$r_1, r_2 \in \mathcal{C}^u_i.$$ But $p$ and $q$ were chosen so that the area of $\mathcal{R}_{\tilde{p}',\tilde{q}'}$ is minimal. Thus there is some $\ell_1, \ell_2 \geq 0$ such that $$\tilde{r}_1 = \tilde{p}'\sigma^{\ell_1}_i \ \ \text{ and } \ \ \tilde{r}_2 = \tilde{p}'\sigma^{\ell_2}_i \ \ \ \text{(modulo $I$)}.$$ Set $\ell := \ell_1 + \ell_2$. Then $$\label{r p2}
r = r_2r_1 = p^2 \sigma^{\ell_1 + \ell_2}_i = p^2 \sigma^{\ell}_i.$$
Since $A$ is cancellative, the $\bar{\eta}$-image of any nontrivial cycle in $Q^+$ is a positive power of $\sigma$ by Lemma \[here3’\].1 (with $\bar{\eta}$ in place of $\bar{\tau}$). In particular, since $(p_1p_3)^+$ is a nontrivial cycle, there is an $n \geq 1$ such that $$\label{p1p3}
\bar{\eta}(p_1p_3) = \sigma^n.$$
Hence $$\bar{\eta}(p)\bar{\eta}(p_2) \stackrel{\textsc{(i)}}{=} \bar{\eta}(s) = \bar{\eta}(r) \stackrel{\textsc{(ii)}}{=} \bar{\eta}(p^2)\sigma^{\ell} \stackrel{\textsc{(iii)}}{=} \bar{\eta}(p)\bar{\eta}(p_2) \bar{\eta}(p_1p_3) \sigma^{\ell} \stackrel{\textsc{(iv)}}{=} \bar{\eta}(p)\bar{\eta}(p_2) \sigma^{n + \ell}.$$ Indeed, (<span style="font-variant:small-caps;">i</span>) and (<span style="font-variant:small-caps;">iii</span>) hold by Lemma \[tau’A’\], (<span style="font-variant:small-caps;">ii</span>) holds by (\[r p2\]), and (<span style="font-variant:small-caps;">iv</span>) holds by (\[p1p3\]). Thus, since $k\left[ x_D \ | \ D \in \mathcal{P} \right]$ is an integral domain, we have $$\sigma^{n + \ell} = 1.$$ But $n \geq 1$ and $\ell \geq 0$. Whence $\sigma = 1$. Therefore $Q$ has no perfect matchings, contrary to our standing assumption that $Q$ has at least one perfect matching.
$$\xy 0;/r.3pc/:
(0,-17)*{\cdot}="1";
(13,-17)*{\cdot}="2";(0,0)*{\cdot}="3";(13,0)*{\cdot}="4";(0,17)*{\cdot}="5";
(13,17)*{\cdot}="6";
(-13,-17)*{\cdot}="7";(-13,0)*{\cdot}="8";(-13,17)*{\cdot}="9";
{\ar_{q_2}@[blue]"2";"4"};{\ar_{q_2}@[blue]"4";"6"};{\ar^{p_2}@[red]"7";"8"};{\ar^{p_2}@[red]"8";"9"};
{\ar@/^/_{q_1}@[blue]"1";"2"};{\ar@/_/^{p_1}@[red]"1";"7"};
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(0,-17)*{}="10";(0,1)*{}="11";(0,17)*{}="12";
{\ar@{}_{i^+}"10";"10"};
{\ar@{}^{i^+ + u}"11";"11"};
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{\ar@/_2pc/_{\tilde{q}'}@[green]"3";"5"};
{\ar@/^2pc/^{\tilde{p}'}@[green]"3";"5"};
\endxy$$
[ We call the subquiver given in Figure \[figure3\].i a *column*, and the subquiver given in Figure \[figure3\].ii a *pillar*. In the latter case, $\operatorname{h}(a_{\ell})$ and $\operatorname{t}(a_1)$ are either trivial subpaths of $q_{\ell}$ and $p_1$ respectively, or $p_{\ell}$ and $q_1$ respectively. ]{}
$$\begin{array}{cccc}
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(-5,-15)*{}="1";(5,-15)*{\cdot}="1'";
(-5,-10)*{\cdot}="2";(5,-5)*{\cdot}="3";
(0,-2.5)*{\vdots}="";
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(5,10)*{\cdot}="7";
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(5,-10)*{}="9";(-5,10)*{}="10";
{\ar^{p_1}@[red]"1";"2"};{\ar_{a_1}@[brown]"2";"1'"};{\ar@{-}_{q_1}@[blue]"1'";"9"};
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{\ar_{a_1}@[brown]"8";"7"};{\ar_{q_1}@[blue]"7";"8'"};{\ar_{a_{\ell}}@[brown]"5";"6"};
\endxy
&
\xy 0;/r.4pc/:
(0,-15)*{\cdot}="1";(-5,-10)*{\cdot}="2";(5,-5)*{\cdot}="3";
(0,-2.5)*{\vdots}="";
(5,0)*{\cdot}="6";(-5,5)*{\cdot}="5";
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&
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(0,-15)*{\cdot}="1";(-5,-10)*{}="2";(5,-5)*{}="3";
(0,-2.5)*{\vdots}="";
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(-5,-5)*{\cdot}="11";
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{\ar@/_/|-{a_{\ell}}@[brown]"8";"7"};{\ar@/_/_{q_{\ell}}@[blue]"7";"8"};{\ar_{a_{\ell-1}}@[brown]"5";"6"};
\endxy
&
\xy 0;/r.4pc/:
(0,-15)*{\cdot}="1";(-5,-10)*{\cdot}="2";(5,-5)*{\cdot}="3";
(0,-2.5)*{\vdots}="";
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(5,10)*+{\cdot}="7";(0,15)*{\cdot}="8";(5,-10)*{}="9";(-5,10)*{}="10";
(5,5)*{\cdot}="11";
{\ar@/^/^{p_1}@[red]"1";"2"};{\ar@/^/|-{a_1}@[brown]"2";"1"};{\ar@{-}@/_/_{q_1}@[blue]"1";"9"};
{\ar^{}@[blue]"9";"3"};{\ar_{b_1}"3";"2"};{\ar^{}@[red]"2";"5"};{\ar^{}@[blue]"3";"6"};
{\ar_{q_{\ell-1}}@[blue]"6";"11"};
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{\ar@/_/@[green]"7";"11"};{\ar@/_/@[green]"11";"6"};{\ar@/^1.5pc/@[green]"6";"7"};
{\ar|-{b_{\ell-1}}"7";"5"};{\ar@{-}^{p_{\ell}}@[red]"5";"10"};{\ar@/^/^{}@[red]"10";"8"};
{\ar@/_/|-{a_{\ell}}@[brown]"8";"7"};{\ar@/_/_{q_{\ell}}@[blue]"7";"8"};{\ar_{a_{\ell-1}}@[brown]"5";"6"};
\endxy
\\
(i) & (ii) & (iii) & (iv)
\end{array}$$
$$\begin{array}{ccc}
\xy
(-15,-20)*{}="1";(15,-20)*{\cdot}="2";(-15,20)*{}="3";(15,20)*{\cdot}="4";(-15,-5)*{\cdot}="5";(-15,5)*{\cdot}="6";
{\ar_{q_j}@[blue]"2";"4"};{\ar_{b_j}"4";"6"};{\ar@/^/@[green]"6";"5"};{\ar_{a_j}@[brown]"5";"2"};{\ar^{p_j}@[red]"1";"5"};{\ar^{p'_j}@[red]"5";"6"};{\ar^{p_{j+1}}@[red]"6";"3"};
\endxy
& \ \ \ &
\xy
(-15,-20)*{}="1'";
(-15,0)*{\cdot}="6";(-15,20)*{}="3'";
{\ar^{p_j}@[red]"1'";"6"};{\ar^{p_{j+1}}@[red]"6";"3'"};
(15,-20)*+{\cdot}="1";(15,-10)*{\cdot}="2";(15,0)*{\cdot}="3";(15,10)*{\cdot}="4";(15,20)*+{\cdot}="5";
(15,6)*{\vdots}="";
{\ar_{b_j}"5";"6"};{\ar_{a_j}@[brown]"6";"1"};{\ar@/^2.5pc/@[green]"1";"5"};
{\ar@/_/@[green]"5";"4"};{\ar@/_/@[green]"3";"2"};{\ar@/_/@[green]"2";"1"};
{\ar_{q_{j1}}@[blue]"1";"2"};{\ar_{q_{j2}}@[blue]"2";"3"};{\ar_{q_{jm}}@[blue]"4";"5"};
\endxy
\\ \\
(\textsc{i}) & & (\textsc{ii})
\end{array}$$
\[columns and pillars\] Suppose paths $p^+$, $q^+$ have no cyclic subpaths, and bound a region $\mathcal{R}_{p,q}$ which contains no vertices in its interior.
1. If $p$ and $q$ do not intersect, then $p^+$ and $q^+$ bound a column.
2. Otherwise $p^+$ and $q^+$ bound a union of pillars. In particular, if $$\operatorname{t}(p^+) = \operatorname{t}(q^+) \ \ \ \text{ and } \ \ \ \operatorname{h}(p^+) = \operatorname{h}(q^+) \not = \operatorname{t}(p^+),$$ then $p = q$ (modulo $I$).
Since $\mathcal{R}_{p,q}$ contains no vertices in its interior, each path that intersects its interior is an arrow. Thus $p^+$ and $q^+$ bound a union of subquivers given by the four cases in Figure \[figure3\].
In case (i), $$p = p_{\ell} \cdots p_1 \ \ \ \text{ and } \ \ \ q = q_{\ell} \cdots q_1.$$ In cases (ii) - (iv), the paths $p_{\ell} \cdots p_1$ and $q_{\ell} \cdots q_1$ are not-necessarily-proper subpaths of $p$ and $q$ respectively. In cases (ii) and (iv), $\operatorname{h}(a_{\ell})$ and $\operatorname{t}(a_1)$ are either trivial subpaths of $q$ and $p$ respectively, or $p$ and $q$ respectively. In case (iii), $\operatorname{h}(a_{\ell})$ and $\operatorname{t}(a_1)$ are either both trivial subpaths of $q$, or both trivial subpaths of $p$. In all cases, each cycle which bounds a region with no arrows in its interior is a unit cycle. In particular, in cases (i) - (iii), each path $a_jp_jb_{j-1}$, $b_jq_ja_j$ is a unit cycle.
Observe that in cases (iii) and (iv), $q^+$ has a cyclic subpath, contrary to assumption. Generalizations of case (iv) are considered in Figures \[factor2\].<span style="font-variant:small-caps;">i</span> and \[factor2\].<span style="font-variant:small-caps;">ii</span>. Consequently, $p^+$ and $q^+$ bound either a column or a union of pillars.
\[non-crossing\] [ For $u \in \mathbb{Z}^2$, denote by $\widehat{\mathcal{C}}^u$ a maximal set of representatives of cycles in $\hat{\mathcal{C}}^u$ whose lifts do not intersect transversely in $\mathbb{R}^2$ (though they may share common subpaths). ]{}
In the following two lemmas, fix $u \in \mathbb{Z}^2 \setminus 0$ and a subset $\widehat{\mathcal{C}}^u$.
\[figure2lemma\] Suppose $\hat{\mathcal{C}}^u_i \not = \emptyset$ for each $i \in Q_0$. Then $\widehat{\mathcal{C}}^u_i \not = \emptyset$ for each $i \in Q_0$.
Suppose $\tilde{p},\tilde{q}$ are representatives of cycles $p,q$ in $\hat{\mathcal{C}}^u$ that intersect transversely at $k \in Q_0$. Then their lifts $\tilde{p}^+$ and $\tilde{q}^+$ intersect at least twice since $p$ and $q$ are both in $\mathcal{C}^u$. Thus $\tilde{p}$ and $\tilde{q}$ factor into paths $\tilde{p} = p_3p_2p_1$ and $\tilde{q} = q_3q_2q_1$, where $$\operatorname{t}(p_2^+) = \operatorname{t}(q_2^+) \ \ \text{ and } \ \ \operatorname{h}(p_2^+) = \operatorname{h}(q_2^+),$$ and $k^+$ is the tail or head of $p_2^+$. See Figure \[figure2\].
Since $\hat{\mathcal{C}}^u_i \not = \emptyset$ for each $i \in Q_0$, we may suppose that $\mathcal{R}_{p_2,q_2}$ contains no vertices in its interior. Since $p$ and $q$ are in $\hat{\mathcal{C}}$, $p_2^+$ and $q_2^+$ do not have cyclic subpaths. Thus $p_2 = q_2$ (modulo $I$) by Lemma \[columns and pillars\].2. Therefore the paths $$\tilde{s} := p_3q_2p_1 \ \ \ \text{ and } \ \ \ \tilde{t} := q_3p_2q_1$$ equal $\tilde{p}$ and $\tilde{q}$ (modulo $I$) respectively. In particular, $s$ and $t$ are in $\hat{\mathcal{C}}^u$. The lemma then follows since $\tilde{s}^+$ and $\tilde{t}^+$ do not intersect transversely at the vertices $\operatorname{t}(p_2^+)$ or $\operatorname{h}(p_2^+)$.
$$\xy 0;/r.4pc/:
(-3,-9)*+{\text{\scriptsize{$i$}}}="1";(-3,-3)*{\cdot}="2";(-3,3)*{\cdot}="3";(-3,9)*+{\text{\scriptsize{$i$}}}="4";
(3,-9)*+{\text{\scriptsize{$j$}}}="5";(3,9)*+{\text{\scriptsize{$j$}}}="6";
{\ar^{p_1}@[red]"1";"2"};{\ar_{p_2}@[red]"2";"3"};{\ar^{p_3}@[red]"3";"4"};
{\ar@/_.9pc/_{q_1}@[blue]"5";"2"};{\ar@/^1.2pc/^{q_2}@[blue]"2";"3"};{\ar@/_.9pc/_{q_3}@[blue]"3";"6"};
\endxy$$
\[figure3lemma\] Let $u \in \mathbb{Z}^2$, and suppose $\hat{\mathcal{C}}^u_i \not = \emptyset$ for each $i \in Q_0$. Then there is a simple matching $D \in \mathcal{S}$ such that $Q \setminus D$ supports each cycle in $\widehat{\mathcal{C}}^u$.
Furthermore, if $A$ contains a column, then there are two simple matchings $D_1,D_2 \in \mathcal{S}$ such that $Q \setminus (D_1 \cup D_2)$ supports each cycle in $\widehat{\mathcal{C}}^u$.
Suppose the hypotheses hold. By Lemma \[figure2lemma\], $\widehat{\mathcal{C}}^u_i \not = \emptyset$ for each $i \in Q_0$ since $\hat{\mathcal{C}}^u_i \not = \emptyset$ for each $i \in Q_0$. Thus we may consider cycles $p,q \in \widehat{\mathcal{C}}^u$ for which $\pi^{-1}(p)$ and $\pi^{-1}(q)$ bound a region $\mathcal{R}_{p,q}$ with no vertices in its interior.
Recall Figure \[figure3\]. Since $\widehat{\mathcal{C}}^u_i \not = \emptyset$ for each $i \in Q_0$, we may partition $Q$ into columns and pillars, by Lemma \[columns and pillars\]. Consider the subset of arrows $D_1$ (resp. $D_2$) consisting of all the $a_j$ arrows in each pillar of $Q$, and all the $a_j$ arrows (resp. $b_j$ arrows) in each column of $Q$. Note that $D_1$ consists of all the right-pointing arrows in each column, and $D_2$ consists of all the left-pointing arrows in each column. Furthermore, if $Q$ does not contain a column, then $D_1 = D_2$.
Observe that each unit cycle in each column and pillar contains precisely one arrow in $D_1$, and one arrow in $D_2$. (Note that this is not true for cases (iii) and (iv).) Furthermore, no such arrow occurs on the boundary of these regions, that is, as a subpath of $p$ or $q$. Therefore $D_1$ and $D_2$ are perfect matchings of $Q$.
Recall that a simple $A$-module of dimension $1^{Q_0}$ is characterized by the property that there is a non-annihilating path between any two vertices of $Q$. Clearly $Q \setminus D_1$ and $Q \setminus D_2$ each support a path that passes through each vertex of $Q$. Therefore $D_1$ and $D_2$ are simple matchings.
\[at least one\] If $A$ is cancellative, then $Q$ has at least one simple matching.
Follows from Proposition \[here\] and Lemma \[figure3lemma\].
Lemma \[at least one\] will be superseded by Theorem \[AKZ\] below.
\[here3\] Let $u \in \mathbb{Z}^2 \setminus 0$, and suppose $\hat{\mathcal{C}}^u_i \not = \emptyset$ for each $i \in Q_0$. Then $A$ does not have a non-cancellative pair where one of the paths is a vertex.
Suppose $e_i, p$ is a non-cancellative pair. Then there is some $m,n \geq 0$ such that $$p\sigma_i^m = e_i \sigma_i^n = \sigma_i^n,$$ by Lemma \[here2\].1. Whence ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = \sigma^{n-m}$ since $B$ is an integral domain. But $n - m > 0$ since $p \not = e_i$. Furthermore, $$\sigma^{n-m} = {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mue\mkern-1.5mu}\mkern 1.5mu}_i = 1,$$ by Lemma \[here2\].3. Thus $\sigma = 1$. Therefore $Q$ has no simple matchings, $\mathcal{S} = \emptyset$. Consequently, for each $u \in \mathbb{Z}^2 \setminus 0$ there is a vertex $i \in Q_0$ such that $\hat{\mathcal{C}}^u_i = \emptyset$, by Lemma \[figure3lemma\].
\[longlist\] Let $u \in \mathbb{Z}^2 \setminus 0$. If $p,q \in \mathcal{C}^u$, then there is some $n \in \mathbb{Z}$ such that ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}\sigma^n$. In particular, if $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$ and $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}$, then ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}$.
Consider cycles $p,q \in \mathcal{C}^u$. Since $Q$ is a dimer quiver, there is a path $r$ from $\operatorname{t}(p)$ to $\operatorname{t}(q)$. Thus there is some $m, n \geq 0$ such that $$rp \sigma_{\operatorname{t}(p)}^m = qr \sigma_{\operatorname{t}(p)}^n,$$ by Lemma \[here2\].1. Furthermore, $\tau$ is an algebra homomorphism by Lemma \[tau’A’\]. Thus $${\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} \sigma^m = \bar{\tau}\left(rp \sigma_{\operatorname{t}(p)}^n \right) = \bar{\tau}\left(qr \sigma_{\operatorname{t}(p)}^n \right) = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu} \sigma^n.$$ Therefore ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} \sigma^{n-m}$ since $B$ is an integral domain.
\[longlist3\] Suppose $A$ is cancellative.
1. If $a \in Q_1$, $p \in \hat{\mathcal{C}}_{\operatorname{t}(a)}^u$, and $q \in \hat{\mathcal{C}}_{\operatorname{h}(a)}^u$, then $ap = qa$.
2. Each vertex corner ring $e_iAe_i$ is commutative.
\(1) Suppose $a \in Q_1$, $p \in \hat{\mathcal{C}}_{\operatorname{t}(a)}^u$, and $q \in \hat{\mathcal{C}}_{\operatorname{h}(a)}^u$. Then $$\operatorname{t}\left((ap)^+\right) = \operatorname{t}\left((qa)^+\right) \ \ \text{ and } \ \ \operatorname{h}\left((ap)^+\right) = \operatorname{h}\left((qa)^+\right).$$ Let $r^+$ be a path in $Q^+$ from $\operatorname{h}\left( (ap)^+ \right)$ to $\operatorname{t}\left( (ap)^+ \right)$. Then by Lemma \[here3’\].2, there is some $m, n \geq 1$ such that $$rap = \sigma_{\operatorname{t}(a)}^m \ \ \text{ and } \ \ rqa = \sigma_{\operatorname{t}(a)}^n.$$
Assume to the contrary that $m < n$. Then $qa = ap \sigma_{\operatorname{t}(a)}^{n-m}$ since $A$ is cancellative. Let $b$ be a path such that $ab$ is a unit cycle. By Lemma \[sigma\], we have $$q \sigma_{\operatorname{h}(a)} = qab = ap \sigma_{\operatorname{t}(a)}^{n-m} b = apb \sigma^{n-m}_{\operatorname{h}(a)}.$$ Thus, since $A$ is cancellative and $n-m \geq 1$, $$q = apb \sigma^{n-m-1}_{\operatorname{h}(a)}.$$ Furthermore, $(ba)^+$ is cycle in $Q^+$ since $ba$ is a unit cycle. But this is a contradiction since $q$ is in $\hat{\mathcal{C}}^u$. Whence $m = n$. Therefore $qa = ap$.
\(2) Consider cycles $p,q \in e_iAe_i$. Since $I$ is generated by binomials, it suffices to show that $qp = pq$. Let $r^+$ be a path in $Q^+$ from $\operatorname{h}\left((qp)^+\right)$ to $\operatorname{t}\left((qp)^+\right)$. Set $r := \pi(r^+)$. Then $(rqp)^+$ and $(rpq)^+$ are cycles. In particular, there is some $m,n \geq 1$ such that $$rqp = \sigma_i^m \ \ \text{ and } \ \ rpq = \sigma_i^n,$$ by Lemma \[here3’\].2. Thus, since $\tau$ is an algebra homomorphism, $$\sigma^m = {\mkern 1.5mu\overline{\mkern-1.5murqp\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5murpq\mkern-1.5mu}\mkern 1.5mu} = \sigma^n.$$ Furthermore, $\sigma \not = 1$ by Lemma \[at least one\]. Whence $m = n$ since $B$ is an integral domain. Thus $rqp = \sigma_i^m = rpq$. Therefore $qp = pq$ since $A$ is cancellative.
\[circle 1\] Let $u \in \mathbb{Z}^2 \setminus 0$. Suppose (i) $A$ is cancellative, or (ii) $\hat{\mathcal{C}}_i^u \not = \emptyset$ for each $i \in Q_0$.
1. If $p \in \hat{\mathcal{C}}^u$, then $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$.
2. If $p,q \in \hat{\mathcal{C}}^u$, then ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}$.
3. If a cycle $p$ is formed from subpaths of cycles in $\hat{\mathcal{C}}^u$, then $p \in \hat{\mathcal{C}}$.
4. If $p,q \in \hat{\mathcal{C}}_i^u$, then $p = q$.
Note that in Claim (2) the cycles $p$ and $q$ are based at the same vertex $i$, whereas in Claim (3) $p$ and $q$ may be based at different vertices.
If $A$ is cancellative, then $\hat{\mathcal{C}}_i^u \not = \emptyset$ for each $i \in Q_0$, by Proposition \[here\]. Therefore assumption (i) implies assumption (ii), and so it suffices to suppose (ii) holds.
\(1) Fix a maximal set $\widehat{\mathcal{C}}^u$ of representatives of cycles in $\hat{\mathcal{C}}^u$ whose lifts do not intersect transversely in $\mathbb{R}^2$. Consider a cycle $p \in \widehat{\mathcal{C}}^u$. If $p$ has a representative $\tilde{p}$ in $\widehat{\mathcal{C}}^u$, then $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$, by Lemma \[figure3lemma\]. It thus suffices to suppose that $p$ does not have a representative belonging to $\widehat{\mathcal{C}}^u$. In particular, for any representative $\tilde{p}$ of $p$, there are cycles $\tilde{s}, \tilde{t} \in \widehat{\mathcal{C}}^u$ such that $$s = p_3q_2p_1, \ \ \ t = q_3p_2q_1, \ \ \ p = p_3p_2p_1,$$ as in Figure \[figure2\].
Since $\tilde{s},\tilde{t} \in \widehat{\mathcal{C}}^u$, we have $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu}$ and $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mut\mkern-1.5mu}\mkern 1.5mu}$. Thus $$\label{no black holes}
{\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mut\mkern-1.5mu}\mkern 1.5mu}$$ by Lemma \[longlist\]. Set $q := q_3q_2q_1$.
Assume to the contrary that $\sigma \mid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$. Then $$\sigma \mid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}_3{\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}_2{\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}_1 {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}_3{\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}_2{\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}_1 = {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5mut\mkern-1.5mu}\mkern 1.5mu} \stackrel{\textsc{(i)}}{=} {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu}^2,$$ where (<span style="font-variant:small-caps;">i</span>) holds by (\[no black holes\]). Therefore $\sigma \mid {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu}$ since $\sigma = \prod_{D \in \mathcal{S}}x_D$. But this is not possible since $\tilde{s} \in \widehat{\mathcal{C}}^u$.
\(2) Follows from Claim (1) and Lemma \[longlist\].
A direct proof assuming $A$ is cancellative: Suppose $p,q \in \hat{\mathcal{C}}^u$. Let $r$ be a path from ${\operatorname{t}(p)}$ to ${\operatorname{t}(q)}$. Then $rp = qr$ by Lemma \[longlist3\].1. Thus $${\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5murp\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muqr\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}.$$ Therefore ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}$ since $B$ is an integral domain.
\(3) Let $p = p_{\ell} \cdots p_1 \in \mathcal{C}^u$ be a cycle formed from subpaths $p_j$ of cycles $q_j$ in $\hat{\mathcal{C}}^u$. Then $$g := {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}_1 = \cdots = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}_{\ell}$$ by Claim (2). Furthermore, $\sigma \nmid g$ by Claim (1). In particular, there is a simple matching $D \in \mathcal{S}$ for which $x_D \nmid g$. Whence $x_D \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}_j$ for each $1 \leq j \leq \ell$. Thus $x_D \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$. Therefore $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$. Consequently, $p \in \hat{\mathcal{C}}$ by the contrapositive of Lemma \[here3’\].3, with Lemma \[here3\].
\(4) Follows from Claim (3) and Figure \[figure3\].ii.
A direct proof assuming $A$ is cancellative: Suppose $p,q \in \hat{\mathcal{C}}_i^u$. Let $r^+$ be a path in $Q^+$ from $\operatorname{h}\left(p^+ \right)$ to $\operatorname{t}\left( p^+ \right)$. Then there is some $m,n \geq 1$ such that $$rp = \sigma_i^m \ \ \text{ and } \ \ rq = \sigma_i^n,$$ by Lemma \[here3’\].2. Suppose $m \leq n$. Then $rp \sigma_i^{n-m} = \sigma_i^n = rq$. Thus $p\sigma_i^{n-m} = q$ since $A$ is cancellative. But then Claim (1) implies $m = n$ since by assumption $p$ and $q$ are in $\hat{\mathcal{C}}^u$. Therefore $p = q$.
\[area to zero\] Suppose $\hat{\mathcal{C}}^u_i \not = \emptyset$ for each $u \in \mathbb{Z}^2 \setminus 0$ and $i \in Q_0$. Let $\varepsilon_1, \varepsilon_2 \in \mathbb{Z}$. There is an $n \gg 1$ such that if $$p \in \hat{\mathcal{C}}^{(\varepsilon_1,0)}_i \ \ \ \text{ and } \ \ \ q \in \hat{\mathcal{C}}^{(n \varepsilon_1, \varepsilon_2)}_i,$$ then $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mupq\mkern-1.5mu}\mkern 1.5mu}$.
Fix $\varepsilon_1, \varepsilon_2 \in \mathbb{Z}$ and $i \in Q_0$. Consider the set of cycles $$p \in \hat{\mathcal{C}}^{(\varepsilon_1,0)}_i \ \ \ \text{ and } \ \ \ q_n \in \hat{\mathcal{C}}^{(n \varepsilon_1, \varepsilon_2)}_i,$$ with $n \geq 0$.
\(i) We claim that for each $n \geq 1$, $q_n^+$ lies in the region $\mathcal{R}_{p^2q_{n-1},q_{n+1}}$ (modulo $I$). This is shown in Figure \[area to zero fig\].i. Indeed, suppose a representative $\tilde{q}_n^+$ of $q_n^+$ intersects a representative $\tilde{q}_{n-1}^+$ of $q_{n-1}^+$, as shown in Figure \[area to zero fig\].ii. Then $q_{n-1}$ and $q_n$ factor into paths $$q_{n-1} = s_3s_2s_1 \ \ \ \text{ and } \ \ \ q_n = t_3t_2t_1,$$ where $\operatorname{t}(s_2^+) = \operatorname{t}(t_2^+)$ and $\operatorname{h}(s_2^+) = \operatorname{h}(t_2^+)$. In particular, there is some $m \in \mathbb{Z}$ such that $${\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu}_2 = {\mkern 1.5mu\overline{\mkern-1.5mut\mkern-1.5mu}\mkern 1.5mu}_2 \sigma^m,$$ by Lemma \[here2\].2. Set $r := t_3s_2t_1$.
Since $q_{n-1}$ and $q_n$ are in $\hat{\mathcal{C}}$, we have $$\label{sigma not divide}
\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}_{n-1} \ \ \ \text{ and } \ \ \ \sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}_n,$$ by Proposition \[circle 1\].1. Whence $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu}_2$ and $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mut\mkern-1.5mu}\mkern 1.5mu}_2$. Thus $m = 0$. Therefore $${\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mut\mkern-1.5mu}\mkern 1.5mu}_3 {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu}_2 {\mkern 1.5mu\overline{\mkern-1.5mut\mkern-1.5mu}\mkern 1.5mu}_1 = {\mkern 1.5mu\overline{\mkern-1.5mut\mkern-1.5mu}\mkern 1.5mu}_3{\mkern 1.5mu\overline{\mkern-1.5mut\mkern-1.5mu}\mkern 1.5mu}_2{\mkern 1.5mu\overline{\mkern-1.5mut\mkern-1.5mu}\mkern 1.5mu}_1 = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}_n.$$ In particular, $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu}$ by (\[sigma not divide\]). Thus the cycle $r$ is in $\hat{\mathcal{C}}$ by Lemmas \[here3’\].4 and \[here3\]. Furthermore, $r$ is in $\mathcal{C}^{(n \varepsilon_1, \varepsilon_2)}_i$ by construction. Whence $r$ is in $\hat{\mathcal{C}}^{(n \varepsilon_1, \varepsilon_2)}_i$. Therefore $r = q_n$ (modulo $I$) by Proposition \[circle 1\].4. This proves our claim.
\(ii) By Claim (i), there is a cycle $s \in \hat{\mathcal{C}}^{(\varepsilon_1,0)}$ such that for each $n \geq 1$, the area of the region $$\mathcal{R}_{sq'_{n},q'_{n+1}},$$ bounded by a rightmost subpath ${q'}^{+}_{n}$ of $q_{n}^+$, a rightmost subpath ${q'}^{+}_{n+1}$ of $q_{n+1}^+$, and $s^+$, tends to zero (modulo $I$) as $n \to \infty$. See Figure \[area to zero fig\].iii. (The case $r = p$ is shown in Figure \[area to zero fig\].i.) Since $Q$ is finite, there is some $N \gg 1$ such that if $n \geq N$, then $$\label{snq'n}
q'_{n+1} = sq'_{n} \ \ \ \text{(modulo $I$)}.$$
Fix $n \geq N$. There is a simple matching $D \in \mathcal{S}$ such that $x_D \nmid {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}_{n}$, by Proposition \[circle 1\].1. Whence $$x_D \nmid {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}'_{n+1} = {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}'_{n}.$$ In particular, $x_D \nmid {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu}$. But ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu}$ since $p$ and $s$ are both in $\hat{\mathcal{C}}^{(\varepsilon_1,0)}$, by Proposition \[circle 1\].2. Thus $x_D \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$. Therefore $x_D \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}_{n}$, and so $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}{\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}_{n}$.
$$\begin{array}{cc}
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Consider the subset of arrows $$\label{Q dagger def}
Q_1^{\mathcal{S}} := \left\{ a \in Q_1 \ | \ a \not \in D \text{ for each } D \in \mathcal{S} \right\},$$ where $\mathcal{S}$ is the set of simple matchings of $A$.
We will show in Theorem \[AKZ\] below that the two assumptions considered in the following lemma, namely that $Q_1^{\mathcal{S}} \not = \emptyset$ and $\hat{\mathcal{C}}^u_i \not = \emptyset$ for each $u \in \mathbb{Z}^2 \setminus 0$ and $i \in Q_0$, can never both hold.
\[purple lemma\] Suppose $\hat{\mathcal{C}}^u_i \not = \emptyset$ for each $u \in \mathbb{Z}^2 \setminus 0$ and $i \in Q_0$. Let $\delta \in Q_1^{\mathcal{S}}$. There is a cycle $p \in \hat{\mathcal{C}}_{\operatorname{t}(\delta)}$ such that $\delta$ is not the rightmost arrow subpath of any representative of $p$.
Assume to the contrary that there is an arrow $\delta$ which is a rightmost arrow subpath of some representative of each cycle in $\hat{\mathcal{C}}_{\operatorname{t}(\delta)}$.
\(i) We first claim that there is some $u \in \mathbb{Z}^2 \setminus 0$ such that $\operatorname{t}(\delta^+)$ lies in the interior of the region $\mathcal{R}_{\tilde{s},\tilde{t}}$ bounded by the lifts of two representatives $\tilde{s},\tilde{t}$ of the (unique) cycle in $\hat{\mathcal{C}}^u_{\operatorname{h}(\delta)}$. (There is precisely one cycle in $\hat{\mathcal{C}}^u_{\operatorname{h}(\delta)}$ by Proposition \[circle 1\].4.)
Indeed, by assumption and Lemma \[area to zero\], there are distinct vectors $$u_0 = u_{n+1}, u_1, u_2, \ldots, u_n \in \mathbb{Z}^2 \setminus 0,$$ ordered clockwise in $\mathbb{R}^2$, and cycles $$p_m = p'_m \delta \in \hat{\mathcal{C}}^{u_m}_{\operatorname{h}(\delta)},$$ such that for $0 \leq m \leq n$, we have $$\label{m+1 m}
\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mup_{m+1}p_m\mkern-1.5mu}\mkern 1.5mu}.$$
Assume to the contrary that for each $m$, $\delta^+$ is not contained in each region $$\mathcal{R}_{p_{m+1}p_m,p_mp_{m+1}}.$$ Fix $m$. Since $p_m$ is in $\hat{\mathcal{C}}$, its lift $p_m^+$ does not have a cyclic subpath. In particular, the tail of $\delta^+$ only meets $p_m^+$ at its tail. We therefore have the setup given in Figure \[purple fig\].
Set $p := p_0$ and $q := p_n$. Then there is a leftmost subpath $\delta p'$ of $p$ and a rightmost subpath $q'$ of $q$ such that $(\delta p' q')^+$ is a cycle in $Q^+$. In particular, $${\mkern 1.5mu\overline{\mkern-1.5mu\delta p'q'\mkern-1.5mu}\mkern 1.5mu} = \sigma ^{\ell}$$ for some $\ell \geq 1$, by Lemmas \[here3’\].1 and \[here3\]. Whence $\sigma \mid {\mkern 1.5mu\overline{\mkern-1.5mupq\mkern-1.5mu}\mkern 1.5mu}$. But this is a contradiction to (\[m+1 m\]).
Thus there is some $0 \leq m \leq n$ such that $\delta^+$ lies in $\mathcal{R}_{p_{m+1}p_m,p_mp_{m+1}}$. Furthermore, since $\operatorname{t}(\delta^+)$ is not a subpath of $p_m^+$ or $p_{m+1}^+$, we have that $\operatorname{t}(\delta^+)$ lies in the interior of $\mathcal{R}_{p_{m+1}p_m,p_mp_{m+1}}$. The claim then follows by setting $$s = p_{m+1}p_m \ \ \ \text{ and } \ \ \ t = p_m p_{m+1}.$$
\(ii) Let $s$ and $t$ be as in Claim (i). In particular, ${\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mut\mkern-1.5mu}\mkern 1.5mu}$. Assume to the contrary that there is a simple matching $D \in \mathcal{S}$ for which $$x_D \nmid {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mut\mkern-1.5mu}\mkern 1.5mu}.$$ Let $r^+$ be a path in $\mathcal{R}_{\tilde{s},\tilde{t}}$ from a vertex on the boundary of $\mathcal{R}_{\tilde{s},\tilde{t}}$ to $\operatorname{t}(\delta^+)$. It suffices to suppose the tail of $r$ is a trivial subpath of $\tilde{s}$, in which case $\tilde{s}$ factors into paths $$\tilde{s} = \tilde{s}_2e_{\operatorname{t}(r)}\tilde{s}_1.$$ See Figure \[purple fig2\]. Then $(rs_1\delta)^+$ is a cycle in $Q^+$. Whence $${\mkern 1.5mu\overline{\mkern-1.5murs_1\delta\mkern-1.5mu}\mkern 1.5mu} = \sigma^{\ell}$$ for some $\ell \geq 1$, by Lemmas \[here3’\].1 and \[here3\]. In particular, $$x_D \mid {\mkern 1.5mu\overline{\mkern-1.5murs_1\delta\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu}{\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu}_1{\mkern 1.5mu\overline{\mkern-1.5mu\delta\mkern-1.5mu}\mkern 1.5mu}.$$ Furthermore, by assumption, $$x_D \nmid {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu}_1 \ \ \ \text{ and } \ \ \ x_D \nmid {\mkern 1.5mu\overline{\mkern-1.5mu\delta\mkern-1.5mu}\mkern 1.5mu}.$$ Whence $x_D \mid {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu}$. Thus, since $r$ is arbitrary, $x_D$ divides the $\bar{\tau}$-image of each path in $\mathcal{R}_{\tilde{s},\tilde{t}}$ from the boundary of $\mathcal{R}_{\tilde{s},\tilde{t}}$ to $\operatorname{t}(\delta^+)$. But $\operatorname{t}(\delta^+)$ lies in the interior of $\mathcal{R}_{\tilde{s},\tilde{t}}$. Thus the vertex $\operatorname{t}(\delta)$ is a source in $Q \setminus D$. Therefore $D$ is not simple, contrary to assumption. It follows that $x_D \mid {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu}$ for each $D \in \mathcal{S}$.
\(iii) By Claim (ii), $\sigma \mid {\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu}$. But this is a contradiction since $s$ is in $\hat{\mathcal{C}}$, by Lemmas \[here3’\].3 and \[here3\]. Therefore there is a cycle $p$ in $\hat{\mathcal{C}}_{\operatorname{t}(\delta)}$ such that $\delta$ is not the rightmost arrow subpath of any representative of $p$.
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(5,0)*{\cdot}="9";(0,-7)*{\cdot}="10";(-7,0)*{\cdot}="11";(0,7)*{\cdot}="12";
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[ There are dimer algebras that have an arrow $\delta \in Q_1$ which is a rightmost arrow subpath of each cycle in $\hat{\mathcal{C}}_{\operatorname{t}(\delta)}$; see Figure \[AKZfig1\]. Furthermore, if $A$ has center $Z$, admits a cyclic contraction, and $\delta \in Q_1^{\mathcal{S}}$, then $\delta$ is a rightmost arrow subpath of each cycle $p \in Ze_{\operatorname{t}(\delta)}$ for which $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$, by [@B2 Lemma 2.4]. ]{}
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\[AKZ\] Suppose (i) $A$ is cancellative, or (ii) $\hat{\mathcal{C}}_i^u \not = \emptyset$ for each $u \in \mathbb{Z}^2 \setminus 0$ and $i \in Q_0$. Then $Q_1^{\mathcal{S}} = \emptyset$, that is, each arrow annihilates a simple $A$-module of dimension $1^{Q_0}$.
Recall that assumption (i) implies assumption (ii), by Proposition \[here\]. So suppose (ii) holds, and assume to the contrary that there is an arrow $\delta$ in $Q_1^{\mathcal{S}}$.
By Lemma \[purple lemma\], there is a cycle $p \in \hat{\mathcal{C}}_{\operatorname{t}(\delta)}$ whose rightmost arrow subpath is not $\delta$ (modulo $I$). Let $u \in \mathbb{Z}^2$ be such that $p \in \mathcal{C}^u$. By assumption, there is a cycle $q$ in $\hat{\mathcal{C}}^u_{\operatorname{h}(\delta)}$. By Lemma \[figure2lemma\], we may choose representatives $\tilde{p}$, $\tilde{q}$ of $p$, $q$ such that $\mathcal{R}_{\tilde{p},\tilde{q}}$ contains no vertices in its interior. We thus have one of the three cases given in Figure \[AKZfig3\].
First suppose $\tilde{p}^+$ and $\tilde{q}^+$ do not intersect (that is, do not share a common vertex), as shown in case (i). Then $\tilde{p}^+$ and $\tilde{q}^+$ bound a column. By Lemma \[figure3lemma\], the brown arrows belong to a simple matching $D \in \mathcal{S}$. In particular, $\delta$ is in $D$, contrary to assumption.
So suppose $\tilde{p}^+$ and $\tilde{q}^+$ intersect, as shown in cases (ii) and (iii). Then $\tilde{p}^+$ and $\tilde{q}^+$ bound a union of pillars. Again by Lemma \[figure3lemma\], the brown arrows belong to a simple matching $D \in \mathcal{S}$. In particular, in case (ii) $\delta$ is in $D$, contrary to assumption. Therefore case (iii) holds. But then $\delta$ is a rightmost arrow subpath of $p$ (modulo $I$), contrary to our choice of $p$.
In [@B3 Theorem 1.1] we show that the converse of Theorem \[AKZ\] also holds: $A$ is cancellative if and only if each arrow of $Q$ is contained in a simple matching.
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(i) & &
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\[easy injective\] Suppose $A$ is cancellative. Let $p,q \in e_jAe_i$ be paths satisfying $$\operatorname{t}(p^+) = \operatorname{t}(q^+) \ \ \text{ and } \ \ \operatorname{h}(p^+) = \operatorname{h}(q^+).$$ If ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}$, then $p = q$.
Since $A$ is cancellative, $Q$ has at least one simple matching by Lemma \[at least one\]. In particular, $\sigma \not = 1$. Thus we may apply the proof of Lemma \[here2\].3, with $\bar{\tau}$ in place of $\bar{\eta}$.
\[generated by\] Suppose $A$ is cancellative. For each $i \in Q_0$, the corner ring $e_iAe_i$ is generated by $\sigma_i$ and $\hat{\mathcal{C}}_i$.
Since $I$ is generated by binomials, $e_iAe_i$ is generated by $\mathcal{C}_i$. It thus suffices to show that $\mathcal{C}_i$ is generated by $\sigma_i$ and $\hat{\mathcal{C}}_i$.
Let $u \in \mathbb{Z}^2$ and $p \in \mathcal{C}_i^u$. If $u = 0$, then $p = \sigma_i^m$ for some $m \geq 0$ by Lemma \[here3’\].2. So suppose $u \not = 0$. Then there is a cycle $q$ in $\hat{\mathcal{C}}^u_i$ by Proposition \[here\]. In particular, ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}\sigma^m$ for some $m \in \mathbb{Z}$ by Lemma \[longlist\]. Furthermore, $m \geq 0$ by Proposition \[circle 1\].1. Therefore $p = q \sigma_i^m$ by Lemma \[easy injective\].
It is well known that if $A$ is cancellative, then $A$ is a 3-Calabi-Yau algebra [@D; @MR]. In particular, the center $Z$ of $A$ is noetherian, and $A$ is a finitely generated $Z$-module. In the following, we give independent proofs of these facts.
\[i=j\] Suppose $A$ is cancellative, and let $i,j \in Q_0$. Then
1. $e_iAe_i = Ze_i \cong Z$.
2. $\bar{\tau}\left(e_iAe_i \right) = \bar{\tau}\left(e_jAe_j\right)$.
3. $A$ is a finitely generated $Z$-module, and $Z$ is a finitely generated $k$-algebra.
\(1) For each $i \in Q_0$ and $u \in \mathbb{Z}^2 \setminus 0$, there exists a unique cycle $c_{ui} \in \hat{\mathcal{C}}_i^u$ (modulo $I$) by Propositions \[here\] and \[circle 1\].4. Thus the sum $$\sum_{i \in Q_0} c_{ui} \in \bigoplus_{i \in Q_0} e_iAe_i$$ is in $Z$, by Lemma \[longlist3\].1. Whence $e_iAe_i \subseteq Ze_i$ by Lemma \[generated by\]. Furthermore, $$Ze_i = Ze_i^2 = e_iZe_i \subseteq e_iAe_i.$$ Therefore $Ze_i = e_iAe_i$.
We now claim that there is an algebra isomorphism $Z \cong Ze_i$ for each $i \in Q_0$. Indeed, fix $i \in Q_0$ and suppose $z \in Z$ is nonzero. Then there is some $j \in Q_0$ such that $ze_j \not = 0$. Furthermore, since $Q$ is a dimer quiver, there is a path $p$ from $i$ to $j$.
Assume to the contrary that $ze_jp = 0$. Thus, since $I$ is generated by binomials, it suffices to suppose $ze_j = c_1 - c_2$ where $c_1$ and $c_2$ are paths. But since $A$ is cancellative, $ze_jp = 0$ implies $c_1 = c_2$. Whence $ze_j = 0$, a contradiction. Therefore $ze_j p \not = 0$. Consequently, $$p e_i z = pz = zp = ze_j p \not = 0.$$ Whence $ze_i \not = 0$. Thus the algebra homomorphism $Z \to Ze_i$, $z \mapsto ze_i$, is injective, hence an isomorphism. This proves our claim.
\(2) Follows from Proposition \[circle 1\].2 and Lemma \[generated by\].
\(3) $A$ is generated as a $Z$-module by all paths of length at most $|Q_0|$ by Claim (1) and [@B6 second paragraph of proof of Theorem 2.11]. Thus $A$ is a finitely generated $Z$-module. Furthermore, $A$ is a finitely generated $k$-algebra since $Q$ is finite. Therefore $Z$ is also a finitely generated $k$-algebra [@McR 1.1.3].
\[finally!’\] Suppose $A$ is cancellative, and let $p \in \mathcal{C}^u$. Then $u = 0$ if and only if ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = \sigma^m$ for some $m \geq 0$.
($\Rightarrow$) Lemma \[here3’\].1.
($\Leftarrow$) Let $u \in \mathbb{Z}^2 \setminus 0$, and assume to the contrary that $p \in \mathcal{C}^u$ satisfies ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = \sigma^m$ for some $m \geq 0$. Since $A$ is cancellative, there is a cycle $q \in \hat{\mathcal{C}}^u_{\operatorname{t}(p)}$ by Proposition \[here\]. Furthermore, $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}$ by Proposition \[circle 1\].1. Thus there is some $n \geq 0$ such that $${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}\sigma^n,$$ by Lemma \[here2\].2. Whence $\sigma^m = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} \sigma^n$. Furthermore, $\sigma \not = 1$ by Lemma \[at least one\]. Therefore $m = n$ and $$\label{bar eta q = 1}
{\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} = 1,$$ since $B$ is a polynomial ring. But each arrow in $Q$ is contained in a simple matching by Theorem \[AKZ\]. Therefore ${\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} \not = 1$, contrary to (\[bar eta q = 1\]).
\[injective prop\] Suppose $A$ is cancellative. The homomorphisms $$\tau: A \to M_{|Q_0|}(B) \ \ \ \text{ and } \ \ \ \eta: A \to M_{|Q_0|}\left(k[x_D \ | \ D \in \mathcal{P}]\right)$$ are injective.
\(i) We fist claim that $\tau$ is injective on the vertex corner rings $e_iAe_i$, $i \in Q_0$. Fix a vertex $i \in Q_0$ and let $p,q \in e_iAe_i$ be cycles satisfying ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}$. Let $r$ be a path such that $r^+$ is path from $\operatorname{h}(p^+)$ to $\operatorname{t}(p^+)$. Then $rp \in \mathcal{C}^0$. Thus there is some $m \geq 0$ such that $$rp = \sigma_i^m,$$ by Lemma \[here3’\].2. Whence $${\mkern 1.5mu\overline{\mkern-1.5murq\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5murp\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mu\sigma_i^m\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern 1.5mu}_i^m = \sigma^m.$$ Thus $rq \in \mathcal{C}^0$ by Lemma \[finally!’\]. Hence $rq = \sigma_i^m = rp$ by Lemma \[easy injective\]. Therefore $p = q$ since $A$ is cancellative.
\(ii) We now claim that $\tau$ is injective on paths. Let $p,q \in e_jAe_i$ be paths satisfying ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}$. Let $r$ be a path from $j$ to $i$. The two cycles $pr$ and $qr$ at $j$ then satisfy ${\mkern 1.5mu\overline{\mkern-1.5mupr\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muqr\mkern-1.5mu}\mkern 1.5mu}$. Thus $pr = qr$ since $\tau$ is injective on the corner ring $e_jAe_j$ by Claim (i). Therefore $p = q$ since $A$ is cancellative.
\(iii) Since $A$ is generated by paths and $\tau$ is injective on paths, it follows that $\tau$ is injective.
\(iv) Finally, we claim that $\eta$ is injective. Let $g \in A$, and suppose $\eta(g) = 0$. Then $$\label{tau g}
\tau(g) = \eta(g)|_{x_D = 1 \, : \, D \not \in \mathcal{S}} = 0.$$ Therefore $g = 0$ by Claim (iii).
\[arehom\] Cancellative dimer algebras are homotopy algebras.
Let $A = kQ/I$ be cancellative. We want to show that $I = \ker \eta$.
Indeed, $I \subseteq \ker \eta$ by Lemma \[tau’A’\]. So let $g \in kQ$, and suppose $\eta(g) = 0$. Then $\tau(g) = 0$ by (\[tau g\]). Whence $g \in I$ by Proposition \[injective prop\]. Therefore $\ker \eta \subseteq I$.
Proof of main theorem {#Cancellative dimer algebras}
=====================
Throughout, let $A = kQ/I$ be a dimer algebra, and let $\psi: A \to A'$ be a cyclic contraction to a cancellative dimer algebra $A' = kQ'/I'$. If $A$ is cancellative, then we may take $\psi$ to be the identity map.
Let $\tau: A' \to M_{|Q'_0|}(B)$ be the algebra homomorphism defined in Lemma \[tau’A’\], with $B$ the polynomial ring generated by the simple matchings of $A'$, $$B = k[ x_D \ | \ D \in \mathcal{S}' ].$$ To prove Theorem \[main\], we will use the composition of $\psi$ with $\tau$. Specifically, let $$\tau_{\psi}: A \to M_{|Q_0|}(B)$$ be the $k$-linear map defined for each $i,j \in Q_0$ and $p \in e_jAe_i$ by $$\label{tilde tau define}
p \mapsto \bar{\tau}\psi(p) \cdot e_{ji}.$$ Denote by ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} := {\mkern 1.5mu\overline{\mkern-1.5mu\tau\mkern-1.5mu}\mkern 1.5mu}_{\psi}(p) := \bar{\tau}\psi(p)$ the single nonzero matrix entry of $\tau_{\psi}(p)$.
\[homomorphism on corner\] The map $\tau_{\psi}: A \to M_{|Q_0|}(B)$ is an algebra homomorphism.
By Lemma \[tau’A’\], $\tau: A' \to M_{|Q'_0|}(B)$ is an algebra homomorphism. Furthermore, $\psi$ is a $k$-linear map, and an algebra homomorphism when restricted to each vertex corner ring $e_iAe_i$.
\[here4\] Let $p$ be a nontrivial cycle.
1. If $p \in \mathcal{C}^0$, then ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = \sigma^m$ for some $m \geq 1$.
2. If $p \in \mathcal{C} \setminus \hat{\mathcal{C}}$, then $\sigma \mid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$.
If $p^+$ is a cycle (resp. has a cyclic subpath) in $Q^+$, then $\psi(p)^+$ is a cycle (resp. has a cyclic subpath) in $Q'^+$. Furthermore, $A'$ is cancellative. Claims (1) and (2) therefore hold by Lemmas \[here3’\].1 and \[here3’\].3 respectively.
\[coincident\] If $\psi(p) = \psi(q)$, then $p^+$ and $q^+$ have coincident tails and coincident heads.
Suppose $\psi(p) = \psi(q)$. Consider lifts $p^+$ and $q^+$ for which $\operatorname{t}(p^+) = \operatorname{t}(q^+)$. Let $r^+$ be a path from $\operatorname{h}(p^+)$ to $\operatorname{h}(q^+)$. Then $\psi(r^+)$ is a cycle in $Q'^+$ since $\psi(p) = \psi(q)$, by Lemma \[r+\].1. Thus $r^+$ is also a cycle by the contrapositive of Lemma \[positive length cycle\].2. Whence $\operatorname{h}(p^+) = \operatorname{h}(q^+)$.
Denote by $\mathcal{P}$ and $\mathcal{P}'$ the set of perfect matchings of $Q$ and $Q'$ respectively. Consider the algebra homomorphisms defined in (\[eta def\]), $$\eta: kQ \to M_{|Q_0|}\left( k[x_D \ | \ D \in \mathcal{P}] \right) \ \ \ \text{ and } \ \ \ \eta': kQ' \to M_{|Q'_0|}\left( k[y_D \ | \ D \in \mathcal{P}'] \right).$$ By Lemma \[cannot contract\], $\psi$ cannot contract a unit cycle to a vertex. Thus, if $D$ is a perfect matching of $Q'$, then $\psi^{-1}(D)$ is a perfect matching of $Q$. We may therefore view $k[y_D \ | \ D \in \mathcal{P}']$ as a subalgebra of $k[x_D \ | \ D \in \mathcal{P}]$ under the identification $y_D = x_{\psi^{-1}(D)}$ for each $D \in \mathcal{P}'$. For $g \in e_jkQe_i$, we then have $$\label{eta' psi}
\bar{\eta}'(\psi(g)) = \bar{\eta}(g)|_{x_D = 1 \,: \, \psi(D) \not \in \mathcal{P}'}.$$
Denote the $\bar{\eta}$- and $\bar{\eta}'$-images of the unit cycles in $Q$ and $Q'$ by $$\sigma_{\mathcal{P}} := \prod_{D \in \mathcal{P}} x_D \ \ \ \text{ and } \ \ \ \sigma_{\mathcal{P}'} := \prod_{D \in \mathcal{P}'} y_D.$$
The $k$-linear map $\psi: kQ \to kQ'$ induces a $k$-linear map of homotopy algebras $\psi: \Lambda \to \Lambda'$. Furthermore, for each $i,j \in Q_0$, the restriction $$\label{psi: ej}
\psi: e_j\Lambda e_i \hookrightarrow e_{\psi(j)} \Lambda' e_{\psi(i)}$$ is injective.
\(i) If $g \in kQ$ satisfies $\eta(g) = 0$, then $\eta'(\psi(g)) = 0$, by (\[eta’ psi\]). Thus, $$\psi(\ker \eta) \subseteq \ker \eta'.$$ Therefore the map $\psi: kQ \to kQ'$ induces a well-defined map $\psi: \Lambda \to \Lambda'$.
\(ii) Fix $i,j \in Q_0$. We claim that the map (\[psi: ej\]) is injective. Let $p,q \in e_jkQe_i$ be paths satisfying $\bar{\eta}'(\psi(p-q)) = 0$. To prove the claim, it suffices to show that $\bar{\eta}(p - q) = 0$.
Since $\bar{\eta}'(\psi(p-q)) = 0$, we have $\psi(p) = \psi(q)$ by Proposition \[injective prop\]. In particular, $p^+$ and $q^+$ have coincident tails and coincident heads, by Lemma \[coincident\]. Let $r^+$ be a path in $Q^+$ from $\operatorname{h}(p^+)$ to $\operatorname{t}(p^+)$. Then there is some $m, n \geq 0$ such that $$\label{eta p =}
\bar{\eta}(rp) = \sigma^m_{\mathcal{P}} \ \ \ \text{ and } \ \ \ \bar{\eta}(rq) = \sigma^n_{\mathcal{P}},$$ by Lemma \[here4\].1. It follows that $$\label{yippy}
\bar{\eta}(p) = \bar{\eta}(q) \sigma_{\mathcal{P}}^{m-n}.$$ Therefore $$\begin{gathered}
\bar{\eta}'(\psi(q)) \stackrel{\textsc{(i)}}{=} \bar{\eta}'(\psi(p)) \stackrel{\textsc{(ii)}}{=} \bar{\eta}(p)|_{x_D = 1\, : \, \psi(D) \not \in \mathcal{P}'} \\
\stackrel{\textsc{(iii)}}{=} \bar{\eta}(q)\sigma_{\mathcal{P}}^{m-n}|_{x_D = 1\, : \, \psi(D) \not \in \mathcal{P}'} \stackrel{\textsc{(iv)}}{=} \bar{\eta}'(\psi(q))\sigma_{\mathcal{P}'}^{m-n},\end{gathered}$$ where (<span style="font-variant:small-caps;">i</span>) holds by assumption; (<span style="font-variant:small-caps;">ii</span>) and (<span style="font-variant:small-caps;">iv</span>) hold by (\[eta’ psi\]); and (<span style="font-variant:small-caps;">iii</span>) holds by (\[yippy\]). But then $m = n$ since $B$ is an integral domain and $\sigma_{\mathcal{P}'} \not = 1$. Thus $\bar{\eta}(p) = \bar{\eta}(q)$, by (\[eta p =\]). Therefore $\eta(p - q) = 0$.
The following strengthens Lemmas \[here2\].3 and \[here2\].4 for dimer algebras that admit cyclic contractions (specifically, the head and tail of the lifts $p^+$ and $q^+$ are not required to coincide).
\[r in T’\] Let $p,q \in e_jAe_i$ be distinct paths. The following are equivalent:
1. $\psi(p) = \psi(q)$.
2. $p,q$ is a non-cancellative pair.
3. ${\mkern 1.5mu\overline{\mkern-1.5mu\tau\mkern-1.5mu}\mkern 1.5mu}_{\psi}(p) = {\mkern 1.5mu\overline{\mkern-1.5mu\tau\mkern-1.5mu}\mkern 1.5mu}_{\psi}(q)$.
4. $\bar{\eta}(p) = \bar{\eta}(q)$.
\(1) $\Rightarrow$ (2): Suppose $\psi(p) = \psi(q)$. Then $p^+$ and $q^+$ have coincident tails and coincident heads, by Lemma \[coincident\]. Furthermore, $\mathcal{P}' \not = \emptyset$ since $A'$ is cancellative, by Lemma \[at least one\]. Whence $\mathcal{P} \not = \emptyset$, by Lemma \[cannot contract\]. Therefore $p,q$ is a non-cancellative pair, by Lemma \[here2\].4 (with $\bar{\tau}_{\psi}$ in place of $\bar{\eta}$).
\(2) $\Rightarrow$ (3),(4): Holds by Lemma \[here2\].3 (with $\bar{\tau}_{\psi}$ in place of $\bar{\tau}$).
\(3) $\Rightarrow$ (1): Holds since $\bar{\tau}: e_{\psi(j)}A'e_{\psi(i)} \to B$ is injective, by Proposition \[injective prop\].
\(4) $\Rightarrow$ (3): If $\bar{\eta}(p) = \bar{\eta}(q)$, then $$\bar{\tau}_{\psi}(p) = \bar{\eta}(p)|_{x_D = 1 \, : \, \psi(D) \not \in \mathcal{S}'} = \bar{\eta}(q)|_{x_D = 1 \, : \, \psi(D) \not \in \mathcal{S}'} = \bar{\tau}_{\psi}(q).$$
The following theorem establishes the relationship between homotopy algebras and dimer algebras on a torus.
\[first main\] There are algebra isomorphisms $$\begin{array}{rcl}
\Lambda := kQ/\ker \eta & \stackrel{\textsc{(i)}}{=} & kQ/\ker \tau_{\psi} \\
& \stackrel{\textsc{(ii)}}{\cong} & A/\ker \tau_{\psi} \\
& \stackrel{\textsc{(iii)}}{=} & A/\left\langle p - q \ | \ p,q \text{ is a non-cancellative pair} \right\rangle.
\end{array}$$
(<span style="font-variant:small-caps;">i</span>) and (<span style="font-variant:small-caps;">ii</span>) hold since $$I \stackrel{(a)}{\subseteq} \ker \eta \stackrel{(b)}{=} \ker \tau_{\psi}.$$ Indeed, ($a$) holds by Lemma \[tau’A’\], and ($b$) holds from the equivalence (3) $\Leftrightarrow$ (4) in Lemma \[r in T’\]. (<span style="font-variant:small-caps;">iii</span>) holds from the equivalence (2) $\Leftrightarrow$ (3) in Lemma \[r in T’\].
\[injective cor\] The map $\tau_{\psi}: A \to M_{|Q_0|}(B)$ induces an injective algebra homomorphism on the homotopy algebra $\Lambda$, $$\label{tau psi lambda}
\tau_{\psi}: \Lambda \to M_{|Q_0|}(B).$$
[ The ideal $$\left\langle p - q \ | \ p,q \text{ is a non-cancellative pair} \right\rangle \subset A$$ is contained in the kernel of $\psi$, but not conversely. Indeed, if $\psi$ contracts an arrow $\delta$, then $\delta - e_{\operatorname{t}(\delta)}$ is in the kernel of $\psi$, but $\delta$ and $e_{\operatorname{t}(\delta)}$ do not form a non-cancellative pair. ]{}
Since $A'$ is cancellative, $A'$ is equal to its homotopy algebra $\Lambda'$ by Theorem \[arehom\] (or Theorem \[first main\]).
\[impression prop\] Let $\psi: \Lambda \to \Lambda'$ be a cyclic contraction.
1. The algebra homomorphisms $$\tau_{\psi}: \Lambda \to M_{|Q_0|}(B) \ \ \ \text{ and } \ \ \ \tau: \Lambda' \to M_{|Q'_0|}(B)$$ are impressions of $\Lambda$ and $\Lambda'$.
2. Suppose that either $k$ is uncountable, or $Q$ is cancellative. Then $\tau_{\psi}$ classifies all simple $\Lambda$-module isoclasses of maximal $k$-dimension: for each such module $V$, there is some $\mathfrak{b} \in \operatorname{Max}B$ such that $$V \cong (B/\mathfrak{b})^{|Q_0|},$$ where $av := \tau_{\psi}(a)v$ for each $a \in \Lambda$, $v \in (B/\mathfrak{b})^{|Q_0|}$.
3. The centers of $\Lambda$ and $\Lambda'$ are given by the intersection and union of the vertex corner rings of $\Lambda$, $$\label{Z cong R S}
Z(\Lambda) \cong k\left[ \cap_{i \in Q_0} \bar{\tau}_{\psi}\left( e_i \Lambda e_i \right) \right] \subseteq k\left[ \cup_{i \in Q_0} \bar{\tau}_{\psi}\left( e_i\Lambda e_i \right) \right] \cong Z(\Lambda').$$
\(1) We first show that $\tau_{\psi}$ and $\tau$ are impressions of $\Lambda$ and $\Lambda'$.
(1.i) $\tau_{\psi}: \Lambda \to M_{|Q_0|}(B)$ is injective by Corollary \[injective cor\].
(1.ii) For each maximal ideal $$\mathfrak{b} \in \mathcal{Z}_B\left( \sigma \right)^c \subset \operatorname{Max}B = \mathbb{A}_k^{\left|\mathcal{S}\right|},$$ the composition $$\label{composition lambda}
\Lambda \stackrel{\tau_{\psi}}{\longrightarrow} M_{|Q_0|}(B) \stackrel{1}{\longrightarrow} M_{|Q_0|}\left(B/\mathfrak{b} \right) \cong M_{|Q_0|}(k)$$ is a simple representation of $\Lambda$. Indeed, when viewed as a vector space diagram on $Q$ of dimension vector $1^{Q_0}$, each arrow is represented by a nonzero scalar. Thus, since $Q$ has a cycle containing each vertex, the representation is simple. It follows that the composition (\[composition lambda\]) is surjective.
(1.iii) Set $R := \tau_{\psi}(Z(\Lambda))$; Claims (1.i) and (1.ii), together with (\[Ziso\]), imply that each element of $R$ is product of a polynomial in $B$ and the identity matrix $1_{|Q_0|} \in M_{|Q_0|}(B)$. For brevity, we will omit $1_{|Q_0|}$ in our expressions.
We claim that the morphism $$\operatorname{Max}B \rightarrow \operatorname{Max}R, \ \ \ \mathfrak{b} \mapsto \mathfrak{b} \cap R,$$ is surjective. Indeed, for any $\mathfrak{m} \in \operatorname{Max}R$, $B\mathfrak{m}$ is a proper ideal of $B$. Thus there is a maximal ideal $\mathfrak{b} \in \operatorname{Max}B$ containing $B\mathfrak{m}$ since $B$ is noetherian. Furthermore, since $B$ is a finitely generated $k$-algebra and $k$ is algebraically closed, the intersection $\mathfrak{b} \cap R =: \mathfrak{m}'$ is a maximal ideal of $R$. Whence $$\mathfrak{m} \subseteq B\mathfrak{m} \cap R \subseteq \mathfrak{b} \cap R = \mathfrak{m}'.$$ But $\mathfrak{m}$ and $\mathfrak{m}'$ are both maximal ideals of $R$. Thus $\mathfrak{m} = \mathfrak{m}'$. Therefore $\mathfrak{b} \cap R = \mathfrak{m}$, proving our claim.
(1.iv) By Claims (1.i), (1.ii), and (1.iii), $\tau_{\psi}$ is an impression of $\Lambda$. It follows that $\tau$ itself is an impression of $\Lambda'$ by letting $\psi: A = A' \to A'$ be the trivial cyclic contraction given by the identity map.
(2.i) If $Q$ is cancellative, then $\tau$ classifies all simple $\Lambda$-module isoclasses of maximal $k$-dimension, by (\[Vcong\]) and Proposition \[i=j\].3.
(2.ii) Now suppose $k$ is uncountable. Using Claim (2.i), it was shown in [@B2 Proposition 3.10 and Theorem 3.11] that, irrespective of whether $Q$ is cancellative or non-cancellative, $\tau_{\psi}$ classifies all simple $\Lambda$-modules (and $A$-modules) of dimension vector $1^{Q_0}$, up to isomorphism.
We claim that the dimension vector of the simple $\Lambda$-modules of maximal $k$-dimension is $1^{Q_0}$. Let $V$ be a simple $\Lambda$-module. Then for each $i \in Q_0$, $e_iV$ is a simple $e_i\Lambda e_i$-module. But $e_i \Lambda e_i$ is a commutative countably generated $k$-algebra, and $k$ is an uncountable algebraically closed field. Thus, $e_iV = 0$ or $e_iV \cong k$. Whence $\operatorname{dim}_k e_iV \leq 1$. Furthermore, there is a representation of $\Lambda$ where each arrow is represented by $1 \in k$ since we may set each $x_D$ equal to $1$; this representation is simple of dimension $1^{Q_0}$ since $Q$ has a cycle containing each vertex. In particular, there is a simple $\Lambda$-module of dimension $1^{Q_0}$, proving our claim.
\(3) We claim that $$\begin{gathered}
Z(\Lambda) \stackrel{(\textsc{i})}{\cong}
k\left[ \cap_{i \in Q_0} \bar{\tau}_{\psi}\left( e_i \Lambda e_i \right) \right]
\subseteq k\left[ \cup_{i \in Q_0} \bar{\tau}_{\psi}\left( e_i\Lambda e_i \right) \right]
\\ \stackrel{(\textsc{ii})}{=}
k\left[ \cup_{i \in Q'_0} \bar{\tau}\left( e_i\Lambda' e_i \right) \right]
\stackrel{(\textsc{iii})}{=}
k\left[ \cap_{i \in Q'_0} \bar{\tau}\left( e_i\Lambda' e_i \right) \right]
\stackrel{(\textsc{iv})}{\cong} Z(\Lambda').\end{gathered}$$ (<span style="font-variant:small-caps;">i</span>) and (<span style="font-variant:small-caps;">iv</span>) hold by Claims (1.i) and (1.ii), together with (\[Ziso\]). (<span style="font-variant:small-caps;">ii</span>) holds since the contraction $\psi$ is cyclic, and since $A' = \Lambda'$. Finally, (<span style="font-variant:small-caps;">iii</span>) holds by Proposition \[i=j\].2. Therefore (\[Z cong R S\]) holds.
[ In the case of cancellative dimer algebras, the labeling of arrows given by $\bar{\tau}$ in (\[taua\]) agrees with the labeling of arrows in the toric construction of [@CQ Proposition 5.3]. We note, however, that impressions are defined more generally for non-toric algebras and have different implications than the toric construction of [@CQ]. ]{}
\[cr\] A dimer algebra $A$ is cancellative if and only if it admits an impression $\tau: A \to M_{|Q_0|}(B)$, where $B$ is an integral domain and $\tau(e_i) = e_{ii}$ for each $i \in Q_0$.
Suppose $A$ admits an impression $\tau: A \to M_{|Q_0|}(B)$, where $B$ is an integral domain and $\tau(e_i) = e_{ii}$ for each $i \in Q_0$. Consider paths $p,q,r$ satisfying $pr = qr \not = 0$. Then $${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mupr\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muqr\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu}.$$ Thus ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}$ since $B$ is an integral domain. Whence $$\tau(p) = {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}e_{\operatorname{h}(p),\operatorname{h}(r)} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}e_{\operatorname{h}(p),\operatorname{h}(r)} = \tau(q).$$ Therefore $p = q$ by the injectivity of $\tau$.
The converse holds by Theorem \[impression prop\].
Recall that an algebra $A$ is prime if for all $a,b \in A$, $aAb = 0$ implies $a = 0$ or $b = 0$, that is, the zero ideal is a prime ideal.
\[cancellative prime\] Homotopy algebras are prime.
We claim that for nonzero elements $p,q \in A$, we have $qAp \not = 0$. It suffices to suppose that $$p \in e_{j}Ae_{i} \ \ \text{ and } \ \ q \in e_{\ell}Ae_{k}.$$ Since $Q$ is a dimer quiver, there is a path $r$ from $j$ to $k$. Furthermore, the polynomials ${\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$, ${\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu}$, and ${\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu}$ are nonzero since $\tau$ is injective, by Theorem \[impression prop\]. Thus the product ${\mkern 1.5mu\overline{\mkern-1.5muqrp\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5mur\mkern-1.5mu}\mkern 1.5mu} {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu} \in B$ is nonzero since $B$ is an integral domain. Therefore $qrp$ is nonzero.
\[four examples\]
A dimer algebra $A = kQ/I$ is *square* if the underlying graph of its cover $Q^+$ is a square grid graph with vertex set $\mathbb{Z} \times \mathbb{Z}$, and with at most one diagonal edge in each unit square. Square dimer algebras exhibit a remarkable interaction between their torus embedding and their representation theory. Specifically, such an algebra $A$ admits an impression $\tau$ where for each arrow $a \in Q^+_1$, $\bar{\tau}(a)$ is the monomial corresponding to the orientation of $a$ given in Figure \[labels\] [@B7 Theorem 3.7]. In this case, $B = k[x,y,z,w]$ is the polynomial ring in four variables. If $Q$ only possesses three arrow orientations, say up, left, and right-down, then we may label the respective arrows by $x$, $y$, and $z$, and obtain an impression with $B = k[x,y,z]$. In either case, $A$ is cancellative by Corollary \[cr\].
Consider the cyclic contractions $\psi: A \to A'$ given in Figure \[deformation figure\]. In each example, $A'$ is a square dimer algebra (with the 2-cycles removed from $Q'$). By Theorem \[impression prop\], the cycle algebra $S$ and center $R$ of the homotopy algebra $\Lambda = kQ/\ker \eta$ are respectively $$\begin{array}{cclcl}
\text{(i):} & \ \ & S = k\left[ xz, xw, yz, yw \right] & & R = k + \sigma S \\
\text{(ii):} & \ \ & S = k\left[ xz, xw, yz, yw \right] & & R = k + (x^2zw, y^2zw, \sigma)S \\
\text{(iii):} & \ \ & S = k\left[ xz, yz, xw, yw \right] & & R = k + (xz, yz)S \\
\text{(iv):} & \ \ & S = k\left[ xz, yw, x^2w^2, y^2z^2 \right] & & R = k + (yw, x^2w^2, y^2z^2)S
\end{array}$$
$$\begin{array}{cccccccc}
\xy
(-5,0)*{\cdot}="1";(5,0)*{\cdot}="2";{\ar@{->}"1";"2"};
\endxy
&
\xy
(-5,0)*{\cdot}="1";(5,0)*{\cdot}="2";{\ar@{<-}"1";"2"};
\endxy
&
\xy
(0,-5)*{\cdot}="1";(0,5)*{\cdot}="2";{\ar@{->}"1";"2"};
\endxy
&
\xy
(0,5)*{\cdot}="1";(0,-5)*{\cdot}="2";{\ar@{->}"1";"2"};
\endxy
&
\xy
(-5,-5)*{\cdot}="1";(5,5)*{\cdot}="2";{\ar@{->}"1";"2"};
\endxy
&
\xy
(-5,-5)*{\cdot}="1";(5,5)*{\cdot}="2";{\ar@{<-}"1";"2"};
\endxy
&
\xy
(-5,5)*{\cdot}="1";(5,-5)*{\cdot}="2";{\ar@{->}"1";"2"};
\endxy
&
\xy
(-5,5)*{\cdot}="1";(5,-5)*{\cdot}="2";{\ar@{<-}"1";"2"};
\endxy
\\
\ \ \ x \ \ \ & \ \ \ y \ \ \ & \ \ \ z \ \ \ & \ \ \ w \ \ \ & \ \ \ xz \ \ \ & \ \ \ yw \ \ \ & \ \ \ xw \ \ \ & \ \ \ yz \ \ \
\end{array}$$
\[finally!\] Suppose $A$ is cancellative, and let $p \in \mathcal{C}$ be a nontrivial cycle. Then $p \in \hat{\mathcal{C}}$ if and only if $\sigma \nmid {\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu}$. In particular, $\tau(Z) \subseteq B$ is generated over $k$ by $\sigma$ and a set of monomials in $B$ not divisible by $\sigma$.
($\Rightarrow$) Proposition \[circle 1\].1.
($\Leftarrow$) Lemma \[here3’\].3.
A brief account of Higgsing with quivers {#Higgsing}
========================================
*Quiver gauge theories*
According to string theory, our universe is 10 dimensional.[^2]^,^[^3] In many string theories our universe has a product structure $M \times Y$, where $M$ is our usual 4-dimensional spacetime and $Y$ is a 6-dimensional compact Calabi-Yau variety.
Let us consider a special class of gauge theories called ‘quiver gauge theories’, which can often be realized in string theory.[^4] The input for such a theory is a quiver $Q$, a superpotential $W$, a dimension vector $d \in \mathbb{N}^{Q_0}$, and a stability parameter $\theta \in \mathbb{R}^{Q_0}$.
Let $I$ be the ideal in $\mathbb{C}Q$ generated by the partial derivatives of $W$ with respect to the arrows in $Q$. These relations (called ‘F-term relations’) are classical equations of motion from a supersymmetric Lagrangian with superpotential $W$.[^5] Denote by $A$ the quiver algebra $\mathbb{C}Q/I$.
According to these theories, the space $X$ of $\theta$-stable representation isoclasses of dimension $d$ is an affine chart on the compact Calabi-Yau variety $Y$. The ‘gauge group’ of the theory is the isomorphism group (i.e., change of basis) for representations of $A$.
Physicists view the elements of $A$ as fields on $X$. More precisely, $A$ may be viewed as a noncommutative ring of functions on $X$, where the evaluation of a function $f \in A$ at a point $p \in X$ (i.e., representation $p$) is the matrix $f(p) := p(f)$ (up to isomorphism).
*Vacuum expectation values*
Given a path $f \in A$ and a representation $p \in X$, denote by $f\left(\bar{p}\right)$ the matrix representing $f$ in the vector space diagram on $Q$ associated to $p$.
A field $f \in A$ is ‘gauge-invariant’ if $f(p) = f(p')$ whenever $p$ and $p'$ are isomorphic representations (i.e., they differ by a ‘gauge transformation’). If $f$ is a path, then $f$ will necessarily be a cycle in $Q$.
The ‘vacuum expectation value’ of a field is its expected (average) energy in the vacuum (similar to rest mass), and is abbreviated ‘vev’. In our case, the vev of a path $f \in A$ at a point $p \in X$ is the matrix $f\left(\bar{p}\right)$, which is just the expected energy of $f$ in $M \times \{ p \}$.
*Higgsing*
Spontaneous symmetry breaking is a process where the symmetry of a physical system decreases, and a new property (typically mass) emerges.
For example, suppose a magnet is heated to a high temperature. Then all of its molecules, which are each themselves tiny magnets, jostle and wiggle about randomly. In this heated state the material has rotational symmetry and no net magnet field. However, as the material cools, one molecule happens to settle down first. As the neighboring molecules settle down, they align themselves with the first molecule, until all the molecules settle down in alignment with the first.[^6] The orientation of the first settled molecule then determines the direction of magnetization for the whole material, and the material no longer has rotational symmetry. One says that the rotational symmetry of the heated magnet was spontaneously broken as it cooled, and a global magnetic field emerged.[^7]
Higgsing is a way of using spontaneous symmetry breaking to turn a quantum field theory with a massless field and more symmetry into a theory with a massive field and less symmetry. Here mass (vev’s) takes the place of magnetization, gauge symmetry (or the rank of the gauge group) takes the place of rotational symmetry, and energy scale (RG flow) takes the place of temperature.
The recent discovery of the Higgs boson at the Large Hadron Collider is another example of Higgsing.[^8]
*Higgsing in quiver gauge theories*
We now give our main example. Suppose an arrow $a$ in a quiver gauge theory with dimension $1^{Q_0}$ is contracted to a vertex $e$. We make two observations:
1. the rank of the gauge group drops by one since the head and tail of $a$ become identified as the single vertex $e$;
2. $a$ has zero vev at any representation where $a$ is represented by zero, while $e$ can never have zero vev since it is a vertex, and $X$ only consists of representation isoclasses with dimension $1^{Q_0}$.
We therefore see that contracting an arrow to a vertex is a form of Higgsing in quiver gauge theories with dimension $1^{Q_0}$.[^9]
In the context of a 4-dimensional $\mathcal{N} = 1$ quiver gauge theory with quiver $Q$, the Higgsing we consider in this paper is related to RG flow. We start with a non-superconformal (strongly coupled) quiver theory $Q$ which admits a low energy effective description, give nonzero vev’s to a set of bifundamental fields $Q_1^*$, and obtain a new theory $Q'$ that lies at a superconformal fixed point.
*The mesonic chiral ring and the cycle algebra*
The cycle algebra $S$ we introduce in this paper is similar to the mesonic chiral ring in the corresponding quiver gauge theory. In such a theory, the mesonic operators, which are the gauge invariant operators, are generated by the cycles in the quiver. If the gauge group is abelian, then the dimension vector is $1^{Q_0}$. In the case of a dimer theory with abelian gauge group, two disjoint cycles may share the same $\bar{\tau}\psi$-image, but take different values on a point of the vacuum moduli space. These cycles would then be distinct elements in the mesonic chiral ring, although they would be identified in the cycle algebra $S$; see [@B2 Remark 3.17].
\
**Acknowledgments.** The author was supported by the Austrian Science Fund (FWF) grant P 30549-N26. Part of this article is also based on work supported by the Simons Foundation and the Heilbronn Institute for Mathematical Research while the author held postdoctoral positions at the Simons Center for Geometry and Physics at Stony Brook University, and the University of Bristol, respectively.
[10]{} K. Baur, A. King, R. Marsh, Dimer models and cluster categories of Grassmannians, Proc. London Math. Soc. **2** (2016) 113. C. Beil, Cyclic contractions of dimer algebras always exist, Algebras and Representation Theory, accepted. , Morita equivalences and Azumaya loci from Higgsing dimer algebras, J. Algebra **453** (2016) 429-455. , Noetherian criteria for dimer algebras, in preparation. , Nonnoetherian geometry, J. Algebra Appl. **15** (2016). , Nonnoetherian homotopy dimer algebras and noncommutative crepant resolutions, Glasgow Math. J., to appear. , On the central geometry of nonnoetherian dimer algebras, in preparation. , On the noncommutative geometry of square superpotential algebras, J. Algebra **371** (2012) 207-249. N. Broomhead, Dimer models and Calabi-Yau algebras, Memoirs AMS (2012) 1011. A. Craw and A. Quintero Velez, Cellular resolutions of noncommutative toric algebras from superpotentials, Adv. Math. **229** (2012), no. 3, 1516-1554. J. Davey, A. Hanany, J. Pasukonis, On the classification of dimer models, J. High Energy Phys. **1001** (2010) 078. B. Davison, Consistency conditions for dimer models, J. Algebra **338** (2011) 1-23. B. Feng, Y.-H. He, K. Kennaway, C. Vafa, Dimer models from mirror symmetry and quivering amoebae, Adv. Theor. Math. Phys. **12** (2008) no. 3, 489-545. S. Franco, A. Hanany, J. Park, D. Rodriguez-Gomez, Towards M2-brane theories for generic toric singularities, J. High Energy Phys. **0812** (2008) 110. S. Franco, I. Klebanov, D. Rodriguez-Gomez, M2-branes on orbifolds of the cone over $Q^{111}$, J. High Energy Phys. **0908** (2009) 033. Y.-H. He, Calabi-Yau Varieties: from Quiver Representations to Dessins d’Enfants, arXiv:1611.09398. A. Ishii and K. Ueda, Dimer models and the special McKay correspondence, Geom. Topol. **19** (2015) 3405-3466. O. Iyama and Y. Nakajima, On steady non-commutative crepant resolutions, J. Noncommut. Geom., to appear. J. C. McConnell and J. C. Robson, *Noncommutative noetherian rings*, Amer. Math. Soc. volume 30, 1987. S. Mozgovoy and M. Reineke, On the noncommutative Donaldson-Thomas invariants arising from dimer models, Adv. Math. **223** (2010) 1521-1544.
[^1]: In Lemma \[columns and pillars\].1, we will not require (\[exception\]) to hold.
[^2]: Thanks to physicists Francesco Benini, Mike Douglas, Peng Gao, Mauricio Romo, and James Sparks for discussions on the physics of non-cancellative dimers.
[^3]: More correctly, weakly coupled superstring theory requires 10 dimensions.
[^4]: Here we are considering theories with $\mathcal{N} = 1$ supersymmetry.
[^5]: More correctly, the F-term relations plus the D-term relations imply the equations of motion.
[^6]: More precisely, there are domains of magnetization.
[^7]: This is an example of ‘global’ symmetry breaking, meaning the symmetry is physically observable.
[^8]: This is an example ‘gauge’ symmetry breaking, meaning the symmetry is not an actual observable symmetry of a physical system, but only an artifact of the math used to describe it (like a choice of basis for the matrix of a linear transformation).
[^9]: This is another example of gauge symmetry breaking.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
[*A.I. Nazarov, N.N. Ural’tseva*]{}[^1],\
Saint-Petersburg University,\
[e-mail: [email protected]]{}
title: |
**The Harnack inequality and related properties\
for solutions to elliptic and parabolic equations\
with divergence-free lower-order coefficients**
---
To the memory of M.S. Birman
Introduction
============
Qualitative properties of solutions to partial differential equations are intensively studied over last half of century. In this paper we deal with classical properties, namely, strong maximum principle, Hölder estimates, the Harnack inequality and the Liouville Theorem.
We consider elliptic and parabolic equations of divergence type: $${\cal L}u \equiv -
D_i\big(a_{ij}(x)D_ju\big)+
b_i(x)D_iu=0;\eqno({\bf DE})$$ $${\cal M}u\equiv \partial_tu-
D_i\big(a_{ij}(x;t)D_ju\big)+
b_i(x;t)D_iu=0.\eqno({\bf DP})$$ We mostly deal with [*a priori*]{} estimates for Lipschitz generalized (sub/super)solutions. When these estimates are established, we discuss the possibility of their generalization for weak (sub/super)solutions. In this case we assume $Du\in L_{2,loc}(\Omega)$ in ([**DE**]{}) and $u\in L_{2,\infty,loc}(Q)$, $Du\in L_{2,loc}(Q)$ in ([**DP**]{}).
We always suppose that operators under consideration are uniformly elliptic (parabolic), i.e. for all values of arguments $$\label{ell}
\nu|\xi|^2\le a_{ij}(\cdot)\xi_i\xi_j\le\nu^{-1}|\xi|^2,\qquad
\xi\in{\mathbb R}^n,$$ where $\nu$ is a positive constant.
The properties of generalized solutions to the equations ([**DE**]{})–([**DP**]{}) were investigated in a number of papers. Hölder estimates for solutions of ([**DE**]{}) were obtained by E. De Giorgi [@DG] for ${\bf b}\equiv0$ and by C. Morrey [@M] for ${\bf b}$ belonging to the Morrey space lying between $L_n$ and any $L_q$, $q>n$ (${\bf b}$ stands for $(b_i)$). Corresponding result for ([**DP**]{}) was established by J. Nash [@Na] for ${\bf b}\equiv0$ and by O.A. Ladyzhenskaya and N.N. Ural’tseva [@LU1] for ${\bf b}\in L_{q+2}$, $q>n$.
Harnack’s inequality for operators without lower-order coefficients was proved by J. Moser ([@Mo1] for ([**DE**]{}) and [@Mo2] for ([**DP**]{})). N. Trudinger [@Tru] proved it for ([**DE**]{}) with ${\bf b}\in L_q$, $q>n$. G. Lieberman (see [@Li1 Ch. VI]) extended the result of [@Mo2] for ${\bf b}\in L_{q,\ell}$, $\frac nq+\frac 2\ell<1$. Obviously, Harnack’s inequality implies Hölder estimates. Also strong maximum principle follows from Harnack’s inequality and weak maximum principle. Some sharpening of mentioned results, as well as corresponding results for nondivergence equations, are discussed in our preprint [@NU].
In this paper we consider mainly the equations ([**DE**]{}) and ([**DP**]{}) with additional structure condition $$\label{bezdiv}
\mbox{\rm div} ({\bf b})\le0\quad \mbox{in the sense of distributions}.$$ The equations with the lower-order coefficients satisfying this structure condition arise in some applications (see, e.g., [@Z], [@KNSS], [@CSTY], [@SSSZ]). We show that in this case the assumptions on ${\bf b}$ can be considerably weakened in the scale of Morrey spaces.
Our paper is organized as follows. In Section 2 we deal with elliptic equations. Section 3 is devoted to parabolic equations (recall that only two-sided Liouville’s Theorem holds for these equations). In Section 4 we show an application of our results to some equations arising in hydrodynamics. We underline that this Section contains just exemplary instances, and we make no pretence to the novelty of results. In particular, the statements of Theorems 4.1 and 4.3 are in fact obtained in [@KNSS].
Let us recall some notation. $x=(x_1,\dots,x_n)$ is a vector in $\mathbb R^n$, $n\ge2$, with the Euclidean norm $|x|$; $(x;t)$ is a point in $\mathbb R^{n+1}$.
$\Omega$ is a domain in $\mathbb R^n$ and $\partial\Omega$ is its boundary. For a cylinder $Q=\Omega\times]0,T[$ we denote by $\partial''Q=\partial\Omega\times]0,T[$ its lateral surface and by $\partial'Q=\partial''Q\cup\{\overline{\Omega} \times \{0\}\}$ its parabolic boundary.
We define $$\begin{array}{lll}
B_{R}(x^0)=\{x\in \mathbb R^n : |x-x^0|<R\}, & B_R=B_R(0);\\
Q_{R}^{\lambda,\theta}(x^0;t^0)=B_{\lambda R}(x^0)\times]t^0-\theta R^2;t^0[, &
Q_{R}^{\lambda,\theta}=Q_{R}^{\lambda,\theta}(0;0), & Q_{R}=Q_{R}^{1,1}
\end{array}$$ (note that $Q_{\lambda R}=Q_{R}^{\lambda,\lambda^2}$).
The indices $i, j$ vary from 1 to $n$. Repeated indices indicate summation.
The symbol $D_i$ denotes the operator of differentiation with respect to $x_i$; in particular, $Du=(D_1u, \dots, D_nu)$ is the gradient of $u$. $\partial_tu$ stands for the derivative of $u$ with respect to $t$.
The dashed integral stands for the mean value: $\Xint-\limits_Eu=(\mbox{meas}\, E)^{-1}\int\limits_Eu$.
We denote by $\|\cdot\|_{p,\Omega}$ the norm in $L_p(\Omega)$. We introduce a scale of anisotropic spaces $L_{q,\ell}(Q)=L_\ell\big(\,]0,T[\,\to L_q(\Omega)\big)$ with the norm $\|f\|_{q,\ell,Q}=\big\|\|f(\cdot;t)\|_{q,\Omega}\big\|_{\ell,]0,T[}$. Obviously, $L_{q,q}(Q)=L_q(Q)$.
We also introduce a scale of Morrey spaces $$\mathbb M^{\alpha}_q(\Omega)=\{f\in L_q(\Omega):\ \|f\|_{\mathbb M^{\alpha}_q(\Omega)}\equiv
\sup\limits_{B_\rho(x)\subset\Omega}\rho^{-\alpha}\|f\|_{q,B_\rho(x)}<\infty\}.$$ Parabolic Morrey spaces $\mathbb M^{\alpha}_{q,\ell}(Q)$ are introduced in a similar way, using $Q_\rho(x;t)\subset Q$ instead of $B_\rho(x)\subset\Omega$.
Finally, we introduce the space ${\cal V}(Q)$ of weak solutions to ([**DP**]{}) with the norm defined by $$\|f\|^2_{{\cal V}(Q)}=\|f\|^2_{2,\infty,Q}+\|Df\|^2_{2,2,Q}.$$
We set $f_+=\max \{f,0\},\ \ f_-=\max \{-f,0\}$, ${\text{osc}}_{\Omega} f =\sup_{\Omega} f - \inf_{\Omega} f$. For $1\le p<n$, $p^*=\frac {np}{n-p}$ is the Sobolev conjugate to $p$.
We use letters $N$, $C$ (with or without indices) to denote various constants. To indicate that, say, $N$ depends on some parameters, we list them in the parentheses: $N(\dots)$.
Elliptic case
=============
Recall that $u$ is a (Lipschitz) subsolution of the equation ${\cal L}u=0$ in $\Omega$ (here $\cal L$ is an operator of the form ([**DE**]{})), if for any Lipschitz test function $\eta\ge0$, supported in $\Omega$, $$\int\limits_{\Omega}(a_{ij}D_juD_i\eta+b_iD_iu\,\eta)\,dx\le0.$$ We take $\eta=\varphi'(u)\cdot\xi$, where $\xi$ is a nonnegative Lipschitz function, supported in $B_{\lambda R}\subset\Omega$, while $\varphi\in{\cal C}^2(\mathbb R)$ is a convex function vanishing in $\mathbb R_-$. This gives $$\label{Moser}
\int\limits_{B_{\lambda R}\cap\{u>0\}}\Big(a_{ij}D_jvD_i\xi+\frac {\varphi''(u)}
{\varphi^{\prime2}(u)}\,a_{ij}D_jvD_iv\,\xi+b_iD_iv\,\xi\Big)\,dx\le0,$$ где $v=\varphi(u)$.
Then, by mollification at a neighborhood of the origin, one can weaken in (\[Moser\]) the assumption $\varphi\in{\cal C}^2(\mathbb R)$ to $\varphi\in{\cal C}^2(\mathbb R_+\cup\mathbb R_-)$.
[**Lemma 2.1**]{}.
*Let $\cal L$ be an operator of the form ([**DE**]{}) in $B_{\lambda R}(x^0)$, $\lambda>1$, and let the conditions (\[ell\]) and (\[bezdiv\]) be satisfied. Let also ${\bf b}\in L_q(B_{\lambda R}(x^0))$ with some $\frac n2<q\le n$[^2].*
Then there exists a positive constant $N_1$ depending on $n$, $\nu$, $\lambda$, $q$ and the quantity $${\cal N}={\cal N}(R,\lambda)\equiv R^{1-\frac nq}\|{\bf b}\|_{q,B_{\lambda R}(x^0)},$$ such that any Lipschitz subsolution of the equation ${\cal L}u=0$ in $B_{\lambda R}(x^0)$ satisfies
$$\label{estmax}
\sup\limits_{B_R(x^0)} u_+\le N_1
\bigg(\!\!\Xint{\quad\ \,-}\limits_{B_{\lambda R}(x^0)}\!\!u_+^2dx\bigg)^{\frac 12}.$$
We use classical technique of Moser (see, e.g., [@LU2 Ch.IX]). Without loss of generality, we assume $x^0=0$.
We put in (\[Moser\]) $\varphi(\tau)=\tau_+^p$, $p>1$, and $\xi=v\zeta^2$ where $\zeta$ is a smooth cut-off function in $B_{\lambda R}$. Then we obtain $$\label{Moser_power}
\int\limits_{B_{\lambda R}}\Big(\frac {2p-1}{p}\,a_{ij}D_jvD_iv\,\zeta^2
+2a_{ij}D_jv\,vD_i\zeta\,\zeta+b_iD_iv\,v\zeta^2\Big)\,dx\le0.$$ The last term in (\[Moser\_power\]) can be estimated using (\[bezdiv\]) and the Hölder inequality: $$\label{bezdiv1}
-\int\limits_{B_{\lambda R}}b_iD_iv\,v\zeta^2\,dx\le
\int\limits_{B_{\lambda R}}b_iv^2\zeta\,D_i\zeta\,dx
\le\|{\bf b}\|_{q,B_{\lambda R}}\|v\zeta\|^{2-\frac 1s}_{r,B_{\lambda R}}
\|v\zeta^{1-s}|D\zeta|^s\|^{\frac 1s}_{2,B_{\lambda R}},$$ where $s>2$ is defined by $\frac 1s=1-\frac n{2q}$ while $r=\frac {2(2q+n)}{2q+n-4}$. Note that $2<r<2^*$, and, by the embedding theorem, $$\label{embed}
\|v\zeta\|_{r,B_{\lambda R}}\le
C(n)(\lambda R)^{n(\frac 1r-\frac 1{2^*})}\,\|D(v\zeta)\|_{2,B_{\lambda R}}
\le C(n)(\lambda R)^{\frac 1{2s-1}}\,\Big(\|Dv\,\zeta\|_{2,B_{\lambda R}}+\|vD\zeta\|_{2,B_{\lambda R}}\Big).$$
Using (\[ell\]), (\[bezdiv1\]) and (\[embed\]), we obtain from (\[Moser\_power\]) $$\begin{gathered}
%\label{aa}
\|Dv\,\zeta\|^2_{2,B_{\lambda R}}\le
\frac 1\nu\int\limits_{B_{\lambda R}}a_{ij}D_jvD_iv\,\zeta^2\,dx\le C_1(n,\nu,s,\lambda)\times\\
\times\bigg[\|Dv\,\zeta\|_{2,B_{\lambda R}}\|vD\zeta\|_{2,B_{\lambda R}}+
R^{\frac 1s}\|{\bf b}\|_{q,B_{\lambda R}}\Big(\|Dv\,\zeta\|^{2-\frac 1s}_{2,B_{\lambda R}}+
\|vD\zeta\|^{2-\frac 1s}_{2,B_{\lambda R}}\Big)
\|v\zeta^{1-s}|D\zeta|^s\|^{\frac 1s}_{2,B_{\lambda R}}\bigg],\end{gathered}$$ and therefore $$\label{ee}
\|Dv\,\zeta\|_{2,B_{\lambda R}}\le
C_2(n,\nu,s,\lambda)\cdot\bigg[\|vD\zeta\|_{2,B_{\lambda R}}+R\|{\bf b}\|^s_{q,B_{\lambda R}}
\|v\zeta^{1-s}|D\zeta|^s\|_{2,B_{\lambda R}}\bigg].$$
We put $R_m=R(1+2^{-m}(\lambda-1))$, $m\in \mathbb N\cup\{0\}$, and substitute $\zeta=\zeta_m$ such that $$\zeta_m\equiv1\ \ \mbox{in}\ \ B_{R_{m+1}};\quad \zeta_m\equiv0\ \ \mbox{out of}\ \ B_{R_m};
\qquad \frac{|D\zeta_m|}{\zeta_m^{1-\frac 1s}}\le \frac {2^mC_3(s)}{(\lambda-1)R}.$$ Then (\[ee\]) implies $$\label{eee}
\|Dv\,\zeta_m\|_{2,B_{R_m}}\le
\frac {C_4(n,\nu,s,\lambda)}R\cdot\|v\|_{2,B_{R_m}}\cdot
\big(2^m+\big(2^m{\cal N}\big)^s\big).$$
Now for $p=p_m\equiv (\frac r2)^m$ we obtain from (\[embed\]) and (\[eee\]) $$\begin{gathered}
\label{iteration}
\bigg(\Xint{\quad\ \, -}\limits_{B_{R_{m+1}}}\!\! u_+^{2p_{m+1}}dx\bigg)^{\frac {1}{2p_{m+1}}}\!\!\le
\bigg(C(n)\Xint{\ \ -}\limits_{B_{R_m}} (v\zeta_m)^r dx\bigg)^{\frac {1}{rp_m}}\le\\
\le
\bigg(2^{2ms}C_5\Xint{\ \ -}\limits_{B_{R_m}} v^2\,dx\bigg)
^{\frac {1}{2p_m}}\!\!=
\bigg(2^{2ms}C_5\Xint{\ \ -}\limits_{B_{R_m}} u_+^{2p_m}
dx\bigg)^{\frac 1{2p_m}}\!,\end{gathered}$$ where $C_5$ depends only on $n$, $\nu$, $\lambda$, $s$ and ${\cal N}$.
Iterating (\[iteration\]) we arrive at (\[estmax\]).
[**Corollary 2.1**]{}. [*Let $\cal L$ satisfy the assumptions of Lemma 2.1 in $B_{\lambda R}(x^0)$. If a Lipschitz subsolution of ${\cal L}u=0$ in $B_{\lambda R}(x^0)$ satisfies*]{} $$\label{tiny}
\mbox{meas}\, (\{u> k\}\cap B_{\lambda R}(x^0))\le \mu\,\mbox{meas}\,(B_{\lambda R}),
\qquad \mu< N_1^{-2},$$ [*for some $k$, then*]{} $$\label{estmax1}
\sup\limits_{B_R(x^0)} (u-k)\le N_1\sqrt{\mu}\sup\limits_{B_{\lambda R}(x^0)}(u-k),$$ [*(here $N_1$ is the constant from Lemma 2.1).*]{}
We apply Lemma 2.1 to $u-k$.
We need the following variant of the embedding theorem.
[**Proposition A**]{}. [*Let $1\le p<n$. Suppose that a non-negative function $u\in W^1_p(B_R)$ vanishes on a positive measure set ${\cal E}_0$. Let $\eta=\eta(|x|)$ be a non-decreasing function, $0\le\eta\le 1$, and $\eta\big|_{{\cal E}_0}\equiv 1$. Then, for any $1\le q\le p^*$ and for any measurable set ${\cal E}\subset B_R$,*]{} $$\|u\,\eta\|_{q,{\cal E}}\le \frac {C(n)R^n}{\mbox{meas}\,({\cal E}_0)}\
\mbox{meas}^{\frac 1q-\frac 1{p^*}}({\cal E})\cdot\|Du\,\eta\|_{p,B_R}.$$
For $q=p=1$ this Proposition was proved in [@LSU Ch. II, Lemma 5.1]. In this Lemma the following inequality was obtained: $$\mbox{meas}\,({\cal E}_0)\cdot u(x)\,\eta(x)\le
\frac {(2R)^n}{n}\int\limits_{B_R}\frac {|Du(y)|\,\eta(y)}{|y-x|^{n-1}}\,dy.$$ By the Hardy–Littlewood–Sobolev inequality (see, e.g., [@LL Sec. 4.3]), we get $$\mbox{meas}\,({\cal E}_0)\cdot \|u\,\eta\|_{p^*,B_R}\le C(n,p)R^n\cdot \|Du\,\eta\|_{p,B_R},$$ and the statement follows by Hölder inequality.
[**Lemma 2.2**]{}. [*Let $\cal L$ satisfy the assumptions of Lemma 2.1 in $B_{\lambda R}(x^0)$. Then for any $\delta>0$ there exists a positive constant $\beta$ depending on $n$, $\nu$, $\lambda$, $q$, $\delta$ and the quantity ${\cal N}$, such that if a Lipschitz nonnegative supersolution of ${\cal L}V=0$ in $B_{\lambda R}(x^0)$ satisfies*]{} $$\label{tiny1}
\mbox{meas}\, (\{V\ge k\}\cap B_R(x^0))\ge\delta\cdot \mbox{meas}\, (B_R)$$ [*for some $k>0$, then*]{} $$\label{estmin}
\inf\limits_{B_R(x^0)} V\ge \beta k.$$
Without loss of generality, we can assume $V>0$; otherwise we deal with $V+\ep$ and pass to the limit as $\ep\downarrow0$. Also we put $x^0=0$.
We define $u=1-\frac Vk$. Note that $u<1$ is a subsolution, and therefore, we can apply the relation (\[Moser\]) with $\varphi$ defined only for $\tau<1$.
We put in (\[Moser\]) $\varphi(\tau)=\ln_-(1-\tau)$. This gives for $v=\varphi(u)$ $$\label{Moser_ln}
\int\limits_{B_{\lambda R}}\Big(a_{ij}D_jvD_i\xi+a_{ij}D_jvD_iv\,\xi+b_iD_iv\,\xi\Big)\,dx\le0.$$ We substitute into (\[Moser\_ln\]) $\xi=\zeta^2$ where $\zeta$ is a smooth cut-off function that equals $1$ in $B_{\frac{1+\lambda}2R}$. Then, using (\[ell\]), (\[bezdiv\]) and the Hölder inequality, we obtain $$\begin{gathered}
%\label{aa}
\|Dv\,\zeta\|^2_{2,B_{\lambda R}}\le
\frac 1\nu\int\limits_{B_{\lambda R}}a_{ij}D_jvD_iv\,\zeta^2\,dx\le \frac 2\nu
\int\limits_{B_{\lambda R}}\Big(-a_{ij}D_jv\zeta D_i\zeta+b_iv\zeta\,D_i\zeta\Big)\,dx\le\\
\le \frac 2\nu (\|Dv\,\zeta\|_{2,B_{\lambda R}}\|D\zeta\|_{2,B_{\lambda R}}+
\|{\bf b}\|_{q,B_{\lambda R}}\|v\,\zeta\|_{q',B_{\lambda R}}\|D\zeta\|_{\infty,B_{\lambda R}}).\end{gathered}$$ Note that $v$ vanishes on the set $\{V\ge k\}\cap B_R$. Therefore, we can estimate the last term by Proposition A. By (\[tiny1\]), this gives $$\|Dv\,\zeta\|_{2,B_{\lambda R}}\le C_6(n,\nu,\lambda,q,\delta)R^{\frac n2-1}\cdot
\big(1+{\cal N}\big).$$ Applying Proposition A once more, we obtain $$\bigg(\!\!\Xint{\quad\ \,-}\limits_{B_{\frac {1+\lambda}2 R}}\!\!v^2dx\bigg)^{\frac 12}\le
C_7,$$ where $C_7$ depends only on $n$, $\nu$, $\lambda$, $q$, $\delta$ and ${\cal N}$.
Finally, the relation (\[Moser\_ln\]) implies that $v$ is a subsolution. So, we apply Lemma 2.1 to $v$ in $B_{\frac {1+\lambda}2 R}$ and arrive at the estimate $\sup\limits_{B_R} v_+\le C_8\equiv N_1C_7$, which is equivalent to (\[estmin\]) with $\beta=\exp(-C_8)$.
[**Corollary 2.2**]{} (strong maximum principle). [*Let $\cal L$ be an operator of the form ([**DE**]{}) in $\Omega$, and let the conditions (\[ell\]) and (\[bezdiv\]) be satisfied. Let also ${\bf b}\in L_{q,loc}(\Omega)$ with some $\frac n2<q\le n$. Then any Lipschitz nonconstant supersolution of ${\cal L}V=0$ in $\Omega$ cannot attain its minimum at interior point of $\Omega$.*]{}
Assume the converse. Without loss of generality, $\inf\limits_\Omega V=0$. Then there exists $x^0\in\Omega$ which is a frontier point of the set $\{V>0\}$. Choose $R$ such that $\overline{B_{2R}}(x^0)\subset\Omega$. Then the relation (\[tiny1\]) holds for some $k>0$ and $\delta>0$, and we obtain (\[estmin\]), a contradiction.
[**Lemma 2.3**]{}. [*Let $\cal L$ satisfy the assumptions of Lemma 2.1 in $B_{3R}$. Then any Lipschitz solution of ${\cal L}u=0$ in $B_{3R}$ satisfies the estimate*]{} $$\label{osc}
\underset{B_R}{\mbox{osc}}\ u\le \varkappa_0\,\underset{B_{3R}}{\mbox{osc}}\ u,$$ [*where $\varkappa_0<1$ depends on $n$, $\nu$, $q$ and the quantity ${\cal N}$.*]{}
We set $$k=\frac 12\big(\sup\limits_{B_{3R}}u+\inf\limits_{B_{3R}}u\big)$$ and consider two cases.
[**1**]{}. Let the relation (\[tiny\]) hold with $\lambda=2$ and $\mu=\frac 14N_1^{-2}$. Then, by Corollary 2.1, $$\sup\limits_{B_R} u\le \frac 12 \big(\sup\limits_{B_{2R}}u+k\big)\le
\sup\limits_{B_{3R}}u-\frac 14\,\underset{B_{3R}}{\mbox{osc}}\ u.$$
[**2**]{}. In the opposite case we apply Lemma 2.2 (with $\lambda=\frac 32$ and $\delta=\mu$) to the (non-negative) function $V=u-\inf\limits_{B_{3R}}u$. This gives $\inf\limits_{B_R} V\ge\inf\limits_{B_{2R}} V\ge \beta\big(k-\inf\limits_{B_{3R}}u\big)$, and thus, $$\inf\limits_{B_R} u\ge\inf\limits_{B_{3R}} u+\frac {\beta}2\,
\underset{B_{3R}}{\mbox{osc}}\ u.$$
In both cases we arrive at (\[osc\]) with $\varkappa_0=\min\big\{\frac 14,\frac{\beta}2\big\}$.
[**Corollary 2.3**]{} (Hölder estimate).
*Let $\cal L$ satisfy the assumptions of Lemma 2.1 in $B_{R_0}$. Let also $\sup\limits_{R<R_0}{\cal N}(R,1)<\infty$.*
Then any Lipschitz solution of ${\cal L}u=0$ in $B_{R_0}$ satisfies the estimate
$$\label{Holder}
\underset{B_\rho}{\mbox{osc}}\ u\le N_2 \Big(\frac \rho r\Big)^\gamma
\cdot\underset{B_r}{\mbox{osc}}\ u,\qquad 0<\rho<r\le R_0,$$
[*where $N_2$ and $\gamma$ depend on $n$, $\nu$, $q$ and $\sup\limits_{R<R_0}{\cal N}(R,1)$.*]{}
Iterating the estimate (\[osc\]) we arrive at (\[Holder\]) with $\gamma=-\log_3(\varkappa_0)$.
[**Corollary 2.4**]{} (two-sided Liouville’s theorem). [*Let $\cal L$ be an operator of the form ([**DE**]{}) in $\mathbb R^n$, and let the conditions (\[ell\]) and (\[bezdiv\]) be satisfied. Let also ${\bf b}\in L_{q,loc}(\mathbb R^n)$, with some $\frac n2<q\le n$. Finally, assume that*]{} $$\label{Liouville}
\liminf\limits_{R\to\infty}{\widehat{\cal N}}(R,1)<\infty.$$ [*Then any Lipschitz bounded solution of ${\cal L}u=0$ in $\mathbb R^n$ is a constant.*]{}
[**Remark 1**]{}. If ${\bf b}\in L_q(\mathbb R^n)$, then (\[Liouville\]) is obviously satisfied.
Iteration of (\[osc\]) with respect to a suitable sequence $R_m\to\infty$ gives the statement.
[**Lemma 2.4**]{}.
*Let $\cal L$ be an operator of the form ([**DE**]{}) in $B_{2R}$, and let the conditions (\[ell\]) and (\[bezdiv\]) be satisfied. Let also ${\bf b}\in \mathbb M^{\frac nq-1}_q(B_{2R})$ with some $\frac n2<q\le n$.*
Let for a Lipschitz nonnegative supersolution of ${\cal L}V=0$ in $B_{2R}$ and for some $y\in B_{2R}$, the inequality $\inf\limits_{B_\rho(y)}V=k>0$ holds with $\rho=\frac 14\mbox{\rm dist}(y,\partial B_{2R})$. Then
$$\label{estmin1}
\inf\limits_{B_R} V\ge \widehat\beta \Big(\frac \rho R\Big)^{\widehat\gamma} k,$$
[*where $\widehat\beta $ and $\widehat\gamma$ depend on $n$, $\nu$, $q$ and $\|{\bf b}\|_{\mathbb M^{\frac nq-1}_q(B_{2R})}$.*]{}
Denote by ${\mathfrak N}$ an integer number such that $2^{-({\mathfrak N}+1)}R<\rho\le 2^{-{\mathfrak N}}R$ and consider a ball $B_{{\mathfrak r}_0}(y^0)$, where ${\mathfrak r}_0=2^{-{\mathfrak N}}R$, $y^0=2R(1-2^{-{\mathfrak N}}){\bf e}$ and ${\bf e}=\frac y{|y|}$. It is easy to see that $B_{{\mathfrak r}_0}(y^0)\subset B_{3\rho}(y)$, and by Lemma 2.2 (with $\lambda=\frac 43$ and $\delta=\frac 1{3^n}$), $$\label{ff}
\inf\limits_{B_{{\mathfrak r}_0}(y^0)} V\ge\inf\limits_{B_{3\rho}(y)} V\ge \beta k.$$
Now we introduce the sequence of balls $B_{{\mathfrak r}_m}(y^m)$, $m=1,\dots,{\mathfrak N}$, as follows: $${\mathfrak r}_m=2{\mathfrak r}_{m-1},\qquad y^m=y^{m-1}-{\mathfrak r}_m{\bf e}.$$ For all $m=1,\dots,{\mathfrak N}$ one has $$B_{2{\mathfrak r}_m}(y^m)\subset B_{2R};\qquad\mbox{meas}\,(B_{{\mathfrak r}_{m-1}}(y^{m-1})\cap
B_{{\mathfrak r}_m}(y^m))\ge C(n)\cdot\mbox{meas}\,(B_{{\mathfrak r}_m}).$$ Thus, Lemma 2.2 (with $\lambda=2$ and $\delta=C(n)$) gives $$\inf\limits_{B_{{\mathfrak r}_m}(y^m)} V\ge \beta\cdot\inf\limits_{B_{{\mathfrak r}_{m-1}}(y^{m-1})} V.$$
Since $B_{{\mathfrak r}_{\mathfrak N}}(y^{\mathfrak N})=B_R$, we obtain $$\inf\limits_{B_R} V\ge \beta^{\mathfrak N}\cdot\inf\limits_{B_{{\mathfrak r}_0}(y^0)} V\ge
\Big(\frac \rho R\Big)^{\widehat\gamma}\cdot\inf\limits_{B_{{\mathfrak r}_0}(y^0)} V,$$ where $\widehat\gamma=-\log_2(\beta)$.
Combining this estimate with (\[ff\]), we arrive at (\[estmin1\]).
[**Theorem 2.5**]{} (the Harnack inequality). [*Let $\cal L$ satisfy the assumptions of Lemma 2.4 in $B_{2R}$. Then there exists a positive constant $N_3$ depending on $n$, $\nu$, $q$ and $\|{\bf b}\|_{\mathbb M^{\frac nq-1}_q(B_{2R})}$, such that any Lipschitz nonnegative solution of ${\cal L}u=0$ in $B_{2R}$ satisfies*]{} $$\label{Harnack}
\sup\limits_{B_R}u\le N_3\cdot\inf\limits_{B_R}u.$$
We follow the idea of Safonov ([@S2]). Denote by $y\in B_{2R}$ a maximum point of the function $$v(x)=(\mbox{dist}(x,\partial B_{2R}))^{\widehat\gamma}\!\cdot u(x)$$ (here $\widehat\gamma$ is the constant from Lemma 2.4) and set $$\rho=\frac 14\,\mbox{dist}(y,\partial B_{2R});\qquad {\mathfrak M}=v(y)=(4\rho)^{\widehat\gamma}\!\cdot u(y).$$
It is obvious that $$\begin{aligned}
\sup\limits_{B_R}u\le \frac {\mathfrak M}{R^{\widehat\gamma}}=
\Big(\frac {4\rho} R\Big)^{\widehat\gamma}\!\cdot u(y);\label{sup}\\
\sup\limits_{B_{2\rho}(y)}u\le \frac {\mathfrak M}{(2\rho)^{\widehat\gamma}}=
2^{\widehat\gamma}\cdot u(y).\phantom{\rho}\label{sup1}\end{aligned}$$
Denote $k=\frac 12 u(y)$. If $\mbox{meas}\, (\{u> k\}\cap B_{2\rho}(y))\le \mu\,\mbox{meas}\,(B_{2\rho})$, then Corollary 2.1 (with $\lambda=2$) and (\[sup1\]) imply the relation $$k=u(y)-k\le \sup\limits_{B_\rho(y)}(u-k)\le N_1\sqrt{\mu}\sup\limits_{B_{2\rho}(y)}(u-k)
\le N_1\sqrt{\mu}\,(2^{\widehat\gamma+1}-1)k,$$ which is impossible for $\mu\le\mu_0\equiv\frac 1{2^{2\widehat\gamma+2}}\,N_1^{-2}$. Thus, $\mbox{meas}\, (\{u> k\}\cap B_{2\rho}(y))\ge \mu_0\,\mbox{meas}\,(B_{2\rho})$, and Lemma 2.2 (with $\lambda=2$ and $\delta=\mu_0$) gives $$\label{inf}
\inf\limits_{B_\rho(y)}u\ge\inf\limits_{B_{2\rho}(y)}u\ge \beta k=\frac{\beta}2\cdot u(y).$$
Finally, Lemma 2.4 gives $$\label{inf1}
\inf\limits_{B_R}u\ge \widehat\beta \Big(\frac \rho R\Big)^{\widehat\gamma}\inf\limits_{B_\rho(y)}u.$$ Combining (\[sup\]), (\[inf\]) and (\[inf1\]), we arrive at (\[Harnack\]) with $N_3=\frac{2^{2\widehat\gamma+1}}{\beta\widehat\beta }$.
[**Theorem 2.6**]{} (one-sided Liouville’s theorem). [*Let $\cal L$ be an operator of the form ([**DE**]{}) in $\mathbb R^n$, and let the conditions (\[ell\]) and (\[bezdiv\]) be satisfied. Let also ${\bf b}\in \mathbb M^{\frac nq-1}_{q,loc}(\mathbb R^n)$ with some $\frac n2<q\le n$, and for some $\delta>0$*]{} $$\label{Liouville1}
\liminf\limits_{R\to\infty}\sup\limits_{|x|=R}
\|{\bf b}\|_{\mathbb M^{\frac nq-1}_q(B_{\delta R}(x))}<\infty.$$ [*Then any Lipschitz semibounded solution of ${\cal L}u=0$ in $\mathbb R^n$ is a constant.*]{}
[**Remark 2**]{}. If ${\bf b}\in \mathbb M^{\frac nq-1}_q(\mathbb R^n)$, then (\[Liouville1\]) is obviously satisfied.
Without loss of generality, we can assume that $u$ is bounded from below, and $\inf\limits_{\mathbb R^n} u=0$.
We take a sequence $R_m\to\infty$ such that $\mathfrak B\equiv\sup\limits_m\sup\limits_{|x|=R_m}\|{\bf b}\|_{\mathbb M^{\frac nq-1}_q(B_{\delta R}(x))}<\infty$. Further, we cover the sphere $|x|=1$ with a finite set of balls $B_{\frac{\delta} 2}(x)$ and dilate these balls to the covering of the sphere $|x|=R_m$. Applying Theorem 2.5 to all the balls of this covering, we obtain $\sup\limits_{|x|=R_m}u\le C(n,\nu,q,\mathfrak B,\delta)\cdot\!
\inf\limits_{|x|=R_m}\!u$ for any $m$. By Corollary 2.2, $$\sup\limits_{B_R}u=\sup\limits_{|x|=R}u;\qquad
\inf\limits_{|x|=R}u=\inf\limits_{B_{R}}u\to0\quad \mbox{as}\quad R\to\infty,$$ and the statement follows.
Let us discuss briefly the possibility to generalize all previous statements for weak (sub/super)solutions.
The proof of Lemma 2.1 runs without changes[^3] also for weak subsolutions of ${\cal L}u=0$ if the bilinear form $${\cal B}\big\langle u,\eta\big\rangle\equiv \int\limits_{B_{\lambda R}}b_iD_iu\,\eta\,dx$$ can be continuously extended to the pair $(v,v\zeta^2)$ with $Dv\in L_2(B_{\lambda R})$. This is certainly true provided $$\big|{\cal B}\big\langle u,\eta\big\rangle\big|\le
C\|Du\|_{2,B_{2\lambda R}}\|D\eta\|_{2,B_{2\lambda R}},\qquad u,\eta\in {\cal C}^\infty_0(B_{2\lambda R}).$$ It is shown in [@MV] that the last estimate holds if $$\label{mazya}
\Delta^{-1} {\rm rot}({\bf b})\in BMO^{n\times n}(B_{2\lambda R});\qquad
h=|\nabla(\Delta^{-1} {\rm div}({\bf b}))|^2\in{\mathfrak M}^{1,2}_+$$ (here we assume ${\bf b}$ extended by zero); the last notation means a class of so-called admissible weights, i.e. $$\int\limits_{B_{2\lambda R}}h|v|^2\,dx\le C\|Dv\|^2_{2,B_{2\lambda R}},\qquad
v\in{\cal C}^\infty_0(B_{2\lambda R}).$$
If ${\bf b}\in\mathbb M^{\frac nq-1}_q(B_{\lambda R})$, the first relation in (\[mazya\]) follows from elliptic coercive estimates and the Poincaré inequality, see, e.g., [@Tro]. Thus, Lemma 2.1 and, therefore, all subsequent statements hold true for weak (sub/super)solutions of ${\cal L}u=0$ if, for example, ${\bf b}\in\mathbb M^{\frac nq-1}_q$ and ${\rm div}({\bf b})\equiv0$.
In addition, let us consider the case $q=n$. The space $\mathbb M^{\frac nq-1}_q(\Omega)$ now becomes conventional Lebesgue space $L_n(\Omega)$, and we claim that main results of this section hold true without the assumption (\[bezdiv\])[^4]. Note that in this case the relation (\[Moser\_power\]) is fulfilled for any weak subsolution $u$.
First, let $n\ge3$. Then we estimate the last term in (\[Moser\_power\]) by the Hölder inequality and the Sobolev inequality and obtain an analog of (\[ee\]): $$\|Dv\,\zeta\|_{2,B_{\lambda R}}\le
C'_2(n,\nu)\cdot\Big[\|vD\zeta\|_{2,B_{\lambda R}}+
\|{\bf b}\|_{n,B_{\lambda R}}\|Dv\,\zeta\|_{2,B_{\lambda R}}\Big].$$ If $\|{\bf b}\|_{n,B_{\lambda R}}\le\ep(n,\nu)\equiv (2C'_2)^{-1}$, then for $p=p_m\equiv 2\big(\frac n{n-2}\big)^m$ we obtain an analog of (\[iteration\]): $$\bigg(\Xint{\quad\ \, -}\limits_{B_{R_{m+1}}}\!\! u_+^{2p_{m+1}}dx\bigg)^{\frac {1}{2p_{m+1}}}\!\!\le
\bigg(C'_5\Xint{\ \ -}\limits_{B_{R_m}} u_+^{2p_m}dx\bigg)^{\frac 1{2p_m}}\!,$$ where $C'_5$ depends only on $n$, $\nu$, and $\lambda$. The remainder of the proof of Lemma 2.1 runs without changes.
Similarly we prove Lemmas 2.2 and 2.4 for weak supersolutions of ${\cal L}V=0$ under the same assumption $\|{\bf b}\|_{n,B_{\lambda R}}\le\ep(n,\nu)$ (in Lemma 2.4 $\lambda=2$).
In the case $n=2$ we use the Yudovich–Pohozhaev embedding theorem (see, e.g., [@BIN 10.6]) instead of the Sobolev inequality. This gives us Lemmas 2.1, 2.2, 2.4 under the assumption $\|{\bf b}\ln^{\frac 12}\big(1+\lambda R|{\bf b}|\big)\|_{2,B_{\lambda R}(x^0)}\le \ep(n,\nu)$.
Further, strong maximum principle holds without smallness assumptions on ${\bf b}$. Indeed, one can choose $R$ sufficiently small such that these assumptions are fulfilled.
Since the proof of Theorem 2.5 depends only on Lemmas 2.1, 2.2, 2.4, the Harnack inequality evidently holds under smallness assumption on ${\bf b}$. However, we can exclude this assumption using a trick of M.V. Safonov.[@S3]
[**Theorem 2.5$\,'$**]{} (the Harnack inequality). [*Let $\cal L$ be an operator of the form ([**DE**]{}) in $B_{2R}$, and let the condition (\[ell\]) be satisfied. Suppose also that*]{} $$\label{b_large_d}
\|{\bf b}\|_{n,B_{2R}}\le \mathfrak B \quad\mbox{for}\quad n\ge3;
\qquad\qquad \|{\bf b}\ln^{\frac 12}\big(1+R|{\bf b}|\big)\|_{2,B_{2R}}\le \mathfrak B
\quad\mbox{for}\quad n=2.$$ [*Then there exists a positive constant $N'_3$ depending only on $n$, $\nu$ and $\mathfrak B$ such that any nonnegative weak solution of ${\cal L}u=0$ in $B_{2R}$ satisfies*]{} $$\label{Harnack2}
\sup\limits_{B_{R}}u\le N'_3\cdot\inf\limits_{B_{R}}u.$$
We split the spherical layer $B_{2R}\setminus B_{R}$ to $M$ layers of equal thickness $\frac R M$ and put $\delta=\frac{1}{2M}$. Obviously, one can choose $M$ depending only on $n$, $\nu$ and $\mathfrak B$ such that at least for one of these layers (say, $K=\{r-2\delta R<|x|<r+2\delta R\}$) the following estimates hold: $$\|{\bf b}\|_{n,K}\le \ep \quad\mbox{for}\quad n\ge3;
\qquad\qquad \|{\bf b}\ln^{\frac 12}\big(1+2\delta R|{\bf b}|\big)\|_{2,K}\le \ep
\quad\mbox{for}\quad n=2$$ (here $\ep=\ep(n,\nu)$ is the above smallness constant).
We cover the sphere $|x|=r$ with a finite set of balls $B_{\delta R}(x)$ (note that the number of balls depends only on $\delta$). Since all doubled balls $B_{2\delta R}(x)$ lye in $K$, we can apply Harnack’s inequality in these balls. This gives $\sup\limits_{|x|=r}u\le C(n,\nu,\delta)\cdot\inf\limits_{|x|=r}u$. However, by the maximum principle, $$\inf\limits_{B_{R}}u\ge\inf\limits_{|x|=r}u;\qquad\qquad \sup\limits_{B_{R}}u\le\sup\limits_{|x|=r}u,$$ and the statement follows.
The following statement can be proved by verbatim repetition of the proof of Theorem 2.6, using Theorem 2.5$\,'$.
[**Theorem 2.6$\,'$**]{} (the Liouville theorem). [*Let $\cal L$ be an operator of the form ([**DE**]{}) in $\mathbb R^n$. Let the condition (\[ell\]) be satisfied, and*]{} $${\bf b}\in L_{n,loc}(\mathbb R^n)
\quad\mbox{for}\quad n\ge3;
\qquad\qquad {\bf b}\ln^{\frac 12}\big(1+|{\bf b}|\big)\in L_{2,loc}(\mathbb R^2)
\quad\mbox{for}\quad n=2.$$ [*Suppose also that for some $\delta>0$*]{} $$\begin{gathered}
\label{Liouville2}
\liminf\limits_{R\to\infty}\sup\limits_{|x|=R}
\|{\bf b}\|_{n,B_{\delta R}(x)}<\infty,\quad n\ge3;\\
\liminf\limits_{R\to\infty}\sup\limits_{|x|=R}
\|{\bf b}\ln^{\frac 12}\big(1+R|{\bf b}|\big)\|_{2,B_{\delta R}(x)}<\infty,
\ \ n=2.\end{gathered}$$ [*Then any weak semibounded solution of ${\cal L}u=0$ in $\mathbb R^n$ is a constant.*]{}
[**Remark 3**]{}. If ${\bf b}\in L_n(\mathbb R^n)$ (respectively, ${\bf b}\ln^{\frac 12}\big(1+|x|\,|{\bf b}|\big)\in L_2(\mathbb R^2)$), then (\[Liouville2\]) is obviously satisfied.
As for Hölder estimates for solutions, we have two possibilities. The first one is to take in the proof of Lemma 2.3 $R$ sufficiently small, such that the smallness assumptions on ${\bf b}$ are satisfied in $B_{2R}$. This gives the estimate (\[Holder\]) with $\gamma$ depending only on $n$ and $\nu$, while $N_2$ depends also on the moduli of continuity of ${\bf b}$ in $L_n(B_{R_0})$ (respectively, of ${\bf b}\ln^{\frac 12}\big(1+R|{\bf b}|\big)$ in $L_2(B_{R_0})$). The second possibility is to use Theorem 2.5$\,'$. This gives (\[Holder\]) with both $\gamma$ and $N_2$ depending on $n$, $\nu$ and $\|{\bf b}\|_{n,B_{R_0}}$ (respectively, $\|{\bf b}\ln^{\frac 12}\big(1+R_0|{\bf b}|\big)\|_{2,B_{R_0}}$).
Parabolic case
==============
[**Lemma 3.1**]{}.
*Let $\cal M$ be an operator of the form ([**DP**]{}) in $Q_{R}^{\lambda,\theta}(x^0;t^0)$, $\lambda>1$, $\theta>0$, and let the conditions (\[ell\]) and (\[bezdiv\]) be satisfied. Let also ${\bf b}\in L_{q,\ell}(Q_{R}^{\lambda,\theta}(x^0;t^0))$, with some $q$ and $\ell$ such that $$\label{alpha}
\alpha=\alpha(q,\ell)\equiv\frac nq+\frac 2\ell-1\in\,[0,1[$$ ($q$ as well as $\ell$ may be infinite[^5]).*
Then there exists a positive constant $N_4$ depending on $n$, $\nu$, $\lambda$, $\theta$, $q$, $\ell$ and the quantity $$\widehat{\cal N}=\widehat{\cal N}(R,\lambda,\theta)\equiv
R^{-\alpha}\|{\bf b}\|_{q,\ell,Q_{R}^{\lambda,\theta}(x^0;t^0)}$$ such that any Lipschitz subsolution of the equation ${\cal M}u=0$ in $Q_{R}^{\lambda,\theta}(x^0;t^0)$ satisfies
$$\label{estmax'}
\sup\limits_{Q_{R}^{1,\frac \theta2}(x^0;t^0)} u_+\le N_4
\bigg(\!\!\!\!\!\!\Xint{\ \ \qquad-}\limits_{Q_{R}^{\lambda,\theta}(x^0;t^0)}\!\!\!u_+^2dxdt\bigg)^{\frac 12}.$$
[**Remark 4**]{}. The quantity $\widehat{\cal N}$ depends also on $q$ and $\ell$. However, we assume these parameters hold fixed, and we do not indicate this dependence. We also do not indicate the dependence $\widehat{\cal N}$ on $x^0$ and $t^0$.
We proceed similarly to Lemma 2.1. Without loss of generality, we assume $(x^0;t^0)=(0;0)$.
For a nonnegative test function $\eta$ we have $$\int\limits_{Q_{R}^{\lambda,\theta}}(\partial_tu\eta+ a_{ij}D_juD_i\eta+b_iD_iu\,\eta)\,dxdt\le0.$$ We take $\eta=\varphi'(u)\cdot\xi$, where $\xi$ is a cut-off function, Lipschitz in $x$ and vanishing at the neighborhood of $\partial'Q_{R}^{\lambda,\theta}$, while $\varphi\in{\cal C}^2(\mathbb R)$ is a convex function vanishing in $\mathbb R_-$. This gives $$\label{Moser'}
\int\limits_{Q_{R}^{\lambda,\theta}\cap\{u>0\}}\Big(\partial_tv\xi+a_{ij}D_jvD_i\xi+\frac {\varphi''(u)}
{\varphi^{\prime2}(u)}\,a_{ij}D_jvD_iv\,\xi+b_iD_iv\,\xi\Big)\,dxdt\le0,$$ where $v=\varphi(u)$.
As in Section 2, by mollification at a neighborhood of the origin, one can weaken in (\[Moser\]) the assumption $\varphi\in{\cal C}^2(\mathbb R)$ to $\varphi\in{\cal C}^2(\mathbb R_+\cup\mathbb R_-)$.
Now we put in (\[Moser’\]) $\varphi(\tau)=\tau_+^p$, $p>1$, and $\xi=\chi_{\{t<\bar t\}}\cdot v\zeta^2$ where $\zeta$ is a smooth cut-off function in $Q_{R}^{\lambda,\theta}$, $\bar t \in\,]-\theta R^2,0[$. Then we obtain $$\begin{gathered}
\label{Moser_power'}
\frac 12\int\limits_{B_{\lambda R}}(v\zeta)^2\big|^{t=\bar t}dx+\\
+\int\limits_{Q_{R}^{\lambda,\theta}}\chi_{\{t<\bar t\}}\,\Big(\frac {2p-1}{p}\,
a_{ij}D_jvD_iv\,\zeta^2+2a_{ij}D_jv\,vD_i\zeta\,\zeta-v^2\zeta\partial_t\zeta+b_iD_iv\,v\zeta^2\Big)
\,dxdt\le0.\end{gathered}$$
The last term in (\[Moser\_power’\]) can be estimated using (\[bezdiv\]) and the Hölder inequality: $$\begin{gathered}
\label{bezdiv1'}
-\int\limits_{Q_{R}^{\lambda,\theta}}\chi_{\{t<\bar t\}}\,b_iD_iv\,v\zeta^2\,dxdt\le
\int\limits_{Q_{R}^{\lambda,\theta}}\chi_{\{t<\bar t\}}\,b_iv^2\zeta\,D_i\zeta\,dxdt\le\\
\le\|{\bf b}\|_{q,\ell,Q_{R}^{\lambda,\theta}}\|v\zeta\|^{2-\frac 1s}_{r,l,Q_{R}^{\lambda,\theta}}
\|v\zeta^{1-s}|D\zeta|^s\|^{\frac 1s}_{2,2,Q_{R}^{\lambda,\theta}},\end{gathered}$$ where $s>2$ is defined by $\frac 1s=1-\frac n{2q}-\frac 1\ell$ while $r$ and $l$ are defined by $$\frac 1{2s}+\frac 1q+\frac {2-\frac 1s}r=1;\qquad \frac 1{2s}+\frac 1\ell+\frac {2-\frac 1s}l=1.$$ Note that $\frac n2<\frac nr+\frac 2l<\frac n2+1$, and, by the embedding theorem [@LSU Ch. II, (3.4)], $$\label{embed'}
\|v\zeta\|_{r,l,Q_{R}^{\lambda,\theta}}\le
C_9(n,r,l,\lambda,\theta)R^{\frac nr+\frac 2l-\frac n2}\,\|v\zeta\|_{{\cal V}(Q_{R}^{\lambda,\theta})}.$$ Using (\[ell\]), (\[bezdiv1’\]) and (\[embed’\]) and the Young inequality, we obtain from (\[Moser\_power’\]) $$\label{ee'}
\|v\,\zeta\|^2_{{\cal V}(Q_{R}^{\lambda,\theta})}\le
C_{10}(n,\nu,q,\ell,\lambda,\theta)\cdot \int\limits_{Q_{R}^{\lambda,\theta}}
v^2\big(|D\zeta|^2+\zeta|\partial_t\zeta|+
R^2\|{\bf b}\|^{2s}_{q,\ell,Q_{R}^{\lambda,\theta}}\zeta^{2-2s}|D\zeta|^{2s}\big)\,dxdt.$$
We put $\lambda_m=1+2^{-m}(\lambda-1)$, $\theta_m=\frac \theta2 (1+4^{-m})$, $m\in \mathbb N\cup\{0\}$, and substitute $\zeta=\zeta_m$ such that $$\zeta_m\equiv1\ \ \mbox{in}\ \ Q_{R}^{\lambda_{m+1},\theta_{m+1}};\quad \zeta_m\equiv0\ \ \mbox{out of}\ \ Q_{R}^{\lambda_m,\theta_m};
\qquad |\partial_t\zeta_m|\le \frac {4^mC}{\theta R^2};\quad \frac{|D\zeta_m|}{\zeta_m^{1-\frac 1s}}\le \frac {2^mC_3(s)}{(\lambda-1)R}.$$ Then (\[ee’\]) implies $$\label{eee'}
\|v\,\zeta_m\|_{{\cal V}(Q_{R}^{\lambda_m,\theta_m})}\le
\frac {C_{11}(n,\nu,q,\ell,\lambda,\theta)}R\cdot\|v\|_{2,2,Q_{R}^{\lambda_m,\theta_m}}\cdot
\big(2^m+\big(2^m\widehat{\cal N}\big)^s\big).$$
Now for $p=p_m\equiv (\frac {n+2}n)^m$ we obtain from (\[embed’\]) (with $r=l=\frac{2(n+2)}{n}$) and (\[eee’\]) $$\begin{gathered}
\label{iteration'}
\bigg(\!\!\!\!\!\!\!\Xint{\qquad\quad -}\limits_{Q_{R}^{\lambda_{m+1,\theta_{m+1}}}}\!\!\!
u_+^{2p_{m+1}}dxdt\bigg)^{\frac {1}{2p_{m+1}}}\!\le
\bigg(C(n)\!\!\Xint{\quad\ \, -}\limits_{Q_{R}^{\lambda_m,\theta_m}} (v\zeta_m)^r dxdt\bigg)^{\frac {1}{rp_m}}\le\\
\le \bigg(2^{2ms}C_{12}\!\!\Xint{\quad\ \,-}\limits_{Q_{R}^{\lambda_m,\theta_m}}\! v^2\,dxdt\bigg)
^{\frac {1}{2p_m}}\!\!=
\bigg(2^{2ms}C_{12}\!\!\Xint{\quad\ \,-}\limits_{Q_{R}^{\lambda_m,\theta_m}}\! u_+^{2p_m}
dxdt\bigg)^{\frac 1{2p_m}}\!,\end{gathered}$$ where $C_{12}$ depends only on $n$, $\nu$, $q$, $\ell$, $\lambda$, $\theta$ and $\widehat{\cal N}$.
Iterating (\[iteration’\]) we arrive at (\[estmax’\]).
[**Remark 5**]{}. If ${\bf b}={\bf b}^{(1)}+{\bf b}^{(2)}$, and ${\bf b}^{(j)}\in L_{q_j,\ell_j}(Q_{R}^{\lambda,\theta}(x^0;t^0))$, $j=1,2$, with $\alpha(q_j,\ell_j)\in\,[0,1[$, then the proof of Lemma 3.1 does not change. The same is true for other statements of this Section.
[**Remark 6**]{}. If, under assumptions of Lemma 3.1, $u$ satisfies additionally $$\label{negat}
u(\cdot;t^0-\theta R^2)\le 0\qquad\mbox{in}\quad B_{\lambda R}(x^0),$$ then we can estimate $u$ up to the bottom of the cilinder, i.e. one can replace the left-hand side in (\[estmax’\]) by $\sup\limits_{Q_{R}^{1,\theta}(x^0;t^0)} u_+$. Indeed, one may simply put $\theta_m\equiv\theta$ and take $\zeta_m$ independent on $t$.
[**Corollary 3.1**]{}.
*Let $\cal M$ satisfy the assumptions of Lemma 3.1 in $Q_{R}^{\lambda,\theta}(x^0;t^0)$.*
1\. If a Lipschitz subsolution of ${\cal M}u=0$ in $Q_{R}^{\lambda,\theta}(x^0;t^0)$ satisfies
$$\label{tiny'}
\mbox{meas}\, (\{u> k\}\cap Q_{R}^{\lambda,\theta}(x^0;t^0))\le \mu\,\mbox{meas}\,(Q_{R}^{\lambda,\theta}),
\qquad \mu< N_4^{-2},$$
[*for some $k$, then*]{} $$\label{estmax1'}
\sup\limits_{Q_{R}^{1,\frac \theta2}(x^0;t^0)} (u-k)\le
N_4\sqrt{\mu}\sup\limits_{Q_{R}^{\lambda,\theta}(x^0;t^0)}(u-k),$$
*(here $N_4$ is the constant from Lemma 3.1).*
2\. If a Lipschitz nonnegative supersolution of ${\cal M}V=0$ in $Q_{R}^{\lambda,\theta}(x^0;t^0)$ satisfies
$$\label{tiny1'}
\mbox{meas}\, (\{V< k\}\cap Q_{R}^{\lambda,\theta}(x^0;t^0))\le \mu\,\mbox{meas}\,(Q_{R}^{\lambda,\theta}),
\qquad \mu\le \mu_1\equiv (2N_4)^{-2},$$
[*for some $k>0$, then*]{} $$\label{estmin'}
V\ge \frac k2\qquad\mbox{in}\quad Q_{R}^{1,\frac \theta2}(x^0;t^0).$$ [*If $V$ additionally satisfies*]{} $$V(\cdot;t^0-\theta R^2)\ge k\qquad\mbox{in}\quad B_{\lambda R}(x^0),$$ [*then the estimate (\[estmin’\]) holds in $Q_{R}^{1,\theta}(x^0;t^0)$.*]{}
1\. We apply Lemma 3.1 to $u-k$.
2\. We apply Lemma 3.1 and Remark 6 to $u=k-V$.
[**Lemma 3.2**]{}. [*Let $\cal M$ satisfy the assumptions of Lemma 3.1 in $Q_{R}(x^0;t^0)$. For any $\delta_0\in\,]0,1]$ there exists $\theta_0\in\,]0,1[$ such that if a Lipschitz nonnegative supersolution of ${\cal M}V=0$ in $Q_{R}^{1,\theta_0}(x^0;t^0)$ satisfies*]{} $$\mbox{meas}\, (\{V(\cdot;t^0-\theta_0 R^2)\ge k\}\cap B_{R}(x^0))\ge \delta_0\,\mbox{meas}\,(B_{R})$$[*for some $k>0$, then*]{} $$\mbox{meas}\, (\{V(\cdot;\bar t)\ge \frac {\delta_0}3 k\}\cap B_{R}(x^0))\ge
\frac {\delta_0}3 \,\mbox{meas}\,(B_{R})
\qquad\mbox{for any}\quad \bar t\in\,[t^0-\theta_0 R^2,t^0].$$[*Moreover, $\theta_0$ is completely determined by ${\delta_0}$, $n$, $\nu$, $q$, $\ell$ and the quantity $\widehat{\cal N}$.*]{}
Without loss of generality, we assume $(x^0;t^0)=(0;0)$. For a nonnegative test function $\eta$ we have $$\label{super}
\int\limits_{Q_{R}^{\lambda,\theta_0}}(\partial_tV\eta+ a_{ij}D_jVD_i\eta+b_iD_iV\,\eta)\,dxdt\ge0.$$ We take $\eta=\chi_{\{t<\bar t\}}\cdot(V-k)_-\zeta^2(x)$, where $\zeta$ is a smooth cut-off function in $B_{R}$, $\bar t \in\,]-\theta_0 R^2,0]$. Using (\[ell\]), (\[bezdiv\]) and the Young inequality, we obtain $$\begin{gathered}
\label{measure}
\int\limits_{B_{R}}(V-k)_-^2\zeta^2\big|^{t=\bar t}dx
+\nu\int\limits_{Q_{R}^{1,\theta_0}}\chi_{\{t<\bar t\}}\,
|D(V-k)_-|^2\zeta^2dxdt%\le\\
\le \int\limits_{B_{R}}(V-k)_-^2\zeta^2\big|^{t=-\theta_0 R^2}dx+\\
+\int\limits_{Q_{R}^{1,\theta_0}}\chi_{\{t<\bar t\}}\,\Big(C_{13}(n,\nu)(V-k)_-^2|D\zeta|^2dxdt+
2b_i(V-k)_-^2\,\zeta\,D_i\zeta\Big)dxdt.\end{gathered}$$ Now we choose $\zeta$ such that $\zeta\equiv1$ in $B_{(1-\sigma)R}$ and $|D\zeta|\le \frac {2}{\sigma R}$ where $\sigma<1$ is a parameter to be chosen later. Observing that $(V-k)_-^2\le k^2$, we estimate the right-hand side of (\[measure\]) by $$k^2\Big[(1-{\delta_0})\,\mbox{meas}\,(B_R)+C_{13}\theta_0 R^2\cdot \frac{4\mbox{meas}\,(B_R)}{(\sigma R)^2}+
\frac{2}{\sigma R}\,\|{\bf b}\|_{q,\ell,Q_{R}}\|{\bf 1}\|_{q',\ell',Q_{R}^{1,\theta_0}}\Big].$$ On the another hand, $$\int\limits_{B_{R}}(V-k)_-^2\zeta^2\big|^{t=\bar t}dx\ge\!\!
\int\limits_{\{V<\frac {\delta_0}3 k\}\cap B_{(1-\sigma)R}}\!\!\!\!(V-k)_-^2\big|^{t=\bar t}dx\ge
\big(1-\frac {\delta_0}3\big)^2 k^2\, \mbox{meas}\, (\{V(\cdot;\bar t)< \frac {\delta_0}3 k\}\cap B_{(1-\sigma)R}).$$ Thus, $$\mbox{meas}\, (\{V(\cdot;\bar t)< \frac {\delta_0}3 k\}\cap B_{(1-\sigma)R})\le
\frac {\mbox{meas}\,(B_R)}{(1-\frac {\delta_0}3)^2}\cdot\Big[(1-{\delta_0})+\frac{4C_{13}\theta_0}{\sigma^2}+
\frac{C(n)\theta_0^{\frac 2{\ell'}}{\widehat{\cal N}}}{\sigma}\,\Big],$$ and therefore, $$\mbox{meas}\, (\{V(\cdot;\bar t)< \frac {\delta_0}3 k\}\cap B_{R})\le
\frac {\mbox{meas}\,(B_R)}{(1-\frac {\delta_0}3)^2}\cdot
\Big[(1-{\delta_0})+C(n)\sigma+\frac{4C_{13}\theta_0}{\sigma^2}+
\frac{C(n)\theta_0^{\frac 2{\ell'}}{\widehat{\cal N}}}{\sigma}\,\Big].$$ Since $1-{\delta_0}\le(1-\frac {\delta_0}3)^3-\frac 8{27}{\delta_0}^2$, one can choose $\sigma$ and then $\theta_0$ small enough such that the right-hand side is not greater that $(1-\frac {\delta_0}3)\, \mbox{meas}\,(B_R)$, and the Lemma follows.
[**Lemma 3.3**]{}. [*Let $\cal M$ satisfy the assumptions of Lemma 3.1 in $Q_{R}^{\lambda,\theta}(x^0;t^0)$ with $\lambda>1$. Let a Lipschitz nonnegative supersolution of ${\cal M}V=0$ in $Q_{R}^{\lambda,\theta}(x^0;t^0)$ satisfy*]{} $$\label{measure1}
\mbox{meas}\, (\{V(\cdot;t)\ge k_0\}\cap B_{R}(x^0))\ge \delta_1\,\mbox{meas}\,(B_{R})
\qquad\mbox{for any}\quad t\in\,[t^0-\theta R^2,t^0]$$ [*for some $k_0>0$ and $\delta_1>0$. Then for any $\mu\in\,]0,1[$ there exists $s>1$ such that*]{} $$\mbox{meas}\, (\{V< 2^{-s} k_0\}\cap Q_{R}^{1,\theta}(x^0;t^0))\le \mu\,\mbox{meas}\,(Q_{R}^{1,\theta}).$$ [*Moreover, $s$ is completely determined by $n$, $\nu$, $\lambda$, $\theta$, $\mu$, $\delta_1$, $q$, $\ell$, and the quantity $\widehat{\cal N}$.*]{}
Without loss of generality, we assume $(x^0;t^0)=(0;0)$. For $m\in{\mathbb Z}_+$ we put $k_m=2^{-m}k_0$, $${\cal E}_m(t)=\{x\in B_R:\ k_{m+1}\le V(x,t)< k_m\};\qquad
{\cal E}_m=\{(x;t):\ t\in\,[-\theta R^2,0],\ x\in{\cal E}_m(t)\}.$$ We take in (\[super\]) $\eta=(V-k_m)_-\zeta^2(x)$, where $\zeta$ is a smooth cut-off function, vanishing at the neighborhood of $\partial B_{\lambda R}$ and satisfying $\zeta\equiv 1$ in $B_R$, $|D\zeta|\le \frac 2{(\lambda-1)R}$. Similarly to the proof of Lemma 3.2, we derive $$\label{fff}
\int\limits_{\{V<k_m\}}
|DV|^2\zeta^2dxdt
=\int\limits_{Q_{R}^{\lambda,\theta}}
|D(V-k_m)_-|^2\zeta^2dxdt\le C_{14}(n,\nu,\lambda, \theta,\ell,\widehat{\cal N}) k_m^2 R^n.$$
Further, De Giorgi’s inequality (see, e.g., [@LSU Ch. II, (5.6)]) and the assumption (\[measure1\]) give $$(k_m-k_{m+1})\cdot \mbox{meas}\, (\{V(\cdot;t)< k_{m+1}\}\cap B_{R})\le
\frac{C(n)R}{\delta_1}\int\limits_{{\cal E}_m(t)}|DV(\cdot;t)|\,dx,\qquad t\in\,[-\theta R^2,0].$$ We integrate this relation w.r.t $t$ and then square both parts, arriving at $$k_{m+1}^2\,\mbox{meas}^2 (\{V< k_{m+1}\}\cap Q_{R}^{1,\theta})\le
\frac{C(n)R^2}{\delta_1^2}\int\limits_{{\cal E}_m}|DV|^2 dxdt\cdot\mbox{meas}\,({\cal E}_m).$$ Together with (\[fff\]), this gives $$\mbox{meas}^2 (\{V< k_{m+1}\}\cap Q_{R}^{1,\theta})\le
C(n)C_{14}\delta_1^{-2}R^{n+2}\cdot\mbox{meas}\,({\cal E}_m).$$ Therefore, $$\begin{gathered}
s\cdot\mbox{meas}^2 (\{V< k_s\}\cap Q_{R}^{1,\theta})\le
\sum\limits_{m=0}^{s-1}\mbox{meas}^2 (\{V< k_{m+1}\}\cap Q_{R}^{1,\theta})\le \\
\le C_{15}\delta_1^{-2}\cdot\mbox{meas}\,(Q_{R}^{1,\theta})\cdot\sum\limits_{m=0}^{s-1}\mbox{meas}\,({\cal E}_m)\le
C_{15}\delta_1^{-2}\cdot\mbox{meas}^2(Q_{R}^{1,\theta})\end{gathered}$$ (here $C_{15}$ depends on the same quantities as $C_{14}$), and the Lemma follows.
[**Corollary 3.2**]{}.
*Let $\cal M$ satisfy the assumptions of Lemma 3.1 in $Q_{R}^{2,1}$.*
1\. If a Lipschitz nonnegative supersolution of ${\cal M}V=0$ in $Q_{R}^{2,1}$ satisfies
$$\label{measure2}
\mbox{meas}\, (\{V(\cdot;\bar t)\ge k\}\cap B_{R})\ge \delta\,\mbox{meas}\,(B_{R})$$
[*for some $\bar t\in\,[-R^2, -\Theta R^2]$, $k>0$, $\delta,\Theta\in\,]0,1]$, then*]{} $$\label{estmin1'}
V\ge \beta_1 k\qquad\mbox{in}\quad Q_{R}^{1,\frac {\theta_1}2}(0;\bar t+\theta_1 R^2).$$
*Here $\theta_1=\min\{\Theta,\theta_0\}$, where $\theta_0=\theta_0(\delta,n,\nu,q,\ell,\widehat{\cal N})$ is the constant from Lemma 3.2, while $\beta_1$ depends on the same quantities as $\theta_0$.*
2\. If the relation (\[measure2\]) holds with $\delta=1$, i.e.
$$\label{estmin2'}
V(\cdot;\bar t)\ge k\qquad\mbox{in}\quad B_{R},$$
[*then for any $\sigma\in\,]0,1[$ $$\label{estmin1''}
V\ge\beta_1 k\qquad\mbox{in}\quad Q_{R}^{\sigma,\theta_1}(0;\bar t+\theta_1 R^2),$$ In this case $\beta_1$ depends additionally on $\sigma$.*]{}
First, we use Lemma 3.2 with $t^0=\bar t+\theta_1 R^2$. Then, in the case 1, we apply Lemma 3.3 with $R\to\frac 32 R$, $\lambda=\frac 43$, $\theta=\frac 49\theta_1$, $\delta_1=\frac{2^n\delta}{3^{n+1}}$, $k_0=\frac{\delta}3 k$ and $\mu=\mu_1$, where $\mu_1=\mu_1(n,\nu,\frac 32, \theta_1,q,\ell,\widehat{\cal N})$ is the constant from Corollary 3.1, part 2. Finally, Corollary 3.1 with $\lambda=\frac 32$ and $\theta=\theta_1$ gives (\[estmin1’\]) with $\beta_1=\frac {\delta}{3\cdot 2^{s+1}}$, where $s=s(n,\nu,\frac 32, \theta_1, \mu_1,\delta_1,q,\ell,\widehat{\cal N})$ is the constant from Lemma 3.3.
In the case 2, we apply Lemma 3.3 with $\lambda=2$, $\theta=\theta_1$, $\delta_1=\frac{\delta}3$, $k_0=\frac{\delta}3 k$ and $\mu=\mu_1(n,\nu,\sigma^{-1}, \sigma^{-2}\theta_1,q,\ell,\widehat{\cal N})$. Finally, the last statement of Corollary 3.1 with $R\to\sigma R$, $\lambda=\sigma^{-1}$ and $\theta=\sigma^{-2}\theta_1$ gives (\[estmin1”\]) with $\beta_1=\frac {\delta}{3\cdot 2^{s+1}}$, where $s=s(n,\nu,\sigma^{-1}, \sigma^{-2}\theta_1, \mu_1,\delta_1,q,\ell,\widehat{\cal N})$.
[**Lemma 3.4**]{}. [*Let $\cal M$ satisfy the assumptions of Lemma 3.1 in $Q_{R}^{2,1}$. Let a Lipschitz nonnegative supersolution of ${\cal M}V=0$ in $Q_{R}^{2,1}$ satisfy (\[estmin2’\]) for some $k>0$ and $\bar t\in\,[-R^2,-\Theta R^2]$, $\Theta\in\,]0,1]$. Then*]{} $$\label{estmin3'}
V\ge \beta_2 k\qquad\mbox{in}\quad \widehat Q=B_{\frac R2}\times[\bar t,0].$$ [*Moreover, $\beta_2$ is completely determined by $\Theta$, $n$, $\nu$, $q$, $\ell$, and the quantity $\widehat{\cal N}$.*]{}
We set $M=\mbox{entier}\big(\frac{|\bar t|}{\theta_1 R^2}\big)+1$ and $\widehat\theta_1=\frac {|\bar t|}{MR^2}$. Now let us consider cylinders $$Q^{(m)}=Q_{R}^{1-\frac m{2M},\widehat\theta_1}(0;\bar t+m\widehat\theta_1R^2), \qquad m=1,\dots,M.$$ By (\[estmin1”\]) we consequently obtain $$V\ge\widehat\beta_1\cdot\inf\limits_{Q^{(m)}} V \qquad\mbox{in}\quad Q^{(m+1)},$$ where $\widehat\beta_1=\widehat\beta_1(n,\Theta,\nu,q,\ell,\widehat{\cal N})>0$.
Since $\widehat Q\subset\bigcup\limits_m Q^{(m)}$, this ensures (\[estmin3’\]) with $\beta_2=\widehat\beta_1^M$.
[**Corollary 3.3**]{}. [*Let $\cal M$ satisfy the assumptions of Lemma 3.1 in $Q_{R}^{2,1}$. If a Lipschitz nonnegative supersolution of ${\cal M}V=0$ in $Q_{R}^{2,1}$ satisfies (\[measure2\]) for some $k>0$, $\delta\in\,]0,1]$ and $\bar t\in\,[-R^2, -\frac 34 R^2]$, then*]{} $$\label{estmin4'}
V\ge \beta_3 k\qquad\mbox{in}\quad Q_{\frac R2},$$ [*where $\beta_3$ is completely determined by $\delta$, $n$, $\nu$, $q$, $\ell$, and the quantity $\widehat{\cal N}$.*]{}
It suffices to apply consequently Corollary 3.2, part 1, and Lemma 3.4.
[**Corollary 3.4**]{}. [*Let $\cal M$ satisfy the assumptions of Lemma 3.1 in $Q_{R}^{2,1}$. If a Lipschitz nonnegative supersolution of ${\cal M}V=0$ in $Q_{R}^{2,1}$ satisfies*]{} $$\label{measure3}
\mbox{meas}\, (\{V>k\}\cap Q_{R})\ge \widehat\delta\,\mbox{meas}\,(Q_{R}),$$ [*for some $k>0$, $\widehat\delta\in\,]0,1]$, then*]{} $$\label{estmin5'}
V\ge \beta_4 k\qquad\mbox{in}\quad Q_{R}^{\frac 12,\frac {\widehat\delta}4},$$ [*where $\beta_4$ is completely determined by $\widehat\delta$, $n$, $\nu$, $q$, $\ell$, and the quantity $\widehat{\cal N}$.*]{}
The inequality (\[measure3\]) obviously implies $$\mbox{meas}\, (\{V>k\}\cap Q_{R}^{1,1-\frac{\widehat\delta}2}(0;-\frac{\widehat\delta}2R^2))\ge
\frac {\widehat\delta}2\,\mbox{meas}\,(Q_{R}).$$ Therefore, there exists $\bar t\in\,[-R^2,-\frac{\widehat\delta}2R^2]$, such that (\[measure2\]) holds with $\delta=\frac{\widehat\delta}2$. By Corollary 3.2, part 1, $V(\cdot;\bar t+\frac{\theta_1}2R^2)\ge \beta_1 k$ in $B_{R}$. Finally, we observe that $\bar t+\frac{\theta_1}2R^2\le-\frac{\widehat\delta}4R^2$, and Lemma 3.4 provides (\[estmin5’\]).
[**Corollary 3.5**]{} (strong maximum principle). [*Let $\cal M$ satisfy the assumption of Lemma 3.1 in $Q$. Then any Lipschitz nonconstant supersolution of ${\cal M}V=0$ in $Q$ cannot attain its minimum at a point of $\partial Q\setminus\partial'Q$.*]{}
Without loss of generality, $\inf\limits_Q V=0$.
Assume the converse. Then there exists $(x^0;t^0)\in\overline Q\setminus\partial'Q$ such that $V(x^0;t^0)=0$ but $V\not\equiv0$ in $Q_R(x^0;t^0)\subset Q_{R}^{2,1}(x^0;t^0)\subset Q$ with some $R$. Then the relation (\[measure3\]) holds for some $k>0$ and $\delta>0$, and we obtain (\[estmin5’\]), a contradiction.
[**Lemma 3.5**]{}. [*Let $\cal M$ satisfy the assumptions of Lemma 3.1 in $Q_{2R}$. Then any Lipschitz solution of ${\cal M}u=0$ in $Q_{2R}$ satisfies the estimate*]{} $$\label{osc1}
\underset{Q_{\frac R2}}{\mbox{osc}}\ u\le \varkappa_1\,\underset{Q_{2R}}{\mbox{osc}}\ u,$$ [*where $\varkappa_1<1$ depends on $n$, $\nu$, $q$ $\ell$ and the quantity $\widehat{\cal N}$.*]{}
We set $k=\frac 12\underset{Q_{2R}}{\mbox{osc}}\ u$ and consider two functions $V_1=u-\inf\limits_{Q_{2R}}u$ and $V_2=\sup\limits_{Q_{2R}}u-u$. At least one of them satisfies (\[measure2\]) with $\delta=\frac 12$ and $\bar t=-R^2$. Therefore, Corollary 3.3 gives for this function the estimate (\[estmin4’\]), which implies (\[osc1\]) with $\varkappa_1=1-\frac 12\beta_3(\frac 12, n,\nu,q,\ell,\widehat{\cal N})$.
[**Corollary 3.6**]{} (Hölder estimate). [*Let $\cal M$ satisfy the assumption of Lemma 3.1 in $Q_{R_0}$. Let also $\sup\limits_{R<R_0}{\widehat{\cal N}}(R,1,1)<\infty$. Then any Lipschitz solution of ${\cal M}u=0$ in $Q_{R_0}$ satisfies the estimate*]{} $$\label{Holder'}
\underset{Q_\rho}{\mbox{osc}}\ u\le N_5 \Big(\frac \rho r\Big)^{\gamma_1}
\cdot\underset{Q_r}{\mbox{osc}}\ u,\qquad 0<\rho<r\le R_0,$$ [*where $N_5$ and $\gamma_1$ depend on $n$, $\nu$, $q$ and $\sup\limits_{R<R_0}{\widehat{\cal N}}(R,1,1)$.*]{}
Iteration of (\[osc1\]) gives (\[Holder’\]) with $\gamma_1=-\log_4(\varkappa_1)$.
[**Corollary 3.7**]{} (the Liouville theorem). [*Let $\cal M$ be an operator of the form ([**DP**]{}) in $\mathbb R^n\times \mathbb R_-$, and let the conditions (\[ell\]) and (\[bezdiv\]) be satisfied. Let also ${\bf b}\in L_{q,\ell,loc}(\mathbb R^n\times\mathbb R_-)$, with some $q$ and $\ell$ satisfying (\[alpha\]) ($q$ as well as $\ell$ may be infinite). Finally, assume that*]{} $$\label{Liouville'}
\liminf\limits_{R\to\infty}{\widehat{\cal N}}(R,1,1)<\infty.$$ [*Then any Lipschitz bounded solution of ${\cal M}u=0$ in $\mathbb R^n\times\mathbb R_-$ is a constant.*]{}
[**Remark 7**]{}. If ${\bf b}\in L_{q,\ell}(\mathbb R^n\times\mathbb R_-)$, then (\[Liouville’\]) is obviously satisfied.
Iteration of (\[osc1\]) with respect to a suitable sequence $R_m\to\infty$ gives the statement.
To prove the Harnack inequality, we need the following modification of Lemma 3.4.
[**Lemma 3.4$\,'$**]{} (slant cylinder). [*Let $\cal M$ satisfy the assumptions of Lemma 3.1 in $Q_{R}^{\lambda,1}$, $\lambda>2$. Let a Lipschitz nonnegative supersolution of ${\cal M}V=0$ in $Q_{R}^{\lambda,1}$ satisfy*]{} $$V(\cdot;-R^2)\ge k\qquad\mbox{in}\quad B_{R}(x^0),$$ [*for some $k>0$ and $x^0\in B_{(\lambda-2)R}$. Then for any $x^1\in B_{(\lambda-2)R}$*]{} $$\label{estmin3''}
V\ge \widehat\beta_2 k\qquad\mbox{in}\quad
\widetilde Q=\{(x;t):\ t\in\,[-R^2,0],\ x\in B_{\frac R2}\big(x^1+(x^1-x^0)\frac t{R^2}\big)\}.$$ [*Moreover, $\widehat\beta_2$ is completely determined by $\lambda$, $n$, $\nu$, $q$, $\ell$, and the quantity $\widehat{\cal N}$.*]{}
We put $\widehat x(t)=x^1+(x^1-x^0)\frac t{R^2}$. Then it is easy to see that the function $\widetilde V(x;t)=V(x-\widehat x(t);t)$ is a Lipschitz nonnegative supersolution of $$\widetilde{\cal M}V\equiv \partial_tV-D_i\big(\widetilde a_{ij}(x;t)D_jV\big)+\widetilde b_i(x;t)D_iV=0$$ in $Q_{R}^{2,1}$, where $$\widetilde a_{ij}(x;t)=a_{ij}(x-\widehat x(t);t);\qquad
\widetilde b_i(x;t)=b_i(x-\widehat x(t);t)+\frac{x^1_i-x^0_i}{R^2}.$$ Note that $\widetilde{\cal M}$ satisfies the assumptions of Lemma 3.1 in $Q_{R}^{2,1}$, and the quantity $\widetilde{\widehat{\cal N}}$ is bounded by $\widehat{\cal N}+C_{16}(\lambda,n)$. By Lemma 3.4 (with $\Theta=1$), we obtain (\[estmin3”\]).
The next statement is a parabolic analog of Lemma 2.4. For $(x;t)\in Q$ we introduce the notation $d_{\mbox{par}}((x;t),\partial'Q)=\inf \{\rho>0:\ Q_\rho(x;t)\subset Q\}$.
[**Lemma 3.6**]{}. [*Let $\cal M$ satisfy the assumptions of Lemma 3.1 in $Q_{2R}$, and let ${\bf b}\in\mathbb M^{\alpha}_{q,\ell}(Q_{2R})$. Let for a Lipschitz nonnegative supersolution of ${\cal M}V=0$ in $Q_{2R}$ and some $(y;s)\in Q_{R}^{2,2}(0;-2R^2)$, the inequality $\inf\limits_{B_\rho(y)}V(\cdot;s)=k>0$ holds with $\rho=\frac 14 d_{\mbox{\rm par}}((y;s),\partial' Q_{2R})$. Then*]{} $$\label{estmin6'}
\inf\limits_{Q_{R}} V\ge N_6\Big(\frac \rho R\Big)^{\widehat\gamma_1} k,$$ [*where $N_6$ and $\widehat\gamma_1$ depend on $n$, $\nu$, $q$, $\ell$ and $\|{\bf b}\|_{\mathbb M^{\alpha}_{q,\ell}(Q_{2R})}$.*]{}
We denote by ${\mathfrak N}$ an integer number such that $2^{-({\mathfrak N}+1)}R\le\rho< 2^{-{\mathfrak N}}R$ and introduce a sequence of cylinders $Q_{{\mathfrak r}_m}^{4,1}(y^m;t^m)$, $m=0,\dots,{\mathfrak N}$, as follows: $$\begin{array}{lll}
{\mathfrak r}_0=2^{-({\mathfrak N}+1)}R,&y^0=y,& t^0=s+{\mathfrak r}_0^2;\\
{\mathfrak r}_m=2{\mathfrak r}_{m-1},&y^m=y^{m-1}-\min\{2{\mathfrak r}_m,|y^{m-1}|\}\,{\bf e},&
t^m=t^{m-1}+{\mathfrak r}_m^2
\end{array}$$ (here ${\bf e}=\frac y{|y|}$). Also we denote $y^{-1}=y$, $t^{-1}=s$.
Direct computation shows that $Q_{{\mathfrak r}_m}^{4,1}(y^m;t^m)\subset Q_{2R}$ for all $m=0,\dots,{\mathfrak N}$. Therefore, the assumptions of Lemma 3.4$\,'$ (with $\lambda=4$, $x^0=y^{m-1}$) are fulfilled in $Q_{{\mathfrak r}_m}^{4,1}(y^m;t^m)$. Using Lemma 3.4$\,'$ with $x^1\in B_{2{\mathfrak r}_m}(y^m)$, we obtain the inequality $$V(\cdot;t^m)\ge \widehat\beta_2(4,n,\nu,q,\ell, \|{\bf b}\|_{\mathbb M^{\alpha}_{q,\ell}(Q_{2R})})
\cdot \inf\limits_{B_{{\mathfrak r}_m}(y^{m-1})} V(\cdot;t^{m-1})
\qquad\mbox{in}\quad B_{\frac 52{\mathfrak r}_m}(y^m)$$ and, in particular, $$\inf\limits_{B_{{\mathfrak r}_{m+1}}(y^m)} V(\cdot;t^m)\ge
\widehat\beta_2\cdot\inf\limits_{B_{{\mathfrak r}_m}(y^{m-1})} V(\cdot;t^{m-1}).$$
Since ${\mathfrak r}_{{\mathfrak N}+1}=R$ and $y^{\mathfrak N}=0$, we obtain $$\inf\limits_{B_{R}} V(\cdot;t^{\mathfrak N})\ge
\widehat\beta_2^{{\mathfrak N}+1}\cdot\inf\limits_{B_{{\mathfrak r}_0}(y)} V(\cdot;s)\ge
\Big(\frac \rho {2R}\Big)^{\widehat\gamma_1}\cdot k,$$ where $\widehat\gamma_1=-\log_2(\widehat\beta_2)$.
It is easy to estimate $t^{\mathfrak N}$: $$t^{\mathfrak N}=s+\sum\limits_{m=0}^{\mathfrak N}{\mathfrak r}_m^2=
s+{\mathfrak r}_0^2\cdot \frac{2^{2{\mathfrak N}+2}-1}3<s+\frac{R^2}3\le -\frac 53 R^2.$$
Now we use Lemma 3.4$\,'$ (with $\lambda=4$, $x^0\in B_{\frac R2}$, $x^1\in B_R$) in $Q_{\frac R2}^{4,1}(0;t^{\mathfrak N}+\frac{R^2}4)$. Since slant cylinders for $x^0\in B_{\frac R2}$, $x^1\in B_R$ cover $Q_{R}^{1,\frac 18}(0;t^{\mathfrak N}+\frac{R^2}4)$, we obtain $$\inf\limits_{Q_{R}^{1,\frac 18}}(0;t^{\mathfrak N}+\frac{R^2}4) V\ge
\widehat\beta_2\Big(\frac \rho {2R}\Big)^{\widehat\gamma_1}\cdot k.$$ Repeating this process, we cover $Q_{R}$. This gives (\[estmin6’\]) with $N_6=2^{-\widehat\gamma_1}\widehat\beta_2^{31}$.
[**Theorem 3.7**]{} (the Harnack inequality).
*Let $\cal M$ be an operator of the form ([**DP**]{}) in $Q_{2R}$, and let the conditions (\[ell\]) and (\[bezdiv\]) be satisfied. Let also ${\bf b}\in \mathbb M^{\alpha}_{q,\ell}(Q_{2R})$, with some $q$ and $\ell$ satisfying (\[alpha\]) (as $q$ as $\ell$ may be infinite).*
Then there exists a positive constant $N_7$ depending on $n$, $\nu$, $q$, $\ell$ and $\|{\bf b}\|_{\mathbb M^{\alpha}_{q,\ell}(Q_{2R})}$, such that any Lipschitz nonnegative solution of ${\cal M}u=0$ in $Q_{2R}$ satisfies
$$\label{Harnack'}
\sup\limits_{Q_{R}(0;-2R^2)}u\le N_7\cdot\inf\limits_{Q_{R}}u.$$
Similarly to Theorem 2.5, we denote by $(y;s)$ a maximum point of the function $$v(x;t)=(d_{\mbox{par}}((x;t),\partial'Q_{2R}))^{\widehat\gamma_1}\!\cdot u(x;t);
\qquad (x;t)\in Q_{R}^{2,2}(0;-2R^2)$$ (here $\widehat\gamma_1$ is the constant from Lemma 3.6) and set $$\rho=\frac 14\,d_{\mbox{par}}((y;s),\partial'Q_{2R});\qquad
{\mathfrak M}=v(y;s)=(4\rho)^{\widehat\gamma_1}\!\cdot u(y;s).$$
It is obvious that $$\begin{aligned}
\sup\limits_{Q_{R}(0;-2R^2)}u\le \frac {\mathfrak M}{R^{\widehat\gamma_1}}=
\Big(\frac {4\rho} R\Big)^{\widehat\gamma_1}\!\cdot u(y;s);\label{sup'}\\
\sup\limits_{Q_{2\rho}(y;s)}u\le \frac {\mathfrak M}{(2\rho)^{\widehat\gamma_1}}=
2^{\widehat\gamma_1}\cdot u(y;s).\phantom{u(y)}\label{sup1'}\end{aligned}$$
Denote $k=\frac 12 u(y;s)$. If $\mbox{meas}\, (\{u> k\}\cap Q_{2\rho}(y;s))\le \mu\,\mbox{meas}\,(Q_{2\rho})$, then Corollary 3.1, part 1 (with $\lambda=2$ and $\theta=4$) and (\[sup1’\]) imply the relation $$k=u(y;s)-k\le \sup\limits_{Q_\rho(y;s)}(u-k)\le N_4\sqrt{\mu}\sup\limits_{Q_{2\rho}(y;s)}(u-k)
\le N_4\sqrt{\mu}\,(2^{\widehat\gamma_1+1}-1)k,$$ which is impossible for $\mu\le\mu_2\equiv\frac 1{2^{2\widehat\gamma_1+2}}\,N_4^{-2}$. Thus, $\mbox{meas}\, (\{u> k\}\cap Q_{2\rho}(y;s))\ge \mu_2\,\mbox{meas}\,(Q_{2\rho})$, and Corollary 3.4 (with $\widehat\delta=\mu_2$) gives $$\label{inf'}
\inf\limits_{B_\rho(y)}u(\cdot;s)\ge \beta_4k=\frac{\beta_4}2\cdot u(y;s).$$
Finally, Lemma 3.6 gives $$\label{inf1'}
\inf\limits_{Q_{R}}u\ge N_6\Big(\frac \rho R\Big)^{\widehat\gamma_1}\inf\limits_{B_\rho(y)}u(\cdot;s).$$ Combining (\[sup’\]), (\[inf’\]) and (\[inf1’\]), we arrive at (\[Harnack’\]) with $N_7=\frac{2^{2\widehat\gamma_1+1}}{\beta_4N_6}$.
As in Section 2, the proofs of Lemmas 3.1–3.3 run without changes also for weak sub/supersolutions of ${\cal M}u=0$ if the bilinear form $$\widehat{\cal B}\big\langle u,\eta\big\rangle\equiv \int\limits_{Q_{R}^{\lambda,\theta}}b_iD_iu\,\eta\,dxdt$$ can be continuously extended to the pair $(v,v\zeta^2)$ with $v\in {\cal V}(Q_{R}^{\lambda,\theta})$. Unfortunately, we have no parabolic analog of sharp results by Maz’ya–Verbitsky, so we can give only rather rough sufficient conditions. The simplest one is $${\bf b}\in \mathbb M^{\frac nq-1}_{q,\infty}(Q_{R}^{\lambda,\theta}),\quad \frac n2<q\le n,\quad
\mbox{div}({\bf b})=0.$$ If Lemmas 3.1–3.3 are proved, all subsequent statements obviously hold true.
In addition, let us consider the case $\alpha(q,\ell)=0$, i.e. $\frac nq+\frac 2\ell=1$. As in the elliptic case, main results of this section hold true for weak (sub/super)solutions without the assumption (\[bezdiv\])[^6]. The only exceptional situation is $q=n$, where the assumption (\[bezdiv\]) seems to be unavoidable without the smallness restriction on ${\bf b}$ [^7]. We explain briefly changes in the proofs.
Similarly to Lemma 2.1, Lemma 3.1 in this case can be proved under additional assumption $\|{\bf b}\|_{q,\ell,Q_{R}^{\lambda,\theta}}\le\ep(n,\nu)$.
Lemmas 3.2 and 3.3 are proved without changes. Therefore, all subsequent statements hold true under assumption of sufficient smallness of ${\bf b}$.
In what follows we will assume $q>n$. Then, as in the elliptic case, strong maximum principle holds without smallness assumption on ${\bf b}$.
When proving the Harnack inequality, one can exclude the smallness assumption on ${\bf b}$ similarly to Theorem 2.5$\,'$. The result reads as follows.
[**Theorem 3.7$\,'$**]{} (the Harnack inequality). [*Let $\cal M$ be an operator of the form ([**DE**]{}) in $Q_{2R}$, and let the condition (\[ell\]) be satisfied. Suppose also that*]{} $$\|{\bf b}\|_{q,\ell,Q_{2R}}\le \mathfrak B,\qquad \frac nq+\frac 2\ell=1,\quad q>n.$$ [*Then there exists a positive constant $N'_7$ depending only on $n$, $\nu$, $q$ and $\mathfrak B$ such that any nonnegative weak solution of ${\cal M}u=0$ in $Q_{2R}$ satisfies*]{} $$\sup\limits_{Q_{R}(0;-2R^2)}u\le N'_7\cdot\inf\limits_{Q_{R}}u.$$
As in the elliptic case, we have two possibilities to prove the Hölder estimates for solutions. The first one is to take in Lemma 3.5 $R$ sufficiently small, such that the smallness assumptions on ${\bf b}$ are satisfied in $Q_{2R}$. This gives the estimate (\[Holder’\]) with $\gamma_1$ depending only on $n$, $\nu$ and $q$, while $N_5$ depends also on the moduli of integral continuity of ${\bf b}$ in $L_{q,\ell}(Q_{R_0})$. The second possibility is to use Theorem 3.7$\,'$. This gives (\[Holder’\]) with both $\gamma_1$ and $N_5$ depending on $n$, $\nu$, $q$ and $\|{\bf b}\|_{q,\ell,Q_{R_0}}$.
Finally, the next statement directly follows from the second variant of the Hölder estimate.
[**Corollary 3.7$\,'$**]{} (the Liouville theorem). [*Let $\cal M$ be an operator of the form ([**DP**]{}) in $\mathbb R^n\times \mathbb R_-$, and let the conditions (\[ell\]) be satisfied. Let also ${\bf b}\in L_{q,\ell}(\mathbb R^n\times\mathbb R_-)$, with some $q$ and $\ell$ such that $\frac nq+\frac 2\ell=1$, $q>n$. Then any weak bounded solution of ${\cal M}u=0$ in $\mathbb R^n\times\mathbb R_-$ is a constant.*]{}
Application to a problem of hydrodynamics
=========================================
When considering axisymmetric flows of viscous incompressible liquid, the following equation of ([**DP**]{}) form arises: $$\label{NS1}
{\cal M}u\equiv \partial_tu-\Delta u+b_i(x',x_3;t)D_iu=0 \qquad\mbox{in}\quad \mathbb R^3\times \mathbb R_-.$$ Here we denote $x'=(x_1,x_2)$; $$\label{NS2}
{\bf b}={\bf v}+\widehat{\bf b}=\Big(v^1+\ep\frac{2x_1}{|x'|^2},v^2+\ep\frac{2x_2}{|x'|^2},v^3\Big),$$ where ${\bf v}=(v^1,v^2,v^3)$ is a solution to the Navier–Stokes system (NSE) while $\ep=\pm1$.
Namely, see [@KNSS], the function $u=v_2x_1-v_1x_2\equiv|x'|v_\vartheta$ satisfies the equation (\[NS1\]) with $\ep=+1$ (here $v_\vartheta$ is the angular component of the velocity). Next, if $v_\vartheta=0$, then the function $u=|x'|^{-2}(({\rm rot}({\bf v}))_2x_1-({\rm rot}({\bf v}))_1x_2)$ satisfies the equation (\[NS1\]) with $\ep=-1$.
Since, by the NSE, $\mbox{div}({\bf v})=0$, it is easy to see that $$\label{div}
\mbox{div}({\bf b})=4\pi\ep\delta_{\Gamma},\qquad\Gamma=\{|x'|=0\}.$$ Thus, if $\ep=-1$, then the results of Section 3 are applicable to (\[NS1\])–(\[NS2\]). Namely, we are interested in the Liouville theorem.
Note that $\widehat{\bf b}\in L_{q,\infty,loc}(\mathbb R^3\times \mathbb R_-)$ with any $q<2$, and, moreover, satisfies the assumption (\[Liouville’\]) with $q\in\,]\frac 32,2[$, $\ell=\infty$. Therefore, taking into account Remark 5, we obtain the following result.
[**Theorem 4.1**]{}. [*Let ${\bf v}$ be an axisymmetric solution of the Navier–Stokes system in $\mathbb R^3\times \mathbb R_-$. Suppose also that ${\bf v}$ satisfies (\[Liouville’\]) with some $q$ and $\ell$ such that $\alpha\equiv\frac 3q+\frac 2\ell-1\in\,[0,1[$. Then any Lipschitz bounded solution of (\[NS1\])–(\[NS2\]) with $\ep=-1$ in $\mathbb R^n\times\mathbb R_-$ is a constant.*]{}
[**Remark 8**]{}. The assumption (\[Liouville’\]) is satisfied, for example, if ${\bf v}$ satisfies the estimate $$|{\bf v}(x',x_3;t)|\le \frac C{|x'|}$$ (in this case one can take $q\in\,]\frac 32,2[$, $\ell=\infty$), or $$|{\bf v}(x',x_3;t)|\le \frac C{(-t)^{\frac 12}}$$ (in this case it suffices $q=\infty$, $\ell\in\,]1,2[$).
To deal with more complicated case $\ep=+1$, we need the following observation.
[**Remark 9**]{}. The statement of Lemma 3.1 holds true without assumption (\[bezdiv\]) if $u\le0$ in the set ${\cal F}=\mbox{supp}(\mbox{div}({\bf b}))_+$. Similarly, Lemmas 3.2–3.4 and Corollaries 3.1 (part 2), 3.2, 3.4 hold true if $V\ge k$ in ${\cal F}$. Lemma 3.3 holds true if $V\ge k_0$ in ${\cal F}$.
Now we prove the following variant of Corollary 3.3.
[**Lemma 4.2**]{}. [*Let ${\bf v}$ be an axisymmetric solution of the Navier–Stokes system in $Q_{R}^{2,1}$, and let ${\bf v}\in L_{q,\ell}(Q_{R}^{2,1})$ with some $q$ and $\ell$ such that $\alpha\equiv\frac 3q+\frac 2\ell-1\in\,[0,1[$. Let $V$ be a Lipschitz nonnegative supersolution of (\[NS1\])–(\[NS2\]) with $\ep=+1$ in $Q_{R}^{2,1}$. If $$\label{Gamma}
V|_{\Gamma\cap Q_{R}^{2,1}}\ge k,\qquad V\le {\mathfrak N}k\quad \mbox{in}\ \ Q_{R}^{2,1}$$ for some $k>0$ and ${\mathfrak N}>1$, then*]{} $$\label{estmin4''}
V\ge \widehat\beta_3 k\qquad\mbox{in}\quad Q_{\frac R2},$$ [*where $\widehat\beta_3$ is completely determined by $q$, $\ell$, ${\mathfrak N}$ and the quantity $\widehat{\cal N}=R^{-\alpha}\|{\bf v}\|_{q,\ell,Q_{R}^{2,1}}$.*]{}
We put $$\widehat{\cal E}_{\varkappa}(t)=\{x\in B_R:\ V(x,t)>\varkappa k\};\qquad
\widehat{\cal E}_{\varkappa}=\{(x;t):\ t\in\, [-R^2,-\frac 34 R^2],\ x\in\widehat{\cal E}_{\varkappa}(t)\}$$ and claim that $$\label{measure4}
\mbox{meas}\, (\widehat{\cal E}_{\varkappa})\ge \delta\,\mbox{meas}\,(Q_{R}^{1,\frac 14})$$ for some $\varkappa>0$ and $\delta>0$ depending only on $q$, $\ell$, ${\mathfrak N}$ and $\widehat{\cal N}$.
Indeed[^8], by (\[div\]), we obtain for any Lipschitz test function $\eta\ge0$ $$\label{g}
\int\limits_{Q_{R}^{1,\frac 14}(0;-\frac 34 R^2)}\!\!(\partial_tV\eta+ D_iVD_i\eta-b_iVD_i\eta)\,dxdt\ge
4\pi\!\!\int\limits_{\Gamma\cap Q_{R}^{1,\frac 14}(0;-\frac 34 R^2)}\!\!V\eta\,dx_3dt.$$ We take $\eta$ such that $$\eta\equiv1\ \ \mbox{in}\ \ Q_{R}^{\frac 12,\frac 18}(0;-\frac{13}{16}R^2);\quad
\eta\equiv0\ \ \mbox{out of}\ \ Q_{R}^{1,\frac 14}(0;-\frac 34 R^2);
\qquad |\partial_t\eta|+|D\eta|^2+|\Delta\eta|\le \frac {C}{R^2}.$$ Then (\[g\]) and $V|_{\Gamma\cap Q_{R}^{2,1}}\ge k$ imply $$\begin{gathered}
\frac {\pi}2\,kR^3\le \!\!\int\limits_{Q_{R}^{1,\frac 14}(0;-\frac 34 R^2)}\!\!
V\big(|\partial_t\eta|+|\Delta\eta|+|{\bf b}|\cdot|D\eta|\big) dxdt\le\\
\le \frac {C}{R^2}\!\!\int\limits_{Q_{R}^{1,\frac 14}(0;-\frac 34 R^2)}\!\!V\, dxdt+
\frac {C}{R}\!\!\int\limits_{Q_{R}^{1,\frac 14}(0;-\frac 34 R^2)}\!\!V|{\bf v}|\, dxdt+
\frac {C}{R}\!\!\int\limits_{Q_{R}^{1,\frac 14}(0;-\frac 34 R^2)}\!\!\frac V{|x'|}\, dxdt.\end{gathered}$$ Splitting the integrals in the right-hand side into integrals over $\widehat{\cal E}_{\varkappa}$ and over its complement, we obtain with regard to $V\le {\mathfrak N}k$ and to $\frac 1{|x'|}\in L_{\frac 95,\infty}(Q_{R}^{1,\frac 14}(0;-\frac 34 R^2))$ $$\begin{gathered}
\frac {\pi}2\,kR^3\le \frac{C{\mathfrak N}k}{R^2}\,
\Big[\mbox{meas}\,(\widehat{\cal E}_{\varkappa})+
R^{1+\alpha}\widehat{\cal N}\|{\bf 1}\|_{q',\ell',\,\widehat{\cal E}_{\varkappa}}+
R^{\frac 53}\|{\bf 1}\|_{\frac 94,1,\,\widehat{\cal E}_{\varkappa}}\Big]+\\
+\frac{C\varkappa k}{R^2}\,\Big[\mbox{meas}\,(Q_{R}^{1,\frac 14})+
R^{1+\alpha}\widehat{\cal N}\|{\bf 1}\|_{q',\ell',Q_{R}^{1,\frac 14}}+
R^{\frac 53}\|{\bf 1}\|_{\frac 94,1,Q_{R}^{1,\frac 14}}\Big].\end{gathered}$$ The second term in the right-hand side is easily estimated by $C\varkappa kR^3(1+C_{17}(q,\ell)\widehat{\cal N})$. Therefore, choosing $\varkappa=\varkappa(q,\ell,\widehat{\cal N})$ sufficiently small, we obtain $$\label{gg}
\frac{C\mathfrak N}{R^5}\Big[\mbox{meas}\,(\widehat{\cal E}_{\varkappa})+
R^{1+\alpha}\widehat{\cal N}\|{\bf 1}\|_{q',\ell',\,\widehat{\cal E}_{\varkappa}}+
R^{\frac 53}\|{\bf 1}\|_{\frac 94,1,\,\widehat{\cal E}_{\varkappa}}\Big]\ge1.$$ To estimate the second term in brackets, we rewrite it as follows: $$\|{\bf 1}\|_{q',\ell',\widehat{\cal E}_{\varkappa}}=\bigg(\int\limits_{-R^2}^{-\frac 34 R^2}\!
\mbox{meas}^{\frac{\ell'}{q'}} (\widehat{\cal E}_{\varkappa}(t))\,dt\bigg)^{\frac 1{\ell'}}.$$ If $q\le\ell$ then, by the Hölder inequality, $$\|{\bf 1}\|_{q',\ell',\widehat{\cal E}_{\varkappa}}\le\bigg(\int\limits_{-R^2}^{-\frac 34 R^2}\!
\mbox{meas}\, (\widehat{\cal E}_{\varkappa}(t))\,dt\bigg)^{\frac 1{q'}}\cdot
\Big(\frac 14 R^2\Big)^{\frac 1{\ell'}-\frac 1{q'}}=
\mbox{meas}^{\frac 1{q'}} (\widehat{\cal E}_{\varkappa})\cdot\Big(\frac 14 R^2\Big)^{\frac 1q-\frac 1{\ell}}.$$ In the opposite case, since $\mbox{meas}\, (\widehat{\cal E}_{\varkappa}(t))\le \mbox{meas}\, (B_R)$, we obtain $$\begin{gathered}
\|{\bf 1}\|_{q',\ell',\widehat{\cal E}_{\varkappa}}=\bigg(\int\limits_{-R^2}^{-\frac 34 R^2}\!
\bigg(\frac{\mbox{meas}\, (\widehat{\cal E}_{\varkappa}(t))}{\mbox{meas}\,(B_R)}\bigg)
^{\frac{\ell'}{q'}}dt\bigg)^{\frac 1{\ell'}}\cdot \mbox{meas}^{\frac 1{q'}}(B_R)\le\\
\le \bigg(\int\limits_{-R^2}^{-\frac 34 R^2}\!
\frac{\mbox{meas}\, (\widehat{\cal E}_{\varkappa}(t))}{\mbox{meas}\,(B_R)}
\,dt\bigg)^{\frac 1{\ell'}}\cdot \mbox{meas}^{\frac 1{q'}}(B_R)=
\mbox{meas}^{\frac 1{\ell'}} (\widehat{\cal E}_{\varkappa})\cdot(4\pi R^3)^{\frac 1{\ell}-\frac 1q}.\end{gathered}$$ Similarly we estimate the third term in brackets in (\[gg\]). Thus, we obtain the inequality for ${\cal A}=\frac 1{R^5}\,\mbox{meas}\, (\widehat{\cal E}_{\varkappa})$: $${\cal A}+{\cal A}^{\frac 49}+\widehat{\cal N}{\cal A}^{1-\max\{\frac 1q,\frac 1{\ell}\}}\ge \frac 1{C{\mathfrak N}},$$ and (\[measure4\]) follows.
The inequality (\[measure4\]) provides $$\mbox{meas}\, (\widehat{\cal E}_{\varkappa}(\bar t))\ge \delta\,\mbox{meas}\,(B_{R})$$ for some $\bar t\in\,[-R^2, -\frac 34 R^2]$, and Corollary 3.3 ensures (\[estmin4”\]) with $\widehat\beta_3=\varkappa\cdot\beta_3(\delta, 3, 1, q, \ell, \widehat{\cal N})$.
[**Theorem 4.3**]{}. [*Let ${\bf v}$ be an axisymmetric solution of the Navier–Stokes system in $\mathbb R^3\times \mathbb R_-$. Suppose also that ${\bf v}$ satisfies (\[Liouville’\]) with some $q$ and $\ell$ such that $\alpha\equiv\frac 3q+\frac 2\ell-1\in\,[0,1[$. Let $u$ be a Lipschitz bounded solution of (\[NS1\])–(\[NS2\]) with $\ep=+1$ in $\mathbb R^n\times\mathbb R_-$. If $u|_{\Gamma}=const$, then $u\equiv const$.*]{}
Given $R$, we set $k=\frac 12\underset{Q_{2R}}{\mbox{osc}}\ u$ and consider two functions $V_1=u-\inf\limits_{Q_{2R}}u$ and $V_2=\sup\limits_{Q_{2R}}u-u$. At least one of them satisfies (\[Gamma\]) with ${\mathfrak N}=2$. Therefore, Lemma 4.2 gives for this function the estimate (\[estmin4”\]), which implies (\[osc1\]) with $\varkappa_1=1-\frac 12\widehat\beta_3(q,\ell,2,\widehat{\cal N})$. Iteration of this inequality with respect to a suitable sequence $R_m\to\infty$ completes the proof.
[AFT]{}
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[^1]: Partially supported by RFBR grant 08-01-00748 and by grant NSh.4210.2010.1.
[^2]: For $q=n$, the assumption (\[bezdiv\]) can be removed. We discuss this at the end of this Section.
[^3]: More formally, in this case the inequality (\[Moser\]) holds under additional condition that $\varphi$ is globally Lipschitz. Thus, to derive (\[ee\]) one should take $\varphi'(u_+)=p\,\min\{u_+,N\}^{p-1}$, $\xi=\min\{u_+,N\}^p\zeta^2$, $N>0$, and then pass to the limit as $N\to\infty$.
[^4]: Moreover, in this case the assumption ${\bf b}\in L_n$ can be weakened in the scale of Lorentz spaces to ${\bf b}\in\Lambda_{n,q}$ with any $q<\infty$. We do not discuss it here for the reason of place.
[^5]: For $\alpha=0$, the assumption (\[bezdiv\]) can be removed, with some limitation in the case $q=n$, $\ell=\infty$. We discuss this at the end of this Section.
[^6]: Also the assumptions on ${\bf b}$ can be weakened in the scale of Lorentz spaces.
[^7]: By Remark 4, if the assumption (\[bezdiv\]) holds, the case $q=n$, $\ell=\infty$ is simply included into the case $q=n$, $2<\ell<\infty$.
[^8]: This idea was in a particular case used in [@CSTY].
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
for the Lattice Strong Dynamics (LSD) Collaboration[^1]\
Physics Department & Center for Computational Science, Boston University, Boston, MA 02215\
Department of Physics, University of Colorado, Boulder, CO 80309[^2]\
Email:
bibliography:
- 'Lattice2011\_proc.bib'
title: |
$S$ parameter and parity doubling\
below the conformal window
---
Introduction
============
The application of lattice gauge theory to strongly-interacting physics beyond QCD is at present a very active field [@DelDebbio:2011rc]. While much of the current interest is motivated by the possibility that new strong dynamics may play a role in electroweak symmetry breaking [@Hill:2002ap; @Rychkov:2011br], improving our general understanding of strong dynamics is an important theoretical goal in its own right.
The standard picture of strongly-interacting SU($N$) gauge theories is that as we increase the number $N_f$ of fermions in a given representation, an infrared fixed point will develop at some critical $N_f^{(c)}$. For $N_f \geq N_f^{(c)}$ (up to the loss of asymptotic freedom) the system is IR-conformal. Approximately-conformal systems with $N_f {\ensuremath{\lesssim} }N_f^{(c)}$ may possess the dynamical scale separation that characterizes “walking” theories, as well as parity doubling between vector ($V$) and axial-vector ($A$) spectra that can reduce the electroweak $S$ parameter to phenomenologically viable values [@Peskin:1991sw].
The Lattice Strong Dynamics Collaboration approaches these questions by using QCD as a baseline. We consider SU(3) gauge theory and steadily increase the number of fundamental fermions, comparing our results against the familiar case $N_f = 2$. We use computationally expensive domain wall fermions for better control over lattice artifacts. Our first studies focused on the $N_f = 6$ model, which while not truly walking exhibits some of the associated phenomena: by matching IR scales between $N_f = 2$ and $N_f = 6$ calculations, we observed an enhancement in the $N_f = 6$ chiral condensate [@Appelquist:2009ka] and a reduction of the $S$ parameter relative to scaled-up QCD [@Appelquist:2010xv]. Here I provide additional details of our $S$ parameter calculation that were not discussed in [Ref. [@Appelquist:2010xv]]{}. Results presented here also include additional data, and do not affect the conclusions of [Ref. [@Appelquist:2010xv]]{}.
We can identify three main ingredients in our expression for the $S$ parameter, $$\label{eq:S}
S = 4\pi N_D \lim_{Q^2 \to 0}\frac{d}{dQ^2}\Pi_{V - A}(Q^2) - {\ensuremath{\Delta} }S_{SM}.$$ The term ${\ensuremath{\Delta} }S_{SM}$ accounts for the three Nambu–Goldstone bosons (NGBs) eaten by the $W^{\pm}$ and $Z$, and is discussed in detail by [Ref. [@Appelquist:2010xv]]{}. In [Section \[sec:currents\]]{} I review our calculation of the transverse $V$–$A$ polarization function $\Pi_{V - A}(Q^2)$, and relate it to the vector and axial spectra in [Section \[sec:slopes\]]{}. Finally, $N_D$ is the number of doublets with chiral electroweak couplings; in [Section \[sec:S\]]{} I show how it affects our results for the $S$ parameter.
\[sec:currents\]Currents and correlators
========================================
On the lattice, the transverse $V$–$A$ polarization function $\Pi_{V - A}(Q^2)$ is determined from $$\label{eq:polFunc}
\begin{split}
\Pi_{V - A}^{\mu\nu}(Q) & = \left({\ensuremath{\delta} }^{\mu\nu} - \frac{{\ensuremath{\widehat Q} }^{\mu}{\ensuremath{\widehat Q} }^{\nu}}{{\ensuremath{\widehat Q} }^2}\right)\Pi_{V - A}(Q^2) - \frac{{\ensuremath{\widehat Q} }^{\mu}{\ensuremath{\widehat Q} }^{\nu}}{{\ensuremath{\widehat Q} }^2}\Pi_{V - A}^L(Q^2) \\
& = Z\sum_x e^{iQ\cdot (x + {\ensuremath{\widehat\mu} }/ 2)}{\ensuremath{\mbox{Tr}\left[ {\ensuremath{\left\langle \mathcal V^{\mu a}(x)V^{\nu b}(0) \right\rangle} } - {\ensuremath{\left\langle \mathcal A^{\mu a}(x)A^{\nu b}(0) \right\rangle} } \right]} }.
\end{split}$$ Here ${\ensuremath{\widehat Q} }= 2\sin(\pi n / L)$ are lattice momenta, while $Q = 2\pi n / L$; these are spacelike $Q^2 = -q^2 > 0$. The current correlators mix two types of domain wall currents. $V^{\mu a}$ and $A^{\mu a}$ are non-conserved “local” currents defined on the domain walls; in terms of five-dimensional fermion fields $\Psi(x, s)$, $$\begin{split}
V^{\mu a}(x) & = {\frac{1}{2}}\left\{{\ensuremath{\overline\Psi} }(x, L_s - 1){\ensuremath{\gamma} }^{\mu}(1 + {\ensuremath{\gamma} }^5)\tau^a \Psi(x, L_s - 1) + {\ensuremath{\overline\Psi} }(x, 0){\ensuremath{\gamma} }^{\mu}(1 - {\ensuremath{\gamma} }^5)\tau^a \Psi(x, 0)\right\} \\
A^{\mu a}(x) & = {\frac{1}{2}}\left\{{\ensuremath{\overline\Psi} }(x, L_s - 1){\ensuremath{\gamma} }^{\mu}(1 + {\ensuremath{\gamma} }^5)\tau^a \Psi(x, L_s - 1) - {\ensuremath{\overline\Psi} }(x, 0){\ensuremath{\gamma} }^{\mu}(1 - {\ensuremath{\gamma} }^5)\tau^a \Psi(x, 0)\right\}.
\end{split}$$ The conserved currents $\mathcal V^{\mu a}$ and $\mathcal A^{\mu a}$ are point-split, and summed over the fifth dimension: $$\begin{aligned}
\mathcal V^{\mu a}(x) & = \sum_{s = 0}^{L_s - 1}j^{\mu a}(x, s) &
\mathcal A^{\mu a}(x) & = \sum_{s = 0}^{L_s - 1}\mbox{sign}\left(s - \frac{L_s - 1}{2}\right)j^{\mu a}(x, s),\end{aligned}$$ $$j^{\mu a}(x, s) = {\frac{1}{2}}\left\{{\ensuremath{\overline\Psi} }(x + {\ensuremath{\widehat\mu} }, s)(1 + {\ensuremath{\gamma} }^{\mu})U_{x, \mu}^{\dag}\tau^a \Psi(x, s) - {\ensuremath{\overline\Psi} }(x, s)(1 - {\ensuremath{\gamma} }^{\mu})U_{x, \mu}\tau^a \Psi(x + {\ensuremath{\widehat\mu} }, s)\right\}.$$ The Fourier transform in [Eqn. \[eq:polFunc\]]{} involves $(x + {\ensuremath{\widehat\mu} }/ 2)$ because the conserved currents are point-split on the link $(x, x + {\ensuremath{\widehat\mu} })$. The flavor matrices $\tau^a$ are normalized to ${\ensuremath{\mbox{Tr}\left[ \tau^a\tau^b \right]} } = \delta^{ab} / 2$.
Although the conserved and local currents must agree in the continuum limit, at finite lattice spacing only the former satisfy a Ward identity (${\ensuremath{\widehat Q} }_{\mu} \Pi_{VV}^{\mu\nu} = 0$, [Fig. \[fig:Ward\]]{}). Because the correlators involve both currents, [Eqn. \[eq:polFunc\]]{} includes the renormalization factor $Z$, which we compute non-perturbatively, $Z = 0.85$ (0.73) for $N_f = 2$ (6). Our chiral lattice fermions ensure that $Z = Z_A = Z_V$.
![On every configuration, ${\ensuremath{\widehat Q} }_{\mu} \Pi_{VV}^{\mu\nu} = 0$ when one conserved current is used in each correlator (left), but not when only non-conserved local currents are used (right). The horizontal offsets around each $Q^2$ value distinguish different $\nu$.[]{data-label="fig:Ward"}](ward.pdf "fig:"){width="0.45\linewidth"}![On every configuration, ${\ensuremath{\widehat Q} }_{\mu} \Pi_{VV}^{\mu\nu} = 0$ when one conserved current is used in each correlator (left), but not when only non-conserved local currents are used (right). The horizontal offsets around each $Q^2$ value distinguish different $\nu$.[]{data-label="fig:Ward"}](wardLL.pdf "fig:"){width="0.45\linewidth"}
In principle, it would be best to work entirely with the conserved currents $\mathcal V^{\mu a}$ and $\mathcal A^{\mu a}$ instead of using the mixed correlators in [Eqn. \[eq:polFunc\]]{}. In practice, evaluating conserved–conserved correlators such as ${\ensuremath{\left\langle \mathcal V^{\mu a}(x) \mathcal V^{\nu b}(0) \right\rangle} }$ requires $\mathcal O(L_s)$ inversions, increasing the computational cost of the calculation by roughly an order of magnitude. As emphasized in [Ref. [@Boyle:2009xi]]{}, lattice artifacts cancel in the $V$–$A$ difference of the mixed correlators, allowing us to use these less expensive quantities. This is illustrated in the left panel of [Fig. \[fig:WardVio\]]{}: even though $\Pi^{\mu\nu}{\ensuremath{\widehat Q} }_{\nu} \ne 0$ since $V^{\nu a}$ and $A^{\nu a}$ are not conserved, $\left[\Pi_{VV}^{\mu\nu}(Q^2) - \Pi_{AA}^{\mu\nu}(Q^2)\right]{\ensuremath{\widehat Q} }_{\nu} \approx 0$. In the right panel, we see that this does not hold if we use only local currents in the correlators.
![On every configuration, lattice artifacts $\Pi^{\mu\nu} {\ensuremath{\widehat Q} }_{\nu} \ne 0$ cancel in the $V$–$A$ difference when one conserved current is used in each correlator (left), but not when only non-conserved local currents are used (right). The horizontal offsets around each $Q^2$ value distinguish different $\mu$.[]{data-label="fig:WardVio"}](wardCL.pdf "fig:"){width="0.45\linewidth"}![On every configuration, lattice artifacts $\Pi^{\mu\nu} {\ensuremath{\widehat Q} }_{\nu} \ne 0$ cancel in the $V$–$A$ difference when one conserved current is used in each correlator (left), but not when only non-conserved local currents are used (right). The horizontal offsets around each $Q^2$ value distinguish different $\mu$.[]{data-label="fig:WardVio"}](wardLL2.pdf "fig:"){width="0.45\linewidth"}
\[sec:slopes\]Parity doubling and finite volume effects
=======================================================
Because chiral perturbation theory cannot reliably be applied to our $N_f = 6$ calculations [@Neil:2010sc], we extract the slope $\Pi_{V - A}'(0)$ by fitting our data to a simple four-parameter rational function, $$\label{eq:pade}
\Pi_{V - A}(Q^2) = \frac{a_0 + a_1 Q^2}{1 + b_1 Q^2 + b_2 Q^4}.$$ This “Padé(1,2)” functional form has the correct asymptotic behavior $\Pi_{V - A}(Q^2) \sim Q^{-2}$ at large $Q^2$, and also resembles the single-pole dominance approximation to the $V$–$A$ dispersion relation $$\label{eq:dispersion}
\Pi_{V - A}(Q^2) = -F_P^2 + \frac{Q^2}{12\pi} \int_0^{\infty} \frac{ds}{\pi} \left[\frac{R_V(s) - R_A(s)}{s + Q^2}\right].$$ ($F_P$ is the pseudoscalar decay constant.) That is, with the single-pole dominance approximation $R(s) = 12 \pi^2 F^2 {\ensuremath{\delta} }(s - M^2)$, this dispersion relation becomes $$\label{eq:pole}
\Pi_{V - A}^{(pole)}(Q^2) = -F_P^2 + \frac{Q^2F_V^2}{M_V^2 + Q^2} - \frac{Q^2F_A^2}{M_A^2 + Q^2},$$ which reproduces the form of [Eqn. \[eq:pade\]]{} when we apply the corresponding approximation to the first Weinberg sum rule, $F_P^2 = F_V^2 - F_A^2$. Because the lattice data contain information about the entire spectrum, the fit parameters in [Eqn. \[eq:pade\]]{} do not directly correspond to the combinations of meson masses and decay constants predicted by the pole-dominance [Eqn. \[eq:pole\]]{}.
Uncorrelated fits of our data to [Eqn. \[eq:pade\]]{} produce stable results with $\chi^2 / dof \ll 1$ as we vary the $Q^2$ fit range. Our results for $\Pi_{V - A}'(0)$ are shown as colored points in the left panel of [Fig. \[fig:slopes\]]{}. The black points in that plot are pole-dominance predictions based on [Eqn. \[eq:pole\]]{}. Both the direct fit results and the pole-dominance predictions show a reduction for $N_f = 6$ compared to $N_f = 2$ at light pseudoscalar masses $M_P {\ensuremath{\lesssim} }M_{V0}$, where $M_{V0}$ is the vector meson mass in the chiral limit. The pole-dominance predictions are systematically lower than the direct results, consistent with the expectation that states neglected by the single-pole dominance approximation would provide additional positive contributions.
![The slope of $\Pi_{V - A}(Q^2)$ at $Q^2 = 0$, plotted versus $M_P^2 / M_{V0}^2$. Left: $N_f = 2$ and 6 results on [$32^3\!\times\!64$ ]{}volumes from direct fits to Eqn. 3.1 (colored), compared to pole-dominance predictions (black). Right: $N_f = 6$ results on [$16^3\!\times\!32$ ]{}and [$32^3\!\times\!64$ ]{}volumes.[]{data-label="fig:slopes"}](compareSlopes.pdf "fig:"){width="0.45\linewidth"}![The slope of $\Pi_{V - A}(Q^2)$ at $Q^2 = 0$, plotted versus $M_P^2 / M_{V0}^2$. Left: $N_f = 2$ and 6 results on [$32^3\!\times\!64$ ]{}volumes from direct fits to Eqn. 3.1 (colored), compared to pole-dominance predictions (black). Right: $N_f = 6$ results on [$16^3\!\times\!32$ ]{}and [$32^3\!\times\!64$ ]{}volumes.[]{data-label="fig:slopes"}](16v32_MP.pdf "fig:"){width="0.45\linewidth"}
The lightest $N_f = 2$ points in [Fig. \[fig:slopes\]]{} are empty because they correspond to a fermion mass $m$ so small that finite-volume effects may be significant. Finite-volume effects are a concern for the $S$ parameter calculation because they can produce spurious parity doubling that artificially reduces $\Pi_{V - A}'(0)$. This is illustrated in the right panel of [Fig. \[fig:slopes\]]{} for $N_f = 6$ calculations on [$16^3\!\times\!32$ ]{}volumes: $\Pi_{V - A}'(0) \to 0$ as $m \to 0$, which would naïvely suggest a negative $S$ parameter from [Eqn. \[eq:S\]]{}. The associated distortion of the spectrum provides clear evidence that this is merely a finite-volume effect: as $m$ decreases, the [$16^3\!\times\!32$ ]{}pseudoscalar mass $M_P$ freezes around $M_P^2 \approx 1.2M_{V0}^2$, which is not the case for the [$32^3\!\times\!64$ ]{}results also shown in the plot.
Returning to the lightest $N_f = 2$ points, the pole-dominance prediction for $\Pi_{V - A}'(0)$ decreases due to spurious parity doubling from finite-volume effects. However, we do not see a similar reduction in the direct fit result. Instead, this point clearly continues the trend established at heavier masses, and the corresponding $N_f = 2$ results for $S$ ([Fig. \[fig:S\]]{}, below) reproduce the prediction obtained by scaling up QCD phenomenology, $\lim_{M_P^2 \to 0} S = 0.32(3)$ [@Peskin:1991sw]. This suggests that the Padé fits may be less sensitive than spectral quantities to these finite-volume effects, increasing our confidence that the reduction observed for $N_f = 6$ is physical.
\[sec:S\]$S$ parameter results
==============================
Realistic models of dynamical electroweak symmetry breaking must produce exactly three massless NGBs to be eaten by the $W^{\pm}$ and $Z$. Any additional pseudo-Nambu–Goldstone bosons (PNGBs) must acquire masses from standard-model and other (e.g., extended-technicolor) interactions in order to satisfy experimental constraints. On the lattice, however, we perform calculations with $N_f^2 - 1$ degenerate massive PNGBs. When we use [Eqn. \[eq:S\]]{} to determine the $S$ parameter from the $\Pi_{V - A}'(0)$ results shown in [Fig. \[fig:slopes\]]{}, the ${\ensuremath{\Delta} }S_{SM}$ term removes the contribution only of the three would-be NGBs. (To be more precise, the $I_3 = 0$ NGB does not contribute, and ${\ensuremath{\Delta} }S_{SM}$ cancels the contribution of the $|I_3| = 1$ pair.) The remaining $N_f^2 - 4$ PNGBs introduce chiral-log terms $\propto \log[M_{V0}^2 / M_P^2]$ that would diverge in the chiral limit $M_P^2 \to 0$.
[Fig. \[fig:S\]]{} presents our $S$ parameter results for $N_f = 2$ and 6, considering two possible values of $N_D$ for $N_f = 6$. The plot on the left presents the case in which every fermion possesses chiral electroweak couplings, $N_D = N_f / 2 = 3$. The minimal case in which only a single doublet has chiral couplings ($N_D = 1$) is shown on the right. In both cases the $N_f = 6$ results show a reduction compared to rescaling $N_f = 2$, before diverging in the chiral limit. With $N_D = 1$ the $S$ parameter can be significantly closer to the experimental value $S \approx -0.15(10)$ for $M_H^{(ref)} \sim 1$ TeV [@Nakamura:2010zzi].
![$S$ parameter for $N_f = 2$ and 6, for the maximum $N_D = 3$ (left) and minimum $N_D = 1$ (right). The bands correspond to fits explained in the text.[]{data-label="fig:S"}](S.pdf "fig:"){width="0.45\linewidth"}![$S$ parameter for $N_f = 2$ and 6, for the maximum $N_D = 3$ (left) and minimum $N_D = 1$ (right). The bands correspond to fits explained in the text.[]{data-label="fig:S"}](S-single.pdf "fig:"){width="0.45\linewidth"}
To guide the eye, we include in [Fig. \[fig:S\]]{} simple linear fits accounting for the $N_D$-dependent chiral-log divergence that remains for $N_f > 2$. We fit the lightest three solid points to the form $$S = A + Bx + \frac{\sharp - 1}{12\pi}\log\left(1 / x\right)$$ where $x \equiv M_P^2 / M_{V0}^2$ and $\sharp$ counts the pairs of PNGBs with $I_3 \ne 0$, $$\begin{aligned}
\sharp & = \left(\frac{N_f}{2}\right)^2 \quad \mbox{for } N_D = N_f / 2 &
\sharp & = 2N_f - 3 \quad \mbox{for } N_D = 1.\end{aligned}$$ The blue $N_f = 6$ curves allow us to estimate the fermion mass $m$ at which we could directly observe chiral log effects. The necessary $m$ is too small for us to explore on our present [$32^3\!\times\!64$ ]{}volumes.
Again, in a realistic phenomenological context, we must have only three massless NGBs, with $N_f^2 - 4$ massive PNGBs. To estimate a definite value for the $N_f = 6$ $S$ parameter in this situation, we can imagine freezing the masses of all $N_f^2 - 4$ PNGBs at some finite value (such as $M_P^2 = 0.38 M_{V0}^2$ at the minimum of the $N_D = 1$ blue curve in [Fig. \[fig:S\]]{}), and then taking only the three NGBs to the chiral limit $M_P^2 \to 0$. A qualitative picture of this scenario is sketched in [Fig. \[fig:shenanigans\]]{}.
![$S$ parameter for $N_f = 2$ and 6 with $N_D = 1$, imagining that we freeze the masses of all $N_f^2 - 4$ PNGBs at $M_P^2 = 0.38 M_{V0}^2$, as described in the text.[]{data-label="fig:shenanigans"}](chiral_shenanigans.pdf){width="0.45\linewidth"}
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank the members of the LSD Collaboration for many useful discussions, and review of this contribution: T. Appelquist, R. Babich, R. Brower, M. Buchoff, M. Cheng, M. Clark, S. Cohen, G. Fleming, J. Kiskis, M. Lin, H. Na, E. Neil, J. Osborn, C. Rebbi, S. Syritsyn, P. Vranas, G. Voronov, J. Wasem and O. Witzel. This work was supported by the U.S. Department of Energy (DOE) through grants DE-FG02-91ER40676 and DE-FG02-04ER41290; the Lawrence Livermore National Laboratory Institutional Computing Grand Challenge program; the DOE Scientific Discovery through Advanced Computing program through the USQCD Collaboration;[^3] the U.S. National Science Foundation through TeraGrid resources provided by the National Institute for Computational Sciences under grant number TG-MCA08X008;[^4] and Boston University’s Scientific Computing Facilities.
[^1]: `http://www.yale.edu/LSD`
[^2]: Present address
[^3]: `http://www.usqcd.org`
[^4]: `http://www.xsede.org`
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This is a sequel to our previous work on LHC phenomenology of the type II seesaw model in the nondegenerate case. In this work, we further study the pair and associated production of the neutral scalars $H^0/A^0$. We restrict ourselves to the so-called negative scenario characterized by the mass order $M_{H^{\pm\pm}}>M_{H^\pm}>M_{H^0/A^0}$, in which the $H^0/A^0$ production receives significant enhancement from cascade decays of the charged scalars $H^{\pm\pm},~H^\pm$. We consider three important signal channels—$b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, $b\bar{b}\ell^+\ell^-\cancel{E}_T$—and perform detailed simulations. We find that at the 14 TeV LHC with an integrated luminosity of $3000~{\textrm{fb}}^{-1}$, a $5\sigma$ mass reach of $151$, $150$, and $180~{{\rm GeV}}$, respectively, is possible in the three channels from the pure Drell-Yan $H^0A^0$ production, while the cascade-decay-enhanced $H^0/A^0$ production can push the mass limit further to $164$, $177$, and $200~{{\rm GeV}}$. The neutral scalars in the negative scenario are thus accessible at LHC run II.'
author:
- 'Zhi-Long Han $^{1}$'
- 'Ran Ding $^{2}$'
- 'Yi Liao $^{3,2,1}$'
title: 'LHC phenomenology of the type II seesaw mechanism: Observability of neutral scalars in the nondegenerate case'
---
Introduction
============
In a previous paper [@Han:2015hba], we presented a comprehensive analysis on the LHC signatures of the type II seesaw model of neutrino masses in the nondegenerate case of the triplet scalars. In this companion paper, another important signature—the pair and associated production of the neutral scalars–is explored in great detail. This is correlated to the pair production of the standard model (SM) Higgs boson, $h$, which has attracted lots of theoretical and experimental interest [@Aad:2013wqa; @Chatrchyan:2013lba] since its discovery [@Aad:2012tfa; @Chatrchyan:2012ufa], because the pair production can be used to gain information on the electroweak symmetry breaking sector [@Plehn:1996wb]. Since any new ingredients in the scalar sector can potentially alter the production and decay properties of the Higgs boson, a thorough examination of the properties offers a diagnostic tool to physics effects beyond the SM. The Higgs boson pair production has been well studied for collider phenomenology in the framework of the SM and beyond [@Plehn:1996wb; @Dawson:1998py; @Djouadi:1999rca; @Baur:2002qd; @Asakawa:2010xj; @Dolan:2012rv; @Papaefstathiou:2012qe; @Goertz:2013kp; @Gupta:2013zza; @Barr:2013tda; @deFlorian:2013jea; @Dolan:2013rja; @Barger:2013jfa; @Englert:2014uqa; @Liu:2014rva; @deLima:2014dta; @Barr:2014sga], and extensively studied in various new physics models [@Dolan:2012ac; @Arhrib:2009hc; @Craig:2013hca; @Hespel:2014sla; @Kribs:2012kz; @Cao:2013si; @Nhung:2013lpa; @Ellwanger:2013ova; @Bhattacherjee:2014bca; @Christensen:2012si; @Wu:2015nba; @Cao:2014kya; @Han:2013sga; @Gouzevitch:2013qca; @No:2013wsa; @Grober:2010yv; @Gillioz:2012se; @Liu:2013woa; @Arhrib:2008pw; @Heng:2013cya; @Dawson:2012mk; @Chen:2014xwa; @Dib:2005re; @Yang:2014gca; @Chen:2014ask], as well as in the effective field theory approach of anomalous couplings [@Contino:2012xk; @Nishiwaki:2013cma; @Liu:2014rba; @Dawson:2015oha] and effective operators [@Azatov:2015oxa; @Goertz:2014qta; @Pierce:2006dh; @Kang:2015nga; @He:2015spf].
The pair production of the SM Higgs boson proceeds dominantly through the gluon fusion process [@Plehn:1996wb; @Djouadi:1999rca], and has a cross section at the $14~{{\rm TeV}}$ LHC (LHC14) of about $18~{\textrm{fb}}$ at leading order [@Plehn:1996wb]. [^1] It can be utilized to measure the Higgs trilinear coupling. A series of studies have surveyed its observability in the $b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, $b\bar{b}W^+W^-$, $b\bar{b}b\bar{b}$, and $WW^*WW^*$ signal channels [@Baglio:2012np; @Dolan:2012rv; @Gouzevitch:2013qca; @Papaefstathiou:2012qe; @Goertz:2013kp; @deLima:2014dta; @Barr:2014sga]. For the theoretical and experimental status of the Higgs trilinear coupling and pair production at the LHC, see Refs. [@Baglio:2012np; @Dawson:2013bba]. In summary, at the $14~{{\rm TeV}}$ LHC with an integrated luminosity of $3000~{\textrm{fb}}^{-1}$ (LHC14@3000), the trilinear coupling could be measured at an accuracy of $\sim 40\%$ [@Barger:2013jfa], and thus leaves potential space for new physics.
As we pointed out in Ref. [@Han:2015hba], in the negative scenario of the type II seesaw model where the doubly charged scalars $H^{\pm\pm}$ are the heaviest and the neutral ones $H^0/A^0$ the lightest, i.e., $M_{H^{\pm\pm}}>M_{H^\pm}>M_{H^0/A^0}$, the associated $H^0A^0$ production gives the same signals as the SM Higgs pair production while enjoying a larger cross section. The leading production channel is the Drell-Yan process $pp\to Z^*\to H^0A^0$, with a typical cross section $20$-$500~{\textrm{fb}}$ in the mass region [$130$-$300~{{\rm GeV}}$]{}. Additionally, there exists a sizable enhancement from the cascade decays of the heavier charged scalars, which also gives some indirect evidence for these particles. The purpose of this paper is to examine the importance of the $H^0A^0$ production with an emphasis on the contribution from cascade decays and to explore their observability.
The paper is organized as follows. In Sec. \[decay\], we summarize the relevant part of the type II seesaw and explore the decay properties of $H^0,~A^0$ in the negative scenario. Sections \[Eh\] and \[signal\] contain our systematical analysis of the impact of cascade decays on the $H^0/A^0$ production in the three signal channels, $b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, and $b\bar{b}\ell^+\ell^-\cancel{E}_T$. We discuss the observability of the signals and estimate the required integrated luminosity for a certain mass reach and significance. Discussions and conclusions are presented in Sec. \[Dis\]. In most cases, we will follow the notations and conventions in Ref. [@Han:2015hba].
Decay Properties of Neutral Scalars in the Negative Scenario {#decay}
============================================================
The type II seesaw and its various experimental constraints have been reviewed in our previous work [@Han:2015hba]. Here we recall the most relevant content that is necessary for our study of the decay properties of the scalars in this section and of their detection at the LHC in later sections.
The type II seesaw model introduces an extra scalar triplet $\Delta$ of hypercharge two [@typeII] on top of the SM Higgs doublet $\Phi$ of hypercharge unity. Writing $\Delta$ in matrix form, the most general scalar potential is $$\begin{aligned}
\label{Vpotential}
V(\Phi,\Delta)&=&
m^2\Phi^\dagger\Phi+M^2\text{Tr}(\Delta^\dagger\Delta)+\lambda_1(\Phi^\dagger\Phi)^2
+\lambda_2\left(\text{Tr}(\Delta^\dagger\Delta)\right)^2
+\lambda_3\text{Tr}(\Delta^\dagger\Delta)^2\notag\\
&&+\lambda_4(\Phi^\dagger\Phi)\text{Tr}(\Delta^\dagger\Delta)
+\lambda_5\Phi^\dagger\Delta\Delta^\dagger\Phi+\left(\mu \Phi^T i\tau^2\Delta^\dagger \Phi+\text{H.c.}\right).\end{aligned}$$ As in the SM, $m^2 < 0$ is assumed to trigger spontaneous symmetry breaking, while $M^2 > 0$ sets the mass scale of the new scalars. The vacuum expectation value (vev) $v$ of $\Phi$ then induces via the $\mu$ term a vev $v_\Delta$ for $\Delta$. The components of equal charge (and also of identical $CP$ in the case of neutral components) in $\Delta$ and $\Phi$ then mix into physical scalars $H^\pm$; $A^0$; $H^0,~h$ and would-be Goldstone bosons $G^{\pm;0}$, with the mixing angles specified by (see, for instance, Refs. [@Arhrib:2011uy; @Aoki:2012jj]) $$\begin{aligned}
\tan \theta_+ = \frac{\sqrt{2} v_{\Delta}}{v},~
\tan \alpha = \frac{2 v_{\Delta}}{v},~
\tan 2\theta_0 = \frac{2v_{\Delta}}{v} \frac{v^2(\lambda_4+\lambda_5)-2M_{\Delta}^2}
{2v^2\lambda_1-M_{\Delta}^2-v_\Delta^2(\lambda_2+\lambda_3)},
\label{mixangles}\end{aligned}$$ where an auxiliary parameter is introduced for convenience, $$\begin{aligned}
M_\Delta^2=\frac{v^2\mu}{\sqrt{2}v_\Delta}.\end{aligned}$$ To a good approximation, the SM-like Higgs boson $h$ has the mass $M_h \approx\sqrt{2\lambda_1}v$, the new neutral scalars $H^0,~A^0$ have an equal mass $M_{H^0}\approx M_{A^0} \approx M_{\Delta}$, and the new scalars of various charges are equidistant in squared masses: $$M^2_{H^{\pm\pm}}-M^2_{H^{\pm}}\approx M^2_{H^{\pm}}-M^2_{H^0/A^0}\approx -\frac{1}{4}\lambda_5v^2.
\label{massrelation}$$ There are thus two scenarios of spectra, positive or negative, according to the sign of $\lambda_5$. For convenience, we define $\Delta M\equiv M_{H^\pm}-M_{H^0/A^0}$.
![Branching ratios of $H^{\pm}$ and $H^{\pm\pm}$ versus $M_{\Delta}$ at some benchmark points of $\Delta M$ and $v_{\Delta}$: $(\Delta M,v_\Delta)=(5,0.01),~(10,0.01),~(5,0.001)~{{\rm GeV}}$, from the upper to the lower panels. \[brhp\]](BRHp1_new.pdf "fig:"){width="0.45\linewidth"} ![Branching ratios of $H^{\pm}$ and $H^{\pm\pm}$ versus $M_{\Delta}$ at some benchmark points of $\Delta M$ and $v_{\Delta}$: $(\Delta M,v_\Delta)=(5,0.01),~(10,0.01),~(5,0.001)~{{\rm GeV}}$, from the upper to the lower panels. \[brhp\]](BRHpp1.pdf "fig:"){width="0.45\linewidth"} ![Branching ratios of $H^{\pm}$ and $H^{\pm\pm}$ versus $M_{\Delta}$ at some benchmark points of $\Delta M$ and $v_{\Delta}$: $(\Delta M,v_\Delta)=(5,0.01),~(10,0.01),~(5,0.001)~{{\rm GeV}}$, from the upper to the lower panels. \[brhp\]](BRHp2.pdf "fig:"){width="0.45\linewidth"} ![Branching ratios of $H^{\pm}$ and $H^{\pm\pm}$ versus $M_{\Delta}$ at some benchmark points of $\Delta M$ and $v_{\Delta}$: $(\Delta M,v_\Delta)=(5,0.01),~(10,0.01),~(5,0.001)~{{\rm GeV}}$, from the upper to the lower panels. \[brhp\]](BRHpp2.pdf "fig:"){width="0.45\linewidth"} ![Branching ratios of $H^{\pm}$ and $H^{\pm\pm}$ versus $M_{\Delta}$ at some benchmark points of $\Delta M$ and $v_{\Delta}$: $(\Delta M,v_\Delta)=(5,0.01),~(10,0.01),~(5,0.001)~{{\rm GeV}}$, from the upper to the lower panels. \[brhp\]](BRHp3.pdf "fig:"){width="0.45\linewidth"} ![Branching ratios of $H^{\pm}$ and $H^{\pm\pm}$ versus $M_{\Delta}$ at some benchmark points of $\Delta M$ and $v_{\Delta}$: $(\Delta M,v_\Delta)=(5,0.01),~(10,0.01),~(5,0.001)~{{\rm GeV}}$, from the upper to the lower panels. \[brhp\]](BRHpp3.pdf "fig:"){width="0.45\linewidth"}
In the rest of this section, we discuss the decay properties of the new scalars in the negative scenario with an emphasis on $H^0$ and $A^0$. The explicit expressions for the relevant decay widths can be found in Refs. [@Djouadi:2005gj; @Aoki:2011pz; @Chabab:2014ara]. It has been shown that $H^0/A^0$ decays dominantly into neutrinos for $v_{\Delta}<10^{-4}~{{\rm GeV}}$ [@Perez:2008ha], resulting in totally invisible final states. We will restrict ourselves to $v_{\Delta}\gg 10^{-4}~{{\rm GeV}}$ in this work, where $H^0/A^0$ dominantly decays into visible particles. Before we detail their decay properties, we give a brief account of the cascade decays of the charged scalars. The branching ratios of the cascade decays are controlled by the three parameters, $v_{\Delta}$, $\Delta M$, and $M_{\Delta}$. The cascade decays dominate in the moderate region of $v_{\Delta}$ and for $\Delta M$ not too small, where a minimum value of $\Delta M\sim2~{{\rm GeV}}$ appears around $v_{\Delta}\sim10^{-4}~{{\rm GeV}}$ [@Perez:2008ha; @Aoki:2011pz; @Han:2015hba; @Melfo:2011nx]. In Fig. \[brhp\], the branching ratios of $H^{\pm}$ and $H^{\pm\pm}$ are shown as a function of $M_{\Delta}$ at some benchmark points of $v_{\Delta}$ and $\Delta M$. Basically speaking, in the mass region $M_\Delta=130$-$300~{{\rm GeV}}$, the cascade decays are dominant for a relatively large mass splitting $\Delta M$ (as shown in the middle panel of Fig. \[brhp\]) or a relatively small $v_{\Delta}$ (in the lower panel).
$H^0$ decays
------------
At tree level, $H^0$ can decay to $f\bar{f}~(f=q,l)$, $\nu\nu$, $W^{+}W^{-}$, $ZZ$, and $hh$. It can also decay to $gg$, $\gamma\gamma$, and $Z\gamma$ through radiative effects. Similarly, $A^0 \to f\bar{f}$, $\nu\nu$, $Zh$ at tree level, and it has the same decay modes as $H^0$ at the loop level. Since we have chosen $v_{\Delta}\gg 10^{-4}~{{\rm GeV}}$, the neutrino mode can be safely neglected for both $H^0$ and $A^0$. Previous work usually concentrated on the decoupling region where the neutral scalars $H^0/A^0$ are much heaver than the light $CP$-even Higgs $h$ and the scalar self-couplings $\lambda_i$ are taken to be zero for simplicity [@Perez:2008ha]. In this case, the mixing angle $\theta_0\approx\alpha$, and the $H^0W^+W^-$ coupling \[being proportional to $\sin(\alpha-\theta_0)$\] tends to vanish. As a consequence, the $W$-pair mode is absent and the dominant channels are $H^0 \to hh$, $ZZ$ for a heavy $H^0$. In contrast, we take into account the effect of scalar self-interactions and focus on the nondecoupling regime, i.e., $H^0/A^0$ are not much heavier than $h$.
For illustration, we choose the benchmark values $v_{\Delta}=10^{-3}~{{\rm GeV}}$, $\Delta M=5~{{\rm GeV}}$; then, $\lambda_5$ is determined by Eq. (\[massrelation\]) upon specifying $M_\Delta$. [^2] To investigate the effect of the scalar self-interactions, we note the following features in the decays of $H^0$. 1) The decay widths of $H^0 \to f\bar{f},~gg$ differ from those of $h$ only by a factor of $\sin^2\theta_0$, which leads to similar behavior for $H^0$ and $h$. 2) The only free parameter for the mixing between $H^0$ and $h$ is $\lambda_4$, because \[as shown in Eq. (\[mixangles\])\] the impact of $\lambda_{2,3}$ is suppressed by a small $v_{\Delta}$ and a relatively large mass difference between $M_{\Delta}$ and $M_h$ while $\lambda_1$ is fixed by $M_h$. 3) $\lambda_4$ enters the $H^0W^+W^-$ and $H^0ZZ$ couplings and thus affects the decays $H^0\to W^+W^-,~ZZ$. 4) The $H^0hh$ coupling simplifies for $v_{\Delta}\ll v$ such that the only free parameter in the decay $H^0\to hh$ is again $\lambda_4$. As a consequence of these features, we shall choose $\lambda_4$ as a free parameter and vary it in the range $[-1.0,1.0]$, and fix the couplings $\lambda_2=\lambda_3=0.1$ which are involved in loop-induced decays.
![Branching ratios of $H^0\to b\bar{b}$ and $H^0\to t\bar{t}$ as a function of $M_{H^0}$ for various values of $\lambda_4$. \[brh0tt\]](BRH0bb.pdf "fig:"){width="0.43\linewidth"} ![Branching ratios of $H^0\to b\bar{b}$ and $H^0\to t\bar{t}$ as a function of $M_{H^0}$ for various values of $\lambda_4$. \[brh0tt\]](BRH0tt.pdf "fig:"){width="0.45\linewidth"}
We first examine the branching ratios of $H^0\to f\bar{f}$. BR($H^0\to b\bar{b}$) and BR($H^0\to t\bar{t}$) are plotted in Fig. \[brh0tt\] for different mass regions of $H^0$. [^3] It is clear that the variation of BR($H^0\to b\bar{b}$) is more dramatic for $\lambda_4>0$. The maximum of BR($H^0\to b\bar{b}$) appears at $\lambda_4\approx0.5$. Obviously, BR($H^0\to b\bar{b}$) is a nonmonotonic function of $\lambda_4$, while BR($H^0\to t\bar{t}$) monotonically increases with $\lambda_4$. As will be discussed later, this different behavior in the two mass regions is due mainly to a zero in the $H^0ZZ$ coupling.
![Left: Branching ratios of $H^0\to W^+W^-,~ZZ$ as a function of $M_{H^0}$ in the mass region $130$-$300~{{\rm GeV}}$. Right: Branching ratios of $H^0\to hh,~ZZ$ as a function of $M_{H^0}$ in the mass region $200$-$1000~{{\rm GeV}}$. \[brh0WW\]](BRH0WW.pdf "fig:"){width="0.45\linewidth"} ![Left: Branching ratios of $H^0\to W^+W^-,~ZZ$ as a function of $M_{H^0}$ in the mass region $130$-$300~{{\rm GeV}}$. Right: Branching ratios of $H^0\to hh,~ZZ$ as a function of $M_{H^0}$ in the mass region $200$-$1000~{{\rm GeV}}$. \[brh0WW\]](BRH0hh.pdf "fig:"){width="0.45\linewidth"} ![Left: Branching ratios of $H^0\to W^+W^-,~ZZ$ as a function of $M_{H^0}$ in the mass region $130$-$300~{{\rm GeV}}$. Right: Branching ratios of $H^0\to hh,~ZZ$ as a function of $M_{H^0}$ in the mass region $200$-$1000~{{\rm GeV}}$. \[brh0WW\]](BRH0ZZ1.pdf "fig:"){width="0.45\linewidth"} ![Left: Branching ratios of $H^0\to W^+W^-,~ZZ$ as a function of $M_{H^0}$ in the mass region $130$-$300~{{\rm GeV}}$. Right: Branching ratios of $H^0\to hh,~ZZ$ as a function of $M_{H^0}$ in the mass region $200$-$1000~{{\rm GeV}}$. \[brh0WW\]](BRH0ZZ2.pdf "fig:"){width="0.45\linewidth"}
Now we study the bosonic decays $H^0\to W^+W^-,~ZZ,~hh$. In the left panel of Fig. \[brh0WW\], we present the branching ratios of $H^0\to W^+W^-,~ZZ$ in the mass region $130$-$300~{{\rm GeV}}$. For most values of $\lambda_4$, BR($H^0\to W^+W^-$) increases with $M_{H^0}$ when $M_{H^0}<2M_{W}$, and varying $\lambda_4$ for $\lambda_4>0$ changes it considerably. $\lambda_4$ has a strong impact on BR($H^0\to W^+W^-$) in the mass region $2M_{Z}<M_{H^0}<2M_{h}$ where the decay channel dominates overwhelmingly for $\lambda_4<0$ but becomes negligible for $\lambda_4$ approaching about $0.5$. However, once the $H^0\to hh$ channel is opened, $H^0\to W^+W^-$ is suppressed significantly independent of $\lambda_4$. The decay $H^0\to ZZ$ cannot dominate when $M_{H^0}<2M_{W}$. In the mass region $2M_{Z}<M_{H^0}<2M_{h}$, it is complementary with the $W^+W^-$ channel, so their behavior is just opposite. More interestingly, there is a zero point for the $H^0ZZ$ coupling, which is proportional to $(v\sin\theta_0-4v_{\Delta}\cos\theta_0)$. According to Eq. (\[mixangles\]), one obtains the corresponding $M_{\Delta}$ at the zero: $$M_{\Delta}^0(ZZ)=\sqrt{2M_h^2-\frac{1}{2}(\lambda_4+\lambda_5)v^2}.$$ Note that the above relation only holds for $\lambda_4+\lambda_5<2M_h^2/v^2\approx0.5$, since we are working in the scenario where $M_{\Delta}>M_h$. The existence of the zero coupling explains the presence of the nodes in BR($H^0\to ZZ$) for $\lambda_4\leq0$.
![Branching ratios of $H^0\to \gamma\gamma,~Z\gamma$ as a function of $M_{H^0}$ for various sets of $\lambda_{2,4}$ values. \[brh0AA\]](BRH0AA.pdf "fig:"){width="0.45\linewidth"} ![Branching ratios of $H^0\to \gamma\gamma,~Z\gamma$ as a function of $M_{H^0}$ for various sets of $\lambda_{2,4}$ values. \[brh0AA\]](BRH0ZA.pdf "fig:"){width="0.44\linewidth"}
In the right panel of Fig. \[brh0WW\], BR($H^0\to hh,ZZ$) are shown in the mass region $200$-$1000~{{\rm GeV}}$. When $M_{H^0}>2M_{h}$, the dependence on $\lambda_4$ is simple: a larger $\lambda_4$ corresponds to a smaller BR($H^0\to hh$) and a larger BR($H^0\to ZZ$). It is clear that $\lambda_4$ has a more significant impact in the mass region $200\sim350~{{\rm GeV}}$, and varying $\lambda_4$ could change BR($H^0\to ZZ$) from $0$ to $0.9$. Once $M_{H^0}$ exceeds $2M_t$, the evolution of Br($H^0\to hh,ZZ$) becomes smooth with the increase of $M_{H^0}$. There also exists a zero point for the $H^0hh$ coupling, which can be obtained as for the $ZZ$ channel: $$M_{\Delta}^0(hh)=\sqrt{2(\lambda_4+\lambda_5)v^2-2M_h^2}~,$$ which is valid for $\lambda_4+\lambda_5>3M_h^2/2v^2\approx0.375$.
Finally, we investigate the loop-induced decays, $H^0\to \gamma\gamma,~Z\gamma$. In addition to the usual contributions from the top quark and $W$ boson, the new charged scalars $H^{\pm}$ and $H^{\pm\pm}$ also contribute to the decays. These new terms involve the $H^0H^+H^-$ and $H^0H^{++}H^{--}$ couplings, which are proportional to $$\begin{aligned}
H^0H^+H^-&:& [(2\lambda_2+2\lambda_3-\lambda_5)\sin\alpha\cos\theta_0-(2\lambda_4+\lambda_5)\cos\alpha\sin\theta_0],
\nonumber
\\
H^0H^{++}H^{--}&:&(\lambda_2\sin\alpha\cos\theta_0-\lambda_4\cos\alpha\sin\theta_0).\end{aligned}$$ One therefore has to consider the scalar self-couplings $\lambda_{2,3}$. For simplicity, we set $\lambda_2=\lambda_3$ and vary them from $-3.0$ to $3.0$. In Fig. \[brh0AA\], we display BR($H^0\to \gamma\gamma$) and BR($H^0\to Z\gamma$) versus $M_{H^0}$ for some typical sets of $\lambda_{2,4}$ values. The evolution of both branching ratios crosses 3 orders of magnitude in this parameter region. The resulting enhancement compared with $h\to\gamma\gamma$ in the SM looks significant: the maximal enhancement can be achieved at the level of $9\%$ for the $H^0\to \gamma\gamma$ channel at $M_{H^0}=130~{{\rm GeV}}$, and of $0.7\%$ for the $H^0\to Z\gamma$ channel at $M_{H^0}\approx140~{{\rm GeV}}$.
$A^0$ decays
------------
![Branching ratios of $A^0\to b\bar{b},~t\bar{t}$ as a function of $M_{A^0}$ for various values of $\lambda_4$. \[bra0tt\]](BRA0bb.pdf "fig:"){width="0.43\linewidth"} ![Branching ratios of $A^0\to b\bar{b},~t\bar{t}$ as a function of $M_{A^0}$ for various values of $\lambda_4$. \[bra0tt\]](BRA0tt.pdf "fig:"){width="0.45\linewidth"}
Similar to $H^0$, the decay widths of $A^0 \to f\bar{f},~gg$ differ from those of $h$ by a factor of $\sin^2\alpha$ with $\alpha$ being given in Eq. (\[mixangles\]). Moreover, the only vertex which involves $\lambda_i$ is the $A^0Zh$ coupling proportional to $(\cos\theta_0\sin\alpha-2\sin\theta_0\cos\alpha)$. As a consequence, one can only choose $\lambda_4$ as a free parameter to illustrate the influence of scalar interactions. In this section, we also vary $\lambda_4$ from $-1.0$ to $1.0$ and take the same benchmark values for $v_{\Delta}$ and $\Delta M$ as for the $H^0$ decays.
In the left panel of Fig. \[bra0tt\], we present BR($A^0\to b\bar{b}$) as a function of $M_{A^0}$. [^4] For a fixed value of $\lambda_4$, BR($A^0\to b\bar{b}$) decreases as $M_{A^0}$ increases. The dependence of BR($A^0\to b\bar{b}$) on $\lambda_4$ is simple: The larger $\lambda_4$ is, the larger BR($A^0\to b\bar{b}$) is. And BR($A^0\to b\bar{b}$) can be dominant with $\lambda_4=1.0$ as long as $A^0\to Zh$ is not fully opened. The right panel of Fig. \[bra0tt\] shows BR($A^0\to t\bar{t}$), which is very similar to BR($H^0\to t\bar{t}$).
![Branching ratios of $A^0\to Zh$ as a function of $M_{A^0}$ for various values of $\lambda_4$.. \[bra0zh\]](BRA0Zh1.pdf "fig:"){width="0.44\linewidth"} ![Branching ratios of $A^0\to Zh$ as a function of $M_{A^0}$ for various values of $\lambda_4$.. \[bra0zh\]](BRA0Zh2.pdf "fig:"){width="0.45\linewidth"}
![Branching ratios of $A^0\to \gamma\gamma$ and $A^0\to Z\gamma$ as a function of $M_{A^0}$ for various values of $\lambda_4$. \[bra0aa\]](BRA0AA.pdf "fig:"){width="0.43\linewidth"} ![Branching ratios of $A^0\to \gamma\gamma$ and $A^0\to Z\gamma$ as a function of $M_{A^0}$ for various values of $\lambda_4$. \[bra0aa\]](BRA0ZA.pdf "fig:"){width="0.45\linewidth"}
We then study the most important decay $A^0\to Zh$. In Fig. \[bra0zh\], we present BR($A^0\to Zh$) as a function of $M_{A^0}$ in the low-mass region ($130$-$300~{{\rm GeV}}$) and high-mass region ($300$-$1000~{{\rm GeV}}$), respectively. The evolution of BR($A^0\to Zh$) with $M_{A^0}$ and $\lambda_4$ is just opposite to that of $A^0\to b\bar{b}~(t\bar t)$ in the low- (high-) mass region. The variation of BR($A^0\to Zh$) with $\lambda_4$ is dramatic below the $Zh$ threshold. In particular, near the $Zh$ threshold BR($A^0\to Zh)\sim 1.0$ for $\lambda_4=-1.0$, while BR($A^0\to Zh$) tends to vanish for $\lambda_4=1.0$, which corresponds to the zero point of the $A^0Zh$ coupling: $$M_{\Delta}^0(Zh)=\sqrt{(\lambda_4+\lambda_5)v^2-M_h^2},$$ with $\lambda_4+\lambda_5>2M_h^2/v^2\approx0.5$. BR($A^0\to Zh$) is totally dominant in the mass region between the $Zh$ and $t\bar t$ thresholds, and becomes comparable to BR($A^0\to t\bar{t}$) when $M_{A^0}>2M_t$.
At last, we study the one-loop-induced decays, $A^0\to \gamma\gamma,~Z\gamma$. These two channels can only be induced by the top quark in the loop since the $A^0W^+W^-$, $A^0H^+H^-$, and $A^0H^{++}H^{--}$ couplings are absent in the $CP$-conserving case. In Fig. \[bra0aa\], both BR($A^0\to \gamma\gamma$) and BR($A^0\to Z\gamma$) are displayed. For $M_{A^0}$ below the $Zh$ threshold, the variation in $\lambda_4$ of BR($A^0\to \gamma\gamma$) increases as $M_{A^0}$ increases. BR($A^0\to\gamma\gamma$) could reach $9\times10^{-4}$ for $M_{A^0}\approx 210~{{\rm GeV}}$ and $\lambda_4=1.0$, which is much smaller than the maximum of BR($H^0\to\gamma\gamma$). The variation in $\lambda_4$ of BR($A^0\to Z\gamma$) is slightly steeper, with a maximum of $1.2\times10^{-4}$ at $M_{A^0}\approx 215~{{\rm GeV}}$ and $\lambda_4=1.0$.
![Branching ratios of $H^0/A^0$ as a function of $M_{H^0/A^0}$ at the benchmark point in Eq. (\[BP\]). \[brh0\]](BRH0.pdf "fig:"){width="0.45\linewidth"} ![Branching ratios of $H^0/A^0$ as a function of $M_{H^0/A^0}$ at the benchmark point in Eq. (\[BP\]). \[brh0\]](BRA0.pdf "fig:"){width="0.45\linewidth"}
In the above, we have discussed the decay channels of $H^0$ and $A^0$ separately. We have shown that the scalar self-interactions have a large impact on their branching ratios. In Sec. \[signal\], we will explore their LHC signatures. For this purpose, we choose the following benchmark values: $$\label{BP}
v_{\Delta}=0.001~{{\rm GeV}}, ~\Delta M=5~{{\rm GeV}}, ~\lambda_2=\lambda_3=0.1, ~\lambda_4=0.25.$$ The reason that we set relatively small values of $v_{\Delta}$ and $\Delta M$ is to obtain large cascade decays of charged scalars as well as a large enhancement of neutral scalar production. In Fig. \[brh0\], we display all relevant branching ratios versus $M_{H^0/A^0}$ for this benchmark model, which is to be simulated in Sec. \[signal\] for the LHC in the $b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, and $b\bar{b}W^+W^-$ signal channels.
Production of Neutral Higgs from Cascade Decays {#Eh}
===============================================
We pointed out in Ref. [@Han:2015hba] the importance of the associated $H^0A^0$ production in the nondegenerate case. To estimate the number of signal events, we simulated the signal channel $b\bar{b}\tau^+\tau^-$ at $M_{H^0/A^0}=130~{{\rm GeV}}$. We found that, with a much higher production cross section than the SM Higgs pair ($hh$) production, a $2.9\sigma$ excess in that signal channel is achievable for LHC14@300. In the present work, we are interested in the observability of the associated $H^0A^0$ production in the nondecoupling mass regime $(130$-$200~{{\rm GeV}})$. In Fig. \[cs\] we first show the production cross sections for a pair of various scalars at LHC14 versus $M_{\Delta}$ with a degenerate spectrum. As before, we incorporate the next-to-leading-order (NLO) QCD effects by multiplying a $K$-factor of $1.3$ in all $q\bar{q}$ production channels [@Dawson:1998py]. The $hh$ production through gluon-gluon fusion at NLO ($33~{\textrm{fb}}$) is also indicated (black dashed line) for comparison. One can see that the cross section for $H^0A^0$ is about $20$-$500~{\textrm{fb}}$ in the mass region $130$-$300~{{\rm GeV}}$, which is much larger than the $hh$ production for most of the mass region and thus leads to great discovery potential.
![Production cross sections for a pair of scalars at LHC14 versus $M_{\Delta}$ for a degenerate spectrum. The black dashed line is for the SM $hh$ production. \[cs\]](CS1.pdf "fig:"){width="0.45\linewidth"} ![Production cross sections for a pair of scalars at LHC14 versus $M_{\Delta}$ for a degenerate spectrum. The black dashed line is for the SM $hh$ production. \[cs\]](CS2.pdf "fig:"){width="0.45\linewidth"}
In general, the new scalars are nondegenerate for a nonzero $\lambda_5$. In the positive scenario where $H^{\pm\pm}$ are the lightest, the cascade decays of $H^\pm$ and $H^0/A^0$ can strengthen the observability of $H^{\pm\pm}$ [@Akeroyd:2011zza; @Chun:2013vma]. For the same reason, in the negative scenario where $H^0/A^0$ are the lightest, the charged scalars contribute instead to the production of $H^0/A^0$ through the cascade decays like $H^{\pm}\to H^0/A^0W^*$. In this work, we study these contributions in the same way as was done for the positive scenario in Refs. [@Akeroyd:2011zza; @Chun:2013vma].
We define the reference cross section $X_0$ for the standard Drell-Yan process $$X_0=\sigma(pp\to Z^*\to H^0A^0),$$ which is independent of the cascade decay parameters $v_{\Delta}$ and $\Delta M$. A detailed study on the $b\bar{b}\tau^+\tau^-$ signal for this process with $M_{\Delta}=130~{{\rm GeV}}$ can be found in Ref. [@Han:2015hba]. Besides the above direct production, neutral scalars can also be produced from cascade decays of charged scalars. These extra production channels include $H^\pm H^0/A^0$, $H^+H^-$, $H^{\pm}H^{\mp\mp}$, and $H^{++}H^{--}$ followed by cascade decays of charged scalars. We consider first the associated $H^\pm H^0/A^0$ production followed by cascade decays of $H^\pm$, $$\begin{aligned}
\nonumber
pp\to W^*\to H^{\pm}H^0 \to H^0H^0 W^* &,&~~
pp\to W^*\to H^{\pm}H^0 \to A^0H^0 W^*,\\
pp\to W^*\to H^{\pm}A^0 \to H^0A^0 W^* &,&~~
pp\to W^*\to H^{\pm}A^0 \to A^0A^0 W^*,\end{aligned}$$ resulting in three final states classified by a pair of neutral scalars: $A^0H^0$, $H^0H^0$, and $A^0A^0$. Noting that the last two originate only from cascade decays, any detection of such production channels would be a hint of charged scalars being involved. Using the fact that $$\begin{aligned}
\sigma(pp\to W^*\to H^\pm H^0)&\simeq&\sigma(pp\to W^*\to H^\pm A^0),
\\
\mbox{BR}(H^\pm\to H^0 W^*)&\simeq&\mbox{BR}(H^\pm\to A^0 W^*),\end{aligned}$$ as well as the narrow width approximation, we calculate the production cross sections for these three final states: $$\begin{aligned}
H^0A^0:X_1&=&2[\sigma(pp\to W^+\to H^+ H^0)+\sigma(pp\to W^-\to H^- H^0)]\times \mbox{BR}(H^\pm \to A^0 W^*),
\\
H^0H^0:Y_1&=&[\sigma(pp\to W^+\to H^+ H^0)+\sigma(pp\to W^-\to H^- H^0)]\times \mbox{BR}(H^\pm \to H^0 W^*),
\\
A^0A^0:Z_1&=&[\sigma(pp\to W^+\to H^+ A^0)+\sigma(pp\to W^-\to H^- A^0)]\times \mbox{BR}(H^\pm \to A^0 W^*).\end{aligned}$$ The factor 2 in $X_1$ accounts for the equal contribution from the process with $H^0$ and $A^0$ interchanged. The relations $X_1=2Y_1=2Z_1$ actually hold true for all of the four production channels, since for a given channel the same branching ratios (such as for $H^{\pm}\to H^0/A^0W^*$) are involved, $$\label{XXX}
X_i=2Y_i=2Z_i,~(i=1,2,3,4),$$ where $X_i,~Y_i$, and $Z_i$ refer to the cross sections for $H^0A^0$, $H^0H^0$, and $A^0A^0$ production with the subscript $i=1,2,3,4$ denoting the production channels $H^\pm H^0/A^0$, $H^{+}H^{-}$, $H^{\pm}H^{\mp\mp}$, and $H^{++}H^{--}$, respectively. The relations imply that we may concentrate on the cross section of $H^0A^0$ production.
Naively, one would expect the next important channel to be $H^+H^-$ since it only involves two cascade decays: $$X_2=2\sigma(pp\to \gamma^*/Z^*\to H^+H^-)\times \mbox{BR}(H^{\pm}\to H^0 W^*)\mbox{BR}(H^{\pm}\to A^0 W^*).$$ But as already mentioned in Ref. [@Akeroyd:2011zza], a smaller coupling and destructive interference between the $\gamma^*$ and $Z^*$ exchange make the cross section of $H^+H^-$ production an order of magnitude smaller than that of $H^0A^0$ even for a degenerate spectrum. Considering further suppression due to cascade decays, $X_2$ is not important for the enhancement of $H^0A^0$ production and can be safely neglected in the numerical analysis.
![Production cross sections for a pair of neutral scalars versus $M_{\Delta}$ at LHC14 and with $\Delta M=5~{{\rm GeV}}$, $v_{\Delta}=0.001~{{\rm GeV}}$. Left: The red solid (dashed) line corresponds to $X_0$ ($X$). Right: The red line corresponds to $H^0H^0/A^0A^0$ from cascade decays $Y/Z$, and the green line to $H^0A^0$ from cascade decays ($X_C$). The shaded regions are filled by scanning over $\Delta M$ and $v_\Delta$. \[csx\]](X.pdf "fig:"){width="0.45\linewidth"} ![Production cross sections for a pair of neutral scalars versus $M_{\Delta}$ at LHC14 and with $\Delta M=5~{{\rm GeV}}$, $v_{\Delta}=0.001~{{\rm GeV}}$. Left: The red solid (dashed) line corresponds to $X_0$ ($X$). Right: The red line corresponds to $H^0H^0/A^0A^0$ from cascade decays $Y/Z$, and the green line to $H^0A^0$ from cascade decays ($X_C$). The shaded regions are filled by scanning over $\Delta M$ and $v_\Delta$. \[csx\]](Xe.pdf "fig:"){width="0.45\linewidth"}
The contribution from $H^{\pm}H^{\mp\mp}$ is more important despite the fact that it involves three cascade decays: $$\begin{aligned}
X_3&=& 2[\sigma(pp\to W^{-*}\to H^+H^{--})+\sigma(pp\to W^{+*}\to H^-H^{++})]\times\\ \nonumber
&&\mbox{BR}(H^{\pm\pm}\to H^\pm W^*)\mbox{BR}(H^{\pm}\to H^0 W^*)\mbox{BR}(H^{\pm}\to A^0 W^*).\end{aligned}$$ As shown in Fig. \[cs\], $\sigma(pp\to W^*\to H^\pm H^{\mp\mp})$ is the largest for a degenerate mass spectrum. When cascade decays are dominant, the phase-space suppression of heavy charged scalars will be important. So we expect that the $H^0A^0$ production receives considerable enhancement from $H^{\pm}H^{\mp\mp}$ when the mass splitting is small and cascade decays are dominant.
Finally, the last mechanism is $H^{++}H^{--}$, which involves four cascade decays: $$\begin{aligned}
\nonumber
X_4 &=&2\sigma(pp\to \gamma^*/Z^* \to H^{++}H^{--})\times\mbox{BR}(H^{\pm\pm}\to H^\pm W^*)^2\\
&&\times\mbox{BR}(H^{\pm}\to H^0 W^*)\mbox{BR}(H^{\pm}\to A^0 W^*)~.\end{aligned}$$ This mechanism is also promising since the cross section of $H^{++}H^{--}$ production is slightly larger than $H^{0}A^0$ production for a degenerate mass spectrum. The phase-space suppression of $X_4$ is more severe than that of $X_3$, because a pair of the heaviest $H^{\pm\pm}$ are produced.
Summing over all four of the above channels yields the contribution to the $H^0A^0$ production from cascade decays, $$X_C=X_1+X_2+X_3+X_4,$$ and the total production cross section of $H^0A^0$ is then $X=X_0+X_C$. Using Eq. (\[XXX\]), the total cross sections for the pair production $H^0H^0/A^0A^0$, $Y=\sum_iY_i$, $Z=\sum_iZ_i$, are given by $$Y=Z=\frac{1}{2}X_C.$$ Since the enhancement from cascade decays depends on a not severely suppressed phase space and a larger branching ratio of cascade decays, we choose to work with a relatively smaller mass splitting and triplet vev as shown in Eq. (\[BP\]). Figure \[csx\] displays the cross sections of the $H^0A^0$, $H^0H^0$, and $A^0A^0$ production as a function of $M_{\Delta}$. As can be seen from the figure, the production of $H^0A^0$ can be enhanced by a factor of 3, while the $H^0H^0/A^0A^0$ production at the maximal enhancement can reach the level of $X_0$. This could make the detection of neutral scalar pair productions very promising in the negative scenario.
LHC Signatures of Neutral Scalar Production {#signal}
===========================================
In this section we investigate the signatures of neutral scalar production at the LHC. From previous studies on the SM $hh$ production, we already know that the most promising signal is $b\bar{b}\gamma\gamma$, and $b\bar{b}\tau^+\tau^-$ is next to it, while both semileptonic and dileptonic decays of $W$’s in the $b\bar{b}W^+W^-$ channel are challenging. In this work we analyze all three of the signals—$b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, and $b\bar{b}W^+W^-\to b\bar{b}\ell^+\ell^-2\nu$ ($\ell=e,~\mu$ for collider identification)—as well as their backgrounds based on the benchmark model presented in Eq. (\[BP\]).
In Sec. \[Eh\] we discussed the Drell-Yan production of $H^0A^0$ and the enhanced pair and associated production of neutral scalars $H^0/A^0$ due to cascade decays of charged scalars $H^\pm,~H^{\pm\pm}$. We are now ready to incorporate the branching ratios of $H^0/A^0$ decays for a specific signal channel. For instance, the cross sections for the $b\bar{b}\gamma\gamma$ signal channel can be written as $$\begin{aligned}
\label{S0}
S_0(b\bar{b}\gamma\gamma)& = & X_0\times\left[\mbox{BR}(H^0\to b\bar{b})\mbox{BR}(A^0\to\gamma\gamma)
+\mbox{BR}(H^0\to\gamma\gamma)\mbox{BR}(A^0\to b\bar{b})\right],\\
S(b\bar{b}\gamma\gamma) & = & X\times\left[\mbox{BR}(H^0\to b\bar{b})\mbox{BR}(A^0\to\gamma\gamma)
+\mbox{BR}(H^0\to\gamma\gamma)\mbox{BR}(A^0\to b\bar{b})\right]\\\nonumber
&&+2Y\hspace{-0.25em}\times\mbox{BR}(H^0\to b\bar{b})\mbox{BR}(H^0\to\gamma\gamma)
+2Z\hspace{-0.35em}\times\mbox{BR}(A^0\to b\bar{b})\mbox{BR}(A^0\to\gamma\gamma).\end{aligned}$$ Here $S_0$ denotes the signal from the direct production $pp\to Z^*\to H^0A^0$ alone, and $S$ includes contributions from cascade decays. $S_{(0)}(b\bar{b}\tau^+\tau^-)$ has a similar expression as $S_{(0)}(b\bar{b}\gamma\gamma)$, while $S_{(0)}(b\bar{b}\ell^+\ell^-2\nu)$ is simpler since the decay mode $A^0\to W^+W^-$ is absent.
![Theoretical cross sections of $b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, and $b\bar{b}\ell^+\ell^-2\nu$ signal channels at LHC14. The red solid (dashed) line corresponds to the signal from $X_0$ ($X$), the green (blue) solid line corresponds to the signal from $Y~(Z)$, and the purple dashed line shows the total cross section $S$ for the signal. The SM $hh$ cross section is shown for comparison. The lower right panel shows the enhancement factor $S/S_0$ in the three signal channels. \[sgn\]](CSbbaa.pdf "fig:"){width="0.45\linewidth"} ![Theoretical cross sections of $b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, and $b\bar{b}\ell^+\ell^-2\nu$ signal channels at LHC14. The red solid (dashed) line corresponds to the signal from $X_0$ ($X$), the green (blue) solid line corresponds to the signal from $Y~(Z)$, and the purple dashed line shows the total cross section $S$ for the signal. The SM $hh$ cross section is shown for comparison. The lower right panel shows the enhancement factor $S/S_0$ in the three signal channels. \[sgn\]](CSbbtata.pdf "fig:"){width="0.44\linewidth"} ![Theoretical cross sections of $b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, and $b\bar{b}\ell^+\ell^-2\nu$ signal channels at LHC14. The red solid (dashed) line corresponds to the signal from $X_0$ ($X$), the green (blue) solid line corresponds to the signal from $Y~(Z)$, and the purple dashed line shows the total cross section $S$ for the signal. The SM $hh$ cross section is shown for comparison. The lower right panel shows the enhancement factor $S/S_0$ in the three signal channels. \[sgn\]](CSbbll2v.pdf "fig:"){width="0.44\linewidth"} ![Theoretical cross sections of $b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, and $b\bar{b}\ell^+\ell^-2\nu$ signal channels at LHC14. The red solid (dashed) line corresponds to the signal from $X_0$ ($X$), the green (blue) solid line corresponds to the signal from $Y~(Z)$, and the purple dashed line shows the total cross section $S$ for the signal. The SM $hh$ cross section is shown for comparison. The lower right panel shows the enhancement factor $S/S_0$ in the three signal channels. \[sgn\]](SoverS0.pdf "fig:"){width="0.44\linewidth"}
The theoretical cross sections for the $b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, and $b\bar{b}\ell^+\ell^-2\nu$ signal channels are plotted in Fig. \[sgn\]. The cross section $S_0(b\bar{b}\gamma\gamma)/S_0(b\bar{b}\tau^+\tau^-)$ is larger than that of the SM $hh$ production until $M_{\Delta}=159/161~{{\rm GeV}}$; taking into account cascade enhancement pushes the corresponding $M_{\Delta}$ further to $179/197~{{\rm GeV}}$. $S_{0}(b\bar{b}\ell^+\ell^-2\nu)$ is always larger than that of $hh$ in the mass region $130$-$200~{{\rm GeV}}$, and interestingly, it keeps about the same value when $M_{\Delta}<160~{{\rm GeV}}$. The signal from $H^0H^0$ is comparable with $S_0$ in these three channels only for $M_{\Delta}<160~{{\rm GeV}}$, while in contrast the signal from $A^0A^0$ becomes dominant for the $b\bar{b}\gamma\gamma$ and $b\bar{b}\tau^+\tau^-$ channels when $M_{\Delta}>160~{{\rm GeV}}$. Therefore, we have a chance to probe the $A^0A^0$ pair production in these two channels. Also shown in Fig. \[sgn\] is the enhancement factor $S/S_0$ for the three signal channels at the benchmark point (\[BP\]) as a function of $M_\Delta$, which will help us understand the simulation results.
$b\bar{b}\gamma\gamma$ signal channel {#sec:bbgg}
-------------------------------------
In our simulation, the parton-level signal and background events are generated with [**MADGRAPH5**]{} [@MG5]. We perform parton shower and fast detector simulations with [**PYTHIA**]{} [@Sjostrand:2006za] and [**DELPHES3**]{} [@Delphes]. Finally, [**MADANALYSIS5**]{} [@Conte:2012fm] is responsible for data analysis and plotting. We take a flat $b$-tagging efficiency of 70%, and mistagging rates of 10% for $c$ jets and 1% for light-flavor jets, respectively. Jet reconstruction is done using the anti-$k_T$ algorithm with a radius parameter of $R=0.5$. We further assume a photon identification efficiency of $85\%$ and a jet-faking-photon rate of $1.2\times 10^{-4}$ [@Aad:2009wy].
The main SM backgrounds to the signal are as follows: $$\begin{aligned}
b\bar{b}\gamma\gamma: p p &\to& b\bar{b}\gamma\gamma,\\
t\bar{t}h: p p &\to& t\bar{t}h \to b\ell^+\nu~\bar{b}\ell^-\nu~\gamma\gamma~(\ell^\pm~\mbox{missed}),\\
Zh: pp &\to& Zh \to b\bar{b} \gamma\gamma.\end{aligned}$$ Among them, $b\bar{b}\gamma\gamma$ and $Zh$ are irreducible, while $t\bar{t}h$ is reducible and can be reduced by vetoing the additional $\ell$’s with $p_T^{\ell}>20~{{\rm GeV}}$ and $|\eta_\ell|<2.4$. In addition, there exist many reducible sources of fake $b\bar{b}\gamma\gamma$: $$\begin{aligned}
\nonumber
pp\to b\bar{b}jj\nrightarrow b\bar{b}\gamma\gamma,pp\to b\bar{b}j\gamma \nrightarrow b\bar{b}\gamma\gamma, \ldots \\
pp\to c\bar{c}\gamma\gamma\nrightarrow b\bar{b}\gamma\gamma,pp\to j\bar{j}\gamma\gamma\nrightarrow b\bar{b}\gamma\gamma, \ldots,\end{aligned}$$ where $x\nrightarrow y$ stands for a final-state $x$ misidentified as $y$. The remaining fake sources are subdominant and are thus not included in our simulation. The QCD corrections to the backgrounds are included by a multiplicative $K$-factor of 1.10 and 1.33 for the leading cross sections of $t\bar{t}h$ and $Zh$ at LHC14 [@Dittmaier:2011ti], respectively. The cross section of the $b\bar{b}\gamma\gamma$ background has been normalized to include fake sources and does not take NLO corrections into account.
![Distributions of $p_T^{b,\gamma},~\Delta R_{bb,\gamma\gamma},~M_{bb,\gamma\gamma,H^0A^0}$, and $E_T$ for the signal $b\bar{b}\gamma\gamma$ and its backgrounds before applying any cuts at LHC14. \[fig:bbaa\]](bbaa_PTb.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^{b,\gamma},~\Delta R_{bb,\gamma\gamma},~M_{bb,\gamma\gamma,H^0A^0}$, and $E_T$ for the signal $b\bar{b}\gamma\gamma$ and its backgrounds before applying any cuts at LHC14. \[fig:bbaa\]](bbaa_PTa.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^{b,\gamma},~\Delta R_{bb,\gamma\gamma},~M_{bb,\gamma\gamma,H^0A^0}$, and $E_T$ for the signal $b\bar{b}\gamma\gamma$ and its backgrounds before applying any cuts at LHC14. \[fig:bbaa\]](bbaa_DRbb.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^{b,\gamma},~\Delta R_{bb,\gamma\gamma},~M_{bb,\gamma\gamma,H^0A^0}$, and $E_T$ for the signal $b\bar{b}\gamma\gamma$ and its backgrounds before applying any cuts at LHC14. \[fig:bbaa\]](bbaa_DRaa.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^{b,\gamma},~\Delta R_{bb,\gamma\gamma},~M_{bb,\gamma\gamma,H^0A^0}$, and $E_T$ for the signal $b\bar{b}\gamma\gamma$ and its backgrounds before applying any cuts at LHC14. \[fig:bbaa\]](bbaa_Mbb.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^{b,\gamma},~\Delta R_{bb,\gamma\gamma},~M_{bb,\gamma\gamma,H^0A^0}$, and $E_T$ for the signal $b\bar{b}\gamma\gamma$ and its backgrounds before applying any cuts at LHC14. \[fig:bbaa\]](bbaa_Maa.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^{b,\gamma},~\Delta R_{bb,\gamma\gamma},~M_{bb,\gamma\gamma,H^0A^0}$, and $E_T$ for the signal $b\bar{b}\gamma\gamma$ and its backgrounds before applying any cuts at LHC14. \[fig:bbaa\]](bbaa_Mh0a0.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^{b,\gamma},~\Delta R_{bb,\gamma\gamma},~M_{bb,\gamma\gamma,H^0A^0}$, and $E_T$ for the signal $b\bar{b}\gamma\gamma$ and its backgrounds before applying any cuts at LHC14. \[fig:bbaa\]](bbaa_Et.pdf "fig:"){width="0.45\linewidth"}
The distributions of some kinematical variables before applying any cuts are shown in Fig. \[fig:bbaa\], where we assume $M_{\Delta}=130,~160,~190~{{\rm GeV}}$. In our analysis, we require that the final states include exactly one $b$-jet pair and one $\gamma$ pair and satisfy the following basic cuts: $$\begin{aligned}
p_T^{b,\gamma}>30~{{\rm GeV}},~|\eta_{b,\gamma}|<2.4,~\Delta R_{bb,\gamma\gamma,b\gamma}>0.4,\end{aligned}$$ where $\Delta R=\sqrt{(\Delta \phi)^2+(\Delta \eta)^2}$ is the particle separation, with $\Delta \phi$ and $\Delta \eta$ being the separation in the azimuthal angle and rapidity, respectively. Here we employ a tighter $p_T$ cut than is usually applied to suppress the QCD-electroweak $b\bar{b}\gamma\gamma$ background. The $b$-jet pair and $\gamma$ pair are then required to fall in the following windows on the invariant masses and fulfill the $\Delta R$ cut criteria: $$\begin{aligned}
\Delta R_{bb}<2.5,&&~|M_{bb}-M_{\Delta}|<15~{{\rm GeV}},\\
\nonumber
\Delta R_{\gamma\gamma}<2.5,&&~|M_{\gamma\gamma}-M_{\Delta}|<10~{{\rm GeV}}.\end{aligned}$$
\[!htbp\]
$M_{\Delta}=130~{{\rm GeV}}$ $H^0A^0(S_0)$ $b\bar{b}\gamma\gamma$ $t\bar{t}h$ $Zh$ $S/B$ $\mathcal{S}(S,B)$
------------------------------------ --------------------- ------------------------ --------------------- --------------------- --------------------- ---------------------
Cross section at NLO $8.01\times10^{-1}$ $5.92\times10^3$ 1.18 $2.99\times10^{-1}$ $1.39\times10^{-4}$ $5.75\times10^{-1}$
Basic cuts $1.22\times10^{-1}$ $4.16\times10^{1}$ $1.03\times10^{-1}$ $3.41\times10^{-2}$ $2.92\times10^{-3}$ 1.03
Reconstruct scalars from $b$s $6.99\times10^{-2}$ 7.07 $1.50\times10^{-2}$ $9.61\times10^{-4}$ $9.87\times10^{-3}$ 1.44
Reconstruct scalars from $\gamma$s $5.28\times10^{-2}$ $1.03\times10^{-1}$ $1.08\times10^{-2}$ $7.32\times10^{-4}$ $4.63\times10^{-1}$ 8.01
Cut on $M_{H^0A^0}$ $4.21\times10^{-2}$ $2.04\times10^{-2}$ $4.69\times10^{-3}$ $3.23\times10^{-4}$ 1.65 $12.0$
Cut on $E_T$ $3.31\times10^{-2}$ $6.58\times10^{-3}$ $4.68\times10^{-3}$ $2.27\times10^{-4}$ 2.88 $12.8$
Cascade enhanced $1.51\times10^{-1}$ $-$ $-$ $-$ $13.1$ $41.0$
$M_{\Delta}=160~{{\rm GeV}}$ $H^0A^0(S_0)$ $b\bar{b}\gamma\gamma$ $t\bar{t}h$ $Zh$ $S/B$ $\mathcal{S}(S,B)$
Cross section at NLO $5.10\times10^{-2}$ $5.92\times10^3$ 1.18 $2.99\times10^{-1}$ $8.61\times10^{-6}$ $3.63\times10^{-2}$
Basic cuts $8.78\times10^{-3}$ $4.16\times10^{1}$ $1.03\times10^{-1}$ $3.41\times10^{-2}$ $2.10\times10^{-4}$ $7.44\times10^{-2}$
Reconstruct scalars from $b$s $4.11\times10^{-3}$ $5.06$ $1.34\times10^{-2}$ $2.36\times10^{-4}$ $8.11\times10^{-4}$ $9.99\times10^{-2}$
Reconstruct scalars from $\gamma$s $3.27\times10^{-3}$ $3.42\times10^{-2}$ $1.57\times10^{-5}$ 0.00 $9.56\times10^{-2}$ $9.53\times10^{-1}$
Cut on $M_{H^0A^0}$ $2.57\times10^{-3}$ $1.12\times10^{-2}$ $1.18\times10^{-5}$ 0.00 $2.30\times10^{-1}$ 1.28
Cut on $E_T$ $1.73\times10^{-3}$ $3.95\times10^{-3}$ $1.03\times10^{-5}$ 0.00 $4.37\times10^{-1}$ 1.41
Cascade enhanced $1.10\times10^{-2}$ $-$ $-$ $-$ 2.77 7.29
$M_{\Delta}=190~{{\rm GeV}}$ $H^0A^0(S_0)$ $b\bar{b}\gamma\gamma$ $t\bar{t}h$ $Zh$ $S/B$ $\mathcal{S}(S,B)$
Cross section at NLO $2.68\times10^{-3}$ $5.92\times10^3$ 1.18 $2.99\times10^{-1}$ $4.53\times10^{-7}$ $1.91\times10^{-3}$
Basic cuts $5.33\times10^{-4}$ $4.16\times10^{1}$ $1.03\times10^{-1}$ $3.41\times10^{-2}$ $1.28\times10^{-5}$ $4.52\times10^{-3}$
Reconstruct scalars from $b$s $2.27\times10^{-4}$ 3.61 $1.05\times10^{-2}$ $1.24\times10^{-4}$ $6.27\times10^{-5}$ $6.53\times10^{-3}$
Reconstruct scalars from $\gamma$s $1.81\times10^{-4}$ $2.47\times10^{-2}$ $3.93\times10^{-6}$ 0.00 $7.34\times10^{-3}$ $6.30\times10^{-2}$
Cut on $M_{H^0A^0}$ $1.55\times10^{-4}$ $9.87\times10^{-3}$ $3.93\times10^{-6}$ 0.00 $1.57\times10^{-2}$ $8.52\times10^{-2}$
Cut on $E_T$ $8.35\times10^{-5}$ $1.48\times10^{-3}$ $3.93\times10^{-6}$ 0.00 $5.63\times10^{-2}$ $1.18\times10^{-1}$
Cascade enhanced $1.50\times10^{-3}$ $-$ $-$ $-$ 1.01 1.87
: Evolution of signal and background cross sections (in ${\textrm{fb}}$) at LHC14 for the $b\bar{b}\gamma\gamma$ signal channel upon imposing the cuts one by one. For the cascade-enhanced signal only the cross section passing all the cuts is shown. The last two columns assume an integrated luminosity of $3000~{\textrm{fb}}^{-1}$. \[tab:bbaacut\]
As shown in Fig. \[fig:bbaa\], the $\Delta R_{bb,\gamma\gamma}$ distributions of the signal are clearly more compact as they are more likely coming from the same particles. Thus the $\Delta R$ cuts can effectively suppress the background. More specific cuts are necessary for further analysis. A useful variable is the invariant mass of the neutral scalar pair $M_{H^0A^0}$, and the total transverse energy $E_T$ is also distinctive. The peak of $M_{H^0A^0}$ increases with $M_{\Delta}$, and similarly for $E_T$. For simplicity, we adopt for the cuts a linear shift between $M_{H^0A^0},~E_T$ and $M_{\Delta}$: $$M_{H^0A^0}>2M_{\Delta}+90~{{\rm GeV}},~E_T>2M_{\Delta}-60~{{\rm GeV}}.$$ For instance, we apply $M_{H^0A^0}>350~{{\rm GeV}}$, $E_T>200~{{\rm GeV}}$ at the benchmark point $M_{\Delta}=130~{{\rm GeV}}$.
To estimate the observability quantitatively, we adopt the following significance measurement: $$\mathcal{S}(S,B)=\sqrt{2\left((S\cdot\mathcal{L}+B\cdot\mathcal{L})
\log\left(1+\frac{S}{B}\right)-S\cdot\mathcal{L}\right)},$$ which is more suitable than the usual definition of $S/\sqrt{B}$ or $S/\sqrt{S+B}$ for Monte Carlo analysis [@Cowan:2010js]. Here $S$ and $B$ are the signal and background cross sections, and $\mathcal{L}$ is the integrated luminosity. The survival cross sections of the signal from the Drell-Yan process and of the backgrounds upon imposing cuts step by step are summarized in Table \[tab:bbaacut\] at the benchmark point (\[BP\]) for $M_\Delta=130,~160,~190~{{\rm GeV}}$ respectively. For the cascade-enhanced signal, only the cross section passing all the cuts is shown. The last two columns in the table show the signal-to-background ratio $S/B$ and the statistical significance $\mathcal{S}(S,B)$.
For $M_{\Delta}=130~{{\rm GeV}}$, the $b\bar{b}\gamma\gamma$ channel is very promising. Without (with) cascade enhancement, the final significance can reach 12.8 (41) for LHC14@3000, corresponding to 99 (453) events. For $M_{\Delta}=160~{{\rm GeV}}$, the channel becomes challenging since the cross section has decreased by a factor of 15.7 compared with the case of $M_{\Delta}=130~{{\rm GeV}}$. But the cuts we applied are efficient to suppress the SM background, and with cascade enhancement the significance could still reach 7.29 for $3000~{\textrm{fb}}^{-1}$, corresponding to 33 events in the most optimistic case. For $M_{\Delta}=190~{{\rm GeV}}$, it looks hopeless even with maximal cascade enhancement in our benchmark model: to achieve 10 signal events, an integrated luminosity of at least $6670~{\textrm{fb}}^{-1}$ is required, which is beyond the reach of the future LHC.
$b\bar{b}\tau^+\tau^-$ signal channel
-------------------------------------
For this signal channel, an important part of the analysis depends on the ability to reconstruct the $b$ pair and the $\tau$ pair. Here we consider the hadronic decays of the $\tau$ lepton and assume a $\tau$-tagging efficiency of $70\%$ with a negligible fake rate.
![Distributions of $p_T^\tau,~\Delta R_{\tau\tau},~M_{\tau\tau,H^0A^0},~E_T$, and $\cancel{E}_T$ for the signal $b\bar{b}\tau^+\tau^-$ and its backgrounds before applying any cuts at LHC14. \[fig:bbtata\]](bbtata_PTta.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^\tau,~\Delta R_{\tau\tau},~M_{\tau\tau,H^0A^0},~E_T$, and $\cancel{E}_T$ for the signal $b\bar{b}\tau^+\tau^-$ and its backgrounds before applying any cuts at LHC14. \[fig:bbtata\]](bbtata_DRtata.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^\tau,~\Delta R_{\tau\tau},~M_{\tau\tau,H^0A^0},~E_T$, and $\cancel{E}_T$ for the signal $b\bar{b}\tau^+\tau^-$ and its backgrounds before applying any cuts at LHC14. \[fig:bbtata\]](bbtata_Mtata.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^\tau,~\Delta R_{\tau\tau},~M_{\tau\tau,H^0A^0},~E_T$, and $\cancel{E}_T$ for the signal $b\bar{b}\tau^+\tau^-$ and its backgrounds before applying any cuts at LHC14. \[fig:bbtata\]](bbtata_Mh0a0.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^\tau,~\Delta R_{\tau\tau},~M_{\tau\tau,H^0A^0},~E_T$, and $\cancel{E}_T$ for the signal $b\bar{b}\tau^+\tau^-$ and its backgrounds before applying any cuts at LHC14. \[fig:bbtata\]](bbtata_Et.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^\tau,~\Delta R_{\tau\tau},~M_{\tau\tau,H^0A^0},~E_T$, and $\cancel{E}_T$ for the signal $b\bar{b}\tau^+\tau^-$ and its backgrounds before applying any cuts at LHC14. \[fig:bbtata\]](bbtata_MET.pdf "fig:"){width="0.45\linewidth"}
The main SM backgrounds are as follows: $$\begin{aligned}
b\bar{b}\tau^+\tau^- : p p &\to& b\bar{b}Z/\gamma^*/h \to b\bar{b}\tau^+\tau^-,
\\
b\bar{b}W^+W^- : p p &\to &b\bar{b}W^+W^- \to b\bar{b}\tau^+ \nu_{\tau} \tau^- \bar{\nu}_{\tau},
\\
Zh : p p &\to& Zh \to b\bar{b}\tau^+\tau^-.\end{aligned}$$ The irreducible QCD-electroweak background comes from $b\bar{b}\tau^+\tau^-$, where the $\tau$ pair originates from the decays of $Z/\gamma^*/h$. Since the hadronic decays of $\tau$ always contain neutrinos, we also include the SM background $b\bar{b}W^+W^-$, which contributes to the $b\bar{b}\tau^+ \nu_{\tau} \tau^- \bar{\nu}_{\tau}$ final state. The $b\bar{b}W^+W^-$ background mainly originates from $t\bar{t}$ production with subsequent decays $t\to b W$ and $W\to \tau \nu_{\tau}$. Moreover, the associated $Zh$ production gets involved through the subsequent decays $h\to b\bar{b}$ and $Z\to\tau^+\tau^-$ or vice versa. The QCD corrections to the backgrounds are included by a multiplicative $K$-factor of 1.21, 1.35, and 1.33 to the leading cross section of $b\bar{b}\tau^+\tau^-$ [@Campbell:2000bg], $t\bar{t}$ [@tt], and $Zh$ [@Dittmaier:2011ti] at LHC14.
\[!htbp\]
$M_{\Delta}=130~{{\rm GeV}}$ $H^0A^0(S_0)$ $b\bar{b}\tau^+\tau^-$ $b\bar{b}W^+W^-$ $Zh$ $S/B$ $\mathcal{S}(S,B)$
---------------------------------- ---------------------- ------------------------ ---------------------- ----------------------- --------------------- ---------------------
Cross section at NLO $4.31\times10^{1}$ $3.10\times10^{4}$ $7.92\times10^{3}$ $2.21\times10^{1}$ $1.11\times10^{-3}$ $12.0$
Basic cuts $7.75\times10^{-1}$ $4.49\times10^{1}$ $8.97\times10^{1}$ $2.91\times10^{-1}$ $5.74\times10^{-3}$ 3.65
Reconstruct scalars from $\tau$s $5.14\times10^{-1}$ $1.19\times10^{1}$ $3.57\times10^{1}$ $1.06\times10^{-1}$ $1.08\times10^{-2}$ 4.06
Reconstruct scalars from $b$s $2.14\times10^{-1}$ 4.34 $9.44\times10^{-1}$ $2.28\times10^{-2}$ $4.04\times10^{-2}$ 5.06
Cut on $M_{H^0A^0}$ $1.29\times10^{-1}$ $1.96\times10^{-1}$ $1.51\times10^{-1}$ $8.10\times10^{-3}$ $3.64\times10^{-1}$ $11.3$
Cut on $E_T$ $1.03\times10^{-1}$ $9.87\times10^{-2}$ $7.35\times10^{-2}$ $5.89\times10^{-3}$ $5.81\times10^{-1}$ $12.4$
Cascade enhanced $5.27\times10^{-1}$ $-$ $-$ $-$ $2.96$ $51.6$
$M_{\Delta}=160~{{\rm GeV}}$ $H^0A^0(S_0)$ $b\bar{b}\tau^+\tau^-$ $b\bar{b}W^+W^-$ $Zh$ $S/B$ $\mathcal{S}(S,B)$
Cross section at NLO $3.08$ $3.10\times10^{4}$ $7.92\times10^{3}$ $2.21\times10^{1}$ $7.92\times10^{-5}$ $8.55\times10^{-1}$
Basic cuts $6.81\times10^{-2}$ $4.49\times10^{1}$ $8.97\times10^{1}$ $2.91\times10^{-1}$ $5.05\times10^{-4}$ $3.21\times10^{-1}$
Reconstruct scalars from $\tau$s $3.14\times10^{-2}$ $1.52\times10^{1}$ $2.46\times10^{2}$ $3.17\times10^{-2}$ $1.20\times10^{-3}$ $3.36\times10^{-1}$
Reconstruct scalars from $b$s $1.2\times10^{-2}$ 2.47 $1.06\times10^{-1}$ 0.00 $4.80\times10^{-3}$ $4.10\times10^{-1}$
Cut on $M_{H^0A^0}$ $6.99\times10^{-3}$ $1.22\times10^{-1}$ $2.06\times10^{-2}$ 0.00 $4.89\times10^{-2}$ 1.00
Cut on $E_T$ $5.04\times10^{-3}$ $4.72\times10^{-2}$ $5.88\times10^{-3}$ 0.00 $9.48\times10^{-2}$ 1.18
Cascade enhanced $5.11\times10^{-2}$ $-$ $-$ $-$ $9.63\times10^{-1}$ $10.7$
$M_{\Delta}=190~{{\rm GeV}}$ $H^0A^0(S_0)$ $b\bar{b}\tau^+\tau^-$ $b\bar{b}W^+W^-$ $Zh$ $S/B$ $\mathcal{S}(S,B)$
Cross section at NLO $2.47\times10^{-1}$ $3.10\times10^{4}$ $7.92\times10^{3}$ $2.21\times10^{1}$ $6.34\times10^{-6}$ $6.86\times10^{-2}$
Basic cuts $6.54\times10^{-3}$ $4.49\times10^{1}$ $8.97\times10^{1}$ $2.91\times10^{-1}$ $4.86\times10^{-5}$ $3.09\times10^{-2}$
Reconstruct scalars from $\tau$s $2.32\times10^{-3}$ $4.60\times10^{-1}$ $1.47\times10^{1}$ $5.89\times10^{-3}$ $1.53\times10^{-4}$ $3.26\times10^{-2}$
Reconstruct scalars from $b$s $7.66\times10^{-4}$ $2.06\times10^{-2}$ 1.21 0.00 $6.25\times10^{-4}$ $3.78\times10^{-2}$
Cut on $M_{H^0A^0}$ $6.34\times10^{-4}$ $1.47\times10^{-3}$ $9.03\times10^{-2}$ 0.00 $6.91\times10^{-3}$ $1.14\times10^{-1}$
Cut on $E_T$ $3.87\times10^{-4}$ 0.00 $2.64\times10^{-2}$ 0.00 $1.47\times10^{-2}$ $1.30\times10^{-1}$
Cascade enhanced $6.85\times10^{-3}$ $-$ $-$ $-$ $2.60\times10^{-1}$ 2.22
: Similar to Table \[tab:bbaacut\], but for the $b\bar{b}\tau^+\tau^-$ signal channel.[]{data-label="tab:bbtatacut"}
The kinematical distributions similar to the $b\bar b\gamma\gamma$ channel are shown in Fig. \[fig:bbtata\]. As one can see from the figure, the $\tau$ jets are less energetic than the $b$ jets (similar to those in the $b\bar{b}\gamma\gamma$ signal channel) due to missing neutrinos in the final state. We first employ the following selection cuts to pick up signals with exactly one $b$ pair and one $\tau$ pair: $$\begin{aligned}
p_T^{b,\tau}>30~{{\rm GeV}},~|\eta_{b,\tau}|<2.4,~\Delta R_{bb,b\tau,\tau\tau}>0.4,\end{aligned}$$ and no cut on $\cancel{E}_T$ is adopted. After the selection, the $\tau$ and $b$ pairs are required to fulfill the cuts on the invariant masses and separations: $$\begin{aligned}
\Delta R_{\tau\tau}<2.5,M_{\Delta}-40~{{\rm GeV}}&<&M_{\tau\tau}<M_{\Delta},\\
\nonumber
\Delta R_{bb}<2.5,~|M_{bb}-M_{\Delta}|&<&15~{{\rm GeV}}.\end{aligned}$$ The different mass shift between $M_{\tau\tau}$ and $M_{bb}$ is owing to the missing neutrinos in $\tau$ decays resulting in a wider distribution of $M_{\tau\tau}$. For the reconstructed neutral scalars, we further adopt similar cuts on $M_{H^0A^0}$ and $E_T$ as in the $b\bar{b}\gamma\gamma$ channel: $$M_{H^0A^0}>2M_{\Delta}+70~{{\rm GeV}},~E_T>2M_{\Delta}-80~{{\rm GeV}}.$$ Both $M_{H^0A^0}$ and $E_T$ cuts are reduced by $20~{{\rm GeV}}$ compared with the $b\bar{b}\gamma\gamma$ channel, which again results from neutrinos in the final state. The corresponding results are summarized in Table \[tab:bbtatacut\].
The $b\bar{b}\tau^+\tau^-$ is also promising for $M_{\Delta}=130~{{\rm GeV}}$ even without enhancement from cascade decays. The final significance is 12.4 and the corresponding number of signal events is 309 for LHC14@3000. Including the cascade enhancement, the significance is improved to 51.6, which is even better than the $b\bar{b}\gamma\gamma$ signal. For $M_{\Delta}=160~{{\rm GeV}}$, the biggest challenge is also the small production cross section of the signal. But in the most optimistic case, the cascade decays can increase the signal by a factor of 10.1, making this channel feasible. Finally, neutral scalars as heavy as $190~{{\rm GeV}}$ are difficult to detect at LHC14 in this channel.
$b\bar{b}W^+W^-$ signal channel {#sec:bbWW}
-------------------------------
It is difficult to search for the SM Higgs pair production in this channel due to missing energy brought about by neutrinos in leptonic decays of the $W$ boson, which makes one of the two Higgs bosons not fully reconstructible [@Gouzevitch:2013qca; @Baglio:2012np]. The situation is ameliorated in our scenario because, the production rate of $H^0A^0$ can be an order of magnitude larger than that of $hh$ and the di-$W$ decay branching ratio of $H^0$ can also be larger than $h$ in the vast parameter space. This considerably improves the signal events and partially compensates the deficiency of the detection capability.
![Distributions of $p_T^\ell,~\Delta R_{\ell\ell},~M^{\ell\ell}_C,~\cancel{E}_T,~M_C$, and $E_T$ for the signal $b\bar{b}\ell^+\ell^-\cancel{E}_T$ and its backgrounds before applying any cuts at LHC14. \[fig:bbll2v\]](bbll2v_PTl.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^\ell,~\Delta R_{\ell\ell},~M^{\ell\ell}_C,~\cancel{E}_T,~M_C$, and $E_T$ for the signal $b\bar{b}\ell^+\ell^-\cancel{E}_T$ and its backgrounds before applying any cuts at LHC14. \[fig:bbll2v\]](bbll2v_DRll.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^\ell,~\Delta R_{\ell\ell},~M^{\ell\ell}_C,~\cancel{E}_T,~M_C$, and $E_T$ for the signal $b\bar{b}\ell^+\ell^-\cancel{E}_T$ and its backgrounds before applying any cuts at LHC14. \[fig:bbll2v\]](bbll2v_MCll.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^\ell,~\Delta R_{\ell\ell},~M^{\ell\ell}_C,~\cancel{E}_T,~M_C$, and $E_T$ for the signal $b\bar{b}\ell^+\ell^-\cancel{E}_T$ and its backgrounds before applying any cuts at LHC14. \[fig:bbll2v\]](bbll2v_MET.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^\ell,~\Delta R_{\ell\ell},~M^{\ell\ell}_C,~\cancel{E}_T,~M_C$, and $E_T$ for the signal $b\bar{b}\ell^+\ell^-\cancel{E}_T$ and its backgrounds before applying any cuts at LHC14. \[fig:bbll2v\]](bbll2v_MC.pdf "fig:"){width="0.45\linewidth"} ![Distributions of $p_T^\ell,~\Delta R_{\ell\ell},~M^{\ell\ell}_C,~\cancel{E}_T,~M_C$, and $E_T$ for the signal $b\bar{b}\ell^+\ell^-\cancel{E}_T$ and its backgrounds before applying any cuts at LHC14. \[fig:bbll2v\]](bbll2v_Et.pdf "fig:"){width="0.45\linewidth"}
\[!htbp\]
$M_{\Delta}=130~{{\rm GeV}}$ $H^0A^0(S_0)$ $t\bar{t}$ $S/B$ $\mathcal{S}(S,B)$
------------------------------- --------------------- ---------------------- --------------------- --------------------
Cross section at NLO 3.91 $2.38\times10^4$ $1.69\times10^{-4}$ 1.41
Basic cuts $1.51$ $4.04\times10^3$ $3.74\times10^{-4}$ $1.30$
Reconstruct scalars from $b$s $3.29\times10^{-1}$ $3.35\times10^2$ $9.82\times10^{-4}$ $0.984$
Cut on $M_C^{\ell\ell}$ $3.21\times10^{-1}$ $2.14\times10^2$ $1.50\times10^{-3}$ 1.20
Cut on $\Delta R_{\ell\ell}$ $2.64\times10^{-1}$ $9.26\times10^1$ $2.85\times10^{-3}$ 1.50
Cut on $\cancel{E}_T$ $8.45\times10^{-2}$ $1.48\times10^1$ $5.71\times10^{-3}$ 1.20
Cut on $M_C$ $3.30\times10^{-2}$ $1.69\times10^{-1}$ $1.95\times10^{-1}$ 4.26
Cut on $E_T$ $3.19\times10^{-2}$ $1.47\times10^{-1}$ $2.17\times10^{-1}$ 4.41
Cascade enhanced $1.40\times10^{-1}$ $-$ $9.53\times10^{-1}$ $17.7$
$M_{\Delta}=160~{{\rm GeV}}$ $H^0A^0(S_0)$ $t\bar{t}$ $S/B$ $\mathcal{S}(S,B)$
Cross section at NLO 4.95 $2.38\times10^4$ $2.13\times10^{-4}$ 1.78
Basic cuts $2.13$ $4.04\times10^3$ $5.27\times10^{-4}$ $1.84$
Reconstruct scalars from $b$s $4.25\times10^{-1}$ $2.68\times10^2$ $1.59\times10^{-3}$ 1.42
Cut on $M_C^{\ell\ell}$ $3.97\times10^{-1}$ $1.89\times10^2$ $2.10\times10^{-3}$ 1.58
Cut on $\Delta R_{\ell\ell}$ $3.21\times10^{-1}$ $7.04\times10^1$ $4.56\times10^{-3}$ 2.09
Cut on $\cancel{E}_T$ $9.47\times10^{-2}$ 4.29 $2.21\times10^{-2}$ 2.50
Cut on $M_C$ $3.28\times10^{-2}$ $4.74\times10^{-2}$ $6.92\times10^{-1}$ 7.50
Cut on $E_T$ $3.02\times10^{-2}$ $3.62\times10^{-2}$ $8.34\times10^{-1}$ 7.78
Cascade enhanced $1.01\times10^{-1}$ $-$ 3.24 $23.2$
$M_{\Delta}=190~{{\rm GeV}}$ $H^0A^0(S_0)$ $t\bar{t}$ $S/B$ $\mathcal{S}(S,B)$
Cross section at NLO 1.19 $2.38\times10^4$ $5.00\times10^{-5}$ $0.424$
Basic cuts $6.44\times10^{-1}$ $4.04\times10^3$ $1.59\times10^{-4}$ $0.554$
Reconstruct scalars from $b$s $1.36\times10^{-1}$ $2.26\times10^{2}$ $6.02\times10^{-4}$ $0.495$
Cut on $M_C^{\ell\ell}$ $1.27\times10^{-1}$ $1.79\times10^{2}$ $7.09\times10^{-4}$ $0.520$
Cut on $\Delta R_{\ell\ell}$ $9.70\times10^{-2}$ $6.05\times10^{1}$ $1.60\times10^{-3}$ $0.683$
Cut on $\cancel{E}_T$ $2.57\times10^{-2}$ 1.62 $1.59\times10^{-2}$ 1.10
Cut on $M_C$ $8.85\times10^{-3}$ $1.89\times10^{-2}$ $4.68\times10^{-1}$ 3.29
Cut on $E_T$ $8.37\times10^{-3}$ $1.40\times10^{-2}$ $5.98\times10^{-1}$ 3.56
Cascade enhanced $2.69\times10^{-2}$ $-$ 1.92 $10.1$
: Similar to Table \[tab:bbaacut\], but for the $b\bar{b}\ell^+\ell^-\cancel{E}_T$ signal channel.[]{data-label="tab:bbllcut"}
With both $W$’s decaying leptonically, the final state appears as $b\bar{b}\ell^+\ell^-\cancel{E}_T$. The dominant SM backgrounds are as follows: $$t\bar{t}:pp\to t\bar{t}\to bW^+\bar{b}W^-\to b\bar{b}\ell^+\ell^-\cancel{E}_T.$$ As before, the QCD correction is included by a multiplicative $K$-factor of 1.35 for the $t\bar{t}$ production [@tt]. We pick up the events that include exactly one $b$-jet pair and one opposite-sign lepton pair and filter them with the basic cuts: $$\begin{aligned}
p_T^{b}>30~{{\rm GeV}},~p_T^{\ell}>20~{{\rm GeV}},~|\eta_{b,\ell}|<2.4,\\
\nonumber
\Delta R_{bb,b\ell,\ell\ell}>0.4,~\cancel{E}_T>20~{{\rm GeV}}.\end{aligned}$$ The separation and invariant mass of the $b$-jet pair are required to fulfill $$\Delta R_{bb}<2.5,~|M_{bb}-M_{\Delta}|<15~{{\rm GeV}}.$$ For the lepton pair, we reconstruct the transverse cluster mass $M_C^{\ell\ell}$: $$M_C^{\ell\ell}=\sqrt{\left(\sqrt{p_{T,\ell\ell}^2+M^2_{\ell\ell}}+\cancel{E}_T\right)^2
+\left(\vec{p}_{T,\ell\ell}+\vec{\cancel{E}}_T\right)^2}.$$ The distributions of $M_C^{\ell\ell}$, $\Delta R_{\ell\ell}$, and $\cancel{E}_T$ are shown in Fig. \[fig:bbll2v\]. The peak of $M_C^{\ell\ell}$ is always lower than $M_{\Delta}$ by about $30$-$40~{{\rm GeV}}$, and the lepton separation $\Delta R_{\ell\ell}$ in the signal is much smaller than in the $t\bar{t}$ background. Accordingly, we set a wide window on $M_{C}^{\ell\ell}$ while tightening up the cuts on $\Delta R_{\ell\ell}$ and $\cancel{E}_T$: $$M_{\Delta}-80~{{\rm GeV}}<M_C^{\ell\ell}<M_{\Delta},~\Delta R_{\ell\ell}<1.2,~\cancel{E}_T>0.9M_{\Delta}.$$ We find that $M_{C}^{\ell\ell}$ is least efficient around $M_{\Delta}\sim190~{{\rm GeV}}$, where the peak of $M_{C}^{\ell\ell}$ for the $t\bar{t}$ background is around $150~{{\rm GeV}}$. The very tight cuts on $\Delta R_{\ell\ell}$ and $\cancel{E}_T$ are sufficient to suppress the background by 1 or 2 orders of magnitude, while keeping the number of signal events as large as possible. We further combine the $b$-jet pair and the lepton pair into a cluster and construct the transverse cluster mass: $$M_C=\sqrt{\left(\sqrt{p_{T,bb\ell\ell}^2+M_{bb\ell\ell}^2}+\cancel{E}_T\right)^2-\left( \vec{p}_{T,bb\ell\ell}+\vec{\cancel{E}}_T\right)^2},$$
![Left: Significance $\mathcal{S}(S,B)$ of the $b\bar{b}\gamma\gamma$ channel versus $M_{\Delta}$ reachable at LHC14@300 (red region) and LHC14@3000 (green). Right: Required luminosity to reach a $3\sigma$ (red region) and $5\sigma$ (green) significance in the $b\bar{b}\gamma\gamma$ channel versus $M_{\Delta}$ at LHC14. The solid line corresponds to the signal from $X_0$ alone, and the dashed line corresponds to the total signal including cascade enhancement. \[bbaa\_sen\]](bbaa_sen.pdf "fig:"){width="0.44\linewidth"} ![Left: Significance $\mathcal{S}(S,B)$ of the $b\bar{b}\gamma\gamma$ channel versus $M_{\Delta}$ reachable at LHC14@300 (red region) and LHC14@3000 (green). Right: Required luminosity to reach a $3\sigma$ (red region) and $5\sigma$ (green) significance in the $b\bar{b}\gamma\gamma$ channel versus $M_{\Delta}$ at LHC14. The solid line corresponds to the signal from $X_0$ alone, and the dashed line corresponds to the total signal including cascade enhancement. \[bbaa\_sen\]](bbaa_lum_new.pdf "fig:"){width="0.45\linewidth"}
which is an analog of $M_{H^0A^0}$ in the previous subsection. The distribution of $M_C$ is displayed in Fig. \[fig:bbll2v\], which is very similar to that of $M_{H^0A^0}$ in the $b\bar{b}\gamma\gamma$ channel. Although it looks from the $M_C$ distributions (before any cuts are made) that the $t\bar{t}$ background has a large overlap with the signal, the cuts on $M_C^{\ell\ell}$, $\Delta R_{\ell\ell}$, and $\cancel{E}_T$ actually modify them remarkably, so that a further cut on $M_C$ could improve the significance efficiently. We apply a cut on $M_C$ as we did with $M_{H^0A^0}$, as well as one on $E_T$: $$M_C>2M_{\Delta}+90~{{\rm GeV}},~E_T>2M_{\Delta}-60~{{\rm GeV}}.$$ The results following the cutflow are summarized in Table \[tab:bbllcut\]. For $M_{\Delta}=130~{{\rm GeV}}$, the final significance is 4.41 (17.7) without (with) cascade enhancement. With cascade enhancement this should be enough to discover the neutral scalars. The signal channel is more promising for $M_{\Delta}=160~{{\rm GeV}}$ due to a slightly larger cross section and higher cut efficiencies. The final significance is 7.78 (23.2), which is also better than the $b\bar{b}\gamma\gamma$ and $b\bar{b}\tau^+\tau^-$ channels with the same mass. Finally, for $M_{\Delta}=190~{{\rm GeV}}$, the significance becomes 3.56 (10.1). Therefore, for our benchmark model, the only promising signal for such heavy neutral scalars ($\sim190~{{\rm GeV}}$) comes from the $b\bar{b}W^+W^-$ channel.
Observability
-------------
![Same as Fig. \[bbaa\_sen\], but for the $b\bar{b}\tau^+\tau^-$ channel. \[bbtata\_sen\]](bbtata_sen.pdf "fig:"){width="0.44\linewidth"} ![Same as Fig. \[bbaa\_sen\], but for the $b\bar{b}\tau^+\tau^-$ channel. \[bbtata\_sen\]](bbtata_lum_new.pdf "fig:"){width="0.45\linewidth"}
Based on our elaborate analysis of signal channels in Secs. \[sec:bbgg\]–\[sec:bbWW\], we examine the observability of the neutral scalars $H_0,~A_0$ in the mass region $130\sim200~{{\rm GeV}}$ by adopting essentially the same cuts as before. In the left panel of Figs. \[bbaa\_sen\], \[bbtata\_sen\], and \[bbll2v\_sen\] we present the significance $\mathcal{S}(S,B)$ as a function of $M_{\Delta}$ in the three signal channels $b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, and $b\bar{b}\ell^+\ell^-\cancel{E}_T$ that is reachable for LHC14@300 and LHC14@3000, respectively. The required luminosity to achieve a $3\sigma$ and $5\sigma$ significance is displayed in the right panel of the figures. As was done in our previous analysis, the effect of cascade enhancement is included by a factor $S/S_0$ in the final results.
As shown in Figs. \[bbaa\_sen\] and \[bbtata\_sen\], both the $b\bar{b}\gamma\gamma$ and $b\bar{b}\tau^+\tau^-$ channel are typically sensitive to the low-mass region ($M_{\Delta}\lesssim160~{{\rm GeV}}$). In the absence of cascade enhancement, the $3\sigma$ significance would never be reached for $M_{\Delta}\gtrsim138~(142)~{{\rm GeV}}$ in the $b\bar{b}\gamma\gamma$ ($b\bar{b}\tau^+\tau^-$) channel for LHC14@300. However, a cascade enhancement of $S/S_0\sim 4-6$ (as can be seen from Fig. \[sgn\]) in this mass region can greatly improve the observability, pushing the $3\sigma$ mass limit up to $157~(162)~{{\rm GeV}}$ in the $b\bar{b}\gamma\gamma$ ($b\bar{b}\tau^+\tau^-$) channel. Moreover, with cascade enhancement, one has a good chance to reach a $5\sigma$ significance if $M_{\Delta}\lesssim 153~(155)~{{\rm GeV}}$. In other words, the cascade enhancement significantly reduces the required luminosity. For instance, to achieve a $3\sigma$ and $5\sigma$ significance in the $b\bar{b}\gamma\gamma$ ($b\bar{b}\tau^+\tau^-$) channel with $M_{\Delta}=130~{{\rm GeV}}$, the required luminosity is as low as $16~(10)~{\textrm{fb}}^{-1}$ and $42~(27)~{\textrm{fb}}^{-1}$ at LHC14, respectively. The $b\bar{b}\tau^+\tau^-$ channel is more promising, thanks to a relatively larger production rate.
At the future LHC14 with $3000~{\textrm{fb}}^{-1}$ data, the heavier mass region can also be probed. With a maximal cascade enhancement, the $3\sigma$ and $5\sigma$ mass reach is pushed to $177$ and $164~{{\rm GeV}}$, respectively, in the $b\bar{b}\gamma\gamma$ channel, which should be compared to $156$ and $151~{{\rm GeV}}$ in the absence of enhancement. For the $b\bar{b}\tau^+\tau^-$ channel, the enhancement factor $S/S_0$ can reach about $18$ above the $W$-pair threshold, upshifting the $3\sigma$ and $5\sigma$ mass reach to $189$ and $177~{{\rm GeV}}$, respectively, from $154$ and $150~{{\rm GeV}}$ without the enhancement.
![Same as Fig. \[bbaa\_sen\], but for the $b\bar{b}\ell^+\ell^-\cancel{E}_T$ channel. \[bbll2v\_sen\]](bbll2v_sen.pdf "fig:"){width="0.44\linewidth"} ![Same as Fig. \[bbaa\_sen\], but for the $b\bar{b}\ell^+\ell^-\cancel{E}_T$ channel. \[bbll2v\_sen\]](bbll2v_lum_new.pdf "fig:"){width="0.45\linewidth"}
The $b\bar{b}\ell^+\ell^-\cancel{E}_T$ channel shown in Fig. \[bbll2v\_sen\] is more special, compared with $b\bar{b}\gamma\gamma$ and $b\bar{b}\tau^+\tau^-$. It is relatively more sensitive to a higher mass between $150$-$180~{{\rm GeV}}$, where the decay mode $H^0\to W^+W^-$ dominates, while its observability deteriorates for $M_{\Delta}<150~{{\rm GeV}}$ due to phase-space suppression in the decay. The cascade enhancement $S/S_0$ at our benchmark point (\[BP\]) is typically $3$-$4$ in the mass region $130$-$200~{{\rm GeV}}$, and decreases as $M_{\Delta}$ increases. For LHC14@300, the $3\sigma$ and $5\sigma$ mass reach is, respectively, $190$ and $181~{{\rm GeV}}$ with maximal cascade enhancement. These limits would just increase by $2$-$3~{{\rm GeV}}$ for LHC14@3000 if there were no cascade enhancement, while with cascade enhancement the $5\sigma$ limit, for instance, is pushed up to $200~{{\rm GeV}}$. Finally, a $3\sigma$ or $5\sigma$ reach in the mass region $150$-$180~{{\rm GeV}}$ requires an integrated luminosity of $50~{\textrm{fb}}^{-1}$ ($450~{\textrm{fb}}^{-1}$) or $150~{\textrm{fb}}^{-1}$ ($1300~{\textrm{fb}}^{-1}$) with (without) cascade enhancement.
Discussions and Conclusions {#Dis}
===========================
In this paper, we have systematically investigated the LHC phenomenology of neutral scalar pair production in the negative scenario of the type II seesaw model. To achieve this goal, we first examined the decay properties of the neutral scalars $H_0/A_0$ and found that the scalar self-couplings $\lambda_i$ have a great impact on the branching ratios of $H^0/A^0$. The coupling $\lambda_4$ is important for tree-level decays of $H^0$ and $A^0$, while one-loop-induced decays of $H^0$ further depend on $\lambda_2$ and $\lambda_3$. We found that the decay $H^0\to W^+W^-$ could dominate for $2M_W<M_{H^0}<2M_h$ with $\lambda_4<0$, while it can be neglected once $M_{H^0}$ is above the light scalar pair threshold $2M_h$. Moreover, the branching ratios of the decays $H^0\to \gamma\gamma,~Z\gamma$ can cross 3 orders of magnitude when varying the couplings $\lambda_i$, and there exist zero points for the $H^0ZZ$, $H^0hh$, and $A^0Zh$ couplings.
The cross section of the Drell-Yan process $pp\to Z^*\to H^0A^0$ for $M_{\Delta}<200~{{\rm GeV}}$ is much larger than that of the SM Higgs pair production driven by gluon fusion. In this paper, we studied the contributions to $H^0/A^0$ production from cascade decays of the charged scalars $H^{\pm}$ and $H^{\pm\pm}$. There are actually three different states for the neutral scalar pair: $H^0A^0$, $H^0H^0$, and $A^0A^0$. Here, $H^0H^0$ and $A^0A^0$ can only arise from cascade decays of charged scalars, and their production rates always stay the same to a good approximation. Further, for a fixed value of $M_{\Delta}$, cascade enhancement is determined by the variables $v_{\Delta}$ and $\Delta M$. By tuning these two variables, the associated production rate of $H^0A^0$ can be maximally enhanced by about a factor of 3, while those of the $H^0H^0$ and $A^0A^0$ pair production can reach the value of $H^0A^0$ production through the pure Drell-Yan process.
We implemented detailed collider simulations of the associated $H^0A^0$ production for three typical signal channels ($b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, and $b\bar{b}W^+W^-$ with both $W$’s decaying leptonically). The enhancement from cascade decays of charged scalars is quantified by a multiplicative factor $S/S_0$. Due mainly to a larger production rate, all three channels are more promising than the SM Higgs pair case. If there were no cascade enhancement, the $5\sigma$ mass reach of the $b\bar{b}\gamma\gamma$, $b\bar{b}\tau^+\tau^-$, and $b\bar{b}\ell^+\ell^-\cancel{E}_T$ channels would be, respectively, $151$, $150$, and $180~{{\rm GeV}}$ for LHC14@3000. The cascade enhancement pushes these limits up to $164$, $177$, and $200~{{\rm GeV}}$. The $b\bar{b}\gamma\gamma$ and $b\bar{b}\tau^+\tau^-$ channels are more promising in the mass region below about $150~{{\rm GeV}}$, and the required luminosities for $5\sigma$ significance are $42~{\textrm{fb}}^{-1}$ and $27~{\textrm{fb}}^{-1}$, respectively, at our benchmark point. Compared with these two channels, the $b\bar{b}\ell^+\ell^-\cancel{E}_T$ channel is more advantageous in the relatively higher mass region $150$-$200~{{\rm GeV}}$, and the required luminosity for $5\sigma$ significance is about $150~{\textrm{fb}}^{-1}$ with maximal cascade enhancement. Needless to say, for the purpose of a full investigation on the impact of heavy neutral scalars on the SM Higgs pair production, more sophisticated simulations are necessary. We hope that this work may shed some light on further studies in both the phenomenological and experimental communities.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the Grants No. NSFC-11025525, No. NSFC-11575089 and by the CAS Center for Excellence in Particle Physics (CCEPP).
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[^1]: This number is modified to $33~{\textrm{fb}}$ at next-to-leading order [@Dawson:1998py] and to $40~{\textrm{fb}}$ at next-to-next-to-leading order [@deFlorian:2013jea].
[^2]: As pointed out in Ref. [@Aoki:2011pz], varying $v_{\Delta}$ in the range $10^{-3}$-$1~{{\rm GeV}}$ would not change the branching ratios significantly.
[^3]: The influence of $\lambda_4$ for light fermions $b,c,\tau,\mu$ and gluons is similar, so we only present BR($H^0\to b\bar{b}$) in Fig. \[brh0tt\].
[^4]: As before, the influence of $\lambda_4$ on the $A^0\to f \bar{f},~gg$ channels is similar to the $b\bar{b}$ mode.
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author:
- 'Tomas Jakab$^1$[^1]Ankush Gupta$^1$Hakan Bilen$^2$Andrea Vedaldi$^1$'
bibliography:
- 'longstrings.bib'
- 'refs.bib'
- 'vgg\_other.bib'
- 'vgg\_local.bib'
title: |
Unsupervised Learning of Object Landmarks\
through Conditional Image Generation
---
[^1]: equal contribution.
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{
"pile_set_name": "ArXiv"
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bibliography:
- 'ref.bib'
---
[**The [Teichmüller]{} space of the standard action of $SL(2,Z)$ on $T^2$ is trivial**]{}\
Elise E. Cawley\
Department of Mathematics\
University of California at Berkeley\
Berkeley, CA 94720\
February 12, 1991
[**1**]{} [**Introduction**]{}
The group $SL(n,{\bf Z})$ acts linearly on ${\bf R}^n$, preserving the integer lattice ${\bf Z}^{n} \subset {\bf R}^{n}$. The induced (left) action on the n-torus ${\bf T}^{n} = {\bf R}^{n}/{\bf Z}^{n}$ will be referred to as the “standard action”. It has recently been shown that the standard action of $SL(n,{\bf Z})$ on ${\bf T}^n$, for $n \geq 3$, is both topologically and smoothly rigid. [@Hu], [@KL], [@HKLZ]. That is, nearby actions in the space of representations of $SL(n,{\bf Z})$ into ${\rm Diff}^{+}({\bf T}^{n})$ are smoothly conjugate to the standard action. In fact, this rigidity persists for the standard action of a subgroup of finite index. On the other hand, while the ${\bf Z}$ action on ${\bf T}^{n}$ defined by a single hyperbolic element of $SL(n,{\bf Z})$ is topologically rigid, an infinite dimensional space of smooth conjugacy classes occur in a neighborhood of the linear action. [@C], [@L], [@MM1], [@MM2] The standard action of $SL(2, {\bf Z})$ on ${\bf T}^2$ forms an intermediate case, with different rigidity properties from either extreme. One can construct continuous deformations of the standard action to obtain an (arbritrarily near) action to which it is not topologically conjugate. [@Hu] [@Ta] The purpose of the present paper is to show that if a nearby action, or more generally, an action with some mild Anosov properties, is conjugate to the standard action of $SL(2, {\bf Z})$ on ${\bf T}^2$ by a homeomorphism $h$, then $h$ is smooth. In fact, it will be shown that this rigidity holds for any non-cyclic subgroup of $SL(2, {\bf Z})$.
A smooth action of a group $\Gamma$ on a manifold $M$ is called Anosov if there is a least one element $\gamma \in \Gamma$ which acts by an Anosov diffeomorphism. We define the $C^{r}$ [Teichmüller]{} space of an Anosov action $F:\Gamma \times M \rightarrow M$ to be the space of “marked” $C^r$ smooth structures preserved by the underlying topological dynamics of the action, and such that Anosov group elements for the action $F$ define Anosov diffeomorphisms with respect to the new smooth structure as well. The precise definition is given in the next section.
The main theorem is:
Let $\Gamma \subset SL(2, {\bf Z})$ be a subgroup containing two hyperbolic elements. We assume that the (four) eigenvectors of these two elements are pairwise linearly independent. Let $0 < \alpha < 1$. Then the $C^{1 + \alpha}$ [Teichmüller]{} space ${\rm Teich}^{1 + \alpha}
(F: \Gamma \times {\bf T}^{n}
\rightarrow {\bf T}^{n})$ is trivial, where $F$ is the standard action.
[**Remark.**]{} Any non-cyclic subgroup satisfies the hypotheses of the Theorem.
An immediate corollary of Theorem 1 is:
Let $0 < \alpha < 1$. The $C^{1 + \alpha}$ [Teichmüller]{} space ${\rm Teich}^{1 + \alpha}
(F: SL(2,{\bf Z}) \times {\bf T}^{2} \rightarrow
{\bf T}^2)$ is trivial, where $F$ is the standard action.
[**Acknowledgements.**]{} I thank Dennis Sullivan and Michael Lyubich for helpful conversations, and Geoffrey Mess and Steve Hurder for making me aware of this problem.
[**2**]{} [**The [Teichmüller]{} space of an Anosov action**]{}
We consider $\Gamma$, a locally compact, second countable group, and a $C^{r}$ left action $$F: \Gamma \times M \rightarrow M$$ of $\Gamma$ on a $C^{r}$ manifold $M$, $1 \leq r \leq \omega$. The action is called Anosov if at least one element $\gamma \in
\Gamma$ acts by an Anosov diffeomorphism. That is, there is a continous splitting of the tangent bundle $TM = E^{s} \bigoplus E^{u}$, invariant under $D\gamma$, and such that vectors in $E^{s}$ are exponentially contracted, and vectors in $E^{u}$ are exponentially expanded, by iterates of $D\gamma$.
The [Teichmüller]{} space of $F$ is defined as follows. Consider triples $(h, N, G)$ where $N$ is a $C^{r}$ manifold, $h: M \rightarrow N$ is a homeomorphism, and $$G: \Gamma \times N \rightarrow N$$ is a $C^{r}$ action conjugate to $F$ by $h$. That is, $G(x,\gamma) = h(F( h^{-1}(x), \gamma))$, for every $\gamma \in
\Gamma$. We assume in addition that if $F( \gamma, \cdot)$ is Anosov, then $G( \gamma, \cdot)$ is also Anosov. Such a triple is called a [*marked Anosov action modeled on $F$*]{}. Two triples $(h_{1}, N_{1}, G_{1})$ and $(h_{2}, N_{2}, G_{2})$ are [*equivalent*]{} if the homeomorphism $s : N_{1} \rightarrow N_{2}$ defined by $s \circ h_{1} = s_{2}$ is a $C^{1}$ diffeomorphism.
[**3**]{} [**The main argument**]{}
The subbundles $E^{s}$ and $E^{u}$ in the definition of an Anosov map are integrable, and the corresponding foliations are called the stable, respectively unstable, foliations of the map. The diffeomorphism is called codimension-one if either the stable or the unstable foliation is codimension-one. If the diffeomorphism is $C^{1 + \alpha}$, i.e. the derivative is [Hölder]{}continuous with exponent $0 < \alpha < 1$, then the codimension-one foliation has $C^{1 + \beta}$ transverse regularity for some $0 < \beta < 1$.[@Ho], [@Mn] (A codimension-$k$ Anosov foliation is transversely absolutely continuous, and the holonomy maps have [Hölder]{} Jacobian. [@An]) An Anosov diffeomorphism of ${\bf T}^{2}$ has simultaneous foliation charts $$\phi: D^1 \times D^1 \rightarrow U \subset {\bf T}^{2}$$ where $D^1$ is the one dimensional disk. The intersection of a leaf of the unstable foliation ${\cal W}^{u}$ with the neighborhood $U$ is a union of horizontals $\phi(D^{1} \times {y})$, and that of a leaf of the stable foliation ${\cal W}^{s}$ is a union of verticals $\phi({x} \times D^{1})$. Moreover, we can choose $\phi$ to be smooth along, say $x_{0} \times D^1$ and $D^{1} \times y_{0}$, for some $(x_{0}, y_{0}) \in D^{1} \times D^{1}$. A simple but important observation is the following: since both foliations have $C^{1 + \beta}$ transverse regularity, these charts belong to the $C^{1 + \beta}$ smooth structure on ${\bf T}^{2}$ preserved by the diffeomorphism. Therefore the smooth structure is determined (up to $C^{1 + \beta}$ equivalence) by the pair of transverse smooth structures.
We give the main argument of the proof of Theorem 1. The lemmas used here are proved in the next section. Let $\gamma_{1}$, $\gamma_2 \in
SL(2, {\bf Z})$ be hyperbolic elements such that the eigenvectors $\lbrace v_{1}^{s}, v_{1}^{u}, v_{2}^{s},
v_{2}^{u} \rbrace$ are pairwise linearly independent. Here $v_{i}^s$, respectively $v_{i}^{u}$, is the contracting (or stable), respectively expanding (or unstable), eigenvector of $\gamma_i$, for $i = 1,2$. Let $$F: \Gamma \times {\bf T}^{2} \rightarrow {\bf T}^{2}$$ be the standard linear action. Define $$f_{i}( \cdot ) = F( \gamma_{i}, \cdot)$$ for $i = 1,2$. The lines in ${\bf R}^2$ parallel to $v_{i}^{s}$ project to ${\bf T}^{2} = {\bf R}^{2} / {\bf Z}^{2}$ to give the stable foliation ${\cal W}^{s,F}_{i}$, for $i = 1,2$. The unstable foliation ${\cal W}^{u,F}_{i}$ is obtained analogously.
Consider a segment $\tau$ contained in a leaf $W \in {\cal W}_{1}^{u,F}$. Then $\tau$ is a transversal to both ${\cal W}_{1}^{s,F}$ and ${\cal W}_
{2}^{s,F}$. There is a [*locally defined*]{} holonomy from $\tau$ to itself, “generated” by the pair of foliations as follows. Slide $\tau$ a small distance along the leaves of ${\cal W}_{1}^{s,F}$, remaining transverse to both stable foliations. Then slide $\tau$ up the leaves of ${\cal W}_{2}^{s,F}$, returning to the original leaf $W$ of ${\cal W}_{1}^{u,F}$. This is all done inside a single foliation chart for ${\cal W}_{1}^{u,F}$. The resulting motion is simply rigid translation in the leaf $W$. In other words, the translation group of $W$, restricted to small translations defined on the segment $\tau$, can be canonically factored into a composition of holonomy along ${\cal W}_{1}^{s,F}$, followed by holonomy along ${\cal W}_{2}^{s,F}$.
Let $$G: \Gamma \times {\bf T}^{2} \rightarrow {\bf T}^{2}$$ define a point in the $C^{1 + \alpha}$ [Teichmüller]{} space of the standard action of $\Gamma$ on ${\bf T}^{2}$. In other words, $G$ is a $C^{1 + \alpha}$ action, and there is a homeomorphism $h: {\bf T}^{2} \rightarrow
{\bf T}^{2}$ such that $$G(\gamma,x) = h ( F( \gamma,h^{-1}(x)))$$ for all $\gamma \in \Gamma$. Let $$g_{i}( \cdot) = G( \gamma_{i}, \cdot)$$ for $i = 1,2$. Then $g_{1}$ and $g_{2}$ are Anosov. Let ${\cal W}_{i}^{s,G}$ and ${\cal W}_{i}^{u,G}$ be the stable and unstable foliations respectively of $g_{i}$. We have $h({\cal W}_{i}^{s,F}) = {\cal W}_{i}^{s,G}$ and $h({\cal W}_{i}
^{u,F}) = {\cal W}_{i}^{u,G}$. We consider $W^{\prime} = h(W)$, and $\tau^{\prime} = h(\tau)$. The homeomorphism $h$ conjugates the small translations on $\tau \subset W$ to an action on $\tau^{\prime} \subset W^{\prime}$. The main claim is that this action, denoted $$S: [0,\epsilon] \times \tau^{\prime} \rightarrow \tau^{\prime},$$ is smooth. More precisely,
$\frac{\partial S}{\partial y} (t,y)$ is continuous.
The following rigidity property of the translation pseudogroup of the line implies that $h$ must be $C^1$ along $\tau$.
Let $T: {\bf R} \times {\bf R} \rightarrow {\bf R}$ be the rigid translations, namely $T(t,x) = x + t$. Let $S: {\bf R} \times {\bf R}
\rightarrow {\bf R}$ be an action conjugate to $T$ by a homeomorphism $h: {\bf R} \rightarrow {\bf R}$. So $S(t,y) = h ( T (t, h^{-1}(y)))$. If $S$ is smooth in a weak sense, namely if $\frac{\partial S}{\partial y} (t,y)$ is continuous, then in fact $h$ is a $C^1$ diffeomorphism.
To prove Lemma 1, we need the following important fact.
The foliations [${\cal W}_{1}^{s,G}$]{}, [${\cal W}_{1}^{u,G}$]{}, [${\cal W}_{2}^{s,G}$]{}, [${\cal W}_{2}^{u,G}$]{}are pairwise transverse.
Of course we know [${\cal W}_{1}^{s,G}$]{}and [${\cal W}_{1}^{u,G}$]{}are transverse, and similarly [${\cal W}_{2}^{s,G}$]{}and [${\cal W}_{2}^{u,G}$]{}are transverse. But one doesn’t know a priori that the homeomorphism $h$ did not introduce tangencies between, say, [${\cal W}_{1}^{u,G}$]{} and [${\cal W}_{2}^{u,G}$]{}.
[**Remark.**]{} Lemma 2 is automatically satisfied for actions [*sufficiently near*]{} to the standard action, since the transversality property is stable. The fact that this can be [*proved*]{} without any nearness assumption is the ingredient that globalizes the rigidity result.
[**Proof of Lemma 1, assuming Lemma 2.**]{} The conjugacy $h$ preserves the foliations. Thus we note that the action $S$ on $\tau^{\prime}$ can be factored into holonomy along [${\cal W}_{1}^{s,G}$]{} followed by holonomy along [${\cal W}_{2}^{s,G}$]{}, simply by applying $h$ to the corresponding decomposition for the rigid translations on $\tau$. The foliations [${\cal W}_{1}^{s,G}$]{} and [${\cal W}_{2}^{s,G}$]{} are transversely $C^{1 + \beta}$. The leaves of [${\cal W}_{1}^{u,G}$]{} form a $C^{1 + \beta}$ family of transversals to [${\cal W}_{1}^{s,G}$]{} and [${\cal W}_{2}^{s,G}$]{}. The claimed regularity of $S$ follows.
$\Box$
The argument up to now shows that the transverse smooth structure to the unstable foliation [${\cal W}_{1}^{u,G}$]{} is $C^1$ equivalent by the conjugacy $h$ to the tranverse smooth structure to ${\cal W}^{u,F}_{1}$, the corresponding foliation for the standard action. Similarly we see that the transverse smooth structure to the stable foliation [${\cal W}_{1}^{s,G}$]{} is $C^1$ equivalent by $h$ to the linear one. But the simultaneous foliation charts are charts in the $C^{1 + \beta}$ smooth structure, hence we conclude that $h$ is $C^1$.
[**Remark.**]{} It can be shown that $h$ is in fact as smooth as the mapping $G$.[@L]
[**4**]{} [**The translation pseudogroup of the real line is rigid**]{}
We prove Proposition 1. Let $T: {\bf R} \times {\bf R} \rightarrow {\bf R}$ be the rigid translations, $T(t,x) = x + t$. Let $S: {\bf R} \times {\bf R}
\rightarrow {\bf R}$ be conjugate to $T$ by the homeomorphism $h$. We assume that $\frac{\partial S}{\partial y}(t,y)$ is continuous. We will show that $S$ is $C^1$ conjugate to a rigid translation $L: {\bf R} \times {\bf R} \rightarrow {\bf R}$, $L(t,z) = z + \alpha t$, where $\alpha$ is a fixed real number. Since $L$ is affinely conjugate to $T$, we will conclude that $h$ is $C^1$.
Pick $y_{0} \in {\bf R}$. Let $t(y)$ be defined by $S(t(y),y) = y_{0}$. Define $g: {\bf R} \rightarrow {\bf R}$ $$g(y) = \int_{y_0}^{y} \frac{\partial S}{\partial y^{\prime}}
(t(y^{\prime}), y^{\prime})dy^{\prime}.$$ Conjugate the action of $S$ by the homeomorphism $g$ to obtain an action $L$, $$L(t,z) = g(S(t,g^{-1}(z))).$$ One sees that $\frac{\partial L}{\partial z}(t,z) = 1$. Hence $L(t,y) = y + k(t)$, for some function $k(t)$. But $L( t + s,y) = L(s,L(y,t)) = L(s,y + k(t)) = y + k(t) + k(s)$, so $k(t) = \alpha t$ for some $\alpha$.
[**5**]{} [**The foliations remain transverse**]{}
In this section we prove Lemma 2. There is an equivalence relation naturally associated to an Anosov diffeomorphism, defined by the pair of foliations ${\cal W}^s$ and ${\cal W}^u$. Namely, $x$ is equivalent to $y$ if $x \in
W^{s}(y) \cap W^{u}(y)$. This equivalence relation is generated by a pseudogroup of local homeomorphisms, defined in simultaneous foliation charts as the product of holonomy along ${\cal W}^{u}$ by holonomy along ${\cal W}^{s}$.
Suppose [${\cal W}_{1}^{u,G}$]{} is tangent to [${\cal W}_{2}^{s,G}$]{} at $z$. The [${\cal W}_{1}^{u,G}$]{} is tangent to [${\cal W}_{2}^{s,G}$]{} at every $z^{\prime} \in
W^{u,G}_{1}(z) \cap W^{s,G}_{1}(z)$.
Assuming this, we can prove Lemma 2. The set of $z^{\prime} \in W^{u,G}_{1}(z) \cap W^{s,G}_{1}(z)$ is dense in ${\bf T}^{2}$, and the distributions $E_{1}^{u,G}$ and $E_{2}^{s,G}$ are continuous. But the foliations [${\cal W}_{1}^{u,G}$]{} and [${\cal W}_{2}^{s,G}$]{} do not coincide, since they are homeomorphic images of the transverse foliations ${\cal W}^{u,F}_{1}$ and ${\cal W}^{s,F}_{2}$. Therefore there can be no tangencies, and Lemma 2 is proved.
[**Proof of Lemma 3.**]{} Let $z^{\prime} \in W^{u,G}_{1}(z) \cap W^{s,G}_{1}(z)$. We denote by $W^{u,G}_{1}(z, \epsilon)$ a small neighborhood of $z$ in the leaf of [${\cal W}_{1}^{u,G}$]{} containing it. Let $$hol_{s}: W^{u,G}_{1}(z, \epsilon) \rightarrow
W^{u,G}_{1}(z^{\prime}, \epsilon)$$ be the map defined by the holonomy of $W^{s,G}_{1}$. Similarly, let $$hol_{u}: W^{s,G}_{1}(z, \epsilon) \rightarrow
W^{s,G}_{1}(z^{\prime}, \epsilon)$$ be the holonomy of [${\cal W}_{1}^{u,G}$]{}. We can represent $W^{s,G}_{2}(z, \epsilon)$ as the graph of a map $$\theta_{z}: W^{u,G}_{1}(z, \epsilon) \rightarrow W^{s,G}_{1}(z,\epsilon).$$ Similarly we can represent $W^{s,G}_{2}(z^{\prime},\epsilon)$ as the graph of a map $$\theta_{z^{\prime}}: W^{u,G}_{1}(z^{\prime},\epsilon) \rightarrow
W^{s,G}_{1}(z^{\prime}, \epsilon).$$ We observe that $$\theta_{z^{\prime}} = hol_{s} \circ \theta_{z} \circ hol_{u}^{-1}.$$ To see this, consider the corresponding graphs and holonomy for the linear map. The corresponding equation holds there. This is a consequence of a non-obvious feature of the foliations in the linear case: the local pseudogroup of homeomorphisms that generates the Anosov equivalence relation described above [*preserves the contracting (or expanding) foliation of any other hyperbolic element*]{}. Since the equation is defined by conjugacy invariant objects, it must hold for the non-standard action as well. Since $hol_{s}$ and $hol_{u}$ are $C^{1 + \beta}$ diffeomorphisms, we see that a tangency at $z$ forces a tangency at $z^{\prime}$.
$\Box$
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---
abstract: 'An analytical model for fully developed three-dimensional incompressible turbulence was recently proposed in the hydrodynamics community, based on the concept of multiplicative chaos. It consists of a random field represented by means of a stochastic integral, which, with only a few parameters, shares many properties with experimental and numerical turbulence, including in particular energy transfer through scales (the cascade) and intermittency (non-Gaussianity) which is most conveniently controlled with a single parameter. Here, we propose three models extending this approach to MHD turbulence. Our formulae provide physically motivated 3D models of a turbulent velocity field and magnetic field coupled together. Besides its theoretical value, this work is meant to provide a tool for observers: a dozen of physically meaningful free parameters enter the description, which is useful to characterize astrophysical data.'
author:
- |
Jean-Baptiste Durrive$^{1}$[^1], Pierre Lesaffre$^{2}$, and Katia Ferrière$^{1}$\
$^{1}$ Institut de recherche en astrophysique et planétologie – Université Toulouse III - Paul Sabatier, Observatoire Midi-Pyrénées,\
Centre National de la Recherche Scientifique, UMR5277 – France\
$^2$ Laboratoire de physique de l’ÉNS - ENS Paris – Centre National de la Recherche Scientifique, École normale supérieure - Paris :\
FR684, Université Paris Diderot - Paris 7, Sorbonne Universite, UMR8023 – France
bibliography:
- 'BxC\_theory.bib'
date: 'Accepted 2020 May 22 ; Received 2020 May 22 ; in original form 2020 January 6'
title: Magnetic fields from Multiplicative Chaos
---
\[firstpage\]
turbulence, magnetic fields
Introduction
============
Magnetic fields are ubiquitous in the Universe, with a stunningly wide range of strengths and coherence lengths, from $10^{15}$G in magnetars to $10^{-6}$G in galaxy clusters, or even weaker at even larger scales. Indeed, while historically astrophysical plasmas pertained to solar and stellar physics, magnetism nowadays is also the focus of physicists studying much larger scales. For example, a multitude of high-resolution observations indicate that the interstellar medium of our galaxy is a magnetized, multi-phase, highly turbulent and intermittent medium [e.g. @FalgaroneEtAl91; @MAMDEtAl03; @ElmegreenScalo04; @Planck16_GalacticDust]. Extra-galactic fields, up to cosmological scales (Mpc size filaments, and, as claimed by an increasing number of groups, inside cosmic voids) are about to be routinely observed as well, thanks for example to the radio-telescopes LOFAR and the SKA. For reviews focused on such fields, see for instance [@Widrow02; @RyuEtAl12; @Subramanian19]. The bottom line is that, in order to understand the dynamics of astrophysical media, one needs to be able to model MHD turbulence. For astrophysically-oriented reviews on MHD turbulence see e.g. [@SchekochihinCowley07; @BrandenburgLazarian13; @Ferriere19; @Tobias19].
Astrophysical systems are fascinating for the same reason that they are so difficult to analyze: what we observe is the result of extremely rich dynamics, consisting of the intertwining of numerous processes governed by a variety of timescales, operating simultaneously at interdependent lengthscales. For this reason, solving consistently all the equations relevant for the interstellar medium for instance is not reachable analytically, and the relevant tools for this challenge are numerical simulations. Here our approach is to derive analytical expressions which, while capturing as much of the underlying physics as possible, remain simple enough to be discussed intuitively. For example, in our model intermittency is controlled by a single parameter, while in principle it is the result of complex dynamics, namely turbulent dynamo action and random shock compression. Hence, the description introduced here is phenomenological, and may be seen as a complementary tool to numerical simulations.
This work was carried out with the intention of providing a tool to describe effectively observations, grasping a fair amount of physics with only a dozen of physically meaningful free parameters. Indeed, our formulae are random processes that contain parameters (and functions) corresponding to: integral scales (at which energy is injected), dissipative scales, the large-scale shape of some mean field, the nature of the process (positively or negatively correlated increments), energy cascades (transfer through scales) and intermittency. Another difficult property to take into account in modeling MHD turbulence, is the coupling of velocity and magnetic fields. Our formulae do account for this in a realistic manner, to a certain extent, as they are built from equations embodying the dynamics. To be able to model such features opens up many applications for astrophysics. For instance, in the interstellar medium the propagation of cosmic rays is very sensitive to the intermittency of the magnetic field in which they move [@ShukurovEtAl17]. Nowadays, to explore the correlation between magnetic fields and cosmic rays, one has to run costly numerical simulations [@SetaEtAl18], while with the present model synthetic data may be obtained orders of magnitude faster and controlling the degree of intermittency with a single parameter.
In addition to being a vast subject, astrophysics inherits the intrinsic complexity of the physics itself. Indeed, it is well-known that turbulence per se, even in the incompressible hydrodynamical case, brings serious theoretical challenges, leaving us today with many long-lasting open questions. Exact solutions of the Navier-Stokes equations were found only in quite limited cases, and only a few statistical laws were derived rigorously. Meanwhile, various phenomenologies for MHD turbulence have been suggested in the literature. The study of Alfvénic turbulence has a long history, with pioneering work by [@Iroshnikov63] and [@Kraichnan65] who generalized Kolmogorov’s isotropic model. A quantitative picture of the properties of anisotropic cascade emerged with the seminal papers by [@GoldreichSridhar95_Part2; @NgBhattacharjee96], at the heart of which lies the concept of critical balance. This picture keeps being refined, with additional ingredients such as dynamical alignment [@Boldyrev06; @MalletSchekochihin17] and thanks to important on-going efforts to tackle the problem by means of numerical simulations; for reviews see for example [@Verma04; @Pouquet15; @Rincon19; @MoffattDormy19]. Our work relies on results derived by members of the hydrodynamics community, who themselves adapted ideas developed in the mathematics community. The latter physicists also consolidated their formal results by benefiting from close interactions with laboratory experiments. To the best of our knowledge, their approach has not been extended to MHD[^2], which is the purpose of the present paper, and has not been mentioned in the astrophysics literature yet either.
More specifically, in a series of papers [essentially @RobertVargas08; @ChevillardEtAl10; @ChevillardHDR; @PereiraEtAl16] a method was proposed to construct analytical expressions of 3D random vectorial fields representing the local structure of homogeneous, isotropic, stationary (fully developed) and incompressible turbulence. The random nature of turbulence is dealt with by means of stochastic calculus, i.e. employing a statistical description of turbulence, modeling fluids in terms of random functions [@Panchev72; @MoninYaglom75]. The gist of the method is the following. Starting from a vectorial Gaussian random field, features characteristic of turbulent flows are included one by one, through a series of cunning modifications. Non-Gaussianity, i.e. intermittency, is brought in the model through the (pleasantly named) Gaussian ‘multiplicative chaos’ introduced by [@Kahane85]. Applications of multiplicative chaos go well beyond the theory of turbulence, as it is used not only in mathematics, but also in finance [@DuchonEtAl12] or quantum gravity [@KnizhnikEtAl88; @KupiainenEtAl18] for example [see @RhodesVargas14 for a recent review]. Then to obtain energy cascades, the key is to extend the scalar theory of [@Kahane85] to matrices. This was first done by [@ChevillardEtAl13] in view of applications to turbulence. In fact, one of the important successes of the phenomenology outlined here is that this velocity field is the first stochastic process proposed in the literature that is able to predict a non-vanishing mean energy transfer across scales [@PereiraEtAl16]. The long range correlated nature of turbulence is incorporated heuristically using a non linear transformation inspired by the ‘recent fluid deformation approximation’ proposed by [@ChevillardMeneveau06]. Finally, note that in the aforementioned literature, both the Lagrangian and Eulerian descriptions are adopted. Here we stick to the Eulerian framework.
The spirit and organization of the paper is the following. In the line of [@ChevillardEtAl10], our approach is practical in the sense that we reduce mathematically involved material to intuitive considerations, such that hopefully the reader will feel that the model is built as blocks assembled gradually through a series of intuitive steps. To this end, we begin with the simple case of a scalar field in section \[section:ScalarField\], which introduces basic ideas. In section \[section:VelocityField\], the more involved case of a velocity field is treated, adapting previous works to our purpose. This allows us to prepare the ground for our generalization, makes the paper self-consistent, and advocates for digging in this direction. Indeed, at the end of this section, we emphasize how well the results of this procedure agree with direct numerical simulations and experimental data. Next, in section \[section:MHD\], we present our extension to magnetic fields which should appear natural, having carefully detailed the hydrodynamical case. In section \[section:Examples\], we illustrate some properties of the formulae of the paper through numerical examples. Finally, we conclude by giving hints of the prospects of this work.
General comments on the notations: (i) To lighten the paper, we use everywhere the shorthand notation $$\vec{r} \equiv \vec{x}-\vec{y} \hspace{0.5cm} \text{and} \hspace{0.5cm} r \equiv |\vec{r}|,
\label{shortHandNotations}$$ as most expressions are convolutions, with $r$ corresponding to the distance from a given position $\vec{x}$ when integrating with respect to $\vec{y}$. (ii) We use subscripts $s,v$ and $m$ for quantities associated with scalar fields, velocity fields and magnetic fields respectively.
Scalar field {#section:ScalarField}
============
Many scalar fields in nature (e.g. density or energy dissipation) are observed to be intermittent, so it is important to find an efficient way of modeling intermittent scalar fields. To do so, let us start with a formula that we are all familiar with : the gravitational potential $\phi$ due to a given distribution of mass $\rho$ is given by $$\phi(\vec{x}) = -G \int_{\mathbb{R}^3} \frac{\rho}{r} {\mathrm{d}}V,
\label{gravitationalPotential_rho}$$ where $G$ is Newton’s constant. We may reformulate this in terms of the gravitational field $\vec{g}$, using Poisson’s equation $\vec{\nabla} \cdot \vec{g} = - 4 \pi G \rho$ and the divergence theorem (assuming that the functions vanish sufficiently fast to infinity for surface integrals to be discarded), namely $$\phi(\vec{x}) = \frac{1}{4 \pi} \int_{\mathbb{R}^3} \frac{\vec{r} \cdot \vec{g}}{r^3} {\mathrm{d}}V.
\label{gravitationalPotential}$$ Since $\vec{g} = - \vec{\nabla} \phi$ the integral operator in acting on $\vec{g}$ may be seen as the inverse of the gradient operator, and similarly since $\Delta \phi = 4 \pi G \rho$ the integral operator in acting on $\rho$ may be seen as the inverse of the Laplacian.
Expression is an example of a physically relevant scalar field. Now, in order to construct a larger class of such scalar fields, we are going to apply several substitutions to this expression, while keeping its overall structure. First, rather than integrating up to infinite distances, let us consider an arbitrary boundary in the integral, i.e. in make the substitution $$\mathbb{R}^3 \rightarrow \mathcal{R}_s,
\label{substitution_integrationDomain_sf}$$ where $\mathcal{R}_s$ will now denote the region of integration. Second, let us generalize the power of the power-law kernel $$r^{-3} \rightarrow r^{-2 h_s},$$ where $h_s$ is a free parameter. Third, to avoid the singularity at the origin $r=0$ in the integrand brought by this power-law behavior, let us make the substitution $$r \rightarrow (r^2+\epsilon_s^2)^{1/2},
\label{substitution_regularization}$$ where $\epsilon_s$ is another free parameter. This is called a *regularization*. Mathematically it will spare us difficulties due to divergences and ensure differentiability, while physically the parameter $\epsilon_s$ will efficiently model dissipative effects at small scales, hence this choice of notation traditionally used for small parameters.
The above substitutions transform the integral into the following stochastic representation of a scalar random field $$\widetilde{s}_g(\vec{x}) = \int_{\mathcal{R}_s} \frac{\widetilde{\vec{\eta}}_g \cdot \vec{r}}{(r^2+\epsilon_s^2)^{h_s}} {\mathrm{d}}V.
\label{s_gaussian}$$
because it is defined as a linear operation (a convolution) over a Gaussian white noise $\widetilde{\vec{\eta}}_g$. In fact, this form is basically a spatial version of the fractional Brownian motion [cf. @MandelbrotVanNess68] in which the exponent of the power-law is called the Hurst exponent, hence our choice of notation $h$ in and in all similar integrals throughout this paper. The parameter $h_s$ controls the severeness of the singularity and consequently modifies the statistics of the resulting field, as it appears in the exponent of power-law behaving structure functions. More intuitively, while in the classical Brownian motion (also called Wiener process) the increments are independent, in fractional Brownian motion when the Hurst exponent is larger (smaller) than a critical value the increments are positively (negatively) correlated, i.e. there is an increasing (decreasing) pattern in the previous steps .
The fact that is a Gaussian field severely restricts its domain of relevance for modeling natural phenomena. Fortunately however, it was shown that from it one may easily construct an intermittent field, by simply applying the exponential function. Indeed, consider the scalar field $$\widetilde{s}(\vec{x}) = s_0(\vec{x}) \ \! \exp \left[\tau_s \ \! \widetilde{s}_g(\vec{x})\right],
\label{sf}$$ where $\tau_s$ is a constant and $s_0(\vec{x})$ is a non-random (i.e. a usual, smooth) scalar field. As illustrated in figures[^3] \[fig\_sf\] and \[fig\_pdf\_sf\], $\widetilde{s}$ is intermittent, its degree of intermittency being controlled by $\tau_s$, hence called the intermittency parameter. Stated in simple words, the field is indeed intermittent because the exponential is a non linear function that increases the contrast of the field $\widetilde{s}_g$, large values becoming larger and small ones becoming smaller, making large increments more frequent than in the Gaussian case, and resulting in the non-Gaussian wings in the probability density functions of figure \[fig\_pdf\_sf\]. This exponential factor then blurs locally, on small scales, the smooth field $s_0(\vec{x})$ that bears the large-scale behavior.
In a more technical wording, this manner of modeling intermittency (i.e. multifractality) by defining a lognormal process with a long range correlation structure as the exponential of a Gaussian field[^4], was introduced in [@Kahane85] and is called ‘Gaussian multiplicative chaos’. This terminology refers to Wiener chaos, not to the notion of chaos in dynamical systems, since this theory may be seen as a multiplicative counterpart of the additive Wiener chaos theory [cf @RhodesVargas14]. Such a construction is used for example in hydrodynamics as a stochastic representation of energy dissipation, i.e. a scalar field that shares the same statistical properties as the dissipation field (same expectation value and moments of the filtered fields at a given scale), see [@Kolmogorov62; @Mandelbrot72; @ChevillardHDR] and the Kolmogorov-Obhukov model. Physically this description corresponds to a continuous version of discrete cascades [cf Richardson’s picture @Richardson1922] referred to as log-infinitely divisible multifractal processes in the mathematically oriented literature [e.g. @BacryMuzy03; @SchmittChainais07].
Velocity field {#section:VelocityField}
==============
The take home message of the previous section is that applying the simple transformation to a Gaussian scalar field is enough to mimic a real scalar field. Let us now investigate how this may be generalized to a vector field, in order to describe efficiently fluid turbulence. In the same spirit, we will first consider a Gaussian field, and modify it with a physically motivated and intuitive, yet more involved, transformation.
Inspiration from multiplicative chaos: Introducing matrix $\widetilde{\vec{\mathcal{D}}}_g$ {#section:MultiplicativeChaos}
-------------------------------------------------------------------------------------------
The vectorial version of the Gaussian scalar field is obtained by simply substituting a vector product for the scalar product, i.e. symbolically $$\cdot \rightarrow \times.$$ Indeed, [@RobertVargas08] showed that a random vector field that is Gaussian, incompressible, homogeneous, isotropic, and singular can be expressed as the stochastic integral $$\widetilde{\vec{v}}_g(\vec{x}) = \int_{\mathcal{R}_v} \frac{\widetilde{\vec{\eta}}_g \times \vec{r}}{(r^2+\epsilon_v^2)^{h_v}} {\mathrm{d}}V,
\label{vGaussian}$$ where $\epsilon_v$ and $h_v$ are constants, and $\mathcal{R}_v$ denotes the region of integration. Roughly speaking, the singular power-law shape of the kernel in ensures the scale invariance property (structure functions, i.e. moments of velocity increments, behaving as power-laws) that is needed to satisfy Kolmogorov’s $2/3$-law. For this reason is a good starting point to model a turbulent velocity field, but it is not enough per se because it lacks two important features. Firstly, it is not intermittent. Secondly, it has a vanishing third-order moment of velocity increments (and actually all odd-order correlators). Therefore it does not reproduce the $4/5$-law and is not enough to model energy transfer across scales.
While this indeed provides an intermittent behavior, it does not yield energy transfer: the skewness of the resulting vector field vanishes at all scales. This is because the velocity increments do not exhibit asymmetry. Soon afterwards, [@ChevillardEtAl10] pointed out that if instead of using a scalar field ($\widetilde{d}_g(\vec{y})$ above) one uses a matrix field ($\widetilde{\vec{\mathcal{D}}}_g(\vec{y})$ below), the symmetry is broken and this limitation is bypassed: the resulting vector field contains both intermittency and an energy cascade. Based on these works, let us consider the velocity field $$\widetilde{\vec{v}}(\vec{x}) = \int_{\mathcal{R}_v} \frac{[e^{\tau_\omega \widetilde{\vec{\mathcal{D}}}_g} \ \! \widetilde{\vec{\eta}}_g] \times \vec{r}}{(r^2+\epsilon_v^2)^{h_v}} {\mathrm{d}}V,
\label{velocity_field_AvantExplicationsDhat}$$ where $h_v$ and $\tau_\omega$ are parameters, whose values will be discussed in section \[section:Examples\], and $\widetilde{\vec{\mathcal{D}}}_g$ is the matrix below. The purpose of the next section is to explain the reasons leading to this expression for $\widetilde{\vec{\mathcal{D}}}_g$.
Inspiration from Biot-Savart’s law: Linking $\widetilde{\vec{\mathcal{D}}}_g$ to vorticity $\vec{\omega}$ {#section:BiotSavart}
---------------------------------------------------------------------------------------------------------
Expression should look familiar to the reader: it can simply be seen as a modified Biot-Savart’s law. Indeed, recall that for an incompressible fluid, velocity may be expressed in terms of vorticity ($\vec{\omega} \equiv \vec{\nabla} \times \vec{v}$) according to Biot-Savart’s law [@MajdaBertozzi01] $$\vec{v}(\vec{x}) = \frac{1}{4 \pi} \int_{\mathbb{R}^3} \frac{\vec{\omega} \times \vec{r}}{r^3} {\mathrm{d}}V,
\label{BiotSavart_Hydro}$$ and starting with , one obtains with the four following, physically meaningful, substitutions. Firstly, the integration domain is limited $$\mathbb{R}^3 \rightarrow \mathcal{R}_v.
\label{substitution_integrationDomain}$$ Physically, this introduces correlation lengths, since the region $\mathcal{R}_v$ controls what points in space (i.e. what $\vec{y}$ in the integrand, ) influence the dynamics at position $\vec{x}$. This phenomenologically models the injection scale of the turbulent cascade operating in a given fluid state. In section \[section:Examples\], we will take $\mathcal{R}_v$ to be a sphere of radius equal to the injection scale, and eddies of that size will be visible (see figures \[fig\_vf\] or \[fig\_mfJ0\] for example). Secondly, the kernel is generalized: $$r^{-3} \rightarrow r^{-2 h_v},
\label{substitution_kernelGeneralized}$$ which allows us to model various types of correlations, as does the Hurst exponent $H$ in fractional Brownian motion. Thirdly, the norm is regularized, as was done in in the case of a scalar field. This again mathematically guarantees that is differentiable, and physically gives a degree of freedom to model dissipation at small scales (see section \[sec:Assets\] for more information on dissipation in this model). Finally, in order to complete this analogy, we need the following fourth substitution $$\vec{\omega} \rightarrow e^{\tau_\omega \widetilde{\vec{\mathcal{D}}}_g} \ \! \widetilde{\vec{\eta}}_g,
\label{substitution_vorticity}$$ but as such, giving a physical meaning to this is not obvious at all, especially because at this stage we do not even have an expression for $\widetilde{\vec{\mathcal{D}}}_g$. However, exhibiting the substitution hints towards a connection between vorticity and matrix $\widetilde{\vec{\mathcal{D}}}_g$. This is surprising a priori: $\widetilde{\vec{\mathcal{D}}}_g$ originates from the concept of multiplicative chaos, which was suggested in works unrelated to fluid turbulence. Still, led by this intuition, let us in this section have a closer look at the dynamics of vorticity. As we will see in the next section, interpreting expression as stemming from will pay off, as it will lead us to a relevant explicit expression for matrix $\widetilde{\vec{\mathcal{D}}}_g$, and bring a physical meaning to .
Momentum conservation for an inviscid fluid of pressure $p$ and constant density $\rho_0$ in a gravitational field $\vec{g}$ reads $$\frac{{\mathrm{d}}\vec{v}}{{\mathrm{d}}t} \equiv \partial_t \vec{v} + \vec{v} \cdot \vec{\nabla} \vec{v} = - \frac{1}{\rho_0} \vec{\nabla} p + \vec{g}.
\label{MomentumConservation_Hydro}$$ For now we are working in the inviscid limit because ultimately the substitution will take dissipative effects into account phenomenologically. The transport equation for vorticity is obtained by taking the curl of , recalling that $\vec{\omega}$ and $\vec{v}$ are divergence-free under the present assumptions. Doing so yields $$\frac{{\mathrm{d}}\vec{\omega}}{{\mathrm{d}}t} = \vec{G} \cdot \vec{\omega},
\label{transportEquation1}$$ where, to improve readability, we denote by $\vec{G}$ the gradient of velocity: $$\vec{G} \equiv (\vec{\nabla} \vec{v})^\textsc{T},$$ with $\textsc{T}$ indicating transposition.
Before going further, let us build some intuition of the underlying dynamics, by analyzing the flow locally. Taylor expanding the velocity field at a given point $\vec{x}_0$ and a given time $t$, to linear order in $|\vec{x}-\vec{x}_0|$ we have $$\vec{v}(\vec{x})=\vec{v}(\vec{x}_0)+\vec{G}(\vec{x}_0)(\vec{x}-\vec{x}_0).
\label{Taylor}$$ Next, let us decompose the matrix $\vec{G}$ into its symmetric and antisymmetric parts as $$\vec{G} = \vec{\mathcal{D}} + \vec{\Omega},
\label{SymmAntisymmDecomposition_partialivj}$$ where $$\renewcommand\arraystretch{1.2} \begin{array}{l}
\mathcal{D}_{ij} \equiv (\partial_i v_j + \partial_j v_i)/2,\\
\Omega_{ij} \equiv (\partial_i v_j - \partial_j v_i)/2.
\end{array}
\label{Def_D_and_Omega}$$ The symmetric part $\vec{\mathcal{D}}$ is called the deformation (or rate-of-strain) matrix. Being real and symmetric, it may be diagonalized by an orthogonal matrix, which corresponds physically to stretching and compression along its eigenvectors at a rate given by its eigenvalues. Stretching and compression occur jointly because of incompressibility. Indeed, the condition $\partial_i v_i=0$ in implies that $\vec{\mathcal{D}}$ is traceless, and thus the sum of its eigenvalues vanishes. Positive eigenvalues correspond to stretching and negatives eigenvalues to compression. The antisymmetric part $\vec{\Omega}$ is called the rotation matrix. Indeed, using the identity $\epsilon_{ijk} \epsilon_{kab} = \delta_{ia} \delta_{jb} - \delta_{ib} \delta_{ja}$, it can be written in terms of the vorticity as $$\Omega_{ij} = \epsilon_{ijk} \omega_k/2,
\label{Omega}$$ i.e. it is a cross product: In matrix form $$\vec{\Omega}=\frac{1}{2} \vec{\omega}\times.
\label{Omega_matrixForm}$$ This reveals that it corresponds to an instantaneous rotation in the direction of -$\vec{\omega}$ with angular velocity $|\vec{\omega}|/2$. As a result, the expansion becomes $$\vec{v}(\vec{x})=\vec{v}(\vec{x}_0)+\vec{\mathcal{D}}(\vec{x}_0)(\vec{x}-\vec{x}_0)+\frac{1}{2}\vec{\omega}\times(\vec{x}-\vec{x}_0),$$ from which we see that the flow is locally the sum of a translation, a deformation (stretching/compression) and a rotation [see e.g. @MajdaBertozzi01].
Introducing , together with , into directly gives $$\frac{{\mathrm{d}}\vec{\omega}}{{\mathrm{d}}t} = \vec{\mathcal{D}} \cdot \vec{\omega}.
\label{transportEquation2}$$ Physically, this equation describes the effect of vortex stretching: Stretching a vortex along its axis reduces its cross-section due to incompressibility, and this increases its angular momentum due to conservation of angular momentum.
Hence, in order to have a closed evolution equation for $\vec{\omega}$, it remains to explicit $\vec{\mathcal{D}}$ in terms of $\vec{\omega}$. Given the definition of $\vec{\mathcal{D}}$, we need to compute the gradient of $\vec{v}$. It is tempting to simply differentiate under the integral sign in , but unfortunately this yields a singularity and for the purpose of the present work we may simply accept the result [@MajdaBertozzi01] $$\vec{\mathcal{D}}(t,\vec{x}) = \frac{3}{8 \pi} \mathrm{P.V.} \! \int_{\mathbb{R}^3} \! \frac{(\vec{r} \times \vec{\omega}) \ \! \vec{r} + \vec{r} \ \! (\vec{r} \times \vec{\omega})}{r^5} {\mathrm{d}}V,
\label{VelocityGradient_SymmPart}$$ where products like $\vec{x} \ \! \vec{y}$ are tensor products ($\vec{x} \ \! \vec{y}|_{ij}=x_i y_j$), and ‘P.V.’ indicates a Cauchy Principal Value integral, defined as $$\mathrm{P.V.} \int_{\mathbb{R}^3} f(\vec{y}) \ \! {\mathrm{d}}V \equiv \lim_{\epsilon \rightarrow 0} \int_{|\vec{y}|\geq \epsilon} f(\vec{y}) \ \! {\mathrm{d}}V.$$ Equation together with constitutes the equation satisfied by $\vec{\omega}$. Rather than trying to solve it exactly, [@ChevillardEtAl10] expose a cunning method to obtain a very efficient approximate solution, based on the short-time evolution as follows.
First, recall that equation is written in Lagrangian form, with the material derivative ${\mathrm{d}}/{\mathrm{d}}t = \partial_t + \vec{v} \cdot \vec{\nabla}$. Consider a Lagrangian particle at position $\vec{X}$ at an initial time $t_0$, and at position $\vec{x}(t,\vec{X})$ at a later time $t$. The vorticity $\vec{\omega}(t,\vec{X})$ felt by this particle at time $t$ satisfies where in general $\vec{\mathcal{D}}(t)$ differs from its value at $t_0$. However, after a short enough period of time, say $t-t_0 = \tau_K$ where $\tau_K$ is the Kolmogorov timescale which is the characteristic timescale of variation of the correlation of gradients, one may consider that $\vec{\mathcal{D}}(t) \approx \vec{\mathcal{D}}(t_0)$ is a good approximation. Hence, over that period of time, the matrix in equation is roughly constant, so we can approximate its solution as $$\vec{\omega}(t, \vec{x}(t,\vec{X})) \approx e^{(t-t_0) \vec{\mathcal{D}}(t_0,\vec{x})} \vec{\omega}(t_0,\vec{X}).$$ This equation shows the two processes at play: the transport of particles from position $\vec{X}$ to $\vec{x}$ (see the arguments of $\vec{\omega}$ in the right and left hand sides), and the stretching of the vorticity field by the initial deformation tensor $\vec{\mathcal{D}}(t_0)$. Now, it turns out that in the dynamics of turbulence, the components of the matrix $\tau_K \vec{\mathcal{D}}(t_0)$ are typically of order unity, while the distance traveled by particles due to advection $\tau_K v(t_0)$ is of order $\mathcal{R}_e^{-1/2}$ where $\mathcal{R}_e$ is the Reynolds number [@ChevillardHDR]. Besides, since in astrophysics we are mainly dealing with high Reynolds number fluids, in the following we will consider that the advection of vorticity is negligible compared to its stretching, and finally write $$\vec{\omega}(t, \vec{x}) = e^{(t-t_0) \vec{\mathcal{D}}(t_0,\vec{x})} \vec{\omega}(t_0,\vec{x}).
\label{ShortTimeDynamics_Vorticity}$$ Since this expression stems from an expansion of equation near a given time $t_0$, in the literature this procedure is referred to as the ‘recent-fluid-deformation’ procedure [@ChevillardMeneveau06; @Meneveau11].
Putting it all together {#sec:buildingDgtilde}
-----------------------
Let us now put the final result of section \[section:MultiplicativeChaos\] together with that of section \[section:BiotSavart\]: The key feature of expression is that this matrix exponential is reminiscent of the matrix exponential appearing in expression , which completes the analogy between multiplicative chaos and vorticity. Hence, let us finally consider the velocity field $\mathcal{R}_{\omega}$ denotes the region of integration, and $h_{\omega}$ and $\epsilon_{\omega}$ are constants, with similar roles as $h_s$ and $\epsilon_s$ of the scalar case . By suggesting this form, a few additional substitutions have been made, which we now detail.
Clearly $\widetilde{\vec{\mathcal{D}}}_g$ in stems from $\vec{\mathcal{D}}$ in , with a few modifications. Firstly, we have introduced a parameter $\epsilon_{\omega}$ in the denominator, i.e. the singular kernel has been regularized, as was done in the scalar field case and in the expression of the velocity field . Thanks to this, we may discard the Cauchy Principal Value. Secondly we have generalized the power-law to some free parameter $h_{\omega}$, in order to gain an additional degree of freedom in the modeling and to have a larger parameter space for data fitting. The hydrodynamics literature we take the above formulae from is dedicated to the local structure of turbulence and ultimately aims at obtaining exact solutions of the Navier-Stokes equations. In contrast, our own purpose is different, and we do not fix a priori the parameters, e.g. we do not place constraints on the integration regions $\mathcal{R}_v$ and $\mathcal{R}_{\omega}$. This will give us more degrees of freedom, as illustrated in the scalar field case in figure \[fig\_EllipsoidalIntegrationRegion\], to fit astrophysical data which include large-scale features. This is somewhat of a poor man’s way, as a first approach, to model correlations between large and small scales, in the same spirit as we included a large-scale position dependent amplitude $s_0(\vec{x})$ in the scalar field . More refined modeling, which would for instance take into account the advection of vorticity, which is neglected in equation , is left for future work. Finally, and most importantly as it is at the heart of the physical interpretation of this model, the velocity field is independent of time (it describes a stationary state), which seems in contradiction with the fact that its construction is inspired from dynamics. Indeed, completing the analogy outlined in section \[section:BiotSavart\] led us to make this last substitution $$t-t_0 \rightarrow \tau_\omega,
\label{stationarization}$$ when going from to . Since this amounts to substituting some parameter for the time variable characterizing the stationary state (more precisely, its intermittency), is referred to as *stationarization*.
Assets of the adopted method {#sec:Assets}
----------------------------
While part of the procedure presented above is based on rigorous derivations, partly taken from the mathematics literature, some points rely on phenomenological arguments. Yet, by means of both analytical and numerical arguments, previous authors have shown that the model gives a surprising amount of correct predictions when confronted to both direct numerical simulations and experimental data [e.g. @ChevillardEtAl12]. What is even more encouraging is that some of its realistic properties are not ingredients introduced by hand but by-products of this construction, which suggests that it really does capture an important part of the underlying physics. In this section we briefly review these successes.
\(i) A first important feature is the existence of intermittent corrections. Let us define the velocity increment of lag $\vec{\ell}$ as $$\delta_{\vec{\ell}} \vec{v}(\vec{x}) \equiv \vec{v}(\vec{x}+\vec{\ell}) - \vec{v}(\vec{x}).
\label{def:increment}$$ Longitudinal and transverse increments are the projections of respectively along and perpendicular to the direction of $\vec{\ell}$. One common way to identify intermittency is to compare the probability density functions (PDFs) of the considered field to those of a Gaussian field. And indeed the PDFs of longitudinal and transverse increments of undergo a continuous deformation as the norm of the lag is decreased, from a Gaussian shape at large lags towards large tails at small lags. These often called ‘non-Gaussian wings’ are typical signatures of intermittency. Through several examples, we show in section \[section:Examples\] that these wings become increasingly significant as the parameter $\tau_\omega$ increases, which supports naming it the intermittency parameter. More details may be found in the above-mentioned literature.
\(ii) Another classical way to characterize intermittency in isotropic turbulence studies is by analyzing the power-law behavior of structure functions in the inertial range. By definition, the $n^\text{th}$ order structure function is the $n^\text{th}$ moment of velocity increments $S_n(\ell) \equiv \langle (\delta_\ell \vec{v})^n \rangle$ where brackets $\langle \rangle$ indicate the expectation value [@Frisch95]. In the inertial range, $S_n \propto \ell^{\tau_n}$, where the dependence on $n$ of $\tau_n$ quantifies the intermittency: the field is intermittent if and only if $\tau_n$ depends non-linearly on $n$. For the velocity , this non-linearity was confirmed numerically, and also partially analytically: under the simplifying assumption of independence of $\widetilde{\vec{\mathcal{D}}}_g$ with the white noise in , [@PereiraEtAl16] showed that $\tau_n = a n + b n^2$, where $a$ and $b$ are constants, with $b \propto \tau_\omega^2$, i.e. the quadratic behavior is determined by the intermittency parameter. This again justifies the name given to $\tau_\omega$.
\(iii) [@PereiraEtAl16] further characterized the non-Gaussian behavior of fluctuations by analyzing in details the skewness $\mathcal{S}_\ell \equiv \langle (\delta_\ell \vec{v})^3 \rangle/[\langle (\delta_\ell \vec{v})^2 \rangle]^{3/2}$ and flatness $\mathcal{F}_\ell \equiv \langle (\delta_\ell \vec{v})^4 \rangle/[\langle (\delta_\ell \vec{v})^2 \rangle]^2$ of velocity increments [@Frisch95]. The propreties of both longitudinal and transverse increments match the results of direct numerical simulations. In particular, they showed that this velocity field has a higher level of intermittency in the transverse case than in the longitudinal case, which is a feature observed in experiments [see @PereiraEtAl16 and references therein].
\(iv) The degree of freedom provided by the Hurst parameter $h_v$ allows the velocity field to satisfy the experimental $2/3$ law of turbulence.
\(v) Another aspect that has been scrutinized in the literature is the statistics of the velocity gradient $\vec{G}$, which regulates the dynamics of the turbulent flow. Its eigenvalues are particularly meaningful physically (see section \[section:BiotSavart\]) and have peculiar statistical properties that reproduces remarquably well. More specifically, these eigenvalues are the solutions of the characteristic polynomial with coefficients given by the two invariants $$\renewcommand\arraystretch{1.2} \begin{array}{l}
\displaystyle Q \equiv - \tfrac{1}{2} \text{tr}(\vec{G}^2) = \tfrac{1}{4} |\vec{\omega}|^2 - \tfrac{1}{2} \text{tr}(\vec{\mathcal{D}}^2),\\
\displaystyle R \equiv - \tfrac{1}{3} \text{tr}(\vec{G}^3) = - \tfrac{1}{4} \vec{\omega} \cdot \vec{\mathcal{D}} \cdot \vec{\omega} - \tfrac{1}{3} \text{tr}(\vec{\mathcal{D}}^3).
\end{array}$$ Physically, $Q$ when $Q>0$ rotation dominates locally in the flow, and when $Q<0$ the region is dissipation-dominated. $R$ when $R>0$ the flow tends to create dissipation, and when $R<0$ more enstrophy is produced. Now, it turns out that when representing the joint probability density of $R$ and $Q$, a characteristic shape appears, known as the ‘teardrop shape in the RQ-plane’. See for instance [@Tsinober01; @Wallace09; @Meneveau11] for both experimental and theoretical aspects of this feature. In particular, this shape is symmetric with respect to the $R=0$ line in the Gaussian case ($\tau_\omega = 0$) and tilted anticlockwise otherwise. As detailed notably in [@ChevillardHDR; @PereiraEtAl16], this geometry is indeed recovered with . This diagnostic tool is yet another piece of evidence of its intermittent behavior, and of the local dissipation occurring in a pretty realistic manner.
\(vi) Other essential and non-trivial features are the orientation properties of vorticity with respect to the eigenframe of the rate-of-strain matrix $\vec{D}$. In particular, the preferential alignement of vorticity with the eigenvector associated with the intermediate eigenvalue of $\vec{D}$ does not occur in the Gaussian case, but gets stronger as the intermittency parameter increases. All this is correctly reproduced by , cf e.g. [@ChevillardHDR; @PereiraEtAl16; @PereiraEtAl18].
\(vii) Equation is the first stochastic process proposed in the literature to have a non-vanishing mean energy transfer across scales, as observed numerically in [@ChevillardEtAl10] and then, by means of a perturbative expansion in the intermittency parameter, [@PereiraEtAl16] confirmed analytically the existence of this cascade. As this feature stems from the matrix nature of the multiplicative chaos used, for more details on mathematical aspects we refer the reader to [@ChevillardEtAl13] who constructed the theory of matrix multiplicative chaos.
\(viii) These formulae are well behaved, notably they are differentiable thanks to the regularization. This aspect is not obvious since they are built from irregular mathematical objects (e.g. sample-paths of Wiener processes are almost nowhere differentiable).
\(ix) Last but not least, another success of this approach is that it results in fields that are easy to obtain numerically. Because most equations in this work are convolutions, their implementation is both simple and fast, thanks to Fast Fourier Transform algorithms and because convolutions in Fourier space are just products. To give an order of magnitude, with only 32 CPUs one may generate a $2048^3$ resolution cube in only 5 minutes.
As a closing remark, let us add that more achievements related to this description may be found in the above-mentioned literature, notably in the Lagrangian framework, studying the time evolution. Here we stick to stationary fields in the Eulerian description. Also, here we aim at constructing expressions that are as handy as possible. For this reason, we chose the simplest regularization, namely , but more elaborate regularized norms are used and discussed in the literature. Similarly, instead of our integration regions, it is more common to introduce large-scale cutoff functions in the kernels, which is more convenient to analyze the statistics of the field depending on the properties of these functions.
Magnetic field {#section:MHD}
==============
The above substitutions constitute building blocks to be assembled to construct physically motivated turbulent vector fields. The successes mentioned in the previous section motivate us to extend this approach to magnetized fluids, which are ubiquitous in astrophysics. Per se, this is a very ambitious project, and we will limit ourselves to proposing three straightforward extensions. To do so, we need to adapt the matrix and ensure that the magnetic fields remain divergence-free.
Matrix field of the underlying dynamics
---------------------------------------
Compared to the hydrodynamical case , the momentum equation is now coupled to the induction equation, and contains the additional contribution from the Lorentz force, $\vec{j} \times \vec{B}$, where the current density $\vec{j}$ is given by Ampère’s law $\vec{j} = \left( \vec{\nabla} \times \vec{B} \right)/\mu_0$ and the constant $\mu_0$ is vacuum permeability. In the same spirit as previously, we may omit viscosity and resistivity for now, since dissipative effects will ultimately be taken into account through regularizations similar to . We are then left with $$\renewcommand\arraystretch{1.7} \begin{array}{l}
\displaystyle \partial_t \vec{v} + \vec{v} \cdot \vec{\nabla} \vec{v} = - \frac{1}{\rho_0} \vec{\nabla} p + \vec{g} + \frac{1}{\mu_0 \rho_0} \left( \vec{\nabla} \times \vec{B} \right) \times \vec{B},\\
\displaystyle \partial_t \vec{B} = \vec{\nabla} \times \left(\vec{v} \times \vec{B}\right),
\end{array}
\label{IncompMHD}$$ together with the constraint $\vec{\nabla} \cdot \vec{B} = 0$. As a first approach, in this paper we consider incompressible MHD and the case when the back-reaction of the magnetic field, due to the Lorentz force, is negligible. This approximation, called kinematic MHD, is valid as long as magnetic energy is small compared to kinetic energy, i.e. $B^2/(2 \mu_0) \ll 1/2 \rho_0 v^2$. This condition is often not satisfied in astrophysics. However, building our models in this framework is also motivated [^5] by 3D MHD simulations such as those of [@ServidioEtAl08] which show a natural tendency towards force-free behaviors in MHD turbulence. Indeed, as nonlinearity develops, strong alignments spontaneously appear notably between $\vec{j}$ and $\vec{B}$, which is called ‘beltramization’ in reference to Beltrami vector fields which are fields parallel to their own curl.
In this framework, the induction equation reads $$\frac{{\mathrm{d}}\vec{B}}{{\mathrm{d}}t} = \vec{G} \cdot \vec{B}.
\label{EvolutionOfB}$$ Introducing , together with , into leads to $$\frac{{\mathrm{d}}\vec{B}}{{\mathrm{d}}t} = \vec{\mathcal{D}} \cdot \vec{B} + \frac{1}{2} \vec{\omega} \times \vec{B}.
\label{EvolutionOfB_2}$$ This equation for the magnetic field is the counterpart of equation for the vorticity. The two terms on the right hand side describe flux tube stretching (similar to vortex stretching in ) and shear, respectively. By analogy with the hydrodynamical case
Three models
------------
Assuming that the matrix field $\widetilde{\vec{G}}_g$ in is an accurate representation of the local dynamics in the magnetized turbulent flow, let us now propose three models for MHD turbulence, that are in principle complementary.\
*Model 1)* By analogy with the short time expression of vorticity from the hydrodynamical case, it is natural to consider the short time solution of , namely the matrix exponential of $\widetilde{\vec{G}}_g$. Doing so, the initial value of the magnetic field appears, just like $\vec{\omega}(t_0,\vec{x})$ in . When suggesting the velocity field expression , the initial value of vorticity was replaced by a Gaussian field through the substitution . Here instead, we take advantage of this degree of freedom to introduce some ordered magnetic field $\vec{B}_0$. Introducing this is important to model astrophysical fluids since the observed magnetic fields in the Universe are usually turbulent with large-scale mean fields [e.g. @Ferriere07; @Jaffe19]. Physically, the magnetic field $\widetilde{\vec{B}}^{(1)}$ is the result of the random stretching/compression and shearing (encoded in the matrix exponential, the strength of which is controlled by $\tau_m$) of a large-scale ordered magnetic field $\vec{B}_0(\vec{x})$, due to the turbulent velocity field that it is coupled to.\
*Model 2)* By analogy with Biot-Savart’s law from the hydrodynamical case, it is natural to consider Biot-Savart’s law for magnetic fields $$\vec{B} = \frac{\mu_0}{4 \pi} \int_{\mathbb{R}^3} \frac{\vec{j} \times \vec{r}}{r^3} {\mathrm{d}}V.
\label{BiotSavart_Magnetic}$$ To strictly follow the procedure performed with $\vec{\omega}$, the next step would be to analyze the evolution equation of $\vec{j}$, and its short time dynamics would reveal a relevant matrix. However, here we propose a simpler and faster approach. We already found a seemingly relevant generalization of $\widetilde{\vec{\mathcal{D}}}_g$, namely $\widetilde{\vec{G}}_g$, costlessly obtained from the governing equations . And fortunately, the aforementioned beltramization exhibited in simulations of [@ServidioEtAl08] prompts us to indeed use $\widetilde{\vec{G}}_g$: since $\vec{j}$ tends to be aligned with $\vec{B}$, the matrix governing its evolution should in some way resemble the matrix governing the evolution of $\vec{B}$, namely . Hence, rather than digging further to look for the precise matrix out of the evolution of $\vec{j}$, we will directly work with $\widetilde{\vec{G}}_g$. Admittedly, this is also very convenient. In addition, the simulations of [@ServidioEtAl08] also indicate that the alignments are directly related to the self-organization of the magnetofluid giving rise to spatial intermittency, which our formulae do take into account through the intermittency parameters $\tau_\omega$ and $\tau_m$. [@ServidioEtAl08] furthermore suggest that these dynamically generated correlations suppress nonlinearity, i.e. the equations tend to be quasilinear rather than fully nonlinear. This is reminiscent of the interpretation mentioned in section \[sec:buildingDgtilde\] to account for the intriguing fact that our formulae to describe a fully developed turbulent state.
From the above considerations, we propose as a second model $$\widetilde{\vec{B}}^{(2)} = \frac{\mu_0}{4 \pi} \int_{\mathcal{R}_m} \frac{\widetilde{\vec{j}}^{(2)} \times \vec{r}}{(r^2+\epsilon_m^2)^{h_m}} {\mathrm{d}}V,
\label{Bmodel2}$$ where $\mathcal{R}_m$, $h_m$ and $\epsilon_m$ have similar physical meanings as $\mathcal{R}_v$, $h_v$ and $\epsilon_v$ in the velocity field (for examples see section \[section:Examples\]).
Because is formally similar to the velocity field , we may give to the magnetic field $\widetilde{\vec{B}}^{(2)}$ an analogous physical meaning to that presented in the end of section \[sec:buildingDgtilde\], replacing vorticity with current density, and adding the effect of shear (contribution from the antisymmetric part of $\widetilde{\vec{G}}_g$, due to local rotation). Hence, introduces an additional regime compared to its hydrodynamical counterpart , namely the regime of shear dominated turbulence, when the local rotation $\hat{\vec{\Omega}}$ dominates over vortex stretching $\widetilde{\vec{\mathcal{D}}}_g$. Compared to model 1, in this model there is no large-scale feature, such as $s_0$ and $\vec{B}_0$. Therefore, in principle this second model is relevant to describe the local, small-scale structure of a turbulent magnetic field. However, given its phenomenological construction, it may turn out to have a wider range of application depending on what it is used for.\
*Model 3)* Finally, let us propose a model that combines the features of models 1 and 2, by considering again a Biot-Savart law form, but inserting some large-scale feature,
As in model 2, the Biot-Savart form introduces additional degrees of freedom ($\mathcal{R}_m$, $h_m$ and $\epsilon_m$) compared to model 1, while at the same time, as in model 1, it contains a large-scale field ($\vec{j}_0$). Physically, the magnetic field $\widetilde{\vec{B}}^{(3)}$ is the result of the random stretching/compression and shearing (encoded in the matrix exponential, the strength of which is controlled by $\tau_m$) of a large-scale ordered current density field $\vec{j}_0$, due to the turbulent velocity field that it is coupled to.\
We summarize in table \[table\_DefaultParameters\] the many free parameters (and functions) introduced through all the above formulae. These constitute degrees of freedom that may be adjusted to data, in order for example to quantify the degree of intermittency of an observed magnetic field. They also provide a handy tool to model analytically a wide variety of astrophysical environments. Furthermore, this diversity opens a vast parameter space, corresponding to many regimes of the dynamics. For example, since the hydrodynamical and the magnetic regularization parameters $\epsilon_v$ and $\epsilon_m$ correspond to dissipation parameters, it is reasonable to consider that the ratio $\epsilon_v/\epsilon_m$ corresponds to the magnetic Prandtl number $\text{Pr}_m$ (we remain cautious while stating this simply because their precise link with $\text{Pr}_m$ has not been rigorously established yet). Therefore, varying this ratio we may generate turbulent magnetic fields with $\text{Pr}_m \gg 1$ or $\text{Pr}_m \ll 1$, which is out of reach of numerical simulations, as it requires a huge dynamical range. In this sense our approach is complementary to numerical simulations.
$
\begin{array}{|c|cc|lllll|}
\cline{2-8}
\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{\text{Expressions}} & \multicolumn{5}{c|}{\text{Degrees of freedom}}\\
\hline
\text{Scalar} & \widetilde{\vec{s}}_g & \eqref{s_gaussian} & \epsilon_s & L_s & h_s & & \\
\text{field} & \widetilde{\vec{s}} & \eqref{sf} & & & & \tau_s & s_0(\vec{x})\\
\hline
\text{Velocity} & \widetilde{\vec{\mathcal{D}}}_g & \eqref{Dg_tilde} & \epsilon_{\omega} & L_{\omega} & h_{\omega} & & \\
\text{field} & \widetilde{\vec{v}} & \eqref{v_tilde} & \epsilon_v & L_v & h_v & \tau_\omega & \\
\hline
& \widetilde{\vec{G}}_g & \eqref{Ggtilde} & \epsilon_{\omega} & L_{\omega} & h_{\omega} & & \\
\text{Magnetic} & \widetilde{\vec{B}}^{(1)} & \eqref{Bmodel1} & & & & \tau_m & \vec{B}_0(\vec{x}) \\
\text{fields} & \widetilde{\vec{B}}^{(2)} & \eqref{Bmodel2} & \epsilon_m & L_m & h_m & \tau_m & \\
& \widetilde{\vec{B}}^{(3)} & \eqref{Bmodel3} & \epsilon_m & L_m & h_m & \tau_m & \vec{j}_0(\vec{x})\\
\hline
\end{array}
$
Examples {#section:Examples}
========
The many degrees of freedom summarized in table \[table\_DefaultParameters\] suggest rich and interesting interplays between the hydrodynamical and magnetic parameters. But an exhaustive analysis of the effects of each of them is also beyond the scope of this paper. Instead, in this section we expose just a couple of selected examples, intending to give the reader a flavor of some key features.
To begin with, let us provide some relevant values for the free parameters, based on [@ChevillardEtAl10; @ChevillardEtAl11; @ChevillardEtAl12; @ChevillardHDR]: (i) in the Gaussian scalar field one may take $h_s = 3/4$ typically, (ii) in the works just quoted, it is shown that $h_v=13/12$, corresponding to a Hurst exponent $H=1/3$ in their notation, makes the velocity field satisfy the experimental $2/3$ law of turbulence, (iii) in addition, it is argued that in order for the process to have a finite variance and convenient mathematical properties, one should take $3/4<h_v<5/4$ (corresponding to $0<H<1$), a constraint that may thus be taken as a prior in data analyses, (iv) in order for the components of $\widetilde{\vec{\mathcal{D}}}_g$ to be correlated logarithmically in space, $h_{\omega}=7/4$, according to [@Mandelbrot72] and [@Kahane85], (v) in the context of laboratory fluids, the value $\tau_\omega = 0.1$ is obtained from experiments, after translating the $\gamma^2=6.7 \times 10^{-2}$ of [@PereiraEtAl16] into our notations. Indeed, in laboratory and numerical flows, these authors find a universal behavior, independent of the flow geometry and the Reynolds number, for the longitudinal increments, with data well reproduced for that particular value of $\tau_\omega$. Finally (vi) regarding the choices of $L_v$ and $\epsilon_v$, in order to reach a large inertial range, one should maximize the $L_v/\epsilon_v$ ratio. The small-scale cut-off $\epsilon_v$ is typically taken equal to a few times the resolution length. For instance for a simulation performed in a 1-periodic box with $N^3$ collocation points, $L_v=1/2$ and $\epsilon_v=3/N$.
We now show examples of various scalar fields, velocity fields and magnetic fields based on the above formulae. Most figures were produced using Mayavi, an application and library for interactive scientific data visualization and 3D plotting in Python [@RamachandranVaroquaux11].
Firstly, we need to generate the Gaussian vectorial field $\widetilde{\vec{\omega}}_g$, from which all the other fields shown in this paper were generated, making them by construction highly correlated. We display this white noise in figure \[fig\_wn\]. What is interesting to bear in mind when visualizing this realization, is that obviously it does not have any particular shape or structure, neither in vector form nor in norm. $\widetilde{\vec{\omega}}_g$ is really only the building block that gives all the other fields their randomness: the structures and non-Gaussianity are produced by the non-linear transformations , , , , and applied to it. Thanks to this, if one varies continuously the free parameters, despite the randomness, the fields also vary continuously: we do not generate new realizations for each value of the free parameters.
Secondly, let us illustrate the effect of the regions of integration, in the case of a scalar field, which is easier to visualize. $\mathcal{R}_s$ determines the large-scale shape, which may depend on position. In figure \[fig\_EllipsoidalIntegrationRegion\], we consider a simple ellipsoid, and vary its elongation. The shape of this region roughly determines the trend appearing in the spatial distribution: a spherical region yields rather isotropic ‘clouds’, a cigare-like region yields filamentary structures and a pancake-like region generates more sheetlike structures. In the rest of this paper, we will stick to spherical regions, namely spheres of radius $L_i$ centered at position $\vec{x}$: $$\mathcal{R}_i(\vec{x}) = \left\{\vec{y} \in \mathbb{R}^3 / |\vec{r}| \leq L_i \right\},
\label{SphericalIntegrationRegion}$$ where $i=s,v,m$ for respectively a scalar field, a velocity field, and a magnetic field. We do so both for simplicity and because in the literature this has been proven to be relevant at least for the local structure of isotropic turbulence.
Thirdly, we analyze how the scalar field changes as we vary the integral scale $L_s$ together with the intermittency parameter $\tau_s$. In figure \[fig\_sf\], we show 2D slices of 3D realizations of the scalar field, for various values of $L_s$ and $\tau_s$, in order to show qualitatively how these modify the shape of the structures and the intermittency. Focusing on the modifications along a column, one clearly sees that $L_s$ modifies the global size of the structures formed, i.e. the correlation length. Focusing on the modifications along a row, it appears that basically $\tau_s$ modifies the contrast, and hence controls the intermittency of the field. To complement these intuitive pictures, in figure \[fig\_pdf\_sf\] we provide more quantitative considerations by plotting the PDFs of the increments corresponding to each of the cases shown in figure \[fig\_sf\] for three different lags. We normalize the PDFs to unit variance and arbitrarily shift vertically the curves for the sake of clarity [as in @PereiraEtAl18 for example]. Focusing on a given row, on the left the PDFs are very close to those of a Gaussian random field, while as we move to the right, the characteristic non-Gaussian wings appear as $\tau_s$ increases. This confirms quantitatively the feeling from figure \[fig\_sf\] that $\tau_s$ controls the intermittency of the field. Then considering the modifications along the columns as well, it appears that the ‘efficiency’ of $\tau_s$ to provide intermittency increases with increasing $L_s$.
Fourthly, we analyze how the velocity field changes as we vary the integral scale $L_v$ together with the intermittency parameter $\tau_\omega$. In figure \[fig\_vf\], we show 2D slices of 3D realizations of the velocity field, for various values of $L_v$ and $\tau_\omega$, in order to show qualitatively how these modify the shape of the structures and the intermittency. Focusing on the modifications along a column, one clearly sees that $L_v$ modifies the global size of the structures formed, i.e. the correlation length. Focusing on the modifications along a row, it appears that basically $\tau_\omega$ modifies the contrast of the norm of the vectors, and hence controls the intermittency of the field. To complete these intuitive pictures, in figure \[fig\_pdf\_vf\] we provide more quantitative considerations by plotting the PDFs of longitudinal increments corresponding to each of the cases shown in figure \[fig\_vf\] for three different lags. As in the scalar field case, non-Gaussian wings appear as we increase the intermittency parameter $\tau_\omega$. These PDFs are very similar to what is usually obtained in direct numerical simulations, but here we may directly control the steepness of the non-Gaussian wings with $\tau_\omega$ and the fields are generated at very little computational cost. Focusing on a given row, on the left the PDFs are very close to those of a Gaussian random field, while as we move to the right, non-Gaussian wings appear as $\tau_\omega$ increases. This confirms quantitatively the feeling from figure \[fig\_vf\] that $\tau_\omega$ controls the intermittency. Then considering the modifications along the columns as well, it appears that the ‘efficiency’ of $\tau_\omega$ to provide intermittency depends on $L_v$.
Finally, let us focus on the magnetic fields. To illustrate model 1, we take the following simple divergence-free ordered magnetic field $$\vec{B}_0=\hat{\vec{x}},
\label{B0_model1}$$ where $\hat{\vec{x}}$ is the unit vector along the $x$-direction. This field is then distorted according to , as shown in figure \[fig\_mfB0\] for three values of the magnetic intermittency parameter $\tau_m$. The choice of representation for the arrows is the same as in figure \[fig\_vf\]. In the left panel $\tau_m$ is very small, such that we almost recover the ordered field . Then as $\tau_m$ is increased, the vector field is randomized but it is still clear by eye that its statistics are biased, with a tendency of vectors to be pointing to the right, i.e. along the underlying $\vec{B}_0$.
For the third model, to exemplify the intertwining of the large-scale (ordered) feature with the smaller-scale turbulence, let us consider the following simple case. Obviously we do not claim that it is realistic, but its simplicity helps the visualization. We leave the modeling with more astrophysically relevant ordered fields for future work. Hence, as an ordered current density field, let us consider a uniform current density wire of radius $r_0$ and norm $j_0$, i.e. $$\vec{j}_0=
\left\{
\begin{array}{lr}
j_0 \hat{\vec{z}} & \text{ for } r \leq r_0,\\
\vec{0} & \text{ for } r>r_0,
\end{array}
\right.
\label{J0_model3}$$ where $r^2 \equiv x^2+y^2$. Pleasingly, in this case the Biot-Savart law can be integrated, so that we know that the corresponding ordered magnetic field is $$\vec{B}_0= \frac{\mu_0 j_0}{2} (- y \hat{x} + x \hat{y})
\left\{
\begin{array}{cr}
1 & \text{ for } r \leq r_0,\\
\left(r_0/r\right)^2 & \text{ for } r>r_0.
\end{array}
\right.
\label{B0_model3}$$ In other words, the vectors $\vec{B}_0$ form concentric circles around the $\hat{\vec{z}}$ axis, with a norm increasing linearly with $r$ below radius $r_0$ and decreasing as $r^{-1}$ beyond $r_0$. In figure \[fig\_mfJ0\] we show 3D plots of examples of the turbulent magnetic field using the ordered current density , for three different values of the magnetic intermittency parameter $\tau_m$. This figure is meant to give a visual impression of how the large-scale features of the ordered magnetic field and the turbulence-related scales are intertwined in this model. On the left, $\tau_m$ is very small, such that we recover the ordered magnetic field , as expected in the absence of significant distortion. Then as we increase $\tau_m$, the vectors are more and more randomized. These plots show qualitatively how the statistics of the magnetic field are impacted. The red region in the left panel corresponds to a cylinder of radius $r_0$ (because this is where $\vec{B}_0$ is maximal, cf ), while in the middle panel for example we can see that the red region is made of lumps. Their typical size basically (in fact there is a non-trivial dependence on $\tau_m$) corresponds to the integral scale $L_m$. Hence we may visualize how the turbulent velocity field distorts the ordered magnetic field $\vec{B}_0$ and modifies its large-scale properties as $\tau_m$ increases. The right panel shows how this information even gets lost at some point once the distortion becomes really strong and one may discern nothing but eddies.
Conclusion $\&$ Prospects
=========================
As an extension to a recent phenomenology of 3D hydrodynamical turbulence based on the so-called multiplicative chaos, we proposed three models for MHD turbulence. These provide compact, physically motivated formulae for a scalar field, a velocity field and three magnetic fields, which are *by construction correlated* to one another, because they all derive from the same white noise. Compared to previous works who focused on the local structure of turbulence, we also introduced large-scale features in these fields, to enlarge their possible range of use. Given the successes exhibited in the hydrodynamics literature concerning the velocity field thus constructed, we have hope that our magnetic field formulae will be useful (i) for data analyses, thanks to the free parameters which provide as many fitting degrees of freedom, (ii) for numerical simulations, providing at low computational cost fairly realistic fields to initialize simulation runs, (iii) as building blocks for analytical toy models of astrophysical environments, and (iv) from a deeper theoretical point of view, the phenomenological steps used here may contribute to improve our understanding of MHD turbulence per se. We will soon make publicly available a Python code to generate realizations of the fields presented in this paper. Written with the purpose of being user-friendly, it will also include scripts to conveniently analyze and view the 3D fields using Mayavi.
The present work may be improved and extended in numerous ways. In future works, we will analyze both analytically and numerically the statistics of the magnetic fields introduced here, in the line of [@PereiraEtAl16]. Then, to better appreciate the scope of these formulae, we will explore more thoroughly the parameter space rather than only a few examples as in section \[section:Examples\]. It will also be informative to inspect the various regimes of the dynamics. We will in particular look for conditions under which an inverse energy cascade occurs. Earlier, we stressed that it is important to systematically use the same white noise throughout the formulae, because it was shown to guarantee the existence of a direct cascade in the velocity field. Nevertheless, it remains to assess what happens in the magnetized case. Considering different white noises inside the various formulae, the properties of energy transfers through scales will vary depending on the assumed correlations between the noises. Furthermore, from a practical viewpoint for data analysis, this would provide an additional degree of freedom. Another prospect is to compare the properties of our analytical expressions to direct numerical simulations, notably the correlations between the velocity and magnetic fields,
In parallel, it is important to improve our physical understanding of the present phenomenology, and crucial to compare it to others [e.g. @GoldreichSridhar95_Part1; @GoldreichSridhar95_Part2; @PolitanoPouquet95; @Nazarenko11; @MalaraEtAl16; @MalletSchekochihin17]. This may possibly require delving back into the references to the mathematics literature that we mentioned in this paper, to acquire a deeper theoretical knowledge about the random measures we handle here. For instance $\exp(\tau_m \widetilde{\vec{G}}_g) \ \! {\mathrm{d}}\vec{W}$ is the product of two random distributions, which is a subtle mathematical object per se.
Finally, after surveying the scope and limitations of our description, one may add more ingredients to improve it. The on-going developments in the hydrodynamics community [e.g. @PereiraEtAl18 who study further the dynamics of the velocity gradient] remain a wealth of information. Otherwise, so far we have introduced large-scale features by inputting some ordered magnetic and current density fields, but another important large-scale aspect to take into account are the boundaries of the system. A promising starting point to extend our work in that respect is the Biot-Savart operator of a bounded domain, as studied in [@EncisoEtAl2018] for example. On the longer term, we will also propose generalizations to compressible turbulence, and go beyond the kinematic regime, to analyse magnetic field dynamos in this formalism.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research is supported by the Agence Nationale de la Recherche (project BxB: ANR-17-CE31-0022).
Technical aspects {#appendix:TechnicalAspects}
=================
Figures {#appendix:Figures}
=======
\[lastpage\]
[^1]: E-mail:[email protected]
[^2]: The closest to our work we found is [@HaterEtAl11], who also have as a starting point ideas from L.Chevillard and his collaborators, but they generalize to magnetic fields other aspects than we do, namely the time evolution of Lagrangian fluctuations.
[^3]:
[^4]: Which has a logarithmic covariance, i.e. $\mathbb{E}[\widetilde{s}_g(\vec{x})\widetilde{s}_g(\vec{y})] \sim \ln(L/|\vec{x}-\vec{y}|)$, with $L$ the integral length scale and $\mathbb{E}$ the expectation value.
[^5]:
|
{
"pile_set_name": "ArXiv"
}
|
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